Handbook of Geomathematics [2 ed.] 9783642545504, 9783642545511, 9783642545528, 2015943671

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Willi Freeden M. Zuhair Nashed Thomas Sonar Editors

Handbook of Geomathematics Second Edition

1 3Reference

Handbook of Geomathematics

Willi Freeden M. Zuhair Nashed Thomas Sonar Editors

Handbook of Geomathematics Second Edition

With 785 Figures and 102 Tables

Editors Willi Freeden Geomathematics Group University of Kaiserslautern Kaiserslautern Germany M. Zuhair Nashed Department of Mathematics University of Central Florida Orlando, FL USA Thomas Sonar Computational Mathematics Technische Universität Braunschweig Braunschweig Germany

ISBN 978-3-642-54550-4 ISBN 978-3-642-54551-1 ISBN 978-3-642-54552-8 (print and electronic bundle) DOI 10.1007/ 978-3-642-54551-1

(eBook)

Library of Congress Control Number: 2015943671 Mathematics Subject Classification: 01A55, 35R25, 65N20, 86A05, 86A20, 86A22, 86A25, 86A30 Springer Heidelberg New York Dordrecht London © Springer-Verlag Berlin Heidelberg 2010, 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer. com)

Foreword

With the first edition of the Handbook of Geomathematics, the foundation was laid to provide some meaning to the notion “Geomathematics” and as to what this new interdisciplinary field comprises. Since then, about 5 years ago, a vast number of publications appeared in various journals and books, where one could classify them to treat topics in the field of Geomathematics—and this number is still continuously increasing! Thus, one can infer that Geomathematics as an interdisciplinary research area has grown, in depth as well as in width, and has (re)defined and sharpened its boundaries. There could be no better reason to collect and publish the current state of the art of geomathematical works—hence this extended second edition. Comparing the two editions, one can immediately see that the progress made in these 5 years is enormous. And to make this progress and the new results more accessible, the editors of Handbook of Geomathematics have decided to utilize Springer’s dynamic reference platform. In this way, the Handbook and all its chapters became alive! Now, authors can update their (still peer-reviewed) contributions anytime to include new results. The editors have the possibility to invite new authors and acquire new chapters whenever they feel a new topic or publication has emerged, without having to wait years for a new edition to make sense. Every version of every chapter is associated with a date and time, to track changes and the currency of the work, such that it can be referred to accordingly in other scientific works. For the active researchers and users, the advantages of such a reference platform comprises all what modern technology has to offer: search functions, pdf download and storage, online/offline availability, HTML formatting for screen adaptations, and many more. Finally, we would like to thank all our collaborators and contributors who made this project possible and we appreciate all the support from the community to continue this exciting and fruitful endeavor in promoting Geomathematics. Kaiserslautern, Germany May 2015

Mario Aigner Willi Freeden

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Preface

The aim of Handbook of Geomathematics, published in 2010, was twofold: The first aim was to offer appropriate means of assimilating, assessing, and reducing to comprehensible form the readily increasing flow of data from geochemical, geodetic, geological, geophysical, and satellite sources. The second aim was to provide an objective basis for scientific interpretation, classification, testing of concepts, modeling, simulation, and solution of problems in the geosciences. The handbook was the first authoritative mathematical forum of geoscientifically relevant research. The editors and the publisher are delighted that the handbook was very successful and well received. In recognition of information changing quickly these days and researchers demanding the most current information possible, it seems that the stage is set for the second edition as a central reference work for academics, policymakers, and practitioners internationally. Thus, the editors and the publisher Springer have decided it is time to update and substantially expand the first edition. The handbook fills the gap of a basic reference work on mathematics concerned with methods and problems of geoscientific significance, i.e., geomathematics. It consolidates the current knowledge by providing succinct summaries of concepts and theories, definitions of terms, biographical entries, organizational profiles, a guide to sources of information, and an overview of the landscapes and contours of today’s geomathematics. Contributions are written in an easy-to-understand and informative style for a general readership, typically from areas outside the particular research field. The second edition of Handbook of Geomathematics comprises the following scientific fields: 1. General Issues, Historical Background, and Future Perspectives 2. Observational and Measurement Key Technologies 3. Modeling of the System Earth (Geosphere, Cryosphere, Hydrosphere, Atmosphere, Biosphere, Anthroposphere) 4. Analytic, Algebraic, and Operator Theoretical Methods 5. Statistical and Stochastic Methods 6. Spherical Function Systems and Methods 7. Computational and Numerical Methods 8. Cartographic, Photogrammetric, Information Systems and Methods vii

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Preface

We, as editors, wish to express our particular gratitude to the people who not only made this work possible but also made it an extremely satisfying project:

• The contributors to the handbook, who dedicated much time, effort, and creative energy to the project. The contributions of the handbook open new opportunities to contribute significantly to the understanding of our planet, its climate, its environment, and about an expected shortage of natural resources. Even more, the editors are convinced that they offer the key parameters for the study of the Earth’s dynamics such that interactions of its solid part with ice, oceans, atmosphere, etc., become more and more accessible. During the process of reviewing and editing, authors as well as editors became exposed to exciting new information that, in some cases, strongly increased activity, creativity, and novelty. At the end, as far as we are aware, this second edition of the handbook presents the appropriate ground in dealing generically with fundamental problems and “key technologies” as well as exploring the wide geomathematical range of consequences and interactions for woman or mankind. • The folks at Springer, particularly Clemens Heine, who initiated the first edition, and Mario Aigner, who was the production interface of the second edition with authors and editors. Indeed, the editors have had a number of editorial experiences during a long time, but working with Mario was extraordinarily enjoyable. Thanks also go to Annalea Manalili and Michael Hermann as the responsible persons of this work within Springer’s innovative, twenty-first century digitalfirst reference publishing platform.

Thank you very much for all your exceptional efforts and support in creating and continuing a work summarizing exciting discoveries and impressive research achievements. We hope that the second edition of Handbook of Geomathematics will stimulate and inspire new research efforts and the intensive exploration of very promising directions. Kaiserslautern, Germany Orlando, USA Braunschweig, Germany

Willi Freeden M. Zuhair Nashed Thomas Sonar

Contents

Volume 1 Part I

General Issues, Historical Background, and Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Geomathematics: Its Role, Its Aim, and Its Potential . . . . . . . . . . . . . . . . . Willi Freeden

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Navigation on Sea: Topics in the History of Geomathematics . . . . . . . . . . Thomas Sonar

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Gauss’ and Weber’s “Atlas of Geomagnetism” (1840) Was not the first: the History of the Geomagnetic Atlases . . . . . . . . . . . . . . . . . . . . . 107 Karin Reich and Elena Roussanova Part II Observational and Measurement Key Technologies . . . . . . . . . . 145 Earth Observation Satellite Missions and Data Access . . . . . . . . . . . . . . . . 147 Henri Laur and Volker Liebig Satellite-to-Satellite Tracking (Low-Low/High-Low SST) . . . . . . . . . . . . . . 171 Wolfgang Keller GOCE: Gravitational Gradiometry in a Satellite . . . . . . . . . . . . . . . . . . . . . 211 Reiner Rummel Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Nils Olsen, Gauthier Hulot, and Terence J. Sabaka Part III Modeling of the System Earth (Geosphere, Cryosphere, Hydrosphere, Atmosphere, Biosphere, Anthroposphere) . . . . 251 Classical Physical Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Helmut Moritz Geodetic Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 F. Sansò ix

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Time-Variable Gravity Field and Global Deformation of the Earth . . . . . 321 Jürgen Kusche Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Willi Freeden and Michael Schreiner Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Erik W. Grafarend, Matthias Klapp, and Zdenˇek Martinec Multiresolution Analysis of Hydrology and Satellite Gravitational Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Helga Nutz and Kerstin Wolf Time-Varying Mean Sea Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Luciana Fenoglio-Marc and Erwin Groten Self-Attraction and Loading of Oceanic Masses . . . . . . . . . . . . . . . . . . . . . . 545 Julian Kuhlmann, Maik Thomas, and Harald Schuh Unstructured Meshes in Large-Scale Ocean Modeling . . . . . . . . . . . . . . . . 567 Sergey Danilov and Jens Schröter Numerical Methods in Support of Advanced Tsunami Early Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 Jörn Behrens Gravitational Viscoelastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Detlef Wolf Elastic and Viscoelastic Response of the Lithosphere to Surface Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Volker Klemann, Maik Thomas, and Harald Schuh Multiscale Model Reduction with Generalized Multiscale Finite Element Methods in Geomathematics . . . . . . . . . . . . . . . . . . . . . . . . . 679 Yalchin Efendiev and Michael Presho Efficient Modeling of Flow and Transport in Porous Media Using Multi-physics and Multi-scale Approaches . . . . . . . . . . . . . . . . . . . . . 703 Rainer Helmig, Bernd Flemisch, Markus Wolff, and Benjamin Faigle Convection Structures of Binary Fluid Mixtures in Porous Media . . . . . . 751 Matthias Augustin, Rudolf Umla, and Manfred Lücke Numerical Dynamo Simulations: From Basic Concepts to Realistic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 Johannes Wicht, Stephan Stellmach, and Helmut Harder Mathematical Properties Relevant to Geomagnetic Field Modeling . . . . . 835 Terence J. Sabaka, Gauthier Hulot, and Nils Olsen

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Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 Christian Gerhards Toroidal-Poloidal Decompositions of Electromagnetic Green’s Functions in Geomagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921 Jin Sun Using B-Spline Expansions for Ionosphere Modeling . . . . . . . . . . . . . . . . . 939 Michael Schmidt, Denise Dettmering, and Florian Seitz The Forward and Adjoint Methods of Global Electromagnetic Induction for CHAMP Magnetic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 Zdenˇek Martinec Modern Techniques for Numerical Weather Prediction: A Picture Drawn from Kyrill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Nils Dorband, Martin Fengler, Andreas Gumann, and Stefan Laps

Volume 2 Radio Occultation via Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 Christian Blick and Sarah Eberle Asymptotic Models for Atmospheric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1127 Rupert Klein Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 Carsten Mayer and Willi Freeden On High Reynolds Number Aerodynamics: Separated Flows . . . . . . . . . . 1255 Mario Aigner Turbulence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297 Steffen Schön and Gaël Kermarrec Forest Fire Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349 Sarah Eberle, Willi Freeden, and Ulrich Matthes Phosphorus Cycles in Lakes and Rivers: Modeling, Analysis, and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387 Andreas Meister and Joachim Benz Model-Based Visualization of Instationary Geo-Data with Application to Volcano Ash Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 Martin Baumann, Jochen Förstner, Vincent Heuveline, Jonas Kratzke, Sebastian Ritterbusch, Bernhard Vogel, and Heike Vogel Modeling of Fluid Transport in Geothermal Research . . . . . . . . . . . . . . . . 1443 Jörg Renner and Holger Steeb

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Fractional Diffusion and Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 1507 Yuri Luchko Modeling Deep Geothermal Reservoirs: Recent Advances and Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547 Matthias Augustin, Mathias Bauer, Christian Blick, Sarah Eberle, Willi Freeden, Christian Gerhards, Maxim Ilyasov, René Kahnt, Matthias Klug, Sandra Möhringer, Thomas Neu, Helga Nutz, Isabel Michel née Ostermann, and Alessandro Punzi Part IV Analytic, Algebraic, and Operator Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1631 Noise Models for Ill-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633 Paul N. Eggermont, Vincent LaRiccia, and M. Zuhair Nashed Sparsity in Inverse Geophysical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659 Markus Grasmair, Markus Haltmeier, and Otmar Scherzer Multiparameter Regularization in Downward Continuation of Satellite Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689 Shuai Lu and Sergei V. Pereverzev Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713 Frank Bauer, Martin Gutting, and Mark A. Lukas Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 1775 Yanfei Wang Correlation Modeling of the Gravity Field in Classical Geodesy . . . . . . . . 1807 Christopher Jekeli Inverse Resistivity Problems in Computational Geoscience . . . . . . . . . . . . 1845 Alemdar Hasanov (Hasanoˇglu) and Balgaisha Mukanova Identification of Current Sources in 3D Electrostatics . . . . . . . . . . . . . . . . . 1863 Aron Sommer, Andreas Helfrich-Schkarbanenko, and Vincent Heuveline Transmission Tomography in Seismology . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887 Guust Nolet Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1905 Ignace Loris Strategies in Adjoint Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1943 Yang Luo, Ryan Modrak, and Jeroen Tromp

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Potential-Field Estimation Using Scalar and Vector Slepian Functions at Satellite Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2003 Alain Plattner and Frederik J. Simons Multidimensional Seismic Compression by Hybrid Transform with Multiscale-Based Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057 Amir Z. Averbuch, Valery A. Zheludev, and Dan D. Kosloff Tomography: Problems and Multiscale Solutions . . . . . . . . . . . . . . . . . . . . 2087 Volker Michel RFMP: An Iterative Best Basis Algorithm for Inverse Problems in the Geosciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2121 Volker Michel Material Behavior: Texture and Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . 2149 Ralf Hielscher, David Mainprice, and Helmut Schaeben Rayleigh Wave Dispersive Properties of a Vector Displacement as a Tool for P- and S-Wave Velocities Near Surface Profiling . . . . . . . . . . 2189 Andrey Konkov, Andrey Lebedev, and Sergey Manakov Simulation of Land Management Effects on Soil N2 O Emissions Using a Coupled Hydrology-Biogeochemistry Model on the Landscape Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207 Martin Wlotzka, Vincent Heuveline, Steffen Klatt, Edwin Haas, David Kraus, Klaus Butterbach-Bahl, Philipp Kraft, and Lutz Breuer

Volume 3 Part V

Statistical and Stochastic Methods . . . . . . . . . . . . . . . . . . . . . . . . 2233

An Introduction to Prediction Methods in Geostatistics . . . . . . . . . . . . . . . 2235 Ralf Korn and Alexandra Kochendörfer Statistical Analysis of Climate Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2257 Helmut Pruscha Oblique Stochastic Boundary-Value Problem . . . . . . . . . . . . . . . . . . . . . . . . 2285 Martin Grothaus and Thomas Raskop Geodetic Deformation Analysis with Respect to an Extended Uncertainty Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317 Hansjörg Kutterer It’s All About Statistics: Global Gravity Field Modeling from GOCE and Complementary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2345 Roland Pail

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Mixed Integer Estimation and Validation for Next Generation GNSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2373 Peter J.G. Teunissen Mixed Integer Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2405 Peiliang Xu Part VI Special Function Systems and Methods . . . . . . . . . . . . . . . . . . . 2453 Special Functions in Mathematical Geosciences: An Attempt at a Categorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2455 Willi Freeden and Michael Schreiner Clifford Analysis and Harmonic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 2483 Klaus Gürlebeck and Wolfgang Sprößig Splines and Wavelets on Geophysically Relevant Manifolds . . . . . . . . . . . . 2527 Isaac Pesenson Scalar and Vector Slepian Functions, Spherical Signal Estimation and Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563 Frederik J. Simons and Alain Plattner Dimension Reduction and Remote Sensing Using Modern Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2609 John J. Benedetto and Wojciech Czaja Part VII Computational and Numerical Methods . . . . . . . . . . . . . . . . . . 2633 Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences . . . . . . . . . . . . . . . . . 2635 Natasha Flyer, Grady B. Wright, and Bengt Fornberg Numerical Integration on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2671 Kerstin Hesse, Ian H. Sloan, and Robert S. Womersley Fast Spherical/Harmonic Spline Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 2711 Martin Gutting Multiscale Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2747 Stephan Dahlke Sparse Solutions of Underdetermined Linear Systems . . . . . . . . . . . . . . . . 2773 Inna Kozlov and Alexander Petukhov Nonlinear Methods for Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . 2799 Charles K. Chui and Jianzhong Wang

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Part VIII Cartographic, Photogrammetric, Information Systems and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2853 Cartography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2855 Liqiu Meng Theory of Map Projection: From Riemann Manifolds to Riemann Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2883 Erik W. Grafarend Modeling Uncertainty of Complex Earth Systems in Metric Space . . . . . . 2965 Jef Caers, Kwangwon Park, and Céline Scheidt Geometrical Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995 Manuela Seitz, Detlef Angermann, Michael Gerstl, Mathis Bloßfeld, Laura Sánchez, and Florian Seitz Analysis of Data from Multi-satellite Geospace Missions . . . . . . . . . . . . . . 3035 Joachim Vogt Geodetic World Height System Unification . . . . . . . . . . . . . . . . . . . . . . . . . . 3067 Michael Sideris Mathematical Foundations of Photogrammetry . . . . . . . . . . . . . . . . . . . . . . 3087 Konrad Schindler Potential Methods and Geoinformation Systems . . . . . . . . . . . . . . . . . . . . . 3105 Hans-Jürgen Götze Geoinformatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3131 Monika Sester Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3151

About the Editors

Willi Freeden Geomathematics Group University of Kaiserslautern P.O. Box 3049 67663 Kaiserslautern Germany Studies in Mathematics, Geography, and Philosophy at the RWTH Aachen, 1971 “Diplom” in Mathematics, 1972 “Staatsexamen” in Mathematics and Geography, 1975 Ph.D. in Mathematics with distinction, 1979 “Habilitation” in Mathematics at the RWTH Aachen, 1981/1982 Visiting Research Professor at the Ohio State University, Columbus (Department of Geodetic Science and Surveying), 1984 Professor of Mathematics at the RWTH Aachen (Institute of Pure and Applied Mathematics), 1989 Professor of Technomathematics (Industrial Mathematics), 1994 Head of the Geomathematics Group, and 2002–2006 Vice-President for Research and Technology at the University of Kaiserslautern, published over 145 papers, several book chapters, and 10 books. Editor-in-Chief of the Springer International Journal on Geomathematics (GEM) and Editor-in-Chief of the Birkhäuser book series Geosystems Mathematics and the Birkhäuser lecture notes series Geosystems Mathematics and Computing.

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About the Editors

M. Zuhair Nashed Department of Mathematics University of Central Florida Orlando, FL 32816 USA S.B. and S.M. degrees in electrical engineering from MIT and Ph.D. in Mathematics from the University of Michigan. Served for many years as Professor at Georgia Tech and the University of Delaware. Held visiting professor positions at the University of Michigan, University of Wisconsin, AUB, and KFUPM and distinguished visiting scholar positions at various universities worldwide. Recipient of the Lester Ford Award of the Mathematical Association of America, the Sigma Xi Faculty Research Award and sustained Research Award in Science, Dr. Zakir Husain Award of the Indian Society of Industrial and Applied Mathematics, Fellow of the American Mathematical Society (Inaugural Class of 2013), and several international awards. Published over 140 papers and several expository papers and book chapters. Editor of two journals and member of editorial board of 30 journals, including four Springer journals. Editor-in-Chief of the Birkhäuser book series Geosystems Mathematics. Editor or coeditor of 11 books. Plenary lectures at meetings of 10 mathematical and engineering societies, and over 400 invited hour talks at conferences and colloquia. Organized over 30 international conferences and mini-symposia. Thomas Sonar Computational Mathematics Technische Universität Braunschweig 38106 Braunschweig Germany Study of Mechanical Engineering at the Fachhochschule Hannover, “Diplom” in 1980. Work as a Laboratory Engineer at the Fachhochschule Hannover from 1980 to 1981, founder of an engineering company, and consulting work from 1981 to 1984. Studies in Mathematics and Computer Science at the University of Hannover, “Diplom” with distinction in 1987. Research Scientist at the DLR in Braunschweig from 1987 to 1989 and then Ph.D. studies in Stuttgart and Oxford, 1991 Ph.D. in Mathematics with distinction, from 1991 to 1996 “Hausmathematiker” at the Institute for Theoretical Fluid Dynamics of DLR at Göttingen. 1995 “Habilitation” in Mathematics at the TH Darmstadt and 1996 Professor of Applied Mathematics at the University of Hamburg. Since 1999 Professor of Technomathematics at the TU Braunschweig. Head of the Group “Partial Differential Equations.” Member of the “Braunschweigische Wissenschaftliche Gesellschaft” and Corresponding Member of the Hamburg Academy of Sciences. Published more than 100 papers, several book chapters, and 10 books.

Contributors

Mario Aigner Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Vienna, Austria Detlef Angermann Deutsches Geodätisches Forschungsinstitut, Munich, Germany Matthias Augustin Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany Amir Z. Averbuch School of Computer Science, Tel Aviv University, Tel Aviv, Israel Frank Bauer DZ Bank AG, Kapitalmärkte Handel, Quantitative Modelle, F/KHSQ, Frankfurt, Germany Mathias Bauer CBM GmbH, Bexbach, Germany Martin Baumann Engineering Mathematics and Computing Lab (EMCL), Karlsruhe Institute of Technology, Karlsruhe, Germany Jörn Behrens Numerical Methods in Geosciences, University of Hamburg, CliSAP, Hamburg, Germany John J. Benedetto Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, MD, USA Joachim Benz Faculty of Organic Agricultural Sciences, Work-Group DataProcessing and Computer Facilities, University of Kassel, Witzenhausen, Germany Christian Blick Geomathematics Kaiserslautern, Germany

Group,

University

of

Kaiserslautern,

Mathis Bloßfeld Deutsches Geodätisches Forschungsinstitut, Munich, Germany Lutz Breuer Institute of Landscape Ecology and Resources Management, JustusLiebig-University of Giessen, Giessen, Germany Klaus Butterbach-Bahl Karlsruhe Institute of Technology (KIT), Institute of Meteorology and Climate Research, Garmisch-Partenkirchen, Germany

xix

xx

Contributors

Jef Caers Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA Charles K. Chui Department of Statistics, Stanford University, Stanford, CA, USA Wojciech Czaja Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, MD, USA Stephan Dahlke FB 12 Mathematics and Computer Sciences, Philipps-University of Marburg, Marburg, Germany Sergey Danilov Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany Denise Dettmering Deutsches Germany

Geodätisches

Forschungsinstitut,

Munich,

Nils Dorband Meteomatics GmbH, St. Gallen, Switzerland Sarah Eberle Geomathematics Kaiserslautern, Germany

Group,

University

of

Kaiserslautern,

Yalchin Efendiev Department of Mathematics, Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX, USA Numerical Porous Media SRI Center, King Abdullah University of Science and Technology (KAUST), Makkah Province, Kingdom of Saudi Arabia Paul N. Eggermont Food and Resource Economics, University of Delaware, Newark, DE, USA Benjamin Faigle Department of Hydromechanics and Modeling of Hydrosystems, Institute of Hydraulic Engineering, University of Stuttgart, Stuttgart, Germany Martin Fengler Meteomatics GmbH, St. Gallen, Switzerland Luciana Fenoglio-Marc Institute of Geodesy, Technical University Darmstadt, Darmstadt, Germany Bernd Flemisch Department of Hydromechanics and Modeling of Hydrosystems, Institute of Hydraulic Engineering, University of Stuttgart, Stuttgart, Germany Natasha Flyer Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, CO, USA Bengt Fornberg Department of Applied Mathematics, University of Colorado, Boulder, CO, USA Jochen Förstner German Weather Service (DWD), Offenbach, Germany Willi Freeden Geomathematics Kaiserslautern, Germany

Group,

University

of

Kaiserslautern,

Contributors

xxi

Christian Gerhards Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany Michael Gerstl Deutsches Geodätisches Forschungsinstitut, Munich, Germany Hans-Jürgen Götze Institut für Geowissenschaften, Geophysik, ChristianAlbrechts-Universität zu Kiel, Kiel, Germany Erik W. Grafarend Department of Geodesy and Geoinformatics, Stuttgart University, Stuttgart, Germany Markus Grasmair Department of Mathematics, Norwegian University of Science and Technology, Trondheim, Norway Erwin Groten Institute of Physical Geodesy, Technical University Darmstadt, Darmstadt, Germany Martin Grothaus Functional Analysis Group, University of Kaiserslautern, Kaiserlautern, Germany Andreas Gumann Zurich, Switzerland Klaus Gürlebeck Bauhaus-Universität Weimar, Weimar, Germany Martin Gutting Geomathematics Group, University of Siegen, Siegen, Germany Edwin Haas Karlsruhe Institute of Technology (KIT), Institute of Meteorology and Climate Research, Garmisch-Partenkirchen, Germany Markus Haltmeier Institute of Mathematics, University of Innsbruck, Innsbruck, Austria Helmut Harder Institut für Geophysik, Westfählische Wilhelms-Universität, Münster, Germany Alemdar Hasanov (Hasanoˇglu) Mathematics and Computer Science, Izmir University, Izmir, Turkey Andreas Helfrich-Schkarbanenko Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany Rainer Helmig Department of Hydromechanics and Modeling of Hydrosystems, Institute of Hydraulic Engineering, University of Stuttgart, Stuttgart, Germany Kerstin Hesse Department of Mathematics, University of Paderborn, Paderborn, Germany Vincent Heuveline Engineering Mathematics and Computing Lab (EMCL), Karlsruhe Institute of Technology, Karlsruhe, Germany Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany University of Heidelberg, Interdisciplinary Center for Scientific Computing, Engineering Mathematics and Computing Lab, Heidelberg, Germany

xxii

Contributors

Ralf Hielscher Applied Functional Analysis, Technical University Chemnitz, Chemnitz, Germany Gauthier Hulot Equipe de Géomagnétisme, Institut de Physique du Globe de Paris, Sorbonne Paris Cité, Université Paris Diderot, Paris, France Maxim Ilyasov Geomathematics Kaiserslautern, Germany

Group,

University

of

Kaiserslautern,

Christopher Jekeli Division of Geodetic Science, School of Earth Sciences, Ohio State University, Columbus, OH, USA René Kahnt G.E.O.S. Ingenieurgesellschaft mbH, Freiberg, Germany Wolfgang Keller Geodätisches Institut, Universität Stuttgart, Stuttgart, Germany Gaël Kermarrec Institut für Erdmessung, Leibniz Universität Hannover, Hannover, Germany Matthias Klapp Asperg, Germany Steffen Klatt Karlsruhe Institute of Technology (KIT), Institute of Meteorology and Climate Research, Garmisch-Partenkirchen, Germany Rupert Klein FB Mathematik und Informatik, Institut für Mathematik, Freie Universität Berlin, Berlin, Germany Volker Klemann Geodesy and Remote Sensing, Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany Matthias Klug Geomathematics Kaiserslautern, Germany

Group,

University

of

Kaiserslautern,

Alexandra Kochendörfer Department of Financial Mathematics, Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany Andrey Konkov The Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia Ralf Korn Department Kaiserslautern, Germany

of

Mathematics,

University

of

Kaiserslautern,

Department of Financial Mathematics, Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany Dan D. Kosloff Department of Earth and Planetary Sciences, Tel Aviv University, Tel Aviv, Israel Inna Kozlov Department of Computer Science, Holon Institute of Technology, Holon, Israel Philipp Kraft Institute of Landscape Ecology and Resources Management, JustusLiebig-University of Giessen, Giessen, Germany

Contributors

xxiii

Jonas Kratzke Engineering Mathematics and Computing Lab (EMCL), Karlsruhe Institute of Technology, Karlsruhe, Germany David Kraus Karlsruhe Institute of Technology (KIT), Institute of Meteorology and Climate Research, Garmisch-Partenkirchen, Germany Julian Kuhlmann Earth System Modelling, Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany Institute of Meteorology, Freie Universität Berlin, Berlin, Germany Jürgen Kusche Astronomical, Physical and Mathematical Geodesy Group, Bonn University, Bonn, Germany Hansjörg Kutterer Bundesamt für Kartographie und Geodäsie, Frankfurt am Main, Germany Vincent LaRiccia Food and Resource Economics, University of Delaware, Newark, DE, USA Stefan Laps Bochum, Germany Henri Laur European Space Agency (ESA), Head of Earth Observation Mission Management Division, ESRIN, Frascati, Italy Andrey Lebedev The Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia Volker Liebig European Space Agency (ESA), Director of Earth Observation Programmes, ESRIN, Frascati, Italy Ignace Loris Université libre de Bruxelles, Bruxelles, Belgium Shuai Lu Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria Yuri Luchko Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Berlin, Germany Manfred Lücke Institut für Theoretische Physik, Universität des Saarlandes, Saarbrücken, Germany Mark A. Lukas Mathematics and Statistics, School of Engineering and Information Technology, Murdoch University, Murdoch, WA, Australia Yang Luo Department of Geosciences, Princeton University, Princeton, NJ, USA Sandra Möhringer Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany David Mainprice Géosciences UMR CNRS 5243, Université Montpellier 2, Montpellier, France

xxiv

Contributors

Sergey Manakov The Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia Zdenˇek Martinec Department of Geophysics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic Dublin Institute for Advanced Studies, Dublin, Ireland Ulrich Matthes Rhineland-Palatinate Center of Excellence for Climate Change Impacts, Trippstadt, Germany Carsten Mayer Geomathematics Kaiserslautern, Germany

Group,

University

of

Kaiserslautern,

Andreas Meister Department of Mathematics, Work-Group of Analysis and Applied Mathematics, University of Kassel, Kassel, Germany Liqiu Meng Institute for Photogrammetry and Cartography, TU Munich, Munich, Germany Isabel Michel née Ostermann Fraunhofer ITWM, Kaiserslautern, Germany Volker Michel Geomathematics Group, University of Siegen, Siegen, Germany Ryan Modrak Department of Geosciences, Princeton University, Princeton, NJ, USA Helmut Moritz Institute of Navigation, Graz University of Technology, Graz, Austria Balgaisha Mukanova Eurasian National University, Astana, Kazakhstan M. Zuhair Nashed Department of Mathematics, University of Central Florida, Orlando, FL, USA Thomas Neu Tiefe Geothermie Saar GmbH, Saarbrücken, Germany Guust Nolet Geosciences Azur, Université de Nice, Sophia Antipolis, France Helga Nutz Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany Nils Olsen DTU Space, Technical University of Denmark, Kgs. Lyngby, Denmark Roland Pail Institute of Astronomical and Physical Geodesy, TU Munich, Munich, Germany Kwangwon Park Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA Sergei V. Pereverzev Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Linz, Austria

Contributors

xxv

Isaac Pesenson Department of Mathematics, Temple University, Philadelphia, PA, USA Alexander Petukhov Department of Mathematics, University of Georgia, Athens, GA, USA Alain Plattner Department of Geosciences, Princeton University, Princeton, NJ, USA Department of Earth and Environmental Science, California State University, Fresno, CA, USA Michael Presho The Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX, USA Helmut Pruscha Mathematical Institute, University of Munich, Munich, Germany Alessandro Punzi Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany Thomas Raskop Functional Analysis Group, University of Kaiserslautern, Kaiserlautern, Germany Karin Reich Department of Mathematics, University of Hamburg, Germany Jörg Renner Experimentelle Geophysik - Institut für Geologie, Mineralogie und Geophysik, Ruhr-Universität Bochum, Bochum, Germany Sebastian Ritterbusch Engineering Mathematics and Computing Lab (EMCL), Karlsruhe Institute of Technology, Karlsruhe, Germany Elena Roussanova Saxonian Academy of Sciences and Humanities in Leipzig, Leipzig, Germany Reiner Rummel Institut für Astronomische und Physikalische Geodäsie, TU Munich, Munich, Germany Laura Sánchez Deutsches Geodätisches Forschungsinstitut, Munich, Germany Terence J. Sabaka Planetary Geodynamics Laboratory, Code 698, NASA Goddard Space Flight Center, Greenbelt, MD, USA F. Sansò Politecnico di Milano – Polo Regionale di Como, Como, Italy Otmar Scherzer Computational Science Center, University of Vienna, Vienna, Austria Steffen Schön Institut für Erdmessung, Leibniz Universität Hannover, Hannover, Germany Helmut Schaeben Geophysics and Geoinformatics, TU Bergakademie Freiberg, Freiberg, Germany

xxvi

Contributors

Céline Scheidt Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA Konrad Schindler Photogrammetry and Remote Sensing, ETH Zürich, Zürich, Switzerland Michael Schmidt Deutsches Geodätisches Forschungsinstitut, Munich, Germany Jens Schröter Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany Michael Schreiner Institute for Computational Engineering, University of Buchs, Buchs, Switzerland Harald Schuh Geodesy and Remote Sensing, Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany TU Berlin, Berlin, Germany Florian Seitz Deutsches Geodätisches Forschungsinstitut, Munich, Germany Manuela Seitz Deutsches Geodätisches Forschungsinstitut, Munich, Germany Monika Sester Institute of Cartography and Geoinformatics, Leibniz University Hannover, Hannover, Germany Michael Sideris Department of Geomatics Engineering, University of Calgary, Calgary, AB, Canada Frederik J. Simons Department of Geosciences, Princeton University, Princeton, NJ, USA Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA Ian H. Sloan School of Mathematics and Statistics, University of New South Wales, Sydney, Australia Aron Sommer Institut für Informationsverarbeitung (TNT), Leibniz Universität Hannover, Hannover, Germany Thomas Sonar Computational Mathematics, Braunschweig, Braunschweig, Germany

Technische

Universität

Wolfgang Sprößig TU Bergakademie Freiberg, Freiberg, Germany Holger Steeb Kontinuumsmechanik - Mechanik, Ruhr-Universität Bochum, Bochum, Germany Stephan Stellmach Institut für Geophysik, Westfählische Wilhelms-Universität, Münster, Germany Jin Sun Institute of Geophysics, ETH Zürich, Zürich, Switzerland

Contributors

xxvii

Peter J.G. Teunissen Department of Spatial Sciences, Curtin University of Technology, Perth, Australia Department of Geoscience and Remote Sensing, Delft University of Technology, Delft, Netherlands Maik Thomas Geodesy and Remote Sensing, Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany Institute of Meteorology, Freie Universität Berlin, Berlin, Germany Jeroen Tromp Department of Geosciences, Princeton University, Princeton, NJ, USA Program in Applied & Computational Mathematics, Princeton University, Princeton, NJ, USA Rudolf Umla BP Exploration Operating Company Limited, Sunbury on Thames, UK Bernhard Vogel Institute for Meteorology and Climate Research (IMK), Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany Heike Vogel Institute for Meteorology and Climate Research (IMK), Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany Joachim Vogt School of Engineering & Science – SES, Jacobs University, Bremen, Germany Jianzhong Wang Department of Mathematics, Sam Houston State University, Huntsville, TX, USA Yanfei Wang Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, People’s Republic of China Johannes Wicht Max-Planck Intitut für Sonnensystemforschung, KaltenburgLindau, Germany Martin Wlotzka University of Heidelberg, Interdisciplinary Center for Scientific Computing, Engineering Mathematics and Computing Lab, Heidelberg, Germany Detlef Wolf Department of Geodesy and Remote Sensing, German Research Center for Geosciences GFZ, Telegrafenberg, Potsdam, Germany Kerstin Wolf Geomathematics Kaiserslautern, Germany

Group,

University

of

Kaiserslautern,

Markus Wolff Department of Hydromechanics and Modeling of Hydrosystems, Institute of Hydraulic Engineering, University of Stuttgart, Stuttgart, Germany

xxviii

Contributors

Robert S. Womersley School of Mathematics and Statistics, University of New South Wales, Sydney, Australia Grady B. Wright Department of Mathematics, Boise State University, Boise, ID, USA Peiliang Xu Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto, Japan Valery A. Zheludev School of Computer Science, Tel Aviv University, Tel Aviv, Israel

Part I General Issues, Historical Background, and Future Perspectives

Geomathematics: Its Role, Its Aim, and Its Potential Willi Freeden

Contents 1 2 3 4 5 6 7 8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomathematics as a Cultural Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomathematics as Task and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomathematics as Interdisciplinary Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomathematics as a Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomathematics as Solution Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomathematics as Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geomathematics: Three Exemplary “Circuits” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Circuit: Gravity Field from Deflections of the Vertical . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Circuit: Oceanic Circulation from Ocean Topography . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Circuit: Seismic Processing from Acoustic Wave Tomography . . . . . . . . . . . . . . . . . 9 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 6 9 10 12 14 22 23 47 57 75 76

Abstract

During the last decades, geosciences and geoengineering were influenced by two essential scenarios: First, the technological progress has completely changed the observational and measurement techniques. Modern high-speed computers and satellite-based techniques are more and more entering all geodisciplines. Second, there is a growing public concern about the future of our planet, its climate, and its environment and about an expected shortage of natural resources. Obviously, both aspects, viz., efficient strategies of protection against threats of a changing Earth and the exceptional situation of getting terrestrial, airborne, as well as

W. Freeden () Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_1

3

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W. Freeden

spaceborne data of better and better quality, explain the strong need of new mathematical structures, tools, and methods, i.e., geomathematics. This paper deals with geomathematics, its role, its aim, and its potential. Moreover, the “circuit” geomathematics is exemplified by three problems involving the Earth’s structure, namely, gravity field determination from terrestrial deflections of the vertical, ocean flow modeling from satellite (altimeter measured) ocean topography, and reservoir detection from (acoustic) wave tomography.

1

Introduction

Geophysics is an important branch of physics; it differs from the other physical disciplines due to its restriction to objects of geophysical character. Why shouldn’t the same hold for mathematics what physics regards as its canonical right since the times of Emil Wiechert in the late nineteenth century? More than ever before, there are significant reasons for a well-defined position of geomathematics as a branch of mathematics and simultaneously of geosciences (cf. Fig. 1). On the one hand, these

There exists a growing public concern about Modern high speed computers are the future of our planet, its climate, its envientering more and more all geodis- ronment, and about an expected shortage of natural resources. ciplines.

There is a strong need for strategies of protection against threats of a changing Earth.

There is an exceptional situation of getting data of better and better quality.

Fig. 1 Four significant reasons for the increasing importance of geomathematics

Geomathematics: Its Role, Its Aim, and Its Potential

5

reasons are intrinsically based on the self-conception of mathematics; on the other hand, they are explainable from the current situation of geosciences. In the following, I would like to explain my thoughts on geomathematics in more detail. My objective is to convince not only the geoscientists but also a broader audience: “Geomathematics is essential and important. As geophysics within physics it has an adequate forum within mathematics, and it should have a fully acknowledged position within geosciences!”

2

Geomathematics as a Cultural Asset

In chapter  Navigation on Sea: Topics in the History of Geomathematics of this Handbook of Geomathematics, T. Sonar starts his contribution with the sentence: “Geomathematics in our times is thought of being a very young science and a modern area in the realms of mathematics. Nothing is farer from the truth. Geomathematics began as man realized that he walked across a sphere–like Earth and that this observation has to be taken into account in measurements and calculations.” In consequence, we can only do justice to geomathematics if we look at its historic importance, at least shortly. According to the oldest evidence which has survived in written form, geomathematics was developed in Sumerian Babylon and ancient Egypt (see Fig. 2) on the basis of practical tasks concerning measuring, counting, and calculation for reasons of agriculture and stock keeping.

Fig. 2 Papyrus scroll containing indications of algebra, geometry, and trigonometry due to Ahmose (nineteenth century BC) [Department of Ancient Egypt and Sudan, British Museum EA 10057, London, Creative Commons Lizenz CC-BY-SA 2.0] (taken from Sonar 2011)

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W. Freeden

In the ancient world, mathematics dealing with problems of geoscientific relevance flourished for the first time, for example, when Eratosthenes (276–195 BC) of Alexandria calculated the radius of the Earth. We also have evidence that the Arabs carried out an arc measurement northwest of Bagdad in the year 827 AD. Further key results of geomathematical research lead us from the Orient across the occidental Middle Ages to modern times. Copernicus (1473–1543) successfully made the transition from the Ptolemaic geocentric system to the heliocentric system. Kepler (1571–1630) determined the laws of planetary motion. Further milestones from a historical point of view are, for example, the theory of geomagnetism developed by Gilbert (1544–1608), the development of triangulation methods for the determination of meridians by Brahe (1547–1601) and Snellius (1580–1626), the laws of falling bodies by Galilei (1564–1642), and the basic theory on the propagation of seismic waves by Huygens (1629–1695). The laws of gravitation formulated by the English scientist Newton (1643–1727) have taught us that gravitation decreases with an increasing distance from the Earth. In the seventeenth and eighteenth centuries, France took over an essential role through the foundation of the Academy in Paris (1666). Successful discoveries were the theory of the isostatic balance of mass distribution in the Earth’s crust by Bouguer (1698– 1758), the calculation of the Earth’s shape and especially of the pole flattening by Maupertuis (1698–1759) and Clairaut (1713–1765), and the development of the calculus of spherical harmonics by Legendre (1752–1833) and Laplace (1749– 1829). The nineteenth century was essentially characterized by Gauß (1777–1855). Especially important was the calculation of the lower Fourier coefficients of the Earth’s magnetic field, the hypothesis of electric currents in the ionosphere, as well as the definition of the level set of the geoid (however, the term “geoid” was defined by Listing (1808–1882), a disciple of Gauß). Riemann (1826–1866) made lasting contributions to differential geometry, some of them enabling the later development of general relativity. Helmert (1843–1917) laid the mathematical foundation of modern geodesy. At the end of the nineteenth century, the basic idea of the dynamo theory in geomagnetics was developed by Elsasser (1904–1981), Bullard (1907– 1980), etc. This very incomplete list (which does not even include essential facets of the last century) already shows that geomathematics is one of the large achievements of mankind from a historic point of view.

3

Geomathematics as Task and Objective

Modern geomathematics deals with the qualitative and quantitative properties of the current or possible structures of the system Earth. It guarantees concepts of scientific research concerning the system Earth, and it is simultaneously the force behind it. The system Earth (see Fig. 3) consists of a number of elements which represent individual systems themselves. The complexity of the entire system Earth is determined by interacting physical, biological, and chemical processes transforming and transporting energy, material, and information (cf. Emmermann and Raiser 1997). It is characterized by natural, social, and economic processes influencing

Geomathematics: Its Role, Its Aim, and Its Potential

7

Fig. 3 Geosystems Mathematics as key technology penetrating the complex system Earth (modified illustration following Emmermann and Raiser 1997)

one another. In most instances, a simple theory of cause and effect is therefore completely inappropriate if we want to understand the system. We have to think in dynamical structures and to account for multiple, unforeseen, and of course sometimes even undesired effects in the case of interventions. Inherent networks must be recognized and made use of, and self-regulation must be accounted for. All these aspects require a type of mathematics which must be more than a mere collection of theories and numerical methods. Mathematics dedicated to geosciences, i.e., geomathematics, deals with nothing more than the organization of the complexity of the system Earth. Descriptive thinking is required in order to clarify abstract complex situations. We also need a correct simplification of complicated interactions, an appropriate system of mathematical concepts for their description, and exact thinking and formulations. Geomathematics has thus become the key science of the complex system Earth. Wherever there are data and observations to be processed, e.g., the diverse scalar, vectorial, and tensorial clusters of satellite data, we need mathematics. For example, statistics serves for noise reduction, constructive approximation serves for compression and evaluation, and the theory of special function systems yields georelevant graphical and numerical representations – there are mathematical

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W. Freeden

algorithms everywhere. The specific task of geomathematics is to build a bridge between mathematical theory and geophysical as well as geotechnical applications. The special attraction of this branch of mathematics is therefore based on the vivid communication between applied mathematicians more interested in model development, theoretical foundation, and the approximate as well as computational solution of problems and geoengineers and physicists more familiar with measuring technology, methods of data analysis, implementation of routines, and software applications. There is a very wide range of modern geosciences on which geomathematics is focused (see Fig. 4), not least because of the continuously increasing observation diversity. Simultaneously, the mathematical “toolbox” is becoming larger. A special feature is that geomathematics primarily deals with those regions of the Earth which are only insufficiently or not at all accessible for direct measurements (even by remote sensing methods (as discussed, e.g., in chapters  Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment,  Inverse Resistivity Problems in Computational Geoscience,  Analysis of Data from Multi-satellite Geospace Missions, and  Potential Methods and Geoinformation Systems of this work)). Inverse methods (see, for instance, chapters  Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution,  Multiresolution Analysis of Hydrology and Satellite Gravitational Data,  Elastic and Viscoelastic Response of the Lithosphere to Surface Loading,  Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents,  The Forward and Adjoint Methods of Global Electromagnetic Induction for CHAMP Magnetic Data,  Modeling Deep Geother-

Fig. 4 Geomathematics, its range of fields, and its disciplines

Geomathematics: Its Role, Its Aim, and Its Potential

9

mal Reservoirs: Recent Advances and Future Perspectives,  Noise Models for Ill-Posed Problems,  Sparsity in Inverse Geophysical Problems,  Multiparameter Regularization in Downward Continuation of Satellite Data,  Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics,  Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment,  Inverse Resistivity Problems in Computational Geoscience,  Identification of Current Sources in 3D Electrostatics,  Transmission Tomography in Seismology,  Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery,  Strategies in Adjoint Tomography,  Potential-Field Estimation Using Scalar and Vector Slepian Functions at Satellite Altitude,  Multidimensional Seismic Compression by Hybrid Transform with Multiscale-Based Coding,  Tomography: Problems and Multiscale Solutions,  RFMP: An Iterative Best Basis Algorithm for Inverse Problems in the Geosciences,  Material Behavior: Texture and Anisotropy, and  Rayleigh Wave Dispersive Properties of a Vector Displacement as a Tool for P- and S-Wave Velocities Near Surface Profiling of this handbook) are absolutely essential for mathematical evaluation in these cases (see, e.g., Nashed (1981) for a deeper insight). Mostly, a physical quantity is measured in the vicinity of the Earth’s surface, and it is then continued downward or upward by mathematical methods until one reaches the interesting depths or heights.

4

Geomathematics as Interdisciplinary Science

Once more it should be mentioned that, today, computers and measurement technology have resulted in an explosive propagation of mathematics in almost every area of society. Mathematics as an interdisciplinary science can be found in almost every area of our lives. Consequently, mathematics is closely interacting with almost every other science, even medicine and parts of the arts (mathematization of sciences). The use of computers allows for the handling of complicated models for real data sets. Modeling, computation, and visualization yield reliable simulations of processes and products. Mathematics is the “raw material” for the models and the essence of each computer simulation. As the key technology, it translates the images of the real world to models of the virtual world, and vice versa (cf. Fig. 5 for an example in seismic tomography). The special importance of mathematics as an interdisciplinary science (see also Neuzert and Rosenberger 1991; Beutelspacher 2001; Pesch 2002) has been acknowledged increasingly within the last few years in technology, economy, and commerce. However, this process does not remain without effects on mathematics itself. New mathematical disciplines, such as scientific computing, financial and business mathematics, industrial mathematics, biomathematics, and also geomathematics, have complemented the traditional disciplines. Interdisciplinarity also implies the interdisciplinary character of mathematics at school. Relations and references to other disciplines (especially informatics, physics, chemistry, biology, and also economy and geography) become more impor-

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W. Freeden

Fig. 5 Geomathematics as a key technology bridging the real and virtual world: An example from seismic tomography

tant, more interesting, and more expandable. Problem areas of mathematics become explicit and observable, and they can be visualized. Of course, this undoubtedly also holds for the system Earth.

5

Geomathematics as a Challenge

From a scientific and technological point of view, the past twentieth century was a period with two entirely different faces concerning research and its consequences. The first two thirds of the century were characterized by a movement toward a seemingly inexhaustible future of science and technology; they were marked by the absolute belief in technical progress which would make everything achievable in the end. Up to the 1960s, mankind believed to have become the master of the Earth (note that, in geosciences as well as other sciences, to master is also a synonym for to understand). Geoscience was able to understand plate tectonics on the basis of Wegener’s theory of continental drift, geoscientific research began to deal with the Arctic and Antarctic, and man started to conquer the universe by satellites, so that for the first time in mankind’s history, the Earth became measurable on a global scale. Then, during the last third of the past century, there was a growing skepticism as to the question whether scientific and technical progress had really brought us forth and whether the effects of our achievements were responsible. As a consequence of the specter of a shortage in raw materials (mineral oil and natural gas reserves), predicted by the Club of Rome, geological/geophysical research with the objective of exploring new reservoirs was stimulated during the 1970s (see, e.g., Jakobs and Meyer 1992). Moreover, the last two decades of the century have sensitized us

Geomathematics: Its Role, Its Aim, and Its Potential

11

Fig. 6 (Potato) “Earth” of radius 6,371 km

for the global problems resulting from our behavior with respect to climate and environment. Our senses have been sharpened as to the dangers caused by the forces of nature, from earthquakes and volcanic eruptions to the temperature development and the hole in the ozone layer, etc. Man has become aware of his environment. The image of the Earth as a potatodrenched by rainfall (which is sometimes drawn by oceanographers) is not a false one (see Fig. 6). The humid layer on this potato, maybe only a fraction of a millimeter thick, is the ocean. The entire atmosphere hosting the weather and climate events is only a little bit thicker. Flat bumps protruding from the humid layer represent the continents. The entire human life takes place in a very narrow region of the outer peel (only a few kilometers in vertical extension). However, the basically excellent comparison of the Earth with a huge potato does not give explicit information about essential ingredients and processes of the system Earth, for example, gravitation, magnetic field, deformation, wind and heat distribution, ocean currents, internal structures, etc. In our twenty-first century, geoproblems currently seem to overwhelm the scientific programs and solution proposals. “How much more will our planet Earth be able to take?” has become an appropriate and very urgent question. Indeed, there have been a large number of far-reaching changes during the last few decades, e.g., species extinction, climate change, formation of deserts, ocean currents, structure of the atmosphere, transition of the dipole structure of the magnetic field to a quadrupole structure, etc. These changes have been accelerated dramatically. The main reasons for most of these phenomena are the unrestricted growth in the industrial societies (population and consumption, especially of resources, territory, and energy) and severe poverty in the developing and newly industrialized countries. The dangerous aspect is that extreme changes have taken place within a very short time; there has been no comparable development in the dynamics of the system Earth in the past. Changes brought about by man are much faster than changes due to natural fluctuations. Besides, the current financial crisis shows that our western model of affluence (which holds for approximately 1 billion people) cannot be transferred globally to 5–8 billion people. Massive effects on mankind are inevitable. The appalling résumé is that the geoscientific problems collected over the decades must now all be solved simultaneously. Interdisci-

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plinary solutions including intensive mathematical research are urgently required as answers to an increasingly complex world. Geomathematics is absolutely essential for a sustainable development in the future. However, the scientific challenge does not only consist of increasing the leading role of mathematics within the current “scientific consortium Earth”. The significance of the subject “Earth” must also be acknowledged (again) within mathematics itself, so that mathematicians will become more enthusiastic about it. Up to now, it has become usual and good practice in application-oriented mathematical departments and research institutions to present applications in technology, economy, finances, and even medicine as being very career enhancing for young scientists. Geomathematics can be integrated smoothly into this phalanx with current subjects like exploration, geothermal research, navigation, and so on. Mathematics should be the leading science for the solution of these complex and economically very interesting problems, instead of fulfilling mere service functions. Of course, basic research is indispensable. We should not hide behind the other geosciences! Neither should we wait for the next horrible natural disaster! Now is the time to turn expressly toward georelevant applications. The Earth as a complex, however limited system (with its global problems concerning climate, environment, resources, and population) needs new political strategies. Consequently, these will step-by-step also demand changes in research and teaching at the universities due to our modified concept of “well-being” (e.g., concerning milieu, health, work, independence, financial situation, security, etc.). This will be a very difficult process. We dare to make the prognosis that, finally, it will also result in a modified appointment practice at the universities. Chairs in the field of geomathematics must increase in number and importance, in order to promote attractiveness, but also to accept a general responsibility for society. The time has come to realize that geomathematics is indispensable as a constituting discipline within a mathematical faculty (instead of “ivory tower-like” parity thinking following traditional structures). Additionally, the curricular standards and models for school lessons in mathematics (see, e.g., Sonar 2001; Bach et al. 2004) have also to change. We will not be able to afford any jealousies or objections on our way in that direction.

6

Geomathematics as Solution Potential

Current methods of applied measurement and evaluation processes vary strongly, depending on the examined measurement parameters (gravity, electric or magnetic field force, temperature and heat flow, etc.), the observed frequency domain, and the occurring basic “field characteristic” (potential field, diffusion field, or wave field, depending on the basic differential equations). In particular, the differential equation strongly influences the evaluation processes. The typical mathematical methods are therefore listed here according to the respective “field characteristic” – as it is usually done in geomathematics:

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13

• Potential methods (potential fields, elliptic differential equations) in geomagnetics, geoelectrics, gravimetry, etc. • Diffusion methods (diffusion fields, parabolic differential equations) in flow and heat transport, magnetotellurics, geoelectromagnetics, etc. • Wave methods (wave fields, hyperbolic differential equations) in seismics, georadar, etc. The diversity of mathematical methods will increase in the future due to new technological developments in computer and measurement technology. More intensively than before, we must aim for the creation of models and simulations for combinations and networks of data and observable structures. The process (i.e., the “circuit”) for the solution of practical problems usually has the following components: • Mathematical modeling: the practical problem is translated into the language of mathematics, requiring the cooperation between application-oriented scientists and mathematicians. • Mathematical analysis: the resulting mathematical problem is examined as to its “well-posedness” (i.e., existence, uniqueness, dependence on the input data). • Development of a mathematical solution method: appropriate analytical, algebraic, statistic/stochastic, and/or numerical methods and processes for a specific solution must be adapted to the problem; if necessary, new methods must be developed. The solution process is carried out efficiently and economically by the decomposition into individual operations, usually on computers. • “Back-transfer” from the language of mathematics to applications: the results are illustrated adequately in order to ensure their evaluation. The mathematical model is validated on the basis of real data and modified, if necessary. We aim for good accordance of model and reality. Often, the circuit must be applied several times in an iterative way in order to get a sufficient insight into the system Earth. Nonetheless, the advantage and benefit of the mathematical processes are a better, faster, cheaper, and more secure problem solution on the basis of the mentioned means of simulation, visualization, and reduction of large amounts of data. So, what is it exactly that enables mathematicians to build a bridge between the different disciplines? The mathematics’ world of numbers and shapes contains very efficient tokens by which we can describe the rule-like aspect of real problems. This description includes a simplification by abstraction: essential properties of a problem are separated from unimportant ones and included into a solution scheme. Their “eye for similarities” often enables mathematicians to recognize a posteriori that an adequately reduced problem may also arise from very different situations, so that the resulting solutions may be applicable to multiple cases after an appropriate adaptation or concretization. Without this second step, abstraction remains essentially useless.

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The interaction between abstraction and concretization characterizes the history of mathematics and its current rapid development as a common language and independent science. A problem reduced by abstraction is considered as a new “concrete” problem to be solved within a general framework, which determines the validity of a possible solution. The more examples one knows, the more one recognizes the causality between the abstractness of mathematical concepts and their impact and cross-sectional importance. Of course, geomathematics is closely interconnected with geoinformatics, geoengineering, and geophysics. However, geomathematics basically differs from these disciplines (cf. Kümmerer 2002). Engineers and physicists need the mathematical language as a tool. In contrast to this, geomathematics also deals with the further development of the language itself. Geoinformatics concentrates on the design and architecture of processors and computers, databases and programming languages, etc., in a georeflecting environment (cf. chapters  Potential Methods and Geoinformation Systems and  Geoinformatics). In geomathematics, computers do not represent the objects to be studied, but instead represent technical auxiliaries for the solution of mathematical problems of georeality. Statistics (usually seen as a basic subdiscipline of mathematics) is generally devoted to the analysis and interpretation of uncertainties caused by limited sampling of a property under study. In consequence, the focus of geostatistics is the development and statistical validation of models to describe the distribution in time and space of Earth sciences phenomena. Geostatistical realizations (see, e.g., chapters  An Introduction to Prediction Methods in Geostatistics,  Statistical Analysis of Climate Series,  Geodetic Deformation Analysis with Respect to an Extended Uncertainty Budget, and  It’s All About Statistics: Global Gravity Field Modeling from GOCE and Complementary Data) aim at integrating physical constraints, combining heterogeneous data sources, and characterizing uncertainty. Applications include a large palette of areas, for example, groundwater hydrology, air quality, and land use change using terrestrial as well as satellite data. Because of both the statistical distribution of sample data and the spatial correlation among the sample data, a large variety of Earth science problems are effectively addressed using statistical procedures. Stochastic systems and processes play an important role in mathematical models of phenomena in geosciences.

7

Geomathematics as Solution Method

Up to now, ansatz functions for the description of geoscientifically relevant parameters have been frequently based on the almost spherical shape of the Earth. By modern satellite positioning methods, the maximum deviation of the actual Earth’s surface from the average Earth’s radius (6,371 km) can be proved as being less than 0.4 %. Although a mathematical formulation in a spherical context may be a restricted simplification, it is at least acceptable for a large number of problems (see chapter  Special Functions in Mathematical Geosciences: An Attempt at a Categorization of this handbook for more details). In fact, ellipsoidal nomenclature

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(see chapter  Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics) is much closer to geophysical and/or geodetic purposes, but the computational and numerical amount of work is a tremendous obstacle that could not be sufficiently cleared so far. Usually, in geosciences, we consider a separable Hilbert space such as the L2 space (of finite signature energy) with a (known) polynomial basis as reference space for ansatz functions. However, there is a striking difference between the L2 space and the Earth’s body and surface. Continuous “surface functions” can be described in arbitrary accuracy, for example, with respect to C- and L2 -topology by restrictions of harmonic functions (such as spherical harmonics), whereas “volume functions” contain anharmonic ingredients (for more details see, e.g., Freeden and Gerhards 2013; Michel 2013). This fact has serious consequences for the reconstruction of signatures. Since the times of C.F. Gauß (1863, Werke, Bd. 5), a standard method for globally reflected surface approximation involving equidistributed data has been the Fourier series in an orthogonal basis in terms of spherical harmonics. It is characteristic for such an approach that these polynomial ansatz functions do not show any localization in space (cf. Fig. 7). In the momentum domain (throughout this work called frequency domain), each spherical function corresponds to exactly one single Fourier coefficient reflecting a certain frequency. We call this ideal frequency localization. Due to the ideal frequency localization and the simultaneous dispensation with localization in space, local data modifications influence all the Fourier coefficients (that have to be determined by global integration). Consequently, this also leads to global modifications of the data representations in case of local changes. Fourier expansions provide approximation by oscillation, i.e., the oscillations grow in number, while the amplitudes become smaller and

Fig. 7 Uncertainty principle and its consequences for space–frequency localization

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Fig. 8 Spherical harmonics of low degrees leading to a zonal function, i.e., a sum of Legendre functions

smaller. Nevertheless, we may state that ideal frequency localization has proved to be extraordinarily advantageous due to the important physical interpretability (as multipole moments) of the model and due to the simple comparability of the Fourier coefficients with observables in geophysical and/or geodetic interrelations (see, e.g., the Meissl diagrams discussed by Meissl (1971), Rummel and van Gelderen (1995), Grafarend (2001), and Nutz (2002)). From a mathematical and physical point of view, however, certain kinds of ansatz functions would be desirable which show ideal frequency localization as well as localization in space. Such an ideal system of ansatz functions would allow for models of highest resolution in space; simultaneously, individual frequencies would remain interpretable. However, the principle of uncertainty, which connects frequency localization and space localization qualitatively and quantitatively, teaches us that both properties are mutually exclusive (except for the trivial case).

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Fig. 9 Weighted summation of spherical harmonics leading to the generation of space-localized zonal kernels

Extreme ansatz functions in the sense of such an uncertainty principle are, on the one hand, spherical polynomials (see Fig. 8), i.e., spherical harmonics (no space localization, ideal frequency localization), and, on the other hand, the Dirac (kernel) function(als) (ideal space localization, no frequency localization). In consequence (see also chapter  Special Functions in Mathematical Geosciences: An Attempt at a Categorization of this handbook, and for further details Freeden 1998, 2011; Freeden et al. 1998; Freeden and Maier 2002; Freeden and Schreiner 2009; Freeden and Gutting 2013), (spherical harmonic) Fourier methods are surely well suited to resolve low- and medium-frequency phenomena, while their application is critical to obtain high-resolution models. This difficulty is also well known to theoretical physics, e.g., when describing monochromatic electromagnetic waves or considering the quantum-mechanical treatment of free particles. In this case, plane waves with fixed frequencies (ideal frequency localization, no space localization) are the solutions of the corresponding differential equations, but do certainly not reflect the physical reality. As a remedy, plane waves of different frequencies are superposed to the so-called wave packages, which gain a certain amount of space localization while losing their ideal frequency (spectral) localization.

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A suitable superposition of polynomial ansatz functions (see Freeden and Schreiner 2009) leads to the so-called kernel functions/kernels with a reduced frequency but increased space localization (cf. Fig. 9). These kernels can be constructed as to cover various spectral bands and, hence, can show all intermediate stages of frequency and space localization. The width of the corresponding frequency and space localization is usually controlled using a so-called scale parameters. If the kernel is given by a finite superposition of polynomial ansatz functions, it is said to be bandlimited, while in the case of infinitely many ansatz functions, the kernel is called non-bandlimited. It turns out that due to their higher-frequency localization (short frequency band), the bandlimited kernels show less space localization than their non-bandlimited counterparts (infinite frequency band). This leads to the following characterization of ansatz functions: Fourier methods with polynomial trial functions are the canonical point of departure for approximations of low-frequency phenomena (global to regional modeling). Because of their excellent localization properties in the space domain, bandlimited and non-bandlimited kernels with increasing space localization properties can be used for stronger and stronger modeling of short-wavelength phenomena (local modeling). Using kernels of different scales, the modeling approach can be adapted to the localization properties of the physical process under consideration. By use of sequences of scale-dependent kernels tending to the Dirac kernel, i.e., the so-called Dirac sequences, a multiscale approximation (i.e., “zooming-in” process) can be established appropriately. Later on in this treatise (see also chapters  Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution,  Mathematical Properties Relevant to Geomagnetic Field Modeling,  Toroidal-Poloidal Decompositions of Electromagnetic Green’s Functions in Geomagnetic Induction,  The Forward and Adjoint Methods of Global Electromagnetic Induction for CHAMP Magnetic Data,  Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications,  Potential-Field Estimation Using Scalar and Vector Slepian Functions at Satellite Altitude, and  Rayleigh Wave Dispersive Properties of a Vector Displacement as a Tool for P- and S-wave Velocities Near Surface Profiling), we deal with simple (scalar and/or vectorial) wavelet techniques, i.e., with multiscale techniques based on special kernel functions: (spherical) scaling functions and wavelets. Typically, the generating functions of scaling functions have the characteristics of lowpass filters, i.e., the polynomial basis functions of higher frequencies are attenuated or even completely left out. The generating functions of wavelets, however, have the typical properties of band-pass filters, i.e., the polynomial basis functions of low and high frequencies are attenuated or even completely left out when constructing the wavelet. Thus, wavelet techniques usually lead to a multiresolution of the Hilbert space under consideration, i.e., a certain two-parameter splitting with respect to scale and space. To be more concrete, the Hilbert space under consideration can be decomposed into a nested sequence of approximating subspaces – the scale spaces – corresponding to the scale parameter. In each scale space, a model of the data function can usually be calculated using the respective scaling functions, thus leading to an approximation of the data at a certain resolution. For increasing scales, the approximation improves and the information obtained on coarse levels

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is contained in all levels of approximation above. The difference between two successive approximations is called the detail information, and it is contained in the so-called detail spaces. The wavelets constitute the basis functions of the detail spaces, and summarizing the subject, every element of the Hilbert space can be represented as a structured linear combination of scaling functions and wavelets of different scales and at different positions (“multiscale approximation”) (cf. Freeden et al. (1998); Freeden and Michel (2004); Michel (2002); Michel (2013), and the references therein). Hence, we have found a possibility to break up complicated functions like geomagnetic field, electric currents, gravitational field, deformation field, oceanic currents, propagation speed of seismic waves, etc., into single pieces of different resolutions and to analyze these pieces separately and to decorrelate certain features. This helps to find adaptive methods (cf. Fig. 10) that take into account the specific structure of the data, i.e., in areas where the data show only a few coarse spatial structures, the resolution of the model can be chosen to be rather low; in areas of complicated data structures, the resolution can be increased accordingly. In areas where the accuracy inherent in the measurements is reached, the solution process can be stopped by some kind of thresholding. That is, using scaling functions and wavelets at different scales, the corresponding approximation techniques can be constructed as to be suitable for the particular data situation. Consequently, although most data show correlation in space as well as in frequency, the kernel functions with their simultaneous space and frequency localization allow for the efficient detection and approximation of essential features in the data structure by only using fractions of the original information (decorrelation of signatures) (Fig. 11). Finally, it is worth mentioning that future spaceborne observation combined with terrestrial and airborne activities will provide huge data sets of the order of millions of data to be continued downward to the Earth’s surface (see chapters  Earth Observation Satellite Missions and Data Access,  Satellite-to-Satellite Tracking (Low-Low/High-Low SST),  GOCE: Gravitational Gradiometry in a Satellite,  Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation,  Mathematical Properties Relevant to Geomagnetic Field Modeling,  Using B-Spline Expansions for Ionosphere Modeling,  Radio Occultation via Satellites, and  Analysis of Data from Multi-satellite Geospace Missions concerning different fields of observation). Standard mathematical theory and numerical methods are not at all adequate for the solution of data systems with a structure such as this, because these methods are simply not adapted to the specific character of the spaceborne problems. They quickly reach their capacity limit even on very powerful computers. In our opinion, a reconstruction of significant geophysical quantities from future data material requires much more: for example, it requires a careful analysis, fast solution techniques, and a proper stabilization of the solution, usually including procedures of regularization (see Freeden 1999; Freeden and Mayer 2003 and the references therein). In order to achieve these objectives, various strategies and structures must be introduced reflecting different aspects (cf. Fig. 12). As already pointed out, while global long-wavelength modeling can be adequately done by the use of polynomial expansions, it becomes more and more obvious

Fig. 10 Scaling functions (upper row) and wavelet functions (lower row) in mutual relation (“tree structure”) within a multiscale reconstruction

20 W. Freeden

Fig. 11 Global multiscale reconstruction of the Earth’s Gravitational Model EGM96 (based on data taken from F. G. Lemoine et al. 1998)

Geomathematics: Its Role, Its Aim, and Its Potential 21

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Fig. 12 Structural principles and methods

that splines and/or wavelets are most likely the candidates for medium- and shortwavelength approximation. But the multiscale concept of wavelets demands its own nature which – in most cases – cannot be developed from the well-known theory in Euclidean spaces. In consequence, the stage is also set to present the essential ideas and results involving a multiscale framework to the geoscientific community (see chapters  Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution,  Multiresolution Analysis of Hydrology and Satellite Gravitational Data,  Multiscale Model Reduction with Generalized Multiscale Finite Element Methods in Geomathematics,  Efficient Modeling of Flow and Transport in Porous Media Using Multi-physics and Multi-scale Approaches,  Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents,  Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications,  Multiparameter Regularization in Downward Continuation of Satellite Data,  Multidimensional Seismic Compression by Hybrid Transform with Multiscale-Based Coding,  Tomography: Problems and Multiscale Solutions,  Splines and Wavelets on Geophysically Relevant Manifolds, and  Multiscale Approximation for different realizations).

8

Geomathematics: Three Exemplary “Circuits”

In the sequel, the “circuit” geomathematics as solution potential will be demonstrated with respect to contents, origin, and intention on three completely different geoscientifically relevant examples, namely, the determination of the gravity field from terrestrial deflections of the vertical, the computation of (geostrophic) oceanic

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circulation from altimeter satellite data (i.e., RADAR data), and seismic processing from acoustic wave propagation. Their solution can be performed within the same mathematical context; it can be actually described within the same apparatus of concepts and formulas based on the use of fundamental solutions of associated partial differential equations and their appropriate regularizations. Nevertheless, the numerical methods must be specifically adapted to the concrete solution of any of the three problems; in this respect they are basically dissimilar. In the first example (i.e., the gravity field determination), we are confronted with a process of integrating the vectorial surface gradient equation; in the second example, (i.e., modeling of the oceanic circulation), we have to execute a process of differentiation following the vectorial surface curl gradient equation – in both cases to discretely given data which are assumed to be available on spheres, for simplicity. The first example uses vectorial data on the Earth’s surface, and the second example utilizes scalar satellite altimetry data. The essential goal of the third example concerned with seismic (post)processing is to transfer the existing signal, which resulted from acoustic wave propagation from seismograms, under the expectation that designated properties of the target bedrock such as a migration result or a velocity field model can be interpreted from the transformed information. In more detail, the acoustic waves are reflected at the places of impedance contrast (rapid changes of the medium density), propagated back, and then recorded on the surface and/or in available boreholes by receivers of seismic energy. The recorded seismograms are carefully processed to detect fractures along with their location, orientation, and aperture, which are needed for interpretation and definition of the target reservoir. In all our three examples, our work is based on a simple regularization procedure of fundamental solutions to associated partial differential equations and the realization of a multiscale approach leading to locally supported wavelets.

8.1

Circuit: Gravity Field from Deflections of the Vertical

The modeling of the gravity field and its equipotential surfaces, especially the geoid of the Earth (see chapters  Earth Observation Satellite Missions and Data Access,  Satellite-to-Satellite Tracking (Low-Low/High-Low SST),  GOCE: Gravitational Gradiometry in a Satellite,  Classical Physical Geodesy,  Geodetic Boundary Value Problem,  Time-Variable Gravity Field and Global Deformation of the Earth,  Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution,  Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics,  Multiresolution Analysis of Hydrology and Satellite Gravitational Data,  Time-Varying Mean Sea Level,  Gravitational Viscoelastodynamics,  Oblique Stochastic Boundary-Value Problem,  It’s All About Statistics: Global Gravity Field Modeling from GOCE and Complementary Data,  Analysis of Data from Multi-satellite Geospace Missions, and  Geodetic World Height System Unification), is essential for many applications (see Fig. 13 for a graphical illustration), from

Fig. 13 Gravity-involved processes (From “German Priority Research Program: Mass transport and mass distribution in the system Earth, DFG–SPP 1257”)

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which we only mention some significant examples (essentially following Rummel 2002): Earth system: there is a growing awareness of global environmental problems (e.g., the CO2 question, the rapid decrease of rain forests, global sea level changes, etc.). What is the role of the future terrestrial campaigns, airborne duties, and satellite activities in this context? They do not tell us the reasons for physical processes, but it is essential to bring the phenomena into one system (e.g., to make sea level records comparable in different parts of the world). In other words, the geoid, i.e., the equipotential surface at sea level as defined by Listing (1873), is viewed as an almost static reference for many rapidly changing processes and at the same time as a “frozen picture” of tectonic processes that evolved over geological time spans. Solid earth physics: the gravity anomaly field has its origin mainly in mass inhomogeneities of the continental and oceanic lithosphere. Together with height information and regional tomography, a much deeper understanding of tectonic processes should be obtainable in the future. Physical oceanography: the altimeter satellites in combination with a precise geoid will deliver global dynamic ocean topography. Global surface circulation can be computed resulting in a completely new dimension of ocean modeling. Circulation allows the determination of transport processes of, e.g., polluted material. Satellite orbits: for any positioning from space, the uncertainty in the orbit of the spacecraft is the limiting factor. The future spaceborne techniques will basically eliminate all gravitational uncertainties in satellite orbits. Geodesy and civil engineering: accurate heights are needed for civil constructions, mapping, etc. They are obtained by leveling, a very time-consuming and expensive procedure. Nowadays geometric heights can be obtained fast and efficiently from space positioning (e.g., GPS, GLONASS, and (future) GALILEO). The geometric heights are convertible to leveled heights by subtracting the precise geoid (see Fig. 22), which is implied by a high-resolution gravitational potential. To be more specific, in those areas where good gravity information is already available, the future data information will eliminate all mediumand long-wavelength distortions in unsurveyed areas. GPS, GLONASS, and GALILEO together with the planned explorer satellite missions for the past 2015 time frame will provide extremely high-quality height information at the global scale. Exploration geophysics and prospecting: airborne gravity measurements have usually been used together with aeromagnetic surveys, but the poor precision of airborne gravity measurements has hindered a wider use of this type of measurements. Strong improvements can be expected by combination with terrestrial and spaceborne observations in the future scenario. The basic interest (see, e.g., Bauer et al. (2014)) in gravitational methods in exploration is based on the small variations in the gravitational field anomalies in relation to, e.g., an ellipsoidal reference model, i.e., the so-called “normal” gravitational field.

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Mathematical Modeling of the Gravity Field (Classical Approach) Gravity as provided on the Earth’s surface by absolute and/or relative measurements (see Fig. 14) is the combined effect of the gravitational mass attraction and the centrifugal force due to the Earth’s rotation. The force of gravity provides a directional structure to the space above the Earth’s surface. It is tangential to the vertical plumb lines and perpendicular to all level surfaces. Any water surface at rest is part of a level surface. As if the Earth were a homogeneous, spherical body, gravity turns out to be constant all over the Earth’s surface, the well-known quantity 9:8 ms2 . The plumb lines are directed toward the Earth’s center of mass, and this implies that all level surfaces are nearly spherical, too. The gravity decreases from the poles to the equator by about 0:05 ms2 (see Fig. 15). This is caused by the flattening of the Earth’s figure and the negative effect of the centrifugal force, which is maximal at the equator. High mountains and deep ocean trenches (cf. Fig. 16) cause the gravity to vary. Materials within the Earth’s interior are not uniformly distributed. The irregular gravity field shapes the geoid as virtual surface. The level surfaces are ideal reference surfaces, for example, for heights. The gravity acceleration (gravity) w is the resultant of gravitation v and centrifugal acceleration c (see chapters  Classical Physical Geodesy,  Geodetic Boundary Value Problem,  Time-Varying Mean Sea Level, and  Geodetic World Height System Unification): w D v C c:

(1)

The centrifugal force c arises as a result of the rotation of the Earth about its axis. Here, we assume a rotation of constant angular velocity  around the rotational axis

Fig. 14 The falling apple: Newton’s approach to absolute (left) and relative (right) gravity observation

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Fig. 15 Illustration of the gravity intensity: constant, i.e., 9:8 ms2 (left), decreasing to the poles by about 0:05 ms2 (mid), and real simulation (right)

Fig. 16 Illustration of the constituents of the gravity intensity g (ESA medialab, ESA communication production SP–1314) Fig. 17 Gravitation v; centrifugal acceleration c; gravity acceleration w

direction of plumb line

x v

c

w

center of mass

x3 , which is further assumed to be fixed with respect to the Earth. The centrifugal acceleration acting on a unit mass is directed outward, perpendicularly to the spin axis (see Fig. 17). If the  3 -axis of an Earth-fixed coordinate system coincides with the axis of rotation, then we have c D rC; where C is the so-called centrifugal potential (with f 1 ;  2 ;  3 g the canonical orthonormal system in Euclidean space R3 ). The direction of the gravity w is known as the direction of the plumb line, and the quantity g D jwj is called the gravity intensity (often just gravity). The gravity potential of the Earth can be expressed in the form

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W D V C C:

(2)

The (vectorial) gravity acceleration w (see chapter  Gravitational Viscoelastodynamics for more details) is given by w D rW D rV C rC:

(3)

The surfaces of constant gravity potential W .x/ D const, x 2 R3 , are designated as equipotential (level, or geopotential) surfaces of gravity. The gravity potential W of the Earth is the sum of the gravitational potential V and the centrifugal potential C , i.e., W D V C C . In an Earth-fixed coordinate system, the centrifugal potential C is explicitly known. Hence, the determination of equipotential surfaces of the potential W is strongly related to the knowledge of the gravitational potential V . The gravity vector w given by w D rW is normal to the equipotential surface passing through the same point. Thus, equipotential surfaces intuitively express the notion of tangential surfaces, as they are normal to the plumb lines given by the direction of the gravity vector (cf. Fig. 18). The traditional concept in gravitational field modeling is based on the assumption that all over the Earth the position (i.e., latitude and longitude) and the scalar gravity g are available. Moreover, it is common practice that the gravitational effects of the sun, moon, Earth’s atmosphere, etc., are accounted for by means of corrections. The gravitational part of the gravity potential can then be regarded as a harmonic function in the exterior of the Earth. A classical approach to gravity field modeling was conceived by Stokes (1849), Helmert (1981), Neumann (1887). They proposed to reduce the given gravity accelerations from the Earth’s surface to the geoid. As the geoid is a level surface, its potential value is constant. The difference between the reduced gravity on the geoid and the reference gravity on the so-called normal ellipsoid is called the gravity anomaly. The disturbing potential, i.e., the difference between the actual and the reference potential, can be obtained from a (third) boundary value problem of potential theory. Its solution is representable in integral form, i.e., by the Stokes integral. The disadvantage of the Stokes approach is that the reduction to the geoid requires the introduction of assumptions concerning the unknown mass distribution between the Earth’s surface and the geoid (for more details concerning the classical theory, the reader is referred, e.g., to chapter  Gravitational Viscoelastodynamics and the references therein).

Fig. 18 Level surface and plumb line

(x) x w(x) level surface plumb line

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Fig. 19 Regularity at infinity

y

0

x 2

x

Next, we briefly recapitulate the classical approach to global gravity field determination by formulating the differential/integral relations between gravity disturbance, gravity anomaly, and deflections of the vertical on the one hand, and the disturbing potential and the geoidal undulations on the other hand. The representations of the disturbing potential in terms of gravity disturbances, gravity anomalies, and deflections of the vertical are written in terms of wellknown integral representations over the geoid. For practical purposes the integrals are replaced by approximate formulas using certain integration weights and knots conventionally within a spherical framework (see chapter  Numerical Integration on the Sphere for a survey paper on numerical integration on the sphere). Equipotential surfaces of the gravity potential W allow in general no simple representation (see Figs. 20, 21, 22). This is the reason why a reference surface – in physical geodesy usually an ellipsoid of revolution – is chosen for the (approximate) construction of the geoid (see Fig. 22). As a matter of fact, the deviations of the gravity field of the Earth from the normal field of such an ellipsoid are small. The remaining parts of the gravity field are gathered in a so-called disturbing gravity field rT corresponding to the disturbing potential T . Knowing the gravity potential, all equipotential surfaces – including the geoid – are given by an equation of the form W .x/ D const. By introducing U as the normal gravity potential corresponding to the ellipsoidal field and T as the disturbing potential, we are led to a decomposition of the gravity potential in the form W DU CT

(4)

such that (C1) (C2)

the center of the ellipsoid coincides with the center of gravity of the Earth, the difference of the mass of the Earth and the mass of the ellipsoid is zero.

According to the classical Newton law of gravitation (1687), knowing the density distribution of a body, the gravitational potential can be computed everywhere in R3 (see chapter  Time-Varying Mean Sea Level for more information in time–spacedependent relation). More explicitly, the gravitational potential V of the Earth’s R G exterior is given by V .x/ D 4 .y/jx  yj1 dV .y/; x 2 R3 nEarth; where Earth G is the gravitational constant (G D 6:67421011 m3 kg1 s2 ) and  is the density

30

W. Freeden

Fig. 20 Geoidal surface over oceans (top) and over the whole Earth (bottom), modeled by smoothed Haar wavelets, Freeden et al. (2009), Geomathematics Group, Kaiserslautern

function. The properties of the gravitational potential V in the Earth’s exterior are easily described as follows: V is harmonic in x 2 R3 nEarth; i.e., V .x/ D 0; x 2 R3 nEarth. Moreover, the gravitational potential V is regular at infinity, i.e.,  1 ; jxj ! 1; jxj   1 jrV .x/j DD O ; jxj ! 1: jxj2 

jV .x/j DD O

(5) (6)

Geomathematics: Its Role, Its Aim, and Its Potential

31

Fig. 21 Geoidal surface (provided by R. Haagmans, European Space Agency, Earth Surfaces and Interior Section, ESTEC, Noordwijk, ESA ID number SEMLXEOA90E)

Fig. 22 Illustration of the gravity anomaly vector ˛.x/ D w.x/  u.y/ and the gravity disturbance vector ı.x/ D w.x/  u.x/

(x)

’(x) geoid W = const = W0

x

w(x)

u(x)

N(x)

geoidal height

y

u(y)

reference ellipsoid U = const = U0

Note that for suitably large values jxj (see Fig. 19), we have jyj  12 jxj, hence, jx  yj  jjxj  jyjj  12 jxj. However, the actual problem is that in reality the density distribution  is very irregular and only known for parts of the upper crust of the Earth. Actually, geoscientists would like to know it from measuring the gravitational field (gravimetry problem). Even if the Earth is supposed to be spherical, the determination of the gravitational potential by integrating Newton’s potential is not achievable (see chapters  Classical Physical Geodesy,  Modeling Deep Geothermal Reservoirs: Recent Advances and Future Perspectives,  Sparsity in Inverse Geophysical Problems, and  RFMP: An Iterative Best Basis Algorithm for Inverse Problems in the Geosciences for inversion methods of Newton’s law). Remark 1. As already mentioned of, the classical remedy avoiding any knowledge of the density inside the Earth is the formulation boundary value problems of potential theory to determine the external gravitational potential from terrestrial

32

W. Freeden

data. For reasons of demonstration, however, here we do not follow the standard (Vening–Meinesz) approach in physical geodesy of obtaining the disturbing potential from deflections of the vertical as terrestrial data (as explained in chapter  Classical Physical Geodesy). Our approach is based on the context as developed by Freeden and Schreiner (2009) and Freeden and Wolf (2008). Furthermore, it should be noted that the determination of the Earth’s gravitational potential under the assumptions of nonspherical geometry and terrestrial oblique derivatives in form of a (deterministic or stochastic) boundary value problem of potential theory is discussed, e.g., in chapters  Geodetic Boundary Value Problem and  Oblique Stochastic Boundary-Value Problem. Further basic details concerning oblique derivative problems can be found in Freeden and Michel (2004), Gutting (2007) and in chapters  Geodetic Boundary Value Problem,  Oblique Stochastic Boundary-Value Problem, and  Fast Spherical/Harmonic Spline Modeling. A point x of the geoid is projected onto the point y of the ellipsoid by means of the ellipsoidal normal (see Fig. 22). The distance between x and y is called the geoidal height or geoidal undulation. The gravity anomaly vector is defined as the difference between the gravity vector w.x/ and the normal gravity vector u.y/, u D rU , i.e., ˛.x/ D w.x/  u.y/ (see Fig. 22). It is also possible to distinguish the vectors w and u at the same point x to get the gravity disturbance vector ı.x/ D w.x/  u.x/: Of course, several basic mathematical relations between the quantities just mentioned are known (see, e.g., chapters  Classical Physical Geodesy and  Geodetic World Height System Unification). In what follows, we only heuristically describe the fundamental relations. We start by observing that the gravity disturbance vector at the point x can be written as ı.x/ D w.x/  u.x/ D r.W .x/  U .x// D rT .x/:

(7)

Expanding the potential U at x according to Taylor’s theorem and truncating the series at the linear term, we get @U : U .x/ D U .y/ C 0 .y/N .x/ @

(8)

: (D means approximation in linearized sense). Here,  0 .y/ is the ellipsoidal normal at y, i.e.,  0 .y/ D u.y/=.y/, .y/ D ju.y/j, and the geoid undulation N .x/, as indicated in Fig. 22, is the aforementioned distance between x and y, i.e., between the geoid and the reference ellipsoid. Using .y/ D ju.y/j D  0 .y/  u.y/ D  0 .y/  rU .y/ D 

@U .y/ @ 0

(9)

Geomathematics: Its Role, Its Aim, and Its Potential

33

we arrive at N .x/ D

T .x/  .W .x/  U .y// T .x/  .W .x/  U .y// D : ju.y/j .y/

Letting U .y/ D W .x/ D const D W0 , we obtain the so-called Bruns’ formula N .x/ D

T .x/ : .y/

(10)

It should be noted that Bruns’ formula (10) relates the physical quantity T to the geometric quantity N (cf. Bruns 1878). In what follows we are interested in introducing the deflections of the vertical of the gravity disturbing potential T . For this purpose, let us consider the vector field .x/ D w.x/=jw.x/j. This gives us the identity (with g.x/ D jw.x/j and .x/ D ju.x/j) w.x/ D rW .x/ D jw.x/j .x/ D g.x/.x/:

(11)

Furthermore, we have u.x/ D rU .x/ D ju.x/j  0 .x/ D .x/ 0 .x/:

(12)

The deflection of the vertical ‚.x/ at the point x on the geoid is defined to be the angular (i.e., tangential) difference between the directions .x/ and  0 .x/, i.e., the plumb line and the ellipsoidal normal in the same point:   ‚.x/ D .x/   0 .x/  ..x/   0 .x//  .x/ .x/:

(13)

Clearly, because of (13), ‚.x/ is orthogonal to .x/, i.e., ‚.x/  .x/ D 0. Since the plumb lines are orthogonal to the level surfaces of the geoid and the ellipsoid, respectively, the deflections of the vertical give briefly spoken a measure of the gradient of the level surfaces. This aspect will be described in more detail below: from (11), we obtain, in connection with (13), w.x/ D rW .x/



0

0



(14)

D jw.x/j ‚.x/ C  .x/ C ...x/   .x//  .x//.x/ : Altogether, we get for the gravity disturbance vector w.x/  u.x/ D rT .x/



   D jw.x/j ‚.x/ C ..x/   0 .x//  .x/ .x/  .jw.x/j  ju.x/j/  0 .x/:

(15)

34

W. Freeden

The magnitude D.x/ D jw.x/j  ju.x/j D g.x/  .x/

(16)

is called the gravity disturbance, while A.x/ D jw.x/j  ju.y/j D g.x/  .y/

(17)

is called the gravity anomaly. Since the vector .x/   0 .x/ is (almost) orthogonal to  0 .x/, it can be neglected in (15). Hence, it follows that w.x/  u.x/ D rT .x/ : D jw.x/j‚.x/  .jw.x/j  ju.x/j/  0 .x/:

(18)

The gradient rT .x/ can be split into a normal part (pointing in the direction of  0 .x/) and an angular (tangential) part (characterized by the surface gradient r  ). It follows that rT .x/ D

@T 1  r T .x/: .x/ 0 .x/ C 0 @ jxj

(19)

By comparison of (18) and (19), we therefore obtain D.x/ D g.x/  .x/ D jw.x/j  ju.x/j D 

@T .x/; @ 0

(20)

i.e., the gravity disturbance, besides being the difference in magnitude of the actual and the normal gravity vector, is also the normal component of the gravity disturbance vector. In addition, we are led to the angular, i.e., (tangential) differential, equation 1  r T .x/ D jw.x/j ‚.x/: jxj

(21)

Remark 2. The reference ellipsoid deviates from a sphere only by quantities of the order of the flattening. Therefore, in numerical calculations, we treat the reference ellipsoid as a sphere around the origin with (certain) mean radius R. This may cause a relative error of the same order (for more details the reader is referred to standard textbooks of physical geodesy). In this way, together with suitable prereduction processes of gravity, formulas are obtained that are rigorously valid for the sphere. Remark 3. Since j‚.x/j is a small quantity, it may be (without loss of precision) multiplied either by jw.x/j or by ju.x/j, i.e., g.x/ or by .x/.

Geomathematics: Its Role, Its Aim, and Its Potential

35

In well-known spherical approximation, we have .y/ D ju.y/j D

GM ; jyj2

(22)

y GM @  ry .y/ D 2 3 .y/ D 0 @ jyj jyj

(23)

1 @ 2 .y/ D  ; 0 .y/ @ jyj

(24)

and

where G is the gravitational constant and M is the mass. If now – as explained above – the relative error between normal ellipsoid and mean sphere of radius R is permissible, we are allowed to go over to the spherical nomenclature x D R ; R D jxj; 2 S2 with S2 being the unit sphere in R3 : Replacing ju.R /j by its spherical approximation GM =R2, we find r  T .R / D 

GM ‚.R /; R

2 S2 :

(25)

By virtue of Bruns’ formula (10), we finally find the relation between geoidal undulations and deflections of the vertical GM  GM ‚.R /; r N .R / D  2 R R

2 S2 ;

(26)

i.e., r  N .R / D R ‚.R /;

2 S2 :

(27)

In other words, the knowledge of the geoidal undulations allows the determination of the deflections of the vertical by taking the surface gradient on the unit sphere. From the identity (20), it follows that 

@T .x/ D D.x/ D jw.x/j  j.x/j @ 0

(28)

@ : D jw.x/j  j.y/j  0 .y/ N .x/ @ @ D A.x/  0 .y/ N .x/; @ where A represents the scalar gravity anomaly as defined by (17). Observing Bruns’ formula, we get

36

W. Freeden

A.x/ D 

@T 1 @ .x/ C .y/ T .x/: @ 0 .y/ @ 0

(29)

In the sense of physical geodesy, the meaning of the spherical approximation should be carefully kept in mind. It is used only for expressions relating to small quantities of the disturbing potential, the geoidal undulations, the gravity disturbances, the gravity anomalies, etc. Together with the Laplace equation in the exterior of the sphere around the origin with radius R and the regularity at infinity, the equations  D.x/ D

x  rT .x/ jxj

(30)

and  A.x/ D

x 2  rT .x/ C T .x/ jxj R

(31)

represent the so-called fundamental equations of physical geodesy in spherical approximation (see, e.g., Heiskanen and Moritz 1967; Groten 1979; Torge 1991). Actually, the identities (30) and (31), respectively, serve as boundary conditions of boundary value problems of potential theory, which are known as Neumann and Stokes problem (see also chapters  Classical Physical Geodesy and  Geodetic Boundary Value Problem of this handbook for more details). The study of Figs. 23–26 leads us to the following remarks: the gravity disturbances, which enable a physically oriented comparison of the real Earth with the ellipsoidal Earth model, are consequences of the imbalance of forces inside the Earth according to Newton’s law of gravitation. It leads to the suggestion of density anomalies. In analogy the difference between the actual level surfaces of the gravity potential and the level surfaces of the model body forms a measure for the deviation of the Earth from a hydrostatic status of balance. In particular, the geoidal undulations (geoidal heights) represent the deflections from the equipotential surface on the mean level of the ellipsoid. The geoidal anomalies generally show no essential correlation to the distribution of the continents (see Fig. 20). It is conjecturable that the geoidal undulations mainly depend on the reciprocal distance of the density anomaly. They are influenced by lateral density variations of large vertical extension, from the core–mantle layer to the crustal layers. In fact, the direct geoidal signal, which would result from the attraction of the continental and the oceanic bottom, would be several hundreds of meters (for more details, see Rummel (2002) and the references therein). In consequence, the weights of the continental and oceanic masses in the Earth’s interior are almost perfectly balanced. This is the phenomenon of isostatic compensation that was observed by P. Bouguer already in the year 1750.

Geomathematics: Its Role, Its Aim, and Its Potential

37

Fig. 23 Absolute values of the deflections of the vertical and their directions computed from EGM96 (cf. Lemoine et al. 1996) from degree 2 up to degree 360 (reconstructed by use of spacelimited (locally supported) scaling functions (from the PhD-thesis Wolf (2009), Geomathematics Group, University of Kaiserslautern)) (min D 0, max D 3:0  104 )

75° N

180° w

120° w

60° w

0°

60° E

120° E

180° E

60° N 45° N 30° N 15° N 0° 15° S 30° S 45° S 60° S 75° S

–1000

–800

–600

–400

–200

0

200

400

600

800

[m2/s2]

Fig. 24 Disturbing potential computed from EGM96 from degree 2 up to degree 360 (reconstructed by use of space-limited (locally supported) scaling functions (from the PhD-thesis Wolf (2009), Geomathematics Group, University of Kaiserslautern)) (min  1 038 m2 =s2 , max  833 m2 =s2 )

38

W. Freeden

75° N

180° w

120° w

60° w

0°

60° E 120° E 180° E

60° N 45° N 30° N 15° N 0° 15° S 30° S 45° S 60° S 75° S

–1.5

–1

–0.5

0

0.5

1

[m/s2]

1.5 x10–3]

Fig. 25 Gravity disturbances computed from EGM96 from degree 2 up to degree 360 (reconstructed by the use of space-limited (locally supported) scaling functions (from the PhD-thesis Wolf (2009), Geomathematics Group, University of Kaiserslautern)) (min D 3:6  103 m=s2 , max D 4:4  103 m=s2 )

75° N

180° w 120° w

60° w

0°

60° E

120° E 180° E

60° N 45° N 30° N 15° N 0° 15° S 30° S 45° S 60° S 75° S

–1.5

–1

–0.5

0

0.5 2

[m/s ]

1

1.5 –3]

x10

Fig. 26 Gravity anomalies computed from EGM96 from degree 2 up to degree 360 (reconstructed by use of space-limited (locally supported) scaling functions (from the PhD-thesis Wolf (2009), Geomathematics Group, University of Kaiserslautern)) (min D 3:4  103 m=s2 , max D 4:4  103 m=s2 )

Geomathematics: Its Role, Its Aim, and Its Potential

39

Mathematical Analysis Equation (27) under consideration is of vectorial tangential type with the unit sphere S2 in Euclidian space R3 as the domain of definition (see also chapters  Gauss’ and Weber’s “Atlas of Geomagnetism” (1840) Was not the first: the History of the Geomagnetic Atlases,  Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation,  Convection Structures of Binary Fluid Mixtures in Porous Media,  Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents,  Toroidal-Poloidal Decompositions of Electromagnetic Green’s Functions in Geomagnetic Induction,  Using B-Spline Expansions for Ionosphere Modeling, and  Potential-Field Estimation Using Scalar and Vector Slepian Functions at Satellite Altitude for relations to geomagnetism). More precisely, we are confronted with a surface gradient equation r  P D p with p as given continuous vector field (i.e., p. / D R‚.R /; 2 S2 ) and P as continuously differentiable scalar field that is desired to be reconstructed. In the previously formulated abstraction, the determination of the surface potential function P via the equation r  P D p is certainly not uniquely solvable. We are able to add an arbitrary constant to P without changing the equation. For our problem, however, this argument is not valid, since we have to observe additional integration relations resulting from the conditions (C1) and (C2), namely, Z P . /. i  /k dS . / D 0I k D 0; 1I i D 1; 2; 3:

(32)

S2

Consequently, we are able to guarantee uniqueness. The solution theory is based on the Green theorem (see Freeden et al. 1998; Freeden and Schreiner 2009; Freeden and Gerhards 2010; Gerhards 2011; Freeden and Gerhards 2013)

P . / D

1 4

Z

Z P . / dS . /  S2

S2

r  G.  /  r  P . / dS . /;

(33)

where t 7! G.t/ D

1 1 ln.1  t/ C .1  ln 2/; t 2 Œ1; 1/; 4 4

(34)

is the Green function (i.e., the fundamental solution) with respect to the (Laplace)– Beltrami operator on the unit sphere S2 . This leads us to the following result: suppose that p is a given continuous tangential, curl-free vector field on the unit sphere S2 : Then P given by P . / D

1 4

Z S2

1 .  .  / /  p. / dS . /; 1 

2 S2 ;

(35)

40

W. Freeden

is the uniquely determined solution of the surface gradient equation r  P D p with Z P . / dS . / D 0:

(36)

S2

Consequently, the existence and uniqueness of the equation is assured. Even more, the solution admits a representation in the form of the singular integral (35).

Development of a Mathematical Solution Method Precise terrestrial data of the deflections of the vertical are not available all over the Earth in dense distribution. They exist, e.g., on continental areas in much larger density than on oceanic ones. In order to exhaust the existent scattered data reservoir, we are not allowed to apply a Fourier technique in terms of spherical harmonics (note that, according to Weyl’s concept, the integrability is equivalently interrelated to the equidistribution of the data points). For more details see Weyl (1916) (onedimensional theory); Freeden et al. (1998); Freeden and Wolf (2008); Freeden and Schreiner (2009); Freeden and Gerhards (2010) (Spherical Theory). Instead, we have to use an appropriate zooming-in procedure which starts globally from (initial) rough data width (low scale) and proceeds to more and more finer local data width (higher scales). A simple solution for a multiscale approximation consists of appropriate “regularization” of the singular kernel. ˆ W t 7! ˆ.t/ D .1  t/1 ;

1  t < 1;

(37)

in (35) by the continuous kernel ˆ W t 7! ˆ .t/, t 2 Œ1; 1 ,  2 .0; 2/, given by  ˆ .t/ D

1 1t 1 

; 1  t > ; ; 1  t  :

(38)

In fact, it is not difficult to verify (see Freeden and Schreiner (2009) and the references therein) that 1 P . / D 4

Z S2

ˆ .  /.  .  / /  p. / dS . /; 2 S2 ;

(39)

where the surface curl-free space-regularized Green vector scaling kernel is given by . ; / 7! ˆ .  /.  .  / /; ; 2 S2 ;

(40)

satisfies the following limit relation: lim sup jP . /  P . /j D 0:

 !0 2S2  >0

(41)

Geomathematics: Its Role, Its Aim, and Its Potential

41

Furthermore, in the scale discrete formulation using a strictly monotonically decreasing sequence .j /j 2N0 converging to zero with j 2 .0; 2 (for instance, j D 21j or j D 1  cos.2j /, j 2 N0 ), we are allowed to write the surface curl-free space-regularized Green vector scaling kernel as follows:

j . ; / D ˆj .  /.  .  / /; ; 2 S2 :

(42)

The surface curl-free space-regularized Green vector wavelet kernel (cf. Fig. 27) then reads as j . ; /

D ‰j .  /.  .  / /; ; 2 S2 ;

(43)

where ‰j D ˆj C1  ˆj

(44)

is explicitly given by 8 ˆ < ‰j .t/ D

ˆ :

0 1 1 1t  j 1  1j j C1

; ;

1  t > j ; j  1  t > j C1 ;

;

j C1 > 1  t:

j . ; /

 p. / dS . /;

(45)

Wj defined by Wj . / D

1 4

Z S2

2 S2 ;

(46)

leads us to the recursion Pj C1 D Pj C Wj :

(47)

Hence, we immediately get by elementary manipulations for all m 2 N (cf. Fig. 28) Pj Cm D Pj C

m1 X

Wj Ck :

(48)

kD0

The functions Pj represent low-pass filters of P . Obviously, Pj is improved by the band-pass filter Wj in order to obtain Pj C1 , while Pj C1 is improved by the band-pass filter Wj C1 in order to obtain Pj C2 , etc. Summarizing our results, we are allowed to formulate the following conclusion: three features are incorporated in our way of thinking about multiscale approximation by the use of locally supported wavelets, namely, basis property, decorrelation, and fast computation. More concretely, our vector wavelets are “building blocks” for huge discrete data sets. By virtue of the basis property, the function P can be better and better approximated from p with increasing scale j . Our wavelets have the

42

W. Freeden

75° N

180° w

90° w

0°

90° E

180° E

60° N 45° N 30° N 15° N 0° 15° S 30° S 45° S 60° S

0

75° S

10

20

30

40

50

60

[m/s2]

60° N

75° N

180° w

0°

90° w

90° E

180° E

45° N 30° N 15° N 0° 15° S 30° S 45° S 60° S

75° S

10

0

75° N 60° N

20 180° w

30 [m/s2]

40

0°

90° E

90° w

50

60

50

60

180° E

45° N 30° N 15° N 0° 15° S 30° S 45° S 60° S

0

75° S

10

20

30

40 [m/s2]

Fig. 27 Absolute values and the directions of the surface curl-free space-regularized Green vector scaling kernel j . ; /; ; 2 S2 , as defined by Eq. (40) with fixed and located at 00 N, 00 W and j D 21j for scales j D 1; 2; 3; respectively

2

2.5

3

3.5

4

4.5

5

90° W

85° W

60° W

3

250

4.5

5 60° W

250

95° W 5° N

90° W

95° W 5° N

90° W

35

40

45

0.5

100

150

200

5° S

2.5° S

0°

2.5° N

50

100

150

200

5° S

2.5° S

0°

2.5° N

85° W

30° S

50

100

150

200

5° S

2.5° S

0°

2.5° N

5

5

5

30° S

10

10

10

15° S

15

15

15

15° S

20

20

20

250

50

85° W

90° W

25

0° 0°

15° N

120° W 30° N

85° W

1

30° S 45° S 60° S 75° S

15° S

25

35

40

45

4

75° N 60° N 45° N 30° N

25

90° W

60° W

1 3.5

180° E

30

95° W 5° N

90° W

2.5

90° E

30

15° N

120° W 30° N

2

0°

30

35

40

45

1.5

90° w

50

100

150

200

250

5° S

2.5° S

0°

2.5° N

95° W 5° N

30° S

15° S

0°

0°

2.5

120° W 30° N

2

90° w

15° N

1.5

180° w

3

90° W

90° W

4

180° E

3.5

90° E

4.5

85° W

60° W

5

10

15

20

25

30

35

40

45

50

100

150

200

250

5

Fig. 28 “Zooming-in” strategy choosing a hotspot (here, Galapagos (00 N,910 W)) as target area. The colored areas illustrate the local support of the wavelets for increasing scale levels (from the PhD-thesis Wolf (2009), Geomathematics Group, University of Kaiserslautern)

5° S

2.5° S

0°

2.5° N

95° W 5° N

30° S

15° S

0°

15° N

90° W

0.5

1

0.5

120° W 30° N

60° S 75° S

30° S 45° S

60° S 75° S

30° S 45° S

1.5

180° w

0°

75° N 60° N 45° N 30° N

15° S

180° E

15° S

90° E

15° N

0°

0°

90° w

15° N

180° w

15° N 0°

75° N 60° N 45° N 30° N

Geomathematics: Its Role, Its Aim, and Its Potential 43

44

W. Freeden

power to decorrelate. In other words, the representation of data in terms of wavelets is somehow “more compact” than the original representation. We search for an accurate approximation by only using a small fraction of the original information of a potential. Typically, our decorrelation is achieved by vector wavelets which have a compact support (localization in space) and which show a decay toward high frequencies (see Fig. 28). The main calamity in multiscale approximation is how to decompose the function under consideration into wavelet coefficients and how to efficiently reconstruct the function from the coefficients. There is a “tree algorithm” (cf. also Fig. 11) that makes these steps simple and fast (see Freeden et al. 1998): Wj

Wj C1

& & Pj ! ˚ ! Pj C1 ! ˚ ! Pj C2 : : : : The fast decorrelation power of wavelets is the key to applications such as data compression, fast data transmission, noise cancellation, signal recovering, etc. With increasing scales j ! 1, the supports become smaller and smaller. This is the reason why the calculation of the integrals has to be extended over smaller and smaller caps (“zooming in”). Of course, downsizing spherical caps and increasing data widths are in strong correlation. Thus, the variable width of the caps with increasing scale parameter j enables the integration of data sets of heterogeneous data width for local areas without violating Weyl’s law of equidistribution.

“Back-Transfer” to Application The multiscale techniques as presented here will be used to investigate the anomalous gravity field particularly for areas in which mantle plumes and hotspots occur. In this respect it should be noted that mantle plume is a geoscientifical term which denotes an upwelling of abnormally hot rocks within the Earth’s mantle (cf. Fig. 13). Plumes (cf. Ritter and Christensen 2007) are envisioned to be vertical conduits in which the hot mantle material rises buoyantly from the lower mantle to the lithosphere at velocities as large as 1 m yr1 , and these quasicylindrical regions have a typical diameter of about 100–200 km. In mantle convection theory, mantle plumes are related to hotspots which describe centers of surface volcanism that are not directly caused by plate tectonic processes. A hotspot is a long-term source of volcanism which is fixed relative to the plate overriding it. A classical example is Hawaii. Due to the local nature of plumes and hotspots such as Hawaii, we have to use high-resolution gravity models. Because of the lack of terrestrial-only data, the GFZ-combined gravity model EIGEN–GLO4C is used, which consists of satellite data, gravimetry, and altimetry surface data. The “zooming-in” property of our analysis is of great advantage. Especially the locally compact wavelet turns out to be an essential tool of the (vectorial) multiscale decomposition of the deflections of the vertical and correspondingly the (scalar) multiscale approximation of the disturbing potential. In the multiscale analysis (cf. Figs. 29 and 30), several interesting observations can be detected. By comparing the different positions and different scales, we can

Geomathematics: Its Role, Its Aim, and Its Potential

45

Fig. 29 Approximation of the vector-valued vertical deflections ‚ in [ms2 ] of the Hawaiian region with smoothed Haar wavelets (a rough low-pass filtering at scale 6 is improved by several band-pass filters of scale j D 6; : : : ; 11; where the last picture shows the multiscale approximation at scale j D 12) (from the PhD-thesis Fehlinger (2009), Geomathematics Group, University of Kaiserslautern)

46

Fig. 30 (continued)

W. Freeden

Geomathematics: Its Role, Its Aim, and Its Potential

47

see that the maximum of the “energy” contained in the signal of the disturbing potential – measured in the so-called wavelet variances (see Freeden and Michel 2004 for more details) – starts in the North West of the Hawaiian islands for scale j D 2 and travels in east southern direction with increasing scale. It ends up, for scale j D 12, in a position at the geologically youngest island under which the mantle plume is assumed to exist. Moreover, in the multiscale resolution with increasing scale, more and more local details of the disturbing potential appear. In particular, the structure of the Hawaiian island chain is clearly reflected in the scale and space decomposition. Obviously, the “energy peak” observed at the youngest island of Hawaii is highly above the “energy intensity level” of the rest of the island chain. This seems to strongly corroborate the belief of a stationary mantle plume, which is located beneath the Hawaiian islands and that is responsible for the creation of the Hawaii–Emperor seamount chain, while the oceanic lithosphere is continuously passing over it. An interesting area is the southeastern part of the chain, situated on the Hawaiian swell, a 1,200 km broad anomalously shallow region of the ocean floor, extending from the island of Hawaii to the Midway atoll. Here a distinct geoidal anomaly occurs that has its maximum around the youngest island that coincides with the maximum topography, and both decrease in northwestern direction.

8.2

Circuit: Oceanic Circulation from Ocean Topography

Ocean flow (see chapters  Time-Varying Mean Sea Level,  Self-Attraction and Loading of Oceanic Masses,  Unstructured Meshes in Large-Scale Ocean Modeling, and  Asymptotic Models for Atmospheric Flows) has a great influence on mass transport and heat exchange. By modeling oceanic currents, we therefore gain, for instance, a better understanding of weather and climate. In what follows we devote our attention to the geostrophic oceanic circulation on bounded regions on the sphere (and in a first approximation, the oceanic surfaces under consideration may be assumed to be parts of the boundary of a spherical Earth model), i.e., to oceanic flow under the simplifying assumptions of stationarity, spherically reflected horizontal velocity, and strict neglect of inner frictions. This leads us to inneroceanic long-scale currents, which still give meaningful results – as, for example, for the phenomenon of El Niño.

J Fig. 30 (continued) Multiscale reconstruction of the disturbing potential T in Œm2 s2 from vertical deflections for the Hawaiian (plume) area using regularized vector Green functions (a rough low-pass filtering at scale j D 6 is improved by several band-pass filters of scale j D 6; : : : ; 11; where the last illustration shows the approximation of the disturbing potential T at scale j D 12) (from the PhD-thesis Fehlinger (2009), Geomathematics Group, University of Kaiserslautern)

48

W. Freeden

Mathematical Modeling of Ocean Flow The numerical simulation of ocean currents is based on the Navier–Stokes equation. Its formulation (see, e.g., Ansorge and Sonar 2009) is well known: let us consider a fluid occupying an arbitrary (open and bounded) subdomain G0  R3 at time t D 0. The vector function v W Œ0; tend G0 ! Gt  R3 describes the motion of the particle positions  2 G0 with time, so that at times t  0 the fluid occupies the domain Gt D fv.tI /j 2 G0 g, respectively. Hence, Gt is a closed system in the sense that no fluid particle flows across its boundaries. The path of a particle  2 G0 is given by the graph of the function t 7! v.tI /; and the velocity of the fluid at a fixed location x D v.tI / 2 Gt by the derivative u.tI x/ D @t@ v.tI /: The derivation of the governing equations relies on the conservation of mass and momentum. The essential tool is the transport theorem, which shows how the time derivative of an integral over a domain changing with the time may be computed. The mass of a fluid occupying a domain is determined by the integral over the density of the fluid : Since the same amount of fluid occupying the R domain at time 0 later occupies the domain at time t > 0, we must have that G0 .0I x/ dV .x/ coincides with R Gt .tI x/ dV .x/ for all t 2 .0; tend : Therefore, the derivative of mass with respect to time must vanish, upon which the transport theorem yields for all t and Gt 

Z Gt

@ .tI x/ C div .u/.tI x/ @t

 dV .x/ D 0:

(49)

Since this is valid for arbitrary regions (in particular, for arbitrarily small ones), this implies that the integrand itself vanishes, which yields the continuity equation for compressible fluids @  C div .u/ D 0: @t

(50)

The momentum of a solid body is the product of its mass with its velocity Z .tI x/u.tI x/ dV .x/:

(51)

Gt

According to Newton’s second law, the same rate of change of (linear) momentum is equal to the sum of the forces acting on the fluid. We distinguish two types of forces, viz., body forces k (e.g., gravity, Coriolis force), which can be expressed as R .tI x/k.tI x/ dV .x/ with a given force density k per unit volume, and surface Gt R forces (e.g., pressure, internal friction) representable as @Gt  .tI x/.x/ dS .x/, which includes the stress tensor  .tI x/: Thus, Newton’s law reads d dt

Z

Z .tI x/u.tI x/ dV .x/ D Gt

Gt

Z .tI x/k.tI x/ dV .x/C

 .tI x/.x/ dS .x/: @Gt

(52)

Geomathematics: Its Role, Its Aim, and Its Potential

49

If we now apply the product rule and the transport theorem componentwise to the term on the left and apply the divergence theorem to the second term on the right, we obtain the momentum equation @ .u/ C .u  r/.u/ C .u/r  u D k C r   : @t

(53)

The nature of the oceanic flow equation depends heavily on the model used for the stress tensor. In the special case of incompressible fluids (here, ocean water) that is characterized by a density .tI x/ D 0 D const dependent neither on space nor on time, we find r  u D 0, i.e., u is divergence-free (for a discussion of (53) on the unit sphere, the reader is also referred to Fengler and Freeden (2005) and the references therein). When modeling an inviscid fluid, internal friction is neglected and the stress tensor is determined solely by the pressure  .tI x/ D P .tI x/i (i is the unit matrix). In the absence of inner friction (in consequence of, e.g., effects of wind and surface influences), we are able to ignore the derivative @u @t and, hence, the dependence on time. As relevant volume forces k, the gravity field w and the Coriolis force c D 2u ^ ! remain valid; they have to be observed. Finally, for largescale currents of the ocean, the nonlinear part does not play any role, i.e., the term .u  r/u is negligible. Under all these very restrictive assumptions, the equation of motion (53) reduces to the following identity : 2! ^ u D 

rP C w: 0

(54)

Even more, we suppose a velocity field of a spherical layer model. For each layer, i.e., for each sphere around the origin 0 with radius r. R/; the velocity field u can be decomposed into a normal field unor and a tangential field utan . The normal part is negligibly small in comparison to the tangential part (see the considerations in Pedlovsky (1979)). Therefore, we obtain with ! D  3 (note that the expression C . / D 2  . 3  /

(55)

is called the Coriolis parameter) the following separations of Eq. (54) (observe that  utan .r / D 0; 2 S2 ) : C . / ^ utan .r / D 

1  r P .r /; 0 r

(56)

and .2 ! ^ utan .r //  D 

1 @ P .r /  gr : 0 @r

(57)

Equation (56) essentially tells us that the tangential surface gradient is balanced by the Coriolis force. For simplicity, in our approach, we regard the gravity

50

W. Freeden

acceleration as a normal field: w.r / D gr ; 2 S2 (with gr as mean gravity intensity). Moreover, the vertical Coriolis acceleration in comparison to the tangential motion is very small, that is, we are allowed to assume that .2 ! ^ utan .R //  D 0 with C given by (55). On the surface of the Earth (here, r D R), we then obtain from (57) a direct relation of the product of the mean density and the mean gravity acceleration to the normal pressure gradient (hydrostatic approximation): @ P .R / D 0 gr : @r

(58)

This is the reason why we obtain the pressure by integration (see also chapters  Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution and  Toroidal-Poloidal Decompositions of Electromagnetic Green’s Functions in Geo-

magnetic Induction of this handbook) as follows: P .R / D 0 gR „.R / C PAtm ;

(59)

where PAtm denotes the mean atmospheric pressure. The quantity „.R / (cf. Fig. 31) is the difference between the heights of the ocean surface and the geoid at the point 2 S2 . The scalar function 7! „.R /; 2 S2 ; is called ocean topography. By use of altimeter satellites, we are able to measure the difference H between the known satellite height HSat and the (unknown) height of the ocean surface HOcean : H D HSat  HOcean . After calculation of HOcean , we then get the ocean topography „ D HOcean  HGeoid from the known geoidal height HGeoid . In connection with (56), this finally leads us to the equation 2  . 3  / ^ utan .R / D

gR  r „.R /: R

(60)

Remembering the surface curl gradient L D ^ r  , we are able to conclude ^ .! ^ utan .R // D

gR  L „.R /; 2R

(61)

i.e., C . /utan .R / D 

gR  L „.R /: R

(62)

This is the equation of the geostrophic oceanic flow.

Mathematical Analysis Again, we have an equation of vectorial tangential type given on the unit sphere S2 : This time, however, we have to deal with an equation of the surface curl 3 gradient L S D s (with s. / D  2R gR .  /utan .R / and S D „). The solution

Geomathematics: Its Role, Its Aim, and Its Potential

51

Fig. 31 Ocean topography (confer “German Priority Research Program: Mass transport and Mass distribution in the system Earth, DFG-SPP 1257”)

theory providing the surface stream function S from the surface divergence-free vector field s would be in accordance with our considerations above (with r  replaced by L ). The computation of the geostrophic oceanic flow simply is a problem of differentiation, namely, the computation of the derivative L S . / D ^ r  S . /; 2 S2 :

Development of a Mathematical Solution Method The point of departure for our intention to determine the geostrophic oceanic flow (as derived above from the basic hydrodynamic equation) is the ocean topography which is obtainable via satellite altimetry (see chapter  Classical Physical Geodesy for observational details). As a scalar field on the spherical Earth, the ocean topography consists of two ingredients. First, on an Earth at rest, the water masses would align along the geoid related to a (standard) reference ellipsoid. Second, satellite measurements provide altimetric data of the actual ocean surface height which is also used in relation to the (standard) reference ellipsoid. The difference between these quantities is understood to be the actual ocean topography. In other words, the ocean topography is defined as the deviation of the ocean surface from the

52

W. Freeden

geoidal surface, which is here assumed to be due to the geostrophic component of the ocean currents. The data used for our demonstration are extracted from the French CLS01 model (in combination with the EGM96 model). The calculation of the derivative L S is not realizable, at least not directly. Also in this case, we are confronted with discrete data material that, in addition, is only available for oceanic areas. In the geodetic literature a spherical harmonic approach is usually used in the form of a Fourier series. The vectorial isotropic operator L (cf. Freeden and Schreiner 2009 for further details) is then applied – unfortunately under the leakage of its vectorial isotropy when decomposed in terms of scalar components – to the resulting Fourier series. The results are scalar components of the geostrophic flow. The serious difficulty with global polynomial structures such as spherical harmonics (with „ D 0 on continents!) is the occurrence of the Gibb’s phenomenon close to the coast lines (see, e.g., Nerem and Koblinski 1994; Albertella et al. 2008). In this respect it should be mentioned additionally that the equation of the geostrophic oceanic flow cannot be regarded as adequate for coastal areas; hence, our modeling fails for these areas. An alternative approach avoiding numerically generated oscillations in coastal areas is the application of kernels with local support, as, e.g., smoothed Haar kernels (see Freeden et al. (1998) and the references therein): ( ˆ.k/  .t/

D

0; kC1 .t .1 //k ; 2  kC1

 < 1  t  2; 0  1  t  :

(63)

.k/

For  2 .0; 2 , k 2 N; the function ˆ as introduced by (63) is .k  1/ times .k/ continuously differentiable on the interval Œ0; 2 . .ˆj /j 2N0 is a sequence tending to the Dirac function(al), i.e., a Dirac sequence (cf. Figs. 32 and 33). For a strictly monotonically decreasing sequence .j /j 2N0 satisfying j ! 0 for j ! 1, we obtain for the convolution integrals (low-pass filters) Z H.k/ . / D j

S2

2 ˆ.k/ j .  /H . /dS . /; 2 S ;

(64)

the limit relation ˇ ˇ ˇ ˇ lim sup ˇH . /  H.k/ . / ˇ D 0: j

j !1 2S2

(65)

An easy calculation yields

L ˆ.k/ j .  / D

8 < 0; :

k.kC1/ ..  /.1j //k1 2 jkC1

j < 1    2; ^ ; 0  1    j :

It is not hard to show (cf. Freeden and Schreiner 2009) that

(66)

Geomathematics: Its Role, Its Aim, and Its Potential 8

53

8 k=0 k=3

7

k=2 k=5

6

6

5

5

4

4

3

3

2

2

1

1

0

0

−1

−1

−2 −4

−3

−2

−1

0

1

2

scale j = 3, k = 0,2,3,5

3

j=0 j=2

7

4

−2 −4

−3

−2

−1

0

1

2

j=1 j=3

3

4

scale j = 0,1,2,3, k = 5 .k/

Fig. 32 Sectional illustration of the smoothed Haar wavelets ˆj .cos. // with  Œ: ; j D 2j

2

Fig. 33 Illustration of the first members of the wavelet sequence for the smoothed Haar scaling function on the sphere (j D 2j , j D 2; 3; 4, k D 5)

ˇ ˇ ˇ ˇ lim sup ˇL H . /  L H.k/ . / ˇ D 0: j

j !1 2S2

(67)

The multiscale approach by smoothed Haar wavelets can be formulated in a standard way. For example, the Haar wavelets can be understood as differences of two successive scaling functions. In doing so, an economical and efficient algorithm in a tree structure (Fast Wavelet Transform (FWT)) can be implemented appropriately.

“Back-Transfer” to Application Ocean currents are subject to different influence factors, such as wind field, warming of the atmosphere, salinity of the water, etc., which are not accounted for in our

54

W. Freeden Haar scaling function (k=3), scale 1

Haar wavelet (k=3), scale 1

0°30° E60° E90° E120° E150° E180° W150° W120° W90° W60° W30°W

0°30°E60° E90° E120° E150° E180°W150°W120°W90°W60°W30°W

75° N 60° N

75° N 60° N

45° N 30° N

45° N 30° N

15° N 0°

15° N

15° S 30° S

15° S

0° 30° S

45° S

45° S

60° S

60° S

75° S

75° S

+ –100

–50

cm

0

50

100

–10

0 cm

low–pass filtering (scale j = 1)

band–pass filtering (scale j = 1)

–20

Haar wavelet (k=3), scale 2

10

20

Haar wavelet (k=3), scale 2 0°30°E60° E90° E120° E150° E180°W150°W120°W90°W60°W30°W

0°30° E60° E90° E120° E150° E180° W150° W120° W90° W60° W30°W 75° N 60° N 45° N 30° N

75° N 60° N 45° N 30° N

15° N 0°

15° N

15° S 30° S

15° S

0° 30° S 45° S

45° S

60° S

60° S

75° S

75° S

+

+ –20

–10

0

10

20

cm

–20

–10

0 cm

band–pass filtering (scale j = 2)

band–pass filtering (scale j = 3)

Haar wavelet (k=3), scale 4

10

20

Haar wavelet (k=3), scale 5

0°30° E60° E90° E120° E150° E180° W150° W120° W90° W60° W30°W

0°30°E60° E90° E120° E150° E180°W150°W120°W90°W60°W30°W

75° N 60° N 45° N ° 30 N

75° N 60° N 45° N 30° N

15° N 0°

15° N

15° S 30° S

15° S

0° 30° S

45° S

45° S

60° S

60° S

75° S

75° S

+

+ –20

–10 cm

0

10

20

–15

–10

0 cm

band–pass filtering (scale j = 4)

band–pass filtering (scale j = 5)

Haar wavelet (k=3), scale 6

–5

5

10

15

Haar scaling function (k=3), scale 7

0°30° E60° E90° E120° E150° E180° W150° W120° W90° W60° W30°W

0°30°E60° E90° E120° E150° E180°W150°W120°W90°W60°W30°W

75° N 60° N

75° N

60° N

45° N

45° N 30° N

30° N 15° N 0°

15° N

15° S 30° S

15° S

0° 30° S

45° S

45° S

60° S

60° S

75° S

=

+ –10

–5

0

5

10

75° S

–100

–50

0

50

100

cm

cm

band–pass filtering (scale j = 6)

low–pass filtering (scale j = 7)

Fig. 34 Multiscale approximation of the ocean topography [cm] (a rough low-pass filtering at scale j D 1 is improved with several band-pass filters of scale j D 1; : : : ; 6; where the last picture shows the multiscale approximation at scale j D 7) (numerical and graphical illustration in cooperation with D. Michel and V. Michel 2006)

Geomathematics: Its Role, Its Aim, and Its Potential

55

Fig. 35 Ocean topography [cm] (top) and geostrophic oceanic flow [cm/s] (bottom) of the gulf stream computed by the use of smoothed Haar wavelets .j D 8; k D 5/ (numerical and graphical illustration in cooperation with D. Michel and V. Michel 2006)

56

W. Freeden August 1997

May 1997

June 1997

06.97

September 1997

09.97

July 1997

07.97

11.97

08.97

05.97

November 1997

02.98

Decamber 1997

12.97

October 1997

10.97

February 1998

March 1998

03.98

Januray 1997

01.98

April 1998

04.98

Fig. 36 Ocean topography during the El Niño period (May 1997–April 1998) from data of the satellite CLS01 model (numerical realization and graphical illustration in cooperation with V. Michel and S. Maßmann 2006)

modeling. Our approximation (see Figs. 34 and 35) must be understood in the sense of a geostrophic balance. An analysis shows that its validity may be considered as given on spatial scales of an approximate expansion of a little more than 30 km and on time scales longer than approximately 1 week. Indeed, the geostrophic velocity field is perpendicular to the tangential gradient of the ocean topography (i.e., perpendicular to the tangential pressure gradient). This is a remarkable property. The water flows along the curves of constant ocean topography, i.e., along isobars (see Fig. 34 for a multiscale modeling). Despite the essentially restricting assumptions necessary for the modeling, we obtain instructive circulation models for the internal ocean surface current for the northern or southern hemisphere, respectively (however, difficulties for the computation of the flow arise from the fact that the Coriolis parameter vanishes on the equator). An especially positive result is that the modeling of the ocean topography has made an essential contribution to the research in exceptional phenomena of internal ocean currents, such as El Niño. El Niño is an anomaly of the ocean–atmosphere system. It causes the occurrence of modified currents in the equatorial Pacific, i.e., the surface water usually flowing in western direction suddenly flows to the east. Geographically speaking, the cold Humboldt Current is weakened and finally ceases. Within only a few months, the water layer moves from Southeast Asia to South America. Water circulation has reversed. As a consequence, the Eastern Pacific is warmed up, whereas the water temperature decreases off the shores of Australia and Indonesia. This phenomenon has worldwide consequences on the weather, in the form of extreme droughts and thunderstorms. Our computations do not only help to visualize these modifications graphically, but they also offer the basis for future predictions of El Niño characteristics and effects (Fig. 36).

Geomathematics: Its Role, Its Aim, and Its Potential

8.3

57

Circuit: Seismic Processing from Acoustic Wave Tomography

The essential goal of seismic processing (see chapters  Modeling Deep Geothermal Reservoirs: Recent Advances and Future Perspectives,  Transmission Tomography in Seismology,  Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery,  Strategies in Adjoint Tomography,  Multidimensional Seismic Compression by Hybrid Transform with Multiscale-Based Coding, and  Tomography: Problems and Multiscale Solutions) is to transfer the existing signal that resulted from the integration of the wave equation under the expectation that designated properties of the target bedrock like the velocity field can be interpreted from the transformed signal. In this work, based on the regularization of Green’s functions (fundamental solutions), new wavelet techniques for a detailed bandpass filtering of acoustic seismic phenomena are formulated in order to get a local understanding and interpretability of scattered wave field potentials in deep geothermal research. More material can be found in Augustin (2014), Augustin et al. (2012), Bauer et al. (2014), Freeden and Nutz (2014), and Ostermann (2011).

Mathematical Modeling in Reservoir Detection In order to determine the structure, depth, and thickness of the target reservoir (see, e.g., Dahlen and Tromp 1998; Nolet 2008; Tarantola 1984; Yilmaz 1987), the standard seismic methods are applied to two- and/or three-dimensional seismic sections. All methods in use can be distinguished between time- and depthmigration strategies and between applications to post- and pre-stack data sets (see chapter  Modeling Deep Geothermal Reservoirs: Recent Advances and Future Perspectives). The time-migration strategy is used to resolve conflicts in dipping events with different velocities. The depth-migration strategy handles strong lateral velocity variations associated to complex overburden structures. The numerical techniques used to solve the migration problem can generally be separated into three broad categories: (i) integral discretization methods such as Kirchhoff migration based on the solution of the eikonal equation; (ii) methods based on finitedifference schemes, e.g., depth continuation methods and reverse-time migration; and (iii) transform methods based on frequency–wave number implementations, e.g., frequency–space and frequency–wave number migration. All these migration methods usually rely on a certain approximation of the scalar acoustic or vectorial elastic wave equation (for more details, the reader is referred, e.g., to Nolet 2008; Yilmaz 1987 and the references therein). According to the geophysical requirements, highly accurate approximations and efficient numerical techniques must be realized in order to handle steep dipping events and complex velocity models with strong lateral and vertical variations, as well as to construct the subsurface image in a locally defined region with high resolution on available computational resources. In consequence, migration algorithms require an accurate velocity model. The adaptation of the interval velocity by use of an inversion by comparing the measured travel times with simulated travel times is called reflection tomography. There are many versions of reflection

58

W. Freeden recording truck

vibrator truck

sandstone P

limestone

P

P

Fig. 37 Principle of seismic reflection (from the PhD-thesis Ilyasov (2011), Geomathematics Group, University of Kaiserslautern)

tomography, but they all use ray-tracing techniques and they are formulated usually as a mathematical optimization problem. The most popular and efficient methods are ray-based travel time tomography, waveform and full-wave inversion tomography (FWI), and Gaussian beam tomography (GBT). The “true” velocity estimation is often obtained by an iterative process called migration velocity analysis (MVA), which uses the kinematic information gained by the migration and consists of the following steps: (initial step) perform a reflection tomography of the coarse velocity structure using a priori knowledge about the subsurface; (iterative step) migrate the seismic data sets and apply the imaging condition; and update the velocity function by tomography inversion. These approaches often have rather poor quality in the sense of the interpretability of the migration result. Since the interest of, e.g., geothermal projects is not only focused on structure heights and traps as in oil field practice but also on fault zones and karst structures under recent stress conditions, the interpretation for geothermal needs is significantly complicated (Fig. 37).

Mathematical Analysis of Acoustic Wave Propagation In the context of seismic processing imaging (see also chapters  Modeling Deep Geothermal Reservoirs: Recent Advances and Future Perspectives,  Identification of Current Sources in 3D Electrostatics,  Transmission Tomography in Seismology,  Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery,  Strategies in Adjoint Tomography, and  Multidimensional Seismic Compression by Hybrid Transform with Multiscale-Based Coding), it is usually assumed that shear stresses generated by the wave impulse and other kinds of damping can be neglected. As a consequence, wave propagation is treated as an acoustic phenomenon: pressure changes P .x; t/ imply volume changes dV

Geomathematics: Its Role, Its Aim, and Its Potential

59

that generate a displacement u.x; t/ which yield further pressure changes in the neighborhood of the volume. The relation between pressure changes and volume changes is assumed to be governed by Hooke’s law (see, e.g., Skudrzyk 1972; Achenbach 1973) P D K

dV V

(68)

as the basis of a linear elastic relation. Here, K is the bulk modulus of the material. In order to connect pressure changes to the displacement (see also Freeden and Nutz 2014), we observe that a small volume V may be written as V D dx1 dx2 dx3 . It is transformed to V 0 D dx10 dx20 dx30 . As the displacement ıu is defined via dxi0 D dxi C ıui ;

i 2 f1; 2; 3g;

(69)

we formally get 

V V0 dV D V V D

dx1 dx2 dx3  .dx1 C ıu1 /.dx2 C ıu2 /.dx3 C ıu3 / dx1 dx2 dx3

D

dx1 dx3 ıu2 C dx2 dx3 ıu1 C dx1 dx2 ıu3 dx1 dx2 dx3

dx1 ıu2 ıu3 C dx2 ıu1 ıu3 C dx3 ıu1 ıu2 ıu1 ıu2 ıu3  dx1 dx2 dx3 dx1 dx2 dx3   ıu2 ıu3 ıu1 CR D C C dx1 dx2 dx3 

D  r  u C R;

(70)

with R summarizing all terms of higher order in ıui which is negligible as ıui is assumed to be small. We obtain P D Kr  u C S

(71)

with the source term S as a constitutive equation. Moreover, we assume the balance of linear momentum, or equivalently Newton’s second law, which in our case reads P .x C ıxi  i ; t/  P .x; t/ D .x/ıxi ai ;

i 2 f1; 2; 3g ;

(72)

with the acceleration a and the canonical orthonormal basis of Euclidean space R3 :  1 ;  2 ;  3 . Under the further assumption that the acceleration is given by the secondorder time derivative of the displacement u, we get

60

W. Freeden

P .x C ıxi  i ; t/  P .x; t/ @2 D .x/ 2 ui .x; t/ ıxi @t

(73)

and finally by considering the limit ıxi ! 0, @2 @ P .x; t/ D .x/ 2 ui .x; t/: @xi @t

(74)

These component equations can be summarized in vectorial form as rx P .x; t/ D .x/

@2 u.x; t/: @t 2

(75)

Assuming that P and u are sufficiently often differentiable, (71) and (75) can be combined by applying the second-order time derivative to (71) to gain @2 P .x; t/ D K.x/rx  @t 2



 @2 @2 u.x; t/ C 2 S .x; t/: 2 @t @t

(76)

Using (75) in (76), we obtain @2 P .x; t/ D K.x/ rx  @t 2



 @2 1 rx P .x; t/ C 2 S .x; t/: .x/ @t

(77)

Applying the product rule yields the identity  rx 

1 0  1 1 @ 1 rx P .x; t/ D .rx .x//  rx  rx P .x; t/A  2 .x/ .x/ „ ƒ‚ …  .x/ Dx

.rx P .x; t// :

(78)

provided that  is smooth enough. If the gradient of  is negligibly small, we arrive at the non-divergence form of the acoustic wave equation @2 @2 2 P .x; t/ D c .x/  P .x; t/ C S .x; t/; x @t 2 @t 2

(79)

where the quantity s c.x/ D

K.x/ .x/

(80)

is called the propagation speed of a wave, which results in the purely divergence form of the acoustic wave equation

Geomathematics: Its Role, Its Aim, and Its Potential



 1 @2 1 @2 P .x; t/ D   S .x; t/: x c 2 .x/ @t 2 c 2 .x/ @t 2

61

(81)

The identity (81) ends the standard approach to the acoustic wave equation, thereby assuming a compressible, viscous (i.e., no attenuation) medium with no shear strength and no internal forces (i.e., in equilibrium). Remark 4. There is a large literature dealing with existence and uniqueness of the forward formulation of the acoustic wave equation (see, e.g., Evans (2002) and the references therein).

Development of a Mathematical Solution Method As is well known, a standard approach to acoustic wave equation (81) is a Fourier transform with respect to time: 1 U .x/ D p 2

Z P .x; t/ exp.i !t/ dt

(82)

R

leading to the reduced wave equation   !2 U .x/ D W .x/; x C 2 c .x/

(83)

usually called the Helmholtz equation. Obviously, W is given by 1 W .x/ D  p 2

Z

1 R

c 2 .x/

@2 S .x; t/ exp.i !t/ dt @t 2

(84)

for all x. We are interested in two solution procedures of the Helmholtz equation (83), namely: (1) Postprocessing: the decorrelation of U and c from a preprocessed (sufficiently suitable) solution U , (2) Inverse modeling: the determination of c under the a priori knowledge of a lowpass filtered “trend solution.” More precisely, we do not make the attempt here to solve the inverse problem of determining c as a whole. Instead, we base our investigations on already successful prework, for example, (i) integral discretization methods such as Kirchhoff migration based on the solution of the eikonal equation (see, e.g., Yilmaz 1987 and the references therein) and Gaussian beam procedures (see, e.g., Popov et al. 2006, 2008); (ii) methods based on the finite-difference schemes, e.g., depth continuation methods (cf. Claerbout 2009) and reverse-time migration (see, e.g., Baysal et al.

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Fig. 38 Scheme of seismic processing

Fig. 39 Results of seismic processing

1984; Yilmaz 1987; Popov et al. 2008; Nolet 2008); and (iii) transform methods based on frequency–wave number implementations, e.g., frequency–space and frequency–wave number migration (see, e.g., Yilmaz 1987; Claerbout 2009). In other words, in order to get information about the structure, depth, and thickness of a target reservoir, we start from today’s realistic assumption that standard seismic tomography results are available and meaningful, at least to some extent. We focus our attention on the interpretability of a “true” migration result obtained from elsewhere and/or the (local) improvement of a trend result (see Figs. 38 and 39). An essential tool is a new class of locally supported wavelets (see Freeden and Blick 2013; Freeden and Nutz 2014) derived from regularizations of Green functions for the Helmholtz operator:

Geomathematics: Its Role, Its Aim, and Its Potential

63

(1) Postprocessing: The Helmholtz equation (83) leads to the definition of the wave number k.x/ and the refraction index N .x/ as N .x/ D

! ! c0 c0 ; k.x/ D D D k0 N .x/; c.x/ c.x/ c0 c.x/

(85)

with c0 being a suitable constant reference velocity (see, e.g., Engl et al. 1996; Snieder 2002; Biondi 2006, and the references therein). Accordingly, the Helmholtz equation (83) can be rewritten as 

 x C k02 N 2 .x/ U .x/ D 0:

(86)

The region where N .x/ ¤ 1 represents the scattering object such that N .x/  1 may be supposed to have compact support. Another standard assumption is that the difference between c.x/ and c0 should be sufficiently small. As a consequence, N 2 .x/ may be developed into a Taylor series up to order one with a center such that c.x0 / D c0 . This yields N 2 .x/ ' 1 C .x/

(87)

with a small perturbation parameter . Consequently, we have k 2 .x/ D k02 N 2 .x/ D k02 .1 C .x// :

(88)

With the same argument as explained before, the unknown function .x/ may be 2 supposed to have compact support. The wave operator x C c 2!.x/ may be separated in the following way: Ax D x C

!2 D x C k02 N 2 .x/ D x C k02 .1 C .x// c 2 .x/

D x C k02 C k02 .x/ D A.0/ C A.1/ ;

(89)

where we have used the abbreviations A.0/ D x C k02

(90)

A.1/ D k02 .x/:

(91)

and

Hence, the wave field U .x/ may be split into an incident wave field UI , corresponding to the wave propagating in the absence of the scatterer, and the scattered wave field US such that U D UI C US :

(92)

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W. Freeden

This splitting leads us to   A.0/ UI D x C k02 UI D 0;   A.0/ US D x C k02 US D k02 .UI C US / D k02 U D A.1/ U:

(93) (94)

It should be mentioned that Eq. (93) formalizes that UI corresponds to the wave propagating in the absence of the scatterer. As the fundamental solution to the Helmholtz operator  C k02 is known to be G. C k02 I jx  yj/ D

1 e i k0 jxyj ; 4 jx  yj

x ¤ y;

(95)

the functions UI and US ; respectively, can be represented as volume potentials Z UI .x/ D Z US .x/ D

R3

B

G. C k02 I jx  yj/ W .y/ dV .y/;

(96)

  G. C k02 I jx  yj/ k02 .y/ U .y/ dV .y/ ƒ‚ … „

(97)

DF .y/

with the volume element dV and B D supp./ being the support of ; where W is given by (84). Remark 5. In seismic reflection modeling, point sources with certain spectra are usually chosen as unperturbed wave (for more details, see, e.g., Nolet 2008). Only if UI and US are available in the compact support B D supp./, a direct computation of  by applying the Helmholtz operator  C k02 to (97) is possible. However, it should be mentioned that US can be usually measured only away from the support B of . Nevertheless, in exceptional cases, local information is available inside boreholes, which is of particular interest. In the sense of the perturbation theory and with the conventional setting U 0 D UI ; U can be formally written as a series U D

1 X

 k U .k/ ;

(98)

kD0

which yields 1 X kD0

   k A.0/ C A.1/ U .k/ D W:

(99)

Geomathematics: Its Role, Its Aim, and Its Potential

65

By collecting terms which are of the same order in , we therefore get A.0/ U .0/ D W;

(100)

A.0/ U .k/ D  A.1/ U .k1/ ;

k 2 N:

(101)

The scattered wave field is then given by

US D U  UI D

1 X

 k U .k/ :

(102)

kD1

This procedure is known as Born approximation (see, e.g., Snieder (2002) for more details). Considering only the first-order approximation, we obtain A.0/ UI DW;

(103)

A.0/ US D  A.1/ UI :

(104)

The difference between (94) and (104) is crucial. On the right-hand side of (97), we find the sum of UI and US which makes this a nonlinear equation. On the right-hand side of (104), only UI appears, which is determined by (103), making the relation between scattered wave and perturbation of the medium linear. Remark 6. UI and the first-order, i.e., Born approximation of US can be represented by the potentials Z UI .x/ D

R3

G. C k02 I jx  yj/ W .y/ dV .y/;

(105)

Z USI .x/ D

  G. C k02 I jx  yj/ k02 .y/ UI .y/ dV .y/: ƒ‚ … „ B

(106)

DFI .y/

The basic properties of volume integrals of type (97) can be summarized as follows (see, e.g., Müller 1969): the volume potential US is a metaharmonic function in R3 nB under the assumption of boundedness of F (i.e., . C k02 /US D 0 in R3 nB). For a continuous F in B, the potential (97) is of class C .1/ .R3 /, and under the assumption of Hölder continuity of F , we have . C k02 /USI .x/ D FI .x/

(107)

for all x 2 B. This equation indicates the direct relation between USI and FI in B. It is actually the key point of Born modeling as discussed in the literature (see, e.g., Marks 2006).

66

W. Freeden

Our postprocessing procedure starts from the nonlinear integral relation (96). The essential idea is to use a sequence of regularizations fGj . C k 2 I /g; j 2 N, for the kernels (95) given by 8 i k0 jxyj ˆ < e4jxyj ;  Gj . C k02 I jx  yj/ D ei k0 jxyj ˆ 3 : 8j

jxyj2 j2



jx  yj > j ; ; jx  yj  j ;

(108)

where fj gj 2N denotes a positive monotonically decreasing sequence converging to 0 (e.g., a dyadic sequence given by j D 2j , j 2 N). The regularization (108) is constructed in such a way that each kernel Gj . C k02 I / is continuously differentiable and only dependent on the distance between two points x and y. Furthermore, under the assumption that F is bounded in B, we obtain for the “regularized version of the potential” (97) Z .US /j .x/ D

B

Gj . C k02 I jx  yj/ F .y/ dV .y/

(109)

the limit relation US .x/ D .US /j .x/ C O.j2 /; j ! 0; for all x 2 R3 . It should be noted that the aforementioned approach of replacing the fundamental solution G. C k02 I jx  yj/ by its regularized versions (109) initiates a multiscale method in canonical way (cf. Freeden and Blick 2013) based on ˇ ˇ Z ˇ ˇ lim sup ˇˇUS .x/  Gj . C k02 I jx  yj/F .y/ dV .y/ˇˇ D 0: j !1 x2B

(110)

B

Each scaling function Gj . C k02 I / provides low-pass filtering of the signature US . In order to obtain multiscale components, we simply calculate the difference of two consecutive scaling functions and get the wavelet functions with respect to the scale parameter j . Of course, other types of regularizations can be chosen. In our approach, however, we restrict ourselves to Haar-related kernel functions (see Haar 1910 for one-dimensional theory). As a matter of fact, by using the regularizations of the fundamental solution of the Helmholtz operator  C k02 , we are immediately led to locally supported wavelets via the scale discrete scaling equation

‰j . C k02 I jx  yj/ D Gj C1 . C k02 I jx  yj/  Gj . C k02 I jx  yj/; j 2 N0 : (111)

Geomathematics: Its Role, Its Aim, and Its Potential

67

0.03

0.06

0.02

0.04

0.01

0.02

0

0

–0.01

–0.02

–0.02

–0.04

–0.03 –5

0

5

–0.06 –5

0

5

0

5

0.3

0.12 0.1

0.25

0.08 0.2 0.06 0.04

0.15

0.02

0.1

0

0.05

–0.02 0

–0.04 –0.06 –5

0

5

–0.05 –5

Fig. 40 Wavelet function ‰j in sectional illustration for k0 D 5 and j D

4 2j

; j D 0; : : : ; 3

Explicitly, we have (see Fig. 40 for graphical illustration) ! ! 8 ˆ e i k0 jxyj jx  yj2 e i k0 jxyj jx  yj2 ˆ ˆ 3  3 ; ˆ ˆ ˆ 8j C1 8j j2C1 j2 ˆ ˆ ˆ ˆ jx  yj  j C1 ; ˆ ˆ < 2 ! (112) ‰j .Ck0 I jx  yj/ D e i k0 jxyj jx  yj2 e i k0 jxyj ˆ ˆ ˆ C 3 ; ˆ ˆ 4jx  yj 8j j2 ˆ ˆ ˆ ˆ ˆ j C1 < jx  yj  j ; ˆ ˆ : 0; j < jx  yj:

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W. Freeden

The convolution integral (113) indicates the difference of two sequential lowpass filters, i.e., it represents a band-pass filtering at the position x with respect to the scale parameter j : Z Wj .x/ D

B

‰j . C k02 I jx  yj/ F .y/ dV .y/:

(113)

Wj includes all detail information contained in .US /j C1 but not in .US /j . In accordance with our construction, we therefore obtain for every L 2 N X

j CL1

.US /j CL D .US /j C

Wn :

(114)

nDj

The formula (114) shows the progress in the “zooming-in process,” which proceeds from scale j to scale j C L. Hence, the identity (114) describes the amount of improvement in the accuracy from level j to level j C L. Indeed, it uniformly follows for each position x and for each scale value j that US .x/ D .US /j .x/ C

1 X

Wn .x/;

(115)

nDj

i.e., the signal US consists of a (coarse) low-pass filtering and an infinite number of successive band-pass convolutions. Of course, in practice, only a finite number of band-pass filters has to be calculated to satisfy a certain error tolerance (for more multiscale aspects in constructive approximation, see Freeden and Michel 2004; Freeden and Gerhards 2013). Until now, we have only reconstructed the quantity US by virtue of a multiscale technique using building blocks. For practical purposes, the decorrelation of both F and  are of interest. An elementary calculation using (108) yields the Helmholtz derivative     Kj . C k02 I jx  yj/ D x C k02 Gj  C k02 I jx  yj

D

8 j :

fKj g is a “Haar-type sequence,” and it approximately reduces to the Haar sequence fHj g ( Hj .jx  yj/ D

1 3 4 3 ; jx  yj  j ; j

0;

jx  yj > j :

Geomathematics: Its Role, Its Aim, and Its Potential

69

for sufficiently large j , i.e., for each k0 Kj . C k02 I jx  yj/ D Hj .jx  yj/ C O.j /; for j ! 1: If F is bounded, then it is clear that   x C k02 Z



D B

Z B

Gj . C k02 I jx  yj/ F .y/ dV .y/

(116)

   x C k02 Gj  C k02 I jx  yj F .y/ dV .y/;

such that Z  F .x/ D lim

j !1 B

Kj . C k02 I jx  yj/ F .y/ dV .y/

(117)

Z

D lim

j !1 B

Hj .jx  yj/ F .y/ dV .y/

for all x 2 B. Moreover, we have under the assumption of Hölder continuity of F (see, e.g., Müller 1969) Z  F .x/ D .x C k02 / US .x/ D .x C k02 /

B

G. C k02 I jx  yj/F .y/ dV .y/

(118) for all x 2 B. In other words, the “Helmholtz derivative”  C k02 of the regularization of the fundamental solution leads back to a “Haar-type” singular integral (117) for detecting F .x/ D " k02  .UI C US /.x/ D " k02  U .x/

(119)

for all x 2 B. Vice versa, our approach offers the possibility to introduce alternative regularizations of the fundamental solution such that their “Helmholtz derivatives” represent singular-type kernels constituting Dirac-type sequences in B. All in all, the multiscale technique by regularized fundamental solutions enables us simultaneously to decompose the signal information of the wave field as well as the refraction index N based on the interrelation (116), however, under the “postprocessing assumption” that US is discretely known (to some extent) inside B. Once more, our understanding of the multiscale technique here does not preferably aim at the reconstruction of the signals, but instead in working out characteristic detail information that emerge from the difference of two consecutive scale-space representations. This practice (see also Freeden and Blick 2013) comes across as the decorrelation of signatures, such that postprocessing of density structures is the key element.

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(2) Inverse Modeling: In the following (cf. Freeden and Gerhards 2013), we make the attempt to apply the Haar philosophy to an approximate determination of F from US inside B. The regularization procedure of the volume potential as proposed for postprocessing is the essential tool. Let fj gj 2N0 be a monotonically decreasing sequence of positive values j such that limj !1 j D 0 (e.g., j D 2j ). Then, in accordance with (117), we are able to specify a sufficiently large integer J such that, for all x 2 B; Z  F .x/ '  FJ .x/ D KJ . C k02 I jx  yj/ F .y/ dV .y/ (120) B

as well as

Z

.US /.x/ ' .US /J .x/ D

B

GJ . C k02 I jx  yj/ F .y/ dV .y/

(121)

(“' ”means that the error is negligible). If F is bounded, then we already know that   (122) x C k02 .US /J .x/ D FJ .x/ for all x 2 B: In order to realize a fully discrete approximation of F , we have to apply approximate integration formulas over B leading to .US /.x/ '

NJ X

NJ J GJ . C k02 I jx  yiNJ j/ wN i F .yi /;

(123)

i D1 NJ J where wN i , yi , i D 1; : : : ; NJ , are the known weights and knots, respectively. Using an appropriate integration formula, we are therefore led to a linear system to be solved in order to obtain insight into approximate information about . As already explained, for numerical realization, we may assume that .US /J 1 is available from elsewhere, at least discretely in points xkMJ 2 B; k D 1; : : : ; MJ ; to be needed in the solution process of the linear system. Obviously, we have to calculate all unknown coefficients NJ J aiNJ D wN i F .yi /;

i D 1; : : : ; NJ ;

(124)

    from discrete values US xkMJ  .US /J 1 xkMJ ; k D 1; : : : ; MJ : Then we have to solve a linear system NJ     X ‰J 1 . C k02 I jxkMJ  yiNJ j/ aiNJ ; US xkMJ  .US /J 1 xkMJ ' i D1

k D 1; : : : ; MJ ; in order to determine the required coefficients aiNJ ; i D 1; : : : ; NJ .

(125)

Geomathematics: Its Role, Its Aim, and Its Potential

71

Remark 7. The linear system (125) is the bottleneck in inverse modeling, although ‰J 1 as constructed here possesses a local support. Formally, (125) can be written in a general matrix notation of the form Ad D u. In fact, there is a large literature for solving such a system by minimizing a penalty term of generic form P .d / D 12 kAd  uk C R.d /; where R.d / is a measure of the size and/or complexity (see, e.g., chapters  Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment,  Transmission Tomography in Seismology,  Numerical Algorithms for Non-smooth Optimization Applicable to Seismic Recovery, and  Strategies in Adjoint Tomography for related problems). Even more, sparsity comes into play (see, e.g., chapters  Sparsity in Inverse Geophysical Problems,  Sparse Solutions of Underdetermined Linear Systems and the references therein). Once all coefficients aiNJ ; i D 1; : : : ; NJ are available (note that the integration J weights wN i ; i D 1; : : : ; NJ ; are known), the function F  FJ 1 can be obtained in obvious way. From the knowledge of US ; .US /J 1 ; we are therefore able via (119) to model , hence, the wanted refraction index N .

“Back-Transfer” to Application A seismic prototype for test investigations in geoexploration is the 2D Marmousi model (see Martin et al. 2002). A velocity model is shown in Fig. 39 (taken from the PhD-thesis due to Ilyasov (2011)). The source wave field involving the Marmousi model in direct time (i.e., snapshots of the wave propagation after 0.6, 1.1, 1.6, 2.1, and 2.6 s) is illustrated in Figs. 41–45. A result of a reverse-time migration (RT) in the context of a finite-difference scheme (FDS) applied to the Marmousi data set using the velocity model (Fig. 39) is illustrated in Fig. 46.

Fig. 41 Wave propagation in the Marmousi velocity model after 0.6 s

Fig. 42 Wave propagation in the Marmousi velocity model after 1.1 s

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W. Freeden

Fig. 43 Wave propagation in the Marmousi velocity model after 1.6 s

Fig. 44 Wave propagation in the Marmousi velocity model after 2.1 s

Fig. 45 Wave propagation in the Marmousi velocity model after 2.6 s (all pictures taken from the PhD-thesis Ilyasov (2011), Geomathematics Group, University of Kaiserslautern)

Fig. 46 A migration result of the Marmousi model by FDS (taken from the PhD-thesis Ilyasov (2011), Geomathematics Group, University of Kaiserslautern)

Geomathematics: Its Role, Its Aim, and Its Potential

73

−600 −800 −1000 −1200 −1400 −1600 −1800 −2000 −2200

3800 3600 3400 3200 3000 2800 2600 2400 2200 2000 1800

[m]

[m]

Fig. 47 Interpretation of the Marmousi model due to Martin et al. (2002)

−600 −800 −1000 −1200 −1400 −1600 −1800 −2000 −2200 1000 2000 3000 4000 5000 6000 7000 8000 [m]

1000 2000 3000 4000 5000 6000 7000 8000 [m]

low-pass filtering (scale j =4)

[m]

−1000 −1500 −2000 −2500 1000 2000 3000 4000 5000 6000 7000 8000 [m]

band-pass filtering (scale j =4) 600 500 400 300 200 100 0 −100 −200 −300 −400

400

−500

[m]

−500

400 300 200 100 0 −100 −200 −300 −400

300

−1000

200

−1500

100

−2000

−100

0 −200

−2500 1000 2000 3000 4000 5000 6000 7000 8000 9000 [m]

band-pass filtering (scale j =5)

−300

band-pass filtering (scale j =6) 100 150

−500

100

−1000

50

−1500

0

−2000

−50

−2500

−100 1000 2000 3000 4000 5000 6000 7000 8000 9000 [m]

band-pass filtering (scale j =7)

[m]

[m]

−1000

80 60

−500

40 20 0 −20 −40

−1500 −2000 −2500 1000 2000 3000 4000 5000 6000 7000 8000 9000

−60

[m]

band-pass filtering (scale j =8)

Fig. 48 Wavelet approximation of the velocity field in [m/s] by Helmholtz derivatives (following Freeden and Blick 2013)

74

W. Freeden

Fig. 49 Wavelet decorrelation (band-pass filtering) of the Marmousi migration model in [m] for scales j D 2; : : : ; 6 (following Freeden and Blick 2013)

Geomathematics: Its Role, Its Aim, and Its Potential

75

For the decorrelation behavior of the Fourier-transformed 3D Helmholtz wavelets, we limit our research to wave numbers k0 in the interval from k0 D 0:010 to k0 D 0:120. Our calculations show two illustrations, namely, for the wave members k0 D 0:047 and k0 D 0:099. Figure 49 shows the details for the scale parameters corresponding to a dyadic sequence. Relevant structural differences during the scale-dependent wavelet convolution become obvious for scale j D 3; : : : ; 6. Indeed, we are able to show that our decorrelation method highlights specific rock formations. By wavelet filtering of the migration result, we are not only able to specify the salt formation but also to dampen other rock formations as well as undesired noise phenomena caused by erroneous migration (see Fig. 47). Finally, in accordance with our construction (119), the Helmholtz derivative simultaneously leads back to a multiscale approximation of the velocity field using Haar-type trial functions. The wave number chosen for the illustrations in Fig. 48 is k0 D 0:099.

9

Final Remarks

The Earth is a dynamic planet in permanent change, due to large-scale internal convective material and energy rearrangement processes, as well as manifold external effects. We can, therefore, only understand the Earth as our living environment if we consider it as a complex system of all its interacting components. The processes running on the Earth are coupled with one another, forming ramified chains of cause and effect which are additionally influenced by man who intervenes into the natural balances and circuits. However, knowledge of these chains of cause and effect has currently still remained incomplete to a large extent. In adequate time, substantial improvements can only be reached by the exploitation of new measurement and observation methods, e.g., by satellite missions and by innovative mathematical concepts of modeling and simulation, all in all by geomathematics. As far as data evaluation is concerned in the future, traditional mathematical methods will not be able to master the new amounts of data neither theoretically nor numerically – especially considering the important aspect of a more intensively localized treatment with respect to space and time, embedded into a global concept. Instead, geoscientifically relevant parameters must be integrated into constituting modules; the integration must be characterized by three essential characteristics: good approximation property, appropriate decorrelation ability, and fast algorithms. These characteristics are the key for a variety of abilities and new research directions. Acknowledgements This introductory chapter is based on the German note “W. Freeden (2009): Geomathematik, was ist das überhaupt?, Jahresbericht der Deutschen Mathematiker Vereinigung (DMV), JB.111, Heft 3, 125–152.” I am obliged to the publisher Vieweg+Teubner for giving the permission for an English translation of essential parts of the original version. Particular thanks go to Dr. Helga Nutz for reading an earlier version and eliminating some inconsistencies.

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Furthermore, I would like to thank my Geomathematics Group, Kaiserslautern, for the assistance in numerical calculation as well as graphical illustration concerning the three exemplary circuits.

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Navigation on Sea: Topics in the History of Geomathematics Thomas Sonar

Contents 1 General Remarks on the History of Geomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The History of the Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Early Modern England . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Gresham Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 William Gilberts Dip Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Briggsian Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Computation of the Dip Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter, we review the development of the magnet as a means for navigational purposes. Around 1600, knowledge of the properties and behavior of magnetic needles began to grow in England mainly through the publication of William Gilbert’s influential book De Magnete. Inspired by the rapid advancement of knowledge on one side and of the English fleet on the other, scientists associated with Gresham College began thinking of using magnetic instruments to measure the degree of latitude without being dependent on a clear sky, a quiet sea, or complicated navigational tables. The construction and actual use of these magnetic instruments, called dip rings, is a tragic episode in the history of seafaring since the latitude does not depend on the magnetic field of the Earth

T. Sonar () Computational Mathematics, Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_2

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but the construction of a table enabling seafarers to take the degree of latitude is certainly a highlight in the history of geomathematics.

1

General Remarks on the History of Geomathematics

Geomathematics in our times is thought of being a very young science and a modern area in the realms of mathematics. Nothing is farer from the truth. Geomathematics began as man realized that he walked across a sphere-like Earth and that this had to be taken into account in measurements and computations. Hence, Eratosthenes can be seen as an early geomathematician when he tried to determine the circumference of the Earth by measurements of the sun’s position from the ground of a well and the length of shadows farther away at midday. Other important topics in the history of geomathematics are the struggles for an understanding of the true shape of the Earth which led to the development of potential theory and much of multidimensional calculus (see Greenberg 1995), the mathematical developments around the research of the Earth’s magnetic field, and the history of navigation.

2

Introduction

The history of navigation is one of the most exciting stories in the history of mankind and one of the most important topics in the history of geomathematics. The notion of navigation thereby spans the whole range from the ethnomathematics of Polynesian stick charts via the compass to modern mathematical developments in understanding the Earth’s magnetic field and satellite navigation via GPS. We shall concentrate here on the use of the magnetic needle for navigational purposes and in particular on developments having taken place in early modern England. However, we begin our investigations with a short overview on the history of magnetism following Balmer (1956).

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The History of the Magnet

The earliest sources on the use of magnets for the purpose of navigation stem from China. During the Han epoche 202–220, we find the description of carriages equipped with compass-like devices so that the early Chinese imperators were able to navigate on their journeys through their enormous empire. These carriages were called tschinan-tsche, meaning “carriages that show noon.” The compasslike devices consisted of little humanlike figures which swam on water in a bowl, the finger of the stretched arm pointing always straight to the south. We do not know nowadays why the ancient Chinese preferred the southward direction instead of a northbound one. In a book on historical memoirs written by Sse-ma-tsien (or Schumatsian), we find a report dating back to the first half of the second century about a present that imperator Tsching-wang gave 1100 before Christ

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to the ambassadors of the cities of Tonking and Cochinchina. The ambassadors received five “magnetic carriages” in order to guide them safely back to their cities even through sand storms in the desert. Since the ancient Chinese knew about the attracting forces of a magnet, they called them “loving stones.” In a work on natural sciences written by Tschin-tsang-ki from the year 727, we read: The loving stone attracts the iron like an affectionate mother attracts their children around her; this is the reason where the name comes from.

It was also very early known that a magnet could transfer its attracting properties to iron when it was swept over the piece of iron. In a dictionary of the year 121, the magnet is called a stone “with which a needle can be given a direction,” and hence, it is not surprising that a magnetic needle mounted on a piece of cork and swimming in a bowl of water belonged to the standard equipment of larger Chinese ship as early as the fourth century. Such simple devices were called “bussola” by the Italians and are still known under that name. A magnetic needle does not point precisely to the geographic poles but to the magnetic ones. The locus of the magnetic poles moves in time so that a deviation has always to be taken into account. Around the year 1115, the problem of deviation was known in China. The word “magnet” comes from the Greek word magnes describing a sebacious rock which, according to the Greek philosopher and natural scientist Theophrastos (ca. 371–287 BC), was a forgeable and silver white rock. The philosopher Plato (428/27–348/47 BC) called magnetic rock the “stone of Heracles,” and the poet Lucretius (ca. 99–55 BC) used the word “magnes” in the sense of attracting stone. He attributed the name to a place named “Magnesia” where this rock could be found. Other classical Greek anecdotes call a shepherd named “Magnes” to account for the name. It is said that he wore shoes with iron nails and while accompanying his sheep suddenly could no longer move because he stood on magnetic rock. Homer wrote about the force of the magnet as early as 800 BC. It seems typical for the ancient Greek culture that one sought for an explanation of this force fairly early on. Plato thought of this force as being simply “divine.” Philosopher Epicurus (341–270) had the hypothesis that magnets radiate tiny particles – atoms. Eventually Lucretius exploited this hypothesis and explained the attracting force of a magnet by the property of the radiated atoms to clear the space between the magnet and the iron. Into the free space then iron atoms could penetrate, and since iron atoms try hard to stay together (says Lucretius), the iron piece would follow them. The pressure of the air also played some minor role in this theory. Lucretius knew that he had to answer the question why iron would follow but other materials would not. He simply declared that gold would be too heavy and timber would show too large porosities so that the atoms of the magnet would simply go through. The first news on the magnet in Western Europe came from Paris around the year 1200. A magnetic needle was used to determine the orientation. We do not know how the magnet came to Western Europe and how it was received but it is almost certainly true that the crusades and the associated contact with the peoples in the Mediterranean played a crucial role. Before William Gilbert around 1600

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Fig. 1 The magnetic perpetuum mobile of Peregrinus

Fig. 2 Elizabeth I (Armada portrait)

came up with a “magnetick philosophy,” it was the crusader, astronomer, chemist, and physician Peter Peregrinus De Maricourt who developed a theory of the magnet in a famous “letter on the magnet” dating back to 1269. He describes experiments with magnetic stones which are valid even nowadays. Peregrinus grinds a magnetic stone in the form of a sphere, places it in a wooden plate, and puts this plate in a bowl with water. Then he observes that the sphere moves according to the poles. He develops ideas of magnetic clocks and describes the meaning of the magnet with respect to the compass. He also develops a magnetic perpetuum mobile according to Fig. 1. A magnet is mounted at the tip of a hand which is periodically moving (says Peregrinus) because of iron nails on the circumference.

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Fig. 3 De Magnete

Peregrinus’ work was so influential in Western Europe that even 300 years after his death, he is still accepted as the authority on the magnet.

4

Early Modern England

In the sixteenth century, Spain and Portugal developed into the leading sea powers. Currents of gold, spices, and gemstones regorge from the South Americans into the home countries. England had missed connection. When Henry VIII died in 1547, only a handful of decaying ships were lying in the English sea harbors. His successor, his son Edward VI, could only rule for 6 years before he died young. Henry’s daughter Mary, a devout Catholic, tried hard to re-catholize the country her father had steered into protestantism and married Philipp II, King of Spain. Mary was fairly brutal in the means of the re-catholization and many of the protestant intelligentsia left the country in fear of their lives. “Bloody Mary” died in 1558 at the age of 47 and the way opened to her stepsister Elizabeth. Within one generation itself, Elizabeth I transformed rural England to the leading sea power on Earth. She

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Unmagnetized needle

Magnetized needle

Fig. 4 Norman’s discovery of the magnetic dip

Fig. 5 Dip rings: (a) the dip ring after Gilbert in De Magnete. (b) A dip ring used in the seventeenth century

was advised very well by Sir Walter Raleigh who clearly saw the future of England on the seas. New ships were built for the navy and in 1588 the small English fleet was able to drown the famous Spanish armada – by chance and with good luck; but this incident served to boost not only the feeling of self-worth of a whole nation but also the realization of the need of a navy and the need of efficient navigational tools. English mariners realized on longer voyages that the magnetic needle inside a compass lost its magnetic power. If that was detected, the needle had to be magnetized afresh – it had to be “loaded” afresh. However, this is not the reason why the magnet is called loadstone in the English language but only a mistranslation. The correct word should be lodstone – “leading stone” – but that word was actually never used (Pumfrey 2002).

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Fig. 6 Measuring the dip on the terrella in De Magnete

5

The Gresham Circle

In 1592, Henry Briggs (1561–1639), chief mathematician in his country, was elected examiner and lecturer in mathematics at St John’s College, Cambridge, which nowadays corresponds to a professorship. In the same year, he was elected Reader of the Physics Lecture founded by Dr. Linacre in London. One hundred years before the birth of Briggs, Thomas Linacre was horrified by the pseudomedical treatment of sick people by hairdressers and vicars who did not shrink back from chirurgical operations without a trace of medical instruction. He founded the Royal College of Physicians of London and Briggs was now asked to deliver lectures with medical contents. The Royal College of Physicians was the first important domain for Briggs to make contact with men outside the spheres of the two great universities, and, indeed most important, he met William Gilbert (1544–1603) who was working on the wonders of the magnetical forces and who revolutionized modern science only a few years later.

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While England was on its way to become the world’s leading sea force, the two old English universities Oxford and Cambridge were in an alarming state of sleepiness (Hill 1997, p. 16ff). Instead of working and teaching on the forefront of modern research in important topics like navigation, geometry, and astronomy, the curricula were directly rooted in the ancient Greek tradition. Mathematics included reading of the first four or five books of Euclid, and medicine was read after Galen and Ptolemy ruled in astronomy. When the founder of the English stock exchange (Royal Exchange) in London, Thomas Gresham, died, he left in his last will money and buildings in order to found a new form of university, the Gresham College, which is still in function. He ordered the employment of seven lecturers to give public lectures in theology, astronomy, geometry, music, law, medicine, and rhetorics mostly in English language. The salary of the Gresham professors was determined to be £50 a year which was an enormous sum as compared to the salary of the Regius professors in Oxford and Cambridge (Hill 1997, p. 34). The only conditions on the candidates for the Gresham professorships were brilliance in their field and an unmarried style of life. Briggs must have been already well known as a mathematician of the first rank since he was chosen to be the first Gresham professor of Geometry in 1596. Modern mathematics was needed badly in the art of navigation, and public lectures on mathematics were in fact already given in 1588 on behalf of the East India Company, the Muscovy Company, and the Virginia Company. Even before 1588, there were attempts by Richard Hakluyt to establish public lectures and none less than Francis Drake had promised £20 (Hill 1997, p. 34), but it needed the national shock of the attack of the Armada in 1588 to make such lectures come true. During his time in Gresham College, Briggs became the center of what we can doubtlessly call the Briggsian circle. Hill writes (1997, p. 37): He [i.e. BRIGGS] was a man of the first importance in the intellectual history of his age, . . . . Under him Gresham at once became a centre of scientific studies. He introduced there the modern method of teaching long division, and popularized the use of decimals.

The Briggsian circle consisted of true Copernicans: men like William Gilbert who wrote De Magnete; the able applied mathematician Edward Wright who is famous for his book on the errors in navigation; William Barlow, a fine instrument maker and men of experiments; and the great popularizer of scientific knowledge, Thomas Blundeville. Gilbert and Blundeville were protégés of the Earl of Leicester, and we know about connections with the circle of Raleigh in which the brilliant mathematician Thomas Harriot worked. Blundeville held contacts with John Dee who introduced modern continental mathematics and the Mercator maps in England (Hill 1997, p. 42). Hence, we can think of a scientific sub-net in England in which important work could be done which was impossible to do in the great universities. It was this time in Gresham College in which Briggs and his circle were most productive in the calculation of tables of astronomical and navigational importance. In the center of their activities was Gilbert’s “magnetick philosophy.”

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William Gilberts Dip Theory

The role of William Gilbert in shaping modern natural sciences cannot be overestimated, and a recent biography of Gilbert (Pumfrey 2002) emphasizes his importance in England and abroad. Gilbert, a physician and member of the Royal College of Physicians in London, became interested in navigational matters and the properties of the magnetic needle in particular by his contacts to seamen and famous navigators of his time alike. As a result of years of experiments, thought, and discussions with his Gresham friends, the book De Magnete, magneticisque corporibus, et de magno magnete tellure; Physiologia nova, plurimis & argumentis, & experimentis demonstrata was published in 1600. (I refer to the English translation Gilbert (1958) by P. Fleury Mottelay which is a reprint of the original of 1893. There is a better translation by Sylvanus P. Thompson from 1900, but while the latter is rare, the former is still in print.) It contained many magnetic experiments with what Gilbert called his terrella – the little Earth – which was a magnetical sphere. In the spirit of the true Copernican, Gilbert deduced the rotation of the Earth from the assumption of it being a magnetic sphere. Concerning navigation, Briggs, and the Gresham circle, the most interesting chapter in De Magnete is Book V: On the dip of the magnetic needle. Already in 1581, the instrument maker Robert Norman had discovered the magnetic dip in his attempts to straighten magnetic needles in a fitting on a table. He had observed that an unmagnetized needle could be fitted in a parallel position with regard to the surface of a table but when the same needle was magnetized and fitted again, it made an angle with the table. Norman published his results already in 1581 (R NORMAN – The New Attractive London, 1581). Even before Norman, the dip was reported by the German astronomer and instrument maker Georg Hartmann from Nuremberg in a letter to the Duke Albrecht of Prussia from 4th of March, 1544, see Balmer 1956, pp. 290–292), but nobody read the letter. In modern notion, the phenomenon of the dip is called inclination in contrast to the declination or variation of the needle. A word of warning is appropriate here: in Gilbert’s time, many authors used the word declination for the inclination. Anyway, Norman was the first to build a dip ring in order to measure the inclination. This ring is nothing else but a vertical compass. Already Norman had discovered that the dip varied with time! However, Gilbert believed that he had found the secrets of magnetic navigation. He explained the variation of the needle by land masses acting on the compass which fitted nicely with the measurements of seamen but is wrong, as we now know. Concerning the dip, let me give a summary of Gilbert’s work in modern terms. Gilbert must have measured the dip on his terrella many, many times before he was led to his: First hypothesis: There is an invertible mapping between the lines of latitude and the lines of constant dip. Hence, Gilbert believed to have found a possibility of determining the latitude on Earth from the degree of the dip. Let ˇ be the latitude and ˛ the dip. He then formulated his:

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Fig. 7 Third hypothesis

Second hypothesis: At the equator, the needle is parallel to the horizon, i.e., ˛ D 0ı . At the north pole, the needle is perpendicular to the surface of the Earth, i.e., ˛ D 90ı . He then draws a conclusion, but in our modern eyes, this is nothing but another: Third hypothesis: If ˇ D 45ı , then the needle points exactly to the second equatorial point. What he meant by this is best described in Fig. 7. Gilbert himself writes . . . points to the equator F as the mean of the two poles. (Gilbert 1958, p. 293)

Note that in Fig. 7, the equator is given by the line A  F and the poles are B (north) and C so that our implicit (modern) assumption that the north pole is always shown on top of a figure is not satisfied. From his three hypotheses, Gilbert concludes correctly: First conclusion: The rotation of the needle has to be faster on its way from A to L than from L to B. Or, in Gilbert’s words, . . . the movement of this rotation is quick in the first degrees from the equator, from A to L, but slower in the subsequent degrees, from L to B, that is, with reference to the equatorial point F , toward C . (Gilbert 1958, p. 293)

And on the same page, we read: . . . it dips; yet, not in ratio to the number of degrees or the arc of the latitude does the magnetic needle dip so many degrees or over a like arc; but over a very different one, for this movement is in truth not a dipping movement, but really a revolution movement, and it describes an arc of revolution proportioned to the arc of latitude.

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Fig. 8 Gilbert’s geometrical construction of the mapping in De Magnete

This is simply the lengthy description of the following: Second conclusion: The mapping between latitude and dip cannot be linear. Now that Gilbert had made up his mind concerning the behavior of the mapping at ˇ D 0ı ; 45ı ; and 90ı , a construction of the general mapping was sought. It is exactly here where De Magnete shows strange weaknesses; in fact, we witness a qualitative jump from a geometric construction to a dip instrument. Gilbert’s geometrical description can be seen in Fig. 8. I do not intend to comment on this construction because this was done in detail elsewhere (Sonar 2001), but it is not possible to understand the construction from Gilbert’s writings in De Magnete. Even more surprising, while all figures in De Magnete are raw wood cuts in the quality shown before, suddenly there is a fine technical drawing of the resulting construction as shown in Fig. 9. The difference between this drawing and all other figures in De Magnete and the weakness in the description of the construction of the mapping between latitudes and dip angles suggest that at least this part was not written by Gilbert alone but by some of his friends in the Gresham circle. Pumfrey speaks of the dark secret of De Magnete (Pumfrey 2002, p. 173ff) and gives evidence that Edward Wright, whose On Certain Errors in Navigation had appeared a year before De Magnete, had his hands in some parts of Gilbert’s book. In Parsons and Morris (1939, pp. 61–67), we find the following remarks: Wright, and his circle of friends, which included Dr. W. Gilbert, Thomas Blundeville, William Barlow, Henry Briggs, as well as Hakluyt and Davis, formed the centre of scientific thought at the turn of the century. Between these men there existed an excellent spirit of co-operation, each sharing his own discoveries with the others. In 1600 Wright assisted Gilbert in the compilation of De Magnete. He wrote a long preface to the work, in which he proclaimed his belief in the rotation of the earth, a theory which Gilbert was explaining, and also contributed chapter 12 of Book IV, which dealt with the method of finding the amount of the variation of the compass. Gilbert devoted his final chapters to practical problems of navigation, in which he knew many of his friends were interested.

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Fig. 9 The fine drawing in De Magnete

There is no written evidence that Briggs was involved too but it seems very unlikely that the chief mathematician of the Gresham circle should not have been in charge in so important a development as the dip theory. We shall see later on that the involvement of Briggs is highly likely when we study his contributions to dip theory in books of other authors.

7

The Briggsian Tables

If we trust Ward (1740, pp. 120–129), the first published table of Henry Briggs is the table which represents Gilbert’s mapping between latitude and dip angles in Thomas Blundeville’s book The Theoriques of the seuen Planets, shewing all their diuerse motions, and all other Accidents, called Passions, thereunto belonging. Whereunto is added by the said Master Blundeuile, a breefe Extract by him made, of Magnus his Theoriques, for the better vnderstanding of the Prutenicall Tables, to calculate thereby the diuerse motions of the seuen Planets. There is also hereto added, The making, description, and vse, of the two most ingenious and necessarie Instruments for Sea-men, to find out therebye the latitude of any place vpon the Sea or Land, in the darkest night that is, without the helpe of Sunne, Moone, or Starre.

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Fig. 10 Title page of The Theoriques by Blundeville

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T. Sonar First inuented by M. Doctor Gilbert, a most excellent Philosopher, and one of the ordinarie Physicians to her Maiestie: and now here plainely set down in our mother tongue by Master Blundeuile. London Printed by Adam Islip. 1602.

Blundeville is an important figure in his own right; see Taylor (1954, p. 173) and Waters (1958, pp. 212–214). He was one of the first and most influential popularizers of scientific knowledge. He did not write for the expert, but for the layman, i.e., the young gentlemen interested in so diverse questions of science, writing of history, mapmaking, logic, seamenship, or horse riding. We do not know much about his life (Campling 1921–1922), but his role in the Gresham circle is apparent through his writings. In The Theoriques, Gilbert’s dip theory is explained in detail, and a step-by-step description of the construction of the dip instrument is given. I have followed Blundeville’s instructions and constructed the dip instrument again elsewhere; see Sonar (2002). See also Sonar (2001). The final result is shown in Blundeville’s book as in Fig. 11. In order to understand the geometrical details, it is necessary to give a condensed description of the actual construction in Fig. 8 which is given in detail in The Theoriques. We start with a circle ACDL representing planet Earth as in Fig. 12. Note that A is an equatorial point while C is a pole. The navigator (and hence the dip instrument) is assumed to be in point N which corresponds to the latitude ˇ D 45ı . In the first step of the construction, a horizon line is sought, i.e., the line from the navigator in N to the horizon. Now a circle is drawn around A with radius AM (the Earth’s radius). This marks the point F on a line through A parallel to CL. A circle around M through F now gives the arc F M . The point H is constructed by drawing a circle around C with radius AM. The point of intersection of this circle with the outer circle through F is H . If the dip instrument is at A, the navigator’s horizon point is F. If it is in C, the navigator’s horizon will be in H. Correspondingly, drawing a circle through N with radius AM gives the point S ; hence, S is the point at the horizon seen from N. Hence, to every position N of the needle, there is a quadrant of dip which is the arc from M to a corresponding point on the outer circle through F . If N is at ˇ D 45ı latitude as in our example, we know from Gilbert’s third hypothesis that the needle points to D. The angle between S and the intersection point of the quadrant of dip with the direction of the needle is the dip angle. The remaining missing information is the point to which the needle points for a general latitude ˇ. This is accomplished by quadrants of rotation which implement Gilbert’s idea of the needle rotating on its way from A to C . The construction of these quadrants is shown in Fig. 13. We need a second outer circle which is constructed by drawing a circle around A through L. The intersection point of this circle with the line AF is B and the second outer circle is then the circle through B around M . Drawing a circle around C through L defines the point G on the second outer circle. These arcs, GL and BL, are the quadrants of

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Fig. 11 The dip instrument in The Theoriques

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Fig. 12 The quadrant of dip

Fig. 13 The quadrant of rotation

rotation corresponding to the positions C and A of the needle, respectively. Assume again that the dip instrument is in N at ˇ D 45ı . Then the corresponding quadrant of rotation is constructed by a circle around N through L and is the arc OL. This arc is now divided in 90 parts, starting from the second outer circle (0ı ) and ending at L.90ı /. Obviously, in our example, according to Gilbert, the 45ı mark is exactly at D.

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Fig. 14 The final steps

Now we are ready for the final step. Putting together all our quadrants and lines, we arrive at Fig. 14. The needle at N points to the mark 45ı on the arc of rotation OL and hence intersects the quadrant of dip (arc S M ) in the point S . The angle of c is the dip angle ı. the arc SR We can now proceed in this manner for all latitudes from ˇ D 0ı to ˇ D 90ı in steps of 5ı . Each latitude gives a new quadrant of dip, a new quadrant of rotation, and a new intersection point R. The final construction is shown in Fig. 15. However, in Fig. 15, the construction is shown in the lower right quadrant instead of in the upper left and uses already the notation of Blundeville instead of those of William Gilbert. The main goal of the construction, however, is a spiral line which appears after the removal of all the construction lines, as in Fig. 16, and can already be seen in the upper left picture in Blundeville’s drawing in Fig. 11. The spiral line consists of all intersection points R. Together with a quadrant which can rotate around the point C of the mater, the instrument is ready to use. In order to illustrate its use, we give an example. Consider a seaman who has used a dip ring and measured a dip angle of 60ı . Then he would rotate the quadrant until the spiral line intersects the quadrant at the point 60ı on the inner side of the quadrant. Then the line A  B on the quadrant intersects the scale on the mater at the degree of latitude; in our case 36ı , see Fig. 18. However, accurate reading of the scales becomes nearly impossible for angles of dip larger than 60ı , and the reading depends heavily on the accuracy of the construction of the spiral line. Therefore, Henry Briggs was asked to compute a table in order to replace the dip instrument by a simple table look-up. At the very end of Blundeville’s The Theoriques, we find the following appendix; see Fig. 19:

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Fig. 15 The construction of Gilbert’s mapping

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Fig. 16 The mater of the dip instrument in The Theoriques E

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A short appendix annexed to the former Treatise by Edward Wright, at the motion of the right Worshipful M. Doctor Gilbert Because of the making and using of the foresaid Instrument, for finding the latitude by the declination of the Magneticall Needle, will bee too troublesome for the most part of Seamen, being notwithstanding a thing most worthie to be put in daily practise, especially by such as undertake long voyages: it was thought meet by my worshipfull friend M. Doctor Gilbert, that (according to M. Blundeuile’s earnest request) this Table following should be

Navigation on Sea: Topics in the History of Geomathematics

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Fig. 17 The quadrant

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F Fig. 18 Determining the latitude for 60ı dip

hereunto adioned; which M. Henry Briggs (professor of Geometrie in Gresham Colledge at London) calculated and made out of the doctrine and tables of Triangles, according to the Geometricall grounds and reason of this Instrument, appearing in the 7 and 8 Chapter of M. Doctor Gilberts fift[h] booke of the Loadstone. By helpe of which Table, the Magneticall declination being giuen, the height of the Pole may most easily be found, after this manner. With the Instrument of Declination before described, find out what the Magneticall declination is at the place where you are: Then look that Magneticall declination in the second Collum[n]e of this Table, and in the same line immediatly towards the left hand, you shall find the height of the Pole at the same place, unleße there be some variation of the declination, which must be found out by particular obseruation in euery place.

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Fig. 19 The appendix as found in Edward Wrights On errors in Navigation

The next page (which is the final page of The Theoriques) indeed shows the table. Fig. 19 shows the appendix. In order to make the numbers in the table more visible, I have retyped the table. We shall not discuss this table in detail, but it is again worthwhile to review the relations between Gilbert, Briggs, Blundeville, and Wright (Hill 1997), p. 36.: Briggs was at the center of Gilbert’s group. At Gilbert’s request he calculated a table of magnetic dip and variation. Their mutual friend Edward Wright recorded and tabulated much of the information which Gilbert used and helped in the production of De Magnete. Thomas Blundeville, another member of Brigg’s group, and, like Gilbert, a former protégé of the Earl of Leicester, popularized Gilbert’s discoveries in The Theoriques of the Seven Planets (1602), a book in which Briggs and Wright again collaborated.

It took Blundeville’s The Theoriques to describe the construction of the dip instrument accurately which nebulously appeared in Gilbert’s De Magnete. However, even Blundeville does not say a word concerning the computation of the table. Another friend in the Gresham circle, famous Edward Wright, included all of the necessary details in the second edition of his On Errors in Navigation

Navigation on Sea: Topics in the History of Geomathematics First column Heighs of the pole Degrees 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Second column Magnetical declination Deg. Min. 2 11 4 20 6 27 8 31 10 34 12 34 14 32 16 28 18 22 20 14 22 4 23 52 25 38 27 22 29 4 30 45 32 24 34 0 35 36 37 9 38 41 40 11 41 39 43 6 44 30 45 54 47 15 48 36 49 54 51 11

First column Heighs of the pole Degrees 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Second column Magnetical declination Deg. Min. 52 27 53 41 54 53 56 4 57 13 58 21 59 28 60 33 61 37 62 39 63 40 64 39 65 38 66 35 67 30 68 24 69 17 70 9 70 59 71 48 72 36 73 23 74 8 74 52 75 35 76 17 76 57 77 37 78 15 78 53

99 First column Heighs of the pole Degrees 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

Second column Magnetical declination Deg. Min. 79 29 80 4 80 38 81 11 81 43 82 13 82 43 83 12 83 40 84 7 84 32 84 57 85 21 85 44 86 7 86 28 86 48 87 8 87 26 87 44 88 1 88 17 88 33 88 47 89 1 89 14 89 27 89 39 89 50 90 0

(Wright 1610), the first edition of which appeared 1599. Much has been said about the importance of Edward Wright, (see, for instance, Parsons and Morris 1939), and he was certainly one of the first – if not the first – who was fully aware of the mathematical background of Mercator’s mapping; see Sonar (2001, p. 131ff). It is in Wright’s On Errors in Navigation where Wright and Briggs explain the details of the computation of the dip table which was actually computed by Briggs showing superb mastership of trigonometry. We shall now turn to this computation.

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Fig. 20 Figure A: CHAP. XIIII to find the inclination or dipping of the magneticall needle under the Horizon

8

The Computation of the Dip Table

In the second edition of Wright’s book On Errors in Navigation, we find in chapter XIIII: Let OBR be a meridian of the Earth, wherein let O be the pole, B the æquinoctal, and R the latitude (suppose 60 degrees), and let BD be perpendicular to AB in B and equal to the subtense OB; and drawing the line AD, describe therwith the arch DSV. Then draw the subtense OR, wherewith (taking R for the center) draw the lines RS equal to RO and parts AS equal to AD. Also because BR is assumed to be 60 deg., therefore let ST be 60 90 of the arch STO, and draw the line RT, for the angle ART shall be the cõplement of the magnetical needles inclinatiõ under the horizon, which may be found by the solution of the two triangles OAR and RAS after this manner:

Although here again other notation is used as in Blundeville’s book as well as in De Magnete, we can easily see the situation as described by Gilbert. Now the actual computation starts: First the triangle OAR is given because of the arch OBR, measuring the same 150 degr. and consequently the angle at R 15 degr. being equall to the equall legged angle at O; both which together are 30 degr. because they are the complement of the angle OAR (150 degr.) to a semicircle of 180 degr.

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Fig. 21 Figure B: The first step

The first step in the computation hence concerns the triangle OAR in Fig. 21. Since point R lies at 60ı (measured from B), the arc OBR corresponds to an angle of 90ı C 60ı D 180ı  30ı D 150ı. Hence, the angle at A in the triangle OAR is just 90ı C .90ı  30ı / D 150ı. Since OAR is isosceles, the angles at O and R are identical, and each is 15ı . Let us go on with Wright: Secondly, in the triangle ARS all the sides are given AR the Radius or semidiameter 10,000,000: RS equal to RO the subtense of 150 deg. 19,318,516: and AS equall to AD triple in power to AB, because it is equal in power to AB and BD, that is BO, which is double in power to AB.

The triangle ARS in Fig. 21 is looked at where S lies on the circle around A with radius AD and on the circle around R with radius OR. The segment AR is the radius of the Earth or the “whole sine.” Wright takes this value to be 107 . We have to clarify what is meant by subtense and where the number 19,318,516 comes from. Employing the law of sines in triangle OAR, we get sin 150ı OR D ; AR sin 15ı and therefore, it follows that OR D RO D AR 

OR sin 150ı D AR  D 19;318;516.:5257 : : :/: AR sin 15ı

Since O lies on the circle around R with radius OR as S does, we also have RO D RS. Furthermore, AS D AD since D as S lies on the circle around A with radius AD. Per constructionem, we have BD D OB, and using the theorem of Pythagoras, we conclude OB 2 D BD 2 D 2AB 2

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as well as AD 2 D AB 2 C BD 2 D AB 2 C 2AB 2 D 3AB 2 : This reveals the meaning of the phrase triple in power to AB: “the square is three times as big as AB.” Hence, it follows for AS: p AS D AD0 3  AB D 17;320;508.:0757 : : :/: It is somewhat interesting that Wright does not compute the square root but gives an alternative mode of computation as follows: Or else thus: The arch OB being 90 degrees, the subtense therof OB, that is, the tangent BD is 14,142,136, which sought in the table of Tangents, shall giue you the angle BAD 54 degr. 44 min. 8 sec. the secant whereof is the line AD that is AS 17,320,508.

p In the triangle ABD, we know the lengths of the segments AB and BD D OB D 2  AB D 14;142;135.:6237 : : :/. Hence, for the angle at A, we get tan †A D

Fig. 22 Figure C: Second step

BD D AB

p p 2  AB D 2; AB

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which results in †A D 54:7356 : : :ı D 54ı 440 800 . Using this value, it follows from sin †A D

OB BD D AD AD

that AD D AS D

p OB AB D 2 D 17;320;508.:0757 : : :/: sin †A sin 54ı 440 800

Wright goes on: Now then by 4 Axiom of the 2 booke of Pitisc.1 as the base or greatest side SR 19,318,516 is to ye summe of the two other sides SA and AR 27,320,508; so is the difference of them SX 7,320,508 to the segment of the greatest side SY 10,352,762; which being taken out of SR 19,318,516, there remaineth YR 8,965,754, the halfe whereof RZ 4,482,877, is the Sine of the angle RAZ 26 degr. 38 min. 2 sec. the complement whereof 63 degr. 21 min. 58 sec. is the angle ARZ, which added to the angle ARO 15 degr. maketh the whole angle ORS, 78 make 52 degr. 14 min. 38 sec. which taken out of ARZ 63 degr. 21 min. 58 sec. wherof 60 90 degr. 21 min. 58 sec. there remaineth the angle TRA 11 deg. 7 min. 20 sec. the cõplement whereof is the inclination sought for 78 degrees, 52 minutes, 40 seconds.

The “Axiom 4” mentiod is nothing but the Theorem of chords: If two chords in a circle intersect then the product of the segments of the first chord equals the product of the segments of the other.

Looking at Fig. 23, the theorem of chords is M S  SX D SR  S Y and since MS D AS C AB, it follows .AS C AB/  SX D SR  S Y; resulting in SX SR D AS C AB SY Now the computations should be fully intelligible. Given are AS D 17;320;508; AB D 107 ; SR D OR D 19;318;516, and SX D AS  AX D 7;320;508. Hence, SY D

SX  .AS C AB/ D 10;352;762: SR

1 The Silesian Bartholomäus Pitiscus (1561–1613) authored the first useful text book on trigonometry: Trigonometriae sive dimensione triangulorum libre quinque, Frankfurt 1595, which was published as an appendix to a book on astronomy by Abraham Scultetus. First independent editions were published in Frankfurt 1599, 1608, 1612 and in Augsburg 1600. The first English translation appeared in 1630.

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Fig. 23 Figure D: The final step

Fig. 24 Figure E

The segment YR has length YR D SR  SY D 8;965;754. Per constructionem, the point Z is the midpoint of YR. Half of YR is RZ D 4;482;877. From sin †RAZ D RZ=AR D 4;482;877=107 D 0:4482877, we get †RAZ D 26:6339ı D 26ı 380 200 . In the right-angled triangle ARZ, we see from Fig. 24 that †ARZ D 90ı  26ı 380 200 D 63ı 210 5800 . At a degree of latitude of 60ı , the angle ARO at R is 15ı since the obtuse angle in the isosceles triangle ORA is 90ı C 60ı D 150ı. Therefore, †ORS D †ARO C †ARZ D 78ı 210 5800 . The part TRS of this angle is 60/90 of it; hence, †TRS D 52ı 140 3800 . We arrive at †TRA D †ARZ  †TRA D 63ı 210 5800  52ı 140 3800 D 11ı 70 2000 :

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Fig. 25 Figure F

Fig. 26 Figure G

The dip angle ı is the complement of the angle TRA, ı D 90ı  †TRA D 78ı 520 4000 : Although the task of computing the dip if the degree of latitude is given is now accomplished, we find a final remark on saving of labor: The Summe and difference of the sides SA and AR being alwaies the same, viz. 27,320,508 and 7,320,508, the product of them shall likewise be alwaies the same, viz. 199,999,997,378,064 to be diuided by ye side SR, that is RO the subtense of RBO. Therefore there may be some labour saued in making the table of magneticall inclination, if in stead of the said product you take continually but ye halfe thereof, that is 99,999,998,689,032, and

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so diuide it by halfe the subtense RO, that is, by the sine of halfe the arch OBR. Or rather thus: As halfe the base RS (that is, as the sine of halfe the arch OBR) is to halfe the summe of the other two sides SA & AR 13,660,254, so is half the difference of the e 3,660,254 to halfe of the segment SY, which taken out of half the base, there remaineth RZ ye sine of RAZ, whose cõplement to a quadrãt is ye angle sought for ARZ. According to this Diagramme and demonstration was calulated the table here following; the first columne whereof conteineth the height of the pole for euery whole degree; the second columne sheweth the inclination or dipping of the magnetical needle answerable thereto in degr. and minutes.

Although we have taken these computations from Edward Wright’s book, there is no doubt that the author was Henry Briggs as is also clear from the foreword of Wright.

9

Conclusion

The story of the use of magnetic needles for the purposes of navigation is fascinating and gives deep insight into the nature of scientific inventions. Gilbert’s dip theory and the unhappy idea to link latitude to dip is a paradigm of what can go wrong in mathematical modeling. The computation of the dip table is, however, a brilliant piece of mathematics and shows clearly the mastery of Henry Briggs.

References Balmer H (1956) Beiträge zur Geschichte der Erkenntnis des Erdmagnetismus. Verlag H.R. Sauerländer, Aarau Campling A (1922) Thomas Blundeville of Newton Flotman, co. Norfolk (1522–1606). Norfolk Archaeol 21:336–360 Gilbert W (1958) De Magnete. Dover, New York Greenberg JL (1995) The problem of the Earth’s shape from Newton to Clairault. Cambridge University Press, Cambridge Hill Ch (1997) Intellectual origins of the English revolution revisited. Clarendon Press, Oxford Parsons EJS, Morris WF (1939) Edward Wright and his work. Imago Mundi 3:61–71 Pumfrey S (2002) Latitude and the magnetic Earth. Icon Books, Cambridge Sonar Th (2001) Der fromme Tafelmacher. Logos Verlag, Berlin Sonar Th (2002) William Gilberts Neigungsinstrument I: Geschichte und Theorie der magnetischen Neigung. Mitteilungen der Math. Gesellschaft in Hamburg, Band XXI/2, 45–68 Taylor EGR (1954) The mathematical practioneers of Tudor & Stuart England. Cambridge University Press, Cambridge Ward J (1740) The lives of the professors of Gresham College. Johnson Reprint Corporation, London Waters DJ (1958) The art of navigation. Yale University Press, New Haven Wright E (1610) Certaine errors in navigation detected and corrected with many additions that were not in the former edition as appeareth in the next pages. Printed by Felix Knights, London

Gauss’ and Weber’s “Atlas of Geomagnetism” (1840) Was not the first: the History of the Geomagnetic Atlases Karin Reich and Elena Roussanova

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Samuel Dunn (1723–1794) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dunn’s Geomagnetic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 John Churchman (1753–1805) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Magnetic Atlas (Philadelphia 1790) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Magnetic Atlas (2. Edition, London 1794) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Magnetic Atlas (3. Edition, New York, 1800; 4. Edition, London, 1804) . . . . . 4 Christopher Hansteen (1784–1873) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Magnetic Atlas (Christiania 1819) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Expedition to Russia (1828–1830) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Hansteen’s Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Carl Friedrich Gauss (1777–1855) and Wilhelm Weber (1804–1891) . . . . . . . . . . . . . . . . 5.1 The Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The “Magnetic Association” in Göttingen (1834–1843) . . . . . . . . . . . . . . . . . . . . . . 5.3 Gauss’ “General Theory of Geomagnetism” (1839) . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Gauss’ and Weber’s “Atlas of Geomagnetism” (Leipzig 1840) . . . . . . . . . . . . . . . . . 5.5 The End of the “Magnetic Association” in Göttingen . . . . . . . . . . . . . . . . . . . . . . . . 6 Excursus: Berghaus’ “Physical Atlas” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Term “Atlas” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108 109 109 110 116 116 118 121 124 126 126 128 130 130 131 131 132 133 133 138 138 139 141

K. Reich () Department of Mathematics, University of Hamburg, Germany e-mail: [email protected] E. Roussanova Saxonian Academy of Sciences and Humanities in Leipzig, Leipzig, Germany e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_94

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K. Reich and E. Roussanova

Abstract

In the beginning there were geomagnetic charts which were interesting mainly for seafaring nations. The first geomagnetic atlas was printed in London in 1776; its author was the mathematician, cartographer, and astronomer Samuel Dunn, whose aim had been to ameliorate the navigation especially to support the trading of England with the East Indies. The American John Churchman, however, was mainly surveyor; his magnetic atlas was published in four editions, in 1790, 1794, 1800, and 1804. Churchman was in contact with George Washington and with Thomas Jefferson, as far as his geomagnetic charts were concerned; he also became a member of the Academy of Sciences in St. Petersburg. Churchman was convinced that the magnetic pole in the north could be found in northern Canada. The Norwegian astronomer and physicist Christopher Hansteen was convinced that there were two magnetic poles in the north and two in the south; his atlas was published in 1819. One of the magnetic poles in the north should be in Siberia. Hansteen found support by the king of Sweden and Norway so that he undertook an expedition to Siberia (1828–1830). Carl Friedrich Gauss and Wilhelm Weber began to study geomagnetism in 1831: They believed that there were only two magnetic poles, one in the north and one in the south. They were able to calculate their positions by means of Gauss’ new theory of geomagnetism (1839); as sailors found out, their coordinates turned out to be nearly correct. Gauss’ and Weber’s Atlas is without doubt the most famous; it was published in Leipzig in 1840, including 18 geomagnetic charts. On two of these charts, equipotential lines were presented for the first time in history.

1

Introduction

It was Gerhard Mercator (1512–1594) who made the word “atlas” common use. His posthumously published work “Atlas sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati Figura” (Mercator 1595) paved the way for it to become a part of our everyday language. His atlas contained maps of both the earth and the sky. Since this time, the word “atlas” has been understood as a collection of terrestrial and celestial maps. Declination (deviation of the compass needle) has always been particularly important for seafarers. One of the earliest maps to show declination lines, the famous “Tabula nautica,” was drawn by Edmond Halley (1656–1742) and published in 1701. This map was based on a special form of projection known as Mercator projection, first introduced by Mercator in 1569 in his map of the world “Nova et aucta orbis terrae descriptio ad usum navigantium emendate accommodata.” As the title of Mercator’s world map indicated, this kind of projection was extremely important to seafarers, as it was conformal. It quickly also became clear that magnetic declination is not constant but subject to continual change. As early as the eighteenth century, numerous maps were published which were updated versions of the “Tabula nautica,” for example, by

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James Dodson (ca.1705–1757), William Mountaine (ca.1700–1779), Johann Gustaf Zegollström (1724–1787), etc. Moreover, both world maps and regional maps were published in which declination lines were only marked in certain areas. Inclination also played a role in geomagnetic research. In the eighteenth century, maps were therefore also published with the inclination marked in lines. Intensity maps came into being in the nineteenth century; one of the first ones was published by Alexander von Humboldt (1769–1859) in 1804 (Hellmann 1895). The very existence of maps with geomagnetic lines suggested the idea of publishing an atlas containing maps of the world and special maps with declination lines, inclination lines, etc. The “Atlas of Geomagnetism” presented by Carl Friedrich Gauss and Wilhelm Weber in 1840 is without a doubt the most wellknown and famous; the maps published in it were exceptionally important. However, mention should also be made of its predecessors, which could hardly have been more different: the atlas published in 1776 by the Englishman Samuel Dunn, which contained 9 maps; the atlas by the American geodesist John Churchman, which contained one or two maps, four editions of which appeared by 1804; and finally the historical atlas published by the Norwegian physicist and astronomer Christopher Hansteen in 1819, which contained 15 maps. However, the aims of the authors varied just as much as the atlases themselves. These will be explained later.

2

Samuel Dunn (1723–1794)

2.1

Biographical Notes

Apparently for a long time, there was only one biography of Samuel Dunn in existence, which however is based on later descriptions, i.e., those published in the “Dictionary of National Biography” (Godwin 1888). Recently a revised and enlarged biography was published in the Internet (Heard 2013). According to these sources, Dunn was born in 1723 in Crediton in Devonshire, where his father died in 1744. Dunn initially earned his living by directing a school, where he also taught “writing, accounts, navigation, and other mathematical science.” He then moved to the school in Bowdown Hill, where he taught until Christmas 1751. Finally Dunn moved to London, where he taught at various schools and also gave private tuition. In 1757, he invented the “universal planisphere, or terrestrial and celestial globes in plano.” In the year 1758 Dunn became “master of an academy, for boarding and qualifying young gentlemen in arts, sciences, and languages, and for business at Chelsea.” Dunn was able to use the Ormond Observatory for astronomical observations and saw a comet there in 1760.1 Dunn informed the Royal Society of his astronomical observations. In 1763, he stopped working as a schoolteacher in Chelsea and moved to Brompton Park near Kensington, where he gave private

1

Comet 1759 III [sic], Great Comet, visible from January 7, 1760, to February 11, 1760.

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tuition. In 1764, he went on a journey to France. In 1769 the Astronomer Royal Nevil Maskelyne (1732–1811) invited him, to observe the transit of Venus. When Dunn published his “New Atlas of the Mundane System; or of Geography and Cosmography” in 1774, he lived at 6 Clement’s Inn, close to Temple Bar. His scientific reputation was by now so great that he was appointed “mathematical examiner of the candidates for the East India Company Service.” This position enabled him to publish more of his work; in 1777 he lived at Covent Garden and in 1780 at 1 Boar’s Head Court, Fleet Street. He died in January 1794; his will was dated January 5, 1794; he was buried on January 23 at St. Dunstan-in-the-West. Dunn published nine treatises in the “Philosophical Transactions of the Royal Society” in London along with numerous astronomical monographs, particularly in the field of practical astronomy; other publications included works and maps (atlases) for seafaring and navigation.

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Dunn’s Geomagnetic Maps

In the year 1774 Dunn published in London for the first time his work: “A New Atlas of the Mundane System, or of Geography and Cosmography: Describing the Heavens and the Earth, the Distances, Motions, and Magnitudes, of the Celestial Bodies; The Various Empires, Kingdoms, States, and Republics, Throughout the Known World: With a Particular Description of the Latest Discoveries; The Whole Elegantly Engraved On Sixty-Two Copper Plates; To These Is Prefixed a General Introduction to Geography and Cosmography, in Which the Elements of These Sciences Are Compendiously Deduced from Original Principles, and Traced from Their Invention to the Latest Improvements. With a general introduction.” This atlas was a great success; the sixth edition was published in 1810, by which time the number of copper plates had grown to 64 (Dunn 1810). Despite being primarily geographical, geomagnetism also played a role. The work contains a chapter titled “Magnetic Needle,” which presents Edmond Halley’s “Tabula nautica” from the year 1701 right at the beginning. Dunn then distinguished between three different types of declination lines: § 206 Lines of Variation, which are usually delineated on the chart, may be considered as one or the other of these three kinds, namely 1st Lines of Equal Variation, which run nearly Eastward and Westward on the chart. 2dly Crooked or Worm Lines, which run nearly northward and southward on the chart. 3dly Parabolic Lines, which do somewhat resemble the path which a body describes when it is thrown otherwise than perpendicular to the horizon. According to their usefulness Dunn mentioned: § 207 “The first kind of lines [: : :] can [: : :] be but of little use to mariners.” § 208 “The second kind [: : :] are of excellent use.” § 209 “The third kind: parabolic lines, are useful in the same manner as the worm lines before described” (Dunn 1810, p. 13).

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Fig. 1 Dunn, Samuel: A new Atlas of the Mundane system. 6 Ed. with additions and considerable improvements (London 1810): cosmography epitomised, in six copperplate delineations. Chart after page 22, right column. State Library Berlin, shelf mark 2ı Kart. B 902

Dunn’s “A New Atlas” contains a plate “Cosmography Epitomised” with pictures of a compass, the worm lines, and the parabolic lines on the variation chart (right column) (Fig. 1). His work “The Navigator’s Guide to the Oriental or Indian Seas: Or, the Description and Use of a Variation Chart of the Magnetic Needle, Designed for Shewing the Longitude, Throughout the Principal Parts of the Atlantic, Ethiopic, and Southern Oceans, Within a Degree, or Sixty Miles. With an Introductory Discourse, Concerning the Discovery of the Magnetic Variation, the finding the Longitude Thereby, and Several Useful Tables” was published in London the following year (Dunn 1775). It did not contain any maps; however, two folios prepared by Dunn, i.e., variation maps of the North and South Atlantic oceans, respectively, appeared that same year (Hellmann 1895, p. 22). These maps were also included in Dunn’s groundbreaking book of geomagnetic maps, “A New Atlas of Variations of the Magnetic Needle for the Atlantic, Ethiopic, Southern and Indian Oceans, Drawn from a Theory of the Magnetic System, Discovered and Applied to Navigation” (London 1776, Fig. 2). Even in the title, Dunn emphasized that his maps were based on a new “theory of the magnetic system” he had evolved himself. He also utilized the astronomical and

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Fig. 2 Title page of Samuel Dunn’s “A New Atlas” (London 1776). The royal library of Copenhagen, shelf mark KBK 2–852, x-2013/28 (Photography by Henrik Dupont)

magnetic observations made by the captains of the ships in the service of the East India Company. The atlas was intended to make it easier for seafarers to navigate to the East Indies, as the declination lines enabled them to establish their longitude on these oceans to the nearest degree or 60 miles. This work was not published by a publishing company; instead, the author had it printed and distributed via “Maiden Lane, Covent Garden.” This is probably why only a few copies found their way into libraries. This atlas begins with a letter “To the Honourable the Court of Directors of the United Company of Merchants of England trading to the EAST INDIES” which dates from November 6, 1776. Further Dunn reports: Under whose Predecessors, near two Centuries since, the British Mathematician, Edward Wright of London, published his Invention of the true Sea Chart, commonly called Mercator’s, changing the Angles made by the Merdidians and Rhumb-lines2 into Rectilinear ones and thereby reducing the whole Process of Navigation, or the Art of Sailing on the Oceans, to the Doctrine of Plane Triangles; AND WHOSE PATRONAGE HATH ENCOURAGED THE PUBLICATION OF THIS BOOK. (Dunn 1776, p. IV)

2

Rhumb line, i.e., loxodrome.

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The Edward Wright (1561–1615) already quoted here was a mathematician and geodesist; he was one of the authors to whom Dunn felt he owed a particular debt. In 1589, Wright took part in an expedition to the Azores; he published his findings in 1599 in his work “Certaine Errors of Navigation,” a second edition of which was published in 1610 and a third in 1657. In this work, Wright gave a detailed explanation of what he understood by “Mercator projection” and described the five most important errors which were critical for establishing a ship’s exact position at sea. As Dunn also used this type of projection, he provided a detailed description in his “Introduction”: The first good Effect arising from this Invention of the true Sea Chart is, that in it the Rhumb-line Bearings are straight Lines, and consequently, by Help of a straight Rule and a Pair of Compasses, those Bearings are easily shewn by the Chart. [: : :] The second good Effect is, that all the Cases of Sailing are solved by Proportions [: : :] Another Advantage which this Chart has: the Course may be accurately set off on it, as also the Distance sailed, by observing a proper Method. (Dunn 1776, pp. V–VI)

Mercator projection is indeed conformal, which made it particularly important for navigation at sea. The first map reproduced in Dunn’s atlas is therefore nothing other than Wright’s map; like Wright’s, it contains no magnetic declination lines. Dunn’s “Introduction” consists of 18 chapters. Dunn finally gets to the point in the ninth chapter: The Variation of the Magnetic Needle, or its Horizontal Deflection from the true Meridian of the Place of Observation, hath been considered as of the greatest Importance in Navigation, by every Man of either Learning or Ingenuity in Mathematics, Philosophy, or Nautical Affairs, who hath given his Opinion concerning it. Hypotheses have been formed, but none of them have agreed with Observations, nor hat it been possible to draw the Variation-lines by them.

Dunn based his maps on a new theory which he described as follows: By the Word Theory, I mean what indicates the Cause of the Variation at different Places and Times, and plainly demonstrates it by Mathematical Principles and Philosophical Laws. The drawing of accurate Variation Charts of the Magnetic Needle by a Theory must sound very strangely to Philosophers, as nothing of this Kind hath hitherto been thought possible.

Dunn mentioned another advantage of his method (chapter 14): Another Advantage which ariseth from the Discovery of a Theory of the Variation is, that the Variation Charts may not only be drawn from a few Observations, but they may be drawn for Years past and to come; from which ariseth, an easy Method of making them applicable for the intermediate Years, with Errors which are very inconsiderable.

Ultimately Dunn came to the conclusion (chapter 17): Charts properly constructed to lesser and greater Scales, after Wright’s Manner, will be of Use to Navigators, in indicating the course to be sailed from one Place to another, whether Distances be great or small. Variation Charts of the Magnetic Needle, accurately and properly drawn, will be of Use in allowing for the Variation. Both of these being applied at Sea, will enable the Navigator to pursue his Voyage in cloudy Weather, or when no Astronomical Observations can be made, with that Certainty which otherwise he cannot expect. It is the Plan and Design of this Work to institute such, but to complete it will require some Time and Judgement. (Dunn 1776, p. VI)

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Fig. 3 Chart 2: Variation chart of the Atlantic Ethiopic and Indian Oceans for the year 1770 delineated according to Mercator’s or Wright’s projections agreeable with the latest and best observations by S. Dunn (November 6, 1776). The royal library of Copenhagen, shelf mark KBK 2–852, x-2013/28. N.B. This is the first variation chart of those seas that has ever been drawn by a theory and found to agree nearly with observations (Photography by Henrik Dupont)

The introduction is followed by 9 maps measuring about 50  60 cm each: 1. A Wright’s Chart of the Atlantic Ethiopic and Indian Oceans (November 6, 1776) (Fig. 3). 2. Variation Chart of the Atlantic Ethiopic and Indian Oceans for the year 1770, delineated according to Mercator’s or Wright’s projections agreeable with the latest and best observations by S. Dunn (November 6, 1776). 3. A Variation Chart of the Atlantic Ethiopic and Indian Oceans for the year 1800 (November 6, 1776). 4. A Variation Chart of the Atlantic Ocean for the year 1776 (November 6, 1776). 5. A Variation Chart of the Atlantic Ethiopic and Indian Oceans for the year 1776 (November 6, 1776). 6. Continuation of plate (5). This chart is designed for determining the longitude in those seas within a degree or 60 miles (November 10, 1775). 7. A Variation Chart of the Indian Ocean for South of the Line for 1776 (November 6, 1776) (Fig. 4).

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Fig. 4 Chart 7: A variation chart of the Indian Ocean for South of the line for 1776 (November 6, 1776). The royal library of Copenhagen, shelf mark KBK 2–852, x-2013/28 (Photography by Henrik Dupont)

8. Continuation of plate (7). The lines in the west part of this chart are designed for determining the longitude (November 6, 1776). 9. India with the magnetic variations for 1776 (November 6, 1776). Hellman’s comment on this work “A New Atlas” was: “This rare atlas contains seven [sic] large-scale declination maps of superb technical execution; magnetic maps on a larger scale have possibly never been published before” (Hellmann 1895, p. 22).3 Similar maps on a much smaller scale are found in Dunn’s work “A new epitome of practical navigation; or guide to the Indian Seas,” published in London in 1777 (Dunn 1777): Plate 12: A miniature variation chart of the Atlantic Ocean for the years from 1770 to 1820 3

In the original German: “Dieser seltene Atlas enthält sieben Deklinationskarten grossen Maassstabes in vorzüglicher technischer Ausführung; vielleicht sind magnetische Karten in grösserem Maassstabe niemals publicirt worden” (Hellmann 1895, p. 22).

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Plate 13: A miniature variation chart of the Ethiopic Ocean for the years 1770 to 1820 Plate 14: A miniature variation chart of the Indian Ocean for the years from 1770 to 1820

3

John Churchman (1753–1805)

3.1

Biographical Notes

Unless otherwise mentioned, the following description is based on Silvio Bedini’s two-part article (Bedini 2000). Churchman came from a Quaker family which settled in Nottingham, Maryland, in 1704. Three generations of this family were surveyors: born in 1753, John was John III of his family and the most famous of its surveyors. John III, from now on referred to merely as John, learnt surveying from his father George (1730–1814). John published numerous maps, including a map of the peninsula between Delaware and Chesapeake; these in particular helped to make him famous. In 1779, John presented a “memorial” to the “American Philosophical Society” (only founded in 1743) with a request to have his map published. This petition was granted, and two editions of the map were published in the years 1786 and 1787 with the following inscription “To the American Philosophical Society this Map of the Peninsula between Delaware and Chesopeak Bays with said Bays and Shored adjacent drawn from the most Accurate Survey is humbly inscribed by John Churchman.” This map was reprinted several times by US Geological Survey with the title “Delaware at the Time of the Ratification of the Constitution.” Besides working as a surveyor, John also ran numerous businesses, particularly with land, i.e., he bought land in grand style and saw that it was resold. John Churchman occupied himself with geomagnetism from the late 1770s on. He developed a magnetic needle theory which aimed to improve the accuracy with which longitude could be established at sea. This was mainly to be facilitated by new maps marked with declination lines. In 1777, an early exposition of his geomagnetic theory appeared in the Philadelphia Press. Here he assumed that there were two satellites, moons, orbiting the earth, one around the North Pole and the other around the South Pole. In 1785, he made a perpetuum mobile which was driven by magnetic forces. In 1787, he presented a new theory of magnetic needle declination to the “American Philosophical Society.” However, this theory was not supported by the respective scholars; David Rittenhouse (1732–1796) was one of those who rejected it.4

4 David Rittenhouse became a member of the “American Philosophical Society” in 1768; he was its president from 1791 to 1796. He was also an astronomer at the University of the State of Pennsylvania, from where the College of Philadelphia was founded in 1791.

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However, Churchman was not discouraged; instead, he sent letters aiming to obtain the support of prominent personages and scientists. This strategy was successful. He published some of the replies he received himself (Churchman (1790), Appendix, pp. 1–5 as well as Churchman (1794), p. 65–76); others were reproduced in exchanges of letters which have only recently been published. April 10, 1787, is the date of Churchman’s first letter to Joseph Banks (1743– 1820), who became President of the Royal Society in 1788 and held this office until 1820. We know of seven letters exchanged by Churchman and Banks (Banks 2007): – Churchman to Banks, April 10, 1787, Philadelphia (letter no. 723) – Banks to Churchman, September 1, 1787, Soho Square (letter no. 768) – Churchman to Banks, including a “Memorial To the Honourable Commissioners of Longitude for the Nation of England,” September 29, 1787, Philadelphia (letter no. 776) – Banks to Churchman, without date, Soho Square (letter no. 777) – Churchman to Banks, May 8, 1788, Philadelphia (letter no. 840) – Churchman to Banks, January 7, 1792, Philadelphia (letter no. 1086) – Churchman to Banks, March 8, 1804, Boston (letter no. 1767) Banks’ response was very positive; he invited Churchman to visit England. Churchman accepted this invitation and stayed in London from 1792 to 1796. In 1787, Churchman also contacted his compatriot Thomas Jefferson (1743– 1826), whom he initially addressed as “Dear Friend,” “Esteemed Friend,” and later as “My Honourable Friend.” Jefferson was stationed in Paris as a diplomat from 1785 to 1789; in 1802, he was elected the third President of the USA, an office which he held until 1809. Ten letters are known to have passed between Churchman and Jefferson between 1787 and 1802 (Jefferson 1950– 2013): – Churchman to Jefferson, June 6, 1787, with an enclosure from April 10, 1787, Philadelphia (vol. 11, pp. 397–399) – Jefferson to Churchman, August 8, 1787, Paris (vol. 12, pp. 5–6) – Churchman to Jefferson, November 22, 1787, Philadelphia (vol. 11, pp. 374–375) – Churchman to Jefferson, May 15, 1789, Philadelphia (vol. 15, pp. 129–130) – Jefferson to Churchman, September 18, 1789, Paris (vol. 15, pp. 439–440) – Jefferson to Churchman, [November 24, 1790] (vol. 18, p. 68) – Churchman to Jefferson, January 13, 1791, South 2nd Street No.183 (vol. 18, pp. 492–493) – Churchman to Jefferson, April 2, 1792, No.183 South 2nd Street (vol. 23, pp. 363–364) – Jefferson to Churchman, April 4, 1792, Philadelphia (vol. 23, pp. 369–370) – Churchman to Jefferson, May 7, 1802, Boston (vol. 37, pp. 424–426)

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The Magnetic Atlas (Philadelphia 1790)

Churchman first published his work “An Explanation of the Magnetic Atlas, or Variation Chart, Hereunto Annexed Projected on a Plan Entirely New, by Which the Magnetic Variation on any Part of the Globe May Be Precisely Determined” in Philadelphia in 1790. This work was accompanied by a magnificent large map with the inscription “George Washington, President of the United States this magnetic Atlas or variation Chart Is humbly inscribed by John Churchman” (Fig. 5). This map, which showed only the northern hemisphere, had a diameter of 60.5 cm and consisted of 12 strips which were probably meant to be affixed to a globe; the diameter of the globe would probably have been about 39 cm. According to the great magnetic map expert Gustav Hellmann, this was the first map with declination

Fig. 5 Dedication: “George Washington, President of the United States of America. This magnetic Atlas or variation chart is humbly inscribed by John Churchman” (Philadelphia 1790). State library Berlin, shelf mark W 780

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lines ever published in America (Hellmann 1895, p. 22). Churchman had already contacted Washington 1 year previously, in 1789; George Washington (1732–1799) was the first President of the newly founded USA from 1789 to 1797. In all, 8 letters from the period between 1789 and 1792 have survived; however, rather than answering himself, Washington left his secretary Tobias Lear (1762–1816) to write his replies (Washington 1983–2011): – Churchman to Washington, May 7, 1789, New York, Water Street No.66 (vol. 2, pp. 225–227) – Churchman to Washington, August 9, 1790, Philadelphia (vol. 6, p. 222) – Lear to Churchman, August 28, 1790 (vol. 6, p. 222) – Churchman to Lear, September 8, 1791, Philadelphia (vol. 8, pp. 512–514) – Lear to Churchman, September 10, 1791, Philadelphia (vol. 8, pp. 512–514) – Churchman to Washington, December 29, 1791 (vol. 9, pp. 342–344) – Churchman an Washington, July 14, 1792, Bank Street Baltimore (vol. 10, p. 540) – Churchman an Washington, September 5, 1792, Baltimore (vol. 11, pp. 71–74) At that time, a map with declination lines, i.e., with geomagnetic data, was so important that it was presented to the President of the USA as a gift, which was accepted with all due consequence. On August 9, 1790, Churchman wrote to Washington from Philadelphia, enclosing his recent publication as a token of his best respects for the president: Being convinced that no name would be likely to stamp so great a value on the work as that of the personage to whom it was dedicated, he hopes to be pardoned for the Liberty which he has taken in this respect.

Secretary Lear responded on Washington’s behalf: The President of the United States has received a Copy of the Magnetic Atlas or Variation Chart, together with the book of explanation which you have been so polite as to send him and requests your acceptance of this thanks for the same. “I am, moreover, ordered by the President to inform you, that being ever desirous of encouraging such publications as tend to promote useful knowledge, he requests you will consider him as a subscriber of your work.” (Washington (1983–2011) vol. 6, p. 222, also in Churchman (1794), pp 71–72)

On this map (Fig. 5), the declination lines are drawn in quite thickly on the oceans but only at large intervals on the continents; none of the declination lines are specially marked, nor is the zero line. There is one and only one magnetic point close to Baffin Bay, at a latitude of about 76ı and a longitude about 78ı west of Greenwich. In his letter to George Washington dated December 29, 1791, Churchman outlined the following plan: he wanted to organize an expedition to Baffin Bay to find the magnetic North Pole which was presumably located there (Washington 1983–2011, vol. 9, pp. 342–344). However, Washington did not respond to this proposal.

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In his document “Explanation of the Magnetic Atlas,” Churchman quoted his predecessors, including Dunn, but only his “A New Atlas of the Mundane System” in the edition of 1788. Moreover, Churchman gave a particularly detailed explanation of the geomagnetic theory published by Leonhard Euler (1707–1783); Euler was the first scientist to produce a map with declination lines projected stereographically (Reich and Roussanova 2012, pp. 147–148). Churchman apparently had sent his essay “Explanation of the Magnetic Atlas” also to the Academy of Sciences in St. Petersburg. At that time, Johann Albrecht Euler (1734–1800), oldest son of the mathematician Leonhard Euler, held the post of Permanent Secretary. Like his father, Johann Albrecht was particularly interested in geomagnetism. On January 31, 1791, the minutes of the meetings held at the Academy in St. Petersburg recorded the following: Monsieur le Conseiller de Collèges Roumovsky5 remit un extrait en langue russe que Monsieur l’Adjoint Konoff6 a fait d’un imprimé anglois de Monsieur Churchman: An Explanation of the Magnetic Atlas or variation chart etc. by John Churchman Philadelphia 1790; que Son Altesse Madame la Princesse de Daschkow7 avoit reçu pendant les dernières vacances, et envoyé à Messieurs les Adadémiciens, pour qu’ils l’examinent et en disent leur sentiment. Monsieur Churchman donne dans cet imprimé une méthode aisée de déterminer par le moyen de quelques tables et d’une sphère la déclinaison de l’aiguille magnétique pour un lieu proposé quelconque et à chaque temps donné: il s’agissoit donc d’examiner si cet imprimé répond pleinement à la question physico-mathématique que Madame la Princesse avoit choisie pour le prix académique de 17938 et s’il n’en rend pas la publication superflue. Messieurs les Académiciens chargés de cet examen rapportèrent donc en conformité de cet ordre, qu’ayant trouvé que les temps périodiques des deux points magnétiques de la Terre, sur lesquels Monsieur Churchman a dressé ses tables et sa carte magnétique, ne sont fondés que sur des observations faites en deux temps, années 1657 et 1790, et qu’ils en ont été même déterminés par un calcul très sujet à caution, ils sont tous d’avis que l’imprimé de Monsieur Churchman ne doit pas empêcher l’Acadmémie de proposer la question telle qu’elle a été donnée par Monsieur l’Academicien Krafft.9 Le Secrétaire10

5

Stepan Jakovlevich Rumovskij (1734–1812), astronomer; in 1753 he became assistant at the Academy of Sciences in St. Petersburg; during the years 1754 to 1756 he was guest scholar of Leonhard Euler in Berlin; in 1756 he succeeded Michail Lomonosov as director of the Geographical Department at the Academy of Sciences in St. Petersburg; in 1763 he became extraordinary and in 1767 ordinary professor at the Academy; during the years 1800–1803, he acted as its vice-president. 6 His correct name his Kononov and not Konoff. Aleksej Kononovich Kononov (1766–1795), physicist; since 1789 he was assistant and since 1795 extraordinary professor at the Academy of Sciences in St. Petersburg. 7 The princess Yekaterina Romanovna Dashkova (1743–1810) was directress of the Academy of Sciences in St. Petersburg from 1783 to 1796. 8 The Academy of Sciences in St. Petersburg had posed the following prize question for the year 1793: to present a magnetic chart of the world for the beginning of the nineteenth century, where the magnetic poles were indicated. This chart should be similar to the “Tabula nautica,” published by Edmond Halley in 1701 (Procès-verbaux 1911, pp 256–258). The original text was published in Latin and in Russian; a German translation in Reich and Roussanova (2012), p. 142. 9 Wolfgang Ludwig Krafft (1743–1814), physicist at the Academy of Sciences in St. Petersburg. 10 Johann Albrecht Euler.

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fut en conséquence chargé d’en dresser le programme et de le soumettre à l’approbation de la Conférence pour être imprimé et publié. (Procès-verbaux 1911, pp. 251–252)

Churchman received information about this meeting; Yekaterina Dashkova notified him on February 27/March 11, 1791: The Contents of your letter, which we received with the enclosed Magnetic Atlas, and its explanation, in due time, were the more interesting and agreeable to the Imperial Academy of Sciences, at the same matter is the subject of a Premium even now proposed by our Academy, as you will see by the printed advertisement I send you herewith. The progress you have already made gives me a pleasant hope, this important matter will derive no small increase from your ingenious works; and I make no doubt but your labours will greatly contribute to the final solution of this question. By the communication of your further enquiries and discoveries, especially relating to the southern hemisphere, the calculation of an universal set of tables, and the ascertaining of the exact revolutions of the two magnetic points round the poles of the earth, by a greater number of observations, you will very much oblige your humble servant, princess of Daschkaw. (Churchman 1794, p. 74)

However, Churchman did not receive the prize awarded in 1793; instead, it went to the Copenhagen-based physicist Christian Gottlieb Kratzenstein (1723– 1795). The most important scientific institution in Philadelphia was that later known as the “American Philosophical Society,” cofounded by Benjamin Franklin (1706– 1790) in 1743. It should be mentioned that Yekaterina Dashkova and Benjamin Franklin met in Paris on February 3, 1781. In consequence, Dashkova became the first female and first Russian member of the “American Philosophical Society,” while Franklin was the first American to become a member of the Academy of Sciences in St. Petersburg. Franklin and Dashkova maintained a constant correspondence, which was one of the highlights presented at the exhibition “The Princess and the Patriot: Ekaterina Dashkova, Benjamin Franklin, and the Age of Enlightenment” in Philadelphia in 2006 (Prince 2006). After his magnetic atlas, Churchman wanted to publish a book about gravity. He sent an exposition, a prospectus, to St. Petersburg: Son Altesse Madame la Princesse de Daschkow envoya pour être présenté de la part de Monsieur Churchman, auteur de l’Atlas magnétique, une fueille imprimée en anglois contenant Proposals for publishing a Dissertation on Gravitation containing conjectures concerning the case of the several kinds of attraction. Le Secrétaire fera circuler ce prospectus parmi Messieurs les Académiciens et Adjoints. (Procès-verbaux 1911, p. 286, January 16, 1792)

However, this work was never published.

3.3

The Magnetic Atlas (2. Edition, London 1794)

While Churchman was staying in England between 1792 and 1796, a second edition of his “Magnetic Atlas” appeared in London in 1794. This second edition contained two geomagnetic maps, one of the northern hemisphere (Fig. 6) and the other of the southern hemisphere (Fig. 7).

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Fig. 6 John Churchman: The magnetic Atlas, or variation charts, 2. edition (London 1794): Northern Hemisphere. State library Berlin, shelf mark 4ı My 3506

Both maps are dated July 1, 1794, measure approx. 59.5 cm in diameter, and – like the map published in 1790 – consisted of 12 strips which would have covered a complete globe. It must be noted that the map portraying the northern hemisphere (Fig. 6) is not identical with the map drawn for George Washington in 1790 (Fig. 5). The map of the northern hemisphere indicates a “magnetic pole” located in Northern Canada; a circle has been drawn around it on which the magnetic point and the magnetic nadir have been marked. The migration of the magnetic North Pole is also marked, i.e., its position in 1600, 1650, 1700, 1750, 1800, 1850, and 1900. The “first magnetic meridian for 1700” and the “first magnetic meridian for 1794” both pass through West Africa. A circle called the “magnetic orbit” was again drawn around the magnetic North Pole, and the magnetic point and magnetic nadir were marked on it.

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Fig. 7 John Churchman: The magnetic Atlas, or variation charts, 2. edition (London 1794): Southern Hemisphere. State library Berlin, shelf mark 4ı My 3506

With regard to the southern hemisphere, the magnetic South Pole is located south of West Australia between 100ı and 110ı east of Greenwich; as in the map of the northern hemisphere, a magnetic orbit is drawn and a magnetic point and magnetic nadir marked. Close to the magnetic point, there is a text stating “This Magnetic Point is fixed by more authentic observations on the Chart than in page 40 & 41.” Churchman sent copies of the second edition of his “Magnetic Atlas” to President Washington and his friend Jefferson. Jefferson gave his volume to the “American Philosophical Society” as a gift (Washington (1983–2011), vol. 2, p. 222 and Jefferson (1950–2013), vol. 37, p. 425). Yet another copy of the second edition went to the Academy of Sciences in St. Petersburg, whereupon he was elected an honorary member of the Academy in 1795, winning 7:4 of the votes. It seems he was the second American after Benjamin Franklin to be honored in this way. The minutes record:

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Monsieur le Directeur en fonction et gentil-homme de la Chambre de S. M. J. Paul de Bacounin,11 fit communiquer une lettre angloise adressée à Madame la Princesse de Daschkow par Monsieur Churchman et datée de Londre le 21 octobre, laquelle Son Altesse lui a envoyée de Moscou pour la faire lire à la Conférence, et lui proposer ensuite ce physicien américain, dont elle a déjà reçu diverses choses, pour être reçu au nombre de ces associés externes. Monsieur Churchman rapporte avoir envoyé à l’Académie un atlas magnétique, relativement auquel il fait quelques remarques sur la situation des pôles magnétiques du globe terrestre; il propose ensuite d’envoyer dans la partie occidentale de l’Amérique septentrionale un observateur pour y determiner la déclinaison et l’inclinaison de l’aiguille aimantée et il s’offre de s’y rendre lui-même si l’adadémie trouve bon de lui bonifier les frais du voyage; il raconte à cette occasion qu’un jeune négociant anglois a déjà fait par terre le trajet remarquable de l’Amérique septentrionale. Enfin Madame la Princesse de Daschkow lui ayant fait présent d’un exemplaire complet des Actes académiques, il indique un comptoir marchand à St- Pétersbourg, auquel l’Académie est priée de le remettre. La Conférence n’yant aucune résolution à prendre ni réponse à faire aux propositions de Monsieur Churchman, elle procéda à son élection par voie ordinaire du scrutin et Monsieur John Churchman fut reçu au nombre des Académiciens étrangers par sept voix contre quatre. (Procès-verbaux 1911, p. 409, January 8, 1795)

Like Benjamin Franklin, Churchman too owed his nomination as a member of the Academy of Sciences in St. Petersburg to Yekaterina Dashkova (Prince 2006, pp. 15–16). This second edition of Churchman’s “The Magnetic Atlas” was sent to a list of subscribers, including the scientists Joseph Banks and William Herschel (1738– 1822) in England and the Academies in Berlin, Lisbon, Copenhagen, etc.

3.4

The Magnetic Atlas (3. Edition, New York, 1800; 4. Edition, London, 1804)

After Churchman’s return to the USA, a third edition of his “Magnetic Atlas or Variation Charts” was published in 1800 in New York with some additions; the title page, for example, referred to the author as “Fellow of the Russian Imperial Academy.” According to a letter to Banks dated February 4, 1801, Churchman sent a copy of this edition to the Royal Society (Banks 2007, vol. 5, p. 343). He also sent a copy to his friend Jefferson, who was now the President of the USA (Jefferson 1950–2013, vol. 37, p. 425). Still another copy is located in Carl Friedrich Gauss’ private library in Göttingen (Fig. 8): In 1802, Churchman went to Europe again; his itinerary included both Copenhagen and St. Petersburg. Nikolaus Fuss (1755–1826), who was married to one of Leonhard Euler’s granddaughters, had been Permanent Secretary in St. Petersburg since 1800. The minutes of the Academy of Sciences in St. Petersburg record: Le Secrétaire présente de la part de Mr. John Churchman, membre externe de l’Académie, arrivé de Philadelphie: A variation chart, by John Churchman, Imperial Russian Academi-

11

Pavel Petrovich Bakunin (1776–1805), since 1794 vice-director of the Academy of Sciences in St. Petersburg, from 1796 to 1798 director.

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Fig. 8 Title page of John Churchman: The magnetic Atlas or variation charts of the whole terraqueous globe: comprising a system of the variation and dip of the needle by which the observations being truly made, the longitude may be ascertained. New York 1800. State and University Library Göttingen, Gauss library no. 130

cian. Le Secrétaire notifia en même tems qu’il s’est fait délivrer, avec la permission de Son Excellence Mr. le Président, six exemplaires de la carte magnétique de feu Mr. Kratzenstein, publiée par l’Académie, afin de les remettre à ce physicien américain qui travaille depuis dix ans au perfectionnement d’une carte pareille. (Procès-verbaux 1911, p. 1011, August 18, 1802)

Churchman was accordingly given Kratzenstein’s map, which had been published in St. Petersburg in 1795, as a gift.12 Churchman probably spent the winter in St. Petersburg, as the “National Cyclopedia” reports: He spent the winter in that high latitude perfecting his observations and corresponding with several European philosophers, the main object with all being the discovery of the law governing the constant variation, dip and declination of the magnetic needle in different parts of the earth. (Anonymus 1907)

12

Christian Gottlieb Kratzenstein: “Mappa exhibens declinationes acus magneticae ad initium saeculi decimi noni pro obtinendo praemio ab Academia Scientiarum imperiali Petropolitana ad annum 1793 proposito.” Kept at the State Library in Berlin, shelf mark 2ı Kart. W 750. This map was accompanied by Kratzenstein’s essay “Tentamen, resolvendi problema geographicomagneticum a perillustri Academia imperiali Petropolitana in annum 1793 propositum” (St. Petersburg 1798).

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Another journey to England was the occasion for the publication of the fourth and final edition of Churchman’s “Magnetic Atlas” in London in 1804; it included yet more additions. Churchman died on July 17, 1805, on board the ship taking him back to the USA; he was accordingly buried at sea.

4

Christopher Hansteen (1784–1873)

4.1

Biographical Notes

Born on September 26, 1784, in Christiania (now Oslo), Hansteen studied at Copenhagen University, where his most important teachers were the astronomer Thomas Bugge (1740–1815) and the physicist Hans Christian Oersted (1777– 1851). In 1806, Hansteen started teaching at the Latin School in Frederiksborg at the north of the island of Zealand in Denmark. Even then, he was already making his first observations and compiling his first thoughts on geomagnetism. A map with declination lines which Hansteen drew by hand and is dated April 2, 1810, still exists: “Mappa exhibens declinationis magneticas pro anno 1730; ad Meridianum Londini secundum tabulas Mountainiis et Dodsonis et observationes Middletonii constructa”13 (Enebakk and Johansen 2011, between the pages 16 and 17). In 1811, the Danish Academy of Sciences in Copenhagen posed the following prize question: “Are we obliged to assume that magnetic phenomena can only be explained by the existence of several magnetic axes in the earth, or is one enough?” Hansteen was awarded the prize in 1812. In 1814, Hansteen became a lecturer at the new university in Christiania, which had only been founded in 1811. In 1816, he was appointed Professor of Astronomy and Applied Mathematics there and also became the director of the observatory. In 1819, he had the opportunity to take a long journey and visited London and Paris (Enebakk and Johansen 2011). Hansteen’s radical, monumental work “Investigations into Terrestrial Magnetism” (“Untersuchungen über den Magnetismus der Erde”) (Hansteen 1819, Fig. 9) was a detailed answer to the prize question posed by the Danish Academy in 1811. In the preface, Hansteen explained what had inspired his interest in geomagnetism: this was a globe at the school in Frederiksborg, on which an area close to the South Pole was marked with an ellipse; its focal points were the earth’s two magnetic South Poles, one stronger (regio fortior), one weaker (regio debilior) (Hansteen 1819, p. VII–XII). Hansteen himself remained a lifelong advocate of Halley’s theory that the earth had two magnetic axes, i.e., that it had four magnetic poles, two in the north and two in the south. Hansteen’s vast work deals with

13 William Mountaine (ca.1700–1779); James Dodson (ca.1705–1757); Christopher Middleton (died in 1770).

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Fig. 9 Title page of Hansteen’s “Investigations into Terrestrial Magnetism.” State and University Library Göttingen, Gauss library no. 856

the magnetic phenomena known at that time; one lengthy chapter is devoted to Leonhard Euler’s geomagnetic theory. The “Investigations” has an appendix in which Hansteen recorded all the magnetic observation data obtained at sea and on land which was available to him. This appendix, i.e., the observation data, is the basis on which the maps published in his atlas were drawn. The observation data included data recorded by ship’s captains at sea and data obtained by scientists. It is worth noting that Hansteen also mentioned the observation data obtained by Edward Wright on his voyage to the Azores in 1589 (Hansteen 1819, Appendix, p. 41). Hansteen’s work triggered many discussions and reactions, for example, from David Brewster (1781–1868) in Edinburgh, Johann Tobias Mayer (1752–1830) in Göttingen, Ludwig Wilhelm Gilbert (1769–1824) in Leipzig, Caspar Horner (1774– 1834) in Zurich, and Edward Sabine (1788–1883) in Britain (Reich and Roussanova 2015, Chap. 2.6). The “Investigations into Terrestrial Magnetism” was the first detailed work devoted solely to geomagnetism; it therefore played a significant role which can by no means be overestimated. It was largely responsible for many physicists and astronomers beginning to investigate the topic of geomagnetism during the ensuing period. One of these was Gauss.

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The Magnetic Atlas (Christiania 1819)

Hansteen’s “Investigations into Terrestrial Magnetism” was accompanied by a “Magnetic Atlas” (Magnetischer Atlas gehörig zum Magnetismus der Erde) which appeared the same year (1819): Neither Dunn’s “A new atlas of variations of the magnetic needle” nor Churchman’s atlas was particularly important for Hansteen; however, he is sure to have been familiar with Dunn’s work, which was available in Copenhagen. Hansteen also quoted Samuel Dunn in a letter to Wilhelm Ludwig Gilbert dated April 15, 1823 (Hansteen 1823, p. 159).

Unlike its predecessors, Hansteen’s “Magnetic Atlas” is purely historical, i.e., it only contains geomagnetic maps depicting situations in previous years. This “Magnetic Atlas” consists solely of tables with no text at all. The 7 plates presented here encompass 15 maps including both declination and inclination maps. Hansteen’s atlas was therefore the first to include inclination maps. Another remarkable feature is that Hansteen’s maps show geomagnetic lines not only on the oceans but also on the continents. Like his predecessors, Hansteen used Mercator projection; however, some of his maps were drawn using stereographic projection. Plate I: three variation maps for the year 1600 (no. I), 1700 (no. II), and 1756 (no. III) Plate II: five variation maps for the year 1770 (no. IV), 1710 (no. V), 1720 (no. VI), and 1730 (no. VII); two inclination maps for the year 1700 (no. VIII) and 1800 (no. IX) Plate III: two variation maps for the year 1800 (no. X) and 1744 (no. XI) Plate IV: polar projection of a segment of the northern and southern hemispheres to clarify the location and movement of the magnetic poles between the years 1600 and 1800 (Fig. 10) Plate V: map of both hemispheres with complete drawings of the magnetic equator and the variation lines for both magnetic axes as per Euler’s first theory14 Plate VI: Mappa Hydrographica sistens Declinationes Magneticas anni 1787 (Fig. 11) Tafel VII: Mappa Hydrographica sistens Inclinationes Magneticas anni 1780 (Fig. 12) The print plates for all these maps still actually exist (Enebakk and Johansen 2011, pp. 66–67). All Hansteen’s maps are based on observation data and not on some kind of theory, as was the case with Dunn and Churchman.

14

A hand-drawn map still exists; see Enebakk and Johansen (2011, between the pages 32 and 33).

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Fig. 10 Hansteen’s “Atlas,” plate IV. State and University Library Göttingen, shelf mark 2 PHYS III, 8480

Fig. 11 Hansteen’s “Atlas,” plate VI. State and University Library Göttingen, shelf mark 2 PHYS III, 8480

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Fig. 12 Hansteen’s “Atlas,” Plate VII State and University Library Göttingen, shelf mark 2 PHYS III, 8480

4.3

Expedition to Russia (1828–1830)

Churchman conceived a plan of searching for the magnetic North Pole close to Baffin Bay; however, he was unable to realize it (see here. chapter 3.2) Hansteen believed the magnetic North Pole was located in Siberia, which is why he considered organizing an expedition to Russia. This expedition started in 1828 and lasted until 1830; it was totally devoted to geomagnetic research. Alexander von Humboldt’s trip to Russia took place in 1829 while Hansteen was still there. Humboldt was also researching geomagnetism, but his expedition was also dedicated to other topics. Hansteen returned to Norway in 1830.

4.4

Hansteen’s Maps

Hansteen’s cartographic work mainly centered around the years from 1819 to 1833. His “Magnetic Atlas” was created at the beginning of this period, but it is important to note that Hansteen did not publish any other atlas after 1819. However, several of the maps in his “Magnetic Atlas” were later reproduced on several occasions (Reich and Roussanova 2015, Appendix 1). Hansteen initially only published declination and inclination maps, but from 1824/5 onwards, he started publishing maps with intensity lines. These mainly comprised the wonderful intensity and isodynamic maps for which he became

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famous. It was Hansteen who coined the term “isodynamic lines” for intensity lines. The terms “isogonic lines” for declination lines and “isoclinic lines” for lines with the same angle of inclination also came from Hansteen (Hellmann 1895, pp. 14–15, 25).

5

Carl Friedrich Gauss (1777–1855) and Wilhelm Weber (1804–1891)

5.1

The Beginning

Born on April 30, 1777, in Brunswick, Carl Friedrich Gauss went down in history as a mathematician, astronomer, geodesist, and physicist. In 1807, he was appointed Professor of Astronomy at Göttingen University, where he was also the director of the observatory. Gauss had already developed an interest in geomagnetism in his younger years; his private library included the third edition of Churchman’s “Magnetic Atlas,” published in New York in 1800, as well as Hansteen’s “Investigations into Terrestrial Magnetism” (Hansteen 1819).15 Whether Gauss owned also Hansteen’s “Atlas” is unclear, because the Gauss library is no longer complete. However, Gauss only adopted geomagnetism as his area of research when Wilhelm Weber became his colleague and friend in 1831. Wilhelm Weber was born on October 24, 1804, in Wittenberg; in 1822, he began studying physics at the university in Halle, where he obtained his doctorate in 1827. The area of research with which he mainly occupied his time in Halle was acoustics; he published numerous works on this subject. Gauss and Weber met for the first time at the seventh symposium of the “Association of German Nature Researchers and Physicians” (Gesellschaft Deutscher Naturforscher und Ärzte) held in Berlin in September 1828; this event was held under the aegis of Alexander von Humboldt, in whose private establishment Gauss was accommodated during the symposium. At the symposium, Weber presented his latest findings, all in the field of acoustics; although acoustics was not his topic, Gauss was deeply impressed and seems to have quickly developed a great affection for the young man. When the post of Professor of Physics at Göttingen University fell vacant in 1830, Weber was appointed to the post, not without Gauss’ intervention. Wilhelm Weber moved to Göttingen in September 1831. This signaled the beginning of a new era in the field of geomagnetic research. Gauss and Weber complemented each other perfectly – Gauss the great theoretician and Weber the great experimenter. The field of electrodynamics and geomagnetism was new ground for both scientists in terms of physics research. Breathtaking results were quickly obtained both theoretically and experimentally. Gauss was able to present

15

State and University Library Göttingen, Gauss library no. 130 (Churchman) and no. 856 (Hansteen).

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his “Intensitas vis magneticae terrestris ad mensuram “Intensitas vis magneticae ad mensuram absolutam absolutam revocata” to the Royal Society of Sciences in Göttingen as early as December 15, 1832 (Gauss 1832); the Latin original did not appear until 1841 (Gauss 1841). However, the physicist Johann Christian Poggendorff (1796–1877) succeeded in publishing a German translation in 1833 in his famous journal “Annalen der Physik” (Gauss 1833). The first electromagnetic telegraph was installed in Göttingen that same year (1833); an aerial magnetic double line ran across Göttingen from Weber’s physics cabinet to Gauss’ observatory. The first telegram was sent at Easter 1833. A plan was soon conceived to set up a magnetic observatory on the site of the astronomical observatory. This was finished by the end of 1833; the new institution was equipped with a new magnetometer, a device which had only recently been developed. Observations began in January 1834.

5.2

The “Magnetic Association” in Göttingen (1834–1843)

The “Magnetic Association” (Magnetischer Verein) in Göttingen was founded at the time the new magnetic observatory was first put to use. The principles developed by Alexander von Humboldt in particular were applied and developed further in Göttingen. Humboldt already had a small network in 1829; all these stations recorded the intensity on specified dates at prescribed intervals with the aid of a Gambey’s instrument. The method consisted of making corresponding observations and graphically depicting the data thus obtained (Reich and Roussanova 2012/2013, Part I). Gauss and Weber succeeded in quickly expanding the network, which came to involve a lot of observation sites all working in cooperation. Humboldt’s method of making corresponding observations was expanded and perfected (Reich and Roussanova 2012/2013, Part II). The success quickly achieved by Gauss and Weber attracted numerous renowned physicists and astronomers to Göttingen and led to the construction of many new magnetic observatories throughout the world, not only in Europe but also in Russia, the USA, India, and various places in the British Empire. While the first publications by Gauss and Weber appeared in the “Annalen der Physik” and the “Astronomische Nachrichten,” a new journal, the “Results of the Observations by the Magnetic Association” (Resultate aus den Beobachtungen des Magnetischen Vereins), was published since 1836; volumes 1 and 2 were published in Göttingen and volumes 3, 4, 5, and 6 in Leipzig. Scientific essays by various authors were published, new instruments presented, new magnetic observatories described, etc. However, the core of the journal was the presentation of the data received and collected in Göttingen, which was commented on in detail in the “Explanatory notes on the scheduled drawings and observation data” (Erläuterungen zu den Terminszeichnungen und den Beobachtungszahlen). Gauss was the author for the first two volumes, Weber for the others. Each volume was accompanied by numerous plates, 50 in all, 25 of which illustrated the corresponding observations.

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5.3

133

Gauss’ “General Theory of Geomagnetism” (1839)

The publication of Gauss’ treatise “General Theory of Geomagnetism” (Allgemeine Theorie des Erdmagnetismus) in the “Results” in 1839 was a milestone in the development of geomagnetic science (Gauss 1839). Here Gauss introduced the term “potential,” which revolutionized the further history of geomagnetic research and was also used in other areas of physics (Reich 2011, pp. 48–50). It became possible to calculate data and draw maps using the potential equation V/R. These maps were completely new; nothing like them had ever been seen before, based as they were on a mathematical theory. The quality of the theory was established by comparing these maps with those based on observation data; Gauss was very satisfied with their conformity. Gauss’ “General Theory” was accompanied by six calculated maps of the world; these were the first maps he published: – Maps I and II: “map for the values of V/R” using both Mercator and stereographic projection – Maps III and IV: “map for the calculated values of declination” using both Mercator and stereographic projection – Maps V and VI: “map for the calculated values of the whole intensity” using both Mercator and stereographic projection

Maps I and II are the most remarkable; they are completely new inasmuch as they show equipotential lines, known by Gauss as “balance lines” (Gleichgewichtslinien). Such a thing had never existed before. From then on, maps were to include declination, inclination, intensity, and equipotential lines. The “Leipzig Allgemeine Zeitung” of August 6, 1839, included an extensive discussion titled “On the general theory of geomagnetism discovered by Gauss.”16

5.4

Gauss’ and Weber’s “Atlas of Geomagnetism” (Leipzig 1840)

The publication of several maps naturally gave rise to the idea of publishing an atlas. This appeared just 1 year later with the title “Atlas of Geomagnetism, traced according to the elements of the theory. Supplement from the Observations of the Magnetic Association, with the collaboration of C. W. B. Goldschmidt edited by Carl Friedrich Gauss and Wilhelm Weber” (Atlas des Erdmagnetismus nach den Elementen der Theorie entworfen. Supplement aus den Beobachtungen des Magnetischen Vereins unter Mitwirkung von C. W. B. Goldschmidt, herausgegeben von Carl Friedrich Gauß und Wilhelm Weber) (Gauss and Weber 1840; Fig. 13).

16

Anonymus: Über die von Gauß entdeckte allgemeine Theorie des Erdmagnetismus [Review]. In: Leipziger Allgemeine Zeitung, August 6, 1839, no. 218, supplement, p. 2566.

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Fig. 13 Title page of the “Atlas of Geomagnetism” (Leipzig 1840). State Library Berlin, shelf mark Kart LS HM 4ı My 3677

In the Preface, we read: My honoured friend Professor Weber, who balks at no sacrifice when it comes to rendering a service to science, undertook to realise the data with great completeness in a number of maps which depict all magnetic conditions everywhere on earth as determined by this theory. [: : :] This explanation exhausts everything required to understand the maps and evaluate their usefulness so completely that nothing remains for me to do but to express the wish that this laborious and meritorious work will be duly recognised by friends of the natural sciences.17 (Gauss Werke 12, pp. 377–378)

17

In the original German: “Mein verehrter Freund, Herr Professor Weber, der keine Aufopferung scheuet, wo es gilt der Wissenschaft einen Dienst zu leisten, unternahm es, eine solche Versinnlichung durch eine Anzahl von Karten zu veranstalten, die in grösster Vollständigkeit alle

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The observation-based map used for comparison was/were: – In the case of declination, Peter Barlow’s map (Barlow 1833) – In the case of inclination, Caspar Horner’s maps (Horner 1836/1842) – In the case of intensity, Edward Sabine’s map (Sabine 1838) The “Atlas” published by Gauss and Weber in 1840 begins with a very comprehensive text and is accompanied by 18 maps. This text explains each of the maps in detail. The maps were based on the geomagnetic parameters calculated for 1,262 sites around the terrestrial globe; 10,096 values were accordingly calculated and recorded in tables (Gauss Werke 12, p. 386, including the last four tables). The scientists who contributed to the maps in the atlas were Wilhelm Weber, the observer Benjamin Goldschmidt (1807–1851) who worked in Göttingen, the Russian astronomer Aleksandr Nikolaevich Drashusov (1816–1890) who was staying in Göttingen at that time, and the mathematician Heinrich Eduard Heine (1821–1881), who was studying with Gauss at that time and later became famous in his own right. This “Atlas” was also completely new inasmuch as it was not a historical atlas, but one which depicted the present situation. It contained 18 maps, 9 with Mercator projection and 9 with stereographic projection; maps XIII, XIV, XVII, and XVIII conform with the corresponding maps already published in 1839: maps I and II “map for the values of V/R” (see Fig. 14), maps VII and VIII “map for the calculated values of western intensity Y.” In detail: – The maps I (Fig. 14) und II: “map for the values of V/R” (Karte für die Werthe von V/R) – The maps III und IV: “ideal distribution of magnetism on the earth’s surface” (Ideale Vertheilung des Magnetismus auf der Erdoberfläche) – The maps V und VI (Fig. 15): “map for the calculated values of the northern intensity X” (Karte für die berechneten Werthe der nördlichen Intensität X) – The maps VII und VIII: “map for the calculated values of the western intensity Y” (Karte für die berechneten Werthe der westlichen Intensität Y) – The maps IX und X: “map for the calculated values of the vertical intensity Z” (Karte für die berechneten Werthe der verticalen Intensität Z) – The maps XI und XII: “map for the calculated values of the horizontal intensity” (Karte für die berechneten Werthe der horizontalen Intensität)

magnetischen Verhältnisse für die ganze Erdoberfläche, so wie jene Theorie sie ergiebt, graphisch darzustellen. [: : :] Diese Erklärung erschöpft alles, was zum Verständnis der Karten und zur Beurtheilung des Nutzens, welchen sie leisten können, nöthig ist, so vollständig, dass mir nichts hinzuzusetzen übrig bleibt als der Wunsch, dass diese mühsame und verdienstliche Arbeit bei den Freunden der Naturwissenschaften gerechte Anerkennung finden möge” (Gauss Werke vol. 12, pp. 377–378).

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Fig. 14 Map I: Map for the values of V/R – equipotential lines (Gauss and Weber 1840). State library Berlin, shelf mark Kart LS HM 4ı My 3677

Fig. 15 Map VI: Map for the calculated values of the northern intensity X, in stereographic projection (Gauss and Weber 1840) state library Berlin, shelf mark Kart LS HM 4ı My 3677

– The maps XIII (Fig. 16) und XIV: “map for the calculated values of the declination” (Karte für die berechneten Werthe der Declination) – The maps XV (Fig. 17) und XVI: “map for the calculated values of the inclination” (Karte für die berechneten Werthe der Inclination) – The maps XVII und XVIII: “map for the calculated values of the whole intensity” (Karte für die berechneten Werthe der ganzen Intensität)

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Fig. 16 Map XIII: Map for the calculated values of the declination in Mercator projection, (Gauss 1839; Gauss and Weber 1840) state library Berlin, shelf mark Kart LS HM 4ı My 3677

Fig. 17 Map XV: Map for the calculated values of the inclination in Mercator projection state library in Berlin, shelf mark Kart LS HM 4ı My 3677

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Map I, the map with the equipotential lines (Fig. 14), is the pièce de résistance; in the original atlas, it was printed on special paper. The version of map I published in 1839 contained considerably fewer lines; the version published in 1840 was a significant improvement. The inclination lines were new to the atlas, as none of the maps in Gauss’ “General Theory” included them. All four geomagnetic parameters, i.e., declination, inclination, intensity, and potential, were now determined theoretically for the first time, “like planets and comet trails through their elements.” The following summary states: “This geomagnetic atlas thus opens the series of atlases which will appear at suitable intervals to clearly present the fundamental data of the history of geomagnetism. No excurse into the history of past times can be made here”18 (Gauss Werke 12, pp. 404–405).

5.5

The End of the “Magnetic Association” in Göttingen

The exceptionally fruitful collaboration between Gauss and Weber was shaken to its core when Wilhelm Weber lost his professorial post in Göttingen in December 1837. He was one of the so-called Göttingen Seven who cosigned the protest of professors in Göttingen against the breach in the constitution made by the new king of Hanover, Ernst August I. (1771–1851, king since 1837). Weber initially remained in Göttingen even though he no longer had a job; however, at Easter 1843 he went to Leipzig University, where he became Professor of Physics. Weber’s departure signaled the end of the “Magnetic Association”; the “Results of the Observations by the Magnetic Association” was discontinued. Both Gauss in Göttingen and Weber in Leipzig turned to other areas of research; geomagnetism ceased to be a topic of interest to either. When Weber returned to Göttingen in 1849 and was restored to his former professorial post, it was too late to build on past successes. It was politics that put an abrupt end to the first large-scale global research network.

6

Excursus: Berghaus’ “Physical Atlas”

Heinrich Karl Berghaus (1797–1884) was a geographer and cartographer; he studied in Berlin and was a professor at the “Building Academy” (Bauakademie) in Berlin from 1824 to 1855. Berghaus and Humboldt were friends for more than 40 years and maintained a lively correspondence. In 1845, Humboldt published the first volume of his acclaimed “Cosmos” (Humboldt 1845–1862). Heinrich

18 In the original German: “[: : :] (wie Planeten- und Cometenbahnen durch ihre Elemente) [: : :]. Der gegenwärtige Atlas des Erdmagnetismus eröffnet also die Reihe von Atlassen, welche in angemessenen Zwischenzeiten erscheinen sollen, um von nun an die Grunddata der Geschichte des Erdmagnetismus vollständig und übersichtlich vor Augen zu legen. Auf die Geschichte der vergangenen Zeit kann hier nicht eingegangen werden” (Gauss Werke vol. 12, pp. 404–405).

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Berghaus was the author and originator of the atlas which accompanied it. Berghaus’ “Physical Atlas or collection of maps depicting the main organic and inorganic natural phenomena according to their geographical dispersion and distribution” (Physikalischer Atlas oder Sammlung von Karten, auf denen die hauptsächlichsten Erscheinungen der anorganischen und organischen Natur nach ihrer geographischen Verbreitung und Vertheilung bildlich dargestellt sind) was published between 1837 and 1843 (Berghaus 1837–1843). At that time, this “Atlas” was the first of its kind; Berghaus’ “Atlas” included maps of the following eight areas: 1. 2. 3. 4. 5. 6. 7. 8.

Meteorology and climatology Hydrology and hydrographics Geology Magnetism of the earth Geography of plants Geography of animals Anthropography Ethnography

Part 4 of the “Physical Atlas” was accordingly devoted to geomagnetism; it contained five maps by different cartographers: three declination maps with both Mercator and stereographic projection and two intensity maps, also with Mercator and stereographic projection. Geomagnetism consequently only accounts for a comparatively small part of the “Physical Atlas,” which comprises altogether 90 pages. A second edition of Berghaus’ “Physical Atlas” also appeared in Gotha between 1849 and 1863. The third edition published in Gotha between 1886 and 1892 by Hermann Berghaus (1828–1890), a nephew of Heinrich Berghaus, is particularly interesting (Berghaus 1886–1892). The author of the section “Geomagnetism” was Georg Neumayer (1826–1909). This section included maps previously published by Hansteen, Gauss, and Weber and a map with equipotential lines calculated for 1885 (Fig. 18); it was the second of its kind, following the Atlas published by Gauss and Weber in 1840 (see Reich and Roussanova 2015, chapter 3.11).

7

The Term “Atlas”

Tobias Mayer’s “Mathematical Atlas” (Mathematischer Atlas), published in Augsburg in 1745, is particularly important. Mayer’s aim was to “extract the most necessary and useful information and place it in the hands of lovers of these marvellous [mathematical] sciences briefly yet in an easy, clear manner”19 (Mayer

19

In the original German: das “nötighste und nützlichste auszulesen, und auf eine kurze, jedoch leichte und deutliche Art denen Geneigten Liebhabern dieser herrlichen Wissenschaften (sc. der mathematischen) in die Hände zu liefern” (Mayer 1745, Vorwort).

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Fig. 18 Map with declination and equipotential lines, calculated for 1885, print 1891. Berghaus’ “Physical Atlas, section “Geomagnetism,” 3. edition (Gotha 1892). State library Berlin, shelf mark 2ı W 183

1745, preface). This was followed by all mathematical areas customary at that time, twelve in all, presented with the help of 60 + 8 plates. Mayer’s “Mathematical Atlas” facilitated the use of the word “atlas” in other areas concerned with collections of maps and tables. The term “atlas” was again used just a short time later for geomagnetic maps; geomagnetic atlases presented terrestrial maps with geomagnetic details, mostly lines, initially only declination lines. In 1776, Dunn’s atlas was in its early stages. The subsequent history of geomagnetic atlases reflects the development of geomagnetic science; ultimately all four types of geomagnetic lines were included in the magnetic atlases. During the first half of the nineteenth century, the term atlas came into use in yet more areas, i.e., in both physics and medicine. While the term “Physical Atlas” was still used in physics with reference to the earth and illustrations of the earth’s surface, this was no longer the case in medicine. Here the word “atlas” was used to mean merely a collection of plates, e.g., “Obstetrical Atlas” (Geburtshülflicher Atlas) (Kilian) or “Atlas of Pathological Anatomy for Practising Physicians” (Atlas

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der pathologischen Anatomie für praktische Ärzte) (Albers 1832–1862). Thereafter, the word “atlas” was no longer restricted to a collection of any kind of images of the earth but was also used to describe any pictorial collection. Acknowledgements The authors want to say thank you very much to the following persons and institutions: Henrik Dupont, the Royal Library of Copenhagen; Wolfgang Crom, Steffi Mittenzwei, and Holger Scheerschmidt, the State Library in Berlin, department of maps; and Bärbel Mund and Helmuth Rohlfing, the State and University Library Göttingen, department of manuscripts and rare books.

References Albers FH (1832–1862) Atlas der pathologischen Anatomie für praktische Aerzte. Bonn 1832–1862 Anonymus (1907) Churchman, John, scientist. Natl Cyclopedia Am Biogr 9:287. New York Banks J (2007) The Scientific Correspondence of Sir Joseph Banks, 1765-1820. Ed. by Neil Chambers, vols 1–6. London Barlow P (1833) On the present situation of the magnetic lines of equal variation, and their changes on the terrestrial surface. Philos Trans R Soc Lond Teil 2:667–673. London Bedini S (2000) John Churchman and his magnetic Atlas, Part I and II. In: History corner, professional surveyor magazine, vol 20. Issue November and December 2000. OnlineRessource, Part 1: http://www.profsurv.com/magazine/article.aspx?i=669; Part 2: http://www. profsurv.com/magazine/article.aspx?i=681 Berghaus H (1837–1843) Physikalischer Atlas oder Sammlung von Karten, auf denen die hauptsächlichsten Erscheinungen der anorganischen und organischen Natur nach ihrer geographischen Verbreitung und Vertheilung bildlich dargestellt sind. Gotha 1837–1843 Berghaus H (1886–1892) Physikalischer Atlas. Dritte Ausgabe. 75 Karten in sieben Abteilungen. Vollständig neu bearbeitet. Gotha 1886–1892 Churchman J (1790) An explanation of the magnetic atlas, or variation chart, hereunto annexed projected on a plan entirely new; by which the magnetic variation on any part of the globe may be precisely determined. Philadelphia 1790 Churchman J (1794) The magnetic atlas, or Variation charts of the whole terraqueous globe: comprising a system of the variation & dip of the needle by which the observations being truly made, the longitude may be ascertained. Second edition of (Churchman 1790): London 1794 (VII, 76 S., 3 charts). 3rd edn, New York 1800 (with additions, p 82). 4th edn, London 1804 (with considerable additions, p 86) Dunn S (1775) The Navigator’s guide to the oriental or Indian seas: or, the description and use of a variation chart of the magnetic needle, designed for shewing the longitude, throughout the principal parts of the Atlantic, Ethiopic, and Southern Oceans, within a degree, or sixty miles. With an introductory discourse, concerning the discovery of the magnetic variation, the finding the longitude thereby, and several useful tables. London 1775 Dunn S (1776) A new atlas of variations of the magnetic needle for the Atlantic, Ethiopic, Southern and Indian Ocean; drawn from a theory of the magnetic system, discovered and applied to navigation. London 1776. (p VI + 9 Charts) Dunn S (1777) A new epitome of practical navigation: or guide to the Indian Seas. Containing, I. The elements of mathematical learning used and applied in the theory and practice of nautical affairs. II. The theory of navigation, deduced from original principles. III. The method of correcting and determining the longitude at sea; by the variation of the magnetic needle. IV. The practice of navigation, in all kinds of sailing. The whole illustrated with a variety of copperplates. London 1777

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Dunn S (1810) A new Atlas of the Mundane system; or, of geography and cosmography: describing the heavens and the earth, the distances, motions, and magnitudes, of the celestial bodies. (First edition London 1774). The Sixth Edition With Additions and Considerable Improvements. London 1810 Enebakk V, Johansen NV (2011) Christopher Hansteens annus mirabilis. Reisen til London og Paris 1819. Oslo 2011 Gauss CF (Werke) 1 edition = vol 1, 2 Göttingen 1863; vol 3 Göttingen 1866; vol 4 Göttingen 1873; vol 5 Göttingen 1867; vol 6 Göttingen 1874; vol 7 Gotha 1871; 2nd edition and new edition = 12 volumes, 1870–1933. Reproductions Hildesheim, New York (1973 and 1981) Gauss CF (1832) Anzeige der “Intensitas vis magneticae terrestris ad mensuram absolutam revocata.” Göttingische Gelehrte Anzeigen 1832, pp 2041–2048, 2049–2058 (24. December, Nr. 205). In: Gauss Werke 5, pp 293–304. Corrected version in: Astronomische Nachrichten 10, 1833, Nr. 238, col. 349–360 Gauss CF (1833) Die Intensität der erdmagnetischen Kraft zurückgeführt auf absolutes Maaß. Annalen der Physik und Chemie 28 (= 104), 1833, pp 241–272, 591–615 Gauss CF (1839) Allgemeine Theorie des Erdmagnetismus. Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1838. Leipzig 1839, pp 1–57. In: Gauss Werke 5, pp 119–175 Gauss CF (1841) Intensitas vis magneticae terrestris ad mensuram absolutam revocata. Commentantiones societatis regiae scientiarum Gottingensis recentiores 8, (1832–1837) 1841, commentationes classis mathematicae pp 3–44. In: Gauss Werke 5, pp 79–118 Gauss CF, Weber W (1840) Atlas des Erdmagnetismus nach den Elementen der Theorie entworfen. Supplement zu den Resultaten aus den Beobachtungen des magnetischen Vereins unter Mitwirkung von C. W. B. Goldschmidt. Leipzig 1840. In: Gauss Werke 12, pp 335–408 Godwin G (1888) Dunn, Samuel (d. 1794). Dictionary of National Biography 16, 1888, pp 210– 212 Hansteen Chr (1819) Untersuchungen über den Magnetismus der Erde. Translation by P. Treschow Hanson. Christiania 1819. Accompanied by: Magnetischer Atlas gehörig zum Magnetismus der Erde. Christiania 1819 Hansteen Chr (1823) Zur Geschichte und zur Vertheidigung seiner Untersuchungen über den Magnetismus der Erde, und kritische Bemerkungen über die hierher gehörigen Arbeiten der Herren Biot und Morlet. (= Schreiben an Gilbert vom 13.4.1823). Annalen der Physik und der physikalischen Chemie 15 (= 75), 1823, pp 145–196 Heard N (2013) Samuel Dunn 1723–1794. Online Resource: http://www.heardfamilyhistory.org. uk/SamuelDunn.html Hellmann G (1895) E. Halley, W. Whiston, J. C. Wilcke, A. von Humboldt, C. Hansteen. Die ältesten Karten der Isogonen, Isoklinen, Isodynamen 1701, 1721, 1768, 1804, 1825, 1826. Berlin 1895. Reprint Nendeln; Liechtenstein 1969 Horner JC (1836/1842) [Kapitel] XVII. Magnetismus der Erde. Physikalisches Wörterbuch, herausgegeben von Johann Samuel Traugott Gehler, neu bearbeitet von Brandes, Gmelin, Horner, Muncke, Pfaff, vol. 6, second part, Leipzig 1836, pp 1023–1147. Charts in: KupferAtlas zu Johann Samuel Traugott Gehler’s Physikalischem Wörterbuch, Leipzig 1842 Jefferson Th (1950–2013) The papers of Thomas Jefferson, vols. 1–40. Princeton University Press, Princeton Kilian HF (ca. 1835-1838) Geburtshülflicher Atlas. Düsseldorf [sine anno] Mayer T (1745) Mathematischer Atlas, in welchem auf 60 Tabellen alle Theile der Mathematik vorgestellet; und nicht allein überhaupt zu bequemer Wiederholung, sondern auch den Anfängern besonders zur Aufmunterung durch deutliche Beschreibung u. Figuren entworfen werden. Augsburg 1745. In: Tobias Mayer: Schriften zur Astronomie, Kartographie, Mathematik und Farbenlehre, vol 4 (Historia scientiarum), ed. by Erhard Anthes and others, Hildesheim 2009 Mercator G (1595) Atlas sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati Figura. Duisburg Prince SA (2006) The princess & the patriot. Ekaterina Dashkova, Benjamin Franklin, and the age of enlightenment. American Philosophical Society, Philadelphia

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Procès-verbaux (1911) Procès verbaux des séances de l’académie Impériale des sciences depuis sa fondation jusqu’à 1803. Tome 4,1: 1786–1803. St. Petersburg 1911 Reich K (2011) Alexander von Humboldt und Carl Friedrich Gauß als Wegbereiter der neuen Disziplin Erdmagnetismus. HiN – Alexander von Humboldt im Netz 22, 2011, pp 35–55. Online-Ressource: http://www.uni-potsdam.de/u/romanistik/humboldt/hin/hin22/reich.htm Reich K, Roussanova E (2012) Meilensteine in der Darstellung von erdmagnetischen Beobachtungen in der Zeit von 1701 bis 1849 unter besonderer Berücksichtigung des Beitrages von Russland. In: Beschreibung, Vermessung und Visualisierung der Welt, ed. by Ingrid Kästner und Jürgen Kiefer. Aachen 2012, pp 137–160 Reich K, Roussanova E (2012/2013) Visualising geomagnetic data by means of corresponding observations. Alexander von Humboldt, Carl Friedrich Gauß and Adolph Theodor Kupffer [part 1]. GEM – International Journal on Geomathematics 3, 2012, 1, pp 1–16. Online Ressource: http://www.springerlink.com/content/x807664661171577. [part 2]. GEM – International Journal on Geomathematics 4, 2013, 1. pp 1–25. Online Ressource: http://link.springer.com/content/ pdf/10.1007/s13137-012-0043-4 Reich K, Roussanova E (2015) Die Erforschung des Erdmagnetismus durch Christopher Hansteen und Carl Friedrich Gauß. Der Briefwechsel beider Gelehrten im historischen Kontext. Manuscript in print Sabine E (1838) Report on the variations of the magnetic intensity abserved at different Points of the Earth’s surface. Report of the seventh meeting of the British Association for the advancement of science held at Liverpool in September 1837, vol VI, London 1838, pp 1–35 von Humboldt A (1845–1862) Kosmos. Entwurf einer physischen Weltbeschreibung. Stuttgart und Tübingen, vol 1, 1845; vol 2, 1847; vol 3, 1850; vol 4, 1858; vol 5, 1862. Further edition by Magnus Enzensberger, mit einem Nachwort versehen von Ottmar Ette und Oliver Lubrich, Frankfurt am Main 2004 Washington G (1983–2011) The papers of George Washington, Presidential Series, vols 1–58, Charlottesville and others, 1983–2011

Part II Observational and Measurement Key Technologies

Earth Observation Satellite Missions and Data Access Henri Laur and Volker Liebig

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 End-to-End Earth Observation Satellite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Space Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ground Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Overview on ESA Earth Observation Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 ERS-1 and ERS-2 Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Envisat Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Proba-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Earth Explorers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 GMES and Sentinels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Meteorological Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 ESA Third Party Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Some Major Results of ESA Earth Observation Missions . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Land . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Cryosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 ESA Programs for Data Exploitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 User Access to ESA Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 How to Access the EO Data at ESA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148 149 150 150 153 153 154 155 155 156 158 160 161 161 162 163 165 165 167 168 169 169 170

H. Laur () European Space Agency (ESA), Head of Earth Observation Mission Management Division, ESRIN, Frascati, Italy e-mail: [email protected] V. Liebig European Space Agency (ESA), Director of Earth Observation Programmes, ESRIN, Frascati, Italy e-mail: [email protected]

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Abstract

This article provides an overview on Earth Observation satellites, describing the end-to-end elements of an Earth Observation mission and then focusing on the European Earth Observation programs. Some significant results obtained using data from European missions (ERS, Envisat) are provided. Finally the access to Earth Observation data through the European Space Agency, free of charge, is described.

1

Introduction

Early pictures of the Earth seen from space became icons of the Space Age and encouraged an increased awareness of the precious nature of our common home. Today, images of our planet from orbit are acquired continuously and have become powerful scientific tools to enable better understanding and improved management of the Earth. Satellites provide clear and global views of the various components of the Earth system – its land, ice, oceans, and atmosphere – and how these processes interact and influence each other. Space-derived information about the Earth provides a whole new dimension of knowledge and services which can benefit our lives on a day-to-day basis. Earth Observation satellites supply a consistent set of continuously updated global data which can offer support to policies related to environmental security by providing accurate information on various environmental issues, including global change. Meteorological satellites have radically improved the accuracy of weather forecasts and have become a crucial part of our daily life. Earth Observation data are gradually integrated within many economic activities, including exploitation of natural resources, land-use efficiency, or transport routing. The European Space Agency (ESA) has been dedicated to observing the Earth from space ever since the launch of its first meteorological mission Meteosat in 1977. Following the success of this first mission, the subsequent series of Meteosat satellites, the ERS, and Envisat missions have been providing a growing number of users with a wealth of valuable data about the Earth, its climate, and changing environment. It is critical, however, to continue learning about our planet. As our quest for scientific knowledge continues to grow, so does our demand for accurate satellite data to be used for numerous practical applications related to protecting and securing the environment. Responding to these needs, ESA’s Earth Observation programs comprise a science and research element, which includes the Earth Explorer missions, and an applications element, which is designed to facilitate the delivery of Earth Observation data for use in operational services, including the well-established meteorological missions with the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT). In addition, the GMES (Global Monitoring for Environment and Security) Sentinel missions, which form part of the GMES Space Component, will collect robust, long-term relevant

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datasets. Together with other satellites, their combined data archives are used to produce Essential Climate Variables for climate monitoring, modeling, and prediction. Mathematics is of crucial importance in Earth Observation: onboard data compression and signal processing are common features of many Earth Observation satellites, inverse problems for data applications (e.g., the determination of the gravitational field from ESA GOCE satellite measurements), data assimilation to combine Earth Observation data with numerical models, etc.

2

End-to-End Earth Observation Satellite Systems

In this chapter, the end-to-end structure of a typical satellite system shall be explained. This will enable the user of Earth Observation (EO) data to better understand the process of gathering the information he or she is using. Earth Observation satellites usually fly on the so-called low Earth orbit (LEO) which means an orbit altitude of some 250–1,000 km above the Earth’s surface and an orbit inclination close to 90ı , i.e., a polar orbit. Another orbit used mainly for weather satellites like Meteosat is the geostationary orbit (GEO). This is the orbit in which the angular velocity of the satellite is exactly the same as the one of Earth (360ı in 24 h). If the satellite is positioned exactly over the equator (inclination 0ı ), then the satellite has always the same position relative to the Earth’s surface. This is why the orbit is called geostationary. Using this position, the satellite’s instruments can always observe the same area on Earth. The price to pay for this position is that the orbit altitude is approx. 36,000 km which is very far compared to a LEO. A special LEO is the sun-synchronous orbit which is used for all observation systems which need the same surface illumination angle of the sun. This is important for imaging systems like optical cameras. For this special orbit, typically 600– 800 km in altitude, the angle between the satellite orbit and the line between sun and Earth is kept equal. As the Earth rotates once per year around the sun, the satellite orbit has also to rotate by 360ı in 365 days. This can be reached by letting the rotational axis of the satellite orbit precess, i.e., rotate, by 360/365 degrees per day to keep exactly pace with the Earth’s rotation around the sun. As the Earth is not a perfect sphere, the excess mass at the equator creates an angular momentum which lets a rotating system precess like a gyroscope. The satellite has to fly with an inclination of approx. 98ı which is close to an orbit flying over the poles. As the Earth is rotating under this orbit, the satellite almost “sees” the Earth’s entire surface during several orbits. Typical LEO satellites are ERS, Envisat, or SPOT (see Sect. 3). All Earth Observation satellite systems consist of a space segment and a ground segment. If we regard the whole chain of technical infrastructures to acquire, downlink, process, and distribute the EO data, we speak about an end-to-end system.

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Fig. 1 Envisat satellite with its platform (service module and solar array) and the payload composed of ten instruments

2.1

Space Segment

The space segment (S/S) mainly consists of the spacecraft, i.e., the EO satellite which has classically a modular design and is subdivided into satellite platform (or bus) and satellite payload. Figure 1 shows as example the Envisat satellite. The satellite bus offers usually all interfaces (mechanical, thermal, energy, data handling) which are necessary to run the payload instruments mounted on it. In addition, it contains usually all housekeeping, positioning, and attitude control functions as well as a propulsion system. It should be mentioned that small satellites often have a more integrated approach. In some cases, the space segment can consist of more than one satellite, e.g., if two or more satellites fly in tandem, such as Swarm (see Sect. 3). The space segment may also include a data relay satellite (DRS), positioned in a geostationary orbit and used to relay data between the EO satellite and the Earth when the satellite is out of visibility of the ground stations, such as ESA Artemis satellite data relay for the Envisat satellite.

2.2

Ground Segment

The ground segment (G/S) provides the means and resources to manage and control the EO satellite, to receive and process the data produced by the payload

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Fig. 2 Main elements of an EO ground segment including the Flight Operations Segment (FOS) and the Payload Data Segment (PGS)

instruments, and to disseminate and archive the generated products. In general, the ground segment can be split into two major elements (see Fig. 2): – The Flight Operations Segment (FOS), which is responsible for the command and control of the satellite – The Payload Data Segment (PDS), which is responsible for the exploitation of the instruments data Both G/S elements have different communication paths: satellite control and telecommand uplink use normally S-band, whereas instrument data is downlinked to ground stations via X-band either directly or after onboard recording. In addition, Ka-band or laser links with very high bandwidth are used for inter-satellite links needed by data relay satellites like Artemis or the future EDRS (European Data Relay Satellite).

Flight Operations Segment The Flight Operations Segment (FOS) at ESA is under the responsibility of the European Space Operation Centre (ESOC) located in Darmstadt, Germany. The mandate of ESOC is to conduct mission operations for ESA satellites and to establish, operate, and maintain the related ground segment infrastructure. Most of ESA EO satellites are controlled and commanded from ESOC control rooms. ESOC’s involvement in a new mission normally begins with the analysis of possible operational orbits or trajectories and the calculation of the corresponding launch windows – selected to make sure that the conditions encountered in this early

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phase remain within the spacecraft capabilities. The selection of the operational orbit is a complex task with many trade-offs involving the scientific objectives of the mission, the launch vehicle, the spacecraft, and the ground stations. ESOC’s activity culminates during the Launch and Early Orbit Phase (LEOP), with the first steps after the satellite separates from the launcher’s uppermost stage, including the deployment of antennas and solar arrays as well as critical orbit and attitude control maneuvers. After the LEOP, ESOC starts the operations of the FOS, including generally the command and control of the satellite; the satellite operations uploading, based on the observation plans prepared by the Payload Data Segment; the satellite configuration and performance monitoring; the orbit prediction, restitution, and maintenance; and the contingency and recovery operations following satellite anomalies. ESOC does also provide a valuable service for avoidance of collision with space debris, a risk particularly high for LEO satellites orbiting around 800 km altitude. More information about ESOC activities can be found at http://www.esa.int/esoc.

Payload Data Segment The EO Payload Data Segment (PDS) at ESA is under the responsibility of ESRIN, located in Frascati, Italy, and known as the ESA Centre for Earth Observation, because it also includes the ESA activities for developing the EO data exploitation. The PDS provides all services related to the exploitation of the data produced by the instruments embarked on board the ESA EO satellites and is therefore the gateway for EO data users. The activities performed within a PDS are: – The payload data acquisition using a network of worldwide acquisition stations as well as data relay satellites such as Artemis – The processing of products either in near real time from the above data acquisitions or on demand from the archives – The monitoring of product quality and instrument performance, as well as the regular upgrade of data processing algorithms – The archiving and long-term data preservation and the data reprocessing following the upgrade of processing algorithms – The interfaces to the user communities from user order handling to product delivery, including planning of instrument observations requested by the users – The availability of data products on Internet servers or through dedicated satellite communication (see Sect. 5 for data access) – The development of new data products and new services in response to evolving user demand The ESA PDS is based on a decentralized network of acquisition stations and archiving centers. The data acquisition stations located close to the poles can “see” most of the LEO satellite passes and are therefore used for acquiring the data recorded on board during the previous orbit. Acquisition stations located away from polar areas are used essentially to acquire data collected over their station mask

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Fig. 3 Example of a network of acquisition stations for the data transmitted by the ESA Envisat satellite. In addition to the above ground stations, the Envisat data was also transmitted to ground through the ESA data relay satellite Artemis

(usually 4,000 km diameter). The archiving centers are generally duplicated to avoid data loss in case of fire or accidents (Fig. 3). In carrying out the PDS activities, ESA works closely with other space agencies as well as with coordination and standardization bodies. This includes a strong interagency collaboration to acquire relevant EO data following a natural disaster. More information about ESRIN activities can be found at http://www.esa.int/ esrin.

3

Overview on ESA Earth Observation Programs

3.1

Background

The European Space Agency (ESA) is Europe’s gateway to space. Its mission is to shape the development of Europe’s space capability and ensure that investment in space continues to deliver benefits to the citizens of Europe and the world. ESA is an international organization with 20 Member States. By coordinating the financial and intellectual resources of its members, it can undertake programs and activities far beyond the scope of any single European country. ESA’s programs are designed to find out more about Earth, its immediate space environment, our Solar System, and the Universe, as well as to develop satellite-based technologies and services, and to promote European industries. ESA works closely with space organizations inside and outside Europe. ESA’s Earth Observation program, known as the ESA Living Planet program, embodies the fundamental goals of Earth Observation in developing our knowledge

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of Earth, preserving our planet and its environment, and managing life on Earth in a more efficient way. It aims to satisfy the needs and constraints of two groups, namely, the scientific community and also the providers of operational satellitebased services. The ESA Living Planet program is therefore composed of two main components: a science and research element in the form of Earth Explorer missions and an operational element known as Earth Watch designed to facilitate the delivery of Earth Observation data for use in operational services. The Earth Watch element includes the future development of meteorological missions in partnership with EUMETSAT and also new missions under the European Union’s GMES initiative, where ESA is the partner responsible for developing the Space Component. The past ERS and Envisat missions contributed both to the science and to the applications elements of the ESA Living Planet program. General information on ESA Earth Observation programs can be found at http:// www.esa.int/esaEO.

3.2

ERS-1 and ERS-2 Missions

The ERS-1 (European remote sensing) satellite, launched in 1991, was ESA’s first Earth Observation satellite on low Earth orbit; it carried a comprehensive payload including an imaging C-band Synthetic Aperture Radar (SAR), a radar altimeter, and other instruments to measure ocean surface temperature and winds at sea. ERS2, which overlapped with ERS-1, was launched in 1995 with an additional sensor for atmospheric ozone research. At their time of launch, the two ERS satellites were the most sophisticated Earth Observation spacecraft ever developed and launched in Europe. These highly successful ESA satellites have collected a wealth of valuable data on the Earth’s land surfaces, oceans, and polar caps and have been called upon to monitor natural disasters such as severe flooding or earthquakes in remote parts of the world. ERS-1 was unique in its systematic and repetitive global coverage of the Earth’s oceans, coastal zones, and polar ice caps, monitoring wave heights and wavelengths, wind speeds and directions, sea levels and currents, sea surface temperatures, and sea ice parameters. Until ERS-1 appeared, such information was sparse over the polar regions and the southern oceans. The ERS missions were both an experimental and a preoperational system, since it has had to demonstrate that the concept and the technology had matured sufficiently for successors such as Envisat and that the system could routinely deliver to end users some data products such as reliable sea ice distribution charts and wind maps within a few hours of the satellite observations. The experimental nature of the ERS missions was outlined shortly after the launch of ERS-2 in 1995 when ESA decided to link the two spacecrafts in the first ever “tandem” mission which lasted for 9 months. During this time the increased frequency and level of data available to scientists offered a unique opportunity to observe changes over a very short space of time, as both satellites orbited Earth only 24 h apart and to experiment innovative SAR measurement techniques.

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The ERS-1 satellite ended its operations in 2000, far exceeding its 3 years planned lifetime. The ERS-2 satellite operated until 2011, i.e., 16 years after its launch, maximizing the benefit of the past investment. More information on ERS missions can be found at http://earth.esa.int/ers/.

3.3

Envisat Mission

Envisat, ESA’s second-generation remote-sensing satellite, launched in 2002, not only provided continuity of many ERS observations – notably the ice and ocean elements – but added important new capabilities for understanding and monitoring our environment, particularly in the areas of atmospheric chemistry and ocean biological processes. Envisat was the largest and most complex satellite ever built in Europe. Its package of ten instruments made major contributions to the global study and monitoring of the Earth and its environment, including global warming, climate change, ozone depletion, and ocean and ice monitoring. Secondary objectives were more effective monitoring and management of the Earth’s resources and a better understanding of the solid Earth. As a total package, Envisat capabilities exceeded those of any previous or planned Earth Observation satellite. The payload included three new atmospheric sounding instruments designed primarily for atmospheric chemistry, including measurement of ozone in the stratosphere. The advanced C-band Synthetic Aperture Radar collected high-resolution images with a variable viewing geometry, with wide swath and selectable dual polarization capabilities. A new imaging spectrometer was included for ocean color and vegetation monitoring, and there were improved versions of the ERS radar altimeter, microwave radiometer, and visible/near-infrared radiometers, together with a new very precise orbit measurement system. Combined with ERS-1 and ERS-2 missions, the Envisat mission is an essential element in providing long-term continuous data sets that are crucial for addressing environmental and climatological issues. In addition, the Envisat mission further promoted the gradual transfer of applications of Earth Observation data from experimental to preoperational and operational exploitation. Although its nominal lifetime was 5 years, ESA could operate the Envisat satellite for 10 years, i.e., until 2012 when a failure in the power subsystem abruptly ended the satellite control. More information on Envisat mission can be found at http://envisat.esa.int.

3.4

Proba-1

Launched in 2001, the small Project for On-Board Autonomy (Proba) satellite was intended as a 1-year ESA technology demonstrator. Once in orbit, however, its unique capabilities and performance made it evident that it could make big contributions to science so its operational lifetime is currently extended until 2012 to

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serve as an Earth Observation mission. Its main payload is a spectrometer (CHRIS), designed to acquire hyperspectral images with up to 63 spectral bands. Also aboard is the high-resolution camera (HRC), which acquires 5 m black and white images.

3.5

Earth Explorers

The Earth Explorer missions form the science and research element of ESA’s Living Planet program and focus on the atmosphere, biosphere, hydrosphere, cryosphere, and the Earth’s interior with the overall emphasis on learning more about the interactions between these components and the impact that human activity is having on natural Earth processes. The Earth Explorer missions are designed to address key scientific challenges identified by the science community while demonstrating breakthrough technology in observing techniques. By involving the science community right from the beginning in the definition of new missions and introducing a peer-reviewed selection process, it is ensured that a resulting mission is developed efficiently and provides the exact data required by the user. The process of mission selection has given the Earth science community an efficient tool for advancing the understanding of the Earth system. The science questions addressed also form the basis for development of new applications of Earth Observation. This approach also gives Europe an excellent opportunity for international cooperation, both within the wide scientific domain and also in the technological development of new missions (ESA 2006). The family of Earth Explorer missions is a result of this strategy. Currently, there are seven Earth Explorers missions and a further three undergoing feasibility study:

GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) GOCE was dedicated to measuring the Earth’s gravity field and modeling the geoid with unprecedented accuracy and spatial resolution to advance our knowledge of ocean circulation, which plays a crucial role in energy exchanges around the globe, sea-level change and Earth interior processes. GOCE also made significant advances in the field of geodesy and surveying. Launched in 2009 with a 2-years planned lifetime, the mission operations were extended by another 2.5 years including a lowering of the satellite in order to obtain an even better geoid spatial resolution. After 4.5 years of successful operations, the GOCE satellite run out of propellant and the satellite re-entered and burned into the atmosphere in November 2013. [This book includes a chapter dedicated to Geodesy and the GOCE mission, written by R Rummel] SMOS (Soil Moisture and Ocean Salinity) Launched in 2009, SMOS observes soil moisture over the Earth’s landmasses and salinity over the oceans. Soil moisture and sea-surface salinity are two variables in Earth’s water cycle that scientists need on a global scale for a variety of applications, such as oceanographic, meteorological, and hydrological forecasting, as well as research into climate change.

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CryoSat Launched in 2010, CryoSat acquires accurate measurements of the thickness of floating sea ice, so that seasonal to interannual variations can be detected, and also surveys the surface of continental ice sheets to detect small elevation changes. CryoSat’s main objective is to provide regional trends in Arctic perennial sea ice thickness and mass and to determine the contribution that the Antarctic and Greenland ice sheets are making to mean global rise in sea level. After 2 years of operations, CryoSat’s results confirmed, for the first time, that the decline in sea ice coverage in the polar region has been accompanied by a substantial decline in ice volume. Swarm Launched in 2013, Swarm is a constellation of three satellites that will provide high-precision and high-resolution measurements of the strength and direction of the Earth’s magnetic field. The geomagnetic field models resulting from the Swarm mission will provide new insights into the Earth’s interior, will further our understanding of atmospheric processes related to climate and weather, and will also have practical applications in many different areas such as space weather and radiation hazards. ADM-Aeolus (Atmospheric Dynamics Mission) Due for launch in 2016, ADM-Aeolus will be the first space mission to measure wind profiles on a global scale. It will improve the accuracy of numerical weather forecasting and advance our understanding of atmospheric dynamics and processes relevant to climate variability and climate modeling. ADM-Aeolus is seen as a mission that will pave the way for future operational meteorological satellites dedicated to measuring the Earth’s wind fields. EarthCARE (Earth Clouds Aerosols and Radiation Explorer) Due for launch in 2018, EarthCARE is being implemented in cooperation with the Japanese Aerospace Exploration Agency (JAXA). The mission addresses the need for a better understanding of the interactions between cloud, radiative, and aerosol processes that play a role in climate regulation. Future Earth Explorers In 2005, ESA released the latest opportunity for scientists from ESA Member States to submit proposals for ideas to be assessed for the next in the series of Earth Explorer missions. As a result, 24 proposals were evaluated and a shortlist of six missions underwent assessment study. Subsequently, three proposals were selected for the next stage of development (feasibility study). This process led to the selection of the BIOMASS mission as ESA’s seventh Earth Explorer mission. The BIOMASS mission aims to take measurements of forest biomass to assess terrestrial carbon stocks and fluxes.

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The process for selecting the eighth Earth Explorer mission is on going. More information on Earth Explorer missions can be found at: http://www.esa.int/esaLP/ LPearthexp.html.

3.6

GMES and Sentinels

The Global Monitoring for Environment and Security (GMES) program, now called Copernicus, has been established to fulfill the growing need among European policy-makers to access accurate and timely information services to better manage the environment, understand and mitigate the effects of climate change, and ensure civil security. Under the leadership of the European Commission, Copernicus relies largely on data from satellites observing the Earth. Hence, ESA is developing and managing the Copernicus Space Component. The European Commission, acting on behalf of the European Union, is responsible for the overall initiative, setting requirements, and managing the Copernicus services. To ensure the operational provision of the Earth Observation data, the Space Component includes a series of five space missions called “Sentinels,” which are being developed by ESA specifically for GMES. In addition, data from satellites that are already in orbit or are planned will also be used for the initiative. These “Contributing Missions” include both existing and new satellites, whether owned and operated at European level by the EU, ESA, and EUMETSAT or on a national basis. They also include data acquired from non-European partners. The GMES Space Component forms part of the European contribution to the worldwide Global Earth Observation System of Systems (GEOSS). The acquisition of reliable information and the provision of services form the backbone of Europe’s GMES initiative. Services are based on data from a host of existing and planned EO satellites from European and national missions, as well as a wealth of measurements taken in situ from instruments carried on aircraft, floating in the oceans, or positioned on the ground. Services provided through GMES fall so far into five main categories: services for land management, services for the marine environment, services relating to the atmosphere, services to aid emergency response, and services associated with security. The service component of GMES is under the responsibility of the European Commission. More information on GMES can be found at http://www.esa.int/esaLP/LPgmes. html.

Sentinel Missions The success of Copernicus will be achieved largely through a well-engineered Space Component for the provision of EO data to turn them into services for continuous monitoring the environment. The GMES Space Component comprises five types of new satellites called Sentinels that are being developed by ESA specifically to meet the needs of GMES.

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The Sentinel missions include radar and super-spectral imaging for land, ocean, and atmospheric monitoring. The so-called a and b models of the first three Sentinels (Sentinel-1a/1b, Sentinel-2a/2b, and Sentinel-3a/3b) are currently under industrial development, with the first satellite, Sentinel-1A, launched in 2013. Sentinel-1 will provide all-weather, day and night C-band radar imaging for land and ocean services and will allow to continue SAR measurements initiated with ERS and Envisat missions; Sentinel-2 will provide high-resolution optical imaging for land services; Sentinel-3 will carry an altimeter, optical, and infrared radiometers for ocean and global land monitoring as a continuation of Envisat measurements; and Sentinel-4 and Sentinel-5 will provide data for atmospheric composition monitoring (Sentinel-4 from geostationary orbit and Sentinel-5 from low Earth polar orbit). Sentinel-4 and Sentinel-5 will be instruments carried on the next generation of EUMETSAT meteorological satellites – Meteosat Third Generation (MSG) and post-EUMETSAT Polar System (EPS), respectively. However a dedicated Sentinel5 precursor mission is planned to be launched in 2015 to reduce the data gap between Envisat and Sentinel-5.

Copernicus Contributing Missions Before data from the Sentinel satellites is available, missions contributing to Copernicus play a crucial role ensuring that an adequate dataset is provided for the Copernicus services. The role of the Contributing Missions will, however, continue to be essential once the Sentinels are operational by complementing Sentinel data and ensuring that the whole range of observational requirements is satisfied. Contributing Missions are operated by national agencies or commercial entities within ESA’s Member States, EUMETSAT, or other third parties. GMES Contributing Missions data initially address services for land and ice and also focus on ocean and atmosphere. Current services mainly concentrate on the following observation techniques: – Synthetic Aperture Radar (SAR) sensors, for all-weather day/night observations of land, ocean, and ice surfaces – Medium- and low-resolution optical sensors for information on land cover, for example, agriculture indicators, ocean monitoring, coastal dynamics, and ecosystems – High-resolution and medium-resolution optical sensors – panchromatic and multispectral – for regional and national land monitoring activities – Very high-resolution optical sensors for targeting specific sites, especially in urban areas as for security applications – High accuracy radar altimeter systems for sea-level measurements and climate applications – Radiometers to monitor land and ocean temperature – Spectrometer measurements for air quality and atmospheric composition monitoring

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A ground segment, facilitating access to both Sentinel and Contributing Missions data, complements the GMES space segment. More information on GMES data can be found at http://gmesdata.esa.int.

3.7

Meteorological Programs

Meteosat With the launch of the first Meteosat satellite into a geostationary orbit in November 1977, Europe gained the ability to gather weather data over its own territory with its own satellite. Meteosat began as a research program for a single satellite by the European Space Research Organisation, a predecessor of ESA. Once the satellite was in orbit, the immense value of the data it provided led to the move from a research to an operational mission. ESA launched three more Meteosat satellites before the founding of EUMETSAT, organization partner of ESA for the meteorological programs. Launched in 1997, Meteosat-7 was the last of the first generation of Meteosat satellites. The first generation of seven Meteosat satellites brought major improvements to weather forecasting. But technological advances and increasingly sophisticated weather forecasting requirements created demand for more frequent, more accurate, and higher-resolution space observation. To meet this demand, EUMETSAT started the Meteosat Second Generation (MSG) program, in coordination with ESA. In 2002, EUMETSAT launched the first MSG satellite, renamed Meteosat-8 when it began routine operations to clearly maintain the link to earlier European weather satellites. It was the first of four MSG satellites, which are gradually replacing the original Meteosat series. The Meteosat Third Generation (MTG) will take the relay in 2018/2019 from Meteosat-11, the last of a series of four MSG satellites. MTG will enhance the accuracy of forecasts by providing additional measurement capability, higher resolution, and more timely provision of data. Like its predecessors, MTG is a joint project between EUMETSAT and ESA that followed the success of the firstgeneration Meteosat satellites. MetOp Launched in 2006, in partnership between ESA and EUMETSAT, MetOp is Europe’s first polar-orbiting satellite dedicated to operational meteorology. It represents the European contribution to a new cooperative venture with the United States providing data to monitor climate and improve weather forecasting. MetOp is a series of three satellites to be launched sequentially over 14 years, forming the space segment of EUMETSAT’s Polar System (EPS). MetOp carries a set of “heritage” instruments provided by the United States and a new generation of European instruments that offer improved remote sensing capabilities to both meteorologists and climatologists. The new instruments augment the accuracy of temperature humidity measurements, readings of wind speed and direction, and atmospheric ozone profiles.

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Preparations have started for the next generation of this EUMETSAT Polar System, the so-called Post-EPS. More information can be found at http://www.eumetsat.int.

3.8

ESA Third Party Missions

ESA uses its multi-mission ground systems to acquire, process, distribute, and archive data from other satellites – known as Third Party Missions (TPM). The data from these missions are distributed under specific agreements with the owners or operators of those missions, which can be either public or private entities outside or within Europe. ESA Third Party Missions are addressing most of the existing observation techniques: – Synthetic Aperture Radar (SAR) sensors, e.g., L-band instrument (PALSAR) on board the Japanese Space Agency ALOS – Low-resolution optical sensors, e.g., MODIS sensors on board the US Terra and Aqua satellites, SeaWIFS sensor on board the US OrbView-2 satellite, etc. – High-resolution optical sensors, e.g., Landsat imagery (15–30 m), SPOT-4 data (10 m), a 10 m radiometer instrument (AVNIR-2) on board the ALOS, medium resolution (20–40 m) sensors on board the Disaster Monitoring Constellation (DMC), etc. – Very high-resolution optical sensors, e.g., US Ikonos-2 (1–4 m), Korean Kompsat-2 (1–4 m), Taiwan Formosat-2 (2–8 m), Japanese PRISM (2.5 m) on board the ALOS, etc. – Atmospheric chemistry sensors, e.g., Swedish Odin, Canadian SciSat-1, and Japanese GOSAT satellites The complete list of Third Party Missions currently supported by ESA is available at http://earth.esa.int/thirdpartymissions/.

4

Some Major Results of ESA Earth Observation Missions

Since 1991, the flow of data provided by the consecutive ERS-1, ERS-2, and Envisat missions has been converted in extensive results, giving new insights into our planet. These results encompass many fields of Earth science, including land, ocean, ice, and atmosphere studies, and have shown the importance of long-term data collection to identify trends such as those associated to climate change. The ERS and Envisat missions are valuable tools not only for Earth scientists but gradually also for public institutions providing operational services such as

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sea ice monitoring for ship routing or UV index forecast. The ERS and Envisat data have also stimulated the emergence of new analysis techniques such as SAR Interferometry.

4.1

Land

One of the most striking results of the ERS and Envisat missions for land studies is the development of the interferometry technique using Synthetic Aperture Radar (SAR) instruments. SAR instruments are microwave imaging systems. Besides their cloud penetrating capabilities and day and night operational capabilities, they have also “interferometric” capabilities, i.e., capabilities to measure very accurately the travel path of the transmitted and received radiation. SAR Interferometry (InSAR) exploits the variations of the radiation travel path over the same area observed at two or more acquisition times with slightly different viewing angles. Using InSAR, scientists are able to measure millimetric surface deformations of the terrain, like the ones associated with earthquake movements, volcano deformation, landslides, or terrain subsidence (ESA 2007). Since the launch of ERS-1 in 1991, InSAR has advanced the fields of tectonics and volcanology by allowing scientists to monitor the terrain deformation anywhere in the world at any time. Some major earthquakes, such as in Landers, California, in 1992 and Bam, Iran, in 2003 or the 2009 L’Aquila earthquake in central Italy, were “imaged” by the ERS and Envisat missions, allowing geologists to better understand the fault rupture mechanisms. Using ERS and Envisat data, scientists have been able to monitor the long-term behavior of some volcanoes such as Mt. Etna, providing crucial information for understanding how the volcano’s surface deformed during the rise, storage, and eruption of magma. Changes in surface deformation, such as sinking, bulging, and rising, are indicators of different stages of volcanic activity, which may result in eruptions. Thus, precise monitoring of a volcano’s surface deformation could lead to predictions of eruptions. The InSAR technique was also exploited by merging SAR data acquired by different satellites. For 9 months in 1995/1996, the two ERS-1 and ERS-2 satellites undertook a “tandem” mission, in which they orbited Earth only 24 h apart. The acquired image pairs provide much greater interferometric coherence than is normally possible, allowing scientists to generate detailed digital elevation maps and observe changes over a very short space of time. The same “tandem” approach was followed by the ERS-2 and Envisat satellites, adding to the ever growing set of SAR interferometric data. Because SAR instruments are able “to see” through clouds or at night time, their ability to map river flooding was quickly recognized and has been gradually exploited by civil protection authorities. Similarly other ERS and Envisat instruments such as the ATSR infrared radiometers have been able to provide relevant forest fires statistics. Using data of the MERIS spectrometer instrument on board the Envisat, the most detailed Earth global land cover map was created. This global map, which is ten

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times sharper than any previous global satellite map, was derived by an automatic and regionally adjusted classification of the MERIS data global composites using land cover classes defined according to the UN Land Cover Classification System (LCCS). The map and its various composites support the international community in modeling climate change impacts, studying ecosystems, and plotting worldwide land-use trends.

4.2

Ocean

Through the availability of ERS and Envisat satellite data, scientists have gained an understanding of the ocean and its interaction with the entire Earth system that they would not have otherwise been able to do. Thanks to their 20 year’s extent, the time series of the ERS and Envisat missions’ data allow scientists to investigate the effects of climate change, in particular on the oceans. Those long-term data measurements allow removing the yearly variability of most of the geophysical parameters, providing results of fundamental significance in the context of climate change. Radar altimeters on board satellites play an important role in those long-term measurements. They work by sending thousands of separate radar pulses down to the Earth per second and then recording how long their echoes take to bounce back to the satellite platform. The sensor times its pulses’ journey down to under a nanosecond to calculate the distance to the planet below to a maximum accuracy of 2 cm. The consecutive availability of altimeters on board the ERS-1, ERS-2, and then Envisat allowed establishing that the global mean sea level raised by about 2–3 mm per year since the early 1990s, with important regional variations. Sea-surface temperature (SST) is one of the most stable of several geographical variables which, when determined globally, helps diagnose the state of the Earth’s climate system. Tracking SST over a long period is a reliable way researchers know of measuring the precise rate at which global temperatures are increasing and improves the accuracy of our climate change models and weather forecasts. There is evidence from measurements made from ERS and Envisat ATSR radiometer instruments that there is a distinctive upward trend in global sea-surface temperatures. The ATSR instruments produced data of unrivaled accuracy on account of its unique dual view of the Earth’s surface, whereby each part of the surface is viewed twice, through two different atmospheric paths. This not only enables scientists to correct for the effects of dust and haze, which degrade measurements of surface temperature from space, but also enables scientists to derive new measurements of the actual dust and haze, which are needed by climate scientists. One of the main assets of the Envisat mission is its multisensor capability, which allows observing geophysical phenomena from various “viewpoints.” A good example is the observation of cyclones: the data returned by Envisat includes cloud structure and height at the top of the cyclone, wind and wave fields at the bottom, sea-surface temperature, and even sea height anomalies, indicative of upper ocean thermal conditions that influence its intensity.

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Fig. 4 This Envisat’s ASAR image acquired on 17 November 2002 shows a double-headed oil spill originating from the stricken Prestige tanker, lying 100 km off the Spanish coast

The ERS and Envisat SAR data did also stimulate the development of maritime applications. They include monitoring of illegal fisheries or monitoring of oil slicks (Fig. 4) which can be natural or the results of human activities. The level of shipping and offshore activities occurring in and around icebound regions is growing steadily and with it the demand for reliable sea ice information. Traditionally, ice-monitoring services were based on data from aircraft, ships, and land stations. But the area coverage available from such sources is always limited and often impeded further by bad weather. Satellite data has begun to fill this performance gap, enabling continuous wide-area ice surveillance. SAR instruments of the type flown on ERS and Envisat were able to pierce through clouds and darkness and therefore are particularly adapted for generating high-quality images of sea ice. Ice classification maps generated from radar imagery are now being supplied to users at sea. The maritime operational applications will be continued with the Sentinel-1 data.

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165

Cryosphere

The cryosphere is both influenced by and has a major influence on climate. Because any increase in the melt rate of ice sheets and glaciers has the potential to greatly increase sea level, researchers are looking to the cryosphere to get a better idea of the likely scale of the impact of climate change. In addition, the melting of sea ice will increase the amount of solar radiation that will be absorbed by icefree polar oceans rather than reflected by ice-covered oceans, increasing the ocean temperature. The remoteness, darkness, and cloudiness of Earth’s polar regions make them difficult to study. Being microwave active instruments, the radar on board the ERS and Envisat missions allowed seeing through clouds and darkness. Since about 30 years, satellites have been observing the Arctic and have witnessed reductions in the minimum Arctic sea ice extent – the lowest amount of ice recorded in the area annually – at the end of summer from around 8 million km2 in the early 1980s to the historic minimum of 3.5 million km2 in 2012, changes widely viewed as a consequence of greenhouse warming. ERS and Envisat radar instruments, i.e., the imaging radar (SAR) and the radar altimeter instruments, witnessed this sharp decline, providing detailed measurements, respectively, on sea ice areas and sea ice thickness. The CryoSat mission adds accurate sea ice thickness measurements to complement the sea ice extent measurements. In addition to mapping sea ice, scientists have used repeat-pass SAR image data to map the flow velocities of glaciers. Using SAR data collected by ERS-1 and ERS-2 during their tandem mission in 1995 and Envisat and Canada’s Radarsat1 in 2005, scientists discovered that the Greenland glaciers are melting at a pace twice as fast as previously thought. Such a rapid pace of melting was not considered in previous simulations of climate change, therefore showing the important role of Earth Observation in advancing our knowledge of climate change and improving climate models. Similar phenomena also take place in Antarctica, with some large glaciers such as the Pine Island Glacier thinning at a constantly accelerating rate as suggested by ERS and Envisat altimetry data. In Antarctica, the stability of the glaciers is also related to their floating terminal platform, the ice shelves. The breakup of large ice shelves (e.g., Larsen ice shelf, Wilkins ice shelf) has been observed by Envisat SAR and is likely a consequence of both sea and air temperature increase around the Antarctic peninsula and West Antarctica.

4.4

Atmosphere

ERS-2 and Envisat were equipped with several atmospheric chemistry instruments, which can look vertically or sideways to map the atmosphere in three dimensions, producing high-resolution horizontal and vertical cross sections of trace chemicals stretching from ground level to 100 km in the air, all across a variety of scales.

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Those instruments could detect holes in the thinning ozone layer, plumes of aerosols and pollutants hanging over major cities or burning forests, and exhaust trails left in the atmosphere by commercial airliners. ERS-2 and Envisat satellites have been maintaining a regular census of global stratospheric ozone levels from 1995 to 2012, mapping yearly Antarctic ozone holes as they appear. The size and precise time of the ozone hole is dependent on the year-to-year variability in temperature and atmospheric dynamics, as established by satellite measurements. Envisat results benefited from improved sensor capabilities. As an example, the high spatial resolution delivered by the Envisat atmospheric instruments means precise maps of atmospheric trace gases, even resolving individual city sources such than the high-resolution global atmospheric map of nitrogen dioxide (NO2), an indicator of air pollution (Fig. 5). By making a link with the measurements started with ERS-2, the scientists could note a significant increase in the NO2 emissions above Eastern China, a tangible sign of the fast economical growth of China. Based on several years of Envisat observations, scientists have produced global distribution maps of the most important greenhouse gases – carbon dioxide (CO2) and methane (CH4) – that contribute to global warming. The importance of cutting emissions from these “anthropogenic,” or man-made gases has been highlighted with European Union leaders endorsing binding targets to cut greenhouse gases in the midterm future. Careful monitoring is essential to ensuring these targets are met,

Fig. 5 Nitrogen dioxide (NO2) concentration map over Europe derived from several years of SCIAMACHY instrument data on board the Envisat satellite (Courtesy Institute of Environmental Physics, Univ. of Heidelberg)

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and space-based instruments are new means contributing to this. The SCIAMACHY instrument on board the Envisat satellite was the first space sensor capable of measuring the most important greenhouse gases with high sensitivity down to the Earth’s surface because it observes the spectrum of sunlight shining through the atmosphere in “nadir”-looking operations. Envisat atmospheric chemistry data are useful for helping build scenarios of greenhouse gas emissions, such as methane – the second most important greenhouse gas after carbon dioxide. Increased methane concentrations induced mainly by human activities were observed. By comparing model results with satellite observations, the model is continually adjusted until it is able to reproduce the satellite observations as closely as possible. Based on this, scientists continually improve models and their knowledge of nature.

4.5

ESA Programs for Data Exploitation

Earth Observation is an inherently multipurpose tool. This means there is no typical Earth Observation user: it might be anyone who requires detailed characterization of any given segment of our planet, across a wide variety of scales from a single city block to a region, or continent, right up to coverage of the entire globe. Earth Observation is already employed by many thousands of users worldwide. However, ESA works to further increase Earth Observation take-up by encouraging development of new science and new applications and services centered on user needs. New applications usually emerge from scientific research. ESA supports scientific research either by providing easy access to high-quality data (see Sect. 5), by organizing dedicated workshops and symposia, by training users, or by taking a proactive role in the formulation of new mission concepts and by providing support to science. Converting basic research and development into an operational service requires the fostering of partnership between research institutions, service companies, and user organizations. ESA’s Data User Element (DUE) program addresses institutional users tasked with collecting specific geographic or environmental data. The Data User Element aims to raise such institutions’ awareness of the applicability of Earth Observation to their day-to-day operations and develop demonstration products tailored to increase their effectiveness. The intention is then to turn these products into sustainable services provided by public or private entities. More information on ESA’s Data User Element (DUE) program can be found at http:// dup.esrin.esa.it. Complementing Data User Element objectives is ESA’s Earth Observation Value Adding program. This provides a supportive framework within which to organize end-to-end service chains capable of leveraging scientific EO data into commercial tools supplied by self-supporting businesses. More information on ESA EO Value Adding program can be found at http://www.eomd.esa.int.

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User Access to ESA Data

ESA endeavors to maximize the beneficial use of Earth Observation data. It does this by fostering the use of this valuable information by as many people as possible, in as many ways as possible. For users, and therefore for ESA, easy access to EO data is of paramount importance. However, the challenges for easy EO data access are many: 1. The ESA EO data policy shall be beneficial to various categories of users, ranging from global change scientists to operational services, and shall have the objective to stimulate a balanced development of Earth science, public services, and valueadding companies. ESA has always pursued an approach of low cost fees for its satellite data, trying to provide free of charge the maximum amount of data. This approach will continue and even be reinforced in the future, by further increasing the amount of data available on the Internet and by reducing the complexity of EO missions. 2. The volume of data transmitted to the ground by ESA EO satellites is particularly high: the Envisat satellite transmitted about 270 GB of data every day; the future Sentinel-1 satellite will transmit about 900 GB of data every day. Once acquired, the data shall be transformed (i.e., processed) into products in which the information is related either to an engineering calibrated parameter (so-called Level 1 products, e.g., a SAR image) or to a geophysical parameter (so-called Level 2 products, e.g., sea surface temperature). Despite the high volume of data, the processing into Level 1 and Level 2 products shall be as fast as possible to serve increasingly demanding operational services. Finally, the data products shall be almost immediately available with users either through a broad use of Internet or through dedicated communication links. 3. The quality of EO data products shall be high so that user can effectively rely on the delivered information content. This means that the processing algorithms (i.e., the transformation of raw data into products) as well as the product calibration and validation shall be given strong attention. ESA has constantly given such attention for their EO missions, investing large efforts in algorithms development, particularly for innovative instruments such as the ones flying on board the Earth Explorer missions. Of equal importance for the credibility of EO data are the validation activities aiming at comparing the geophysical information content of EO data products with similar measurements collected through, e.g., airborne campaigns or ocean buoys setup. 4. Finally, EO data handling tools shall be offered to users. Besides the general assistance given through a centralized ESA EO user service ([email protected]), the ESA user tools include: – Online data information (http://earth.esa.int) providing EO missions news, data product description, processing algorithms documentation, workshop proceedings, etc.

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– Data collection visualization through online catalogues, including request for product generation when the product does not yet exist (e.g., for future data acquisition) or direct download when the product is already available in online archives – EO data software tools, aiming to facilitate the utilization of data products by provision of, e.g., viewing capabilities, innovative processing algorithms, format conversion, etc.

5.1

How to Access the EO Data at ESA

Internet is the main way to access to the EO satellite services and products at ESA. Besides the general description on data access described below, further assistance can be provided by the ESA EO help desk ([email protected]). 1. For most of the ESA EO data, open and free-of-charge access is granted after a simple user registration. The data products are those that are systematically acquired, generated, and available online. This includes all altimetry, sea surface temperature, atmospheric chemistry, and future Earth Explorer data but also large collections of optical (MERIS, Landsat) and SAR datasets. User registration is done at http://eopi.esa.int. The detailed list of open and free-of-charge data products is available on this Web site. For users who are only interested in basic imagery (i.e., false color jpg images), ESA provides access to large galleries of free-of-charge Earth images (http://earth.esa.int/satelliteimages). 2. Some ESA EO data and services cannot be provided free of charge or openly either because of restrictions in the distribution rights granted to ESA (e.g., some ESA Third Party Missions) or because the data/service is restrained by technical capacities and therefore is on demand (i.e., not systematically provided). The restrained dataset essentially includes on-demand SAR data acquisition and production. In this case, users shall describe the intended use of the data within a project proposal. The project proposals are collected at http://eopi.esa.int. ESA analyzes the project proposal to review its scientific objectives, to assess its feasibility, and to establish project quotas for the requested products and services (e.g., instrument tasking). The products and services are provided free of charge. Products are provided on the Internet.

6

Conclusion

During the last three decades, Earth Observation satellites have gradually taken on a fundamental role with respect to understanding and managing our planet.

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Contributing to this trend, the Earth Observation program of the European Space Agency addresses a growing number of scientific issues and operational services, thanks to the continuous development of new satellites and sensors and to a considerable effort in stimulating the use of EO data. Partnerships between satellite operators and strong relations with user organizations are essential at ESA for further improving information retrieval from EO satellite data. The coming decade will see an increasing number of orbiting EO satellites, not only in Europe. This is a natural consequence of the growing user demands and expectations, but also of the gradual decrease of the costs for satellite manufacturing. The main challenge in Earth Observation will therefore be to maximize the synergies between existing satellites, both with respect to combining their respective observations and optimizing their operation concepts.

References ESA (2006) The changing Earth – new scientific challenges for ESA’s living planet programme, ESA SP-1304 ESA (2007) InSAR principles – guidelines for sar interferometry processing and interpretation, ESA TM-19

Satellite-to-Satellite Tracking (Low-Low/High-Low SST) Wolfgang Keller

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Scientific Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Observation Models and Data Processing Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Satellite-to-Satellite Tracking in the High–Low Mode . . . . . . . . . . . . . . . . . . . . . . . 5.2 Satellite-to-Satellite Tracking in the Low–Low Mode . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Integral Equations Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Regional Gravity Field Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Missions and Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 CHAMP Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 GRACE Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172 172 175 175 177 181 183 183 194 200 204 205 206 206 207 208 209

Abstract

This contribution reviews the mathematical ideas behind the most frequently used techniques for the processing of satellite-to-satellite tracking data. Its emphasis is on the model part rather than on all necessary technicalities in data preprocessing and numerical implementation. The main outcomes of these dataprocessing strategies, when applied to data of the satellite missions CHAMP and GRACE, are reviewed.

W. Keller () Geodätisches Institut, Universität Stuttgart, Stuttgart, Germany e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_56

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Introduction

The dedicated gravity field missions CHAMP and GRACE have attracted and are still attracting the attention of many researchers worldwide. An impressive number of algorithms and approaches have been developed to extract information about the gravitational field of the Earth from the original satellite-to-satellite tracking (SST) signal. This contribution aims at a mathematical description of the basic ideas behind the most frequently used and most developed approaches for the processing of SST data. It will not cover the important issue of data preprocessing, though this preprocessing is vital for the extraction of geophysical meaningful results from the SST data.

2

Scientific Relevance

Geophysical processes close to the Earth’s surface, which are connected with mass transports, such as the hydrological cycle, imprint their signature on the time variability of the gravitational field of the Earth. Though gravity and gravity changes can be precisely measured at the Earth’s surface, this data has a very coarse spatial and temporal resolution. The way out of this sparse data coverage is to use artificial satellites as proof masses in the Earth’s gravitational field: The deviation of the orbits of these satellites from their simple Keplerian orbits is due to the deviation of the Earth’s gravitational field from a rotational symmetric field. The orbital heights of those satellites have to be big enough to guarantee that the atmospheric friction of the satellite does not exceed the anomalous gravitational acceleration on the satellite. On the other hand, the intensity of the gravitational signal decreases with increasing distance from the mass center of the Earth. And this decay is the faster the smaller the scales of the gravitational anomalies are. This means a recovery of the gravitational field from orbit observations of artificial satellites smoothes out smaller details of the gravitational field. This smoothing cannot be counteracted by a lower orbit because the atmospheric friction would obscure the gravitational signal. This smoothing only can be counteracted by differential measurements, meaning that instead of the orbits of the satellites themselves the orbit differences between two or more satellites are observed. This concept of relative orbit observation and its implications to the recovery of geophysical phenomena is explained in more detail in Rummel (2003). This satellite-to-satellite tracking (SST) principle can be put into mathematical terms in the following way: Assume that there are n satellites which can “see” each other in a certain way. The orbits of these satellites will be denoted by .xi .t/; xP i .t//; i D 1; : : : ; n. The “things” that the satellites 2; : : : ; n “see” of satellite 1 can be modeled as a vectorial function F of the orbits of all involved satellites:

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

s.t/ WD F.x1 .t/; xP 1 .t/; : : : ; xn .t/; xP n .t// C .t/:

173

(1)

Here  comprises all unavoidable observation errors as well as the errors from imperfect data preprocessing. The signal s is the primary SST signal and all the desired information about the gravitational field and its variability in space and time has to be extracted from this signal. The gravitational field V is hidden in the orbits of the satellites. Hence a more precise notation of the orbits of the involved satellites is .xi .t; V /; xP i .t; V //; i D 1; : : : ; n, which changes the primary SST model (1) to s.t/ WD F.x1 .t; V /; xP 1 .t; V /; : : : ; xn .t; V /; xP n .t; V // C .t/:

(2)

The recovery of the gravitational field from SST data can be modeled as the following minimization problem: Z V D argmin

T



ks.t/  F.x1 .t; V /; xP 1 .t; V /; : : : ; xn .t; V /; xP n .t; V //k2 dt 0



(3)

j V 2 Harm./ : The minimization is carried out over all functions V , which are harmonic in  and regular at infinity (Harm./). In most cases  will be the exterior of a sphere of radius R. Already at this stage two different modes of SST can be distinguished: the high– low SST (hl-SST) and the low–low SST (ll-SST). In the hl-SST mode the satellites 2; : : : ; n have such a high orbital altitude that the uncertainties in the gravitational field do not measurably influence their computed orbits. This means the orbits .xi .t; V0 /; xP i .t; V0 //; i D 2; : : : ; n can be computed using a known reference potential V0 and do not longer depend on the unknown gravitational potential V . The minimization problem simplifies to Z T  2 O V D argmin ks.t/  F.x1 .t; V /; xP 1 .t; V //k dt j V 2 Harm./ (4) 0

with O 1 .t; V /; xP 1 .t; V // W F.x D F.x1 .t; V /; xP 1 .t; V /; x2 .t; V0 /; xP 2 .t; V0 /; : : : ; xn .t; V0 /; xP n .t; V0 //:

(5)

In the ll-SST mode two satellites in the same low orbit are chasing each other and are observing their relative positions and velocities. The corresponding minimization problem is

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Z V  D argmin

T

ks.t/  F.x1 .t; V /; xP 1 .t; V /; x2 .t; V /; xP 2 .t; V //k2 dt 0



j V 2 Harm./ :

(6)

Both the hl-SST (4) and the ll-SST (6) minimization problems are infinitedimensional problems and as such not suitable for numerical implementation. In order to discretize these infinite-dimensional problems, a parametric model for the unknown gravitational potential has to be introduced: X

VQ .x/ D

cz ˆz .x;  z /:

(7)

z2C Z r

In this notation ˆz 2 Harm./ are harmonic basis functions, which besides on the location x can also depend on additional parameters  z . The weights cz in the linear combination (7) can be complex numbers and the multi-index z ranges inside a certain subset C of the r-dimensional integers. As a second discretization step, the SST signal s.t/ has to be sampled equidistantly: si WD s.ih/;

h > 0:

(8)

After that, the infinite-dimensional minimization problems (4) and (6) can be approximated by their finite-dimensional counterparts:

f.cz ;  z / j z 2 C g D argmin

N X

       ksi  FO x1 ih; VQ cz ;  z ; xP 1 ih; VQ cz ;  z k2

i D1

(9) in the hl-SST mode and

f.cz ;  z / j z 2 C g D argmin

N X i D1

          ksi  F x1 ih; VQ cz ;  z ; xP 1 ih; VQ cz ;  z ; x2 ih; VQ cz ;  z ;    xP 2 ih; VQ cz ;  z k2

(10)

in the ll-SST mode. Both (9) and (10) are finite-dimensional nonlinear leastsquares problems and can be solved by certain variants of the Levenberg–Marquardt algorithm (Levenberg 1944; Marquardt 1963):

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.cz ;  z /.i C1/ D .cz ;  z /.i /

!1 @F > @F > @F .i / C˛ I  @ .cz ;  z / @ .cz ;  z / @ .cz ;  z /    s  F .cz ;  z /.i / :

(11)

The different algorithms for SST data processing differ in the way how they in the Jacobian or how they combine the primary compute the entries @.c@F z ; z / observations si to secondary observations i , which then are linearly related to the unknown parameters. These questions will be discussed in detail in Sect. 5.

3

Conventions

3.1

Reference Systems

In geodynamics a well-defined set of reference systems is in use. For the precise definition of these reference systems and the transformation methods between the individual reference systems, the IERS conventions (Petit and Luzum 2010) can be consulted. Since for astronomical standards the time-span of SST observations is rather small, three simplified reference systems can be used: 1. A space-fixed system 2. An Earth-fixed system 3. An orbital system The space-fixed system has its origin in the mass center of the Earth and its x3 -axis points into the direction of the mean rotation axis of the Earth at the initial epoch t0 of then SST observations. Its x1 -axis points to the intersection of the mean equatorial plane with the ecliptic at this epoch t0 and the x2 -axis completes the former two axes to an orthogonal right-handed Cartesian system. Since the origin of the system is not in un-accelerated motion, this system is not a proper inertial system. The acceleration is due to the attracting forces of Sun, Moon, and planets. If these attracting forces are considered in the form of tidal forces in the data preprocessing, the space-fixed system becomes an inertial system. The differences between the simplified space-fixed system used here and the space-fixed system defined in the IERS convention have to be taken into account by precession, nutation, and polar-motion corrections. These corrections can be made part of the data preprocessing and will not be discussed here. The Earth-fixed system has the same x3 -axis but the x1 -axis lies on the meridian of Greenwich. The x2 -axis completes the former two axes to an orthogonal

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right-handed Cartesian system. The angle ‚ between the x1 -axis of the space-fixed and the x1 -axis of the Earth-fixed system is called Greenwich sidereal time. Hence, the coordinates of a point change from the space-fixed to the Earth-fixed system according to 0

1 cos  sin  0 R3 . / WD @ sin  cos  0A : 0 0 1

xEf D R3 .‚/xsf ;

(12)

The orbital system is related to a fictitious satellite with an exactly circular orbit in the gravitational field of a flattened Earth. Its x3 -axis is perpendicular to the orbital plane and its x1 -axis points to the fictitious satellite. The x2 -axis completes again to a right-handed orthogonal Cartesian system. According to Kaula (2000), for such a satellite, the angle  between the x1 -axis of the space-fixed system and the nodal line, i.e., the intersection between equatorial and orbital plane, changes with time as 3n  D 0  J2 2



R a

r

2 cos i  t;

nD

GM a3

with GM being the product of gravitational constant and mass of the Earth; J2 being the dynamic form factor (cf. Petit and Luzum 2010); i being the inclination of the orbit, i.e., the angle between the equatorial and the orbital plane; and a being the radius of the orbital circle (Fig. 1). The argument of latitude u, i.e., the angle between the nodal line and the position of the satellite, is given by

Fig. 1 Relations between reference systems

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)



3n J2 u D u0 C n C 4

R a

177

!

2 2

.3 cos i  1/  t:

(13)

This means the transformation from the space-fixed to the orbital system is given by 1 1 0 0 R1 .˛/ WD @0 cos ˛ sin ˛ A : 0  sin ˛ cos ˛ 0

xorb D R3 .u/R1 .i /R3 ./xsf ;

3.2

(14)

Basis Functions

There are two types of basis functions for the representation of the gravitational potential, which are used in the SST community: spherical harmonics and radial basis functions. Nevertheless, they are used in different normalizations. Here, the standard used in Wolfram MathWorld (Weisstein) will be applied. The surface spherical harmonics are defined as

Yl;m .#; / D

8q < .2lC1/ .l m/Š P m .cos #/e {m

;m  0

:.1/m Y l;m .#; /

;m < 0

4

.lC m/Š

l

;

(15)

where Y l;m is the conjugate complex of Yl;m , the Plm are the Legendre functions, and #;  are spherical coordinates related to the Earth-fixed system. Assume now that the original coordinate system is rotated by the three Eulerian angles ˛; ˇ; and  : 1 cos ˇ 0  sin ˇ R2 .ˇ/ D @ 0 1 0 A sin ˇ 0 cos ˇ 0

R.˛; ˇ;  / WD R3 . /R2 .ˇ/R3 .˛/;

and that the spherical coordinates in the rotated system are denoted by # 0 ; 0 , then a spherical harmonics in the non-rotated coordinates can be expressed as a linear combination of spherical harmonics of the same degree in the rotated coordinates

Yl;m .#; / D

l X

l Dl;m;k .˛; ˇ;  /Yl;k .# 0 ; 0 /;

(16)

kDl

with l l .˛; ˇ;  / D e {m˛ dmk .ˇ/e {k Dl;m;k

(17)

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being the so-called Wigner functions according to Kostelec and Rockmore (2008). The representation of the gravitational potential as spherical harmonics expansion in the Earth-fixed system is given by l 1   GM X R lC1 X V .r; #; / D Kl;m Yl;m .#; /; R r lD0

Kl;m 2 Z:

(18)

mDl

This means, in the case of spherical harmonics in an Earth-fixed system as basis functions, the general notation specializes to ˆl;m .x/ D

GM R



R r

lC1

˚

Yl;m .#; /; .l; m/ 2 C D .i; j / 2 Z2 j i  0; jj j  i : (19)

Besides on the index vector z D .l; m/ and on the position vector x, the basis functions do not depend on additional parameters z . Since the transition from the Earth-fixed system to the orbital system is accomplished by the rotation    R   ‚  ; i; u C ; 2 2

(20)

the gravitational potential at the position of the fictitious satellite is given by

V .r; u; ƒ/ D

l 1   l GM X R lC1 X X k l Kl;m { km dm;k .i /P l .0/e {.kuCmƒ/ ; R r lD0 mDl kDl (21)

with ƒD ‚ and m P l .x/

D

8q < .2lC1/ .l m/Š P m .x/

;m  0

:.1/m P m .x/ l

;m < 0

4

.lC m/Š

l

:

(22)

Hence, in the orbital system, the general notation of a basis function is specialized to 2 3  lC1 1 X R GM k l ˆm;k .u; ƒ/ D 4 { km dm;k .i /P l .0/5 e {.kuCmƒ/ : R r lDmaxfjkj;jmjg

(23)

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This means in the orbital systems the basis functions modify to imaginary exponentials defined on the torus Œ0; 2/  Œ0; 2/. The domain of definition gave this representation its name: torus approach (Sneeuw 2000). A satellite position with respect to the Earth is mapped onto the position ƒ; u on the torus. Since the Earth has the topological genus 0 and the torus has the topological genus 1, there is no one-to-one mapping between the satellite positions in an Earth-fixed system and the positions on a torus. Two different points on the torus can correspond to the same point related to the Earth: once the point is reached on an ascending and once on a descending orbital arc. A second kind of basis functions is the so-called radial basis functions. A radial basis function (RBF) on the sphere is a function which depends only upon the distance of its argument from the north pole of the sphere, i.e.,   Q e> ˆ.x/ D ˆ 3  ;

x ; D kxk

0 1 0 @ e3 D 0 A : 1

(24)

Radial basis functions are used to approximate functions defined on the sphere as linear combinations of rotated versions of ˆ. A RBF, rotated to the position , is a function which only depends on the distance of its argument from that position:   Q >  : ˆ.x; / D ˆ

(25)

Since 1  >   1 holds, any square integrable rotated RBF must have a series expansion in Legendre polynomials: ˆ.x; / D

1 X

  n Pn >  :

(26)

nD0

This means in an Earth-fixed system and for rotated RBFs, the general notation of a basis function specializes to ˆl .x;  l / D

1 X

  n Pn > l  ;

(27)

nD0

with  l D fl ; fn gg :

(28)

So far, RBFs have been defined on the surface of the unit sphere only. Their harmonic continuation to the exterior of the mean Earth sphere is given by ˆl .x;  l / D

1 X nD0

n

 nC1   R Pn > l  : r

(29)

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Since the Legendre polynomials Pn are sums of products of surface spherical harmonics n X .2n C 1/  >  Pn   D Y n;m ./Yn;m ./; 4 mDn

(30)

the rotated RBF ˆ.x;  l / can be expressed as ˆ.x; l / D

1 X nD0

n X 4 n Y n;m ./Yn;m .l /: 2n C 1 mDn

(31)

In this representation the position parameter l and the argument  are separated. Since the position parameter l is the following rotation of the vector e3 1 sin #l cos l l D @ sin #l sin l A ; cos #l 0

l D R3 .l /R2 .#l /e3 ;

(32)

the representation (31) is equivalent to 1 X

n X 4 n ˆ.x; l / D 2n C 1 mDn nD0

n X

! n dm;k .#l /e {kl Yn;k .e3 /

Y n;m ./:

(33)

kDn

A comparison of (31) with (18) allows the conversion of a RBF representation into the corresponding surface spherical harmonics representation. So far, both RBF representations refer to an Earth-fixed system. The representation of a RBF in the orbital system is ˆ.u; ƒ;  l /

!  nC1 X n n X R 4 km l {.kuCmƒ/ k n { dm;k .i /e P l .0/ Yn;m .l / D 2n C 1 r mDn kDn nD0 2 3  nC1 1 1 X X X R 4 k l 4 n D { km dm;k .i /P l .0/Yn;m .l /5 2n C 1 r mD1 1 X

kD1

 e {.kuCmƒ/ :

nDmaxfjkj;jmjg

(34)

This means, as in the case of spherical harmonics also, the RBFs in the orbital system are imaginary exponentials on the torus. A basis function on the torus is of particular importance if the underlying orbit is a so-called “repeat orbit.” For a repeat orbit holds

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

181

ˇ uP D ; P ˛ ƒ

(35)

with ˛; ˇ relative prime. Then

m;k

WD ku C mƒ D

uP .kˇ C m˛/ t D P m;k t ˇ „ ƒ‚ …

(36)

P m;k

holds. Due to this relation between the time t and the location u; ƒ on the torus the basis function, evaluated along a repeat orbit, and changes into a periodic time function: 2 ˆm;k .t/ D 4

GM R

1 X lDmaxfjkj;jmjg

3  lC1 R k P l { km dm;k .i /P l .0/5 e { m;k t ; r

(37)

and ˆm;k .t;  l / 1 X

D

1 X

mD1 kD1

e{

2 4

X nDmaxfjkj;jmjg

4 n 2n C 1



R r

nC1

3 k l { km dm;k .i /P l .0/Yn;m .l /5

P m;k t

(38)

respectively. The period of these functions is given by T D ˛ nodal days.

4

Celestial Mechanics

In an inertial system, the equation of motion of a unit point mass is governed by the Newtonian equations xR .t/ D rV .x.t/; t/ C F.x.t/; xP .t/; t/;

(39)

where x.t/ is the Cartesian position of the unit point mass, representing the satellite, V is the potential of the gravitational force acting on it, and F is the nonconservative

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W. Keller

force exerted, for example, by atmospheric friction, on the satellite. In a rotating system inertial forces have to be added and the equation of motion changes from (39) to       xR0 .t/ D rV 0 x0 .t/; t CF0 x0 .t/; xP 0 .t/; t !  !  x0 .t/  2!  xP 0 .t/ : ƒ‚ … „ ƒ‚ … „

(40)

Coriolisforce

centrifugal force

Here ! is the rotation axis of the system and all the quantities denoted by a prime refer to that rotating system. Since we can assume ! P D 0, the inclusion of the Eulerian force is not necessary. Here, we distinguish two rotating systems: the Earth-fixed and the orbital system. For the Earth-fixed system, ! D 7:292115  105 e3 holds, and V 0 does not longer explicitly depend on t. For the orbital system r !D

1 sin i sin  GM @  sin i cos A a3 cos i 0

holds, but in this case the gravitational potential V 0 still explicitly depends on t. The equations of motion can be solved as an initial value problem or as a boundary value problem. In the case of an initial value problem, position x0 .0/ and velocity xP 0 .0/ at the beginning of the orbital arc have to be known, while in the case of the boundary value problem, the positions x0 .0/ at the beginning and x0 .T / at the end of the orbital arc have to be given. In the case of a boundary value problem, it is convenient to transform the equations of motion into an equivalent integral equation (cmp. Schneider 1968): Z xQ ./ D xQ .0/.1  / C xQ .1/  T

1

2

    K ;  0 f  0 d  0 ;

(41)

0

where  WD

t ; T

xQ ./ WD x0 .T /

and the integral kernel K is given by ( 0

K.;  / WD

 0 .1  / 0

.1   /

; 0   ;  0 > :

:

(42)

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

183

The function f comprises all forces acting on the satellite:             f  0 WD rV 0 x0  0 T ;  0 T C F0 x0  0 T ; xP 0  0 T ;  0 T       !  !  x0  0 T  2!  xP 0  0 T :

5

Observation Models and Data Processing Strategies

5.1

Satellite-to-Satellite Tracking in the High–Low Mode

(43)

High–low modus is the name of scenario where a low-flying satellite is tracked by several high-flying satellites. Since the orbits of the high-flying satellites can be computed sufficiently precise from existing gravity field models, their orbits do not contribute to an improvement of the knowledge about the gravitational field. Hence, the positions of the high-flying satellites are used as known reference positions and the position of the low-flying satellite is determined in reference to these known positions. In general GPS satellites are used as high-flying satellites and the low-flying satellite tracks its position and velocity in an Earth-fixed system, in relation to the known positions and velocities of the GPS satellites, by an onboard GPS receiver. If this is put in relation to the general SST model (4) and (6), we obtain  0 0   0  0  0  0  x .V ; t/ O s.t/ D F x V ; t ; xP V ; t C .t/ WD 0 0 C .t/ xP .V ; t/

(44)

and Z

T



V D argmin 0

  0  0  0  0   2 O s.t/  F x V ; t ; xP V ; t  dt:

(45)

If this problem is discretized according to (7) and (8), it results in a nonlinear leastsquares problem

f.cz ;  z / j z 2 C g D argmin

N  X       2 si  FO x0 V 0 ; ti ; xP 0 V 0 ; ti  : i D1

In order to solve this least-squares problem, the entries of the Jacobian       @FO @FO @x0 VQ cz ;  z @FO @Px0 VQ cz ;  z     D 0 C 0 @ .cz ;  z / @x @Px @ cz ;  z @ cz ;  z

(46)

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have to be known. As a standard approach, they are computed by solving the socalled variational equations (Ballani 1988). Since this technique is numerically costly, some less costly techniques have been developed, which convert the position and velocity information into synthetic observations, which then are directly related to the unknown potential. Those techniques are named: • Acceleration approach • Energy-balance approach–torus approach A technique, which is strongly related to the variational equation approach, is the so-called integral equation approach.

Energy-Balance Approach The energy-balance approach is one of the oldest ideas in dynamical satellite geodesy. It was first discussed in O’Keefe (1957), Reigber (1969), and Bjerhammar (1976). The method acquired practical importance only with availability of timely and spatially dense data, as, for instance, delivered by the satellite mission CHAMP. There are numerous publications about the application of the energy-balance approach to CHAMP data. Without a ranking the following contributions will be mentioned: Visser et al. (2003), Gerlach et al. (2003), and Badura et al. (2006). In an inertial frame the Lagrangian of the low-flying satellite is given as LDT V D

1 > xP xP  V: 2

(47)

In the Earth-fixed system, the Lagrangian changes to LD

 1 >    >  1  0 > 0 xP C 2 xP 0 !  x0 xP !  x0 C !  x0  V: 2 2

(48)

A change from the Lagrangian to the Hamiltonian H D p> xP 0  L;

pD

@L @Px0

(49)

yields H D

>   1  0 > 0 1  xP  !  x0 xP !  x0 C V: 2 2

(50)

Since the Hamiltonian is constant and x0 ; xP 0 are known from orbit integration, this leads to an observation equation for the unknown parameters fcz ;  z g in the basis function representation VQ of V . So far all nonconservative forces acting on the satellite have been ignored. Taking them into account adds another term to the energy-balance equation: Z t >   1  0 > 0 1  H D xP  !  x0 xP !  x0 C V C f> xP 0 dt 0 : (51) 2 2 0

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

185

This last term is the line integral of the nonconservative force f along the orbital path. If the nonconservative force is measured or modeled, this line integral can be evaluated for each observation epoch t D ih. Inserting the base function representation VQ and the known positions x0i WD 0 x .ih/ and velocities xP 0i WD xP 0 .ih/ yields the following observation equation: H

X z2C Z

  >   1  2 1  cz ˆz x0i ;  z D xP 0i   !  x0i C !  x0i 2 2 r

Z

ih

f> xP 0 dt 0 :

0

(52) Hence the observations si in (52) are the sum of kinetic energy, potential energy, and dissipative work

si D

     1 xP 0 2  1 !  x0 > !  x0 C i i i 2 2

Z

ih

f> xP 0 dt 0

(53)

0

and the least-squares problem N X i D0

si  H C

X

!2  0  cz ˆz xi ;  z ! min

(54)

z2C Z r

is partly linear and partly nonlinear in the unknown parameters H; cz ;  z . In order to refer the least-squares problem (54) to the generic setting (9), the following definitions have to be made: X        FO x0 ih; VQ ; xP 0 ih; VQ D H  cz ˆz x0 ;  z : z2C Z r

For the solution of this least-squares problem, the entries of the Jacobian 8 ˆ ˆ

xl .t1 /x0 .t1 / > B  kx0l .t1 /x0 .t1 /k  1 .t1 / : : : kx0l .t1 /x0 .t1 /k  N .t1 / C B C :: C: Xl D B : B C     @ A x0l .tn /x0 .tn / > x0l .tn /x0 .tn / >  kx0 .t /x0 .t /k  1 .tn / : : :  kx0 .t /x0 .t /k  N .tn / l

n

1

l

n

n

4. Collect all GPS pseudo-ranges residuals s D .s1 .t1 /; : : : ; s1 .tn /; s2 .t1 /; : : : ; s2 .tn /; : : : ; s4 .tn //>

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W. Keller

where the residuals are defined as the difference between observed pseudo-ranges and pseudo-ranges computed with the initial guess p0 . 5. Assemble the design matrix 0

1 X1 B C X D @ ::: A : X4 6. Compute parameter corrections p D .X> X/1 Xs:

Colombo’s Modification of Variational Equation Approach Since for each unknown parameter pk one differential equation has to be solved, the total numerical effort is considerable. Therefore, it is desirable to find an at least approximative closed solution of (77). Colombo (1984) solved this problem by treating it in the orbital system. The transformation from the Earth-fixed to the orbital system is accomplished by the rotation (20) and results in the following rotation vector with respect to the orbital system: 0 1 r 0 GM ! D @ 0A ; n D : a3 n We get 0 2 1 n k;1 C 2nPk;2 !  .!   k .t//  2!  Pk .t/ D @ n2 k;2  2nPk;1 A : 0 If now, the gravitational potential V 0 is split in its spherical part U D disturbing potential T 0 , we first obtain 0 1 2k;1 r 2 U k D n2 @k;2 A k;3

GM r

and the

and (77) simplifies to 0

1 Rk;1  3n2 k;1  2nPk;2 0 B C   B C D r 2 T 0 x0  k .t/ C @rT : Rk;2 C 2nPk;1  @ A @pk Rk;3 C n2 k;3

(78)

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

193

Since both  and r 2 T are small, their product can be neglected, and the final form of the variational equations in the orbital system is obtained: 0R 1 k;1  3n2 k;1  2nPk;2 @rT 0 B C : Rk;2 C 2nPk;1 @ AD @pk Rk;3 C n2 k;3

(79)

This is an inhomogeneous ordinary differential equation with constant coefficients, which can be solved in a closed form, provided the inhomogeneity is sufficiently “simple.” In order to make this sure, the orbit of the satellite is approximated by a so-called repeat orbit. If a basis function representation of T is chosen according to (37), its evaluation along the repeat orbit results in a periodic disturbing potential: T D

X m;k

cm;k ˆm;k .t/ D

X

cm;k Am;k .a; i /e {

P m;k t

:

(80)

m;k

Consequently, the gradient of T along the repeat orbit is also a periodic vector function:

rT D

1 @   X @a P @ A cm;k @ a1 @u Am;k .a; i /e { m;k t 1 @ m;k a cos i @i

0

0

D

1

@Am;k X B {k @a C P cm;k @ a Am;k A e { m;k t : @A 1 m;k m;k a cos i @i

(81) This means the inhomogeneity is a Fourier series, and since the differential equation is linear, the superposition principle can be applied and each term of the force function can be treated separately. For each term a differential equation with constant coefficients and a periodic inhomogeneity has to be solved. The solution is elementary and consists in a superposition of two periodic solutions: one with the eigenfrequency n of the homogeneous differential equation and one with the frequency P m;k of the excitation term. Hence, the partial derivatives  k are series of periodic functions with the frequencies n and P m;k . These partial derivatives have to be transformed back from the orbital system to the Earth-fixed system according to    @x0  D R k D R   u; i; C ‚   : k @pk 2 2

(82)

This means in the Colombo modification step 2, the variational equation approach has to be replaced by the closed solution of the variational equations in the orbital frame and the back-transformation of the partial derivatives  k into the Earth-fixed system.

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Colombo’s Modification 1. With an initial guess p0 for the unknown parameters, compute a reference orbit x0 .t/. 2. Find the orbital elements a; i; ; M .t0 / of the best-fitting circular repeat orbit. 3. Find the Fourier series representation of rT according to (81). 4. For each Fourier term, solve Eqs. (79) for the partial derivatives  k in the original system. 5. For each GPS satellite l, build the matrix 0   x0l .t1 /x0 .t1 / > B  kx0l .t1 /x0 .t1 /k R 1 .t1 / B Xl D B B  @  0 x .t /x0 .t / >  kxl0 .tn /x0 .tn /k R 1 .tn / l

n

n

1 R N .t1 / C C C: C   0 A xl .tn /x0 .tn / > : : :  kx0 .t /x0 .t /k R N .tn / 

:::  :: :

 x0l .t1 /x0 .t1 / > kx0l .t1 /x0 .t1 /k

l

n

n

6. Collect all GPS pseudo-ranges residuals s D .s1 .t1 /; : : : ; s1 .tn /; s2 .t1 /; : : : ; s2 .tn /; : : : ; s4 .tn //> where the residuals are defined as the difference between observed pseudo-ranges and pseudo-ranges computed with the initial guess p0 . 7. Assemble the design matrix 0

1 X1 B C X D @ ::: A : X4 8. Compute parameter corrections  1 p D X> X Xs:

5.2

Satellite-to-Satellite Tracking in the Low–Low Mode

For the determination of the gravitational field of the Earth, satellite-to-satellite tracking enfolds its true potential only in the low–low mode. In this mode two satellites measure their relative velocity, the so-called range-rate: >  P WD xP 02 .t; p/  xP 01 .t; p/ e12 ;

(83)

with e12 being the line-of-sight (LOS) unit vector e12 WD

x02  x01 : kx02  x01 k

(84)

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

195

For the determination of the parameters p, describing the gravitational field of the Earth, the following least-squares problem has to be solved: p WD argmin

n  X        > 2 P tj  xP 02 tj ; p  xP 01 tj ; p e12 tj ; p :

(85)

j D1

As in the high–low mode, there are two groups of methods to solve this least-squares problem: 1. Conversion of the measured range-rates into artificial in situ observations as in the • Potential-difference approach • The line-of-sight gradiometry approach 2. The computation of the partial derivatives @@pP as in • The variational equation approach • The integral equation approach

Potential-Difference Approach The basic idea of this approach dates back to an article of Wolff in 1969. It was further developed and tested for the GRACE mission by Jekeli (1999) and Han (2004). The observation quantity is the range-rate P12 D xP > x2  xP 1 /> e12 : 12 e12 WD .P

(86)

Using the energy conservation in the inertial frame V D

1 > xP xP  E0 2

(87)

and forming the along-track derivative da on both side yields da V D xP > da xP :

(88)

If both satellites are close to each other, one can approximate

V12 WD V .x2 /  V .x1 /  kPx1 kP12 :

(89)

So far the developments are made in an inertial system for a static potential V . In reality the Earth rotates and therefore the potential is time dependent. In order to remove this time dependency, the energy balance has to be considered in a rotating system. Doing this, the centrifugal potential has to be added to the energy balance, yielding the following observation equation for the potential differences:

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W. Keller

V 0 .x2 /V 0 .x1 /CE0  kPx1 kP12 

>   1 >   1 !  x02 !  x02 C !  x01 !  x01 : 2 2 (90)

Using the basis function representation of the gravitational field, the following relationship between range-rates and gravitational field parameters can be established: X

      cz ˆz x02 ;  z  ˆz x01 ;  z C E0

z2C Z r

 kPx1 kP12 

>   1 >   1 !  x02 !  x01 !  x02 C !  x01 : 2 2

(91)

If either the basis functions do not depend on the parameters  z or if these parameters are fixed a priori, Eqs. (91) are linear in the unknown coefficients cz , and these coefficients can be determined by a standard linear least-squares technique: n  X

    1  >    !  x02 tj kPx1 tj kP12 tj  !  x02 tj 2 j D1   >    1 !  x01 tj C !  x01 tj 2 !2 X      0   0  C E0 ! min: cz ˆz x2 tj  ˆz x1 tj

(92)

z

Potential-Difference Approach 1. Multiply the measured range-rates with the total velocity of the trailing satellite      s j D P12 tj x01 tj  : 2. Reduce this pseudo-observation by the difference in the centrifugal potential sj D s j 

 >    1   >    1 !  x02 tj !  x01 tj !  x02 tj C !  x01 tj : 2 2

3. Solve the linear least-squares problem n X j D1

sj 

X z

       2 cz ˆz x02 tj  ˆz x01 tj C E0

! ! min :

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

197

Line-of-Sight Gradiometry A different technique to convert range-rates in into situ observations is the lineof-sight gradiometry. This technique is described in Blaha (1992), Heß and Keller (1999), and Keller and Sharifi (2005). The main idea of this approach is to compute the time derivative of the observed range-rates: R D .Rx2  xR 1 /> e12 C

kPx2  xP 1 k2 P2  ;  

(93)

with the measured range-rate P and the inter-satellite range  D kx2  x1 k. Since xR i D rV .xi / holds, Eq. (93) can be recast into an observation equation for the differences in the potential gradients: .rV .x2 /  rV .x1 //> e12 D R 

kPx2  xP 1 k2 P2  :  

(94)

A Taylor expansion of the left-hand side of this equation at the satellite midpoint x WD 12 .x1 C x2 / yields 2 e> 12 r V .x/ e12 

R kPx2  xP 1 k2 P2   2:  2 

(95)

The left side of this equation is the gravity gradient in the direction of the lineof-sight between the two satellites, which gives this approach its name. If now the basis function representation of the gravitational potential is inserted, we arrive at a least-squares problem for the parameters of this representation:

.cz ;  z / D argmin

n X

X       2 cz e> yy tj  12 r ˆz x tj ;  z e12

!2 :

(96)

z2C Z r

j D1

In this equation the quantity yy is an abbreviation for the artificial observation yy WD

R kPx2  xP 1 k2 P2   2:  2 

(97)

Instead in the Earth-fixed system, the problem can also be treated in the orbital system. This has the advantage that the line-of-sight gradient can be expressed as the linear combination of the second-order derivative in u direction and of the firstorder derivative in radial direction: 2 e> 12 r V .x/ e12 D

1 @2 V 1 @V ; C r 2 @u2 r @r

(98)

which makes the computation of all components of the Marussi tensor r 2 V obsolete. The least-squares problem simplifies to

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W. Keller n X      cm;k ;  m;k D argmin yy tj j D1



X

cm;k

m;k

  1 @2 ˆm;k u; ƒ;  m;k   tj a2 @u2

!2   1 @ˆm;k u; ƒ;  m;k   tj C : a @a

(99)

For spherical harmonics as basis functions, their representation (23) in the orbital system leads to the following simplified least-squares problem: 



cm;k ;  m;k D argmin

n X

0

12   X     @yy tj  cm;k ˆm;k uj ; ƒj tj A ;

j D1

(100)

m;k

with 

ˆm;k uj ; ƒj



2 GM D4 3 R

1 X lDmaxfjkj;jmjg

3  lC3   R k l .i /P l .0/5 l  1  k 2 { km dm;k r

e {.kuj Cmƒj / :

(101)

All in all, for spherical harmonics as basis functions, the line-of-sight gradiometry approach consists of the following steps: Line-of-Sight Gradiometry Approach 1. Convert the measured range P rate into synthetic relative accelerations R by some numerical differentiation scheme. 2. Compute the line-of-sight gravity gradient yy .tj / according to (97). 3. Solve the linear least-squares problem n X j D1

12   X     @yy tj  cm;k ˆm;k uj ; ƒj tj A ! min 0

m;k

Variational Equation Approach The variational equation approach is used by several groups concerned with the processing of GRACE data (Beutler et al. 2010a,b). It is centered around the nonlinear least-squares problem

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

199

n  X       > 0  2 P tj  xP 02 tj  xP 01 tj e12 tj ! min :

(102)

j D1

To solve this nonlinear least-squares problem, the computation of the partial derivatives of the model range-rates  > PM .t/ D xP 02 .t/  xP 01 .t/ e012 .t/ with respect to the unknown parameters p is an essential point. Obviously, we have >       P  0  > @ x02  x01 @ xP 02  xP 01 @PM 1 0 0 0 0 x  x1 xP  xP 1  D e12 C @pk @pk  2  2 @pk (103)  >   .2/    P  0  1 0 .1/ .2/ .1/ D e012 P k  P k C k  k x2  x01 xP 2  xP 01    .i /

For each satellite .i / the quantities k are computed as the solutions of the variational equations as described in Sect. 5.1. Hence, the variational equation approach for the determination of the gravity field parameters p from the observed range-rates P consists of the following steps: Variational Equation Approach .i /

1. For each satellite solve the variational equations for the partial derivatives  k according to Sect. 5.1. 2. Compute the partial derivatives of the model range-rates with respect to the parameters pk as  .2/  @PM .1/ D e012 P k  P k C @pk



   P  0  >  .2/ 1 0 .1/ 0 0 xP 2  xP 1  k  k x2  x1  

3. Solve the linear least-squares problem n X j D1

  X @PM P tj  pk @pk

!2 ! min :

k

Basically, also an iterative solution of the nonlinear least-squares problem is possible. But taking into account that rather good a priori values p0 for p are known and that the solution of the variational equations is costly, in most cases only one single step is carried out.

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W. Keller

5.3

Integral Equations Approach

The integral equation approach, fully developed in Mayer-Gürr et al. (2007), MayerGürr (2012), also tackles the least-squares problem (102). Its difference to the .i / .i / variational equation approach lies in the way the quantities  k and P k in the partial derivatives of the model range-rates with respect to the unknown parameters p are computed. In the integral equations approach, the point of departure is not the equation of motion (40) but its equivalent integral equation counterpart (43). If now in (43) the unknown potential V 0 is replaced by its basis function representation, for each satellite .i / we obtain the following approximation of the integral equation describing its motions: Z xQ .i / ./ D xQ .i / .0/.1  / C xQ .i / .1/  T 2

1

    K ;  0 f.i /  0 ; p d  0

(104)

0

with f

.i /

 0   ; p WD r

X



cz ˆz x

0.i /



0



 T ; z



!

z2C Z r

         C F0 x0.i /  0 T ; xP 0.i /  0 T ;  0 T  !  !  x0.i /  0 T   (105)  2!  xP 0.i /  0 T : The vector p collects all unknown parameters cz ;  z in the basis function representation of the unknown potential. For the determination of the parameters p from the observed positions and velocities, their partial derivatives with respect to the parameters are needed. The partial derivatives solve the integral equations @Qx.i / ./ @p Z 1  @f.i /  D T 2 K ;  0 @x0 0  @f.i /  0  C  ; p d0 @p  Z 1  @f.i /  D T 2 K ;  0 @x0 0  @f.i /  0   ; p d0 C @p

 .i / WD

 0  @Qx.i /  0  @f.i /  0  @xPQ .i /  0   C   ;p  ;p @p @Px0 @p



  @f.i /  0  .i /  0    0 ; p  .i /  0 C  ; p P  @Px0 (106)

Satellite-to-Satellite Tracking (Low-Low/High-Low SST)

201

and @xPQ .i / .i / ./ P WD @p Z 1  @f.i /  0  @Qx.i /  0  @f.i /  0  @xPQ .i /  0  @K  ;  0  C  C D T 2  ;p  ;p @x0 @p @Px0 @p 0 @  @f.i /  0   ; p d0 @p  Z 1  @f.i /  0  .i /  0  @f.i /  0  .i /  0  @K  2 0 D T  ;p   C  ; p P  ;  @x0 @Px0 0 @  @f.i /  0   ; p d0 C @p

(107)

respectively. If now the integrals are approximated by quadrature formulas Z

1

I WD 0

   @f.i /  0  .i /  0  @f.i /  0  .i /  0  @f.i /  0   0 P  ;p  ;p   C 0  ;p   C K ;  @x0 @Px @p

d0 

n X

wl K



; l0

lD1

C

@f.i /  0   ;p @p l

Z

1

0

I WD 0

  @f.i /  0  .i /  0  @f.i /  0  .i /  0   ; p  l C  ; p P l @x0 l @Px0 l

 (108)

   @f.i /  0  .i /  0  @f.i /  0  .i /  0  @f.i /  0  @K  0 P  ;p   C  ;p   C ;   ;p @ @x0 @Px0 @p d0



n X lD1

  @f.i /  0  .i /  0  @f.i /  0  .i /  0  @K  0 ; l  ; p  l C  ; p P l wl @ @x0 l @Px0 l  @f.i /  0   ;p ; C @p l

(109)

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W. Keller

we obtain two linear systems of equations for the unknown partial derivatives:



.i /

 n X    @f.i /  0  .i /  0  @f.i /  0  .i /  0   2 0 j D T  ; p  l C  ; p P l wl K j ; l @x0 l @Px0 l lD1  @f.i /  0   ;p (110) C @p l

 n X  @f.i /  0  .i /  0  @f.i /  0  .i /  0  @K  .i /   j D T 2  ; p  l C  ; p P l P wl j ; l0 @ @x0 l @Px0 l lD1

C

 @f.i /  0  l ; p : @p

(111)

If the unknown partial derivatives with respect to the kth parameter pk are assembled into column vectors   .i / .i / .i / .i / .i / Zk WD vec  k .1 /; : : : ;  k .n /; P k .1 /; : : : ; P k .n / ;

(112)

these linear equations can be written in matrix form as   .i / .i /  A B .i / .i /  Zk D bk IC C.i / D.i /

(113)

with 2

A.i /

 .i /  w1 K 1 ; 10 @f@x0 6   6 0 @f.i / 2 6 w1 K 2 ; 1 @x0 DT 6 6 4  .i /  w1 K n ; 10 @f@x0 2

B.i /

 .i /  w1 K 1 ; 10 @f@Px0 6   6 0 @f.i / 2 6w1 K 2 ; 1 @Px0 DT 6 6 4  .i /  w1 K n ; 10 @f@Px0

 .i /   10 ; p : : : wn K 1 ; n0 @f@x0  0   .i /  1 ; p : : : wn K 2 ; n0 @f@x0 :: :  0   .i /  1 ; p : : : wn K n ; n0 @f@x0

 0 3 n ; p  0 7 7 n ; p 7 7; 7 5  0  n ; p

(114)



 0 3 n ; p  0 7 7 n ; p 7 7; 7 5  0  n ; p

(115)



 .i /   10 ; p : : : wn K 1 ; n0 @f@Px0  0   .i /  1 ; p : : : wn K 2 ; n0 @f@Px0 :: :  0   .i /  1 ; p : : : wn K n ; n0 @f@Px0

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C.i /

  .i / w1 @K 1 ; 10 @f@x0 @ 6  6 @K  0 @f.i / 2 6w1 @ 2 ; 1 @x0 DT 6 6 4   0 @f.i / w1 @K @ n ; 1 @x0 2

D.i /

  0 @f.i / w1 @K @ 1 ; 1 @Px0 6  6 @K  0 @f.i / 2 6w1 @ 2 ; 1 @Px0 DT 6 6 4   .i / w1 @K n ; 10 @f@Px0 @

203

 @f.i /  0  3 n ; p @x0    0 7 7 0 @f.i / 2 ; n @x0 n ; p 7 7; 7 5    .i /  @f 0 0 n ; n @x0 n ; p

(116)

   0 3  0 @f.i / 10 ; p : : : wn @K  n ; p ;  1 0 n @Px @ 7  0    .i /  7 @f @K 1 ; p : : : wn @ 2 ; n0 @Px0 n0 ; p 7 7; :: 7 : 5  0      @K 0 @f 0 1 ; p : : : wn @ n ; n @Px0 n ; p

(117)



 10 ; p : : : wn @K @  0  1 ; p : : : wn @K @ :: :  0  1 ; p : : : wn @K @



1 ; n0



and .i / bk

! n X @f.i /  0  @f.i /  0   ;p ;:::;  ;p : D T vec wl K .1 ; l / wl K .n ; l / @pk l @pk l lD1 lD1 (118) 2

n X

The matrices A.i / ; B.i / ; C.i / ; D.i / are independent of the parameter pk . Hence the main numerical effort, the LU-decomposition, has to be carried out only once per satellite, no matter how many parameters pk are to be determined. This leads to the following algorithm:

Integral Equation Approach 1. For each satellite compute the matrices A.i / ; B.i / ; C.i / ; D.i / according to (114)– (117). 2. For each parameter pk .i / • Compute the vector bk according to (118). .i / • Compute the partial derivatives Zk with respect to the parameter pk as the solution of (113). • Compute the partial derivatives of the model range-rates with respect to the parameters pk as  .2/  @PM .1/ P P C D e>    k k 12 @pk



   P  0  >  .2/ 1 0 .1/ k  k x2  x01 xP 2  xP 01   

3. Solve the linear least-squares problem n X j D1

  X @PM P tj  pk @pk k

!2 ! min :

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Regional Gravity Field Models

Spherical harmonics are suitable basis functions for a global recovery of the gravitational field of the Earth. Because of their global support on the sphere, they are best suited for a homogeneous data distribution on the sphere. Since in general SST satellites have almost polar orbits, their orbital arcs converge toward the poles. Therefore, the data distribution at the poles is much denser than around the equator. In this situation it is reasonable to combine a global gravity field model, based on spherical harmonics expansion, with regional improvements of this global field. The latter is represented by basis functions with a local support or by basis functions, which are at least rapidly decaying. A technique relying on basis functions with a local support is the so-called mascons technique. This technique was developed by Muller and Sjogren (1968) for the recovery of the lunar gravitational field. Due to the fact that constraints between the basis functions can be introduced, this technique has been applied to GRACE data by several authors (Luthcke et al. 2008; Rowlands et al. 2010). Nevertheless, the majority of authors use RBFs for the regional improvement of a global gravity field solution. In this respect the contributions of Schmidt et al. (2006), Fengler et al. (2007), Eicker (2012), Klees et al. (2008) and others have to be mentioned. The basic idea of the regional improvement by RBFs is the separation of the data M N set fsi gN i D1 in Eq. (10) into two disjunct subsets fsi gi D1 [fsi gi DM C1 . The first subset is used to derive a global spherical harmonics model

fcl;m g D argmin

M X i D1

       si  F x1 ih; VQ .Kl;m / ; xP 1 ih; VQ .Kl;m / ; x2 ih; VQ .Kn;m / ;  2 xP 2 ih; VQ .Kl;m / 

(119)

with l 1   GM X R lC1 X VQ D Kl;m Yl;m .#; /: R r lD0

(120)

mDl

Once the global model (120) has been determined, the residual observations between the original data and the synthetic data are computed for both subsets:        ri Dsi  F x1 ih; VQ .Kl;m / ; xP 1 ih; VQ .Kl;m / ; x2 ih; VQ .Kn;m / ;   xP 2 ih; VQ .Kl;m / ; i D 1; : : : ; N: (121)

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The residuals then undergo an analysis with respect to RBFs as basis functions

f.cl ;  l /g D argmin

N X i D1

       kri  F x1 ih; VO .cl ;  l / ; xP 1 ih; VO .cl ;  l / ; x2 ih; VO .cl ;  l / ;   xP 2 ih; VO .cl ;  l / k2 (122) with VO D

L X

cl ˆ .x;  l /

(123)

lD1

and ˆ according to (27). In most applications, the parameters  l are assigned fixed a priori values, which makes the regional improvement technique a linear leastsquares problem. One of the few publications, where also the parameters  l are subject to optimization, i.e., where the base functions are allowed to change shape and position during the minimization process, is Antoni (2012).

6.1

Final Remarks

So far, all arguments give only a coarse sketch of the methods applied in the processing of SST data. Only the basic ideas behind these methods have been presented. The practical application differs in the following aspects from the given arguments: 1. An important step in data preprocessing is the so-called de-aliasing. Though this step is absolutely vital for the derivation of meaningful results from the SST data, it could not be discussed here. De-aliasing means that all forces, which do not fit into the simple observation model, have to be either measured or modeled and the genuine SST data has to be reduced for these forces. Hence, the concept of de-aliasing includes the following reductions: • • • •

For atmospheric friction For tidal effects For precession and nutation effects For atmospheric and oceanographic loading effects

A comprehensive description of the de-aliasing procedures is given in Bettadpur (2012). 2. In general not the de-aliased data itself but the de-aliased residual data is the input for the subsequent analysis. Residual data means that synthetic observations,

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which are computed from an a priori gravity field model, are subtracted from the de-aliased data. This step reduces the effect of all approximations made in the derivation of the observation model. The consequence of forming residuals is that not the gravity field parameters themselves but only their differences to the parameters of the a priori model can be determined. 3. In general the normal equation matrices of the least-squares problems have a poor condition. Therefore, in most cases different kinds of regularization are applied. 4. Due to imperfect de-aliasing, the monthly variations in GRACE-derived gravity field models show dominant north–south stripes, which obscure the desired geophysical information. Therefore, a number of filter strategies are applied in post-processing to remove these stripes (Kusche et al. 2009). 5. Since the GRACE satellites also carry onboard GPS receivers, the GPS mission is both a hl and a ll SST mission. While the hl data are insensitive to the short wavelengths in the gravity field, the ll mission is “ignorant” for the long wavelength features. For an optimal resolution both data types have to be combined. Due to their different error budget, a proper combination requires a preceding variance components estimation.

7

Missions and Outcomes

7.1

CHAMP Mission

The CHAMP mission was not only dedicated to the gravity field but also to the magnetic field of the Earth. Another important experiment carried out with CHAMP was the study of the atmosphere by radio-occultation. An overview over the results after 5 years in orbit is given in Reigber et al. (2006). Different institutions processed the CHAMP data for different orbit lengths and derived individual spherical harmonic gravity field models. All these models have been collected by the International Centre for Global Earth Models (ICGEM) and can be accessed via its website (ICGEM). The models differ in the time-span of data and in the resolution limit in degree and order. The highest resolution up to degree and order 140 is given by the model EIGEN-CHAMP03S (Reigber et al. 2004). For its derivation CHAMP data from October 2000 until June 2003 have been analyzed. The normal equation regularization for this model started from degree and order 60 on. An evaluation of the quality of the derived models can be done by a comparison with a model of higher quality, derived from GRACE data, e.g., by a comparison with the GRACE-EIGEN6C2 model. Taking this model for the truth, one can see that the estimation error of the EIGEN-CHAMP03S model matches the signal strengths at about degree 90. Roughly speaking, this means that from this point on, the estimation error of a coefficient exceeds its magnitude. While this is a relative evaluation, an absolute evaluation can be done by a comparison of geoid heights, obtained by a combination of GPS heights and spiritleveling heights, with gravity field-derived geoid heights. In this comparison the

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GPS heights minus leveling heights solution is considered the master solution. The comparison shows a disagreement of about 0.8 m in North America and 1.2 m in Europe for EIGEN-CHAMP03S.

7.2

GRACE Mission

Besides static solutions for the time-invariant part of the gravitational field, the GRACE mission also provides monthly solutions. The static solutions can be accessed via the ICGEM website (ICGEM). The models differ in the time-span of data and in the resolution limit in degree and order. The highest resolution up to degree and order 180 is given by the model ITG-GRACE2010s (Mayer-Gürr et al. 2010). It uses data from August 2002 till August 2009. If evaluated against GRACE-EIGEN6C2, it shows that the estimation error matches the signal strength at about degree 160. This means an improvement by the factor 2 compared to the best CHAMP solution. Also the comparison of the ITG-GRACE201s model with GPS-leveling results in a disagreement of about 0.5 m for all continents, which is also by the factor 2 better than the best CHAMP solution in Europe. The essential outcome of the GRACE missions is the monthly gravity field solutions, because they carry the imprint of mass transport processes at or close to the surface of the Earth. They also can be accessed via the ICGEM website (ICGEM). Besides the unmodified monthly solution, also monthly solution, which is de-striping filtered according to Kusche et al. (2009), is provided. The importance of monthly GRACE solution for geophysical interpretation rests on the assumption that all mass changes, which are observed by GRACE, take place in a thin layer at the surface of the Earth. Under this assumption, observed changes Kl;m in the spherical harmonics coefficients can be converted into changes of surface layer density , i.e., mass change per area (cmp. Chao and Gross 1987):

.#; / D

l 1 R X 2l C 1 X Kl;m Yl;m .#; /: 3 1 C kl lD0

(124)

mDl

In this equation  stands for the average density of the Earth and the coefficients kl are the so-called Love numbers (Han and Wahr 1995). Since the majority of mass changes, which can be observed by GRACE, are related to water and ice redistributions, these density changes can be converted into changes of equivalent water thickness by EW T D

 ; w

(125)

with w being the density of water. This change in EWT corresponds to the vertically integrated mass changes inside aquifers, soil, surface reservoirs, and snow and ice packs. It can be observed by GRACE with an accuracy of a few millimeters for a

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spatial resolution of about 400 km. For this reasons the EWT observed by GRACE has an important impact to: • Hydrology • Oceanography • Glaciology In continental hydrology GRACE estimates of continental water storage improve the understanding of hydrological and atmospheric processes. The great advantage of GRACE estimates is that they are averages over a few hundred of kilometers while traditional ground-based hydrological data refer to scales of 10 km or less. The weakness of GRACE results is that they cannot distinguish between water on the surface and in the soil. Neither they can discriminate between water, snow, and ice. All in all, the GRACE estimates are in good agreement with estimates derived from hydrological models as WGHM or GLDAS. GRACE can give reliable estimates about the total water budget over large regions, and with this information, it can contribute to the improvement of hydrological models. In oceanography the main contribution of GRACE is the separation between steric and non-steric sea-level rise, because only the latter is related to mass transports. Precise information about the steric sea-level rise makes it possible to estimate the change in heat storage in the oceans, which is important for climate change prediction. In the pre-GRACE area of glaciology, information about the mass balance of the arctic and antarctic ice shields could only be derived from measurements at a small number of benchmark glaciers. The parameters derived from these measurements are hardly representative for the entire ice shield. Here the mass change derived from GRACE is an important additional source of information, because it refers to averages over a few hundreds of kilometers. But the total mass change observed by GRACE is not only due to the mass loss or mass gain of ice shields but additionally stems from the effect of postglacial rebound. The reduction of GRACE observation by the influence of postglacial rebound, computed from existing models, gives valuable constraints for ice-dynamics models.

8

Conclusions

Satellite-to-satellite tracking techniques proved capable to monitor both the static and the time-variable gravity field of the Earth with an unprecedented accuracy and resolution. A number of different algorithms have been developed to convert the genuine SST observations into parameters of a basis function representation of the gravitational field of the Earth. Nevertheless, there are still a number of unsolved problems, like the de-striping of monthly GRACE solutions, the spectral leakage, and the aliasing problem, which still constitute a challenge for further research.

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References Antoni M (2012) Nichtlineare Optimierung regionaler Graviationsfeldmodelle aus SST Daten. PhD thesis, Universität Stuttgart Badura T, Sakulin C, Gruber T, Klostius R (2006) Derivation of the CHAMP-only gravity field model TUG-CHAMP04 applying the energy integral approach. Stud Geophys Geod 50:57–74 Ballani L (1988) Partielle Ableitungen und Variationsgleichungen zur Modellierung von Satellitenbahnen und Parameterbestimmung. Vermessungstechnik 36:192–194 Bettadpur S (2012) Level-2 gravity field product user handbook rev. 3.0, May 29. ftp://podaac-ftp. jpl.nasa.gov/GeodeticsGravity/grace/L1B/JPL/RL01/docs/L2-UserHandbook_v3.0.pdf Beutler G, Jäggi A, Mervart L, Meyer U (2010a) The celestial mechanics approach: theoretical foundations. J Geodesy 84:65–624 Beutler G, Jäggi A, Mervart L, Meyer U (2010b) The celestial mechanics approach: application to data of the GRACE mission. J Geodesy 84:661–681 Bjerhammar A (1976) On the energy integral for satellites. Technical report, Report of the Royal Institute of Technology, Stockholm Blaha G (1992) Refinement of the satellite-to-satellite line-of-sight model in residual gravity field. Manuscr Geod 17:321–333 Chao BF, Gross RS (1987) Changes in the Earth’s rotation and low-degree gravitational field induced by earthquakes. J R Astron Soc 91:569–596 Colombo O (1984) Global mapping of gravity with two satellites. Technical report vol 7 Nr 3, Netherlands Geodetic Commission Eicker A (2012) Gravity field Refinement by radial basis functions from in-situ satellite data. Technical report, DGK Reihe C, Bd. 676 Fengler MJ, Freeden W, Kohlhaas A, Michel V, Peters T (2007) Wavelet modeling of regional variations of the Earth’s gravitational potential observed by GRACE. J Geodesy 81:5–15 Gerlach CL, Földvary L, Švehla D, Gruber T, Wermut M, Sneeuw N, Frommknecht B, Oberhofer H, Peters T, Rothacher M, Rummel R, Steigenberger P (2003) A CHAMP-only gravity field model from kinematic orbits using the energy integral. Geophys Res Lett, doi:10.1029/2003GLO18025 Han D, Wahr J (1995) The viscoelastic relaxation of a realistic stratified Earth and further analysis of post-glacial rebound. Geophys J Int 120:287–311 Han S-C (2004) Efficient determination of global gravity field from satellite-to-satellite tracking mission. Celest Mech Dyn Astron 88:69–102 Heß D, Keller W (1999) Gradiometrie mit GRACE. Z Vermess 124:137–144 ICGEM. http://icgem.gfz-potsdam.de/ICGEM/ICGEM.html, 2014 Jekeli C (1999) The determination of gravitational potential differences from satellite-to-satellite tracking. Celest Mech Dyn Astron 75:85–101 Kaula WM (2000) Theory of satellite geodesy. Applications of satellites to geodesy. Dover, New York Keller W, Sharifi MA (2005) Satellite gradiometry using a satellite pair. J Geodesy 78:544–557 Klees R, Liu X, Wittwer T, Gunter BC, Revtona EA, Tenzer R, Ditmar P, Winsemius HC, Savanije HHG (2008) A comparison of global and regional GRACE models for land hydrology. Surv Geophys 29:335–359 Kostelec PJ, Rockmore DN (2008) FFTs on the rotation group. J Fourier Anal Appl 14:145–179 Kusche J, Schmidt R, Petrovic S, Rietbroeck R (2009) Decorrelated GRACE time-variable gravity field solutions by GFZ, and their validation using a hydrological model. J Geodesy 83:903–913 Levenberg KA (1944) A method for the solution of certain problems in least squares. Q Appl Math 2:164–168 Luthcke SB, Arendt AA, Rowlands DD, McCarthy JJ, Larsen CF (2008) Recent glacier mass changes in the Gulf of Alaska region from GRACE mascons solutions. J Glaciol 54:767–777 Marquardt D (1963) An algorithm for least-squares estimation of nonlinear parameters. SIAM J Appl Math 11:431–443

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Mayer-Gürr T (2012) Gravitationsfeldbestimmung ausn der Analyse kurzer bahnbögen am eispiel der Satellitenmissionen CHAMP und GRACE. Technical report, DGK Reihe C, Bd. 675 Mayer-Gürr T, Eicker A, Ilk K-H (2007) ITG-Grace02s: a GRACE gravity field derived from range measurements of short arcs. In: Gravity field of the Earth, proceedings of the 1st international symposium of the international gravity field service (IGFS), Istanbul Mayer-Gürr T, Ilk H, Eicker A, Feuchtinger M (2005) ITG-CHAMP01: a CHAMP gravity field model from short kinematic arcs over a one-year observation period. J Geodesy 78:462–480 Mayer-Gürr T, Kurtenbach E, Eicker A (2010) ITG-grace2010 gravity field model. http://www. igg.uni-bonn.de/apmg/index.php?id=itg-grace2010 Muller PM, Sjogren WL (1968) Mascons: lunar mass concentrations. Science 161:680–684 O’Keefe JA (1957) An application of Jacobi’s integral to the motion of an Earth satellite. Astron J 62:265–266 Petit G, Luzum B (2010) IERS conventions (2010) (IERS technical note 36). Technical report, Verlag des Bundesamtes für Kartographie und Geodäsie, Frankfurt am Main Reigber C (1969) Zur Bestimmung des Gravitationsfeldes der Erde aus Satellitenbeobachtungen. Technical report, DGK Reihe C, Bd. 137 Reigber C, Jochmann H, Wünsch J, Petrovic S, Schwintzer P, Barthelmes F, Neumayer K-H, König R, Förste C, Balmino G, Biancale R, Lemoine J-M, Loyer S, Perosanz F (2004) Earth gravity field and seasonal variability from CHAMP. In: Reigber C, Lühr H, Schwintzer P, Wickert J (eds) Earth observation with CHAMP – results from three years in orbit. Springer, Berlin, pp 25–30 Reigber C, Lühr H, Grunwald L, Förste C, König R (2006) CHAMP mission 5 years in orbit. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the Earth system from space. Springer, Berlin/Heidelberg/New York Reubelt T (2009) Harmonische Gravitationsfeldanalyse aus GPS-vermessenen kinematischen Bahnen niedrig fliegender Satelliten vom Typ CHAMP, GRACE, GOCE mit einem hochauflösenden Beschleunigungsansatz. Technical report, DGK Reihe C, Bd. 632 Reubelt T, Austen G, Grafarend EW (2003) Harmonic analysis of the Earth’s gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP. J Geodesy 77:257–278 Rowlands DD, Luthcke SB, McCarthy JJ, Klosko SM, Chinn DS, Lemoine FG, Boy J-P, Sabaka TS (2010) Global mass-flux solutions from grace: a comparison of parameter estimation strategies – mass concentrations versus Stokes coefficients. J Geophys Res 115:B01403 Rummel R (2003) How to climb the gravity wall. Space Sci Rev 108:1–14 Schmidt M, Han S-C, Kusche J, Sanchez L, Shum CK (2006) Regional high-resolution spatiotemporal gravity modeling from GRACE data using spherical wavelets. Geophys Res Lett 33:L08403 Schneider M (1968) A general method of orbit determination. Technical report, Library Translations, Aircraft Establishment, Ministry of Technology, Farnborough Sneeuw N (2000) A semi-analytical approach to gravity field analysis from satellite observations. Technical report, DGK Reihe C, Bd. 527 Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geodesy 77:207–216 Weisstein E Wolfram mathworld. http:mathworld.wolfram.com, 2014 Wolff M (1969) Direct measurements of the Earth’s gravitational field using a satellite pair. J Geophys Res 74:5295–5300

GOCE: Gravitational Gradiometry in a Satellite Reiner Rummel

Contents 1 Introduction: GOCE and Earth Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 GOCE Gravitational Sensor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Gravitational Gradiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 GOCE Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions: GOCE Science Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212 213 216 221 221 224

Abstract

Spring 2009 the satellite Gravity and steady-state Ocean Circulation Explorer (GOCE), equipped with a gravitational gradiometer, was launched by European Space Agency (ESA). Its purpose is the detailed determination of the spatial variations of the Earth’s gravitational field, with applications in oceanography, geophysics, geodesy, glaciology, and climatology. Gravitational gradients are derived from the differences between the measurements of an ensemble of three orthogonal pairs of accelerometers, located around the center of mass of the spacecraft. Gravitational gradiometry is complemented by gravity analysis from orbit perturbations. The orbits are thereby derived from uninterrupted and threedimensional GPS tracking of GOCE. The gravitational tensor consists of the nine second-derivatives of the Earth’s gravitational potential. Due to its symmetry only six of them are independent. These six components can also be interpreted in terms of the local curvature of the field or in terms of components of the tidal

R. Rummel () Institut für Astronomische und Physikalische Geodäsie, TU Munich, Munich, Germany e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_4

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field generated by the Earth inside the spacecraft. Four of the six components are measured with high precision (1011 s2 per square-root of Hz), the others are less precise. Several strategies exist for the determination of the gravity field at the Earth’s surface from the measured tensor components at altitude. The mission ended in November 2013. Until August 2012 in total 2.3 years of data were collected. They entered into ESA’s fourth release of GOCE gravity models. After August 2012 the orbit altitude was lowered in several steps by altogether 31 km in order to test the enhanced gravitational sensitivity at lower orbit heights.

The fields of application range from solid earth physics, via geodesy and oceanography to atmospheric physics. For example, several studies are concerned with the state of isostatic mass compensation in regions such as South America, Africa, Himalaya, and Antarctica. GOCE will help to unify height systems worldwide and enable the direct conversion of GPS-based ellipsoidal heights to accurate and globally consistent heights above the geoid. For the first time, it became possible to derive mean dynamic ocean topography and geostrophic ocean velocities with high spatial resolution and accuracy directly from space, combining the altimetric mean sea surface and the GOCE geoid. Assimilation into numerical ocean circulation models will help to improve estimates of ocean mass and heat transport. Commonmode accelerations as measured by GOCE lead to improved atmospheric density and wind estimates at GOCE altitudes.

1

Introduction: GOCE and Earth Sciences

On March 17, 2009, the European Space Agency (ESA) launched the satellite Gravity and steady-state Ocean Circulation Explorer (GOCE). It is the first satellite of ESA’s Living Planet Programme (see ESA 1999a, 2006). It is also the first one that is equipped with a gravitational gradiometer. The purpose of the mission is to measure the spatial variations of the Earth’s gravitational field globally with maximum resolution and accuracy. Its scientific purpose is essentially twofold. First, the gravitational field reflects the density distribution of the Earth’s interior. There are no direct ways to probe the deep Earth interior, only indirect ones in particular, seismic tomography, gravimetry, and magnetometry. The gravimetry part is now been taken care of by GOCE. Also, in the field of space magnetometry, an ESA mission was launched in fall 2013; it is denoted “Swarm” and consists of three satellites. Seismic tomography is based on a worldwide integrated network of seismic stations. From a joint analysis of all seismic data, a tomographic image of the spatial variations in the Earth’s interior of the propagation velocity of seismic waves is derived. The three methods together establish the experimental basis for the study of solid Earth physics, or more specifically, of phenomena such as core-mantle topography, mantle convection, mantle plumes, ocean ridges, subduction zones, mountain building, and mass compensation. Inversion of gravity alone is non-unique but joint inversion

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together with seismic tomography, magnetic field measurements and in addition with surface data of plate velocities and topography, and models from mineralogy will lead to a more and more comprehensive picture of the dynamics and structure of the Earth’s interior (see, e.g., Bunge et al. 1998; Hager and Richards 1989; Kaban et al. 2004; Lithgow-Bertelloni and Richards 1998). Second, the gravitational field and therefore the mass distribution of the Earth determines the geometry of level surfaces, plumb lines, and lines of force. This geometry constitutes the natural reference in our physical and technical world. In particular, in cases where small potential differences matter such as in ocean dynamics and large civil constructions, precise knowledge of this reference is an important source of information. The most prominent example is ocean circulation. Dynamic ocean topography, the small one up to 2 m deviation of the actual ocean surface from an equipotential surface, can be directly translated into ocean surface circulation. The equipotential surface at mean ocean level is referred to as geoid and it represents the hypothetical surface of the world oceans at complete rest. GOCE, in conjunction with satellite altimetry missions, like Jason will allow for the first time direct and detailed measurement of ocean circulation (see discussions in Albertella and Rummel 2009; Ganachaud et al. 1997; LeGrand and Minster 1999; Losch et al. 2002; Maximenko et al. 2009; Wunsch and Gaposchkin 1980). GOCE is an important satellite mission for oceanography, solid Earth physics, geodesy, and climate research (compare ESA 1999b; Johannessen et al. 2003; Rummel et al. 2002).

2

GOCE Gravitational Sensor System

In the following, the main characteristics of the GOCE mission which is unique in several ways will be summarized (see also  Gauss’ and Weber’s “Atlas of Geomagnetism” (1840) Was not the first: the History of the Geomagnetic Atlases). The mission consists of two complementary gravity sensing systems. The large-scale spatial variations of the Earth’s gravitational field are derived from its orbit, while the medium to short scales are measured by a so-called gravitational gradiometer. Even though satellite gravitational gradiometry has been proposed already in the late 1950s in Carroll and Savet (1959) (see also Wells 1984), the GOCE gradiometer is the first instrument of its kind to be put into orbit. The principles of satellite gradiometry will be described in  Gauss’ and Weber’s “Atlas of Geomagnetism” (1840) Was Not the First: The History of the Geomagnetic Atlases. The purpose of gravitational gradiometry is the measurement of the gradients of gravitational acceleration or, equivalently, the second-derivatives of the gravitational potential. In total, there exist nine second-derivatives in the orthogonal coordinate system of the instrument. The GOCE gradiometer is a three-axis instrument and its measurements are based on the principle of differential acceleration measurement. It consists of three pairs of accelerometers, mounted orthogonally to each other, each accelerometer having three axes (see Fig. 1). The gradiometer baseline of each oneaxis gradiometer is 50 cm. The precision of each accelerometer is about 21012 m/s2

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Fig. 1 GOCE gravitational gradiometer (courtesy ESA)

per square-root of Hz along two sensitive axes; the third axis has much lower sensitivity. This results in a precision of the gravitational gradients of 1011 s2 or 10 mE per square-root of Hz (1 E = 109 s2 = 1 Eötvös Unit). From the measured gravitational acceleration differences, the three main diagonal terms and one offdiagonal term of the gravitational tensor can be determined with high precision. These are the three diagonal components xx , yy , zz as well as the off-diagonal component xz , while the components xy and yz are less accurate. Thereby the coordinate axes of the instrument are pointing in flight direction (x/, cross

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direction (y/, and radially toward the Earth (z/. The extremely high gradiometric performance of the instrument is confined to the so-called measurement band (MB), while outside the measurement band noise is increasing. Strictly speaking, the derivation of the gradients from accelerometer differences assumes all six accelerometers (three pairs) to be perfect twins and all accelerometer test masses to be perfectly aligned. In real world, small deviations from such an idealization exist. Thus, the calibration of the gradiometer is of high importance. Calibration is essentially the process of determination of a set of scale, misalignment, and angular corrections. They are the parameters of an affine transformation between an ideal and the actual set of six accelerometers. Calibration in orbit is done by random shaking of the satellite by means of a set of cold gas thrusters and comparison of the actual output with the theoretically correct one. Before calibration the nonlinearities of each accelerometer are removed electronically; in other words, the proof mass of each accelerometer inside the electrodes of the capacitive electronic feedback system is brought into its linear range. The gravitational signal is superimposed by the effects of angular velocity and angular acceleration of the satellite in space. Knowledge of the latter is required for the removal of the angular effects from the gradiometer data and for angular control. The separation of angular acceleration from the gravitational signal is possible from a particular combination of the measured nine acceleration differences. The angular rates (in the MB) as derived from the gradiometer data in combination with those deduced from the star sensor readings are used for attitude control of the spacecraft. The satellite has to be well controlled and guided smoothly around the Earth. It is Earth pointing, which implies that it performs one full revolution in inertial space per full orbit cycle. Angular control is attained via magnetic torquers, i.e., using the Earth’s magnetic field lines for orientation. This approach leaves uncontrolled onedirectional degree of freedom at any moment. In order to prevent non-gravitational forces, in particular atmospheric drag, to “sneak” into the measured differential accelerations as secondary effect, the satellite is kept “drag-free” in along-track direction by means of a pair of ion thrusters. The necessary control signal is derived from the available “common-mode” accelerations (sum instead of differences of the measured accelerations) along the three orthogonal axes of the accelerometer pairs of the gradiometer. Some residual angular contribution may also add to the commonmode acceleration, due to the imperfect symmetry of the gradiometer relative to the spacecraft’s center of mass. This effect has to be modeled. The second gravity sensor device is a newly developed European GPS receiver. From its measurements, the orbit trajectory is computed to within a few centimeters, either purely geometrically, the so-called kinematic orbit, or by the method of reduced dynamic orbit determination (compare Bock et al. 2011; Jäggi 2007; Švehla and Rothacher 2004). As the spacecraft is kept in an almost drag-free mode (at least in along-track direction) the orbit motion can be regarded as purely gravitational. It complements the gradiometric gravity field determination and covers the long wavelength part of the gravity signal.

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CESS

Xenon tank

Ion propulsion module

MT

Gradiometer

STR SSTI

CDM LRR

GCD tank

Fig. 2 GOCE satellite and main instruments (courtesy ESA) (CESS coarse earth and Sun sensor, MT magneto torquer, STR star tracker, SSTI satellite to satellite tracking instrument, CDM command and data management unit, LRR laser retro reflector)

The orbit altitude is extremely low, only about 255 km at perigee. This is essential for a high gravitational sensitivity. No scientific satellite has been flown at such low altitudes so far. Its altitude is maintained through the drag-free control and additional orbit maneuvers, which are carried out at regular intervals. As said above, this very low altitude results in high demands on drag-free and attitude control. Finally, any time-varying gravity signal of the spacecraft itself, the so-called self-gravitation, must be excluded. This results in extremely tight requirements on metrical stiffness and thermal control. In summary, GOCE is a technologically very complex and innovative mission. The gravitational field sensor system consists of a gravitational gradiometer and GPS receiver as core instruments. Orientation in inertial space is derived from star sensors. Common-mode and differential-mode accelerations from the gradiometer and orbit positions from GPS are used together with ion thrusters for drag-free control and together with magneto-torquers for angular control. The satellite and its instruments are shown in Fig. 2. The system elements are summarized in Table 1.

3

Gravitational Gradiometry

Gravitational gradiometry is the measurement of the second derivates of the gravitational potential V . Its principles are described in textbooks such as Misner et al. (1970), Falk and Ruppel (1974), and Ohanian and Ruffini (1994) or in articles like Rummel (1986), compare also Colombo (1989) and Rummel (1997). Despite the high precision of the GOCE gradiometer instrument, the theory can still be

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Table 1 Sensor elements and type of measurement delivered by them (approximate orientation of the instrument triad: x = along-track, y = out-of-orbit-plane, z = radially downward) Sensor Three-axis gravity gradiometer

Star sensors (STR) GPS receiver (SSTI) Drag control with two ion thrusters Angular control with magnetic torquers Orbit altitude maintenance Internal calibration (and quadratic factors removal) of gradiometer

Measurements Gravity gradients xx , yy , zz , xz in instrument system and in MBW (measurement bandwidth) Angular accelerations (highly accurate around y-axis, less accurate around x, z axes) Common-mode accelerations High-rate and high-precision inertial orientation Orbit trajectory with centimeter precision Based on common-mode accelerations from gradiometer and GPS orbit Based on angular rates from star sensors and gradiometer Based on GPS orbit Calibration signal from random shaking by cold gas thrusters (and electronic proof mass shaking)

formulated by classical Newton mechanics. Let us denote the gravitational tensor, expressed in the instrument frame as 1 0 @2 V Vxx Vxy Vxz @x 2 B @2 V A @  D Vij D Vyx Vyy Vyz D @ @y@x @2 V Vzx Vzy Vzz 0

@2 V @x@y @2 V @y 2 @2 V @z@x @z@y

@2 V @x@z @2 V @y@z @2 V @z2

1 C A;

(1)

where the gravitational potential represents the integration over all Earth masses (cf.  Classical Physical Geodesy and  Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications) ZZZ VP D G

Q d ˙Q : `PQ

(2)

where G is the gravitational constant, Q the density, `PQ the distance between the mass element in Q and the computation point P , and d ˙ is the infinitesimal volume. We may assume the gravitational effect of the atmosphere to be negligible. Then the space outside of the Earth is empty and it holds r  rV D 0 (source free) apart from P r  rV D 0 (vorticity free). This corresponds to saying in Eq. 1 Vij D Vj i and i Vi i D 0. It leaves only five independent components in each point and offers important cross-checks between the measured components. If the Earth were a homogenous sphere, the off-diagonal terms would be zero and in a local triad {north, east, radial} one would find

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0

1 1 0 Vxx Vxy Vxz 1 0 0 GM  D Vij D @ Vyx Vyy Vyz A D 3 @ 0 1 0 A ; r 0 0 2 Vzx Vzy Vzz

(3)

where M is the mass of the spherical Earth. This simplification gives an idea about the involved orders of magnitude. At GOCE satellite altitude, it is Vzz D 2;740 E. This also implies that at a distance of 0.5 m from the spacecraft’s center of mass, the maximum gravitational acceleration is about 1:5  106 m/s2 . In an alternative interpretation, one can show that the Vij express the local geometric curvature structure of the gravitational field, i.e., 0

1 k1 t1 f1  D Vij D g @ t2 k2 f2 A ; f1 f2 H

(4)

where g is gravity, k1 and k2 express the local curvature of the level surfaces in north and east directions, t1 and t2 are the torsion, f1 and f2 the north and east components of the curvature of the plumb line, and H the mean curvature. For a derivation, refer to Marussi (1985). This interpretation of gravitational gradients in terms of gravitational geometry provides a natural bridge to Einstein’s general relativity, where gravitation is interpreted in terms of space-time curvature for it holds i R0j 0 D

1 Vij c2

(5)

i for the nine components of the tidal force tensor, which are components of Rkj l the Riemann curvature tensor with its indices running from 0, 1, 2, 3 (Ohanian and Ruffini 1994, p. 41; Moritz and Hofmann-Wellenhof 1993, Chap. 5). A third interpretation of gravitational gradiometry is in terms of tides. Sun and moon produce a tidal field on Earth. It is zero at the Earth’s center of mass and maximum at its surface. Analogously, the Earth is producing a tidal field in every Earth-orbiting satellite. At the center of mass of a satellite, the tidal acceleration is zero; i.e., the acceleration relative to the center of mass is zero; this leads to the terminology “zero-g.” The tidal acceleration increases with distance from the satellite’s center of mass like

ai D Vij dx j

(6)

with the measurable components of tidal acceleration ai and of relative position dx j , both taken in the instrument reference frame. Unlike sun and moon relative to the Earth, GOCE is always Earth pointing with its z-axis. This implies that the gradiometer measures permanently “high-tide” in z-direction and “low-tide”

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in x- and y-directions. The gradient components are deduced from taking the difference between the acceleration at two points along one gradiometer axis and symmetrically with respect to the satellite’s center of mass Vij D

ai .1/  ai .2/ : 2dx j

(7)

Remark 1. In addition to the tidal acceleration of the Earth, GOCE is measuring the direct and indirect tidal signal of sun and moon. This signal is much smaller, well-known and taken into account. If the gravitational attraction of the atmosphere is taken care of by an atmospheric model, the Earth’s outer field can be regarded source free and Laplace equation holds. It is common practice to solve Laplace equation in terms of spherical harmonic functions. The use of alternative base functions is discussed, for example, in Schreiner (1994) or Freeden et al. (1998). For a spherical surface, the solution of a Dirichlet boundary value problem yields the gravitational potential of the Earth in terms of normalized spherical harmonic functions Ynm .˝P / of degree n and order m as: V .P / D

X  R nC1 X n

rP

tnm Ynm .P / D Y t

(8)

m

with f˝P , rP g D fP , P , rP g the spherical coordinates of P and tnm the spherical harmonic coefficients. The gravitational tensor at P is then Vij .P / D

(

X X n

m

tnm @ij

R rP

)

nC1 Ynm .P /

D Y fij gt :

(9)

The spherical harmonic coefficients tnm are derived from the measured gradiometric components Vij by least squares adjustment. Remarks 2. In the case of GOCE gradiometry the situation is as follows. Above degree and order n D m D 220, noise starts to dominate signal. Thus, the series has to be truncated in some intelligent manner, minimizing aliasing and leakage effects. GOCE is covering our globe in 61 days with a dense pattern of ground tracks. Original planning of its mission lifetime assumed the completion of at least three times such 61-day cycles. The orbit inclination is 96:7ı . This leaves the two polar areas (opening angle 6:7ı ) free of observations, the so-called polar gaps. Various strategies have been

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suggested for minimization of the effect of the polar gaps on the determination of the global field (compare, e.g., Baur et al. 2009). Instead of dealing in an least-squares adjustment with the analysis of the individual tensor components one could consider the study of particular combinations. A very elegant approach is the use of the invariants of the gravitational tensor. They are independent of the orientation of the gradiometer triad. It is referred to Rummel (1986) and in particular to Baur and Grafarend (2006) and Baur (2007). While the first invariant cannot be used for gravity field analysis, it is the Laplace trace condition, the two others can be used. They are nonlinear and lead to an iterative adjustment. Let us assume for a moment the gradiometer components Vij would be given in an Earth-fixed spherical {north, east, radial}-triad. In that case, the tensor can be expanded in tensor spherical harmonics and decomposed into the irreducible radial, mixed normal-tangential, and pure tangential parts with the corresponding eigenvalues (Martinec 2003; Nutz 2002; Rummel 1997; Rummel and van Gelderen 1992; Schreiner 1994), for zz and xx C yy ; p.n C 1/.n C 2/ for fxx  yy ; 2xy g .n C 2/ n.n C 1/ for fxz ; yz g; .nC2/Š .n2/Š All eigenvalues are of the order of n2 . In Schreiner (1994) and Freeden et al. (1998) as well as  Geodetic Boundary Value Problem of this handbook, it is also shown that {xz , yz } and {xx  yy , – 2xy } are insensitive to degree zero, the latter combination also to degree one. In the case of GOCE, the above properties cannot be employed in a straightforward manner, because (1) not all components are of comparable precision and, more importantly, (2) the gradiometric components are measured in the instrument frame, which is following in its orientation the orbit and the attitude control commands. There exist various competing strategies for the actual determination of the field coefficients tnm , depending on whether the gradients are regarded in situ measurements on a geographical grid, along the orbit tracks, as a time series along the orbit or as Fourier-coefficients derived from the latter (compare, e.g., Brockmann et al. 2009; Migliaccio et al. 2004; Pail and Plank 2004; Stubenvoll et al. 2009). These methods take into account the noise characteristics of the components and their orientation in space. The principles of the methods of gravity modeling are described by Roland Pail in this handbook (Pail 2014). Here, the stochastic as well as the functional model are discussed. It also contains a summary of the methods of determination of angular rates from a joint analysis of star tracking and gradiometry applying Wiener filtering. Because of the polar gap gravity modeling combining gravitational gradiometry and GPS-based kinematic orbits leads to numerical instabilities and requires some form of regularization (Kusche and Klees 2002). A further step will be regional refinement by combination of GOCE with terrestrial data sets (e.g., Eicker et al. 2009; Stubenvoll et al. 2009 or Förste et al. 2011).

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Table 2 ESA release 4 GOCE gravity models DIR4 and TIM4 and their characteristics DIR4 260 01.11.09–01.08.12; 2.3 years(net)

TIM4 250 01.11.09–19.06.12; 2.2 years(net)

Vxx , Vyy , Vzz , Vxz 288 Mio. Obs. Band-pass filter – 2003–2012 GRGS RL02 (d/o 55), GFZ RL05 (d/o 56–180) LAGEOS1/2 (SLR) 1985–2010, 25 years Regularization Iterative spherical cap (d/o 260) based on GRACE/LAGEOS Kaula zero constraint(d/o >200)

Vxx , Vyy , Vzz , Vxz 279 Mio. Obs. ARMA filter per segment Short arc approach (d/o 130) –

Maximum D/O GOCE Data Volume Gravity Gradients Gradient Filter GOCE SST (GPS) GRACE SST(K-Band)

4

– Kaula zero constraint (near zonals and for d/o >180)

GOCE Status

GOCE was launched on March 17, 2009. After a commissioning and calibration phase, the first operational measurement cycle started on November 1, 2009. The mission was originally planned for 20 months only, because of its smooth performance it was ultimately extended until November 2013. All sensor systems worked well with the exception of a slightly higher than nominal noise level in the radial components Vzz and Vxz for reasons still not understood. Three major interruptions occurred: from February 12 to March 2 and from July 2 to September 25, 2010 due to problems with the processor units and from January 1 to January 21, 2011 because of a software problem related to the GPS receiver. The mission ended on November 11, 2013 with the reentry of the spacecraft over the South Atlantic Ocean near the Falkland Islands. By August 2013, altogether 2.3 years of data were available and entered the release 4 models. ESA Release 4 GOCE gravity models are summarized in Table 2. From August 1, 2012 on the orbit, altitude was lowered in four steps with at least one full measurement cycle in between by 9, 6, 5, and 11 km, i.e., altogether by 31km. The lowest altitude was attained on May 31, 2013 with 224 km. This was done in order to test whether the increased gravitational sensitivity can be exploited despite the increase of atmospheric drag at lower altitudes. Preliminary analysis shows an increase in spatial resolution. Release five is expected to be published in summer 2014. It will include the complete GOCE data set, including the data from the lower orbit altitudes.

5

Conclusions: GOCE Science Applications

Data exploitation of GOCE for science and application is far from being completed. We observe that the research fields are essentially those identified already in the

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Fig. 3 Vertical gravity gradient field of Antartica as measured by GOCE (units: milliEötvös D10-12s-2) and some prominent tectonic features; the GOCE polar gap is marked in gray

GOCE mission proposal (ESA 1999b). There is an exception. The expectation was that GOCE will be unable to sense temporal variations of gravity and geoid. This is true in general; however some big mass movements such as those associated with the big Japan earthquake seem to be detectable from the gradiometer data (Bouman et al. 2013). Solid earth physics. The gravity field as sensed by GOCE reflects the density distribution of the earth’s masses, primarily from topography, crust, and lithosphere and in an attenuated form from the upper and lower mantle down to the core. In ocean areas, gravity is well-known already from satellite altimetry. The situation is different in continental areas. Comparisons of GOCE gravity models with EGM2008 (e.g., Yi and Rummel 2014) reveal good agreement in well-surveyed parts of the earth such as North America, Europe, Australia, New Zealand, and Japan but rather poor agreement in large parts of South America, Africa, Himalaya, and South-East Asia. EGM2008 is a combination of GRACE satellite gravimetry with a global set of terrestrial gravity anomalies. Some of the regions with poor terrestrial gravity are of high geodynamic relevance. Currently several studies are underway, looking into the state of isostatic equilibrium and into the elastic thickness of the lithosphere (e.g., Sampietro et al. 2012 or van der Meijde et al. 2013). A special case is Antarctica where terrestrial gravity data is sparse. GRACE delivered gravity and geoid information in Antarctica but confined to rather large spatial scales. GOCE shows now the gravity field and with it tectonic processes hidden under an ice layer with a thickness of several kilometers (e.g., Ferraccioli et al. 2011). Figure 3 shows the vertical gravity gradient field of Antarctica as measured by GOCE and some prominent tectonic features. The polar gap is marked in gray.

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Height systems. Official heights are provided to the user either as gravity potential differences between terrain points or as metric heights derived from the potential differences such as orthometric or normal heights. The official height systems refer to a zero value at some adopted reference marker at a tide gauge, representing mean sea level there. However, mean sea level varies from location to location due to the variations in coastal oceanic conditions. The variations are not big, usually less than 2 m, but they are responsible for unknown height off-sets between the various worldwide official height systems. GOCE provides the best possible geoid surface (ideally refined locally by shorter scales from some regional geoid computation). It represents the theoretical sea surface at rest and can be introduced as an ideal worldwide height reference. It also permits determination of the height off-sets between the various height systems, not detectable in the past. This process of height unification is of great value for mapping, large civil constructions and sea level research. As a demonstration of the value of the GOCE geoid for height determination, Woodworth et al. (2012) revealed the bias of the North American height system which is based on classical spirit leveling. In the near future, GPS positions can be translated to heights above the GOCE geoid yielding globally consistent and physically meaningful heights worldwide. Oceanography. In his classical textbook “Atmosphere-Ocean Dynamics”, A.E. Gill (1982) writes on page 46: “If the sea were at rest, its surface would coincide with the geopotential surface.” This geopotential surface is the geoid and with GOCE its shape is determined with an accuracy of 2–3 cm. In reality the sea is not at rest, it is in motion as driven by winds, atmospheric pressure differences, and tides and as a consequence the ocean surface deviates from a geopotential surface by up to 1 or 2 m at the major ocean currents; the deviation is denoted dynamic ocean topography. The actual sea surface is measured from space by satellite altimetry. More than 20 years of altimetry yield models of the mean sea surface (MSS) accurate to a few centimeters. Subtraction of the GOCE geoid from MSS gives the geodetic mean dynamic ocean topography (MDT). It is maintained by the balance of the pressure differences due to the MDT and Coriolis acceleration.

Fig. 4 Geostropic velocities (in cm/s) derived from drifter measurements (left) and from geodetic mean dynamic ocean topography (right)

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Its slope is proportional to the geostrophic velocities of ocean circulation. GOCE together with altimetry give MDT and the geostrophic velocity field without the use of any oceanographic in situ data. Geodetic MDT and geostrophic velocities represent a new type of input quantity for numerical ocean circulation models and help to improve ocean mass and heat transport estimates, e.g., in the area of the Weddell Sea in the Southern ocean, one of the tipping points of our climate system. Figure 4 shows the magnitude of the geostrophic velocities in the North Atlantic based on the Danish MSS model DTU-10 (right) and geostrophic velocities derived from drifter data (left). We observe higher signal strength of the geodetic estimate. A large number of ocean studies based on GOCE was published already. Examples are Bingham et al. (2011), Janji´c et al. (2012) or Le Traon et al. (2011). Atmosphere. GOCE was kept drag-free in flight direction using ion thrusters. The feedback signal for drag-free control was the measured common-mode accelerations of the six accelerometers of the gradiometer. They are a measure of the nongravitational forces acting on the GOCE spacecraft and open the possibility for studies of atmospheric density and winds (Doornbos et al. 2013). At GOCE altitude, no other data source is available.

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Marussi A (1985) Intrinsic geodesy. Springer, Berlin Maximenko N, Niiler P, Rio M-H, Melnichenko O, Centurioni L, Chambers D, Zlotnicki V, Galperin B (2009) Mean dynamic topography of the ocean derived from satellite and drifting buoy data using three different techniques. J Atmos Ocean Technol 26:1910–1919 Migliaccio F, Reguzzoni M, Sansò F (2004) Space-wise approach to satellite gravity field determination in the presence of colored noise. J Geod 78:304–313 Misner CW, Thorne KS, Wheeler JA (1970) Gravitation. Freeman, San Francisco Moritz H, Hofmann-Wellenhof B (1993) Geometry, relativity, geodesy. Wichmann, Karlsruhe Nutz H (2002) A unified setup of gravitational observables. Dissertation. Shaker Verlag, Aachen Ohanian HC, Ruffini R (1994) Gravitation and spacetime. Norton & Comp., New York Pail R (2014) It is all about statistics: global gravity field modelling from GOCE and complementary data. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics. Springer, Heidelberg Pail R, Plank R (2004) GOCE gravity field processing strategy. Stud Geophys Geod 48:289–309 Rummel R (1986) Satellite gradiometry. In: Sünkel H (ed) Mathematical and numerical techniques in physical geodesy. Lecture notes in earth sciences, vol 7. Springer, Berlin, pp 317–363. ISBN (Print):978-3-540-16809-6, doi:10.1007/BFb0010135 Rummel R (1997) Spherical spectral properties of the earth’s gravitational potential and its first and second-derivatives. In: Sansò F, Rummel R (eds) Geodetic boundary value problems in view of the one centimeter geoid. Lecture notes in earth sciences, vol 65. Springer, Berlin, pp 359–404. ISBN:3-540-62636-0 Rummel R, van Gelderen M (1992) Spectral analysis of the full gravity tensor. Geophys J Int 111:159–169 Rummel R, Balmino G, Johannessen J, Visser P, Woodworth P (2002) Dedicated gravity field missions-principles and aims. J Geodyn 33/1–2:3–20 Sampietro D, Reguzzoni M, Braitenberg C (2012) The GOCE estimated Moho beneath the Tibetan Plateau and Himalaya. In: C Rizos, P Wills (eds) Earth on the edge: science for a substainable planet, International Association of Geodesy Symposia, vol 139. Springer, pp 391–397. doi:10.1007/978-3-642-37222-3_52 Schreiner M (1994) Tensor spherical harmonics and their application in satellite gradiometry. Dissertation, Universität Kaiserslautern Stubenvoll R, Förste Ch, Abrikosov O, Kusche J (2009) GOCE and its use for a high-resolution global gravity combination model. In: Flechtner F, Mandea M, Gruber Th, Rothacher M, Wickert J, Güntner A, Schöne T (eds) System earth via geodetic-geophysical space techniques. Springer, Berlin Svehla D, Rothacher M (2004) Kinematic precise orbit determination for gravity field determination. In: Sansò F (ed) The proceedings of the international association of geodesy: a window on the future of geodesy. Springer, Berlin, pp 181–188 Van der Meijde M, Julia J, Assumpcao M(2013) Gravity derived Moho for South America. Tectonophysics 609:456–467 Wells WC (ed) (1984) Spaceborne gravity gradiometers. NASA conference publication, vol 2305, Greenbelt Woodworth PL, Hughes CW, Bingham RJ, Gruber T(2012) Towards worthwide height system unification using ocean information., J Geodetic Sci 2(4):302–318. doi:10.2478/v10156-0120004-8 Wunsch C, Gaposchkin EM (1980) On using satellite altimetry to determine the general circulation of the oceans with application to geoid improvement. Rev Geophys 18:725–745 Yi W, Rummel R (2014) A comparison of GOCE gravitational models with EGM2008. J Geodyn 73:14–22

Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation Nils Olsen, Gauthier Hulot, and Terence J. Sabaka

Contents 1 Sources of the Earth’s Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Internal Field Sources: Core and Crust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ionospheric, Magnetospheric, and Earth-Induced Field Contributions . . . . . . . . . . . 2 Modern Geomagnetic Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of Magnetic Elements and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ground Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Satellite Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Making the Best of the Data to Investigate the Various Field Contributions: Geomagnetic Field Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The geomagnetic field one can measure at the Earth’s surface or on board satellites is the sum of contributions from many different sources. These sources have different physical origins and can be found both below (in the form of electrical currents and magnetized material) and above (only in the form of electrical currents) the Earth’s surface. Each source happens to produce a contribution

N. Olsen () DTU Space, Technical University of Denmark, Kgs. Lyngby, Denmark e-mail: [email protected] G. Hulot Equipe de Géomagnétisme, Institut de Physique du Globe de Paris, Sorbonne Paris Cité, Université Paris Diderot, Paris, France e-mail: [email protected] T.J. Sabaka Planetary Geodynamics Laboratory, Code 698, NASA Goddard Space Flight Center, Greenbelt, MD, USA e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_5

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with rather specific spatio-temporal properties. This fortunate situation is what makes the identification and investigation of the contribution of each source possible, provided appropriate observational data sets are available and analyzed in an adequate way to produce the so-called geomagnetic field models. Here we provide a general overview of the various sources that contribute to the observed geomagnetic field, and of the modern data that enable their investigation via such procedures. The Earth has a large and complicated magnetic field, a major part of which is produced by a self-sustaining dynamo operating in the fluid outer core. What is measured at or near the surface of the Earth, however, is the superposition of the core field and of additional fields caused by magnetized rocks in the Earth’s crust, by electric currents flowing in the ionosphere, magnetosphere and oceans, and by currents induced in the Earth by the time-varying external fields. The sophisticated separation of these various fields and the accurate determination of their spatial and temporal structure based on magnetic field observations is a significant challenge, which requires advanced modeling techniques (see e.g., Hulot et al. 2007). These techniques rely on a number of mathematical properties which we review in the accompanying chapter by Sabaka et al. (2010), entitled  Mathematical Properties Relevant to Geomagnetic Field Modeling. But as many of those properties have been derived by relying on assumptions motivated by the nature of the various sources of the Earth’s magnetic field and of the available observations, it is important that a general overview of those sources and observations be given. This is precisely the purpose of the present chapter. It will first describe the various sources that contribute to the Earth’s magnetic field (Sect. 1) and next discuss the observations currently available to investigate them (Sect. 2). Special emphasis is given on data collected by satellites, since these are extensively used for modeling the present magnetic field. We will conclude with a few words with respect to the way the fields those sources produce can be identified and investigated, thanks to geomagnetic field modeling.

1

Sources of the Earth’s Magnetic Field

Several sources contribute to the magnetic field that is measured at or above the surface; the most important ones are sketched in Fig. 1. The main part of the field is due to electrical currents in the Earth’s fluid outer core at depths larger than 2,900 km; this is the so-called core field. Its strength at the Earth’s surface varies from less than 30,000 nT near the equator to about 60,000 nT near the poles, which makes the core field responsible for more than 95 % of the observed field at ground. Magnetized material in the crust (the uppermost few kilometers of Earth) causes the crustal field; it is relatively weak and accounts on average only for a few percent of the observed field at ground. Core and crustal fields together make the internal field (since their sources are internal to the Earth’s surface). External magnetic field contributions are caused by electric currents in the ionosphere (at altitudes 90–1,000 km) and magnetosphere (at altitudes of several Earth radii). On average,

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Fig. 1 Sketch of the various sources contributing to the near-Earth magnetic field

their contribution is also relatively weak – a few percent of the total field at ground during geomagnetic quiet conditions. However, if not properly considered, they disturb the precise determination of the internal field. It is therefore of crucial importance to account for external field (by data selection, data correction, and/or field co-estimation) in order to obtain reliable models of the internal fields. Finally, electric currents induced in the Earth’s crust and mantle by the time-varying fields of external origin, and the movement of electrically conducting seawater, cause magnetic field contributions that are of internal origin like the core and crustal field; however, typically only core and crustal field is meant when speaking about “internal sources.” A useful way of characterizing the spatial behavior of the geomagnetic field is to make use of the concept of spatial power spectra (e.g., Lowes 1996 and section 4 of Sabaka et al. 2010). Figure 2 shows the spectrum of the field of internal origin, often referred to as the Lowes-Mauersberger spectrum, which gives the mean square magnetic field at Earth’s surface due to contributions with horizontal wavelength n corresponding to spherical harmonic degree n. The spectrum of the observed magnetic field (based on a combination of the recent field models derived by Olsen et al. 2009 and Maus et al. 2008) is shown by black dots, while theoretical spectra describing core, resp. crustal, field spectra (Voorhies et al. 2002) are shown as blue,

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Fig. 2 Spatial power spectrum of the geomagnetic field at Earth’s surface. Black dots represent the spectrum of a recent field model (Maus et al. 2008; Olsen et al. 2009). Also shown are theoretical spectra (Voorhies et al. 2002) for the core (blue) and crustal (magenta) part of the field, as well as their superposition (red curve)

resp. magenta, curves. Each of these two theoretical spectra has two free parameters which have been fitted to the observed spectra; their sum (red curve) provides a remarkable good fit to the observed spectrum. There is a sharp “knee” at about degree n D 14 which indicates that contributions from the core field are dominant at large scales (n < 14) while those of the crustal field dominate for the smaller scales (n > 14). We now proceed with a more detailed overview of the various field sources and their characteristics.

1.1

Internal Field Sources: Core and Crust

Core Field Although the Earth’s magnetic field has been known for at least several thousands of years (see e.g., Merrill et al. 1998), the nature of its sources has eluded scientific understanding for a very long time. It was not until the nineteenth century that its main source was finally proven to be internal to the Earth. We now know that this

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main source is most likely a self-sustaining dynamo within the Earth’s core (see e.g., Roberts 2007; Wicht et al. 2010). This dynamo is the result of the fact that the liquid electrically conducting outer core (consisting of a Fe-Ni alloy) is cooling down and convecting vigorously enough to maintain electrical currents and a magnetic field. The basic process is one whereby the convective motion of the conducting fluid within the magnetic field induces electromotive forces which maintain electrical currents, and therefore also the magnetic field, against Ohmic dissipation. The Earth’s core dynamo has several specific features. First, the core contains a solid, also conducting, inner core. This inner core is thought to be the solidified part of the core (Jacobs 1953), the growth of which is the result of the cooling of the core. Because the Fe-Ni alloy that makes the core must in fact contain additional light elements, the solidification of the inner core releases so-called compositional buoyancy at the inner core boundary. This buoyancy will add up to the thermal buoyancy and is thought to be a major source of energy for the convection (see e.g., Nimmo 2007). Second, the core, together with the whole Earth, is rotating fast, at a rate of one rotation per day. This leads Coriolis forces to play a major role in organizing the convection, and the way the dynamo works. In particular, spherical symmetry is dynamically broken, and a preferential axial (North-South) dipole field can be produced (see e.g., Gubbins and Zhang 1993). At any given instant, however, the field produced cannot be too simple (a requirement which has been formalized in terms of anti-dynamo theorems, starting with the best-known Cowling theorem (Cowling 1957)). In particular, no fully axisymmetric field can be produced by a dynamo. In effect, all dynamo numerical simulations run so far with conditions approaching that of the core dynamo produce quite complex fields in addition to a dominant axial dipole. This complexity does not only affect the so-called toroidal component of the magnetic field which remains for the most part within the fluid core (such toroidal components are nonzero only where their sources lie (cf. section 3 of Sabaka et al. 2010), and the poorly conducting mantle forces those to essentially remain within the core). It also affects the poloidal component of the field which can escape the core by taking the form of a potential field (such poloidal components can indeed escape their source region in the form of a Laplacian potential field, see again Sabaka et al. (2010)), reach the Earth’s surface and make the core field we can observe. The core field thus has a rich spatial spectrum beyond a dominant axial dipole component. It also has a rich temporal spectrum (with typical time scales from decades to centuries (see e.g., Hulot and Le Mouël 1994)) directly testifying for the turmoil of the poloidal field produced by the dynamo at the core surface. However, what is observable at and above the Earth’s surface is only part of the core field that reaches it. Spatially, its small-scale contributions are masked by the crustal field, as shown in Fig. 2, and therefore only its largest scales (corresponding to spherical harmonic degrees smaller than 14) can be recovered. And temporally, the high frequency part of the core field (corresponding to periods shorter than a few months) is screened by the finite conductivity of the mantle (see e.g., Alexandrescu et al. 1999). This puts severe limitations on the possibility to

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recover the spatiotemporal structure of the core field, regardless of the quality of the magnetic field observations. More information on our present knowledge of the core field based on recent data can be found in e.g., Hulot et al. (2007), Jackson and Finlay (2007), and Finlay et al. (2010).

Crustal Field The material that makes the mantle and crust contains substantial amounts of magnetic minerals. Those minerals can become magnetized in the presence of an applied magnetic field. To produce any significant magnetic signals, this magnetism must however be of ferromagnetic type, which also requires the material to be at a low enough temperature (below the so-called Curie temperature of the minerals, see e.g., Dunlop and Özdemir 2007). Those conditions are only met within the Earth’s upper layers, above the so-called Curie isotherm. Its depth can vary between zero (such as at mid-oceanic ridges) and several tens of kilometers, with a typical value on the order of 20 km in continental regions. Magnetized rocks can thus only be found in those layers. Magnetized rocks essentially carry two types of magnetization, induced magnetization and remanent magnetization. Induced magnetization is one that is proportional, both in strength and direction, to the ambient field within which the rock is embedded. The ability of such a rock to acquire this magnetization is a function of the nature and proportion of the magnetic minerals it contains. It is measured in terms of a proportionality factor known as the magnetic susceptibility. Were the core field (or more correctly the local field experienced by the rock) to disappear, this induced magnetization would also disappear. Then, only the second type of magnetization, remanent magnetization, would remain. This remanent magnetization could have been acquired by the rock in many different ways (see e.g., Dunlop and Özdemir 2007). For instance at times of deposition for a sedimentary rock, or via chemical transformation, if the rock has been chemically altered. The most ubiquitous process, however, which also usually leads to the strongest remanent magnetization, is thermal. It is the way igneous and metamorphic rocks acquire their remanent magnetization when they cool down below their Curie temperature. The rock becomes magnetized in proportion, both in strength and direction, to the ambient magnetic field that the rocks experiences at the time it cools down (the proportionality factor being again a function of the magnetic minerals contained in the rock). Remanent magnetization from a properly sampled rock thus can provide information about the ancient core field (see e.g., Hulot et al. 2010). There is no way one can identify the signature of the present core field without taking the crustal field into account (in fact, modern ways of modeling the core field from satellite data often also involves modeling the crustal field). It is therefore important to also mention some of the most important spatio-temporal characteristics of the field produced by magnetized sources. It should first be recalled that not all magnetized sources will produce observable fields at the Earth’s surface. In particular, if the upper layers of the Earth consisted in a spherical shell

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of uniform magnetic properties magnetized within the core field at a given instant, they would produce no observable field at the Earth’s surface. This is known as Runcorn’s theorem (Runcorn 1975), an important implication of which is that the magnetic field observed at the Earth’s surface is not sensitive to the induced magnetization due to the average susceptibility of a spherical shell best describing the upper magnetic layers of the Earth. It will only sense the departure of those layers from sphericity (see e.g., Lesur and Jackson 2000), either because of the Earth’s flattening, because of the variable depth of the Curie isotherm, or because of the contrasts in magnetization due to the variable nature and susceptibility of rocks within those layers (although even such contrasts can sometimes also fail to produce observable fields (Maus and Haak 2003)). Another issue of importance is that of the relative contributions of induced and remanent magnetization. Induced magnetization is most likely the main source of large scale magnetization, while remanent magnetization plays a significant role only at regional (especially in oceans) and local scales (e.g., Purucker et al. 2002; Hemant and Maus 2005). Finally, it is important that we also briefly mention the poorly known issue of possible temporal changes in crustal magnetization on a human time scale. At a local scale, any dynamic process that can alter the magnetic properties of the rocks, or change the geological setting (such as an active volcano), would produce such changes. On a planetary scale, by contrast, significant changes can only occur in the induced magnetization because of the slowly time-varying core field, as has been recently demonstrated by Hulot et al. (2009) and Thebault et al. (2009). Recent reviews of the crustal field are given by Purucker and Whaler (2007) and Thébault et al. (2010).

1.2

Ionospheric, Magnetospheric, and Earth-Induced Field Contributions

Ionospheric and magnetospheric currents (which produce the field of external origin), as well as Earth-induced currents (which produce externally induced fields), contribute in a nonnegligible way to the observed magnetic field, both at ground and at satellite altitude. It is therefore important to consider them in order to properly identify and separate their signal from that of the field of internal origin.

Ionospheric Contributions Geomagnetic daily variations at nonpolar latitudes (known as Sq variations) are caused by diurnal wind systems in the upper atmosphere: Heating at the dayside and cooling at the nightside generates tidal winds which drive ionospheric plasma against the core field, inducing electric fields and currents in the ionospheric Eregion dynamo region between 90 and 150 km altitude (Richmond 1989; Campbell 1989; Olsen 1997b). The currents are concentrated at an altitude of about 110– 115 km and hence can be represented by a sheet current at that altitude, (cf. section 3.4 and 3.5 of Sabaka et al. 2010). They remain relatively fixed with

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respect to the Earth-Sun line and produce regular daily variations which are directly seen in the magnetograms of magnetically quiet days. On magnetically disturbed days there is an additional variation which includes superimposed magnetic storm signatures of magnetospheric and high-latitude ionospheric origin. Typical peakto-peak Sq amplitudes at middle latitudes are 20–50 nT; amplitudes during solar maximum are about twice as large as those during solar minimum. Sq variations are restricted to the dayside (i.e., sunlit) hemisphere, and thus depend mainly on local time. Selecting data from the nightside when deriving models of the internal field is therefore useful to minimize field contributions from the nonpolar ionospheric E-region. Because the geomagnetic field is strictly horizontal at the dip equator, there is about a fivefold enhancement of the effective (Hall) conductivity in the ionospheric dynamo region, which results in about a fivefold enhanced eastward current, called the Equatorial Electrojet (EEJ), flowing along the dayside dip equator (Rastogi 1989). Its latitudinal width is about 6ı –8ı . In addition, auroral electrojets (AEJ) flow in the auroral belts (near ˙(65ı –70ı ) magnetic latitude) and vary widely in amplitude with different levels of magnetic activity from a few tens nT during quiet periods to several thousand nT during major magnetic storms. As a general rule, ionospheric fields at polar latitudes are present even at magnetically quiet times and on the nightside (i.e., dark) hemisphere, which makes it difficult to avoid their contribution by data selection. Electric currents at altitude above 120 km, i.e., in the so-called ionospheric F region (up to 1,000 km altitude), cause magnetic fields that are detectable at satellite altitude as nonpotential (e.g., toroidal) magnetic fields (Olsen 1997a; Richmond 2002; Maus and Lühr 2006). Their contributions in nonpolar regions are also important during local nighttime when the E-region conductivity vanishes and therefore contributions from Sq and the Equatorial Electrojet are absent.

Magnetospheric Contributions The field originating in Earth’s magnetosphere is due primarily to the ring-current and to currents on the magnetopause and in the magnetotail (Kivelson and Russell 1995). Currents flowing on the outer boundary of the magnetospheric cavity, the magnetopause currents, cancel the Earth’s field outside and distend the field within the cavity. This produces an elongate tail in the antisolar direction within which socalled neutral-sheet currents are established in the equatorial plane. Interaction of these currents with the radiation belts near the Earth produces a ring-current in the dipole equatorial plane which partially encircles the Earth, but achieves closure via field-aligned currents (FAC) (currents which flow along core field lines) into and out of the ionosphere. These resulting fields have magnitudes on the order of 20– 30 nT near the Earth during magnetically quiet periods, but can increase to several hundreds of nT during disturbed times. In polar regions (poleward of, say, ˙65ı dipole latitude), the auroral ionosphere and magnetosphere are coupled by field-aligned currents. The fields from these FAC have magnitudes that vary with the magnetic disturbance level. However, they are

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always present, on the order of 30–100 nT during quiet periods and up to several thousand nT during substorms. There are also currents which couple the ionospheric Sq current systems in the two hemispheres that flow, at least in part, along magnetic field lines. The associated magnetic fields are generally 10 nT or less. Finally, there exists a meridional current system which is connected to the EEJ with upward directed currents at the dip equator and field-aligned downward directed currents at low latitudes. These currents result in magnetic fields of about a few tens of nT at 400 km altitude.

Induction in the Solid Earth and the Oceans Time-varying external fields produce secondary, induced, currents in the oceans and the Earth’s interior; this contribution is what we refer to as externally induced fields, which are the topic of electromagnetic induction studies (see Parkinson and Hutton 1989; Constable 2007; Kuvshinov 2012). In addition, the motion of electrically conducting seawater through the core field, via a process referred to as motional induced induction, also produces secondary currents (e.g., Tyler et al. 2003; Kuvshinov and Olsen 2005; Maus 2007b). The oceans thus contribute twofold to the observed magnetic field: by secondary currents induced by primary current systems in the ionosphere and magnetosphere; and by motion induced currents due to the movement of seawater, for instance by tides. The amplitude of induced contributions generally decreases with the period. As an example, about one third of the observed daily Sq variation in the horizontal components is of induced origin (Schmucker 1985). But induction effects also depend on the scale of the source (i.e., the ionospheric and magnetospheric current systems); as a result, the induced contribution due to the daily variation of the Equatorial Electrojet is, for instance, much smaller than the above mentioned one third, typical of the large-scale Sq currents (Olsen 2007b).

2

Modern Geomagnetic Field Data

2.1

Definition of Magnetic Elements and Coordinates

Measurements of the geomagnetic field taken at ground or in space form the basis for modeling the Earth’s magnetic field. Observations taken at the Earth’s surface are typically given in a local topocentric (geodetic) coordinate system (i.e., relative to a reference ellipsoid as approximation for the geoid). The magnetic elements X; Y; Z are the components of the field vector B in an orthogonal right-handed coordinate system, the axis of which are pointing towards geographic North, geographic East, and vertically down, as shown in Fig. 3. Derived magnetic elements are: the angle between geographic North and the (horizontal) direction in which a compass needle is pointing, denoted as declination D D arctan Y =X ; the angle between the local horizontal plane and the field vector,

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Fig. 3 The magnetic elements in the local topocentric coordinate system, seen from North-East

p 2 2 denoted as inclination p I D arctan Z=H ; horizontal intensity H D X C Y ; and 2 2 2 total intensity F D X C Y C Z . In contrast to magnetic observations taken at or near ground, satellite data are typically provided in the geocentric coordinate system as spherical components Br ; B ; B' where r, , ' are radius, colatitude, and longitude respectively. Equations for transforming between geodetic components X; Y; Z and geocentric components Br ; B ; B' can be found in, e.g., section 5.02.2.1.1 of Hulot et al. (2007). The distribution in space of the observations at a given time determines the spatial resolution to which the field can be determined for that time. Internal sources are often fixed with respect to the Earth (magnetic fields due to induced currents in the Earth’s interior are an exception) and thus follow its rotation. Internal sources are therefore best described in an Earth-Centered-Earth-Fixed (ECEF) coordinate frame like that given by the geocentric coordinates r, , '. In contrast, many external fields are fixed with respect to the position of the Sun, and therefore the use of a coordinate frame that follows the (apparent) movement of the Sun is advantageous. Solar magnetospheric (SM) coordinates for describing near magnetospheric currents like the ring-current, and geocentric solar magnetospheric (GSM) coordinates for describing far magnetospheric current systems like the tail currents have turned out to be useful when determining models of Earth’s magnetic field (Maus and Lühr 2005; Olsen et al. 2006, 2009, 2010b).

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Presently about 150 geomagnetic observatories monitor the time changes of the Earth’s magnetic field. Their global distribution, shown in the lower part of Fig. 4,

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is very uneven, with large uncovered areas especially over the oceans. Red symbols indicate sites that provide data (regardless of the observation time instant and the duration of the time series), while the yellow dots show observatories that have provided hourly mean values for the recent years. These observatories provide data of different temporal resolution which are distributed through the World-DateCenter system (e.g., http://www.ngdc.noaa.gov/wdc, http://wdc.kugi.kyotou.ac.jp, http://www.wdc.bgs.ac.uk). Traditionally, annual mean values have been used for deriving field models, but the availability of hourly mean values (or even 1 min values) in digital form for the recent years allow for a better characterization of external field variations. The upper part of Fig. 4 shows the distribution in time of observatory data of various sampling rate. International campaigns, like the Göttingen Magnetic Union, the 1st and 2nd International Polar Year (IPY), the International Geophysical Year (IGY/C), the International Quiet Solar Year (IQSY), the International Magnetospheric Study (IMS), and the preparation of the Ørsted satellite mission have stimulated observatory data processing and the establishment of new observatories. Geomagnetic observatories aim at measuring the magnetic field in the geodetic reference frame with an absolute accuracy of 1 nT (Jankowski and Sucksdorff 1996). However, it is presently not possible to take advantage of that measurement accuracy due to the (unknown) contribution from nearby crustal sources. When using observatory data for field modeling it is therefore common practice to either use first time differences of the observations (thereby eliminating the static crustal field contribution) or to co-estimate together with the field model an “observatory bias” for each site and element, following a procedure introduced by Langel et al. (1982). A joint analysis of observatory and satellite data allows one to determine these observatory biases. The need for knowledge of the absolute baseline is therefore less important during periods for which satellite data are available. Recognizing this will simplify the observation practice, especially for ocean-bottom magnetometers for which the exact determination of true north is very difficult and expensive. In addition to geomagnetic observatories (which monitor the time changes of the geomagnetic field at a given location), magnetic “repeat stations” are sites where high-quality magnetic measurements are taken every few years for a couple hours or even days (Newitt et al. 1996; Turner et al. 2007). The main purpose of repeat stations is to measure the time changes of the core field (secular variation); they offer better spatial resolution than observatory data but do not provide continuous time series.

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Satellite Data

The possibility to measure the Earth’s magnetic field from space has revolutionized geomagnetic field modeling. Magnetic observations taken by low-earth-orbiting (LEO) satellites at altitudes below 1,000 km form the basis of recent models of the geomagnetic field.

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There are several advantages of using satellite data for field modeling: 1. Satellites sample the magnetic field over the entire Earth (apart from the polar gap, a region around the geographic poles that is left unsampled if the satellite orbit is not perfectly polar). 2. Measuring the magnetic field from an altitude of 400 km or so corresponds roughly to averaging over an area of this dimension. Thus the effect of local heterogeneities, for instance caused by local crustal magnetization, is reduced. 3. The data are obtained over different regions with the same instrumentation, which helps to reduce spurious effects. There are, however, some points to consider when using satellite data instead of ground based data: 1. Since the satellite moves (with a velocity of about 8 km/s at 400 km altitude) it is not possible to decide whether an observed magnetic field variation is due to a temporal or spatial change of the field. Thus, there is risk for time-space aliasing. 2. It is necessary to measure the magnetic field with high accuracy – not only regarding resolution, but also regarding orientation and absolute values. 3. Due to the Earth’s rotation, the satellite revisits a specific region after about 1 day.1 Hence the magnetic field in a selected region is modeled from time series with a sampling rate of 1 day. However, since the measurements were not lowpass filtered before “resampling,” aliasing may occur. 4. Satellites usually acquire data not at one fixed altitude, but over a range of altitudes. The decay of altitude through mission lifetime often leads to time series that are unevenly distributed in altitude. 5. Finally, the satellite moves through an electric plasma, and the existence of electric currents at satellite altitude does, in principle, not allow to describe the observed field as the gradient of a Laplacian potential. An overview of previous and present satellites that have been used for geomagnetic field modeling is given in Table 1 (see also Table 3.3 of Langel and Hinze 1998). The quality of the magnetic field measurements is rather different for the listed satellites, and before the launch of the Ørsted satellite in 1999, the POGO satellite series (Cain 2007) that flew in the second half of the 1960s, and Magsat (Purucker 2007), which flew for 6 months around 1980, were the only high-precision magnetic satellites. A timeline of high-precision missions is shown in Fig. 5. After a gap of almost 20 years with no high-precision satellites in orbit, the launch of the Danish

1 Actually the satellite revisits that region already after about 12 h, but this will be for a different local time. Because of external field contributions – which heavily depend on local time – it is safer to rely on data taken at similar local time conditions, which results in the above stated sampling recurrence of 24 h.

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Table 1 Satellite missions of relevance for geomagnetic field modeling

87ı 86ı 82ı 97ı 90ı 90ı 90ı

413–1,510 412–908 397–1,098 325–550 568–23,290 309–1,012 639–769

6 6 6 6 ? 30(F)/100 ?

UARS Ørsted CHAMP SAC-C Swarm

1991–1994 1999– 2000–2010 2001–2004 2013–

57ı 97ı 87ı 97ı 88ı /87ı

560 650–850 250–450 698–705 530/ < 450

? 4 3 4 2

Scalar only Scalar only Scalar only Vector and scalar Vector (spinning) Low accuracy vector Low accuracy vector, timing problems Vector (spinning) Scalar and vector Scalar and vector Scalar only Scalar and vector

Swarm

Vector and scalar

Ørsted

?

3-sat constellation, vector and scalar

1965–1967 1967–1969 1969–1971 1979–1980 1981–1991 1981–1983 1990–1993

Vector and scalar

Remarks Scalar only

Ørsted, CHAMP, SAC-C

Altitude/km Accuracy/nT 261–488 22

Magsat

Inclination i 50ı

Scalar only

Years 1964

POGO (OGO-2, -4, -6)

Satellite Cosmos 49 POGO OGO-2 OGO-4 OGO-6 Magsat DE-1 DE-2 POGS

CHAMP SAC-C 1965 1970 1975

1980 1985 1990 1995 2000 2005 2010 2015 2020

Fig. 5 Distribution of high-precision satellite missions in time

Ørsted satellite (Olsen 2007a) in February 1999 marked the beginning of a new epoch for exploring the Earth’s magnetic field from space. Ørsted was followed by the German CHAMP satellite (Maus 2007a) and the US/Argentinian/Danish SAC-C satellite, launched in July and November 2000, respectively. All three satellites carry essentially the same instrumentation and provide high-quality and high-resolution magnetic field observations from space. They sense the various internal and external

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Fig. 6 Left: The path of a satellite at inclination i in orbit around the Earth. Right: Ground track of 24 h of the Ørsted satellite on January 2, 2001 (yellow curve). The satellite starts at –57ı N, 72ı E at 00 UT, moves northward on the morning side of the Earth, and crosses the Equator at 58ı E (yellow arrow). After crossing the polar cap it moves southward on the evening side and crosses the equator at 226ı E (yellow open arrow) 50 min after the first equator crossing. The next Equator crossing (after additional 50 min) is at 33ı E (red arrow), 24ı westward of the first crossing 100 min earlier, while moving again northward

field contributions differently, due to their different altitudes and drift rates through local time. A closer look at the characteristics of satellite data sampling is helpful. A satellite moves around Earth in elliptical orbits. However, ellipticity of the orbit is small for many of the satellites used for field modeling, and for illustration purposes we will concentrate on circular orbits. As sketched on the left side of Fig. 6, orbit inclination i is the angle between the orbit plane and the equatorial plane. A perfectly polar orbit implies i D 90ı , but for practical reasons most satellite orbits have inclinations that are different from 90ı . This results in “polar gaps,” which are regions around the geographic poles that are left unsampled. The right part of Fig. 6 shows the ground track of 1 day (January 2, 2001) of Ørsted satellite data. It is obvious that the coverage in latitude and longitude provided even by only a few days of satellites data is much better than that of the present ground based observatory network (cf. Fig. 4). The polar gaps, the regions of half-angle j90ı – i j around the geographic poles, are obvious when looking at the orbits in a polar view in an Earth-fixed coordinate system, as done in the left part of Fig. 7 for the Ørsted (red), resp. CHAMP (blue), satellite tracks of January 2, 2001. The polar gaps are larger for Ørsted (inclination i D 97ı ) compared to CHAMP (i D 87ı ), as confirmed by the figure. As mentioned before, internal magnetic field sources are often fixed (or slowly changing, in the case of the core field) with respect to the Earth while most external fields have relatively fixed geometries with respect to the Sun. A good description of the various field contributions requires a good sampling of the data in the respective coordinate systems.

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Fig. 7 Left: Ground track of 1 day (January 2, 2001) of the Ørsted (red) and CHAMP (blue) satellites in dependence on geographic coordinates. Right: Orbit in the solar-magnetospheric (SM) reference frame

Good coverage in latitude and longitude is essential for modeling the internal field. There are, however, pitfalls due to peculiarities of the satellite orbits, which may result in less optimal sampling. The top panel of Fig. 8 shows the longitude of the ascending node (the equator crossing of the satellite going from south to north) of the Ørsted (left) and CHAMP (right) satellite orbits. Depending on orbit altitude (shown in the middle part of the figure) there are periods with pronounced “revisiting patterns”: In June and July 2003, the CHAMP satellite samples, for instance, only the field near the equator at longitudes ' D 7:6ı ; 19:2ı ; 30:8ı : : :356:0ı . This longitudinal sampling of ' D 11:6ı hardly allows to resolve features of the field of spatial scale corresponding to spherical harmonic degrees above n D 15. Another issue that has to be considered when deriving field models from satellite data concerns satellite altitude. The middle panel of Fig. 8 shows the altitude evolvement for the Ørsted and CHAMP satellites. Various altitude maneuvers are the reason for the sudden increase of altitude of CHAMP. At lower altitudes the magnetic signal of small-scale features of the internal magnetic field (corresponding to higher spherical harmonic degrees) are relatively more amplified compared to large-scale features (represented by low degree spherical harmonics). The crustal field signal measured by a satellite is thus normally stronger towards the end of the mission lifetime due to the lower altitude. However, if the crustal field is not properly accounted for, the decreasing altitude may hamper the determination of the core field time changes. Good sampling in the Earth-Fixed coordinate system, which is essential for determining the internal magnetic field, can, at least in principle, be obtained from a few days of satellite data. However, good sampling in sun-fixed coordinates is required for a reliable determination of the external field contributions. The right part of Fig. 7 shows the distribution of the Ørsted, resp. CHAMP, satellite

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Fig. 8 Some orbit characteristics for the Ørsted (left), resp. CHAMP (right) satellite in dependence on time. Top: Longitude of the ascending node, illustrating longitudinal “revisiting patterns.” Middle: mean altitude. Bottom: Local Time of ascending (red), resp. descending (blue) node

data of January 2, 2001, in the SM coordinate system, i.e., in dependence on the distance from the geomagnetic North pole (which is in the center of the plot) and magnetic local time MLT. Despite the rather good sampling in the geocentric frame (left panel), the distribution in the sun-fixed system (right panel) is rather coarse, especially when only data from one satellite are considered. Data obtained at different local times are essential for a proper description of external fields. The bottom panel of Fig. 8 shows how (geographic) local time of the satellite orbits

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change through mission lifetime. The Ørsted satellite scans all local times within 790 days (2.2 years), while the local time drift rate of CHAMP is much higher: CHAMP covers all local times within 130 days. Combining observations from different spacecraft flying at different local times helps to improve data coverage in the various coordinate systems. Especially the Swarm satellite constellation mission (Friis-Christensen et al. 2006, 2009) consisting of three satellites that were launched in November 2013 has specifically been designed to reduce the time-space ambiguity that is typical of single-satellite missions.

3

Making the Best of the Data to Investigate the Various Field Contributions: Geomagnetic Field Modeling

Using the observations of the magnetic field described in the previous section to identify the various magnetic fields described in Sect. 1 is the main purpose of geomagnetic field modeling. This requires the use of mathematical representations of such fields in both space and time. The mathematical tools that make such a representation in space possible are described in the accompanying chapter by Sabaka et al. (2010). But because the fields vary in time, some temporal representations are also needed. Using such spatio-temporal representations formally makes it possible to represent all the fields that contribute to the observed data in the form of a linear superposition of elementary functions. The set of numerical coefficients that define this linear combination is then what one refers to as a geomagnetic field model. It can be recovered from the data via inverse theory, which next makes it possible to identify the various field contributions. In practice, however, one has to face many pitfalls, not the least because the data are limited in number and not ideally distributed. In particular, although numerous, the usefulness of satellite data is limited by the time needed for satellites to complete an orbit, during which some of the fields can change significantly. This can then translate into some ambiguity in terms of the spatial/temporal representations. But advantage can be taken of the known spatiotemporal properties of the various fields described in Sect. 1 and of the combined use of ground and satellite data. Fast changing fields, with periods up to typically a month, are for instance known to mainly be of external origin (both ionospheric and magnetospheric), but with some electrically induced internal fields. Those are best identified with the help of observatory data, which can be temporally band filtered, and next spatially analyzed with the help of the tools described in Sabaka et al. (2010). This then makes it possible to identify the contribution from sources above and below the Earth’s surface. The relative magnitude and temporal phase shifts between the (induced) internal and (inducing) external fields can then be computed, which provide very useful information with respect to the distribution of the electrical conductivity within the solid Earth (see e.g., Constable 2007; Kuvshinov 2008). Satellite data can also be used for similar purposes, but this is a much more difficult endeavor since, as we already noted, one then has to deal with additional space/time separation

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issues related to the fact that satellites sense both changes due to their motion over stationary sources (such as the crustal field) and true temporal field changes. Much efforts are currently devoted to deal with those issues and make the best of such data for recovering the solid Earth electrical conductivity distribution, with encouraging preliminary successes (see e.g., Kuvshinov and Olsen 2006), especially in view of the Swarm satellite constellation mission (Kuvshinov et al. 2006). As a matter of fact, and for the time being, satellite data turn out to be most useful for the investigation of the field of internal origin (the core field and the crustal field). But even recovering those fields requires considerable care and advanced modeling strategies. In principle, and as explained in Sabaka et al. (2010), full vector measurements can be combined with ground-based vector data to infer both the field of internal origin, the E-region ionospheric field, the local F -region ionospheric field and the magnetospheric field. But there are many practical limits to this possibility, again because satellites do not provide instantaneous sets of measurements on a sphere at all times, and also because the data distribution at the Earth’s surface is quite sparse. This sets a limit on the quality of the E-region ionospheric field, one can possibly hope to recover by a joint use of ground-based and satellite data. But appropriate knowledge of the spatiotemporal behavior of each type of sources can again be used. This basically leads to the two following possible strategies to infer the contribution of each field from satellite data. A first strategy consists in acknowledging that the field due to nonpolar ionospheric E-region is weak at night, especially at so-called magnetic quiet time (as may be inferred from ground-based magnetic data), and selecting satellite data in this way, so as to minimize contributions from the ionosphere. Those satellite data can then be used alone to infer both the field of internal origin and the field of external (then only magnetospheric) origin (though this usually still requires some care when dealing with polar latitude data, because those are always, also at night and during quiet conditions, affected by some ionospheric and local field-aligned currents). This is a strategy that can be used to focus on the field of internal origin, and in particular the crustal field (see e.g., Maus et al. 2008). A second strategy consists in making use of both observatory data and satellite data, and simultaneously parameterizing the spatial and temporal behavior of as many sources as possible. This strategy has been used in particular to improve the recovery of the core field and its slow secular changes (see e.g., Thomson and Lesur 2007; Lesur et al. 2008; Olsen et al. 2009), but can more generally be used to try and recover all field sources simultaneously, using the so-called Comprehensive Modeling approach (Sabaka et al. 2004, 2002), to investigate the temporal evolution of all fields over long periods of times when satellite data are available. This strategy is one that looks particularly promising in view of the Swarm satellite constellation mission (Sabaka and Olsen 2006). Finally, it is worth pointing out that whatever strategy is being used, residuals from the modeled fields may then also always be used to investigate additional nonmodeled sources such as local ionospheric F -region sources (Lühr and Maus 2006; Lühr et al. 2002) to which satellite data are very sensitive.

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Considerable more details about all those strategies and other possible future strategies can be found in e.g., Hulot et al. (2007), Olsen et al. (2010a), Lesur et al. (2011), and Schott and Thebault (2011) to which the reader is referred, and where many more references can be found. Acknowledgements This is IPGP contribution 2595 (updated).

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Part III Modeling of the System Earth (Geosphere, Cryosphere, Hydrosphere, Atmosphere, Biosphere, Anthroposphere)

Classical Physical Geodesy Helmut Moritz

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What is Geodesy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Gravitational Potential and Gravity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Normal Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Geoid and Height Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Gravity Gradients and General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Key Issues of Theory: Harmonicity, Analytical Continuation, and Convergence Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Harmonic Functions and Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Convergence and Analytical Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 More About Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Krarup’s Density Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Key Issues of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Boundary-Value Problems of Physical Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Earth as a Nonrigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Smoothness of the Earth’s Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Geoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Journal Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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H. Moritz () Institute of Navigation, Graz University of Technology, Graz, Austria e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_6

253

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H. Moritz

Abstract

Geodesy can be defined as the science of the figure of the Earth and its gravitational field, as well as their determination. Even though today the figure of the Earth, understood as the visible Earth’s surface, can be determined purely geometrically by satellites, using Global Positioning System (GPS) for the continents and satellite altimetry for the oceans, it would be pretty useless without gravity. One could not even stand upright or walk without being “told” by gravity where the upright direction is. So as soon as one likes to work with the Earth’s surface, one does need the gravitational field. (Not to speak of the fact that, without this gravitational field, no satellites could orbit around the Earth.) To be different from the existing textbooks, a working knowledge of professional mathematics can be taken for granted. In some areas where professors of geodesy are hesitant to enter too deeply, afraid of losing their students, some fundamental problems can be studied. Of course, there is a brief introduction to terrestrial gravitation as treated in the first few chapters of every textbook of geodesy, such as gravitation and gravity (gravitation plus the centrifugal force of the Earth’s rotation), the geoid, and heights above the ellipsoid (now determined directly by GPS) and above the sea level (a surprisingly difficult problem!). But then, as accuracies rise from 106 in 1960 (about 0. However much the modest sand grain tries to be nonobtrusive, it will cause a singularity into the originally regular geopotential model and thus introduce a convergence sphere (Fig. 2). This shows that the possibility of regular analytic continuation of the geopotential and convergence is a highly complicated and unstable problem. A single sand grain may change convergence into divergence, and think of how many mass points, rocks, molecules, and electrons make up the Earth’s body and cause the most fanciful singularities. A counterexample. This would indicate that for all solid bodies, consisting of many mass points, the analytical continuation of the outer potential into its interior is automatically singular. A counterexample is the homogeneous sphere of radius R. Its external potential is well known to be that of a point mass M situated at its center: GM Vext D ; r > R; (37) r whereas the potential inside the Earth is (int = interior)   1 2 2 Vint D 2G R  r ; 3

(38)

which satisfies Poisson’s equation Vint D 4G. The analytical continuation of Vext is evidently the harmonic function Vcont D

GM ; 0 R g and having on SQ a generalized trace  v 2 L2 .S /. In fact, thanks to the unique Q (see Cimmino solvability of the Dirichlet Problem for  in ˝Q with data in L2 .S/ (1955); McLean (2000); Miranda (2008)), we can identify HH0  fv I v D 0 in ˝Q I  v 2 L2 .SQ /g with the space H0 of traces of functions v, also extending the topology (57) to HH0 . So from now we shall agree that HH0  H0 : Now consider in H0 the functionals 1 Lj k .v/ D 4

Z v.R; /Yj k ./d 

(58)

with R > RC as in (33); since the points .R; / are in the harmonicity domain of v, such functionals are bounded on H0 . As a matter of fact, if we put x  .R; / ; y  .R 0 ;  0 / ; F . 0 / D  v and we call G.x; y/ the exterior Green function of SQ , by denoting

(59)

304

F. Sansò

Gny .x; y/  H .;  0 /;

(60)

we have Z 1 H .;  0 /F . 0 /dS 4 Z 1 D H .;  0 /F . 0 /R2 0 J 0 d  0 : 4

v.R; / D 

(61)

Therefore, putting HC D maxH .;  0 /, which is bounded because R > RC , we  ; 0

have 3=2

jv.R; /j 

HC RC JC k F k0  K k v k0 : p 4

(62)

Since 1 4

Z jYj k jd   1;

so that jLj k .v/j  sup jv.R; /j ; 

(62) shows that Lj k are bounded in H0 . More precisely we have k Lj k k K :

(63)

Accordingly, thanks to the Riesz theorem, there are in H0 functions f such that Lj k .v/ 
0 ; k

jk

k0  K :

j k ./g

(64)

We shall not need the explicit form of f j k g, so we will not give it here; rather we need only to be sure that such functions are linearly independent.  nC1 But this is immediate because if we take the functions snm .x/  Rr Ynm ./ Q which are continuous on S, because R > 0, and therefore belong to H0 , we have
0 D

1 4

Z Ynm ./Yj k ./d  D ınj ımk :

Accordingly, if we have for some constants aj k

(65)

Geodetic Boundary Value Problem

305 L X

aj k

jk

D 0;

j;kD0

we also have 8.n; m/ with n  L; jmj  n, 0 D
0 D anm ;

j;kD0

which proves that f j k g are linearly independent. Note that, due to our previous Q convention, we can think of j k as well as functions harmonic in ˝. We note also that Span f j k ; 0  j  Lg  H0 is an .L C 1/2 D N dimensional subspace of H0 , and therefore, as finite subspace, it is naturally closed. We will call PN the orthogonal projector on H0 . We will consider also the complementary subspace H 0 , such that H0 D H 0 ˚ H0 ; H 0 ?H0 I clearly H 0 is closed and the orthogonal projector on H 0 is I  PN . Since u 2 H 0 , PN u D 0 $
D 0 ; 0  j  L ;

the following proposition is elementary to prove. Proposition 2. H 0 is precisely the closed subspace of H0 , of functions such that  uDO

1 jxjLC2

 ; jxj ! 1 :

Now we introduce the solution space H1 , defined as the subspace of H0 Z H1  fu 2 H0 I

SQ

jruj2 R3 d  < C1g :

(66)

Indeed the topology in H1 is defined by the norm Z k u k1 D Πjruj2 R3 d  1=2 :

(67)

SQ

Remark 1. that H1 is a subspace of H0 comes from the following reasoning. H0 is equivalent to L2 .SQ /, and indeed the Sobolev space H1 .SQ / is embedded in it, k u k0  c k u kL2 .SQ /  c k u kH1 .S/ Q :

(68)

306

F. Sansò

On the other hand, known estimates of the Stekelov-Poincaré operator,  @n u, when u is harmonic and SQ is Lipschitz, say that (McLean (2000), p. 145) 0 k u kH1 .S/ Q  c k @n u kL2 .S/ Q :

Therefore, on account of the inequality j@n uj  jruj; we have too 0 00 k u kH1 .S/ Q  c k ru kL2 .S/ Q  c k u k1 :

(69)

The relations (68) and (69) together prove that, under the present hypotheses on SQ , k u k0  A k u k1 for some constant A, i.e., H1  H0 . Since it will be useful in the sequel, a numerical determination of an upper bound of A is performed in the Appendix. Because of the above remark, the following proposition makes sense. Proposition 3. Define the subspace of H1 H 1  fu 2 H1 I
0 D 0 ; 0  j  Lg I

(70)

then H 1 is a closed subspace of H1 , with the following characterization: 

1 jxjLC2

u 2 H1 W u 2 H 1 , u D O

 ; jxj ! 1 :

(71)

With the above definitions, we see that our SiMP can be expressed in a more synthetic form. Namely, recalling the SiMP boundary operator Bu D ru0 C 2u ;

(72) 2

the problem (48) is equivalent to: find u 2 H 1 and faj k g 2 R.LC1/ such that BujSQ D F C

L X

aj k

j;kD0

for every F 2 H0 .

jk

;

(73)

Geodetic Boundary Value Problem

307

With the help of the projector PN , (73) can be even split into two parts. To do that we have to exploit the following lemma. Lemma 1. The operator B maps H 1 into H 0 , whenever L  1. Proof. Let us put v D Br D

X

xi @i u C 2u I

(74)

i

then v is harmonic in ˝. In fact, noting that xi D 0 and @i u D 0, we have, in ˝ v D 2

X

@j xi @j i u C 2u D 4u D 0:

i;j

Furthermore we note that r@r is a homogenous differential operator and Bsnm .r; / D .n  1/snm .r; / !nC1 R snm .r; / D Ynm ./ : r

(75)

Therefore in ˝ D fr  Rg, setting C1 X

uD

unm snm .r; / ;

(76)

n;mDLC1

we derive

v D Bu D 

C1 X

.n  1/unm snm .r; / ;

(77)

n;mDLC1

namely, v nm D .n  1/unm :

(78)

Such a relation shows first of all that  vDO

1 jxjLC2

 ; jxj ! 1

(79)

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F. Sansò

and second that the relation Bu D v is invertible at least in ˝. But then, by the unique continuation property of harmonic functions (see Sansò and Sideris (2013), p. 623), we see that v D 0 in ˝ implies u D 0 in ˝, i.e., B is into. Finally, if u 2 H 1  H1 , since ju0 j  jruj, we see that v D ru0 C2u is in L2 .SQ /, i.e., v 2 H0 . This together with (79) proves that v 2 H 0 . t u On the basis of Lemma 1, we see that 8u 2 H 1

.I  PN /Bu D Bu

(80)

PN Bu D 0 :

(81)

and 8u 2 H 1

So if we multiply (73) by .I  PN / and PN , respectively, we find .I  PN /BujSQ D BujSQ D .I  PN /F  F ; PN Bu D 0 D PN F C

L X aj k

jk

(82)

:

(83)

Given any F 2 H0 (83) determines uniquely faj k g because f j k g are a finite set of linearly independent functions. On the other hand, we can observe that F D .I  PN /F can be seen as a function in H 0 , so our SiMP is reduced to find a solution of BujSQ D F

(84)

with F 2 H 0 and u 2 H 1 . Since B maps H 1 into H 0 , which we have already seen, the problem now is to prove that B is also onto. We will do it in the next paragraph by showing that if v 2 H 0 ; vjSQ D F , then there is a solution u of (84) harmonic in ˝ and such that k u k1 < C1.

4

Solution of the SiMP and of the LSMP

The first part of this section is devoted to prove that the SiMP, represented by (48) and subsequently summarized in (84), has one and only one solution in H 1 for every datum F in H 0 . Then, in the last part of the section, we generalize the theorem of existence and uniqueness to the LSMP, by a simple perturbative argument. Theorem 1. The SiMP, formulated in (48), has one and only one solution 2

u 2 H 1 ; faj k I u  j  L; j  K  j g 2 R.LC1/ ;

(85)

Geodetic Boundary Value Problem

309

for every datum F 2 H 0 , on the condition that 3JC COL < 1 ;

(86)

where (recall (50), (51) and (56) for the definition of the symbols)  COL D

 1 ıR JC : C RC LC2

(87)

Moreover, under (86), the solution depends continuously on the data, since k u k1  CL k F k0 ; CL D

(88)

JC 1  3JC COL

(89)

and p p jaj k j  4 RC

R R

!LC1 k F k0 :

(90)

Proof. We already know from the previous section that the SiMP can be decomposed into a finite system (see (83)) j L X X

aj k

jk

D PN F

(91)

j D0kDj

and the equation u 2 H 1 ; BujSQ D F  .I  PN /F : As for (91) we can exploit the bi-orthogonality of f  j C1 R Yj k ./g, (see (65)), to get r

(92) jkg

with fsj k

D

jmj  n ; 0  n  L I anm D< snm ; PN F >0 : Therefore janm j k snm k0 k F k0 :

(93)

310

F. Sansò

On the other hand Z k

snm k20

D

Ynm ./

 4

2

R R

!2nC2

R R

R d 

!2LC2 RC I

(94)

by using (94) in (93) one gets (90). We come now to (92). We know from Lemma 1 that B maps continuously H 1 into H 0 . We further on prove that 8u 2 H 1 , if (86) holds, the majorization k u k1  CL k Bu k0 D CL k F k0

(95)

is true. Then (95) implies that the image of B is closed in H 0 . In the second part of the theorem, we will show that there is a subset of the image of B, which is dense in H 0 . Therefore B is into and onto H 0 so that the solution of (92) exists and is unique, 8F 2 H 0 . Since F D .I  PN /F , i.e., k F k0 k F k0 , existence and uniqueness of the solution of the SiMP will result; the majorization (88) then is nothing but (95). So we start from proving (95). We will apply the so-called energy identity, already used in Hörmander (1976) and Sansò and Venuti (2008). A short way to get it is to apply the differential identity r  fxjruj2  2ru0 ru  urug D 0

(96)

(see Jerison and Kenig (1982), p. 37), valid for every u harmonic in ˝, and use the Gauss theorem, noting also that r  ndS D R3 d . So we get Z 2 (97) k u k1 D jruj2 R3 d  D Z D

Z 2ru0 un dS C Z

2

uun dS Z

F un dS  3

uundS

where we have denominated ˇ F D ru0 C 2u D BuˇSQ :

(98)

We further note that Z Z j F un dS j D j F ./un ./R2 J d j

(99)

 JC k F k0 k un R k0  JC k F k0 k u k1

Geodetic Boundary Value Problem

311

as well as (see (131) in the Appendix) Z j

uun dS j  JC k u k0 k u k1 

(100)

 JC COL k u k21 : Putting (97), (99), and (100) together, we obtain k u k21  JC k F k0 k u k1 CJC COL k u k21 ;

(101)

which simplified and solved for k u k1 , if (86) is satisfied, yields (95), with CL as in (89). The geometric meaning of (86) will be discussed later on. We come now to the second part of the proof. Let us consider the space of finite linear combinations of the solid spherical harmonics, i.e., Q D Span fsn;m .x/ I jmj  u; :n D 0; : : :g H.˝/

(102)

Since fsnm .x/g and fjrsnm .x/jg are smooth functions in ˝, and in particular bounded on SQ , it is obvious that both relations Q  H0 ; H.˝/ Q  H1 ; H.˝/

(103)

hold true. Q the harmonics up to degree L; namely we define Now we take away from H.˝/ Q D Span fsnm .x/ I jmj  n ; n D L C 1; : : :g I H.˝/

(104)

Q  H 0 ; H.˝/ Q  H1 : H.˝/

(105)

it is clear that

On the other hand, on account of the relation (75), Bsnm D .n  1/snm ;

(106)

Q with itself. we see that, since L > 1 by hypothesis, B is an isomorphism of H.˝/ Q We shall prove that H.˝/ is dense in H 0 , and so, by the above comment, the theorem will be completely proved. Q is dense in H0 , We achieve our result in two steps: first we will prove that H.˝/ which is well known but will be repeated here for the sake of completeness; then we Q is dense in H 0 . shall constructively prove that H.˝/ As for the first part, we note that it is enough to show that if u 2 H0 is also such that u?H, i.e.,

312

F. Sansò

< u; snm >0 D 0 ;

8n; m;

(107)

then necessarily u D 0. In fact (107) can be written too as Z Ynm ./ u.R ; / nC1 R d  D 0 I R

(108)

so if we call u.R ; / ; JR

(109)

Ynm ./ dS D 0 : RnC1

(110)

./ D (108) becomes

Z ./ SQ

Multiplying (110) by Z w.r;  0 / D

rn 0 2nC1 Ynm . /,

with r < R, and adding, we get

1 ./ p 2 2 r C R  2rR cos

0

dS D 0 ;

(111)

meaning that the single layer potential w.r; / is identically zero in a neighborhood of the origin. Whence w.x/ is zero everywhere inside SQ . On the other hand we know that (McLean (2000), Theorem 6.11) w.x/ satisfies the jump relation ˇ ˇ ˇ @w ˇˇ @w ˇˇ ˇ wC jSQ  w SQ D 0 ;  D  ; @nC ˇSQ @n ˇSQ

(112)

if SQ is at least Lipschitz, as we have supposed, and  is at least in H 1=2 .SQ /, what is the case because in the present application  is in L2 .SQ /. The first of (112) tells us that wC jSQ D 0, namely, that w.x/ is zero everywhere, but for SQ itself. The second relation then implies that D0)uD0; as it was to be proved. Q that it is to a distance Now take any u 2 H 0 ; we prove that there is a v 2 H.˝/ Q such that, for closer than a fixed > 0 from u in H0 . We start by finding vQ 2 H.˝/ an arbitrary " > 0, k u  vQ k0 < " ; Q in H0 . what is always possible by virtue of the density of H.˝/

(113)

Geodetic Boundary Value Problem

313

Naturally in general vQ does not belong to H 0 , but we can put vQ D

L X

N X

cnm snm C

n;mD0

cnm snm D vQ L C v

(114)

n;mDLC1

Q with v 2 H.˝/. Now, for all m and n  L, since u 2 H 0 , cnm D 
0 ;

(115)

jcnm j  K k u  vQ k0  K" :

(116)

k snm k0  C

(117)

k vL k .L C 1/2 KC " :

(118)

implying that (recall (57))

Since also

when n  L, we have too

Accordingly we have, with v 2 H 0 , k u  v k0 Dk u  vQ C vQ L k0  .1 C .L C 1/2 KC /" :

(119)

Therefore it is enough to choose " such that .1 C .L C 1/2 KC /" < and the proof of the theorem is complete. t u We can now pass to the analysis of the LSMP, that was formulated by (47) as 8 ˆ < u D 0 P Bu D f C aj k ˆ 1 :u D O jxjLC2

jk

in ˝Q  Du on SQ jxj ! 1 ;

with f ./ D R g./ ; D D R "  r 

j"j  "C j j  C :

(120)

314

F. Sansò

The first remark is that, as we have done for the SiMP, such a problem can be reformulated with the help of the projector PN , as: find u 2 H 1 such that Bu D .I  PN /.f  Du/

(121)

and numbers faj k g such that L X

aj k

jk

D PN .f  Du/ :

(122)

j kD0

The idea is to determine conditions for the existence of u 2 H 1 as solution of (121), when f 2 H 0 and then substitute u in (122), to determine faj k g. Theorem 2. Assume that f 2 H 0 and that the condition JC Œ.3 C C /COL C "C < 1

(123)

is satisfied. Then there is one and only one solution u 2 H 1 of (121) satisfying the inequality k u k1 

CL k f k0 I 1  CL ."C C C COL /

(124)

moreover, since k Du k0 < C1, the Eq. (122) determines uniquely the constants faj k g. Proof. We first observe that (90) is identical with (48) if we put F D f  Du D f  .R "  ru  u/ :

(125)

Accordingly, since if (123) is satisfied, a fortiori (86) is satisfied too, we can claim that the Eq. (92) is equivalent to (121), and its solution, in force of (88), satisfies the inequality k u k1  CL k f k0 CCL k r"  ru k0 CCL k u k0 :

(126)

On the other hand, the following inequalities hold k r"  ru k0  "C k u k1 k u k0  C k u k0  C COL k u k1 : Summarizing we obtain the following a priori majorization for the solution of (121)

Geodetic Boundary Value Problem

315

k u k1  CL k f k0 CCL ."C C C COL / k u k1 :

(127)

Now if the condition CL ."C C C COL / < 1

(128)

is satisfied, we find that the operator 1

B 1 D W H ! H

1

is just a contraction, i.e., k B 1 D k< 1, so that (121) has one and only one solution for every f 2 H 0 . Actually, recalling that CL D

JC ; 1  3JC COL

we see that (128) is equivalent to (123). Finally (124) descends immediately from (127). The proof is complete. t u

5

Conclusions

The proofs reported in Sect. 4 are largely based on the paper Sansò and Venuti (2008). However if one compares the present article with the above, one can realize that the proof of the existence of the solution has been added and, maybe even more important, that almost a factor of 2 has been gained in the number of harmonic coefficients to be provided as input data. Roughly speaking, we can summarize the present understanding of the LSMP by saying that if a telluroid with a maximum inclination of 60ı is given and a datum f 2 L2 .SQ / plus the first 12 degrees of the asymptotic expansion of the potential, Q In fact, going to the key condition (123) then a unique solution exists in H 1;2 .S/. and substituting IC D 60ı ; JC D 2; "C Š 0:0067; C Š 0:0134; L D 12; RC D a C 6 km Š 6;384 km; ıR D a  b C 6 km Š 26:5 km; one obtains     1 ıR JC .3 C C / JC C "C D 0:9244 < 1 I C RC LC2

316

F. Sansò

in reality the limit value is IC D 61ı :5. Note that the 6 km added to the equatorial radius corresponds roughly to the height of the Chimborazo mountain, which is almost on the equator. To the knowledge of the author, this is the best result known on this problem. However it can be argued that, possibly applying some interpolation theory to the Q norm with that of H 1;2 .SQ /, one could obtain geometric majorization of the L2 .S/ Q conditions on S in the form of some norm Lp .SQ / of J D .cos I /1 , as it has been done in Hörmander’s paper (Hörmander (1976)). One has the impression that, although IC  60ı is not very restrictive, a result in the mean could be the best. A final remark on the choice of L2 .SQ / as topology for the data is that with such a space it is easy to translate the result to a space of generalized random field, including a white noise on SQ (see Grothaus and Raskop (2010)). This means that our solution should be stable even under a white noise perturbation. But the analysis of this argument is out of the scope of the present work.

Appendix In this appendix we aim to prove the subsequent proposition. Proposition 4. Let u 2 H 1 , namely, Z k u k1 D

SQ

and

jruj2 R2 d 

12 < C1

(129)

Z
0 D

u.R; /Yj k ./d  D 0

(130)

0  j  L ; j  k  j I then Z k u k0 D

12

2

u .R ; /R d 

 COL k u k1

(131)

where  COL D

 1 ıR JC : C RC LC2

(132)

Proof. Let us put uC ./ D u.RC ; /

(133)

Geodetic Boundary Value Problem

317

and observe that k u k0 k u  uC k0 C k uC k0  12 Z 2 Œu.R ; /  u.RC ; / R d   Z

12

2

C

(134)

u.RC ; / R d 

:

On the other hand, we can write ˇZ ˇ ju  uC j2 D ˇˇ 

RC R

ˇ2 Z ˇ u0 dr ˇˇ 

ıR R RC

Z

RC

RC

 u02 rdr

R

1 1  R RC



jruj2 r 2 dr I

R

therefore multiplying by R and integrating over the unit sphere, we get k u  uC k20 

ıR RC

Z jruj2 d ˝

(135)

˝n˝C

where ˝C  fRC  rg

(136)

˝n˝C  fR  r  RC g :

(137)

Furthermore, we can write k

uC k20 

C1 X

2 RC 4 uC nm n;mDLC1

Z D RC

u.RC ; /2 d  ;

(138)

with uC nm D

1 4

Z u.RC ; /Ynm ./d  :

(139)

Therefore we can claim too that k

uC k20 

C1 X 1 2 .n C 1/uC RC 4 nm LC2 n;mDLC1

! :

(140)

318

F. Sansò

On the other hand it is easy to verify that Z

Z 2

jruj d ˝ D  ˝C

uu SC

0

2 RC d

C1 X

D RC 4

2

.n C 1/uC nm

n;mDLC1

so that (140) can be put into the form k

uC k20 

1 LC2

Z jruj2 d ˝:

(141)

˝C

Collecting (134)–(136) yields s ıR RC

sZ

1

jruj2 d ˝ C p LC2 s sZ ıR 1  C jruj2 d ˝ : RC LC2 ˝

k u k0 

ı˝

sZ jruj2 d ˝ ˝C

So, by applying the Gauss theorem,  k u k20 

1 ıR C RC LC2

 Z   uun R2 Jd 

(142)

S

 COL k u k0 k un R k0  COL k u k0 k u k1 : Dividing both members of (142) by k u k0 , we get (131).

t u

References Cimmino G (1955) Spazi hilbertiani di funzioni armoniche e questioni connesse. In: Equazioni lineari alle derivate parziali. UMI Roma Freeden W, Gerhards C (2013) Geomathematically oriented potential theory. Chapman & Hall/CRC/Taylor Francis, Boca Raton Friedman A (1982) Variational principles and free boundary value problems. Wiley, New York Grothaus M, Raskop T (2010) Oblique stochastic boundary value problems. Handbook of GeoMath, Springer, pp 1052–1076 Hörmander L (1976) The boundary value problems in physical geodesy. Arch Prot Mech Anal 62:51–52 Jerison DS, Kenig C (1982) Boundary value problems on Lipschitz domains. MAA studies in mathematics, University of Pennsylvania, Minneapolis, vol 23, pp 1–68 McLean W (2000) Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge Miranda C (2008) Partial differential equations of elliptic type. Springer, Berlin

Geodetic Boundary Value Problem

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Molodensky MS, Eremeev VF, Yurkina MI (1960) Methods for the study of the gravitational field of the Earth. Russian Israel Program for Scientific Translations, Jerusalem Moser J (1966) A rapidly convergent iteration method and non-linear differential equations. Acc Sc Norm Sup Pisa 20:265–315 Otero J, Sansò F (1999) An analysis of the scalar geodetic boundary value problems with natural regularity results. J Geod 73:437–435 Sacerdote F, Sansò F (1986) The scalar boundary value problem of physical geodesy. Man Geod 11:15–28 Sansò F (1977) The geodetic boundary value problem in gravity space. Memorie Accademia nazionale dei Lincei, Rome, Italy, vol 14, ser 8, no 3 Sansò F, Sideris M (2013) Geoid determination: theory and methods. Springer, Berlin/Heidelberg Sansò F, Venuti G (2008) On the explicit determination of stability constants for the linearized geodetic boundary value problems. J Geod 82:909–916

Time-Variable Gravity Field and Global Deformation of the Earth Jürgen Kusche

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mass and Mass Redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Earth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Analysis of TVG and Deformation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

322 323 329 333 336 337 337

Abstract

The analysis of the Earth’s time-variable gravity field and its changing patterns of deformation plays an important role in Earth system research. These two observables provide, for the first time, a direct measurement of the amount of mass that is redistributed at or near the surface of the Earth by oceanic and atmospheric circulation and through the hydrological cycle. In this chapter, we will first reconsider the relations between gravity and mass change. We will in particular discuss the role of the hypothetical surface mass change that is commonly used to facilitate the inversion of gravity change to density. Then, after a brief discussion of the elastic properties of the Earth, the relation between surface mass change and the three-dimensional deformation field is considered. Both types of observables are then discussed in the framework of inversion. None of our findings are entirely new; we merely aim at a systematic compilation and discuss some frequently made assumptions. Finally, some directions for future research are pointed out.

J. Kusche () Astronomical, Physical and Mathematical Geodesy Group, Bonn University, Bonn, Germany e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_8

321

322

1

J. Kusche

Introduction

Mass transports inside the Earth and at or above its surface generate changes in the external gravity field and in the geometrical shape of the Earth. Depending on their magnitude, spatial and temporal scale, they become visible in the observations of modern space-geodetic and terrestrial techniques. Examples for mass transport processes that are sufficiently large to become observed include changes in continental water storage in greater river basins and catchment areas, large-scale snow coverage, present-day ice mass changes occurring at Greenland and Antarctica and at the continental glacier systems, atmospheric pressure changes, sea level change and the redistribution of ocean circulation systems, land-ocean exchange of water, and glacial-isostatic adjustment (cf. Fig. 1). As many of these processes are directly linked to climate, an improved quantification and understanding of their present-day trends and interannual variations from geodetic data will help to separate them from the long-term evolution, typically inferred from proxy data. The order of magnitude for atmospheric, hydrological, and oceanic loads, in terms of the associated change of the geoid, an equipotential surface of the Earth’s gravity field, is about 1 cm or 1  109 in relative terms. For example, in the Amazon region of South America, this geoid change is largely caused by an annual basin-wide oscillation in surface and ground water that amounts to several dm. The variety of geodetic techniques that are sufficiently sensible include intersatellite tracking as currently conducted with the Gravity Recovery and Climate Experiment (GRACE) mission, satellite laser ranging, superconducting relative and free-fall absolute gravimetry, and the monitoring of deformations with the Global Positioning System (GPS) and with the Very-Long Baseline Interferometry (VLBI) network of radio telescopes, Interferometric Synthetic Aperture Radar (InSAR), and of the ocean surface with satellite altimetry and tide gauges. Here, we will limit ourselves to a discussion of the fundamental relations that concern present-day mass redistributions and their observability in time-variable gravity (TVG) and time-variable deformation of the Earth. This is not intended to form the basis for real-data inversion schemes. Rather, we would like to point out essential assumptions and fundamental limitations in some commonly applied algorithms. To this end, it will be sufficient to consider the response of the solid Earth to the mass loading as purely elastic. In fact, this is a first assumption that will not be allowed anymore in the discussion of sea level, when geological timescales are involved at which the Earth’s response is driven by viscous or viscoelastic behavior. Furthermore, we will implicitly consider only that part of mass redistribution or mass transport that is actually associated with local change of density. Mass transport that does not change the local density (stationary currents in the hydrological cycle; i.e., the “motion term” in Earth’s rotation analysis) is in the null-space of the Newton and deformation operators; it cannot be inferred from observations of time-variable gravity or from displacement data.

Time-Variable Gravity Field and Global Deformationof the Earth 100

Static

10

GRACE

CHAMP

1.000

Ice Bottom Topography Fronts Topographic Control Coastal Currents

Quasi Static’ Ocean Circulation (near Surface) Plumes

Mantle Converction

Plate Boundaries Lithosphere Structure

Secular Glacial Isostatic Adjustment

Decadal

Hydrology

Ice Flow

Ground Water

Hydrology Water Balance

Interannual Annual

Sea Ice Ice Sheet Mass Balance

Sea Level Change

Snow Soil Moisture

Storage Variations

Seasonal

Atmosphere

Basin Scale Ocean Flux

Ocean Bottam Currents

Atmosphere

Monthly

Run Off

Postseismic Deformation

Soild Earth and Ocean Tides

Diurnal Semi-Diurnal Instantaneous Time resolution

[km]

GOCE

10.000

323

Volcanos Co-Seismic Deformation

10.000

1.000

100

10

[km]

Spatial resolution

Fig. 1 Spatial and temporal scales of mass transport processes and the resolution limits of satellite gravity missions CHAMP, GRACE, and GOCE (From Ilk et al. 2005)

2

Mass and Mass Redistribution

In what follows, we will adopt an Eulerian representation of the redistribution of mass, where one considers mass density as a 4D field rather than following the path of individual particles (Lagrangian representation). The mass density inside the Earth, including the oceans and the atmosphere, is then described by    D  x0 ; t :

(1)

Mass density is the source of the external gravitational potential, described by the Newton integral

324

J. Kusche

Z V .x; t/ D G v

 .x0 ; t/ dv; jx  x0 j

(2)

where v is the volume of the Earth including its fluid and gaseous envelope. The inverse distance (Poisson) kernel can be developed into n   1   1 1 X X r 0 nC1 1 D Ynm .e/Ynm e0 ; 0 0 jx  x j r nD0 mDn r .2n C 1/

(3)

where r D jxj, r 0 D jx0 j, and eD

x r

e0 D

x0 : r0

Ynm are the 4-normalized surface spherical harmonics. Using spherical coordinates  (spherical longitude) and  (spherical colatitude),  Ynm D Pnjmj .cos /

cos m; sin jmj;

m  0; m < 0:

Here, the Pnm D …nm Pmn are the 4-normalized associated Legendre functions. 1=2  .nm/Š denotes a normalization factor that depends …nm D .2  ı0m /.2n C 1/ .nCm/Š only on harmonic degree n and order m. The associated Legendre functions relate m=2 d m Pn .u/  to the Legendre polynomials by Pmn .u/ D 1  u2 , and the Legendre d um n d n .u2 1/ polynomials may be expressed by the Rodrigues formula, Pn .u/ D 2n1nŠ d un . On plugging Eq. (3) into Eq. (2), the exterior gravitational potential of the Earth takes on the representation V .x; t/ D G

n 1 X X

1 .2n C 1/r nC1 nD0 mDn

Z

     x0 ; t r 0n Ynm e0 dv

 Ynm .e/;

(4)

v

which converges outside of a sphere that tightly encloses the Earth. On the other hand, by introducing a reference scale a, the exterior potential V can be written as a general solution of the Laplace equation; V .x; t/ D

n 1 GM X X  a nC1 vnm .t/Ynm .e/ a nD0 mDn r

with 4-normalized spherical harmonic coefficients  vnm D

cnm ; snjmj ;

m  0; m < 0:

(5)

Time-Variable Gravity Field and Global Deformationof the Earth

325

A direct comparison of Eqs. (4) and (5) provides the source representation of the spherical harmonic coefficients (see also chapter  Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications): vnm .t/ D

1 1 1 M .2n C 1/ an

Z

    r 0n Ynm e0  x0 ; t dv:

(6)

v

There are 2n C 1 coefficients for each degree n, and each coefficient follows by projecting the density on a single 3D harmonic function (solid spherical harmonic) r n Ynm . An equivalent approach is to expand the inverse distance into a Taylor series: 0 1 3 1 3 X 3 X X X 1 .n/ @  i1 i2 :::in xi01 xi02 : : : xi0n A D jx  x0 j nD0 i D1 i D1 i D1 1

2

(7)

n

with components x10 , x20 , x30 of x0 , and  .n/ i1 i2 :::in

D

1 nŠ

"

1 @n jxx 0j

@xi01 @xi02 : : : @xi0n

# :

(8)

xDx0

This leads to a representation of the exterior potential through mass moments .n/ Mi1 i2 :::in of degree n: 0 1 3 1 3 X 3 X X X .n/ .n/ @ V .x; t/ D G  i1 i2 :::in Mi1 i2 :::in A nD0

i1 D1 i2 D1

(9)

in D1

with .n/

Mi1 i2 :::in D

Z v

  xi01 xi02 : : : xi0n  x0 ; t dv0 :

(10)

Note that the integrals in Eqs. (6) and (10) can be interpreted as inner products (; ) in a Hilbert space with integrable density defined on v. However, whereas the homogeneous Cartesian polynomials in Eq. (10) do provide a complete basis for the approximation of the density, the solid spherical harmonics in Eq. (6) fail to do so (in fact, they allow to represent the “harmonic density” part C for which C D 0). Due to symmetry of Eq. (10), there are .nC1/.nC2/ independent mass moments of 2 degree n. Of those only 2n C 1 become visible in the gravitational potential, which means that the remaining .nC1/.nC2/  .2n C 1/ D .n1/n span the null-space of the 2 2 Newton integral (Chao 2005). Some of the low-degree coefficients and mass moments deserve a special discussion, e.g., obviously

326

J. Kusche

v00 D

1 M

Z

  M .0/  x0 ; t dv D M v

corresponds to the total mass of the Earth, scaled by a reference value M . Since this is largely constant in time, inversion of geodetic data usually assumes ddt v00 D 0 (or ıv00 D 0; see below). Moreover, the degree-1 coefficients directly correspond to the coordinates of the center of mass x0 .t/: p Z .1/ 1 3 M v10 .t/ D x30 dv D 3 D p x30 .t/ 3M v M 3 and 1 v11 .t/ D p x10 .t/ 3

1 v11 .t/ D p x20 .t/ 3

(in the reference system where the spherical coordinates ,  are referring to). Space-geodetic evidence suggests that if the considered reference frame is fixed to the crust of the Earth, the temporal variation of the center of mass is no more than a few mm, with the x30 -component being largest. This is the case for realizations of the International Terrestrial Reference System (ITRS). Yet rather large mass redistributions are required to shift the geocenter for a few mm, and the study of these effects is subject to current research. Similarly, the degree-2 spherical harmonic coefficients v2m can be related to the .2/ tensor of inertia of the Earth. However, of the .2C1/.2C2/ D 6 mass moments Mi1 i2 , 2 only 2  2 C 1 D 5 become visible in the five spherical harmonic coefficients of the gravity field, with the null-space of the Newton integral being of dimension 6  5 D 1 for degree 2. The annual variation of the “flattening” coefficient v20 is of the order 11010 , the linear rate ddt v20 (predominantly due to the continuing rebound of the Earth in response to deglaciation after the last ice age) at the 1  1011 y1 level. A special case is the variation of degree 2, order 1 coefficients (v21 , v21 ), as those correspond to the position of the figure axis of the Earth (in the reference system where the ,  are referring to). Since the mean figure axis is close to the mean rotation axis and since the latter can be determined with high precision from the measurement of Earth’s rotation, additional observational constraints for their time variation (of the order of 1  1011 y1 ) exist. Nowadays, the vnm can be observed from precise satellite tracking with global navigation systems, intersatellite ranging, and satellite accelerometry with temporal resolution of 1 month or below and spatial resolution of up to nmax D 120. However, as mentioned earlier, one cannot uniquely invert gravity change into density change. The question had been raised which physically plausible assumptions nevertheless allow to somehow locate the sources of these gravity changes. A common way to regularize this “gravitational-change inverse problem” (GCIP) is to restrict the solution space to density changes within an infinitely thin spherical shell of radius

Time-Variable Gravity Field and Global Deformationof the Earth

327

a (Wahr et al. 1998). This corresponds to the determination of surface mass from an external potential. The spherical harmonic coefficients vnm of the potential caused by a surface density  on a sphere are 1 a2 vnm .t/ D M .2n C 1/

Z

     e0 ; t Ynm e0 ds:

(11)

s

Since only changes with respect to a reference status, which can be realized through the measurements (e.g., a long-term average), are observable, one defines     ı.e0 ; t/ D  e0 ; t  N e0 and the coefficients of potential change follow from ıvnm .t/ D vnm .t/  vN nm where vN nm D

R t2 t1

1 a2 D M .2n C 1/

Z

    ı e0 ; t Ynm e0 ds;

(12)

s

vnm .t 0 /dt 0 . Or, involving the spherical harmonic expansion of ı,

ıvnm D

1 1 4a2 3 ınm D ınm ; M .2n C 1/ e .2n C 1/

(13)

, va D 34 a2 where we made use of the average density of the Earth, e D M va being the volume of a sphere of radius a. Obviously, this relation is coefficient-bycoefficient invertible. Corresponding to the 2n C 1 (unnormalized) coefficients ıvnm at degree n, there are just 2n C 1 surface mass moments, with components e10 , e20 , e30 of e0 , .n/

Z

ıAi1 i1 :::in D an

s

  ei01 ei02 : : : ei0n ı e0 ; t ds:

(14)

The integrals in Eqs. (12) and (14) can be viewed as inner products (; ı) in a Hilbert space of integrable (density) functions defined on the sphere. We will again look at low-degree terms. Obviously, ıv00 D

4a2 3 a2 .0/ ı00 D ı00 D ıA M e 4

corresponds to the change in total surface mass. This should be zero as long as  P ı considers all subsystems ı D s ı.s/ , whereas a shift of total mass from, e.g., the ocean to the atmosphere may happen very well. For the center-of-mass shift referred to the Earth (strictly speaking, to the center of the sphere where  is located on) with mass M ,

328

J. Kusche

4a2 1 1 1 .1/ ı10 D ı10 D p x30 D p ıA3 3M 3e 3 3e

ıv10 .t/ D

and similar for ıv11 and ıv11 . Restriction of the density to a spherical shell serves for eliminating the nullspace of the problem, thus allowing a unique solution for density determination from gravity. In fact, the surface layer could be located on the surface of an ellipsoid of revolution or any other sufficiently smooth surface as well. This could be implemented in Eqs. (11) and (12) by including an upward continuation integral term; however, it would not allow a simple scaling of the coefficients as with Eq. (13) anymore. On the other hand, in the light of comparing inferred surface densities to modeled densities (e.g., from ocean or hydrology models), one could determine point-wise densities on a nonspherical surface from those on a spherical shell by  n applying an ar term (Chao 2005). The 3D density  and the 2D density  are of course related to each other. This can be best seen by writing the source representation of the spherical harmonic coefficients, Eq. (6), in the following way: 1 a2 vnm .t/ D M .2n C 1/

Z (Z s

rmax 0

)  0 nC2    0  r  x ; t dr Ynm e0 ds a

(15)

and comparing the term in {. . . } brackets to Eq. (11). Surface mass change ı can thus be considered as a column-integrating “mapping” of ı, say, L2 .s/  L1 .Œ0; rmax / ! L2 .s/ if we assume square integrability. A coarse approximation of the integral in Eq. (15) is used if surface density change ı is transformed to “equivalent water height” change ıhw D ı w with a constant reference value w for the density of water, or if (real) water height change (e.g., from ocean model output) is transformed to surface density change ı D w ıhw . In this case, the assumption is implicitly made that ar  1, introducing an error of the order of .n C 2/f ınm , where f is the flattening of the Earth (about 1/300). For computing the contribution of the Earth’s atmosphere to observed gravity change, it is nowadays accepted that 3D integration should be preferred to the simpler approach where surface pressure anomalies are converted to surface density. For example, in the GRACE analysis (Flechtner 2007)  .x0 ; t/ dr in Eq. (15) is dp , assuming hydrostatic equilibrium. Then with g.r/  g  a 2 replaced by  g.r/ r and r D a C N C 1ˆˆ , a

1 1 a2 vnm .t/ D M .2n C 1/ g

Z (Z s

0 ps



N a C aˆ a

nC4

)

  dp Ynm e0 ds:

(16)

Time-Variable Gravity Field and Global Deformationof the Earth

329

In Eq. (16), N is the height of the mean geoid above the ellipsoid, ps is surface pressure, and ˆ is the geopotential height that is derived from vertical level data on temperature, pressure, and humidity. A completely different way to solve the gravitational inverse problem (GIP) (and the GCIP) has been suggested by Michel (2005), who derives the harmonic part C of the density from gravity observations.

3

Earth Model

A caveat must be stated at this point, since so far we have considered the Earth as rigid. In reality, any mass redistribution at the surface or even inside the Earth is accompanied by a deformation of the solid Earth in its surroundings, which may be considered as an elastic, instantaneous response for short timescales and a viscoelastic response for longer timescales. This deformation causes an additional, “indirect” change of the gravity potential, which is not negligible and generally depends on the harmonic degree n of the load ınm . The linear differential equations that describe the deformation and gravity change of an elastic or viscoelastic, symmetric Earth, forced by surface loading of redistributing mass, are usually derived by considering a small perturbation of a radial-symmetric, hydrostatically prestressed background state. In the linearized equation of momentum, r   C r.0 g0 u  er /  0 rıV  ıg0 er 

d2 uD0 dt 2

(17)

the first term describes the contribution of the stresses; the second term the advection of the hydrostatic prestress (related to the Lagrangian description of a displaced particle); the third term represents self-gravitation, expressing the change in gravity due to deformation; and the fourth term describes the change in density if one accounts for compressibility. Here,  is the incremental stress tensor, u D u.x; t/ the displacement of a particle at position x; ıV the perturbation of the gravitational potential, and ı the perturbation of density. The fifth term in Eq. (17) is necessary when one is interested in the (free or forced) eigenoscillations of the Earth or in the body tides of the Earth caused by planetary potentials; however, the dominant timescales for external loads and the “integration times” for observing systems such as GRACE are rather long, with the consequence that one is usually confident with the quasi-static solutions. Assuming the Earth in hydrostatic equilibrium prior to the deformation is certainly a gross simplification; it is however a necessary assumption to find “simple” solutions with the perturbation methods that are usually applied (e.g., Farrell 1972). Generally, the density perturbation can be expressed by the continuity equation: ı D r  .u/:

(18)

330

J. Kusche

For the perturbation of the potential, the Poisson equation ıV D 4Gı

(19)

must hold. These equations are to be completed by a rheological law (e.g.,  D f ./ for elastic behavior,  D f .; P / for viscoelastic Maxwell behavior,  being the strain tensor) and boundary conditions for internal interfaces of a stratified Earth and for the free surface. For an elastic model consisting of Z layers, the rheology is usually prescribed by polynomial functions  D .z/ .r/;  D .z/ .r/;  D .z/ .r/ .z/ of the Lamé parameters ;  and the density within each layer, rmin .r/  r  .z/ rmax . Solutions for potential change and deformation at the surface, obtained for the boundary condition of surface loading, are often expressed through Green’s functions: Z 1Z     KV x; x0 ;  0 ı x0 ; t   0 ds d ıV .x; t/ D  D0

s

.x0 D ae0 / and Z

1

Z

    Ku x; x0 ;  0 ı x0 ; t   0 ds d;

u.x; t/ D  D0

s

where the kernels for general (anisotropic, rotating, ellipsoidal, viscoelastic) Earth models can be represented location- and frequency-dependent coefficients, P through 0 k .; ; z/ k 0 nm .0 ;  0 ; z/, where z is the Laplace e.g., K .x; x0 ;  0 / D nm nm transform parameter with unit s1 . Couplings between the degree-n, order-m terms of the load and potential-change and deformation harmonics of other degrees and orders have to be taken into account (e.g., Wang 1991). It should be noted already here that Green’s functions essentially represent the impulse response of the Earth (i.e., Eqs. (17)–(19) together with a rheological law) and, as such, may be determined from measurements under a known load “forcing.” For symmetric, nonrotating, elastic, and isotropic (SNREI) models of the Earth, 0 the knm .; ; z/ D kn0 are simply a function of the harmonic degree and the Green’s function kernels depend on the spatial distance of x and x0 only. Numerical solutions for the kn0 start with the observation that in Eqs. (17)–(19) the reference quantities 0 ; g0 and the Lamé parameters that enter Hooke’s law all depend on the radius r only. Figure 2 shows the load Love numbers – h0n , nln0 , – nk0n that Farrell (1972) computed, using a Gutenberg-Bullen model for the Earth’s rheology, for surface potential change and displacement. (It should be noted that Love numbers are radius dependent but usually in geodesy only the surface limits are applied.) The solution for the gravitational response to a surface mass distributed on top of an elastic Earth can be best described in spherical harmonics: ıvnm .t/ D

 1 a2  1 C kn0 M .2n C 1/

Z

    ı x0 ; t Ynm e0 ds s

(20)

Time-Variable Gravity Field and Global Deformationof the Earth

331

Fig. 2 Load Love numbers after Farrell (1972), h0n .C/; nln0 . /; nkn0 ./ versus degree n

or ıvnm D

  4a2  3  1 1 ınm D ınm ; 1 C kn0 1 C kn0 M .2n C 1/ e .2n C 1/

(21)

where the “1” term is the potential change caused by ı and the load Love number “kn0 ” term describes the incremental potential due to solid Earth deformation. Again, degree 0 and 1 terms deserve special attention: Due to mass conservation, k00 must be zero. Also, l00 D 0 (see below) is obvious since a uniform load on a spherical Earth cannot cause horizontal displacements due to symmetry reasons. In contrast, h00 corresponds to the average compressibility of the elastic Earth and will not vanish. In the local spherical East-North-Up frame, the solution for u is  u.x; t/ D

u.x; t/ h.x; t/

 Da

n  1 X X ı

nm .t/r

ıhnm .t/

nD0 mDn



 Ynm .e/

(22)

with r  being the tangential part of the gradient operator r. In spherical coordinates, r D

1 @ 1 @ e C e r @ r sin  @

r D r C

@ : @r

The radial displacement function is h0n a2 ıhnm .t/ D gM .2n C 1/

Z

    ı x0 ; t Ynm e0 ds

s

and at location x of the Earth’s surface, the radial displacement is ıh.x; t/ D a

n 1 X X nD0 mDn

ıhnm Ynm .e/:

(23)

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The lateral displacement function is ln0 a2 ı nm .t/ D gM .2n C 1/

Z

    ı x0 ; t Ynm e0 ds:

(24)

s

At location x, the East and North displacement components are n 1 X X   ı ıe x0 ; t D a

nm

@ Ynm .e/ @e

nm

@ Ynm .e/: @n

nD0 mDn

and n 1 X X   ın x0 ; t D a ı nD0 mDn

A reference system fixed to the center-of-mass of the solid Earth (CE system) is the natural system to compute the dynamics of solid Earth deformation and to model load Love numbers (Blewitt 2003). This system is obviously “blind” to mass transports that shift the center-of-mass of the earth (excluding ocean, atmosphere, etc.); hence, by definition for the degree-1 potential Love number k10 D 0. Note that this is not true for the other degree-1 Love numbers, e.g., Farrell (1972) found h01 D 0:290 and l10 D 0:113. However, the CE system is difficult to realize in practice by space-geodetic “markers.” There are two principal ways to compute deformations for other reference systems: (a) first compute in CE, and subsequently apply a translation; (b) transform degree-1 Love numbers, and compute in the new system. Blewitt (2003) showed that a translation of the reference system origin along the direction of the Load moment can be absorbed in the three load Love numbers by subtracting the “isomorphic parameter” ˛ from them (we follow the convention of Blewitt 2003), kQ10 D k10  ˛, hQ 01 D h01  ˛, and lQ10 D l10  ˛. For example, when transforming from the CE system to the center of mass of Earth system (CM system), which is fixed to the center of the mass of the Earth including the atmospheric and oceanographic loads, one has ˛ D 1: However, in reality the network shift (and rotation and scaling) might not entirely be known, as supposed for the mentioned approaches. In this situation, Kusche and Schrama (2005) and Wu et al. (2006) showed that from 3D displacements in a realistic network and for a given set of Love numbers with h01 ¤ l10 , it is possible to separate residual unknown network translation and degree-1 deformation in an inversion approach. For realistic loads, the theory predicts vertical deformation of up to about 1 cm and horizontal deformation of a few mm. The solution in Eq. (22) refers to the local spherical East-North-Up frame, since it refers to an SNREI model. This must be kept in mind since geodetic displacement vectors are commonly referred to a local ellipsoidal East-North-Up frame. Otherwise a small but systematic error will be introduced from projecting height displacements erroneously onto North displacements.

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In reality, the Earth is neither spherically symmetric nor purely elastic or isotropic, and it rotates in space. The magnitude of these effects is generally thought to be small. For example, Métivier et al. (2005) showed that ellipticity of the Earth, under zonal (atmospheric) loading, leads to negligible amplitude and phase perturbations in low-degree geopotential harmonics. Another issue is that, even within the class of SNREI models, differences in rheology would cause differences in the response to loading. Plag et al. (1996) found for the models preliminary reference Earth model (PREM) and parametric Earth model, continental (PEMC), differing in lithospheric properties, differences in the vertical displacement of up to 20 %, in the horizontal displacement of up to 40 %, and in gravity change of up to 20 % in the vicinity (closer than 2ı distance) of the load, but identical responses for distances greater than this value.

4

Analysis of TVG and Deformation Pattern

In many cases, the aim of the analysis of time-variable gravity and deformation patterns is, at least in an intermediate sense, the determination of mass changes within a specific volume: Z    .x; t/  N x0 dv; (25) ıM D v

e.g., an ocean basin or a hydrological catchment area. Under the assumption that the thin-layer hypothesis provides an adequate description, ıM can be found by integrating ı over the (spherical) surface s  of v (Wahr et al. 1998). The integration is usually performed in spectral domain, with the 4-normalized spherical harmonic  coefficients snm of the characteristic function S  for the area s  Z ıM D

s

n 1 X X    ı e0 ; t ds D snm ınm .t/;

(26)

nD0 mDn

or, in brief, ıM D .S  ; ı/. Following the launch of the GRACE mission, the analysis of time-variable surface mass loads from GRACE-derived spherical harmonic coefficients via the inverse relation (Wahr et al. 1998) ınm D

e .2n C 1/   ıvnm 3 1 C kn0

(27)

has been applied in a fast-growing number of studies (for an overview, the publication database of the GRACE project at http://www-app2.gfz-potsdam.de/ pb1/op/grace/ might be considered). Since GRACE-derived monthly or weekly ıvnm D vnm  vN nm are corrupted with low-frequency noise, and since they exhibit longitudinal artifacts when projected in the space domain, isotropic or anisotropic

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filter kernels are applied to these estimates usually. See Kusche (2007) for a review on methods, where in general a set of spectrally weighted coefficients ıQvnm D P filter pq wnm ıvpq is derived. p;q

In principle, spherical harmonic models of surface load can be inferred coefficient-by-coefficient as well once a spherical harmonic model of vertical deformation has been derived: ınm D

e .2n C 1/ ıhnm 3 h0n

(28)

ınm D

e .2n C 1/ ı 3 ln0

(29)

or, of lateral deformation nm :

For the combination of vertical and lateral deformations in an inversion, two approaches exist: (a) either the ınm are estimated directly from discrete u.xi / or (b) in a two-step procedure, first the hnm and nm are estimated separately from discrete height and lateral displacements, and the ınm are subsequently inferred from those. In practice, 3D displacement vectors are available in discrete points of the Earth’s surface. For the network of the International GNSS Service (IGS), a few hundred stations provide continuous data such that a maximum degree and order of well beyond degree and order 20 might be resolved. However, several studies have shown that this rather theoretical resolution cannot be reached. The reason is that spatial aliasing is present in the signal measured at a single site, which measures the sum of all harmonics up to infinity, unless the signal above the theoretical resolution can be removed from other data such as time-variable gravity. Inversion approaches must be seen in the light of positioning accuracies of modern worldwide networks, which are currently of a similar order compared with the deformation signals (few mm horizontally and a factor of 2–3 worse in the vertical direction). Whereas random errors may significantly reduce in the spatial averaging process that is required to form estimates for low-degree harmonics, any systematic errors at the spatial scale of the signals of interest may be potentially dangerous. A particular problem is given by the presence of secular trends in the displacement data that are dominated by nonloading phenomena like glacial isostatic adjustment (GIA), plate motion, and local monument subsidence. If the load ı.t/ is known, from independent measurements (e.g., water level measurements from gauges or surface pressure from barometric measurements and meteorological modeling) the LLN’s kn 0 , hn 0 , ln 0 could, in principle, be determined experimentally from measurements of gravity change and displacement. Plag et al. (1996) coined the term “loading inversion” (and as an alternative “loading tomography,” noting that tomography is usually based on probing along rays and not by body-integrated response) and suggested a method to invert for certain spherical harmonic expansion coefficients of density ınm and Lamé parameter perturbations ınm , ınm , together with polynomial parameters that describe their

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variation in the radial direction. This inverse approach to recover elastic properties from measurements of gravity and deformation is followed in planetary exploration, however, for the body Love numbers k2 and h2 . There, the role of the known load is assumed by the known tidal potentials in the solar system. On the other hand, one has the possibility to eliminate the ınm from the equations and determine the ratio ıhnm h0n D 0 1 C kn ıvnm

for each

ı nm ln0 D 1 C kn0 ıvnm

for each m D n : : : n

m D n : : : n

and

from a combination of gravity and displacement data. If all spherical harmonic coefficients up to degree nmax are well determined from data, these equations provide 2n C 1 relations per degree n. Mendes Cerveira et al. (2006) proposed that h0 this freedom might in principle be used to uniquely solve for the ratio 1Cknm0 and nm

0 lnm 0 1Cknm

of global, “azimuth-dependent” LLNs. Other observations of gravity change and deformation of the Earth’s solid and fluid surface may be considered as well. With the global network of superconducting gravimeters, it is believed that annual and short-time variations in mass loading can be observed and compared with GRACE results, after an appropriate removal of local effects (Neumeyer et al. 2006). Absolute (free-fall) gravimeters provide stable time series from which trends in gravity can be obtained and from which time series of superconducting gravimeters can be calibrated. Gravity change ıg D jrV j  jrV j as sensed by a terrestrial gravimeter, which is situated on the deforming Earth surface, reads (in spherical approximation, where the magnitude of the gradient is replaced by the radial derivative) ıg.x; t/ D

n 1  GM X X  n C 1  2h0n ıvnm .t/Ynm .e/ 2 a nD0 mDn

(30)

and when related to surface mass change, 1 n GM 3 X X n C 1  2h0n ıg.x; t/ D 2 ınm .t/Ynm .e/: a e nD0 mDn 2n C 1

(31)

In principle, observations of the sea level might be considered in a multi-data inversion scheme as well. This requires that the steric sea-level change, which is related to changes in temperature and salinity of the ocean, can be removed from the measured total or volumetric sea level. Sea-level change can be measured using satellite altimeters, as with the current Jason-1 and Jason-2 missions, in an absolute

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a r

n+1

1 + k n′ 2n + 1

Space gravity

h ′n (2n + 1)

r (X′, t )

M (n) ii

m (X′, t)

A(n) ii

l ′n (2n + 1)

Vertical Lateral displacement displacement

1 2.....in

1 2.....in

n + 1 – 2h n′ 2n+1

1 + k ′n 2n+1

1 + k ′n – h ′n (2n+1)

Terrestrial gravity

Absolute sealevel

Relative sealevel

Fig. 3 Spectral operators for space and terrestrial gravity, displacement, and sea-level observables

sense, since altimetric orbits refer to an ITRS-type global reference system. Tide gauge observations, on the other hand, provide sea level in a relative sense since they refer to land benchmarks. Ocean bottom pressure recorders measure the load change directly. If one assumes that the ocean response is largely “passive” (Blewitt and Clarke 2003), i.e., the ocean surface follows an equipotential surface, the absolute sea-level change is related to surface load as ıs.x; t/ D

1 n GM 3 X X 1 C kn0 ınm .t/Ynm .e/ C ıs0 .t/ a e nD0 mDn 2n C 1

(32)

and relative sea level as ı sL.x; t/ D

1 n GM 3 X X 1 C kn0  h0n ınm .t/Ynm .e/ C ı sL0 .t/; a e nD0 mDn 2n C 1

(33)

where ıs0 .t/ and ı sL0 .t/ are spatially uniform terms that account for mass conservation (cf. Blewitt 2003). All spectral operators discussed in this chapter are summarized in Fig. 3.

5

Future Directions

Within the limits of accuracy and spatial resolution of current observing systems, inversions for (surface) mass appear to have almost reached their potential. Limitations are, in particular, the achievable accuracy of spherical harmonic coefficients

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from GRACE, systematic errors nicknamed as “striping,” the presence of systematic errors in displacement vectors from space-geodetic techniques, and the spatial density and inhomogeneous distribution of global networks (space-geodetic techniques, absolute and superconducting gravimetry). However, moderate improvements in data quality and consistency can be expected from reprocessings such as the anticipated GRACE RL05 products or the IGS reprocessing project. In the long run, GRACE Follow-On missions and geometric positioning in the era of GALILEO will provide, hopefully, bright prospects. At the time of writing, some groups focus on the combination of gravity and geometrical observations with a priori models of mass transport (so-called joint inversions; cf. Wu et al. 2006; Jansen et al. 2009; Rietbroek et al. 2009). The benefit of this strategy is that certain weaknesses of individual techniques can be covered by other techniques. For example, it has been shown that the geocenter motion or degree-1 surface load which is not observed with GRACE can be restituted to some extent by GPS and/or ocean modeling. Research is ongoing in this direction. Another issue is that if the space agencies fail to replace the GRACE mission in time with a follow-on satellite mission, a gap in the observation of the time-variable gravity field might occur. To some extent, the very low degrees of mass loading processes might be recovered during the gap from geometrical techniques and loading inversion, provided that their mentioned limitations can be overcome and a proper cross-calibration is facilitated with satellite gravity data in the overlapping periods (Bettadpur et al. 2008; Plag and Gross 2008). The same situation occurs, if one tries to go back in time before the launch of GRACE, e.g., using (reprocessed) GPS solutions. Anyway, the problem remains extremely challenging.

6

Conclusions

We have reviewed concepts that are currently used in the interpretation of timevariable gravity and deformation fields in terms of mass transports and Earth system research. Potential for future research is seen in particular for combinations of different observables.

References Bettadpur S, Ries J, Save H (2008) Time-variable gravity, low Earth orbiters, and bridging gaps. In: GRACE Science Team Meeting 2008, San Francisco, 12–13 Dec 2008 Blewitt G (2003) Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth. J Geophys Res 108(B2):2103. doi:10.1029/2002JB002082 Blewitt G, Clarke P (2003) Inversion of Earth’s changing shape to weigh sea level in static equilibrium with surface mass redistribution. J Geophys Res 108(B6):2311. doi:10.1029/2002JB002290 Chao BF (2005) On inversion for mass distribution from global (time-variable) gravity field. J Geodyn 39:223–230 Farrell W (1972) Deformation of the Earth by surface loads. Rev Geophys Space Phys 10(3):761– 797

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Flechtner F (2007) AOD1B Product Description Document for Product Releases 01 to 04 (Rev. 3.1, 13 Apr 2007), GR-GFZ-AOD-0001, GFZ Potsdam Ilk KH, Flury J, Rummel R, Schwintzer P, Bosch W, Haas C, Schröter J, Stammer D, Zahel W, Miller H, Dietrich R, Huybrechts P, Schmeling H, Wolf D, Götze HJ, Riegger J, Bardossy A, Güntner A, Gruber T (2005) Mass transport and mass distribution in the Earth system. Contribution of the new generation of satellite gravity and altimetry missions to geosciences, GFZ Potsdam Jansen MWF, Gunter BC, Kusche J (2009) The impact of GRACE, GPS and OBP data on estimates of global mass redistribution. Geophys J Int. doi:10.1111/j.1365-246X.2008.04031.x Kusche J (2007) Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity fields. J Geodesy 81(11):733–749 Kusche J, Schrama EJO (2005) Surface mass redistribution inversion from global GPS deformation and Gravity Recovery and Climate Experiment (GRACE) gravity data. J Geophys Res 110:B09409. doi:10.1029/2004JB003556 Mendes Cerveira P, Weber R, Schuh H (2006) Theoretical aspects connecting azimuth-dependent Load Love Numbers, spatiotemporal surface geometry changes, geoid height variations and Earth rotation data. In: WIGFR2006, Smolenice Castle, 8–9 May 2006 Métivier L, Greff-Lefftz M, Diament M (2005) A new approach to computing accurate gravity time variations for a realistic earth model with lateral heterogeneities. Geophys J Int 162:570–574 Michel V (2005) Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth’s gravitational field at satellite height. Inverse Probl 21:997– 1025 Neumeyer J, Barthelmes F, Dierks O, Flechtner F, Harnisch M, Harnisch G, Hinderer J, Imanishi Y, Kroner C, Meurers B, Petrovic S, Reigber C, Schmidt R, Schwintzer P, Sun H-P, Virtanen H (2006) Combination of temporal gravity variations resulting from superconducting gravimeter (SG) recordings, GRACE satellite observations and global hydrology models. J Geodesy 79:573–585 Plag H-P, Gross R (2008) Exploring the link between Earth’s gravity field, rotation and geometry in order to extend the GRACE-determined terrestrial water storage to non-GRACE times. In: GRACE science team meeting 2008, San Francisco, 12–13 Dec 2008 Plag H-P, Jüttner H-U, Rautenberg V (1996) On the possibility of global and regional inversion of exogenic deformations for mechanical properties of the Earth’s interior. J Geodyn 21(3):287– 308 Rietbroek R, Brunnabend S-E, Dahle C, Kusche J, Flechtner F, Schröter J, Timmermann R (2009) Changes in total ocean mass derived from GRACE, GPS, and ocean modeling with weekly resolution. J Geophys Res 114:C11004. doi:10.1029/2009JC005449 Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103(B12):30205– 30229 Wang R (1991) Tidal deformations on a rotating, spherically asymmetric, visco-elastic and laterally heterogeneous Earth. Peter Lang, Frankfurt/Main Wu X, Heflin MB, Ivins ER, Fukumori I (2006) Seasonal and interannual global surface mass variations from multisatellite geodetic data. J Geophys Res 111:B09401. doi:10.1029/2005JB004100

Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution Willi Freeden and Michael Schreiner

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SGG in Potential Theoretic Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition of Tensor Fields by Means of Tensor Spherical Harmonics . . . . . . . . . . . Formulation as Pseudodifferential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 SGG as Pseudodifferential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Upward/Downward Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Operator of the First-Order Radial Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Pseudodifferential Operator for SST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Pseudodifferential Operator of the Second-Order Radial Derivative . . . . . . . . . . . 4.6 Pseudodifferential Operator for Satellite Gravity Gradiometry . . . . . . . . . . . . . . . . 4.7 Survey on Pseudodifferential Operators Relevant in Satellite Technology . . . . . . . 4.8 Classical Boundary Value Problems and Satellite Problems . . . . . . . . . . . . . . . . . . 4.9 A Short Introduction to the Regularization of Ill-Posed Problems . . . . . . . . . . . . . 4.10 Regularization of the Exponentially Ill-Posed SGG-Problem . . . . . . . . . . . . . . . . . 5 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

340 342 348 353 356 357 358 359 360 360 361 363 364 370 371 376 377

Abstract

Satellite gravity gradiometry (SGG) is an ultrasensitive detection technique of the space gravitational gradient (i.e., the Hesse tensor of the Earth’s gravitational potential). In this note, SGG – understood as a spacewise inverse problem of satellite technology – is discussed under three mathematical aspects: First, SGG

W. Freeden () Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany e-mail: [email protected] M. Schreiner Institute for Computational Engineering, University of Buchs, Buchs, Switzerland e-mail: [email protected] © Springer-Verlag Berlin Heidelberg 2015 W. Freeden et al. (eds.), Handbook of Geomathematics, DOI 10.1007/978-3-642-54551-1_9

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is considered from potential theoretic point of view as a continuous problem of “harmonic downward continuation.” The space-borne gravity gradients are assumed to be known continuously over the “satellite (orbit) surface”; the purpose is to specify sufficient conditions under which uniqueness and existence can be guaranteed. In a spherical context, mathematical results are outlined by the decomposition of the Hesse matrix in terms of tensor spherical harmonics. Second, the potential theoretic information leads us to a reformulation of the SGG-problem as an ill-posed pseudodifferential equation. Its solution is dealt within classical regularization methods, based on filtering techniques. Third, a very promising method is worked out for developing an immediate interrelation between the Earth’s gravitational potential at the Earth’s surface and the known gravitational tensor.

1

Introduction

Due to the nonspherical shape, the irregularities of its interior mass density, and the movement of the lithospheric plates, the external gravitational field of the Earth shows significant variations. The recognition of the structure of the Earth’s gravitational potential is of tremendous importance for many questions in geosciences, for example, the analysis of present day tectonic motions, the study of the Earth’s interior, models of deformation analysis, the determination of the sea surface topography, and circulations of the oceans, which, of course, have a great influence on the global climate and its change. Therefore, a detailed knowledge of the global gravitational field including the local high-resolution microstructure is essential for various scientific disciplines. Satellite gravity gradiometry (SGG) is a modern domain of studying the characteristics, the structure, and the variation process of the Earth’s gravitational field. The principle of satellite gradiometry can be explained roughly by the following model (cf. Fig. 1): several test masses in a low orbiting satellite feel, due to their distinct positions and the local changes of the gravitational field, different forces, thus yielding different accelerations. The measurements of the relative accelerations

Fig. 1 The principle of a gradiometer

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between two test masses provide information about the second-order partial derivatives of the gravitational potential. To be more concrete, differences between the displacements of opposite test masses are measured. This yields information on the differences of the forces. Since the gradiometer itself is small, these differences can be identified with differentials so that a so-called full gradiometer gives information on the whole tensor consisting out of all second-order partial derivatives of the gravitational potential, i.e., the Hesse matrix. In an ideal case, the full Hesse matrix can be observed by an array of test masses. On 17 March 2009, the European Space Agency (ESA) began to realize the concept of SGG with the launch of the most sophisticated mission ever to investigate the Earth’s gravitational field, viz. GOCE (Gravity Field and Steady-State Ocean Circulation Explorer). ESA’s 1-ton spacecraft carries a set of six state-of-the-art, high-sensitivity accelerometers to measure the components of the gravity field along all three axes (see the contribution of R. Rummel in this issue for more details on the measuring devices of this satellite). GOCE produced a coverage of the entire Earth with measurements (apart from gaps at the polar regions). For around 20 months, GOCE gathered gravitational data. After running out of propellant, the GOCE satellite begun dropping out of this orbit in October 2013 and made an uncontrolled reentry on 11 November 2013. In order to make this mission successful, ESA and its partners had to overcome an impressive technical challenge by designing a satellite that is orbiting the Earth close enough (at an altitude of only 250 km) to collect high-accuracy gravitational data while being able to filter out disturbances caused, e.g., by the remaining traces of the atmosphere. It is not surprising that, during the last decade, the ambitious mission GOCE motivated many scientific activities such that a huge number of written material is available in different fields concerned with special user group activities, mission synergy, calibration as well as validation procedures, geoscientific progress (in fields like gravity field recovery, ocean circulation, hydrology, glaciology, deformation, climate modeling, etc.), data management, and so on. A survey about the recent status is well demonstrated by the “ESA Living Planet Programme”, which also contains a list on GOCE-publications (see also the contribution by the ESA-Frascati Group in this issue, for information from geodetic point of view the reader is referred, e.g., to the notes (Beutler et al. 2003; ESA 1999, 2007; Rummel et al. 1993), too). Mathematically, the literature dealing with the solution procedures of problems related to SGG can be divided essentially into two classes: the timewise approach and the spacewise approach. The former one considers the measured data as a time series, while the second one supposes that the data are given in advance on a (closed) surface. This chapter is part of the spacewise approach. Its goal is a potential theoretically reflected approach to SGG with strong interest in the characterization of SGG-data types and tensorial oriented solution of the occurring (pseudodifferential) SGGequations by regularization. Particular emphasis is laid on the transition from scalar data types (such as the second-order radial derivative) to full tensor data of the Hesse matrix.

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SGG in Potential Theoretic Perspective

Gravity as observed on the Earth’s surface is the combined effect of the gravitational mass attraction and the centrifugal force due to the Earth’s rotation. The force of gravity provides a directional structure to the space above the Earth’s surface. It is tangential to the vertical plumb lines and perpendicular to all level surfaces. Any water surface at rest is part of a level surface. As if the Earth were a homogeneous, spherical body gravity turns out to be constant all over the Earth’s surface, the wellknown quantity 9.8 ms2 . The plumb lines are directed toward the Earth’s center of mass, and this implies that all level surfaces are nearly spherical, too. However, the gravity decreases from the poles to the equator by about 0.05 ms2 . This is caused by the flattening of the Earth’s figure and the negative effect of the centrifugal force, which is maximal at the equator. Second, high mountains and deep ocean trenches cause the gravity to vary. Third, materials within the Earth’s interior are not uniformly distributed. The irregular gravity field shapes as virtual surface, the geoid. The level surfaces are ideal reference surfaces, for example, for heights. In more detail, the gravity acceleration (gravity) w is the resultant of gravitation v and centrifugal acceleration c, i.e., w D v C c. The centrifugal force c arises as a result of the rotation of the Earth about its axis. We assume here, a rotation of constant angular velocity !0 about the rotational axis x3 , which is further assumed to be fixed with respect to the Earth. The centrifugal acceleration acting on a unit mass is directed outward perpendicularly to the spin axis. If the  3 -axis of an Earthfixed coordinate system coincides with the axis of rotation, then we have c.x/ D !02  3 ^. 3 ^x/. Using the so-called centrifugal potential C.x/ D .1=2/!02 .x12 Cx22 / we can write c D rC . The direction of the gravity w is known as the direction of the plumb line, the quantity jwj is called the gravity intensity (often just gravity). The gravity potential of the Earth can be expressed in the form: W D V CC . The gravity acceleration w is given by w = rW D rV C rC . The surfaces of constant gravity potential W .x/ D const, x 2 R3 , are designated as equipotential (level, or geopotential) surfaces of gravity. The gravity potential W of the Earth is the sum of the gravitational potential V and the centrifugal potential C, i.e., W D V C C . In an Earth’s fixed coordinate system, the centrifugal potential C is explicitly known. Hence, the determination of equipotential surfaces of the potential W is strongly related to the knowledge of the potential V . The gravity vector w given by w.x/ D rx W .x/ where the point x 2 R3 is located outside and on a sphere around the origin with Earth’s radius R, is normal to the equipotential surface passing through the same point. Thus, equipotential surfaces intuitively express the notion of tangential surfaces, as they are normal to the plumb lines given by the direction of the gravity vector (for more details see, for example, Heiskanen and Moritz (1967), Freeden and Schreiner (2009) and the contributions by H. Moritz in this issue). According to the classical Newton’s Law of Gravitation (1687), knowing the density distribution  of a body, the gravitational potential can be computed everywhere in R3 . More explicitly, the gravitational potential V of the Earth’s exterior is given by

Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution

Z

.y/ dV .y/; jx  yj

V .x/ D G Earth

343

x 2 R3 nEarth;

(1)

where G is the gravitational constant .G D 6:6742  1011m3 kg1 s2 / and dV is the (Lebesgue-) volume measure. The properties of the gravitational potential (1) in the Earth’s exterior are appropriately described by the Laplace equation: V .x/ D 0; x 2 R3 nEarth:

(2)

The gravitational potential V as defined by (1) is regular at infinity, i.e.,  jV .x/j D O

 1 ; jxj ! 1: jxj

(3)

For practical purposes, the problem is that in reality the density distribution  is very irregular and known only for parts of the upper crust of the Earth. It is actually so that geoscientists would like to know it from measuring the gravitational field. Even if the Earth is supposed to be spherical, the determination of the gravitational potential by integrating Newton’s potential is not achievable. This is the reason why, in simplifying spherical nomenclature, we first expand the so-called reciprocal distance in terms of harmonics (related to the Earth’s mean radius R) as a series 1 2nC1 X X 4R 1 R R .x/Hn;k .y/; D Hn1;k jx  yj 2n C 1 nD0 j D1

(4)

R is an inner harmonic of degree n and order k given by where Hn;k

R Hn;k .x/ D

1 R



jxj R

n Yn;k . /; x D jxj ; 2 ;

(5)

R and Hn1;k is an outer harmonic of degree n and order k given by

R Hn1;k .x/ D

1 R



R jxj

nC1 Yn;k . /; x D jxj ; 2 

( is the unit sphere in R3 ). Note that the family fYn;k g

nD0;1;::: kD1;:::;2nC1

(6) is an L2 ./-

orthonormal system of scalar spherical harmonics (for more details concerning spherical harmonics see, e.g., Müller (1966), Freeden et al. (1998), Freeden and Schreiner (2009), Freeden and Gerhards (2013), and Freeden and Gutting (2013)). Insertion of the series expansion (4) into Newton’s formula for the external gravitational potential yields

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W. Freeden and M. Schreiner 1 2nC1 X X 4R Z R R V .x/ D G .y/ Hn;k .y/ dV .y/ Hn1;k .x/: int 2n C 1  R nD0

(7)

kD1

The expansion coefficients of the series (7) are not computable, since their determination requires the knowledge of the density function  in the Earth’s interior (see the introductory chapter and the contribution of V. Michel in this issue). In fact, it turns out that there are infinitely many mass distributions, which have the given gravitational potential of the Earth as exterior potential. Nevertheless, collecting the results from potential theory on the Earth’s gravitational field v for the outer space (in spherical approximation) we are confronted with the following (mathematical) characterization: v is an infinitely often differentiable vector field in the exterior of the Earth such that (v1) div v D r  v= 0, curl v = L v = 0 in the Earth’s exterior, (v2) v is regular at infinity: jv.x/j D O 1=.jxj2 / ; jxj ! 1. Seen from mathematical point of view, the properties (v1) and (v2) imply that the Earth’s gravitational field v in the exterior of the Earth is a gradient field v D rV , where the gravitational potential V fulfills the properties: V is an infinitely often differentiable scalar field in the exterior of the Earth such that (V1) V is harmonic in the Earth’s exterior, and vice versa. Moreover, the gradient field of the Earth’s gravitational field (i.e., the Jacobi matrix field) v = r v, obeys the following properties: v is an infinitely often differentiable tensor field in the exterior of the Earth such that (v1) div v = r v = 0, curl  v D 0 in the Earth’s exterior, (v2) v is regular at infinity: jv.x/j D O 1=.jxj3 / ; jxj ! 1, and vice versa. Combining our identities we finally see that v can be represented as the Hesse tensor of the scalar field V, i.e., v D r ˝ rV D r .2/ V. The technological SGG-principle of determining the tensor field v at satellite altitude is illustrated graphically in Fig. 2. The position of a low orbiting satellite is tracked using GPS. Inside the satellite there is a gradiometer. A simplified model of a gradiometer is sketched in Fig. 1. The photo of the GOCE satellite is contained in the contribution of R. Rummel in this issue. An array of test masses is connected with springs. Once more, the measured quantities are the differences between the displacements of opposite test masses. According to Hooke’s law, the mechanical configuration provides information on the differences of the forces. They, however, are due to local differences of rV . Since the gradiometer itself is small, these differences can be identified with differentials, so that a so-called full gradiometer gives information on the whole tensor consisting out of all second-order partial derivatives of V , i.e., the Hesse matrix v of V . From our preparatory remarks, it becomes obvious that the potential theoretic situation for the SGG-problem can be formulated briefly as follows: Suppose the satellite data v = r ˝ r V are known continuously over the “orbital surface,” the satellite gravity gradiometry problem amounts to the problem of determining V from v D r ˝ rV at the “orbital surface.” Mathematically, SGG is a nonstandard problem of potential theory. The reasons are obvious:

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GPS satellites

Gradiometry

Mass anomaly

Earth

Fig. 2 The principle of satellite gravity gradiometry (From ESA (1999))

• SGG is ill-posed since the data are not given on the boundary of the domain of interest, i.e., on the Earth’s surface but on a surface in the exterior domain of the Earth, i.e., at a certain height. • Tensorial SGG-data (or scalar manifestations of them) do not form the standard equipment of potential theory (such as, e.g., Dirichlet or Neumann data). Thus, it is – at first sight – not clear whether these data ensure the uniqueness of the SGG-problem or not. • SGG-data have its natural limit because of the strong damping of the highfrequency parts of the (spherical harmonic expansion of the) gravitational potential with increasing satellite heights. For a heuristic explanation of this calamity, let us start from the assumption that the gravitational potential outside the spherical Earth’s surface R with the mean radius R is given by the ordinary expansion in terms of outer harmonics (confer the identity (7))

V .x/ D

1 2nC1 X XZ nD0 kD1

R

R R V .y/Hn1;k .y/d !.y/Hn1;k .x/

(8)

(d! is the usual surface measure). Then it is not hard to see that those parts of R the gravitational potential belonging to the outer harmonics Hn1;k of order n at height H above the Earth’s surface R are damped by a factor ŒR=.R C H / nC1 . Just a way out of this difficulty is seen in SGG, where, e.g., second-order radial derivatives of the gravitational potential are available at a height of typically about 250 km. The second derivatives cause (roughly speaking) an amplification of the potential coefficients by a factor of order n2 . This compensates the

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damping effect due to the satellite’s height if n is not too large. Nevertheless, in spite of the amplification, the SGG-problem still remains (exponentially) ill-posed. Altogether, the gravitational potential decreases exponentially with increasing height, and therefore the process of transforming, the data down to the Earth surface (usually called “downward continuation”) is unstable. The non-canonical (SGG)-situation of uniqueness within the potential theoretic framework can be demonstrated already by a simple example in spherical context: Suppose that one scalar component of the Hesse tensor is prescribed for all points x at the sphere RCH D fx 2 R3 j jxj D R C H g. Is the gravitational potential V unique on the sphere R D fx 2 R3 j jxj D Rg? The answer is not positive, in general. To see this, we construct a counterexample: If b 2 R3 with jbj = 1 is given, the second-order directional derivative of V at the point x is b T r ˝ rV .x/b. Given a potential V , we construct a vector field b on RCH , such that the second-order directional derivative b T r ˝r Vb is zero: Assume that V is a solution of (2) and (3). For each x 2 RCH , we know that the Hesse tensor r ˝ rV .x/ is symmetric. Thus, there exists an orthogonal matrix A.x/ so that A.x/T .r ˝ rV .x//A.x/ D diag.1 .x/; 2 .x/; 3 .x//, where 1 .x/; 2 .x/; 3 .x/ are the eigenvalues of r ˝ rV .x/. From the harmonicity of V it is clear that 0 D rV .x/ D 1 .x/ C 2 .x/ C 3 .x/. Let 0 D 31=2 .1; 1; 1/T . We define the vector field  W RCH ! R3 by .x/ D A.x/0 ; x 2 RCH . Then we obtain T .x/.r ˝ rV .x//.x/ D T0 A.x/T .r ˝ rV .x//A.x/0

(9)

0

10 1 1 .x/ 0 1 0 1 @ A @ D 3.1 1 1/ 1A 0 2 .x/ 0 1 0 0 3 .x/ D 13 .1 .x/ C 2 .x/ C 3 .x// D 0:

(10)

Hence, we have constructed a vector field  such that the second-order directional derivative of V in the direction of .x/ is zero for every point x 2 RCH . It can be easily seen that, for a given V , there exist many vector fields showing the same properties for uniqueness as the vector field . Observing these arguments we are led to the conclusion that the function V is undetectable from the directional derivatives corresponding to  (see also Schreiner 1994a,b). It is, however, good news that we are not lost here: As a matter of fact, there do exist conditions under which only one quantity of the Hesse tensor yields a unique solution (at least up to low order harmonics). In order to formulate these results, a certain decomposition of the Hesse tensor is necessary, which strongly depends on the separation of the Laplace operator in terms of polar coordinates. In order to follow this path, we start to reformulate the SGG-problem more easily in spherical context. For that purpose we start with some basic facts specifically formulated on the unit sphere  D fx 2 R3 j jxj D 1g. As is well-known, any x 2 R3 ; x ¤ 0,

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can be decomposed uniquely in the form x D r , where the directional part is an element of the unit sphere: 2 . Let fYn;mg W  ! R3 , n D 0, 1, . . . , m D 1, . . . , 2n + 1, be an orthonormal set of spherical harmonics. As is well-known (see, e.g., Freeden and Schreiner 2009), the system is complete in L2 ./, hence, each function F 2 L2 ./ can be represented by the spherical harmonic expansion F . / D

1 2nC1 X X

F ^ .n; m/Yn;m . /; 2 ;

(11)

nD0 mD1

with “Fourier coefficients” given by Z ^

F .n; m/ D .F; Yn;m /L2 ./ D

F . /Yn;m . / d !. /:

(12)



Furthermore, the (outer) harmonics Hn1;m W R3 nf0g ! R related to the unit 1 1 sphere  are denoted by Hn1;m .x/ D Hn1;m .x/, where Hn1;m .x/ D nC1 .1=jxj /Yn;m .x=jxj/. Clearly, they are harmonic functions and their restrictions coincide on  with the corresponding spherical harmonics. Any function F 2 L2 ./ can, thus, be identified with a harmonic potential via the expansion (11), in particular, this holds true for the Earth’s external gravitational potential. This motivates the following mathematical model situation of the SGG-problem to be considered next: (i) Isomorphism: Consider the sphere R  R3 around the origin with radius R > ext 0. int R is the inner space of R , and R is the outer space. By virtue of the isomorphism  3 7! R 2 R we assume functions F W R ! R to be defined on . It is clear that the function spaces defined on  admit their natural generalizations as spaces of functions defined on R . Obviously, an L2 ./-orthonormal system of spherical harmonics forms an orthogonal system on R (with respect to .; /L2 .R / ). Moreover, with the relationship $ R , the differential operators on R can be related to operators on the unit sphere . In more detail, the surface gradient r IR , the surface curl gradient LIR and the Beltrami operator IR on R , respectively, admit the representation r IR D .1=R/r I1 D .1=R/r , LIR D (1/R/LI1 D (1/R/L , IR D (1/R2 /I1 = (1/R2 / , where r  , L ,  are the surface gradient, surface curl gradient, and the Beltrami operator of the unit sphere , respectively. For Yn being a spherical harmonic of degree n we have IR Yn D .1=R2 /n.n C 1/Yn D .1=R2 / Yn . (ii) Runge Property: Instead of looking for a harmonic function outside and on the (real) Earth, we search for a harmonic function outside the unit sphere  (assuming the units are chosen in such a way that the sphere  with radius 1 is inside of the Earth and at the same time not too “far away” from the Earth’s boundary). The justification of this simplification (see Fig. 3) is based on the Runge approach (see, e.g., Freeden 1980a; Freeden and Michel 2004): To any

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Fig. 3 The role of the “Runge sphere” within the spherically reflected SGG-problem ΩR+H

ΩR H

R Earth

harmonic function V outside of the (real) Earth and any given " > 0, there exists a harmonic function U outside of a sphere inside the (real) Earth such that the absolute error jV .x/  U .x/j < " holds true for all points x outside and on the (real) Earth’s surface.

3

Decomposition of Tensor Fields by Means of Tensor Spherical Harmonics

Let us recapitulate that any point 2  may be represented by polar coordinates in a standard way p 1  t 2 .cos '  1 C sin '  2 /; 1  t  1; 0  ' < 2; t D cos #; (13) (# 2 Œ0;  : (co-)latitude, ': longitude, t: polar distance). Consequently, any element 2  may be represented using its coordinates .'; t/ in accordance with (13). For the representation of vector and tensor fields on the unit sphere , we are led to use a local triad of orthonormal unit vectors in the directions r, ', and t as shown by Fig. 4 (for more details the reader is referred to Freeden and Schreiner (2009) and the references therein). As is well known, the second-order tensor fields on the unit sphere, i.e., f W  ! R3 ˝ R3 , can be separated into their tangential and normal parts as follows: D t 3 C

p;nor f D .f / ˝ ;

(14)

pnor; f D ˝ . T f/;

(15)

p;tan f D f  p;nor f D f  .f / ˝ ;

(16)

Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution Fig. 4 Local triads  r ,  ,  t with respect to two different points and on the unit sphere

349

t (ξ)

r (ξ) f (ξ)

t (h)

f (h) r (h)

ptan; f D f  pnor; f D f  ˝ . T f/;

(17)

pnor;tan f D pnor; .p;tan f/ D p;tan .pnor; f/

(18)

D ˝ . T f/  . T f / ˝ : The operators pnor;nor , ptan;nor , and ptan;tan are defined analogously. A vector field f W  ! R3 ˝ R3 is called normal if f = pnor;nor f and tangential if f = ptan;tan f. It is called left normal if f = pnor; f, left normal/right tangential if f = pnor;tan f, and so on. The constant tensor fields itan and jtan can be defined using the local triads by itan D  ' ˝  ' C  t ˝  t ; jtan D ^ itan D  t ˝  '   ' ˝  t :

(19)

Spherical tensor fields can be discussed in an elegant manner by the use of certain differential processes. Let u be a continuously differentiable vector field on , i.e., u 2 C .1/ ./, given in its coordinate form by u. / D

3 X

Ui . / i ; 2 ; Ui 2 C .1/ ./:

(20)

i D1

Then we define the operators r  ˝ and L ˝ by r  ˝ u. / D

3 X

.r  Ui . // ˝  i ; 2 ;

(21)

 L Ui . / ˝  i ; 2 :

(22)

i D1

L ˝ u. / D

3  X i D1

Clearly, r  ˝ u and L ˝ u are left tangential. But it is an important fact, that even if u is tangential, the tensor fields r  ˝ u and L ˝ u are generally not tangential.

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It is obvious, that the product rule is valid. To be specific, let F 2 C .1/ ./ and u 2 C .1/ ./ (once more, note that the notation u 2 c.1/ ./ means that the vector field u is a continuously differentiable on ), then r  ˝ .F . /u. // D r  F . / ˝ u. / C F . /r  ˝ u. /;

2 :

(23)

In view of the above equations and definitions, we accordingly introduce operators o.i;k/:C.2/ () ! c.0/ ./ (note that c.0/ ./ is the class of continuous second-order tensor fields on the unit sphere ) by .1;1/

F . / D ˝ F . /;

(24)

.1;2/

F . / D ˝ r  F . /;

(25)

.1;3/

F . / D ˝ L F . /;

(26)

.2;1/

F . / D r  F . / ˝ ;

(27)

.3;1/

F . / D L F . / ˝ ;

(28)

.2;2/

F . / D itan . /F . /;

(29)

.2;3/

  F . / D r  ˝ r   L ˝ L F . / C 2r  F . / ˝ ;

(30)

.3;2/

  F . / D r  ˝ L C L ˝ r  F . / C 2L F . / ˝ ;

(31)

F . / D jtan . /F . /;

(32)

o o o o o o o o

.3;3/

o

2 : After our preparations involving spherical second-order tensor fields it is not difficult to prove the following lemma. Lemma 1. Let F W  ! R be sufficiently smooth. Then the following statements are valid: 1. 2. 3. 4. 5. 6.

o.1;1/ F is a normal tensor field. o.1;2/ F and o.1;3/ F are left normal/right tangential. o.2;1/ F and o.3;1/ F are left tangential/right normal. o.2;2/ F , o.2;3/ F , o.3;2/ F and o.3;3/ F are tangential. o.1;1/ F , o.2;2/ F , o.2;3/ F and o.3;2/ F are symmetric. o.3;3/ F is skew-symmetric.

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7. .o.1;2/ F /T = o.2;1/ F and .o.1;3/ F /T = o.3;1/ F . 8. For 2  8 < F . / for .i; k/ D .1; 1/; .i;k/ trace o F . / D 2F . / for .i; k/ D .2; 2/; : 0 for .i; k/ ¤ .1; 1/; .2; 2/: The tangent representation theorem (cf. Backus 1966, 1967) asserts that if ptan;tan f is the tangential part of a tensor field f 2 c.2/ (), as defined above, then there exist unique scalar fields F2;2 , F3;3 , F2;3 , F3;2 such that Z

Z F2;2 . / d !. / D 

F3;3 . / d !. / D 0;

(33)



Z

Z F3;2 . /. i  / d !. / D 

F2;3 . /. i  / d !. / D 0;

i D 1; 2; 3;

(34)



and ptan;tan f D o.2;2/ F2;2 C o.2;3/ F2;3 C o.3;2/ F3;2 C o.3;3/ F3;3 :

(35)

Furthermore, the following orthogonality relations may be formulated: Let F; G W .i;k/ .i 0 ;k 0 / F ( )  ! R be sufficiently smooth. Then for all 2 , we have o F ( )  o 0 0 .i;k/ = 0 whenever we have (i , k/ ¤ (i , k /. The adjoint operators O satisfying Z

Z o

.i;k/

F . / O .i;k/ f. / d !. /;

F . /  f. / d !. / D



(36)



for all sufficiently smooth functions F W  ! R and tensor fields f W  ! R3 ˝ R3 can be deduced by elementary calculations. In more detail, for f 2 c(2) (), we find (cf. Freeden and Schreiner 2009) .1;1/

f. / D T f. / ;

(37)

.1;2/

f. / D r   ptan . T f. //;

(38)

.1;3/

f. / D L  ptan . T f. //;

(39)

.2;1/

f. / D r   ptan .f. / /;

(40)

.3;1/

f. / D L  ptan .f. / /;

(41)

.2;2/

f. / D itan . /  f. /;

(42)

O O O O O O

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W. Freeden and M. Schreiner .2;3/

O

    f. / D r   ptan r   ptan; f. /  L  ptan L  ptan; f. / 2r   ptan .f. / /;

.3;2/

O

    f. / D L  ptan r   ptan; f. / C r   ptan L  ptan; f. / 2L  ptan .f. / /;

.3;3/

O

f. / D jtan . /  f. /;

(43)

(44) (45)

2 . Provided that F W  ! R is sufficiently smooth we see that .i 0 ;k 0 / .i;k/ o F . /

O

D 0 if .i; k/ ¤ .i 0 ; k 0 /;

(46)

whereas 8 ˆ ˆ ˆ ˆ ˆ ˆ
0, 2 , we obtain the following decomposition of the Hesse matrix on the sphere RCH , i.e., for x 2 R3 with jxj D R C H : .1;1/

r ˝ rHn1;m ..R C H / / D .n C 1/.n C 2/ .RCH1 /nC3 o

Yn;m . /

.n C 2/ .RCH1 /nC3   .1;2/ .2;1/ o Yn;m . / C o Yn;m . / .2;2/

1 o  .nC1/.nC2/ 2 .RCH /nC3 .2;3/

C 12 .RCH1 /nC3 o

(66)

Yn;m . /

Yn;m . /:

Keeping in mind, that any solution of the SGG-problem can be expressed as a series of outer harmonics and using the completeness of the spherical harmonics in the space of square-integrable functions on the unit sphere, it follows that the SGG-problem is uniquely solvable (up to some low order spherical harmonics) by

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the O .1;1/ , O .1;2/ , O .2;1/ , O .2;2/ , and O .2;3/ components. To be more specific, we are able to formulate the following theorem: Theorem 2. Let V satisfy the following condition V 2 Pot.C .0/ .//; i:e:;

 jV .x/j D O

V 2 C .0/ .ext / \ C (2) .ext /;

(67)

V .x/ D 0; x 2 ext ;

(68)

 1 ; jxj ! 1; uniformly for all directions: jxj

(69)

Then the following statements are valid: 1. O .i;k/ r ˝ r V ..R C H / / = 0 if (i; k/ 2 f.1; 3/; .3; 1/; .3; 2/; .3; 3/g. 2. O .i;k/ r ˝ r V ..R C H / / = 0 for (i; k/ 2 f.1; 1/; .2; 2/g if and only if V = 0. 3. O .i;k/ r ˝ r V ..R C H / / = 0 for (i; k/ 2 f.1; 2/; .2; 1/g if and only if Vj is constant. 4. O .2;3/ r ˝ r V ..R C H / / = 0 if and only if Vj is linear combination of spherical harmonics of degree 0 and 1. This theorem gives detailed information, which tensor components of the Hesse tensor ensure the uniqueness of the SGG-problem (see also the considerations due to Schreiner (1994a) and Freeden et al. (2002), Freeden and Nutz (2011)). Anyway, for a potential V of class P ot.C .0/ .// with vanishing spherical harmonic moments of degree 0 and 1 such as the Earth’s disturbing potential (see, e.g., Heiskanen and Moritz (1967) for its definition) uniqueness is assured in all cases (listed in Theorem 2). Since we now know at least in the spherical setting, which conditions guarantee the uniqueness of an SGG-solution we can turn to the question of how to find a solution and what we mean with a solution, since we have to take into account the ill-posedness. To this end, we are interested here in analyzing the problem step by step. We start with the reformulation of the SGG-problem as pseudodifferential equation on the sphere, give a short overview on regularization, and show how this ingredients can be composed together to regularize the SGG-data. In doing so, we find great help by discussing how classical boundary value problems in gravitational field of the Earth as well as modern satellite problems may be transferred into pseudodifferential equations, thereby always assuming the spherically oriented geometry. Indeed, it is helpful to treat the classical Dirichlet and Neumann boundary value problem as well as significant satellite problems such as satellite-to-satellite tracking (SST) and SGG.

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SGG as Pseudodifferential Equation

Let †  R3 be a regular surface, i.e., we assume the following properties: (i) † divides the Euclidean space R3 into the bounded region †int (inner space) and the unbounded region †ext (outer space) so that †ext D R3 n†int , † D †int \†ext with ø = †int \†ext , (ii) †int contains the origin, (iii) † is a closed and compact surface free of double points, (iv) † is locally of class C (2) (see Freeden and Schreiner (2004), Freeden and Gerhards (2013) for more details concerning regular surfaces). From our preparatory considerations (in particular, from the Introduction), it can be deduced that a gravitational potential of interest may be understood to be a member of the class V 2 Pot.C .0/ .†//, i.e.,

 jV .x/j D O

V 2 C .2/ .†ext / \ C .2/ .†ext /;

(70)

V .x/ D 0; x 2 †ext ;

(71)

 1 ; jxj ! 1; uniformly for all directions: jxj

(72)

Assume that R D fx 2 R3 j jxj D Rg is a (Runge) sphere with radius R around the origin, i.e., a sphere that lies entirely inside †, i.e., R  †int . On the class L2 .R / we impose the inner product .; /L2 .R / . Then we know that the functions  1 form an orthonormal set of functions on R , i.e., given F 2 L2 .R /, its Y R n;m R Fourier expansion reads F .x/ D

1 2nC1    x X X 1  F; Y ; Y n;m n;m R2 R L2 .R / R nD0 mD1

x 2 R :

(73)

Instead of considering potentials that are harmonic outside † and continuous on †, we now consider potentials that are harmonic outside R and that are of class L2 .R /. In accordance with our notation we define ( 2

Pot.L .R // D x 7!

1 2nC1 X X

1 R2

nD0 mD1 RnC1 Y jxjnC1 n;m

  F; Yn;m . R /



L2 .R /

x jxj



o j F 2 L2 .R / :

(74)

Clearly, Pot.L2 .R // is a “subset” of Pot.C 0 .†// in the sense that if V 2 Pot.L2 .R //, then V j†ext 2 Pot.C 0 .†//. The “difference” of these two spaces is not “too large”: Indeed, we know from the Runge approximation theorem (cf. Freeden 1980a), that for every " > 0 and every V 2 Pot.C 0 .†// there exists a VO 2Pot.L2 .R // such that supx2†ext jV .x/  VO .x/j < . Thus, in all geosciences, it is common (but not strictly consistent with the Runge argumentation) to identify

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R with the surface of the Earth and to assume that the restriction V jR is of class L2 .R /. Clearly, we have a canonical isomorphism between L2 .R / and Pot.L2 .R //, which is defined via the trace operator, i.e., the restriction to R and its harmonic continuation, respectively.

4.2

Upward/Downward Continuation

   1 is Let RCH be the sphere with radius R C H . The system RCH Yn;m RCH then orthonormal in L2 .RCH /. (We assume H to be the height of a satellite above the Earth’s surface.) Let F 2 Pot.L2 .R // be represented in the form   1 2nC1    X X 1  x RnC1 x 7! F; Yn;m : Yn;m 2 nC1 2 R R jxj jxj L .R / nD0 mD1

(75)

Then the restriction of F on RCH reads F jRCH

  1 2nC1    X X 1  x RnC1 F; Yn;m : W x 7! Yn;m R2 R L2 .R / .R C H /nC1 RCH nD0 mD1 (76)

  Hence, any element R1 Yn;m R of the orthonormal system in L2 .R / is mapped to a function Rn =.R C H /n 1/.R C H / Yn;m ( / .R C H //. The operation defined in such away is called upward continuation. It is representable by the pseudodifferential operator (for more details on pseudodifferential operators the reader should consult Svensson (1983), Schneider (1997), Freeden et al. (1998), and Freeden (1999), Freeden and Schreiner (2009) ƒR;H W L2 .R / ! L2 .RCH / up with associated symbol  ^ ƒR;H .n/ D up

Rn : .R C H /n

(77)

In other words, we have  ƒR;H up

      ^ 1  1 D ƒR;H Yn;m Y : .n/ n;m up R R RCH RCH

(78)

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W. Freeden and M. Schreiner

The image of ƒR;H up is given by Picard’s criterion (cf. Theorem 4): 

2 ƒR;H up .L .R //

D

1 2nC1 P P  .RCH /n 2 F 2 L2 .RCH /j Rn nD0 mD1      2 1  F; RCH Yn;m RCH 0; so that

Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution

367

lim R˛ ƒx D x for all x 2 H;

˛!0

i.e., the operators R˛ ƒ converge pointwise to the identity. From the theory of inverse problems (see, e.g., Kirsch 1996) it is also clear that if ƒ W H ! K is compact and H has infinite dimension (as it is the case for the application we have in mind), then the operators R˛ are not uniformly bounded, i.e., there exists a sequence (˛ j / with limj !1 ˛ j = 0 and jjR˛j jjL.K;H/ ! 1 for j ! 1:

(106)

Note that the convergence of R˛ ƒx in Definition 1 is based on y D ƒx, i.e., on unperturbed data. In practice, the right-hand side is affected by errors and then no convergence is achieved. Instead, one is (or has to be) satisfied with an approximate solution based on a certain choice of the regularization parameter. Let us discuss the error of the solution. For this purpose, we let y 2 R.ƒ/ be the (unknown) exact right-hand side and y ı 2 K be the measured data with jjy  y ı jjK < ı:

(107)

x ˛;ı D R˛ y ı ;

(108)

For a fixed ˛ > 0, we let

and look at x ˛;ı as an approximation of the solution x of ƒx D y. Then the error can be split as follows: jjx ˛;ı  xjjH D jjR˛ y ı  xjjH  jjR˛ y ı  R˛ yjjH C jjR˛ y  xjjH  jjR˛ jjL.K;H/ jjy ı  yjjK C jjR˛ y  xjjH ;

(109)

such that jjx ˛;ı  xjjH  ıjjR˛ jjL.K;H/ C jjR˛ ƒx  xjjH :

(110)

We see that the error between the exact and the approximate solution consists of two parts: The first term is the product of the bound for the error in the data and the norm of the regularization parameter R˛ . This term will usually tend to infinity for ˛ ! 0 if the inverse ƒ1 is unbounded and ƒ is compact (cf. (106)). The second term denotes the approximation error jj.R˛  ƒ1 /yjjH for the exact right-hand side y = ƒx. This error tends to zero as ˛ ! 0 by the definition of a regularization strategy. Thus, both parts of the error show a diametrically oriented behavior. A typical picture of the errors in dependence on the regularization parameter ˛ is

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Fig. 9 Typical behavior of the total error in a regularization process

Error

Total error δ Ra

Ra Λ x

–x

H

L ( K,H )

a

sketched in Fig. 9. Thus, a strategy is needed to choose ˛ dependent on ı in order to keep the error as small as possible, i.e., we would like to minimize ıjjR˛ jjL.K;H/ C jjR˛ ƒx  xjjH :

(111)

In principle, we distinguish two classes of parameter choice rules: If ˛ = ˛(ı) does not depend on ı, we call ˛ = ˛(ı) an a priori parameter choice rule. Otherwise ˛ also depends on y ı and we call ˛ = ˛(ı, y ı / an a posteriori parameter choice rule. It is usual to say a parameter choice rule is convergent, if for ı ! 0 the rule is such that lim supfjjR˛.ı;y ı / y ı  T C yjjH jy ı 2 K; jjy ı  yjjK  ıg D 0

ı!0

(112)

and lim supf˛.ı; y ı / jy ı 2 K; jjy  y ı jjK  ıg D 0:

ı!0

(113)

We stop here the discussion of parameter choice rules. For more material the interested reader is referred to, e.g., Engl et al. (1996) and Kirsch (1996). The remaining part of this section is devoted to the case that ƒ is compact, since then we gain benefits from the spectral representations of the operators. If ƒ W H ! K is compact, a singular system ( n ; vn , un / is defined as follows: fn2 gn2N are the nonzero eigenvalues of the self-adjoint operator ƒ*ƒ (ƒ* is the adjoint operator of ƒ/, written down in decreasing order with multiplicity. The family fvn gn2N constitutes a corresponding complete orthonormal system of eigenvectors of ƒ*ƒ. We let  n > 0 and define the family fun gn2N via un D ƒvn =jjƒvn jjK . The sequence fun gn2N forms a complete orthonormal system of eigenvectors of ƒƒ*, and the following formulas are valid: ƒvn D n un ;

(114)

Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution

ƒ un D n vn ; ƒx D

1 X

369

(115)

n .x; vn /H un ; x 2 H;

(116)

nD1

ƒ y D

1 X

n .y; un /K vn ; y 2 K:

(117)

nD1

The convergence of the infinite series is understood with respect to the Hilbert space norms under consideration. The identities (116) and (117) are called the singular value expansions of the corresponding operators. If there are infinitely many singular values, they accumulate (only) at 0, i.e., limn!1  n = 0. Theorem 4. Let (n ; vn , un / be a singular system for the compact linear operator ƒ, y 2 K. Then we have y 2 d.ƒC / if and only if

1 X j.y; un /K j2 nD1

n2

< 1;

(118)

and for y 2 D.ƒC / it holds C

ƒ yD

1 X .y; un /K nD1

n

vn :

(119)

The condition (118) is the Picard criterion. It says that a best-approximate solution of ƒx D y exists only if the Fourier coefficients of y decay fastly enough relative to the singular values. The representation (119) of the best-approximate solution motivates a method for the construction of regularization operators, namely by damping the factors 1/ n in such a way that the series converges for all y 2 K. We are looking for filters q W .0; 1/  .0; jjƒjjL.H;K// ! R

(120)

such that

R˛ y D

1 X q.˛; n / nD1

n

.y; un /K vn ;

y 2 K;

is a regularization strategy. The following statement is known, e.g., from Kirsch (1996).

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Theorem 5. Let ƒ W H ! K be compact with singular system (n ; vn , un ). Assume that q from (120) has the following properties: 1. jq(˛,)j  1 for all ˛ > 0 and 0 <   jjƒjjL.H;K/ : 2. For every ˛ > 0 there exists a c(˛) so that jq(˛,)j  c(˛) for all 0 <   jjƒjjL.H;K/ : 3. lim˛!0 q(˛; ) = 1 for every 0    jjƒjjL.H;K/ : Then the operator R˛ W K ! H, ˛ > 0, defined by R˛ y D

1 X q.˛; n /

n

nD1

.y; un /K vn ;

y 2 K;

(121)

is a regularization strategy with jjR˛ jjL.K;H/  c.˛/: The function q is called a regularizing filter for ƒ. Two important examples should be mentioned: q.˛; / D

2 ˛ C 2

(122)

defines the Tikhonov regularization, whereas  q.˛; / D

1;  2  ˛; 0;  2 < ˛;

(123)

leads to the regularization by truncated singular value decomposition.

4.10

Regularization of the Exponentially Ill-Posed SGG-Problem

We are now in the position to have a closer look at the role of the regularization techniques particularly working for the SGG-problem. In (95), the SGG-problem is formulated as pseudodifferential equation: Given G 2 L2 ./, find F 2 L2 ./ so that ƒR;H SGG F D G with ^  ƒR;H .n/ D SGG

.n C 1/.n C 2/ Rn : n .R C H / .R C H /2

(124)

Switching now to a finite dimensional space (which is then the realization of the regularization by a singular value truncation), we are interested in a solution of the representation FN D

N 2nC1 X X nD1 mD1

F ^ .n; m/Yn;m :

(125)

Satellite Gravity Gradiometry (SGG): From Scalar to Tensorial Solution

371

Using a decomposition of G of the form

GD

N 2nC1 X X

G ^ .n; m/Yn;m ;

(126)

nD1 mD1

we end up with the spectral equations  ^ ƒR;H .n/F ^ .n; m/ D G ^ .n; m/; n D 1; : : : ; N; m D 1; : : : ; 2n C 1: SGG (127) In other words, in connection with (125) and (126), we find the relations F ^ .n; m/ D 

G ^ .n; m/ ; n D 1; : : : ; N; m D 1; : : : ; 2n C 1: ^ ƒR;H .n/ SGG

(128)

For the realization of this solution we have to find the coefficients G ^ .n, m/. Of course, we are confronted with the usual problems of integration, aliasing, and so on. The identity (128) also opens the perspective for SGG-applications by bandlimited regularization wavelets in Earth’s gravitational field determination. For more details, we refer to Freeden et al. (1997), Schneider (1996, 1997), Freeden and Schneider (1998), Glockner (2002), Hesse (2003), and Freeden and Nutz (2011). The book written by Freeden (1999) contains non-bandlimited versions of (harmonic) regularization wavelets. Multiscale regularization by use of spherical up functions is the content of the papers by Freeden and Schreiner (2004) and Schreiner (2004).

5

Future Directions

The regularization schemes described above are based on the decomposition of the Hesse tensor at satellite’s height into scalar ingredients due to geometrical properties (normal, tangential, mixed) as well as to analytical properties originated by differentiation processes involving physically defined quantities (such as divergence, curl, etc). SGG-regularization, however, is more suitable and effective if it is based on algorithms involving the full Hesse tensor such as from the GOCE mission (for more insight into the tensorial decomposition of GOCE-data, the reader is referred to the contribution of R. Rummel in this issue). In addition, see Rummel and van Gelderen (1992) and Rummel (1997). Our context initiates another approach to tensor spherical harmonics. Based on cartesian operators (see Freeden and Schreiner 2009), the construction principle .i;k/ starts from operators oQ n ; i; k 2 f1; 2; 3g given by

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W. Freeden and M. Schreiner

    oQ .1;1/ F .x/ D .2n C 3/x  jxj2 rx ˝ .2n C 1/x  jxj2 rx F .x/; n

(129)

  oQ .1;2/ F .x/ D .2n  1/x  jxj2 rx ˝ rx F .x/; n

(130)

  oQ .1;3/ F .x/ D .2n C 1/x  jxj2 rx ˝ .x ^ rx /F .x/; n

(131)

  oQ .2;1/ F .x/ D rx ˝ .2n C 1/x  jxj2 rx F .x/; n

(132)

oQ .2;2/ F .x/ D rx ˝ rx F .x/; n

(133)

oQ .2;3/ F .x/ D rx ˝ .x ^ rx /F .x/; n

(134)

  oQ .3;1/ F .x/ D .x ^ rx / ˝ .2n C 1/x  jxj2 rx F .x/; n

(135)

oQ .3;2/ F .x/ D .x ^ rx / ˝ rx /F .x/; n

(136)

oQ .3;3/ F .x/ D .x ^ rx / ˝ .x ^ rx /F .x/ n

(137)

for x 2 R3 and sufficiently smooth functions F W R3 ! R: Elementary calculations in cartesian coordinates lead us in a straightforward way to the following result. Lemma 2. Let Hn ; n 2 N0 , be a homogeneous harmonic polynomial of degree n. .i;k/ Then, oQ n Hn is a homogeneous harmonic tensor polynomial of degree deg.i;k/ .n/, where 8 ˆ n2 ˆ ˆ ˆ ˆ b0 T Df

for u D b0   1 for u ! 1; T .u/ D O u

(2) (3) (4)

where f is assumed to be a known square-integrable function, f 2 L2 . The asymptotic condition in Eq. (4) means that the harmonic function T approaches zero at infinity. The solution of the Laplace equation (2) can be written in terms of ellipsoidal harmonics as follows (Heiskanen and Moritz 1967; Moon and Spencer 1961; Thong and Grafarend 1989):   Qj m i Eu   Yj m./; T .u; / D Tj m Qj m i bE0 j D0 mDj j 1 X X

(5)

  where Qj m i Eu are the Legendre function of the second kind, Yj m./ are complex spherical harmonics of degree j and order m, and Tj m are coefficients to be determined from the boundary condition of Eq. (3). Substituting Eq. (5) into Eq. (3), and expanding f ./ in a series of spherical harmonics, f ./ D

Z j 1 X X j D0 mDj

0

f .0 /Yjm .0 /d 0 Yj m./

(6)

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E.W. Grafarend et al.

where 0 is the full solid angle and d  D sin#d #d , and comparing the coefficients at spherical harmonics Yj m ./ in the result, one gets Z f .0 /Yjm.0 /d 0

Tj m D

(7)

0

for j D 0; 1; : : : , and m D j; : : : ; j where Yjm is the conjugate complex spherical harmonic of Yj m . Furthermore, substituting coefficients Tj m into Eq. (5) and interchanging the order of summation over j and m and integration over 0 due to the uniform convergence of the series expansion given by Eq. (5), the solution to the ellipsoidal Dirichlet boundary-value problem, Eqs. (2)–(4), formally reads Z

  Qj m i Eu   Yjm .0 /Yj m./d 0 f . / Tj m Qj m i bE0 j D0 mDj 0

T .u; / D 0

j 1 X X

(8)

From a practical point of view, the spectral form of Eq. (8) of the solution to the Dirichlet problem given by Eqs. (2)–(4) may often become inconvenient, since the construction of Qj m.z/ functions and their summation up to high degrees and orders (j 104  105 ) is time consuming and numerically unstable (Sona 1995). Moreover, in the case that the level ellipsoid u D b0 deviates from a sphere by only a tiny amount, which is the case for the Earth, the solution of the problem should be close to the solution to the same problem but formulated on a sphere. One should thus attempt to rewrite T .u; / as a sum of the well-known Poisson integral (Kellogg 1929, Sect. IX.4), which solves the Dirichlet problem on a sphere, plus the corrections due to the ellipticity of the boundary. An evident advantage of such a decomposition is that existing theories as well as numerical codes for solving the Dirichlet problem on a sphere can simply be corrected for the ellipticity of the boundary.

2.2

Power-Series Representation of the Integral Kernel

Thong (1993) and Martinec and (1997) showed that the Legendre  Grafarend  function of the second kind, Qj m i Eu , can be developed in an infinite power series of the first eccentricity e: 1  u .j C m/Š j C1 X e D .1/m.j C1/=2 aj mk e 2k Qj m i E .2j C 1/ŠŠ

(9)

kD0

where coefficients aj mk can, for instance, be defined by the recurrence relation as aj m0 D 1

(10)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

aj mk D

.j C 2k  1/2  m2 aj mk1 2k.2j C 2k C 1/

for k  1:

389

(11)

Throughout this chapter, it is assumed that the eccentricity e0 of the reference ellipsoid u D b0 e0 D q

E b02

(12)

C E2

is less than 1. Then, for points .u; / being outside or on the reference ellipsoid, i.e., when u D b0 , the series in Eq. (9) is convergent. By Eq. (9), the ratio of the Legendre functions of the second kind in Eq. (8) reads 



Qj m i Eu   Qj m i bE0

 D

e e0

j C1

1 P

aj mk e 2k

kD0 1 P kD0

:

(13)

aj mk e02k

Dividing the polynomials in Eq. (13) term by term, one can write 1 P kD0 1 P kD0

aj mk e 2k D1C aj mk e02k

1 X

bj mk ;

(14)

kD1

where the explicit forms of the first few constituents read   (15) bj m1 D aj m1 e 2  e02 ;  4    (16) bj m2 D aj m2 e  e04  aj2 m1 e02 e 2  e02 ;  6      bj m3 D aj m3 e  e06  aj m2 aj m1 e02 e 4 C e 2 e02  2e04 C aj3 m1 e04 e 2  e02 : (17) Generally,   bj mk D O e 2r e02s ; r C s D k:

(18)

To get an analytical expression for the kth term of the series in Eq. (14), some cumbersome algebraic manipulations have to be performed. To avoid them, confine to the case where the computation point ranges in a limited layer above the reference ellipsoid (for instance, topographical layer), namely, b0 < u < b0 C9;000 m, which includes all the actual topographical masses of the Earth. For this restricted case, which is, however, often considered when geodetic boundary-value problems are

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E.W. Grafarend et al.

solved, express the first eccentricity e of the computation point by means of the first eccentricity e0 of the reference ellipsoid and a quantity "; " > 0, as e D e0 .1  "/:

(19)

Assuming b0 < u < b0 C 9;000 m means that " < 1:4  103 , and one can put approximately e 2k D e02k .1  2k"/;

(20)

bj m1 D 2"e02 aj m1 :

(21)

This allows one to write 1 P kD0 1 P kD0

1 P

aj mk e 2k D1 aj mk e02k

kD1 2" 1 P

kaj mk e02k

kD0

:

(22)

aj mk e02k

Expand the fraction on the right-hand side of the last equation (divided by aj m1 ) into an infinite power series e02 : 1 P

1 aj m1

kD1 1 P

kaj mk e02k

kD0

D aj mk e02k

1 X

ˇj mk e02k :

(23)

kD1

To find the coefficients ˇj mk , rewrite the last equation in the form 1 1 X

aj m1

kaj mk e02k D

kD1

1 X

aj mk e02k

kD0

1 X

ˇj ml e02l :

(24)

lD1

Since both the series on the right-hand side are absolutely convergent, their product may be rearranged as 1 X

aj mk e02k

kD0

1 X

ˇj ml e02l D

lD1

1 X k X

ˇj ml aj m;kl e02k :

(25)

kD1 lD1

Substituting Eq. (25) into Eq. (24) and equating coefficients at e02k on both sides of Eq. (24), one obtains X aj mk D ˇj ml aj mkl ; aj m1 k

k

lD1

k D 1; 2 : : :

(26)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

391

which yields the recurrence relation for ˇj mk : aj mk X  ˇj ml aj m;kl ; aj m1 k1

ˇj mk D k

k D 2; 3 : : :

(27)

lD1

with the starting value ˇj m1 D 1:

(28)

With the recurrence relation in Eq. (27), one may easily construct the higher coefficients ˇj mk : aj m2  aj m1 ; aj m1 aj m3 D3  3aj m2 C aj2 m1 : aj m1

ˇj m2 D 2

(29)

ˇj m3

(30)

This process may be continued infinitely. Important properties of the coefficients ˇj mk are 1 D ˇj m1 > ˇj m2 > ˇj m3 >     0;

(31)

ˇjj k > ˇjj 1k >    > ˇj 0k :

(32)

Figure 1 demonstrates these properties for j D 30. Fig. 1 Coefficients ˇj mk for j D 30, m D 0; 20; 30, and k D 1; : : :; 100

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E.W. Grafarend et al.

Finally, substituting the series Eq. (23) into Eq. (22) and using Eqs. (21) and (28) for bj m1 and ˇj m1 , respectively, yields 1 P kD0 1 P kD0

2.3

aj mk e 2k D 1 C bj m1 1 C aj mk e02k

1 X

! ˇj mk e02k2

:

(33)

kD2

The Approximation of O.e20 /

The harmonic upward or downward continuation of the potential or the gravitation between the Geoid to the Earth’s surface is an example of the practical application of the boundary-value problem Eqs. (2)–(4) (Engels et al. 1993; Martinec 1996; Vaniˇcek et al. 1996). To make the theory as simple as possible but still matching the requirements on Geoid height accuracy, one should keep throughout the following derivations the terms of magnitudes of the order of the first eccentricity e02 of the Earth’s level ellipsoid and neglect the term of higher powers of e02 . This approximation is justifiable because the error introduced by this approximation is at most 1. 5  105 , which then causes an error of at most 2 mm in the Geoidal heights. Keeping in mind the inequalities in Eq. (31), and assuming that e02 1, the magnitude of the last term in Eq. (33) is of the order of e02 , 1 X

  ˇj mk e02k2  e02 :

(34)

kD2

Hence, Eq. (13) becomes  u  j C1 Qj m i       e  E D 1 C aj m1 e 2  e02 1 C O e02 : b0 e0 Qj m i E

(35)

Substituting Eq. (35) into Eq. (8), evaluating a j m1 according to Eq. (11) for k D 1, and bearing in mind the classical Laplace addition theorem for spherical harmonics (for instance, Varshalovich et al. 1989, p. 164), Pj .cos / D

j X 4 Yj m./Yjm .0 /; 2j C 1 mDj

(36)

is the angular where Pi .cos / is the Legendre polynomial of degree j and distance between directions  and 0 , one gets Z

1 f .0 / K sph .t; cos /  2e02 k ell .t; ; 0 /.1 C O.e02 // d 0 ; T .u; / D 4 0 (37)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

393

where tD

e : e0

(38)

K sph .t; cos / is the spherical Poisson kernel (Heiskanen and Moritz 1967; Kellogg 1929; Pick et al. 1973), K sph .t; cos / D

1 X

.2j C 1/t j C1 Pj .cos / D

j D0

t.1  t 2 / g3

(39)

with g  g.t; cos / D

p

1 C t 2  2t cos

(40)

.j C 1/2  m2 Yj m ./Yjm.0 /: 2.2j C 3/

(41)

and k ell .t; ; 0 / stands for ell

0

2

k .t; ;  / D 4.1t /

j 1 X X j D0 mDj

t j C1

Moreover, it holds that (t < 1) jk ell .t; ; 0 /j  k ell .t; ; /  k ell .t; ; /j#D0 D

1 2j C 1 1  t 2 X j C1 t .j C 1/2 2 j D0 2j C 3


b0 ;

2 @T C D f for u D b0 @u u   1 c ; u ! 1; T .u/ D C O u u3

(80) (81) (82)

where f is assumed to be a known square-integrable function, i.e., f 2 L2 , and c is a constant. A limiting case E D 0 when the reference ellipsoid of revolution, u D b0 , reduces to a sphere x 2 C y 2 C z2 D b0 , and the boundary-value problem Eqs. (80)–(82) reduces to the spherical Stokes problem (Heiskanen and Moritz 1967, Sect. 2.16) will also be considered. To guarantee the existence of a solution in this particular case, the first-degree harmonics of f ./ have to be removed by means of the postulate (Holota 1995) Z 0

 f ./Ylm ./d ;

m D 1; 0; 1

(83)

Here, Y1m ./ are complex spherical harmonics of the first degree and order m, the asterisk denotes a complex conjugate, 0 is the full solid angle, and d  D sin #d #d . Throughout the chapter it is assumed that conditions (5) are satisfied. Moreover, to guarantee the uniqueness of the solution, the first-degree harmonics have to be removed from the potential as the asymptotic condition (82) indicates. The question of the uniqueness of the problem (80)–(82) for a general case E ¤ 0 will be examined later in Sect. 3.5. The problem (80)–(82) will be called the ellipsoidal Stokes boundary-value problem since it generalizes the traditional Stokes boundary-value problem formulated on a sphere. It approximates the boundary-value problem for Geoid determination (Martinec and Vaniˇcek 1996) or the Molodensky scalar boundaryvalue problem (Heck 1991, Sect. 6.3) by maintaining the two largest terms in boundary condition (81) and omitting two small ellipsoidal correction terms. (The effect of the ellipsoidal correction terms on the solution of the ellipsoidal Stokes boundary-value problem will be treated in a separate chapter.) Alternatively, the boundary-value problem for Geoid determination may be formulated in geodetic

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E.W. Grafarend et al.

coordinates (Otero 1995) which results in a boundary condition of a form slightly different from Eq. (81). However, ellipsoidal coordinates and formulation (80)– (82) will be used since the Laplace operator is separable in ellipsoidal coordinates (Moon and Spencer 1961) which will substantially help in finding a solution to the problem. The solution of the Laplace equation (80) can be written in terms of ellipsoidal harmonics as follows (Moon and Spencer 1961, p. 32; Heiskanen and Moritz 1967, Sect. 1.20; Thong and Grafarend 1989, p. 302):   Qj m i Eu   Yj m./; T .u; / D Tj m Qj m i bE0 j D0 mDj j 1 X X

(84)

  where Qj m i Eu are the Legendre function of the second kind and Tj m are coefficients to be determined from boundary condition (81). Equation A.11, derived in Appendix A (see Thong and Grafarend 1989), shows that    u 1 DO Qj m i ; E uj C1

u ! 1:

(85)

To satisfy asymptotic condition (82), the term with j D 1 must be eliminated from the sum over j in Eq. (84), i.e.,   Qj m i Eu   Yj m ./: T .u; / D Tj m Qj m i bE0 j D0 mDj j 1 X X

(86)

j ¤1

Substituting expansion (86) into boundary conditions (81) yields j 1 X X j D0 mDj j ¤1

"

1   Qj m i bE0

 ˇ dQj m i Eu ˇˇ ˇ ˇ du

uDb0

#  b0 2 Tj mYj m ./ D f ./: C Qj m i b0 E

(87)

Moreover, expanding function f in a series of spherical harmonics

f ./ D

Z j 1 X X j D0 mDj 

0

f .0 /Yjm .0 /d 0 Yj m./;

(88)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

405

where the first-degree spherical harmonics of f ./ are equal to zero by assumption (83), substituting expansion (88) into Eq. (87), and comparing the coefficients at spherical harmonics Yj m./ in the result yields R

Tj m

f .0 /Yjm .0 /d 0 ˇ " # D ˇ   dQj m .i Eu / ˇ b 1 2 0   C b0 Qj m i E ˇ b du Qj m i E0 ˇ 0

(89)

uDb0

for j D 0; 2; : : :, and m D j; : : :; j . Substituting coefficients Tj m into Eq. (86) and changing the order of summation over j and m and of integration over 0 , due to the uniform convergence of the series, one obtains Z f .0 /

T .u; / D 0

j 1 X X

˛j m .u/Yjm.0 /Yj m./d 0 ;

(90)

j D0 mDj

where functions ˛j m .u/ have been introduced: ˛j m .u/ D 

  Qj m i Eu dQj m .i Eu / juDb0 du

C

2 b0 Qj m

  i bE0

(91)

to abbreviate notations.

3.2

The Zero-Degree Harmonic of T

The zero-degree harmonic of potential T .u; / is to be calculated. One first has ˛00 .u/ D

  Q00 i Eu dQ00 .i Eu / juDb0 du

C

2 Q00 b0



i bu0

;

(92)

  where Q00 i Eu can be expressed as (Arfken 1968, Eq. 12.222)    u E D i arctan Q00 i E u with i D gets

(93)

p 1. Taking the derivatives of the last equation with respect to u, one   dQ00 i Eu iE D 2 : du u C E2

(94)

406

E.W. Grafarend et al.

By substituting Eqs. (93) and (94) into Eq. (92), ˛00 .u/ D

E 

arctan E b02 CE 2



2 b0

 :

u

arctan

(95)

E b0

: Since arctan x D x for x 1, one can see that ˛00 .u/ D

c u

for u ! 1

(96)

with cD

3.3

arctan E b02 CE 2



2 b0

E   :

u

arctan

(97)

E b0

Solution on the Reference Ellipsoid of Revolution

Solution (90) can now be expressed in the way indicated above as ˛00 .u/ T .u; / D 4

Z f .0 /d 0 0

Z f .0 /

C

j 1 X X

˛j m .u/Yjm.0 /Yj m ./d 0 :

(98)

j D2 mDj

0

In particular, the interest is in finding potential T .u; / on the reference ellipsoid of revolution u D b0 , i.e., function T .b0 ; /. In this case, the general formulae (91) and (98) reduce to T .b0 ; / D

˛00 .u/ 4

Z f .0 /d 0 0

Z f .0 /

C 0

j 1 X X

˛j m .b0 /Yjm .0 /Yj m ./d 0 ;

(99)

j D2 mDj

where ˛00 .b0 / are given by Eq. (95) for u D b0 and the other ˛j m .b0 / read

˛j m .b0 / D 

  Qj m i bE0 dQj m .i Eu / juDb0 du

C

2 Q b0 j m



i bE0

:

(100)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

407

From the practical point of view, the spectral form Eq. (99) of the solution of the Stokes boundary-value problem Eqs. (80)–(82) is often inconvenient, since constructing the spectral components of f ./ and summing them up to high degrees and orders may become time consuming and numerically unstable. Moreover, for the case in which the reference ellipsoid of revolution u D b0 deviates only slightly from a sphere, which is the case of the Earth, the solution to the problem should be close to the solution for the same problem but formulated for a sphere. It is thus attempted to rewrite T .b0 ; / as a sum of the well-known Stokes integral plus the corrections due to the ellipticity of the boundary. An evident advantage of such a decomposition is that existing theories as well as numerical codes for Geoidal height computations can simply be corrected for the ellipticity of the Geoid. To make the theory as simple as possible, but still maintain the requirements for Geoidal height accuracy throughout the following derivations, one shall retain the terms of magnitudes of the order of the first eccentricity e02 of an ellipsoid of revolution and neglect the term of higher power of e02 . This approximation is justifiable, because the error introduced by this approximation is 1  5  105 at the most which then is expected to cause an error of no more than 2 mm in the Geoid heights.

3.4

The Derivative of the Legendre Function of the Second Kind

Now, look for the derivative of the Legendre function Qj m.i Eu / with respect to variable u. Equation A.11,  derived in Appendix A (see Thong and Grafarend 1989), shows that Qj m i Eu can be expressed as an infinite power series of first eccentricity e, E eDp 2 u C E2

(101)

in the form 1  u .j C m/Š j C1 X e D .1/m.j C1/=2 Qj m i ˛j mk e 2k ; E .2j C 1/ŠŠ

(102)

kD0

where coefficient ˛j mk can, for instance, be defined by recurrence relations (A.14). In particular, one shall need ˛j m0 D 1

(103)

and ˛j m1 D

.j C 1/2  m2 ; 2.2j C 3/

˛j m2 D

.j C 3/2  m2 ˛j m1 : 4.2j C 5/

(104)

408

E.W. Grafarend et al.

Throughout this chapter it is assumed that the eccentricity e0 of the reference ellipsoid of revolution u D b0 , e0 D q

E b02

(105)

C E2

is less than 1. For points .u; / outside, or on the reference ellipsoid of revolution, i.e., when u  b0 , series Eq. (102) then converges. One can take the derivative of this series with respect to u and change the order of integration and summation since the resulting series Eq. (106) is uniformly convergent. Consequently,   1 dQj m i Eu .j C m/Š j C1 X de D .1/m.j C1/=2 e .2k C j C 1/˛j mk e 2k : du .2j C 1/ŠŠ du kD0 (106) The derivative of eccentricity e with respect to u can be easily obtained from Eq. (101): de e D .1  e 2 / : du u

(107)

Substituting Eq. (107) into Eq. (106) yields   1 dQj m i Eu .j C m/Š e j C1 X D .1/m.j C1/=2 .1  e 2 / .2k  j  1/˛j mk e 2k : du .2j C 1/ŠŠ u kD0 (108)

3.5

The Uniqueness of the Solution

Now, one is ready to express the sum in the denominator of Eq. (100)  u   dQj m i b0 2 E j i C Q uDb0 jm du b0 E D .1/m.j C1/=2

j C1 1 P

.j C m/Š e0 .2j C 1/ŠŠ b0

kD0

˛j mk

 

1  e02 .2k  j  1/ C 2 e02k: (109)

Substituting Eqs. (102) and (109) into Eq. (100) yields 1 P

˛j m .b0 / D b0

1 P kD0

kD0

˛j mk

˛j mk e02k

 

1  e02 .2k C j C 1/  2 e02k

:

(110)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

409

The last formula enables one to examine the uniqueness of the solution of boundaryvalue problem Eqs. (80)–(82). Let the denominator of expression Eq. (110) for ˛j m be denoted by dj m .e02 /,

dj m .e02 / D

1 X

˛j mk





1  e02 .2k C j C 1/  2 e02k

(111)

kD0

j D 2; : : :; m D 0; 1; : : :; j , and investigate the dependence of dj m.e02 / on e02 . Numerical examinations have resulted in Fig. 1 showing the behavior of dj m .e02 / for three values of e02 as functions of angular degree j and order m, j D 2; : : :; 8; m D 0; 2; : : :; j ; the combined index j m WD j .j C 1/=2 C m C 1. If e02 < 0:42303, then all coefficients dj m.e02 / are positive which means that the solution of boundaryvalue problem Eqs. (80)–(82) is unique. Once, e02 Š 0:42303; d22.e02 / D 0, and the solution of the problem is not unique. If the size of e02 is increased further, see, for instance, the curve denoted by triangles in Fig. 5 for e02 D 0:7, other dj m .e02 / may then vanish and problem Eqs. (80)–(82) have a nonunique solution. One can conclude that the uniqueness of the solution can only be guaranteed if the square of the first eccentricity is less than 0.42303. Fortunately, this condition is satisfied for the Earth since e02 D 0:006 694 380 (Moritz 1980) for the ellipsoid best fitting the Earth’s figure (the corresponding dj m.e02 / are plotted in Fig. 5 as black dots).

Fig. 5 Functions dj m (e0 2 ) for various e0 2 as functions of degree j and order m; the combined index is as follows j m D j .j C 1/=2 C m C 1

410

E.W. Grafarend et al.

The Approximation up to O.e0 2 /

3.6

Arrange formula Eq. (110) into a form that is more suitable for highlighting the Stokes contribution. After some cumbersome but straightforward algebra, one arrives at 2 ˛j m .b0 / D

1 h P

b0 6 61  j 1 4

kD1

1C

2k ˛ j 1 j mk 1 P

.1 C

kD1

  1C

i 3 ˛j m;k1 e02k 7 7; 5 2k  ˛j m;k1 /e0

2k j 1

2k j 1 /.˛j mk



(112)

where ˛j m0 has been substituted from Eq. (103). The ratio of the two power series in Eq. (112) can further be expanded as a power series of eccentricity e02 . The explicit forms of the first two terms of such a series are as follows: 1 h P kD1

1C

2k j 1 ˛j mk

1 P

kD1

C.1 C

 .1 C

2k j 1 /˛j m;k1

2k j 1 /.˛j mk

i

e02k D dj m1 e02 C dj m2 e04 C R;



(113)

˛j m;k1 /e02k

where 2 j C1 ˛j m1  (114) j 1 j 1 h i  j C 1 2 2 2 2.j  1/˛  D  .j C 1/˛ C .j C 3/˛ j m2 j m1 j m1 .j  1/2 j 1 (115)

dj m1 D dj m2

and R is the residual of the series expansion. It shall now be attempted to estimate the sizes of particular terms on the righthand side of Eq. (113). Substituting for ˛j m1 and ˛j m2 from Eq. (104) into Eqs. (114) and (115) and after some more algebra, one gets

dj m1 D 

j 2 C 3j C 2 C m2 .j  1/.2j C 3/

(116)

and

dj m2 D

  10j 3 C 50j 2 C 100j C 58 C 2m2 .3j C 4/ j C1 2 ˛  : j m1 .j  1/2 .2j C 3/.2j C 5/ j 1

(117)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

411

Hence, jdj m1j 

1 2 16 2j 2 C 3j C 2 D1C C  .j  1/.2j C 3/ j 1 .j  1/.2j C 3/ 7

(118)

for j  2 and jmj  j . It shall continue estimating the maximum size of the first constituent creating coefficients dj m2 for j  2 and jmj  j :

0
0/ < 0/ D 0/;

1 2 ."  .X 2 C Y 2 C Z 2 / 2"2

i1=2 p C .X 2 C Y 2 C Z 2  "2 /2 C 4"2 Z 2 / ;  uD

(220)

p 1 2 .X C Y 2 C Z 2  "2 C .X 2 C Y 2 C Z 2  "2 /2 C 4"2 Z 2 / 2

(221) 1=2 : (222)

(iii) Jacobi matrix 2

3 D X D' X Du X J D 4 D Y D' Y Du Y 5 ; D Z D' Z Du Z p 2 p  u2 C "2 cos ' sin   u2 C "2 sin ' cos  p p 6 J D 4 u2 C "2 cos ' cos   u2 C "2 sin ' sin  0

u cos '

(223)

p u u2 C"2 p u u2 C"2

cos ' cos 

3

7 cos ' sin  5 :

sin ' (224)

(iv) Metric 3 d d u J T J 4 d ' 5 ; du 2

dS 2 D Œd 

d'

(225)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

433

G D JTJ; (226) 2 2 3 .u C "2 / cos2 ' 0 0 2 2 2 5; G D4 0 0 u C " sin ' 2 2 2 2 2 0 0 .u C " sin '/=.u C " / (227) dS 2 D .u2 C "2 / cos2 'd 2 C .u2 C "2 sin2 '/d ' 2 C

u2 C "2 sin2 ' 2 du : u2 C "2 (228)

The derivatives {Dƒ x; D' x,Dux} are presented in order to construct in Eqs. (229) and (230) a local orthonormal frame of reference, namely, E D D x jjD xjj; E' D D' x jjD' xjj; Eu D Du x jjDu xjj, in short {E ; E' ; Eu jP }, also called {ellipsoidal east, ellipsoidal north, ellipsoidal vertical} at P . The local orthonormal frame of reference {E ; E' ; Eu jP } at P is related to the global orthonormal frame of reference {E1 ; E2 ; E3 jO} at O by the orthonormal matrix T1 . Thanks to the transformation Eq. (165) fe1 ; e2 ; e3 g ! feƒ; e'; eug, one has already succeeded to establish the orthonormal matrix T2 . ; ' ), Eq. (233). Such a relation will again be used to transform the ellipsoidal orthonormal frame of reference {E ; E' ; Eu jP } to the reference “east, north, vertical” frame {e ; e' ; eu jP }. Indeed ŒE ; E' ; Eu T D T1 Œe1 ; e2 ; e3 T D T1 T2T Œe ; e' ; eu T or Œe ; e' ; eu T D T2 T1T ŒE ; E' ; Eu T is the compound transformation T of Eqs. (231)–(235). As soon as one develops T close to the identity, namely, by means of the decomposition (Eqs. (236)–(239)) as well as of a special case (Eq. (240)) of the binomial series, ("u/ < 1, one gains the elements of the matrix T (case study t23 /. The matrix T is finally decomposed into the unit matrix I 3 and the incremental antisymmetric matrix A, Eqs. (248) and (249). Construction of the local reference frame {E ; E' ; Eu jP } from the Jacobi ellipsoidal coordinates of point P on the topography:   D X @x1 @x2 @x3 D E1 C E2 C E3 E D jjD Xjj @ @ @ " 2  2   #1=2 @x1 @x2 @x3 2 C C ; @ @ @

E' D

  D' X @x1 @x2 @x3 D E1 C E2 C E3 jjD' Xjj @' @' @' "    #1=2   @x1 2 @x2 2 @x3 2 C C ; @' @' @'

434

E.W. Grafarend et al.

  @x1 @x2 @x3 Du X D E1 C E2 C E3 Eu D jjDu Xjj @u @u @u "    #1=2   @x1 2 @x2 2 @x3 2 C C ; @u @u @u 3 2 3 E1 E 4 E' 5 D T 1 4 E2 5 ; Eu E3

(229)

2

2

 sin 

cos 

(230)

0

3

6 7 6 7 p p 6 7 2 2 2 2 6  u C" 7  u C" u 6q 7 q q cos  sin ' sin  sin ' cos ' 6 7 2' 2' 2' 2 2 2 2 2 2 6 7: u C " sin u C " sin u C " sin T 1 D6 7 6 7 6 7 p 6 7 2 2 u C" u u 6 7 cos  cos ' q sin  cos ' q sin ' 5 4q 2 2 2 2 2 2 2 2 2 u C " sin ' u C " sin ' u C " sin '

Basis transformation from geometry space to gravity space: 2

2 3 2 3 3 e ” E E 4 e' 5 D T 2 T T1 4 E' 5 D T 4 E' 5 ” Eu Eu e”

(231)

: T 1 D T 1 .ƒ; '; u/;

(232)

    : T 2 D T 2 .” ; '” / D R E ” C ;  '” ; 0 ; 2 2

(233)

subject to

T WD T 2 T T1 D T .ƒ; '; uI ” ; '” /; T 1 ; T 2 2 SO.3/;

(234)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

2

p  u2 C "2 sin.  ” / sin ' p u2 C "2 sin2 ' p u2 C "2 cos.  ” / sin '” C p u2 C "2 sin2 '

u sin.  ” / cos ' p u2 C "2 sin2 '

435

3

6cos.  ” / 7 6 7 6 7 6 7 6 sin.  ” / 7 / cos ' sin ' u cos.   ” ” 7 6 sin '” C p 6 7 2 6 7 u2 C "2 sin ' 6 7 p 6 7 2 2 6 u C " sin ' cos '” 7 u cos ' cos '” 6 7 p p C C T D6 7: 6 7 u2 C " sin2 ' u2 C " sin2 ' 6 7 p 6 7 6 sin.   / u2 C "2 cos.  ” / sin ' cos '” 7 / cos ' cos ' u cos.   ” ” 6 cos ' p p C C7 6 7 2 2 2 2 2 2 u C " sin ' u C " sin ' 6 7 6 7 p 6 7 6 u2 C "2 sin ' sin '” 7 u cos ' sin '” 4 5 C p Cp u2 C "2 sin2 ' u2 C "2 sin2 ' (235)

Additive decomposition:  D ” C ı , ” D '  ı;

(236)

' D '” C ı' , '” D '  ı';

(237)

: sin.ƒ  ı/ D sin ƒ   cos ƒı;

(238)

: sin.'  ı'/ D cos ' C sin 'ı':

(239)

Special case of binomial series: 3 1 15 .1 C x/1=2 D 1  x C x 2  x 3 C O.4/: 2 8 48 Condition: u > " D

(240)

p a2  b 2

!1=2 "2 sin2 ' p (241) D 1C u2 u2 C "2 sin2 '       " sin ' 1 " sin ' 2 3 " sin ' 4 D 1 C C O6 2 u 8 u u s  1=2 "2 cos2 ' u2 C "2 D 1 C (242) u2 C "2 u2 C "2 sin2 '  2   " cos ' 1 "2 cos2 ' 3 "2 cos2 ' : D 1 C C O p 6 2 u2 C "2 8 u2 C "2 u2 C "2 u

436

E.W. Grafarend et al.

Case study: "

t23

# "2 sin2 ' D 1 C O.4/ . sin ' cos ' C cos2 'ı'/ u2

(243)

 "2 cos2 ' C O.4/ .sin ' cos ' C sin2 'ı'/; C 1 2 u C "2 

"2 sin ' cos ' : t23 D ı' C 2 2 2u .u C "2 /

(244)

 .u2 C "2 / sin2 ' C u2 cos2 '  .u2 C "2 / sin ' cos 'ı' C u2 sin ' cos 'ı' ; "2 sin ' cos ' : t23 D ı' C 2u2   sin2 ' C

(245)

 u2 u2 2 cos '  sin ' cos 'ı' C 2 sin ' cos 'ı' ; u2 C "2 u C "2

: t23 D ı' C ."2 =4u2 / sin 2':

(246)

Linearized basis transformation from geometry space to gravity space: 3 2 3 32 1  sin 'ıL cos 'ıL E e ” 4 e' 5 D 4 sin 'ıL 1 ı' C ."=4u2/ sin 2' 5 4 E' 5 ” 2 1 Eu  cos 'ıL ı'  ."=4u / sin 2' e” (247) subject to 2

T D I 3 C ıA;

(248)

2

3 0  sin 'ıL cos 'ıL ıA D 4 sin 'ıL 0 ı' C ."=4u2/ sin 2' 5 : 2 0  cos 'ıL ı'  ."=4u / sin 2'

(249)

Based upon the linearized version of the transformation {E ƒ ; E ' ; E u g ! fe ƒ ; e ' ; e u }, Eqs. (231)–(249), one can build up the representation of the incremental gravity vector ı, Eq. (176), in terms of [e ƒ ; e ' ; e u ]: Eqs. (250)– (253) illustrate the third representation of the incremental gravity vector -ı, now in the basis [E ƒ ; E ' ; E u ]. The second basic result has to be interpreted as follows. The third representation of the incremental gravity vector ı contains horizontal components as well as a vertical component which are all functions of ( ; ; ı ). For instance, the east component Eƒ is a function of  (first order) and of ( ; ı ) (second order). Or the vertical component Eu is a function of ı (first order) and

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

437

of ( ;  ) (second order). If one concentrates on first-order terms only, Eqs. (252) and (253) prove the identity of the first and third definition of vertical deflections. Vertical deflections with respect to a basis in geometry space: Jacobi ellipsoidal coordinates (ƒ; '; u).

 ı” D e ” ; e '” ; e ” ;

(250)

2

3 ” C sin 'ı”   cos 'ıı”  6 7 "2 :  sin 'ı” C ”  ı' C 4u 2 sin 2' ı 7 :  ı” D ŒE E' Eu 6 4 5   "2 cos 'ı” C ı' C 4u sin 2' ” C ı 2

(251)

Vertical deflections :

D cos '” ı” D cos 'ı;

(252)

"2 : D ı'” D ı' C 2 sin 2': 4u

(253)

The potential theory of the horizontal and vertical components of the gravity field is reviewed in Eqs. (254)–(262). First, the reference gravity vector  as the gradient of the gravity potential and the incremental gravity vector ı as the gradient of the incremental gravity potential, also called disturbing potential, newly represented in the Jacobi ellipsoidal frame of reference {E  ; E ' ; E u jP }, are presented. The elements (g ; g' ' ; guu / of the matrix of the metric G have to be implemented. Second, the updated highlight is the first-order potential representation ( ; ; ı ), Eqs. (257)–(259), as functionals of type (Dƒ ıw ; D' ıw ; Du ıw /. Potential theory of horizontal and vertical components of the gravity field: 1 1 1 ” D grad w D E p D w C E' p D' w C Eu p Du w; g g' ' guu

(254)

1 1 1 ı” D grad ıw D E p D ıw C E' p D' ıw C Eu p Du ıw: g g' ' guu

(255)

Functionals of the disturbing potential ıw: 3 p 2 3 .1= g /D ıw ”

7 6 7 6 .1=pg' ' /D' ıw 7 D T 6 4 ” 5 ; 5 4 p .1= guu /Du ıw ı” 2

1

D ”

! 1 1 1 D ıw C sin 'ı p D' ıw  cos 'ı p Du ıw ; p g g' ' guu

(256)

(257)

438

E.W. Grafarend et al.

!   1 1 1 "2 1p D   sin 'ı p D ıwC p D' ıw ı' C 2 sin 2' guu Du ıw ; ” g g' ' 4u (258) !   "2 1 1 1 D ıwC ı' C 2 sin 2' p D' ıwC p Du ıw ; ı” D   cos 'ı p g 4u g' ' gu (259) if sin 'ı cos 'ı ı' C ."2 =4u2 / sin 2' D ıw 1; p D ıw 1; D' ıw 1; p p g g g' ' cos 'ı ı' C ."2 =4u2 / sin 2' sin 'ı D' ıw 1; p Du ıw 1; Du ıw 1; p p g' ' guu guu then 1 : 1

D p D ıw; 2 ” u C "2 cos '

(260)

1 : 1 D p D' ıw; ” u2 C "2 sin2 '

(261)

s : ı” D 

6

u2 C "2 sin2 ' Du ıw: u2 C "2

(262)

Potential Theory of Horizontal and Vertical Components of the Gravity Field: Gravity Disturbance and Vertical Deflections

Now the gravitational disturbing potential in terms of Jacobi ellipsoidal harmonics is represented. As soon as one takes reference to a normal potential of SomiglianaPizzetti type, the ellipsoidal harmonics of degree/order (0,0), (1,0), (1, 1), (1,1) and (2,0) are eliminated from the gravitational disturbing potential. In order to present the potential theory of the horizontal and vertical components of the gravity field, namely, ellipsoidal vertical deflections and ellipsoidal gravity disturbance, one has to make a decision about what is the proper choice of the ellipsoidal potential field of reference w.ƒ; '; u/ and about the related ellipsoidal incremental potential field ıw.ƒ; '; u/, also called “disturbing potential.”

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

6.1

439

Ellipsoidal Reference Potential of Type Somigliana-Pizzetti

There has been made three proposals for an ellipsoidal potential field of reference. The first choice is the zero-degree term [arccot.u="/ .GM="/ of the external ellipsoidal harmonic expansion. Indeed, it would correspond to the zero-order term GM =r of the external spherical harmonic expansion. As proven in Grafarend and Ardalan (1999), the equipotential surface w D Œarccot.u="/ .GM="/ D constant where GM is the geocentric gravitational constant. Unfortunately, such an equipotential reference surface does not include the rotation of the Earth, namely, its cen trifugal potential 2 "2 1 C .u="/2 =3 C 2 "2 1 C .u="/2 P2 .sin '/=3. Accordingly, there has been made the proposal for a second choice, namely, to choose [arccot(u="/ GM=" C 2 "2 .u2 C "2 / cos2 '=2. In this approach the zero-degree term of the gravitational potential and the centrifugal potential is superimposed. Unfortunately, the level surface [arccot(u="/ GM=" C .1 C P2 .sin '//2 "2 Œ1 C .u="/2 =3 D constant is not an ellipsoid of revolution. It is for this reason that the proposal for the third choice has been chosen. Superimpose the gravitational potential, which is externally expanded in ellipsoidal harmonics, and the centrifugal potential represented also in ellipsoidal base functions (the centrifugal potential is not a harmonic function) and postulate an equipotential reference surface to be a level ellipsoid. Such a level ellipsoid should be an ellipsoid of revolution. Such an ellipsoidal reference field has been developed by Pizzetti (1894) and Somigliana (1929) and is properly called Somigliana-Pizzetti reference potential. The Euclidean length of its gradient is referred to as the International Gravity Formula, which recently has been developed to the sub-nano Gal level by Ardalan and Grafarend (2001). Here, the recommendations followed are that of the International Association of Geodesy, namely, Moritz 1984, to use the Somigliana-Pizzetti potential of a level ellipsoid as the reference potential summarized in Eqs. (263)–(267). Reference gravity potential field of type Somigliana-Pizzetti. Reference Level Ellipsoid, Semimajor axis a, semiminor p axis b, Absolute eccentricity " D a2  b 2 , E 2a;b D X 2 R3 j.X 2 C Y 2 /=a2 C Z 2 =b 2 D 1:

(263)

The first version of the reference potential field       u 2 3 C 1 arccot u"  3 u" " GM 1 2 2  w.'; u/ D arccot C  a   2 .3 sin2 '  1/   " " 6 3 b" C 1 arccot b"  3 b" u

1 C 2 .u2 C "2 / cos2 ': 2 Input: 4 parameters: GM, , a, " D

(264) p a2  b 2 or b.

440

E.W. Grafarend et al.

Legendre polynomials of the first and second kind:  arccot .u="/ D Q00 .u="/;     u 2 C1 3 "  2 ! b 3 C1 "

  u u 3 " "   b arccot " 3 b"

arccot

D

 Q20 .u="/ ;  Q20 .b="/

2  3 sin2 '  1 D p P20 .sin '/: 5

(265)

(266)

The second version of the reference potential field: p  GM  1 5 2 2 Q20 .u="/  w.'; u/ D Q00 .u="/ C a  P .sin '/ C ! 2 .u2 C "2 / cos2 ': " 15 Q20 .b="/ 20 2 (267) Constraints: reference gravity potential field of type Somigliana-Pizzetti. Conditions for the ellipsoidal terms of degree/order (0,0) and (2,0). The first version: 1 .0; 0/ W u00 C 2 a2 D W0 ; 3 p 5 2 2  a D 0: .2; 0/ W u20  15

(268)

(269)

The first condition in multipole expansion: u00 D

    GM b GM b arccot D p ; arccot p " " a2  b 2 a2  b 2   GM  b u00 D Q00 : " "

(270)

(271)

Corollary: 

GM

b

p arccot p a2  b 2 a2  b 2



1 C 2 a2  W0 D 0: 3

(272)

The second condition in multipole expansion: u20

G1 D " 2

"

# !  2   b b b 3 3 J20 ; C 1 arccot " " "

(273)

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

u20

G  D Q20 "

  b J20 : "

441

(274)

Ellipsoidal multipole of degree/order (2,0): Z2 d 0

J20 D

!  0 2 u 1 3 d u0 .u02 C "2 sin2 ' 0 / C1 2 "

.0 ;' 0 / u0 Z

Z=2 d ' 0 cos ' 0

=2

0

0

p 5 .3 sin2 ' 0  1/.0 ; ' 0 ; u0 /:  2

(275)

Functional of the mass density field .ƒ0 ; ' 0 ; u0 /: 0

J20

0

.;' / uZ  0 Z2 Z=2 u 0 0 0 0 02 2 2 0   P20 D d d ' cos ' d u .u C" sin ' /P20 .sin ' 0 /.; '; u0 /: " 0

0

=2

(276) Cartesian multipole of degree two versus ellipsoidal multipole of degree/order (2,0): J11 D A; J22 D B; J33 D C;

(277)

Z Jpq D

d w3 .jjX jj2 ıpq  Xp Xq /.X; Y; Z/ 8p; q 2 f1; 2; 3g:

(278)

Z d w3 .Y 2 C Z 2 /.X; Y; Z/;

J11 D Z

d w3 .X 2 C Z 2 /.X; Y; Z/;

J22 D

(279)

Z J33 D J20 D

d w3 .Y 2 C Y 2 /.X; Y; Z/;     ACB 1p 1  C C M "2 : 5 2 3 2 " 2

(280)

The second condition in Cartesian multipole expansion: u20

u20

! "   # p    b 3 5 1 b b 2 2 2 3 D C1 arccot 3 G.A C B  2C /C GM" ; 8 "3 " " " 3 (281) p      1 ACB 3 5 1  b 2  C C GM" G : (282) D Q 4 "3 20 " 2 3

442

E.W. Grafarend et al.

Corollary:   G 3 1 1 GM C .A C B  2C / p p "3 4 a 2  b 2 8 3 a2  b 2      b2 b 1 b  3 2  3  2 a2 D 0: C 1 arccot p p 2 2 2 2 a  b2 15 a b a b (283) Corollary (second ellipsoidal condition): u20

  b GM arccot C20 D " "

(284)

is a transformation of the dimensionless ellipsoidal coefficient of degree (2,0) C20 to the nondimensionless ellipsoidal coefficient of degree/order (2,0), and then the second condition in the ellipsoidal harmonic coefficient C20 is 



1 C20  p 2 a2 D 0: p arccot p 2 2 2 2 3 5 a b a b GM

b

(285)

Lemma (World Geodetic Datum, Grafarend and Ardalan 1999) If the parameters {W0 ; GM; C20 ; } are given, the Newton iteration of the nonlinear two condition equations is contractive and leads to a D a (W0 ; GM; C20 ; /; b D b.W0 ; GM; C20 ; /, and " D ".W0 ; GM; C20 ; /.

6.2

Ellipsoidal Reference Gravity Intensity of Type Somigliana-Pizzetti

Vertical deflections { .ƒ; '; u/; .ƒ; '; u/} as defined as the longitudinal and lateral derivatives of the disturbing potential are normalized by means of reference gravity intensity .; '; u/ DW jjgrad.; '; u/jj (Eqs. (254)–(262), (285), and (286)). Here the aim is at computing the modulus of reference gravity with respect to the ellipsoidal reference potential of type Somigliana-Pizzetti. The detailed computation of jj grad.; '; u/jj is presented by means of Eqs. (285)–(291), two lemmas, and two corollaries. Since the reference potential of type Somigliana-Pizzetti w.'; u/ depends only on spheroidal latitude ' and spheroidal height u, the modulus of the reference gravity vector Eq. (286) is a nonlinear operator based upon the lateral derivative D' w and the vertical derivative Du w. As soon as one departs from the standard representation of the gradient operator in orthogonal coordinates, namely, grad.w/ D e .g /1=2 D w C e' .g' ' /1=2 D' w C eu .guu /1=2 Du w, one arrives at the standard form of jjgradwjj of type Eq. (286). Here, for the near-field computation, one shall assume x D .u2 C "2 /1 .D' w=Du w/. Accordingly by means of Eq. (287),

Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

443

p 1 C x is expanded in binomial series and is led to the first-order approximation of  D jjgradwjj by Eq. (288). Obviously up to O.2/, it is sufficient to compute the vertical derivative jDu wj. An explicit version of Du w is given by Eqs. (291) and (292), in the form of ellipsoidal base functions by Eq. (293). Reference gravity intensity of type Somigliana-Pizzetti, International Gravity Formula: ” D jjgradw.'; u/jj D q D q D

p hgradw.'; u/jgradw.'; u/i

.u2 C "2 sin2 '/1 .D' w/2 C .u2 C "2 /.u2 C "2 sin2 '/1 .Du w/2

(286)

q .u2 C "2 /=.u2 C "2 sin2 '/ .Du w/2 C .u2 C "2 /1 .D' w/2 ;

q q ” D .u2 C "2 /=.u2 C "2 sin2 '/jDu wj 1 C .u2 C "2 /1 .D' w=Du w/2 ; (287) x D.u2 C "2 /1 .D' w=Du w/2 ;

p 1 1 1 C x D 1 C x  x 2 C O.3/8jxj < 1: 2 8

If x D .u2 C "2 /1 .D' w=Du w/2 < 1, then q ”D

.u2 C "2 /=.u2 C "2 sin2 '/jDu wj C O.2/:

(288)

The first version of Du w reads   u    u 2 6u " C 1 u2 C" 2 arccot "  3 " 2  GM 1 2 2 " 2    C a .3 sin '  1/ Du w D  2  2   u C "2 6 3 b C 1 arccot b  3 b "

C 2 u cos2 ';

Du w D 

"

(289)

GM 1 1  2 a 2 2 u2 C "2 6 u C "2

sin2 ' 6 1 C 2 a 2 2 2 u C "2

"

3 "

6

  u 2 "

       2 C 1 uarccot u"  2" 3 u" C 2   2    3 b" C 1 arccot b"  3 b"

       2 C 1 uarccot u"  2" 3 u" C 2     C2 u cos2 ': b  u 2 b 3 " C 1 arccot "  3 "

  u 2 "

(290)

444

E.W. Grafarend et al.

The second version of Du w reads Du w D

p ŒQ .u="/ 0 5 2 2  GM  ŒQ00 .u="/ 0 C  a P20 .sin '/ 20 C2 u cos2 ':  " 15 Q20 .b="/

(291)

A more useful closed-form representation of the reference gravity intensity of type Somigliana-Pizzetti will be given by two lemmas and two corollaries. First, if one collects the coefficients of {1; cos2 '; sin2 'g by {0 , c , s }, one is led to the representation of  by Eqs. (292)–(295) expressedpin the Lemma 1. Second, if p one takes advantage of the (a; b) representation of u2 C "2 = u2 C "2 sin2 ' and decompose 0 according to 0 .sin2 ' C cos2 '), one is led to the alternative elegant representation of  by Eqs. (296)–(303), presented in Lemma 2. Lemma 1. Reference gravity intensity of type Somigliana-Pizzetti, International Gravity Formula: If x D .u2 C "2 /1 .D' w=Du w/2 1 holds, where w D w.'; u/ is the reference potential field of type Somigliana-Pizzetti, then its gravity field intensity .'; u/ can be represented up to the order O.2/ by q .u2 C "2 /=.u2 C "2 sin2 '/j”0 C ”c cos2 ' C ”s sin2 'j

”D

(292)

subject to 0 D 

a2

1 1 GM  2 a 2 2 C .u C b/.u  b/ 3 a C .u C b/.u  b/

3..u="/2 C 1/uarccot.u="/  ".3.u="/2 C 2/ ;  .3.b="/2 C 1/arccot.b="/  3.b="/ ”c D 2 u; ”s D 2 a 2

(293)

(294)

3..u="/2 C 1/uarccot.u="/  ".3.u="/2 C 2/ 1 : a2 C .u C b/.u  b/ .3.b="/2 C 1/arccot.b="/  3.b="/ (295)

Lemma 2. Reference gravity intensity of type Somigliana-Pizzetti, International Gravity Formula: If x D .u2 C"2 /1 .D' w=Du w/2 .k  l/Š