Satellite Measurements of Clouds and Precipitation: Theoretical Basis (Springer Remote Sensing/Photogrammetry) 981192242X, 9789811922428

Citation preview

Springer Remote Sensing/Photogrammetry

Hirohiko Masunaga

Satellite Measurements of Clouds and Precipitation Theoretical Basis

Springer Remote Sensing/Photogrammetry

The Springer Remote Sensing/Photogrammetry series seeks to publish a broad portfolio of scientific books, aiming at researchers, students, and everyone interested in the broad field of geospatial science and technologies. The series includes peer-reviewed monographs, edited volumes, textbooks, and conference proceedings. It covers the entire area of Remote Sensing, including, but not limited to, land, ocean, atmospheric science and meteorology, geophysics and tectonics, hydrology and water resources management, earth resources, geography and land information, image processing and analysis, satellite imagery, global positioning systems, archaeological investigations, and geomorphological surveying. Series Advisory Board: Marco Chini, Luxembourg Institute of Science and Technology (LIST), Belvaux, Luxembourg Manfred Ehlers, University of Osnabrueck Venkat Lakshmi, The University of South Carolina, USA Norman Mueller, Geoscience Australia, Symonston, Australia Alberto Refice, CNR-ISSIA, Bari, Italy Fabio Rocca, Politecnico di Milano, Italy Andrew Skidmore, The University of Twente, Enschede, The Netherlands Krishna Vadrevu, The University of Maryland, College Park, USA

More information about this series at https://link.springer.com/bookseries/10182

Hirohiko Masunaga

Satellite Measurements of Clouds and Precipitation Theoretical Basis

Hirohiko Masunaga Institute for Space-Earth Environmental Research Nagoya University Nagoya, Aichi, Japan

ISSN 2198-0721 ISSN 2198-073X (electronic) Springer Remote Sensing/Photogrammetry ISBN 978-981-19-2242-8 ISBN 978-981-19-2243-5 (eBook) https://doi.org/10.1007/978-981-19-2243-5 © Springer Nature Singapore Pte Ltd. 2022 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

A secret plan to write a textbook on satellite meteorology has been haunting my mind for more than a decade. There were at least two reasons. I am occasionally asked by students and colleagues to recommend introductory books on satellite remote sensing that they could learn the basics from. They may have tried with research articles or algorithm theoretical basis documents (ATBDs) in an attempt to figure out the workings of satellite retrieval algorithms, only to find those documents undecipherable for non-expert readers. There are excellent books to recommend such as “Remote Sensing of the Lower Atmosphere: An Introduction” by Stephens and “Satellite Meteorology: An Introduction” by Kidder and Vonder Haar. These books, although still of great value, were published in the mid-1990s, before the new satellite technologies that are taken for granted today: the TRMM satellite having been in operation from 1997 to 2015, the Terra and Aqua launched in 1999 and 2002, respectively; the CloudSat and CALIPSO launched in 2006; and the GPM core observatory sent into orbit in 2014. A new era of earth observations from space has flourished, but vastly diversified measurement capabilities demand more intellectual effort than before to make sense out of observation data. There has been a growing need for a new book addressing the fundamentals of satellite observations in light of recent technological advances. Another reason that motivated me to write this book is more personal. Portions of this book were originally meant to answer my own questions rather than those from students and colleagues. The basic physical concepts critical for the measurement principles of satellite remote sensing, for example, the blackbody spectrum, are in many cases presented without proof in textbooks. Never being happy with the don’t-ask-me-why approach, I often delved into the literature in search of the answer on my own. The outcomes are shared in this book, including thorough derivations of the J2 perturbation, the Planck function, and the Debye relaxation, just as a few examples. The opening chapter following a brief introduction summarizes the past and present satellite programs and instruments. I am aware of the obvious risk that these pages will quickly age as satellite missions change over time. I nonetheless decided not to omit this part for the sake of non-expert readers in need of a lighthouse helping v

vi

Preface

them sail through a flood of mission/satellite/sensor names in unfamiliar acronyms. The subsequent chapters begin with the physical basis and then proceed to more practical details. The chapters are structured to attain an optimal balance between the theoretical and practical aspects of satellite retrieval, which I believe is crucial for this kind of book to be informative and readable at the same time for a broad audience. The theoretical elements would not only elaborate the logic behind retrieval algorithms but, not less importantly, caution the reader to be aware of the potential sources of error and bias in the retrieved geophysical variables. As can be imagined from the book title, the topics of the main chapters are targeted exclusively on cloud and precipitation measurements, in which my expertise resides. Atmospheric chemistry including aerosol and temperature/humidity sounding are among the important subjects of satellite meteorology that are left outside the scope of this book. That being said, I humbly hope that the background physical principles delineated in different chapters (e.g., Mie solutions, gas absorption lines, and radiative transfer) will be of utility beyond the intended audience. I am grateful to many colleagues for their assistance in writing this book. My sincere acknowledgments go to Fumie A. Furuzawa, Husi Letu, Takashi Y. Nakajima, and Moeka Yamaji for kindly offering data or plots that I could not afford to prepare for myself, and to Keiji Imaoka, Maki Kikuchi, and Hajime Okamoto for helping me gather information essential for the book. I greatly appreciate the insightful advice and suggestions of Sandrine Bony, Toshio Iguchi, Chris Kummerow, Johnny Luo, Kenji Nakamura, and Kozo Okamoto. I am particularly indebted to Toshio Iguchi for his careful reading and pointing out numerous errors and readability issues in an earlier version of the manuscript. Finally, I deeply thank my wife Ayumi for her support and patience: not having my own study at home, I occupied the most comfortable corner of our living room for countless hours immersing myself in the book writing project. Nagoya, Japan January 2022

Hirohiko Masunaga

Contents

Part I

General Background

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 6

2

Satellite Missions and Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Satellite Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Operational Satellite Missions . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Research Satellite Missions . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Satellite Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Imagers and Radiometers . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Sounders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Radars and Lidars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 9 12 16 16 17 21 24 26

3

Satellite Orbit and Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Geostationary Earth Orbit (GEO) . . . . . . . . . . . . . . . . . . . 3.1.2 Low Earth Orbit (LEO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Scanning Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Field of View (FOV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 GEO Imager Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 LEO Sensor Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 29 31 45 46 48 49 52

Part II 4

Basic Physics

Principles of Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Particle Distribution Functions in Phase Space . . . . . . . . . . . . . . . . 4.1.1 Phase Space and Density of States . . . . . . . . . . . . . . . . . . . 4.1.2 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 55 58 vii

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4.1.3

Fermi-Dirac and Bose-Einstein Statistics: Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Fermi-Dirac Statistics: Basic Properties . . . . . . . . . . . . . . 4.1.5 Bose-Einstein Statistics: Basic Properties . . . . . . . . . . . . . 4.1.6 Boltzmann Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Boltzmann Equation for Non-relativistic Particles . . . . . 4.2.2 Conservation Laws for Non-relativistic Particles . . . . . . . 4.2.3 Boltzmann Equation for Photons . . . . . . . . . . . . . . . . . . . . 4.2.4 The Conservation Laws for Photons . . . . . . . . . . . . . . . . . 4.2.5 Radiative Transfer Equation and Optical Depth . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 64 66 68 69 69 70 74 77 80 82

5

Principles of Electrodynamics and Geometrical Optics . . . . . . . . . . . . 5.1 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Refractive Index and Dielectric Function . . . . . . . . . . . . . 5.1.3 Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Mie’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Approaches to Non-spherical Particle Scattering . . . . . . 5.2 Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Angles of Reflection and Refraction . . . . . . . . . . . . . . . . . 5.2.2 Amplitudes of Reflected and Transmitted Rays . . . . . . . . 5.2.3 Reflectivity and Transmittivity . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Fraunhofer Diffraction and Airy Pattern . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 83 85 88 89 102 103 103 105 107 110 113

6

General Theory of Radiative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Absorption and Emission of Radiation . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Planck Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Einstein Coefficients and Kirchhoff’s Law . . . . . . . . . . . . 6.2 Gas Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Microwave Water-Vapor Bands . . . . . . . . . . . . . . . . . . . . . 6.2.3 Microwave Oxygen Bands . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Infrared Molecular Bands . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Condensate Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Microwave Properties of Water Condensate . . . . . . . . . . . 6.3.3 Infrared Properties of Water Condensate . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 115 119 123 123 125 130 133 136 136 137 145 147

Contents

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Part III Measurement Principles 7

Infrared Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Radiative Transfer in Non-scattering Atmospheres . . . . . . . . . . . . 7.1.1 Infrared Properties of Cloud Particles . . . . . . . . . . . . . . . . 7.1.2 Mathematical Formulation and Solution . . . . . . . . . . . . . . 7.1.3 Absorption and Emission Lines . . . . . . . . . . . . . . . . . . . . . 7.1.4 Infrared Brightness Temperature . . . . . . . . . . . . . . . . . . . . 7.2 Infrared Spectrum of the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Effects of Water Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Effects of Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Infrared Properties of Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Split-Window BTD Method . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Detection of Cloud Thermodynamic Phase . . . . . . . . . . . 7.3.3 CO2 Slicing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 151 152 155 156 157 158 159 161 161 165 165 167 168

8

Visible/Near-Infrared Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Radiative Transfer in Scattering Atmospheres . . . . . . . . . . . . . . . . 8.1.1 Mathematical Formulation and Solution for ων = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Mathematical Formulation and Solution for ων = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Visible/Near-Infrared Spectrum of the Atmosphere . . . . . . . . . . . . 8.2.1 Clouds with Different Optical Depths . . . . . . . . . . . . . . . . 8.2.2 Clouds with Different Effective Radii . . . . . . . . . . . . . . . . 8.3 Visible/Near-Infrared Properties of Clouds . . . . . . . . . . . . . . . . . . . 8.3.1 Liquid/Ice Water Path and Cloud Effective Radius . . . . . 8.3.2 Visible Versus Infrared Optical Depth of Clouds . . . . . . . 8.3.3 Retrieval of Cloud Optical Depth and Effective Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Scattering Phase Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Rayleigh Scattering Phase Function . . . . . . . . . . . . . . . . . 8.4.2 Phase Function from Mie’s Solution . . . . . . . . . . . . . . . . . 8.4.3 Phase Function of Hexagonal Ice Columns . . . . . . . . . . . 8.4.4 Detection of Cloud Thermodynamic Phase . . . . . . . . . . . 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171

Microwave Radiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Radiative Transfer in Scattering and Emitting Atmospheres . . . . . 9.1.1 Microwave Properties of Precipitating Particles . . . . . . . 9.1.2 Mathematical Formulation and Solution . . . . . . . . . . . . . . 9.1.3 Microwave Brightness Temperature . . . . . . . . . . . . . . . . . 9.2 Surface Microwave Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 193 195 198 199

9

172 176 178 179 181 181 181 184 185 187 187 188 190 190 191 192

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9.2.1 Brief Theoretical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Ocean Surface Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Land Surface Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Regional Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Microwave Spectrum of the Atmosphere . . . . . . . . . . . . . . . . . . . . . 9.3.1 Effects of Water Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Effects of Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Effects of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Microwave Precipitation Measurement . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Surface Rainfall and Microwave Brightness Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Non-uniform Beam-Filling (NUBF) Effect . . . . . . . . . . . 9.4.3 Polarization Corrected Temperature (PCT) . . . . . . . . . . . 9.5 Microwave Sounding of Temperature and Humidity . . . . . . . . . . . 9.5.1 Measuring Principles of Satellite Sounding . . . . . . . . . . . 9.5.2 Precipitation Measurements by Microwave Sounders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200 201 202 204 205 207 209 210 212 212 214 216 218 218 220 221 222

10 Active Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Radar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Antenna Gain and Effective Aperture . . . . . . . . . . . . . . . . 10.1.2 Back-Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Scattering Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Radar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.5 Radar Reflectivity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.6 Effect of Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Radar Measurements of Clouds and Precipitation . . . . . . . . . . . . . 10.2.1 Z -W , Z -R, and k-Z Relations . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Basic Properties of Radar Reflectivity Factor . . . . . . . . . 10.2.3 Attenuation Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Radar Measurements of Ice Clouds and Snow . . . . . . . . . 10.3 Lidar Observations of Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Lidar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Lidar Detection of Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Depolarization Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 226 227 228 229 230 232 234 234 236 242 244 247 247 249 250 251 252

11 Mathematical Basis of Retrieval Algorithms . . . . . . . . . . . . . . . . . . . . . 11.1 Forward and Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Inversion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . .

255 255 257 259 260 262

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11.3.3 Deterministic Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Part IV Applications 12 Global Datasets of Clouds and Precipitation . . . . . . . . . . . . . . . . . . . . . 12.1 Data Processing Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Global Cloud Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Cloud Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Cloud Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Global Precipitation Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Multi-satellite Precipitation Datasets . . . . . . . . . . . . . . . . . 12.3.2 Global Distribution of Precipitation . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269 269 270 271 271 274 274 277 279

13 Satellite Data Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Satellite Data Simulations for This Book . . . . . . . . . . . . . . . . . . . . . 13.2.1 Model Atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Cloud Microphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Simulation Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 283 285 285 286 288 291

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Acronyms

ABI ADEOS AGRI AHI AIRS AMSR AMSR-E AMSU AMV ATLID ATMS ATOVS AVHRR BOA BTD CALIOP CALIPSO CATS CCI CDR CER CERES CFMIP CHIRPS CLARA CM SAF CMAP CMORPH CNES

Advanced Baseline Imager Advanced Earth Observing Satellite Advanced Geosynchronous Radiation Imager Advanced Himawari Imager Atmospheric Infrared Sounder Advanced Microwave Scanning Radiometer Advanced Microwave Scanning Radiometer for EOS Advanced Microwave Sounding Unit Atmospheric Motion Vector Atmospheric Lidar Advanced Technology Microwave Sounder Advanced TIROS Operational Vertical Sounder Advanced Very High Resolution Radiometer Bottom of the atmosphere Brightness temperature difference Cloud-Aerosol Lidar with Orthogonal Polarization Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations Cloud-Aerosol Transport System Climate Change Initiative Climate Data Record Cloud effective radius Clouds and the Earth’s Radiant Energy System Cloud Feedback Model Intercomparison Project Climate Hazards group Infrared Precipitation with Stations CM SAF Cloud, Albedo, And Surface Radiation dataset from AVHRR data Climate Monitoring Satellite Application Facility CPC Merged Analysis of Precipitation CPC morphing technique Centre National d’Études Spatiales xiii

xiv

COMS COSP CPC CPR CrIS CWV DDA DFR DMSP DPR DSD EarthCARE EOS EPS ESA EUMETSAT FDTD FOV FWHM FY GCOM GEO GEO-KOMPSAT GIIRS GLI GMI GMS GNSS GOES GOSAT-GW GPCC GPCP GPI GPM GPROF GPS G-SDSU GSMaP HIRAS HIRS HOAPS HSB

Acronyms

Communication Ocean and Meteorological Satellite CFMIP Observation Simulator Package Climate Prediction Center Cloud Profiling Radar Cross-Track Infrared Sounder Column water vapor Discrete dipole approximation Dual frequency ratio Defense Meteorological Satellite Program Dual-frequency Precipitation Radar Drop size distribution Earth Clouds, Aerosols, and Radiation Explorer Earth Observing System EUMETSAT Polar System European Space Agency European Organisation for the Exploitation of Meteorological Satellites Finite-difference time domain Field of view Full width at half maximum Feng Yun Global Change Observation Mission Geostationary (Geosynchronous) Earth Orbit/Orbiting Geostationary-Korea Multi-Purpose Satellite Geostationary Interferometric Infrared Sounder Global Imager GPM Microwave Imager Geostationary Meteorological Satellite Global Navigation Satellite System Geostationary Operational Environmental Satellite Global Observing Satellite for Greenhouse gases and Water cycle Global Precipitation Climatology Centre Global Precipitation Climatology Project GOES Precipitation Index Global Precipitation Measurement Goddard profiling algorithm Global Positioning System Goddard Satellite Data Simulator Unit Global Satellite Mapping of Precipitation Hyperspectral Infrared Atmospheric Sounder High Resolution Infrared Radiation Sounder Hamburg Ocean Atmosphere Parameters and Fluxes from Satellite Data Humidity Sounder for Brazil

Acronyms

HSRL IASI ICI IFOV IMERG INSAT IRS ISCCP ISRO ISS ITCZ IWC IWP JAXA JPSS KaPR KuPR LDR LEO LT LTE LUT LWC LWP MAC-LWP MADRAS MAP Megha-Tropiques MERSI MetOp MFG MHS MISR MLS MODIS MSG MSU MSWEP MTG MTSAT MVIRI MWHS MWRI MWTS

xv

High spectral resolution lidar Infrared Atmospheric Sounder Interferometer Ice Cloud Imaging Instantaneous field of view Integrated Multi-satellitE Retrievals for GPM Indian National Satellite Infrared Sounder International Satellite Cloud Climatology Project Indian Space Research Organisation International Space Station Inter-tropical convergence zone Ice water content Ice water path Japan Aerospace Exploration Agency Joint Polar Satellite System Ka-band Precipitation Radar Ku-band Precipitation Radar Linear depolarization ratio Low Earth Orbit/Orbiting Local time Local thermodynamic equilibrium Lookup table Liquid water content Liquid water path Multisensor Advanced Climatology of Liquid Water Path Microwave Analysis and Detection of Rain and Atmosphere Systems Maximum a posteriori Meteorological LEO Observations in the Intertropical Zone Medium Resolution Spectral Imager Meteorological Operational Satellite Program of Europe Meteosat First Generation Microwave Humidity Sounder Multi-angle Imaging SpectroRadiometer Microwave Limb Sounder Moderate Resolution Imaging Spectroradiometer Meteosat Second Generation Microwave Sounding Unit Multi-Source Weighted-Ensemble Precipitation Meteosat Third Generation Multifunction Transport Satellite Meteosat Visible and Infrared Imager Microwave Humidity Sounder Microwave Radiation Imager Microwave Temperature Sounder

xvi

NASA NASDA NOAA NPOESS NPP NUBF OLR OPI PARASOL PATMOS-x PCT PERSIANN PERSIANN CCS PIA POES POLDER PR PSD RAAN RainCube RH RO RSS RTTOV SAPHIR ScaRaB SCOPS SDSU SEVIRI SGLI SMR SRF SRT SSM/I SSMIS SST Suomi NPP TAPEER TEMPEST-D TIROS

Acronyms

National Aeronautics and Space Administration National Space Development Agency of Japan National Oceanic and Atmospheric Administration National Polar-orbiting Operational Environmental Satellite System NPOESS Preparatory Project Non-uniform beam filling Outgoing longwave radiation OLR Precipitation Index Polarization and Anisotropy of Reflectance for Atmospheric Sciences coupled with Observations from a Lidar Pathfinder Atmospheres-Extended Polarization Corrected Temperature Precipitation Estimation from Remotely Sensed Information using Artificial Neural Networks PERSIANN Cloud Classification System Path-integrated attenuation Polar Operational Environmental Satellite Polarization and Directionality of the Earth’s Reflectances Precipitation Radar Particle size distribution Right ascension of ascending node Radar in a CubeSat Relative humidity Radio occultation Remote Sensing Systems Radiative Transfer for TOVS Sounder for Atmospheric Profiling of Humidity in the Intertropics by Radiometry Scanner for Radiation Budget Subgrid Cloud Overlap Profile Sampler Satellite Data Simulator Unit Spinning Enhanced Visible and Infrared Imager Second-generation Global Imager Submillimeter Radiometer Spectral response function Surface reference technique Special Sensor Microwave Imager Special Sensor Microwave Imager Sounder Sea surface temperature Suomi National Polar-orbiting Partnership Mission Tropical Amount of Rainfall with Estimation of ERors Temporal Experiment for Storms and Tropical SystemsDemonstration Television Infrared Observation Satellite

Acronyms

TIROS-N TMI TMPA TOA TOVS TRMM TROPICS UTC VIIRS VIRS WMO WRS WVP

xvii

TIROS-Next Generation TRMM Microwave Imager TRMM Multi-satellite Precipitation Analysis Top of the atmosphere TIROS Operational Vertical Sounder Tropical Rainfall Measuring Mission Time-Resolved Observations of Precipitation structure and storm Intensity with a Constellation of Smallsats Coordinated Universal Time Visible/Infrared Imager and Radiometer Suite Visible Infrared Scanner World Meteorological Organisation Worldwide Reference System Water vapor path

Part I

General Background

The three chapters in Part I are dedicated to a general background of satellite remote sensing. A brief introduction to this book is followed by the second chapter offering an overview of the satellite programs and instruments whose main targets include cloud and precipitation observations. The last chapter outlines the basics of orbital mechanics and satellite scanning geometry.

Chapter 1

Introduction

Yuri Gagarin is sometimes quoted as having said “the earth is blue” during his historic flight as the first man who orbited our planet. This is not exactly how he put it. He was awed by the beauty of a bluish halo where the atmosphere dissolved into the darkness of space, but did not really claim that the earth itself was blue. What Gagarin mentioned while looking down upon the earth was decks of clouds elegantly punctuating the landscape. The earth is in fact more of a white planet rather than a blue planet, given that nearly 70% of the earth’s surface is covered with clouds (Stubenrauch et al. 2013). Ubiquitous clouds are among the key elements of the earth’s climate system, modulating the atmospheric radiation budget in a complicated manner. Our growing but limited knowledge of cloud radiative effects, however, remains a source of large uncertainties in climate model simulations (e.g., Sherwood et al. 2020). Reliable measurements of cloud variables are more important than ever for enriching our understanding of the cloud-radiation interactions to sharpen our skills to predict the future of climate change. A rich variety of clouds are formed in association with vertical air motions that occur across a broad spectrum of temporal and spatial scales. Clouds are not just a passive tracer of atmospheric convection and turbulence, but impose perturbations on the environment in which they reside. While cloud radiative effects slowly modify the thermal structure of the ambient atmosphere, the condensation of water vapor into clouds gives rise to a rapid effect of latent heat release that is thermodynamically compensated by vertical atmospheric motion. As such, clouds are deeply intertwined with the atmospheric dynamics governing local and global circulations through their diabatic heating (radiative plus latent heating) effects. The net latent heating (the heating by condensation offset by the cooling by reevaporation) is proportional to the mass of water removed from the air, that is, surface precipitation. Accurate measurements of global precipitation hence provide © Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_1

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Radiative Transfer: Solar radiation (visible/near infrared)

Statistical/Quantum Mechanics: Thermal/Radiative processes of the atmosphere

Electrodynamics: Particle scattering/absorption

Radiative Transfer: Earth ’s thermal radiation (thermal infrared/microwave)

Geometrical Optics: Surface emissivity/reflectivity

Fig. 1.1 Schematic illustration of satellite remote sensing of the atmosphere. Phenomena/processes appear in italic and the relevant fields/theories of physics are in bold

a critical observational constraint on the earth’s energy budget, just as are cloud measurements. Precipitation is not only a natural supplier of fresh water that our lives depend on, but sometimes brings about devastating disasters. Many countries across the world have deployed a dense, nation-wide network of precipitation observation for weather monitoring and forecasting. Nevertheless, a major fraction of the earth’s surface, in particular over oceans, remains beyond the reach of ground observation networks. Satellite remote sensing is a vital means of uniform observations on a global scale, offering invaluable data for research across various disciplines of earth sciences including climatology, meteorology, hydrology, and oceanography. Satellite measurements are indispensable also for operational weather forecasts and reanalysis as input observations to data assimilation systems. There are numerous satellite data products freely accessible that contain a variety of cloud and precipitation parameters. For a user new to satellite remote sensing, the challenge would be probably not the lack of choice but that there are too many options to choose from. The user is advised to be aware that the same variable could be derived from various types of sensors with uncertainties of different nature. Cloud amount from infrared radiance is not identical to lidar-based cloud amount, and precipitation from microwave brightness temperature never precisely agrees with radarderived precipitation. Even the same parameter from the same kind of instruments may give diverse answers when the measurements are made from different orbits or processed through different retrieval algorithms. Expert knowledge is instrumental for disentangling all these complex issues involved in the construction of satellite data products. This book is hoped to be a little fountain of useful knowledge on the basic principles of satellite remote sensing. It is merely electromagnetic wave signals that are received by satellite sensors. Layers of theories are required for building an end-to-end model to translate the satellite-received signals into geophysical variables (Fig. 1.1). At the heart of satellite

1 Introduction

5

remote sensing lies the theory of radiative transfer, which describes the propagation of photon energy over a distance by a simple transport equation known as the radiative transfer equation. The radiative transfer equation is very practical and useful in that it is equally applicable to the whole thickness of (and beyond) the atmosphere over a broad spectral range. The only major limitation is that the radiative transfer equation fails to deal with the wave aspects of light such as interference and diffraction. Such “wavy” phenomena, occurring on a microphysical scale when radiation is scattered or absorbed by particulates, require to invoke electrodynamics. As a practical compromise, the solutions to Maxwell’s equations are obtained off-line and then incorporated into the scattering/absorption cross section for use by the radiative transfer equation. Interference also plays a crucial role when radiation travels through and is reflected by a boundary at which refractive index changes discontinuously. Geometrical optics serves as a convenient theoretical framework for such situations. The emission and reflection of radiation at the earth’s surface are formulated by geometrical optics (the Fresnel equations) and used as the boundary condition to the atmospheric radiative transfer equation. The interactions of thermal radiation with media, the Planck function for a notable example, are understood in terms of statistical mechanics. Origins of gas absorption lines in the atmospheric spectrum are explained by elementary quantum mechanics. These elements are integrated into the prescribed assumptions (the source function and absorption cross section) in the radiative transfer equation. The goal of this book is to outline the physical background of measurement principles central of the cloud and precipitation retrieval from space. The chapters are grouped into four parts. The first part provides a general background, in which Introduction (this chapter) is followed by a summary of the satellite programs and instruments relevant to cloud and precipitation observations (Chap. 2). An overview of the theoretical basis of satellite orbits, scanning geometry, and sensor field of views is given in Chap. 3. Part II (Basic Physics) offers short courses on selected fields of physics: statistical mechanics as the theoretical basis of the blackbody spectrum and radiative transfer equation (Chap. 4), electrodynamics with a derivation of Mie’s solutions and geometrical optics (Chap. 5), general theories of radiative processes such as the absorption and emission of radiation by atmospheric gases and condensates (Chap. 6). These chapters are not necessarily a prerequisite for digesting later chapters, so the reader unwilling to dig so deep (or already proficient) in basic physics may skip the whole second part. Part III (Measurement Principles) constitutes the main contents of this book. The opening chapter is devoted to infrared measurements of clouds (Chap. 7), comprised of an introduction to the theory of non-scattering radiative transfer problems, simulated spectra showing the infrared properties of water vapor and clouds, and an overview of algorithms to retrieve different cloud variables (e.g., top temperature, optical depth, and particle radius) from infrared brightness temperature. The second and third chapters of Part III delineate visible/near-infrared and microwave observations, respectively (Chaps. 8 and 9). These chapters are paral-

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lel in structure to Chap. 7. Both begin with the radiative transfer theory at increasing levels of complexity, dealing with a scattering atmosphere for visible/near-infrared transfer and a scattering and emitting atmosphere for microwave transfer, followed by sections presenting spectral features and retrieval strategies. The subsequent chapter is dedicated to active remote sensing (Chap 10), where the measurement principles of radar and lidar are introduced. The last chapter of Part III offers a concise summary of the mathematical basis of retrieval algorithms (Chap. 11). Although the main focus of this book lies in the theoretical background of satellite remote sensing, some practical aspects of satellite observations are briefly discussed in Part IV (Applications). Chapter 12 offers an introduction to satellite-based cloud and precipitation data products. The final chapter (Chap. 13) shows a selected list of satellite data simulators, along with a summary of parameter setups in the simulator employed for synthetic satellite measurements presented throughout the book.

References Sherwood SC, Webb MJ, Annan JD, Armour KC, Forster PM, Hargreaves JC, Hegerl G, Klein SA, Marvel KD, Rohling EJ, Watanabe M, Andrews T, Braconnot P, Bretherton CS, Foster GL, Hausfather Z, von der Heydt AS, Knutti R, Mauritsen T, Norris JR, Proistosescu C, Rugenstein M, Schmidt GA, Tokarska KB, Zelinka MD (2020) An assessment of earth’s climate sensitivity using multiple lines of evidence. Rev Geophys 58:e2019RG000, 678. https://doi.org/10.1029/ 2019RG000678 Stubenrauch CJ, Rossow WB, Kinne S, Ackerman S, Cesana G, Chepfer H, Girolamo LD, Getzewich B, Guignard A, Heidinger A, Maddux BC, Menzel WP, Minnis P, Pearl C, Platnick S, Poulsen C, Riedi J, Sun-Mack S, Walther A, Winker D, Zeng S, Zhao G (2013) Assessment of global cloud datasets from satellites: project and database initiated by the GEWEX Radiation Panel. Bull Am Meteor Soc 94:1031–1049. https://doi.org/10.1175/BAMS-D-12-00117.1

Chapter 2

Satellite Missions and Instruments

In this chapter, satellite missions targeted on cloud and precipitation measurements are first outlined, followed by an overview of instruments aboard. It is not intended to go through a complete catalog of missions and instruments ever sent into orbit, but is instead to focus on the present and relatively recent satellite programs at the time of this writing. The reader new to satellite remote sensing might be overwhelmed at first by an inundation of acronyms, but there is of course no need to memorize all these abbreviations. The names of missions are not of eternal value anyway in that all satellite programs will soon become outdated (or may have already so for future readers). Nevertheless, the heritage of key mission objectives and instrument capabilities at present time will be passed on to future satellite programs with the underlying basics remaining largely unchanged. The ultimate goal of this chapter is to illustrate such ageless fundamentals of satellite technologies.

2.1 Satellite Missions 2.1.1 Overview The Television Infrared Observation Satellite (TIROS)-1, the very first weather satellite, was launched by the United States in 1960, a mere two and a half years after the advent of the satellite age by Sputnik 1 of the Soviet Union. Although operational for only less than three months, TIROS-1 proved the potential of satellite remote sensing for meteorological applications with thousands of cloud images acquired during its operation. Numerous satellite missions launched since then have vastly improved our capability of measuring clouds and precipitation over the globe.

© Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_2

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Operational Missions

LEO

DMSP (US) NOAA/Suomi NPP/JPSS (US) Meteor-M (RU) FY-1/FY-3 (CN) MetOp (EU)

Sun-synchronous (polar)

Research Missions Terra (US) Aqua (US) CALIPSO (US,FR) CloudSat (US) GCOM-C, GCOM-W (JP) TRMM (US,JP) Megha-Tropiques (FR,IN) GPM (US,JP)

Sun-asynchronous

GEO

GOES (US) GMS/MTSAT/Himawari (JP) Meteosat (EU) Elektro-L (RU) INSAT (IN) FY-2/FY-4 (CN) COMS/GK-2 (KR)

Fig. 2.1 Selected past and current meteorological satellite missions classified by mission goals and orbital configuration. Country codes: CN stands for China, EU for Europe, FR for France, IN for India, JP for Japan, KR for South Korea, RU for Russia, and US for the United States

Figure 2.1 presents a tabulated list of past and current satellite missions widely known for cloud and precipitation observations, classified by mission objectives and orbital configuration. Most of weather satellite programs in early years, except for experimental missions like TIROS-1 to test the feasibility of future satellite technologies, were built for operational purposes such as weather forecast and climate monitoring. Such operational satellite missions are implemented generally in collaboration between space agencies, responsible for spacecraft payloads/buses and launch vehicles, and weather/satellite agencies primarily in charge of the operations after launch. An operational mission typically consists of a series of satellites without substantial changes to instrument design over an extended period of time, because the long-term stability of observation data records (climate data records or CDRs) is a key priority of the operational satellite missions. As opposed to the operational missions, research satellite missions explore a new instrumental technology rather than a continuation of existing capabilities. Research missions are intended to open pathways for a breakthrough in weather and climate sciences by means of advanced satellite sensors that did not exist before. Space

2.1 Satellite Missions

9

agencies typically continue to operate research satellite missions after launch, seeking the assistance of science community for retrieval algorithm development and a variety of application studies to achieve the science goals of the missions. Satellite programs are classified into two major categories in terms of orbital configuration: the geostationary earth orbit (GEO) and low-earth orbits (LEO). A GEO satellite, phase-locked with terrestrial rotation, constantly observes the same side of the planet as the satellite revolves far away at a distance of ∼36,000 km from the earth’s surface. By contrast, LEO satellites rapidly orbit the earth at a much lower altitude between a few 100 and 1000 km.1 LEOs have an orbital period of roughly 90– 105 min, that is, LEO satellites go around the earth 14–16 times each day. GEO has the obvious advantage of conducting fixed-point observations, while LEO satellites enjoy far less restrictions in instrument design. Microwave sensors, for instance, are carried only by LEO spacecrafts since it would require a giant antenna beyond the current technology (and budget) to achieve a reasonable spatial resolution with microwave observations from GEO. Many of LEO meteorological satellites are sun-synchronous, which means that the satellite orbit is chosen so that observations are made always around fixed local hours, one in daytime and the other in nighttime, at any given point on the earth. Solar zenith angle stays nearly constant in the sunlit half of a sun-synchronous orbit, facilitating quality control for visible remote sensing. Another important property of sun-synchronous LEOs is that the inclination is close to 90◦ (see Sect. 3.1.2.3 for reasons behind this). It follows that sun-synchronous orbits are able to cover nearly the entire globe literally “from pole to pole”. For this reason, sun-synchronous satellites are also called polar(-orbiting) satellites. These strengths may be regarded as drawbacks depending on applications. Sunsynchronous satellites are by construction unable to sample diurnal cycle beyond a pair of fixed local hours. Polar orbiters have a relatively low revisiting frequency at low latitudes. Some LEO missions have preference for sun-asynchronous orbits, with priority given to diurnal sampling and dense observations over tropical latitudes. This section outlines the meteorological satellite programs in the past and those currently in operation (as of early 2022). A short summary of individual missions is given below.

2.1.2 Operational Satellite Missions The left column of Fig. 2.1 summarizes the operational satellite programs, constituted of LEO (top left) and GEO missions (bottom left). All operational LEO satellites, with the exception of early generations of the Russian Meteor series, are sun-synchronous.

1

There is no strict definition for the range of “LEO” altitudes.

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2 Satellite Missions and Instruments

Fig. 2.2 Artist’s view of the MetOp (left) and JPSS (right) satellites. The names of selected instruments were added by the author. Image credit: NASA (https://science.nasa.gov/toolkits/spacecrafticons)

2.1.2.1

LEO Satellites

The Defense Meteorological Satellite Program (DMSP), a Department of Defense program operated by the United States Air Force, is one of the most long-lasting weather satellite missions dating back to the 1960s. The DMSP Block 5D series consists of 19 (from F-1 to F-19) satellites, among which F-16, F-17, and F-18 are in operation at the time of this writing. The DMSP satellite instruments important for cloud and precipitation measurements are Special Sensor Microwave Imager (SSM/I) and Special Sensor Microwave Imager/Sounder (SSMIS). The first of a series of the SSM/I instruments was launched with F-8 in 1987. SSMIS, the successor of SSM/I with sounding capabilities integrated together, has been implemented in F-16 and later spacecrafts since 2003. The National Oceanic and Atmospheric Administration (NOAA) Polar Orbiting Environmental Satellite (POES) project was initiated in collaboration with NASA’s TIROS Next-generation (TIROS-N) program. Following the TIROS-N satellite launched in 1978, a series of the POES spacecrafts are sequentially numbered as NOAA-6 and NOAA-7 up to NOAA-19. Advanced Very High Resolution Radiometer (AVHRR) aboard the NOAA platforms is a visible/infrared imager providing a long-term record of cloud observations. The latest (fifth) generation of POES began with NOAA-15 in 1998, which was the first spacecraft equipped with the Advanced Microwave Sounding Unit (AMSU) instruments onboard. There was a plan to integrate DMSP and POES into a next-generation polar satellite program called the National Polar-orbiting Operational Environmental Satellite System (NPOESS). NPOESS, however, faced a difficulty due to budget inflation and was forced to be canceled. The Suomi NPOESS Preparatory Project (NPP), intended originally to be a precursor mission for NPOESS, was launched in 2011 as a transition to the Joint Polar Satellite System (JPSS), a program newly introduced to replace NPOESS.

2.1 Satellite Missions

11

Meteorological Operational Satellite Program of Europe (MetOp) is the operational LEO satellite mission managed by the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) (Klaes et al. 2007). Three MetOp spacecrafts have been put into orbit since 2006, all of which are operated in a “morning” orbit to complement the “afternoon” orbits offered by NOAA’s POES/NPP/JPSS satellites since NOAA-18. Figure 2.2 shows spacecraft images for MetOp and JPSS. Both satellites carry visible/infrared imagers (AVHRR/3 and Visible/Infrared Imager and Radiometer Suite or VIIRS), microwave sounders (AMSU and Advanced Technology Microwave Sounder or ATMS), and hyperspectral infrared sounders (Infrared Atmospheric Sounder Interferometer or IASI and Cross-Track Infrared Sounder or CrIS). Russia’s Meteor-M is the latest, upgraded version of their earlier LEO meteorological satellite missions (Meteor) dating back to the late 1960s and the 1970s. The Meteor-M series, started with its first spacecraft launched in 2009, consists of three satellites deployed with visible/infrared imagers like AVHRR and microwave radiometer/sounder units similar to SSMIS. China has launched a constellation of Fengyun-1 (FY-1)2 spacecrafts into polar orbits since 1988, followed by the secondgeneration polar orbiters (FY-3) since 2008. Instruments aboard the FY-3 platforms include Medium Resolution Spectral Imager (MERSI), Microwave Radiation Imager (MWRI), Hyperspectral Infrared Atmospheric Sounder (HIRAS), Microwave Temperature Sounder (MWTS), and Microwave Humidity Sounder (MWHS) (Yang et al. 2012).

2.1.2.2

GEO Satellites

The first generation of GEO meteorological satellites were dispatched in the late 1970s by the United States, Japan, and Europe. The geographical fortune that these three countries/regions are roughly equally spaced in longitude likely facilitated the WMO-coordinated international partnerships to achieve a complete coverage around the equator by a constellation of geostationary satellites. Since the first spacecraft launched in 1975, 17 GOES satellites have been put into orbit by the United States, among which two of the latest generation, the GOES-R series, are in operation. The two satellites are deployed at 75.2 ◦ W (GOES-East) and 137.2 ◦ W (GOES-West) to together cover the western Atlantic and eastern Pacific as well as the American continents. The primary GOES-R instrument is Advanced Baseline Imager (ABI), a 16-channel visible/infrared imager, equipped with additional near- and thermalinfrared bands that were unavailable from earlier generations of the GOES instruments (Schmit et al. 2005). Japan has been responsible for the Asian, Oceanian, and western Pacific regions in the GEO satellite networks. As a successor to the five GMS and two MTSAT spacecrafts launched into orbit since 1977, Himawari-8 (with Himawari-9 flying

2

Fengyun means “wind cloud” in Chinese.

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2 Satellite Missions and Instruments

together as a backup) is in operation at a longitude of 140.7 ◦ E.3 Both Himawari spacecrafts carry Advanced Himawari Imager (AHI), a visible/infrared sensor very similar to GOES ABI (Bessho et al. 2016). Meteosat, the European GEO satellite series, is a two-satellite system operated with one observing Africa and Europe and the other looking over the Indian Ocean. At the time of this writing, Meteosat-11 and -8 conduct full-disk observations at 0◦ and 41.5 ◦ E, respectively, while Meteosat-9 and -10 offer rapid-scanning services for a limited observational domain containing Europe and northern Africa (Schmetz et al. 2002). These Meteosat spacecrafts, which belong to Meteosat Second Generation (MSG), have Spinning Enhanced Visible and Infrared Imager (SEVIRI) aboard to measure visible and infrared radiation at 12 channels. An experimental GEO satellite launched in 2016 by China (FY-4A) is equipped with 14-channel Advanced Geosynchronous Radiation Imager (AGRI) and the first GEO hyperspectral infrared sounder, Geostationary Interferometric Infrared Sounder (GIIRS) (Yang et al. 2017). Other meteorological (or multi-purpose) satellites participating in the GEO satellite networks are the Elektro-L series of Russia, Indian National Satellite System (INSAT) of India, and the Communication, Ocean and Meteorological Satellite (COMS) and Geostationary-Korea Multi-Purpose Satellite (GEO-KOMPSAT or GK) series of South Korea.

2.1.3 Research Satellite Missions A sequence of satellite instruments that emerged in the late 1990s and onward set new technological standards of earth observations from space. Moderate Resolution Imaging Spectroradiometer (MODIS), for instance, is a visible/infrared imager equipped with 36 bands in contrast to traditional AVHRR-like sensors having typically about five channels. A new generation of microwave radiometers such as Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) and Advanced Microwave Scanning Radiometer-EOS (AMSR-E) have low-frequency (C/X-band) channels unavailable from SSM/I, which helped expand the applicability of passive microwave radiometry. TRMM Precipitation Radar (PR), CloudSat Cloud Profiling Radar (CPR), and Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) marked the advent of spaceborne active sensors, literally adding a new dimension to cloud and precipitation measurements from space. The past and current research missions (right column in Fig. 2.1) are all comprised of LEO satellites, many of which are sun-synchronous just like the operational LEO satellites while some research programs exploit unique benefits of a

3

The first of the latest generation (the Himawari series) is numbered as 8 because “himawari”, sunflower in Japanese, has been the domestic nickname of all the GMS/MTSAT/Himawari satellites throughout their generations.

2.1 Satellite Missions

13

Fig. 2.3 Artist’s view of the Aqua (left) and Terra (right) satellites. The names of selected instruments were added by the author. Image credit: NASA (https://science.nasa.gov/toolkits/spacecrafticons)

sun-asynchronous orbit. No research satellite has been sent into the GEO orbit to date.

2.1.3.1

Sun-Synchronous Satellites

Advanced Earth Observing Satellite (ADEOS) was launched in 1996 by National Space Development Agency of Japan (NASDA, currently JAXA), carrying eight instruments for the purpose of global environmental monitoring. Unfortunately, ADEOS and its follow-on mission ADEOS-II were each terminated 10 months after launch by spacecraft failures. The NASA Earth Observing System (EOS) program proposed a framework for a series of LEO satellites with a variety of observational capabilities. Terra, among the first of the EOS missions, was launched in 1999 with five sensors onboard including MODIS, Clouds and the Earth’s Radiant Energy System (CERES), and Multi-angle Imaging SpectroRadiometer (MISR). CERES is a broad-band imager, in contrast to narrow-band imagers like MODIS, designed to observe radiance integrated over a wide spectral range from visible to infrared wavelengths. MISR has the capability of measuring reflected solar radiation at separate viewing angles from nadir to 70.5◦ . A second MODIS instrument was launched with Aqua in 2002, an another EOS mission, together with AMSR-E, Atmospheric Infrared Sounder (AIRS), AMSU, and CERES. Figure 2.3 shows an artist’s illustration of the Terra and Aqua spacecrafts. These two satellites offer complementary sun-synchronous observations at 10:30/22:30 (Terra) and at 01:30/13:30 (Aqua) LT. The Aqua satellite is a part of the A-Train constellation, or a fleet of sunsynchronous (01:30/13:30) satellites flying closely together in formation (Stephens

14

2 Satellite Missions and Instruments

Fig. 2.4 Artist’s rendition of CloudSat (left) and CALIPSO (right). The instrument names were added by the author. Image credit: NASA (https://science.nasa.gov/toolkits/spacecraft-icons)

et al. 2002). CloudSat and CALIPSO (Fig. 2.4) joined A-Train in 2006.4 Simultaneous observations of clouds by CPR and CALIOP along with Aqua MODIS and other A-Train instruments have greatly enriched our understanding of cloud physical properties and their vertical structure. Other A-Train platforms include Polarization and Anisotropy of Reflectance for Atmospheric Sciences coupled with Observations from a Lidar from a Lidar (PARASOL), which is a small satellite developed by Centre National d’Études Spatiales (CNES), carrying the third instrument of Polarization and Directionality of the Earth’s Reflectances (POLDER). The PARASOL satellite terminated its 9-year-long mission in 2013. Aqua AMSR-E ceased its operation in 2011, and the follow-on instrument AMSR2 on board the Global Change Observation Mission-W (GCOM-W) satellite newly joined the A-Train in 2012. The JAXA GCOM-W program has a sister mission called GCOM-C, which has been in orbit since 2018 conducting visible and infrared observations with Second-generation Global Imager (SGLI).

2.1.3.2

Sun-Asynchronous Satellites

TRMM, a joint United-States and Japan mission, was unprecedented in different ways. The first spaceborne radar designed to measure precipitation, TRMM PR, was launched in late 1997 together with other instruments including TMI and Visible and Infrared Scanner (VIRS) (Fig. 2.5 left) (Kummerow et al. 2000). The TRMM satellite was operated in a sun-asynchronous orbit covering latitudes between roughly 35 ◦ S ad 35 ◦ N, with its orbital plane precessing in a period of about 46 days with respect to the direction toward the Sun. This unique orbital configuration, unlike polar orbits, allows to analyze the diurnal cycle of precipitation from TRMM observations 4

In 2018, CloudSat was forced by reaction wheel anomalies to exit from the A-Train and was later joined by CALIPSO to form a new fleet in a slightly lower orbit. This new formation is called C-Train.

2.1 Satellite Missions

15

Fig. 2.5 Artist’s rendition of the TRMM satellite (left) and the GPM core observatory (right). The names of selected instruments were added by the author. Image credit: NASA (https://science.nasa. gov/toolkits/spacecraft-icons)

accumulated over a duration sufficiently longer than a cycle of the orbital (nodal) precession. The TRMM orbit was boosted from 350 to 402 km in mid-2001 in order to reduce the rate of fuel consumption for spacecraft maneuvers against the atmospheric drag. This helped prolong the TRMM mission until April 2015. The Global Precipitation Measurement (GPM) core observatory was put into orbit in 2014, just in time before its predecessor mission TRMM came to an end. The ultimate goal of GPM is to construct a dense network of global precipitation monitoring with a constellation of existing LEO satellites in tandem with the core observatory (Skofronick-Jackson et al. 2017). The GPM core platform, carrying Dual-frequency Precipitation Radar (DPR) and GPM Microwave Imager (GMI) (Fig. 2.5 right), flies in a sun-asynchronous orbit with an inclination elevated to 65◦ so that it can observe higher latitudes than the TRMM satellite did. The downside, however, is a slower precession requiring 81 days, instead of 46 days for TRMM, to complete a whole cycle relative to the solar direction (see Sect. 3.1.2.5 for further details). The GPM constellation is comprised of microwave radiometers and sounders aboard different satellites provided by international partner agencies such as DMSP SSMIS, GCOMW AMSR2, JPSS ATMS, and MetOp AMSU. Megha-Tropiques, a satellite program launched jointly by France and India, is targeted exclusively on tropical meteorology (Roca et al. 2015). To this end, the orbital inclination is set to be as low as 20◦ , enabling frequent revisits of satellite overpasses over tropical latitudes (approximately 6 times per day between 10◦ and 20◦ in latitude). The Megha-Tropiques spacecraft, in orbit since 2011, has three major sensors aboard: Microwave Analysis and Detection of Rain and Atmosphere Systems

16

2 Satellite Missions and Instruments

(MADRAS), Sounder for Atmospheric Profiling of Humidity in the Intertropics by Radiometry (SAPHIR),5 and Scanner for Radiation Budget (ScaRaB). A mechanical failure forced MADRAS to cease data acquisition in 2013.

2.2 Satellite Instruments 2.2.1 Overview Spaceborne remote sensors may be categorized into four groups based on measuring principles (Table 2.1). Passive remote sensors measure radiation of natural origin, comprised of scattered solar radiation and terrestrial thermal emissions in case of meteorological applications. Insolation dominates visible and near-infrared wavelengths up to roughly 3 µm, while overwhelmed by the earth’s thermal radiation for longer wavelengths containing thermal infrared and microwave bands (Fig. 2.6). A passive instrument is a self-scanning telescope equipped with an onboard detector. The terms imager and radiometer, often used interchangeably, refer to a passive sensor designed to acquire plan-view images as seen from space. The imagers specialized in visible and infrared observations are summarized in Sect. 2.2.2.1 and the radiometers measuring microwave radiation are outlined in Sect. 2.2.2.2. Passive sensors with a capability to spectrally resolve the absorption lines of selected atmospheric gases can be employed to profile their vertical structure. Such instruments are called sounders (Sect. 2.2.3).6 As opposed to passive instruments, an active remote sensor exploits artificial electromagnetic echo transmitted by the sensor itself. A radar, coined to stand for “radio detection and ranging”, measures the intensity of backscattered echo relative to the transmitted radio wave power along with the time elapsed for transmitted echoes to return. A spaceborne down-looking radar observes the vertical profiles of cloud and precipitation in detail, or is even able to retrieve the three-dimensional structure if the radar has a scanning capability as well. A lidar, meaning “light detection and ranging”, is another active instrument similar to radars, except that Table 2.1 Satellite instruments classified by measuring principles Passive Active Plan-view imaging Vertical profiling

5

Imager/Radiometer Sounder

Scatterometer Radar and Lidar

The original name of SAPHIR in French is Sondeur Atmosphérique du Profil d’Humidité Intertropicale par Radiométrie. 6 It is called sounding in meteorological jargon to measure the vertical profiles of temperature and humidity (and of other gaseous components). This term is of course only analogical because acoustic waves play no role in either satellite or radiosonde observations.

2.2 Satellite Instruments

17

IR al m V er U Th 0.4 0.7 m 3 m 15 m R le rI sib ea i N V

Solar radiation

R rI

Fa

100 m

r

ic

M 1 mm

e

av

ow

1 m wavelength

Earth’s thermal emission Rescaled Blackbody Spectra

B

T = 288 K T = 5777 K

0.1

1

100

10

1000

10000

[ m]

Fig. 2.6 Schematic diagram of the electromagnetic spectrum. UV and IR stand for ultraviolet and infrared, respectively. Wavelengths specified are approximate measures to partition different spectral ranges. The 100-µm boundary between the far IR and submillimeter ranges is only tentative because there is no clear terminological distinction between them. Blackbody spectra (see Sect. 6.1.1) representing solar and earth’s thermal radiation, scaled to an identical amplitude, are shown at the bottom

lidars utilize visible light in place of microwave. Lidars have an excellent sensitivity to small cloud particles below radar detectability at the expense of severe attenuation of signals inside a thick cloud layer. The spaceborne radars and lidars ever put into orbit are summarized in Sect. 2.2.4. Radio occultation (RO) techniques, utilizing artificial radio signals from Global Navigation Satellite System (GNSS), may fall into the category of passive remote sensing aimed at vertical profiling because RO satellites only receive existing signals. A scatterometer is a radar designed specifically for estimating wind vector just above ocean surface. Sea surface roughness and waves are largely modulated by near-surface wind, giving rise to a fluctuation in the microwave surface reflectivity in different ways. Scatterometers measure the surface-reflected echoes to derive the wind speed and direction through a semi-empirical relationship between wind and reflectivity. Scatterometers are not explored any further in this book.

2.2.2 Imagers and Radiometers 2.2.2.1

Visible/Infrared Imagers

A vast majority of the operational satellite missions launched to date (Fig. 2.1) have the capability of visible and/or infrared imaging. Imagers covering the visible and infrared spectral ranges are a powerful observational tool for diverse applications such as clouds, aerosols, ocean color, sea surface temperature (SST), snow cover,

18

2 Satellite Missions and Instruments

Table 2.2 Channel central wavelength (λc in µm) and IFOV at nadir (Δx in km) for NOAA/MetOp AVHRR/3 (Robel and Graumann 2014), GOES ABI (Schmit et al. 2005), and Terra/Aqua MODIS (King et al. 1992). MODIS has double 3.959-µm channels having different sensitivities AVHRR/3 ABI MODIS λc (µm) Δx (km) λc (µm) Δx (km) λc (µm) Δx (km) Visible 0.630

Near IR

1.1

0.47 0.64

1.0 0.5

0.865

1.1

0.865, 1.61

1.0

1.61

1.1

1.378, 2.25

2.0

3.90, 6.19

2.0

Thermal IR 3.74

1.1

6.95, 7.34

2.0

10.8

1.1

8.5, 9.61

2.0

12.0

1.1

10.35, 11.2

2.0

12.3, 13.3

2.0

0.659, 0.865 0.470, 0.555 0.415, 0.443, 0.490, 0.531 0.565, 0.653, 0.681, 0.750 0.865, 0.905, 0.936, 0.940 1.240, 1.640, 2.130 3.750, 3.959, 4.050 4.465, 4.515, 4.565, 6.715 7.325, 8.550, 9.730, 11.030, 12.020, 13.335 13.635, 13.935, 14.235

0.25 0.5 1.0

1.0

1.0

0.5 1.0 1.0

1.0

1.0

1.0

and vegetation. Table 2.2 presents a summary of the channel specifications of selected visible/infrared instruments. The central wavelength (λc ) are chosen in light of the spectral properties of observational targets: for example, visible wavelengths (0.6– 0.7 µm) are useful for cloud/aerosol optical thickness, near-infrared bands (1–4 µm) are for cloud particle size, and the split-window channels (10–14 µm) are for cloud top temperature. Instantaneous field of view (IFOV), a measure of the horizontal resolution (Δx) (see Sect. 3.2.1 for details), is typically 1–2 km except for a limited number of channels with an enhanced resolution of 250 and 500 m. It is noted that these numbers are exact only at satellite nadir because sensor footprints become elongated as the viewing angle deviates from nadir. NOAA/MetOp AVHRR/3 is a six-channel radiometer (Robel and Graumann 2014), drawing upon the heritage of the traditional five-channel (without 1.6 µm)

2.2 Satellite Instruments

19

Table 2.3 Channel frequency ( f in GHz) with polarization (H and V for horizontal and vertical polarizations, respectively) and IFOV (Δx in km, along-track × cross-track) for DMSP SSM/I (Hollinger et al. 1987), TRMM TMI (Kummerow et al. 1998), Aqua AMSR-E (Kawanishi et al. 2003), and GPM GMI (Hou et al. 2014). The SSM/I cross-track effective FOVs (EFOVs) listed in Table 2.1 of Hollinger et al. (1987) have been adjusted to the corresponding IFOVs by multiplying the ratio of instantaneous beamwidth to effective beamwidth, while EFOV and IFOV are assumed to be identical along the satellite track. The TMI IFOVs were slightly degraded from the listed values after the TRMM orbit was boosted from 350 to 402.5 km in 2001 SSM/I TMI AMSR-E GMI f (GHz) Δx (km) f (GHz) Δx (km) f (GHz) Δx (km) f (GHz) Δx (km)

19.35 H/V 22.235 V 37.0 H/V 85.5 H/V

69×42 50×36 37×24 15×9

10.65 H/V 19.35 H/V 21.3 V 37.0 H/V 85.5 H/V

60×36 30×18 27×17 16×10 7×4

6.925 H/V 10.65 H/V 18.7 H/V 23.8 H/V 36.5 H/V 89.0 H/V

75×43 51×29 27×16 32×18 14×8 6×4

10.65 H/V 18.7 H/V 23.8 V 36.5 H/V 89.0 H/V 166 H/V 183.31±3 V 183.31±7 V

32×19 18×11 15×9 15×9 7×4 7×4 7×4 7×4

AVHRR instruments on an early series of the NOAA satellites. This AVHRR channel configuration set a minimum baseline for the instrument design of visible/infrared imagers ever since. Visible/infrared imagers aboard the GEO satellites were also equipped typically with three to six bands before upgraded to the latest generation. GOES-R ABI (Schmit et al. 2005) and Himawari AHI (Bessho et al. 2016) each have 16 channels from 0.47 µm to 13.3 µ, and MSG SEVIRI (Schmetz et al. 2002) and FY-4A AGRI (Yang et al. 2017) are similar instruments with slightly fewer channels. The ABI channel specifications are shown in Table 2.2. Some recent LEO satellites carry visible/infrared imagers of enhanced capabilities beyond the AVHRR series. VIIRS on Suomi NPP and the JPSS series is a 16-band imager having somewhat different channel specifications from ABI (Cao et al. 2013). The two MODIS instruments (King et al. 1992) carried by the Terra and Aqua platforms are arguably the most widely used visible/infrared imagers in the science community. MODIS has 36 channels, some of which share an identical wavelength but have different IFOVs and sensitivities to accommodate multiple observational targets.

2.2.2.2

Microwave Radiometers

The microwave radiometry of clouds and precipitation employs centimeter and millimeter waves (10–183 GHz) (Table 2.3). Microwave radiometers are conically scanning instruments, which sweep an arc on the earth’s surface as looking down at a fixed zenith angle (52–55◦ at the surface, see Sect. 3.2.3.2 for more detail). Each

20

2 Satellite Missions and Instruments

frequency is split into a pair of channels specialized for horizontal (H) and vertical (V) polarizations. This is intended mainly to isolate out surface emission from atmospheric emission and absorption because the surface emissivity depends distinctly on the H and V polarizations (Sect. 5.2). Water vapor channels (22 and 183 GHz) often lack a second polarization. A series of the DMSP SSM/I instruments had been in operation since its first launch in 1987 until the last on board DMSP F-15 completed its mission in 2020.7 SSM/I has seven channels of 19.35 GHz H/V, 22.235 GHz V, 37.0 GHz H/V, and 85.5 GHz H/V (Hollinger et al. 1987). Low-frequency channels among them are used mainly for retrieving sea-surface wind speed and liquid precipitation, while high-frequency channels are sensitive to cloud liquid water and solid precipitation particles as well. The 22.235-GHz channel, tuned to match the center of a watervapor absorption line (Sect. 6.2.2), is instrumental for estimating column water vapor (CWV).8 Most of the microwave radiometers developed later draw upon the legacy of these SSM/I channels apart from minor alterations to frequency. TRMM TMI was built with a pair of 10.65-GHz channels in addition to the SSM/I-like configuration (Kummerow et al. 1998). This addition allows to make a reliable estimation of heavy precipitation that could suffer radiative saturation at higher frequencies. Similarly, a slight change was made to the waver-vapor channel frequency from 22.235 to 21.3 GHz to avoid radiative saturation by extreme moisture in the tropics. An even lower frequency of 6.925 GHz was implemented in Aqua AMSR-E for the purpose of expanding the 10-GHz capability of SST estimation to cold ocean surfaces and of enabling soil moisture measurements (Kawanishi et al. 2003). GCOM-W AMSR2 is nearly identical to its predecessor AMSR-E except for an extra channel at 7.3 GHz, which has been added to mitigate the radio frequency interference at 6.925 GHz. GMI aboard the GPM core satellite, a follow-on sensor to TMI, is upgraded with additional high-frequency (G-band millimeter-wave) channels at 166, 183.31±3, and 183.31±7, aimed at improving snow and light-rain detections (Hou et al. 2014). Other microwave radiometers measuring frequencies above 89 GHz include MeghaTropiques MADRAS, which has a pair of channels at 157 GHz in addition to the conventional lower-frequency bands. The AMSR3 instrument, being developed for a near-future launch with the Global Observing Satellite for Greenhouse gases and Water cycle (GOSAT-GW) satellite, will be equipped with GMI-like millimeter-wave channels. An emerging trend is small millimeter-wave radiometers carried by CubeSats such as Temporal Experiment for Storms and Tropical Systems-Demonstration (TEMPEST-D) and Time-Resolved Observations of Precipitation structure and storm Intensity with a Constellation of Smallsats (TROPICS).

7

DMSP F-16 and the later DMSP satellites carry SSMIS in replace of SSM/I. CWV, also known as water vapor path (WVP) or (total) precipitable water, is the vertically integrated mass of atmospheric water vapor per area. CWV has the units of kg m−2 , or mm when expressed as the depth of liquid water of equivalent mass.

8

2.2 Satellite Instruments

21

Microwave radiometers do not have as good a spatial resolution as visible/infrared imagers because IFOV is constrained by the diffraction limit for microwave radiometry (see Sect. 3.2.1 for detailed discussion). In short, the best achievable IFOV is inversely proportional to frequency for a given antenna diameter (or to antenna size for a given frequency). It is confirmed from Table 2.3 that IFOV can exceed 50 km at low frequencies while much smaller than 10 km at high frequencies. AMSR-E has a significantly better IFOV than SSM/I because the 1.6-m AMSR-E dish is more than twice as large as the SSM/I reflector. The two sensors in a sun-asynchronous orbit, TMI and GMI, have an advantage of flying closer to the observational targets than SSM/I and AMSR-E. The orbital altitudes of the TRMM and GPM-core satellites are only about a half of the polar orbiters’ altitudes (705 km for Aqua and 830 km for DMSP). GMI’s 1.2-m dish is twice as big as the TMI antenna, which accounts for the resolution difference between the two instruments.

2.2.3 Sounders 2.2.3.1

Microwave Sounders

A sounder is a radiometer instrument tailored with a set of channel frequencies targeted on selected molecular absorption lines. Microwave sounders are built around the 60-GHz O2 band for temperature sounding and the 183-GHz H2 O band for humidity sounding (consult Sects. 6.2.2 and 6.2.3 for the theoretical background of microwave absorption lines). Table 2.4 is an excerpt from the SSMIS and ATMS specifications. SSMIS, adopted for the F-16 and later DMSP satellites, is an integrated microwave sensor combining an imager (SSM/I) and a sounder that were separate payloads on earlier DMSP platforms. As such, SSMIS is a conically scanning instrument equipped with a number of temperature and humidity sounding channels as well as several window channels inherited from SSM/I. All other existing microwave sounders are cross-track scanning sensors (Sect. 3.2.3.1). AMSU, the first of which was launched aboard the NOAA-15 satellite in 1998, since has been the standard for the NOAA and MetOp sounder series. AMSU consists of two components: AMSU-A for use by temperature sounding and AMSUB, later replaced by Microwave Humidity Sounder (MHS), for water vapor sounding. ATMS, the latest version of the AMSU instruments, is implemented on the Suomi NPP and JPSS satellites (Kim et al. 2014). The channel specifications of ATMS are listed in Table 2.4. AMSU-A/B and MHS share nearly all the channel frequencies with ATMS except for a few minor alterations (Robel and Graumann 2014). Unlike conically scanning radiometers, cross-track scanning sounders typically do not measure horizontal and vertical polarizations separately. Instead, the two polarizations are mixed in a varying proportion with the scan angle because the polarization plane as measured by the instrument rotates with the reflector. In Table 2.4, “QH”

22

2 Satellite Missions and Instruments

Table 2.4 Sensor properties of DMSP SSMIS (Kunkee et al. 2008) and Suomi NPP/JPSS ATMS (Kim et al. 2014). SSMIS: Channel frequency ( f in GHz), IFOV (Δx in km, along-track × crosstrack) and polarization (H for horizontal, V for vertical, and C for circular polarization). Five channels from 50 to 55 GHz have H polarization (denoted by H† ) except for the F-16 SSMIS instrument, for which V is assigned to these channels. ATMS: Channel frequency ( f in GHz), IFOV at nadir (Δx in km), and polarization at nadir. Polarization is designated as QH (QV) in case that the polarization plane accords with the along-track (cross-track) plane when the rotating antenna is pointed at nadir SSMIS Window

Temperature Sounding

Window WV Sounding

ATMS

f (GHz)

Δx (km)

Pol.

19.35

72×24

H/V

f (GHz)

Δx (km)

Pol. QV

22.235

72×24

V

23.8

75

37.0

44×26

H/V

31.4

75

QV

50.3

29×17

H†

50.3

32

QH

51.76

32

QH

52.8

29×17

H†

52.8

32

QH

53.596

29×17

H†

53.596 ± 0.115

32

QH

54.4

29×17

H†

54.4

32

QH

54.95

32

QH QH

55.5

29×17

H†

55.5

32

57.29

26×16

C

57.290344

32

QH

63.283248 ± 0.285271

26×16

C

57.290344 ± 0.217

32

QH

60.792668 ± 0.357892

26×16

C

60.792668 ± 0.357892 ± 0.002

26×16

C

57.290344 ± 0.3222 ± 0.048

32

QH

60.792668 ± 0.357892 ± 0.0055

26×16

C

57.290344 ± 0.3222 ± 0.022

32

QH

60.792668 ± 0.357892 ± 0.016

26×16

C

57.290344 ± 0.3222 ± 0.010

32

QH

60.792668 ± 0.357892 ± 0.050

26×16

C

57.290344 ± 0.3222 ± 0.0045

32

QH

59.4

26×16

C

91.655

15×9

H/V

88.2

32

QV

150

15×9

H

165.5

16

QH

183.31 ± 6.6

15×9

H

183.31 ± 3.0 183.31 ± 1.0

15×9 15×9

H H

183.31 ± 7.0

16

QH

183.31 ± 4.5

16

QH

183.31 ± 3.0

16

QH

183.31 ± 1.8

16

QH

183.31 ± 1.0

16

QH

2.2 Satellite Instruments

23

Table 2.5 Channel wavelength range (λ in µm), the number of channels (# Chnls.), IFOV at nadir (Δx in km) for Aqua AIRS, MetOp IASI, and Suomi NPP/JPSS CrIS. The listed values are adopted from Menzel et al. (2018) AIRS λ (µm)

IASI # Chnls. Δx (km) λ (µm)

CrIS # Chnls. Δx (km) λ (µm)

3.74–4.61

# Chnls. Δx (km)

3.92–4.64

6.20–8.22 2,378

13

3.62–15.5

8.80–15.4

8,464

12

5.71–8.26

1,385

14

9.31–15.38

and “QV” signify the polarization plane parallel to the along-track and cross-track directions, respectively, when the rotating antenna is pointed at nadir. The Aqua satellite carries AMSU-A and Humidity Sounder for Brazil (HSB), the latter of which ceased operation in 2003. SAPHIR on board the Megha-Tropiques satellite is a six-channel humidity sounder at the frequencies of 183.31±0.2, ±1.1, ±2.8, ±4.2, ±6.6, and ±11.0 GHz (Roca et al. 2015).

2.2.3.2

Infrared Sounders

The near- and thermal-infrared spectra are a treasure box of absorption lines since molecular vibrational transitions fall in this spectral range (see Sect. 6.2.4 for more detail). Carbon dioxide and water vapor are the primary source of the infrared absorption, with other trace gases such as ozone making secondary contributions as well. Infrared sounding provides temperature and humidity estimates at a higher vertical resolution than microwave sounding, at the expense of severer susceptibility to cloud contamination. Infrared and microwave soundings are often carried out in tandem, exploiting the fact that the drawbacks and strengths of these individual sounding techniques are largely complementary. TIROS Operational Vertical Sounder (TOVS), developed for the TIROS and NOAA satellite series, was among the earliest attempts to integrate infrared and microwave sounding measurements by High Resolution Infrared Radiation Sounder (HIRS) and Microwave Sounding Unit (MSU). Traditional infrared sounders including HIRS had a limited number (typically 20–30) of channels at discrete wavelengths. These infrared sounders, however, are not optimal in spectral resolution because infrared absorption lines are so rich and fine that the information content in the infrared gas spectrum is not fully exploitable by the conventional sounders. This limitation is overcome by a hyperspectral sounder, which is a spectrometer (grating instrument or interferometer) spanning wide infrared ranges with a drastically improved spectral resolution. The hyperspectral infrared sounders implemented on LEO satellites are Aqua AIRS (Chahine et al. 2006), MetOp IASI (Klaes et al. 2007), Suomi NPP/JPSS CrIS (Han et al. 2013), and FY-3 HIRAS (Yang et al. 2012). Table 2.5 presents the spectral ranges and number of channels for selected instruments. Aqua AIRS is designed to observe three separate spectral ranges with

24

2 Satellite Missions and Instruments

2,378 channels in total. MetOp IASI has a single, continuous band covering the entire thermal infrared range with more than 8,000 channels. GIIRS is the first hyperspectral infrared sounder sent into GEO (Yang et al. 2017). Meteosat Third Generation (MTG) is planned to be launched with Infrared Sounder (IRS) to conduct full-disk hyperspectral sounding from GEO.

2.2.4 Radars and Lidars Three spaceborne radars built for clouds and/or precipitation observations have been put into orbit to date (as of 2022, see Table 2.6), apart from RainCube, a technology demonstration CubeSat carrying a small Ka-band radar, in orbit from 2018 to 2020. TRMM PR is a Ku-band (13.6-GHz) phased-array radar with the capability of crosstrack scanning across a 215-km-wide swath (or 245 km after the orbital boost) (Kummerow et al. 1998). With a minimum detectable echo of about 18 dBZ, PR was able to measure rain rates above ∼0.7 mm h−1 . The targets of PR observations include solid precipitation as well as rainfall, although detecting frozen hydrometers is somewhat challenging because of the differences in microwave dielectric properties between liquid water and ice. GPM DPR consists of two stand-alone phased-array radars called KuPR and KaPR (Hou et al. 2014). KuPR is almost identical to TRMM PR in instrument design. KaPR has a higher frequency of 36 GHz (Ka-band) and initially had a 120-km-wide swath, half as wide as the KuPR swath (245 km). KaPR is expected to (1) detect light rain and snow below the KuPR detectability and (2) refine precipitation estimates when used in combination with KuPR. To accomplish these two goals, KaPR initially had the two observational modes of matched scans (MS) and high-sensitivity scans (HS). In the MS mode, KaPR samples FOVs that are exactly matched in space with KuPR beams. This facilitates simultaneous measurements between the two frequencies for the overlapping scans within the inner half of the KuPR swath. The HS mode, on the other hand, makes observations with an enhanced sensitivity of ∼12 dBZ compared

Table 2.6 Channel frequency ( f in GHz), IFOV (Δx in km), and range (vertical) bin size (Δz in m) for TRMM PR (Kummerow et al. 1998), GPM DPR (Hou et al. 2014), and CloudSat CPR (Stephens et al. 2002). The Ka component of DPR (KaPR) has the vertical resolution of 250 m in the MS mode and of 500 m in the HS mode (see text). The PR IFOV was enlarged to about 5 km by the TRMM orbit boost from 350 to 402.5 km in 2001 TRMM PR GPM DPR CloudSat CPR f (GHz) Δx (km) Δz (m) f (GHz) Δx (km) Δz (m) f (GHz) Δx (km) Δz (m) 13.80 (Ku)

4.3

250

13.60 (Ku) 35.55 (Ka)

5

250

5

250/500

94 (W)

1.4

500

2.2 Satellite Instruments

25

Table 2.7 Horizontal resolution (Δx in km) and vertical resolution (Δz in m) for different ranges of the altitude (z in km) in the downlinked data from CALIPSO CALIOP (Winker et al. 2004) z (km) 532 nm 1,064 nm Δx (km) Δz (m) Δx (km) Δz (m) 30.1–40.0 20.2–30.1 8.2–20.2 −0.5–8.2 −2.0–−0.5

5.0 1.67 1.0 0.33 0.33

300 180 60 30 300

– 1.67 1.0 0.33 0.33

– 180 60 60 300

to the nominal value of ∼18 dBZ at the expense of degraded vertical resolution (500 m). It was decided later, however, to reconfigure the HS scan pattern to match the outer half of the KuPR swath so that the entire 245-km-wide DPR scans are filled with matched FOVs. This new operation began on 21 May 2018. CloudSat CPR is a nadir-looking W-band (94-GHz) radar (Stephens et al. 2002). A 1.4-km IFOV of CPR, along with 500-m-wide vertical bins oversampled at an interval of 250 m, enables observations of the detailed cloud structure within a vertical cross section sliced along the satellite track. CPR, capable of capturing subtle echoes as weak as −28 dBZ, is sensitive to non-precipitating clouds that are left largely undetected by PR and DPR. CloudSat encountered a severe battery anomaly in April 2011, impairing the ability of spacecraft attitude control. CloudSat observations resumed six months later as “daylight only” operations, in which CPR collects data only while flying on the sun-lit side of the earth. CALIPSO CALIOP is a spaceborne lidar with a pair of channels at the wavelengths of 523 and 1,064 nm, the former of which can measure two polarizations perpendicular to each other (Winker et al. 2004). CALIOP observes thin clouds and aerosols that microwave sensors fail to capture, while lidar signals are quickly lost as entering deep in a thick cloud layer because of attenuation. The nominal horizontal and vertical resolutions of CALIOP measurements are 0.33 km and 30 m, respectively, which are averaged onboard over multiple pulses in a manner depending on the altitude to achieve an optimal balance of sensitivity (Table 2.7). The height-dependent resolution allows CALIOP to profile thin, broken clouds in the lower troposphere as well as to detect weak signals from spatially extensive polar stratospheric clouds. Negative altitudes down to −0.5 km account for elevations below sea level (e.g., the Dead Sea) and even lower altitudes between −2.0 and −0.5 km with a degraded vertical resolution are intended for monitoring artificial echoes arising from, for example, multiple scattering by dense clouds near surface. Another CALIOP-like lidar is Cloud-Aerosol Transport System (CATS), operated on the International Space Station (ISS) for 2.5 years from 2015 to 2017. CloudSat and CALIPSO make nearly simultaneous observations as part of the A Train (or the C Train after 2018, see Sect. 2.1.3.1) constellation, offering the opportunities to observe a broad spectrum of clouds from thin cirrus to cumulonimbus.

26

2 Satellite Missions and Instruments

This radar-lidar synergy concept is extended to the Earth Clouds, Aerosols and Radiation Explorer (EarthCARE) mission, planned to be launched in the near future. The EarthCARE payloads include CPR and Atmospheric Lidar (ATLID). EarthCARE CPR is a W-band Doppler radar with an enhanced sensitivity and resolution compared to CloudSat CPR. ATLID is a 355-nm high spectral resolution lidar (HSRL), a lidar with the capability to isolate out cloud and aerosol signals from the molecular return (Illingworth et al. 2015).

References Bessho K, Date K, Hayashi M, Ikeda A, Imai T, Inoue H, Kumagai Y, Miyakawa T, Murata H, Ohno T, Okuyama A, Oyama R, Sasaki Y, Shimazu Y, Shimoji K, Sumida Y, Suzuki M, Taniguchi H, Tsuchiyama H, Uesawa D, Yokota H, Yoshida R (2016) An introduction to Himawari-8/9–Japan’s new-generation geostationary meteorological satellites. J Meteor Soc Jpn 94:151–183. https:// doi.org/10.2151/jmsj.2016-009 Cao C, Xiong J, Blonski S, Liu Q, Uprety S, Shao X, Bai Y, Weng F (2013) Suomi NPP VIIRS sensor data record verification, validation, and long-term performance monitoring. J Geophys Res 118:11,664–11,678. https://doi.org/10.1002/2013JD020418 Chahine MT, Pagano TS, Aumann HH, ATLAS R, Barnet C, Blaisdell J, Chen L, Divakarla M, Fetzer EJ, Goldberg M, Gautier C, Granger S, Hannon S, Irion FW, Kakar R, Kalnay E, Lambrigtsen BH, Lee SY, MARSHALL JL, McMillan WW, McMillin L, Olsen ET, Revercomb H, Rosenkranz P, Smith WL, Staelin D, Strow LL, Susskind J, Tobin D, Wolf W, Zhou L (2006) AIRS: improving weather forecasting and providing new data on greenhouse gases. Bull Am Meteor Soc 87:911– 926. https://doi.org/10.1175/BAMS-87-7-911 Han Y, Revercomb H, Cromp M, Gu D, Johnson D, Mooney D, Scott D, Strow L, Bingham G, Borg L, Chen Y, DeSlover D, Esplin M, Hagan D, Jin X, Knuteson R, Motteler H, Predina J, Suwinski L, Taylor J, Tobin D, Tremblay D, Wang C, Wang L, Wang L, Zavyalov V (2013) Suomi NPP CRIS measurements, sensor data record algorithm, calibration and validation activities, and record data quality. J Geophys Res 118:12,734–12,748. https://doi.org/10.1002/2013JD020344 Hollinger J, Lo R, Poe G, Savage R, Peirce J (1987) Special sensor microwave/imager user’s guide. Naval Research Laboratory, Washington, DC, USA Hou AY, Kakar RK, Neeck S, Azarbarzin AA, Kummerow CD, Kojima M, Oki R, Nakamura K, Iguchi T (2014) The Global Precipitation Measurement mission. Bull Am Meteor Soc 95:701– 722. https://doi.org/10.1175/BAMS-D-13-00164.1 Illingworth AJ, Barker HW, Beljaars A, Ceccaldi M, Chepfer H, Clerbaux N, Cole J, Delanoë J, Domenech C, Donovan DP, Fukuda S, Hirakata M, Hogan RJ, Huenerbein A, Kollias P, Kubota T, Nakajima T, Nakajima TY, Nishizawa T, Ohno Y, Okamoto H, Oki R, Sato K, Satoh M, Shephard MW, Velázzquez-Blázquez A, Wandinger U, Wehr T, van Zadelhoff GJ (2015) The EarthCARE satellite: the next step forward in global measurements of clouds, aerosols, precipitation, and radiation. Bull Am Meteor Soc 96:1311–1332. https://doi.org/10.1175/BAMS-D-12-00227.1 Kawanishi T, Sezai T, Ito Y, Imaoka K, Takeshima T, Ishido Y, Shibata A, Miura M, Inahata H, Spencer R (2003) The advanced microwave scanning radiometer for the earth observing system (AMSR-E), NASDA’s contribution to the EOS for global energy and water cycle studies. IEEE Trans Geosci Remote Sen 41:184–194. https://doi.org/10.1109/TGRS.2002.808331 Kim E, Lyu CHJ, Anderson K, Leslie VR, Blackwell WJ (2014) S-NPP ATMS instrument prelaunch and on-orbit performance evaluation. J Geophys Res 119:5653–5670. https://doi.org/10.1002/ 2013JD020483

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King M, Kaufman Y, Menzel W, Tanre D (1992) Remote sensing of cloud, aerosol, and water vapor properties from the Moderate Resolution Imaging Spectrometer (MODIS). IEEE Trans Geosci Remote Sens 30:2–27. https://doi.org/10.1109/36.124212 Klaes KD, Cohen M, Buhler Y, Schlüssel P, Munro R, Luntama JP, von Engeln A, Clérigh EO, Bonekamp H, Ackermann J, Schmetz J (2007) An introduction to the EUMETSAT polar system. Bull Am Meteor Soc 88:1085–1096. https://doi.org/10.1175/BAMS-88-7-1085 Kummerow C, Simpson J, Thiele O, Barnes W, Chang A, Stocker E, Adler R, Hou A, Kakar R, Wentz F, Ashcroft P, Kozu T, Hong Y, Okamoto K, Iguchi T, Kuroiwa H, Im E, Haddad Z, Huffman G, Ferrier B, Olson W, Zipser E, Smith E, Wilheit T, North G, Krishnamurti T, Nakamura K (2000) The status of the Tropical Rainfall Measuring Mission (TRMM) after two years in orbit. J Appl Meteor 39(12):1965–1982 Kummerow CD, Barnes W, Kozu T, Shiue J, Simpson J (1998) The Tropical Rainfall Measuring Mission (TRMM) sensor package. J Atmos Oceanic Technol 15:809–817 Kunkee DB, Poe GA, Boucher DJ, Swadley SD, Hong Y, Wessel JE, Uliana EA (2008) Design and evaluation of the first Special Sensor Microwave Imager/Sounder. IEEE Trans Geosci Remote Sen 46:863–883. https://doi.org/10.1109/TGRS.2008.917980 Menzel WP, Schmit TJ, Zhang P, Li J (2018) Satellite-based atmospheric infrared sounder development and applications. Bull Am Meteor Soc 99:583–603. https://doi.org/10.1175/BAMS-D16-0293.1 Robel J, Graumann A (2014) NOAA KLM user’s guide. National Oceanic and Atmospheric Administration, Asheville, NC, USA Roca R, Brogniez H, Chambon P, Chomette O, Cloché S, Gosset ME, Mahfouf JF, Raberanto P, Viltard N (2015) The Megha-Tropiques mission: a review after three years in orbit. Front Earth Sci 3:17. https://doi.org/10.3389/feart.2015.00017 Schmetz J, Pili P, Tjemkes S, Just D, Kerkmann J, Rota S, Ratier A (2002) An introduction to Meteosat Second Generation (MSG). Bull Am Meteor Soc 83:977–992. DOI:https://doi.org/10.1175/1520-0477(2002)0832.3.CO;2 Schmit TJ, Gunshor MM, Menzel WP, Gurka JJ, Li J, Bachmeier AS (2005) Introducing the nextgeneration Advanced Baseline Imager on GOES-R. Bull Am Meteor Soc 86(8):1079–1096. https://doi.org/10.1175/BAMS-86-8-1079 Skofronick-Jackson G, Petersen WA, Berg W, Kidd C, Stocker EF, Kirschbaum DB, Kakar R, Braun SA, Huffman GJ, Iguchi T, Kirstetter PE, Kummerow C, Meneghini R, Oki R, Olson WS, Takayabu YN, Furukawa K, Wilheit T (2017) The Global Precipitation Measurement (GPM) mission for science and society. Bull Am Meteor Soc 98(8):1679–1695. https://doi.org/10.1175/ BAMS-D-15-00306.1 Stephens GL, Vane DG, Boain RJ, Mace GG, Sassen K, Wang Z, Illingworth AJ, O’connor EJ, Rossow WB, Durden SL, Miller SD, Austin RT, Benedetti A, Mitrescu C, Team CS (2002) The CloudSat mission and the A-Train: a new dimension of space-based observations of clouds and precipitation. Bull Am Meteor Soc 83:1771–1790. https://doi.org/10.1175/BAMS-83-12-1771 Winker DM, Hunt WH, Hostetler CA (2004) Status and performance of the CALIOP lidar. Laser Radar Tech Atmos Sens SPIE 5575:8–15. https://doi.org/10.1117/12.571955 Yang J, Zhang P, Lu N, Yang Z, Shi J, Dong C (2012) Improvements on global meteorological observations from the current Fengyun 3 satellites and beyond. Int J Digital Earth 5:251–265. https://doi.org/10.1080/17538947.2012.658666 Yang J, Zhang Z, Wei C, Lu F, Guo Q (2017) Introducing the new generation of Chinese geostationary weather satellites, Fengyun-4. Bull Am Meteor Soc 98:165–1637. https://doi.org/10.1175/ BAMS-D-16-0065.1

Chapter 3

Satellite Orbit and Scan

Choosing the optimal orbit is a critical step for designing satellite programs to meet given mission goals. Fundamentals of orbital mechanics, with emphasis on those relevant to meteorological satellites, are reviewed for illustrating the physical background behind different types of satellite orbits mentioned in Sect. 2.1. While GEO stands on a very simple physical principle, the physics underlying LEO, for which a higher-order spherically-asymmetric component of the earth’s gravity field (the J2 perturbation) is crucial, is far more complicated than GEO. Section 3.1 provides a compact but complete derivation of the J2 perturbation formula along with its implications for LEO configurations. The second half of the chapter (Sect. 3.2) is devoted to an overview of the instrument design such as the field of view and scanning geometry.

3.1 Orbital Mechanics We have seen in Sect. 2.1 that satellite missions may be grouped in terms of orbital characteristics. GEO and LEO constitute the major categories, and LEO satellites are further classified into the subcategories of sun-synchronous and asynchronous orbits. This section outlines the theoretical basis of each orbital configuration in light of elementary orbital mechanics.

3.1.1 Geostationary Earth Orbit (GEO) A satellite orbits the earth at a constant altitude when the centrifugal acceleration is balanced against the terrestrial gravity, © Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_3

29

30

3 Satellite Orbit and Scan

Fig. 3.1 Full-disk observations by Himawari-8 AHI at Band 3 (0.64 µm, left) and Band 13 (10.4 µm, right) acquired at 0300 UTC 28 April 2021. The latitude-longitude grid is drawn every 10◦ . Image credit: JMA Meteorological Satellite Center (https://www.data.jma.go.jp/mscweb/data/ himawari/)

r ω2 =

G M⊕ , r2

(3.1)

where r is the distance from the earth’s center of gravity, ω is the angular velocity of satellite revolution, G is the gravitational constant (6.674 × 10−11 m3 kg−1 s −2 ), M⊕ is the mass of the earth (5.974 × 1024 kg). Equation (3.1) immediately leads to  h=

G M⊕ T 2 4π 2

1/3 − R⊕ ,

(3.2)

where R⊕ is the equatorial radius of the earth (6378 km), h ≡ r − R⊕ is the orbital altitude, and T ≡ 2π/ω is the orbital period. Equation (3.2), an equivalent of Kepler’s third law of planetary motion, determines the altitude of a circularly orbiting satellite having a given orbital period, T . The revolution of a GEO satellite is by design synchronized with the earth’s spin, so T is equivalent to the earth’s rotational period (23 h 56 m 4 s or 86,164 s).1 Inserting these numbers into (3.2), one obtains an estimate of the GEO altitude, h GEO . (3.3) h GEO = 35,786 km All GEO satellites are controlled to stay at this altitude and within the equatorial plane so that fixed-point observations are maintained.

The earth completes a 360◦ rotation for a period 4 min short of 24 h. The planet must slightly over-rotate to finish the 24-h (culmination-to-culmination) cycle, since the earth orbits the sun as it spins.

1

3.1 Orbital Mechanics Fig. 3.2 Schematic diagram for θobs . See text for details

31 Earth

R+ obs

O

R+

P

S

h O: Earth’ s center of gravity P: Subsatellite point S: Satellite

Figure 3.1 shows a snapshot of GEO observations at 0.64 µm (left) and 10.4 µm (right) from the Himawari-8 satellite. A seamless, full-disk view is centered around the subsatellite point of 140 ◦ E and 0 ◦ N/S. Note that this particular snapshot was chosen to be about noon time at the subsatellite point so the earth is fully illuminated by sunlight from nearly exactly behind the satellite. The half width of the maximum longitudinal/latitudinal range that is observable at a time from a satellite, θobs , is estimated as (3.4) θobs = cos−1 [R⊕ /(h + R⊕ )] from a simple geometric consideration (Fig. 3.2). For a GEO satellite (h = h GEO ), θobs is found to be approximately 80◦ , which gives the actual radius of a “full disk”. Care must be taken in using the data close to the disk edges because spatial resolution degrades as the viewing angle increases from nadir toward the edges. Other potential complications for observations near the edges include parallax, the effect that clouds at a certain height are horizontally displaced when observed at an angle. These issues challenge the quality of GEO observations in high latitudes and at longitudes distant from the GEO subsatellite point.

3.1.2 Low Earth Orbit (LEO) LEOs, a few 100 to 1000 km in altitude, are by definition so close to the earth that LEO satellites do not provide full-disk observations. The ISS crews, for instance, can overlook only a fraction of the earth’s surface at a time, given that (3.4) yields θobs ≈ ±20◦ for a = 400 km. To collect a whole catalog of the images of Earth taken from the Cupola module, ISS must repeat cycles around the planet until it eventually completes sweeping the entire globe. This exemplifies how LEO satellites make observations. The orbital period T given by Kepler’s third law (3.2),

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3 Satellite Orbit and Scan

Fig. 3.3 True-color image of Terra MODIS observations from descending (morning) orbits on April 19, 2000. Image credit: NASA’s Earth Observatory (https://earthobservatory.nasa.gov/)

 T = 2π

(h + R⊕ )3 G M⊕

1/2 ,

(3.5)

is 99 min for h = 705 km (Aqua and Terra) and 102 min for h = 850 km (DMSP F17/F18), resulting in 14–15 orbits per day. Figure 3.3 is a true-color image from MODIS observations constituted of the descending (10:30 am) branch of Terra orbits accumulated for one day. It takes 15 orbits for the Terra satellite to nearly complete a full coverage of the earth’s surface. The coverage is not uniform across latitudes: narrow gaps are left unfilled in the tropics, while neighboring swaths overlap each other in the polar regions. When combined with the ascending (10:30 pm) branch of the same orbits, a given location on the earth has typically two MODIS overpasses per day although the chance is slightly lower (higher) in low (high) latitudes. These statistics vary with different factors including satellite orbital elements and instrument’s swath width. It is a crucial step in satellite mission planning to decide on an optimal set of orbital elements, especially the altitude and inclination, Since satellite motion is governed almost exclusively by the terrestrial gravity field, the selection of orbital configuration as desired by mission objectives, e.g., sun-synchronous versus asynchronous, is restricted by the laws of celestial mechanics. For instance, a circular orbit with a given altitude is sun-synchronous only when the inclination is fine-tuned to be a

3.1 Orbital Mechanics

33

certain value or, to put it more positively, any satellite would be guaranteed to be sun-synchronous once sent into an orbit having a certain combination of the altitude and inclination. We next outline the basic principles of orbital mechanics behind this. Readers interested in further theoretical details are encouraged to consult Prussing and Conway (1993).

3.1.2.1

Earth’s Oblateness and J2 Perturbation

LEO planes must be inclined (i.e., not in the equatorial plane) unlike GEO, since otherwise regions outside latitudes of ±θobs would never be visited. Satellites in an inclined orbit feel the gravity field slightly undulate as they revolve even if the orbit is a perfect circle, as the result of the oblateness of the earth. The earth may be approximated as a sphere in many cases but in fact is an ellipsoid bulging around the equator by about 0.3% (the polar radius of 6,357 km compared to the equatorial radius of 6,378 km). This effect makes the orbital plane precess, that is, gradually deviate in orientation. A sun-synchronous orbit is obtained by properly tuning the rate of this nodal precession of the orbit. To take into account the oblateness of the earth, the gravity field potential is expanded into a Legendre polynomial series,   ∞ R⊕ n G M⊕  Jn Pn (sin δ) , U (r, φ) = − r n=0 r

(3.6)

where δ is the declination angle (practically equivalent to latitude), Jn is the Legendre expansion coefficient, and Pn is the Legendre polynomials. P0 (x) = 1 P1 (x) = x 1 P2 (x) = (3x 2 − 1) 2 .. . 1 dn 2 (x − 1)n Pn (x) = n 2 n! dx n The azimuthal inhomogeneity is negligible for problems of current interest. The zeroth term represents the spherically symmetric component yielding (3.1), which is by far dominant in magnitude (Table 3.1). The first term would arise when the center of the earth’s gravity is displaced poleward from the origin of the coordinates. In practice J1 always vanishes if the coordinates are set up properly. The leadingorder coefficient of the perturbation series is hence J2 , which has a value of the order of 10−3 due to the aforementioned oblateness of the earth, followed by higher-order terms of O(10−6 ) (Table 3.1). The orbital precession is thus practically dominated

34

3 Satellite Orbit and Scan

Table 3.1 Values of Jn (n = 0–4) for the terrestrial gravity field n Jn 0 1 2 3 4

1 0 1082.6 × 10−6 −2.5 × 10−6 −1.6 × 10−6

N z Descending node

Equatorial plane

R Satellite

T

Orbital plane

r

L

L : Angular momentum i : Inclination

i

x

: Right ascention of ascending node (RAAN)

y Ascending node

Fig. 3.4 Schematic illustration of orbital elements and coordinates. The line of nodes (or the intersection between the orbital and equatorial planes) is indicated by dashed line. The x and y axes lie within the equatorial plane, with the x axis pointing at the vernal equinox (γ )

by the J2 perturbation to the earth’s gravitational potential, G M⊕ J2 U J 2 (r, z) = − r



R⊕ r

2

 2  z 1 3 2 −1 , 2 r

(3.7)

where sin δ has been replaced by z/r with z being the polar axis. The gravity force due to this perturbation potential F J 2 is F J 2 = ∇U J 2

2 G M⊕ R ⊕ = 3J2 r4



5 z2 1 − 2 r2 2



 z er − ez , r

(3.8)

where er and ez are unit direction vectors along the radial and polar axes, respectively.

3.1 Orbital Mechanics

35

To derive the equation to express the rate of nodal precession due to the J2 perturbation to the gravitation force (3.8), it is convenient to project F J 2 onto a moving frame attached to the satellite, comprised of the radial (R), transverse (T), and normal (N) coordinates as illustrated in Fig. 3.4. The rest frame (the x-y-z coordinates in Fig. 3.4) is transformed to this RTN frame by three consecutive rotations as ⎛ ⎞ ⎛ ex cos Ω − sin Ω ⎝e y ⎠ = ⎝ sin Ω cos Ω 0 0 ez

⎞⎛ ⎞⎛ 0 1 0 0 cos θ − sin θ 0⎠ ⎝0 cos i − sin i ⎠ ⎝ sin θ cos θ 1 0 sin i cos i 0 0

⎞⎛ ⎞ 0 eR 0⎠ ⎝ e T ⎠ . 1 eN (3.9)

Substituting the z component of this transformation, ez = sin i sin θ e R + sin i cos θ eT + cos i e N , and z = r sin θ sin i into (3.8), one finds F J 2 = −3J2

2 G M⊕ R ⊕



r4

3 1 − sin2 i sin2 θ 2 2

 e R + sin2 i sin θ cos θ eT  + sin i cos i sin θ e N .

(3.10)

Note that e R = er by definition.

3.1.2.2

Nodal Precession of an Orbital Plane 

We next derive the rate of the nodal precession, Ω ≡ dΩ/dt , where Ω is the right ascension of ascending node (RAAN) (see Fig. 3.4) and · · · designates the temporal average over an orbital cycle. The averaging is intended to extract the secular component of the orbital plane rotation. Bearing in mind that the angular momentum vector per unit mass L is by definition normal to the orbital plane, the projection of L into the rest frame is found to be L x = L sin i sin Ω ,

(3.11a)

L y = −L sin i cos Ω , L z = L cos i ,

(3.11b) (3.11c)

where L ≡ |L|, and hence tan Ω = − The time derivative of (3.12) leads to

Lx . Ly

(3.12)

36

3 Satellite Orbit and Scan 







Lx Ly − Ly Lx Ω= cos2 Ω L 2y 

Lx Ly − Ly Lx = . L 2x + L 2y

(3.13)

Inserting (3.11) and 











L x = ex · ( L N e N + L T eT ) L y = e y · ( L N e N + L T eT ) into (3.13) using (3.9), one obtains 

L T sin θ . Ω=− L sin i 

The equation of motion,



L = r × FJ2 ,

(3.15)

(3.16)



provides an explicit expression of L, 

L = −r FJ 2,N eT + r FJ 2,T e N , so that



L T = −r FJ 2,N .

(3.17)

(3.18)

Inserting (3.18) and (3.10) into (3.15), one finds 

Ω = −3J2

2 G M⊕ R ⊕ cos i sin2 θ . 3 Lr

(3.19)

If r stays constant during a cycle,2 the rate of orbital precession is readily calculated for a circular orbit as  2π 2 G M⊕ R ⊕ 1 Ω = −3J2 cos i · sin2 θ dθ Lr 3 2π 0 2 3 (G M⊕ )1/2 R⊕ = − J2 cos i , 2 r 7/2 

2

(3.20)

The assumption of constant r is of practical use because most meteorological satellites have a circular orbit.

3.1 Orbital Mechanics

37

To the sun Orbital plane Orbital plane

RAAN

Satellite’ s motion

Earth’s motion

Right ascention of the sun

Fig. 3.5 Schematic diagram of the sun-synchronous orbit configuration. The angular difference between RAAN (Ω) and the right ascension of the sun (Ω ), denoted by Ω, is defined as positive (negative) when Ω leads (lags) Ω

where L = r 2 ω = (G M⊕r )1/2 [see (3.1)] has been used. For elliptical orbits, (3.20) is generalized to 2  3 (G M⊕ )1/2 R⊕ Ω = − J2 cos i , (3.21) 2 (1 − e2 )2 r 7/2 where e is orbital eccentricity. The negative sign in (3.21) implies that the line of nodes regresses, or the orbital plane precesses westward, for 0 ≤ i < π/2, while the precession proceeds eastward for π/2 < i ≤ π . No precession occurs when the orbit is exactly polar (i = π/2), in which case the satellite does not feel any torque inducing a precession in an axially symmetric gravity field.

3.1.2.3

Sun-Synchronous Orbit

A satellite orbit is sun-synchronous when the period of nodal precession accords with that of the earth’s revolution around the sun or 365.26 days. Figure 3.5 offers a schematic explanation for this. The local times of twice-daily satellite overpasses, LT1 and LT2 , are Ω + x∗,Ω + 12 h , 2π Ω + x∗,Ω LT2 = 24 h. 2π LT1 = 24

(3.22a) (3.22b)

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3 Satellite Orbit and Scan

Here Ω denotes RAAN minus the right ascension of the sun (Ω ) as defined in Fig. 3.5, and x∗,Ω is the satellite’s longitude measured from the ascending node. It is derived from spherical trigonometry that x∗,Ω = sin

−1



tan y∗ tan i

 ,

(3.23)

where y∗ is the satellite’s latitude. In (3.22), Ω accounts for a combined effect of the slowly progressing orbital precession and the earth’s revolution, while x∗,Ω arises from a rapid motion of the satellite orbiting the earth. Let us first focus on the former component by setting x∗,Ω = 0, that is, the local time of equatorial crossing for ascending tracks. It is then evident from (3.22) that the condition for a sunsynchronous orbit is achieved by making Ω constant over time,

  dΩ = Ω − Ω = 0 . dt

It follows that the orbital precession for a sun-synchronous orbit should be equivalent to the rate of the earth’s progression around the sun. For a circular orbit, this condition is written down from (3.20) as − where

2  3 (G M⊕ )1/2 R⊕ cos i = Ω , J2 7/2 2 r 

Ω =

2π . 365.26 d

(3.24)

(3.25)

This equation yields the relationship between altitude and inclination for a sunsynchronous orbit.  7/2 h + R⊕ cos i = − (3.26) 12, 353 km Since the rhs of (3.26) is always negative, the inclination should exceed 90◦ for a sun-synchronous orbit. The theoretical curve of (3.26) is depicted in the altitude-inclination plane along with selected satellite missions in Fig. 3.6. The theory, despite its simplicity, turns out to be an excellent predictor of the orbital elements of actual sun-synchronous satellites. Typical altitudes for sun-synchronous meteorological satellites are 700– 850 km, corresponding to an inclination of 98–99◦ . The fact that sun-synchronous orbits have an inclination close to 90◦ explains why sun-synchronous satellites are “polar orbiters”.3 We have not considered, up to this point, a fluctuation in local time due to x∗,Ω [see (3.22) above]. In reality, the local time of observations deviates, unless i = π/2, back 3

The inverse is not true: not all high-inclination orbits are sun-synchronous.

3.1 Orbital Mechanics

39

Fig. 3.6 Theoretical relationship between altitude and inclination for sun-synchronous orbits and the corresponding parameters from selected satellite missions. The CloudSat and CALIPSO parameters are those after departure from A-Train

and forth to some degree as the satellite orbits the earth. For an inclination of 99◦ , for example, x∗,Ω is estimated by (3.23) to be −16◦ at a latitude of 60◦ , indicating that the x∗,Ω effect produces a fluctuation of about ±1 h in local time. Lower latitudes have a smaller value of x∗,Ω , so the constancy of local time is guaranteed within ±1 h for the entire tropics and mid-latitudes. For scanning instruments, this estimate would be somewhat raised by an additional local time deviation across the swath. These effects would be greater for non-polar orbits as the denominator decreases with i in (3.23), but this would not be an issue in any case because such an orbit is no longer sun-synchronous (Sect. 3.1.2.5).

3.1.2.4

Revisit Cycle

A sun-synchronous orbit can be realized for any given altitude by tuning the inclination according to (3.26). A “wrong” choice of the altitude, however, could end up in an undesirable result that orbital tracks do not cover the whole planet quickly enough or uniformly enough. This is examined in terms of the revisit period at which the ascending node returns to the same longitude on the equator after the satellite orbits the earth for a certain number of times, 



N T (φ ⊕ − Ω ) = 2Mπ ,

(3.27)

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3 Satellite Orbit and Scan

Fig. 3.7 The revisit period of sun-synchronous orbits for different altitudes. Colors show the number of orbits per cycle. The Aqua/Terra orbit (705-km altitude and 16-day cycle with 233 orbits per cycle) is indicated by a square



where T is the orbital period (3.5), φ ⊕ is the rate of the earth’s spin, 

φ⊕ =

2π rad/s , 86164

(3.28)

and N and M are arbitrary integers. The revisit cycle Trev is determined by finding 

a pair of N and M that satisfy (3.27) for given T and Ω . For a sun-synchronous 



orbit (Ω = Ω ), (3.27) leads to Trev = N T =

2Mπ 



φ ⊕ − Ω

= M Pd ,

(3.29)

where Pd = 86400 s is the culmination-to-culmination period of the earth’s spin (24 h). The revisit period for a sun-synchronous orbit is by construction a multiple of one day. The revisit period for sun-synchronous orbits, computed from (3.5) and (3.29), is plotted as a function of altitude in Fig. 3.7. If Trev is small, satellite overpasses return frequently to certain geographical locations at the expense that other locations may never have chance to be observed. Such an extreme sampling disparity is mitigated

3.1 Orbital Mechanics

41

Fig. 3.8 Simulated satellite tracks for a specified orbital configuration. Fifty consecutive orbital cycles are shown in changing colors indicating different days. a Sun-synchronous with h = 705 km (Terra/Aqua), b Sun-synchronous with h = 658 km (precisely 15 orbits per day), c TRMM after boost (h = 402 km with i = 35◦ ), d GPM core observatory (h = 407 km with i = 65◦ )

as Trev increases. The shortest possible period of one day is found only for a limited number of “zero-drift” orbits whose orbital cycle is phase-locked with the earth’s rotation. Complete 15 orbits bring the satellite back to exactly the same longitude and latitude at one-day interval when the altitude is 568 km. The same is true for 895 km with 14 orbits per day. The Aqua/Terra satellite orbit (h = 705 km) has the revisit period of 16 days, requiring 233 orbits to complete one revisit cycle (marked by square in Fig. 3.7). The nadir tracks of 50 consecutive Terra/Aqua orbits, accounting for a period somewhat longer than three days, are simulated in Fig. 3.8a (see Sect. 3.1.2.6 for technical details behind the simulations). Orbits drift in longitude as days proceed, gradually filling in the gaps between the tracks. This is contrasted by a second example of the one-day recurrent sun-synchronous orbit with 15 cycles per day (h = 568 km), for which orbits follow exactly the same tracks day after day (Fig. 3.8b). In this case, locations that happen to be beneath the tracks are observed every day while the rest of the earth’s surface is never visited. These simulations visually support the expectations of Fig. 3.7. The Terra/Aqua orbital configuration is intended to fit in the Worldwide Reference System-2 (WRS-2), which is a global coordinate system designed for Landsats 4–9. Ground tracks of any satellite flying in this orbit, with LT adequately selected, are

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guaranteed to fall on the WRS-2 grid constituted of 233 paths. The 185-km-wide Landsat swath is just broad enough (with slight overlaps) to cover the whole equator once 233 orbits are completed. In fact, Trev as predicted by (3.29) is a rather conservative estimate for satellite instruments having a wider swath. As visually demonstrated in Fig. 3.3, one day of MODIS observations sweep almost the entire globe with a 2,330-km-wide swath, despite that it still takes 16 days for subsatellite tracks to complete one revisit cycle. In contrast, nadir-looking sensors such as CloudSat and CALIPSO have difficulty achieving a complete coverage of the earth even with months of accumulated data, since for those sensors the 16-day cycle leaves out gaps that are hardly filled in.

3.1.2.5

Sun-Asynchronous Orbit

Any orbit having an altitude and an inclination that do not satisfy (3.24) is sunasynchronous. For LEO altitudes, this is achieved by adjusting the inclination to a value substantially lower than a sun-synchronous inclination of 98◦ –99◦ . A lowered inclination results in an enhanced frequency of observations at low- and mid-latitudes, although any latitude beyond the inclination is entirely left out of the satellite coverage as demonstrated in Fig. 3.8c, d. The local time of observations drifts from one orbit to another for sun-asynchronous satellites, making it possible to sample diurnal cycle. This is a great advantage for precipitation measurements, given that precipitation, particularly over land and coast, is known to have a distinct diurnal variation. Past and current satellite programs targeted on precipitation such as TRMM, GPM, and Megha-Tropiques enjoy this benefit. It takes a certain duration of time, however, for satellite tracks to accomplish a full diurnal cycle at a given location on the earth. The repeat period required for sampling one diurnal cycle, Tdc , is equal to the period of the nodal precession relative to the solar direction, 2π Tdc =  . (3.30)  |Ω − Ω | Equations (3.20), (3.25), and (3.30) are combined into an expression of Tdc in terms of the altitude and inclination as plotted in Fig. 3.9. The TRMM satellite and GPM core observatory have a relatively low altitude of about 400 km (or 350 km for TRMM before the orbital boost). With the inclination of 35◦ , the diurnal sampling from TRMM repeats every 46 days (or 48 days after the boost). If the ascending and descending paths need not to be discriminated from each other, the actual period in diurnal sampling is just half as long (23 or 24 days). The GPM core observatory has a higher inclination of 65◦ so that mid latitudes are fully within reach, but as a result Tdc is as long as 83 days. The repeat cycle is increasingly prolonged as the inclination increases further, suggested by more and more densely spaced contours toward the top in Fig. 3.9. By definition, Tdc is diverged to infinity for sun-synchronous orbits.

3.1 Orbital Mechanics

43

Fig. 3.9 The repeat period in diurnal sampling is contoured every 5 days (in solid curve every 10 days) as a function of the altitude and inclination. TRMM (before and after the orbital boost), the GPM core observatory, and Megha-Tropiques are indicated by symbols

Observations from Megha-Tropiques, a satellite mission focused exclusively on the tropics, are restricted to an equatorial band between 20 ◦ S–20 ◦ N. The inclination of 20◦ , when combined with the Megha-Tropiques altitude of 867 kms, results in a repeat period of 51 days. This value of Tdc exceeds that for TRMM despite the very low inclination of Megha-Tropiques because of the vast difference in altitude between the TRMM and Megha-Tropiques spacecrafts. Figure 3.10 is a chart similar to Fig. 3.7 but is tailored in a format that suits sun-asynchronous orbits. Color shaded is the time required for orbits to sweep the complete coverage on the earth for different altitudes and inclinations. This plot is constructed by repeating orbital simulations as shown by Fig. 3.8 until all 2◦ × 2◦ grid boxes are visited by at least one overpass across the whole observable domain (i.e., all longitudes within a latitudinal band between ±i, see Fig. 3.8c–d). Here the two-square-degree resolution is meant to be comparable to the TRMM PR and GPM DPR swath widths at low latitudes. Figure 3.10 shows that the time for global coverage varies in a highly complex manner from several days to more than months. A complete coverage requires many days at near-vertical (but somewhat slant) “ridges” in the altitude-inclination plane. A global coverage will be never accomplished on two striking ridges around 500 km and 800 km, which are zero-drift orbits (see Sect. 3.1.2.4 above). Satellite altitude and

44

3 Satellite Orbit and Scan

Fig. 3.10 Time required for global coverage as a function of the altitude and inclination. Here “global coverage” refers to the entire observable domain (all longitudes within a latitudinal band bound by ±i) on a 2◦ × 2◦ grid. As in Fig. 3.9, past and current sun-asynchronous satellites are indicated by symbols

inclination are designed to avoid the ridges because otherwise the regional disparity in sampling frequency would not be alleviated even for months of observations. It becomes increasingly challenging as the inclination increases to find an optimal set of the orbital parameters amid a forest of the ridges. GPM (i = 65◦ ) finds itself very close to one of those ridges.

3.1.2.6

Simulation of Orbital Tracks

Before closing Sect. 3.1, a set of equations for use by orbital-track simulations as drawn in Fig. 3.8 are summarized. The orbits are assumed to be circular and no swath is considered (i.e., nadir tracks only). The angular position of the satellite in the orbital plane (θ , see Fig. 3.4) is given as a function of time t by t − t0 θ (t) = 2π , (3.31) T where T is the orbital period (3.5) and t0 is the equatorial crossing time at RAAN. In the present formulation, t is defined between 0 and T and should be reset for each

3.1 Orbital Mechanics

45

orbital cycle. The satellite’s longitude relative to the ascending node is computed as x∗,Ω (t) = tan−1 [cos i tan θ (t)] ,

(3.32)

where i is the inclination (Fig. 3.4). Note that x∗,Ω belongs to the same quadrant as θ : for example, x∗,Ω should be chosen to be between π/2 and π when π/2 < θ < π . The longitude of the ascending node is obtained by taking into consideration an offset arising from the earth’s spin and orbital precession as, 







Ω0 (t) = (m − 1)[±2π − (φ ⊕ − Ω)T ] − (φ ⊕ − Ω)t , 

(3.33) 

for the m-th orbital cycle. Here φ ⊕ is the rate of the earth’s rotation (3.28), Ω is the rate of the nodal precession (3.20), and the sign of 2π is to be positive for cos i ≥ 0 and negative otherwise to ensure the continuity between consecutive orbital cycles. The longitude and latitude of the satellite ground track, denoted by x∗ and y∗ respectively, are x∗ (t) = Ω0 (t) + x∗,Ω (t) mod 2π ,

(3.34a)

y∗ (t) = sin−1 [sin i sin θ (t)] .

(3.34b)

The modulo operation (a mod b) is defined here as the remainder against the largest multiple of b not exceeding a, regardless of whether a is positive or negative.

3.2 Scanning Geometry The optical system most familiar to us is probably our own eyes. The field of view (FOV) of human eyes is rather broad in that you notice something move in the corner of your eye, but you would have to shift your eyeballs toward that direction in order to actually see what it was. Because human eyes perform their best only at the central vision, we constantly move the eyes around, consciously or unconsciously, to fully scan the world surrounding us. This resembles how satellite instruments observe the earth. Each satellite sensor has an FOV of its own and a manner to scan the FOVs one after another by moving the “eyes” around. In this section, the optical basis of FOV and beam pattern is first outlined, followed by a summary of scanning geometries for different spaceborne instruments.

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3 Satellite Orbit and Scan

Fig. 3.11 Schematic diagram of diffraction for waves with a wavelength of λ propagating from left to right through an aperture (left) toward a screen (right). The distance of the screen from the aperture is assumed to be substantially larger than the aperture diameter, D. The diffraction limit θd is found to be comparable to λ/D

3.2.1 Field of View (FOV) In the terminology common to satellite remote sensing, FOV, or more precisely the instantaneous FOV (IFOV), refers to the size of a satellite footprint and hence is a measure of the horizontal resolution.4 Let us first discuss the diffraction limit, which physically restricts the spatial resolution of an instrument. Figure 3.11 illustrates a simple optical system in which waves incident on a small aperture propagate toward a screen. The waves make a diffraction pattern on the screen having an angular width θd that satisfies a destructive interference equation, D λ sin θd = , 2 2

(3.35)

where D is the aperture diameter and λ is wavelength. If the distance between the aperture and screen is much greater than D, (3.35) is approximated by θd ∼

λ . D

(3.36)

This simple relationship captures an essence of the diffraction limit: the angular resolution of an optical instrument is proportional to the ratio of wavelength to the aperture size as will become clearer later. Precise calculations of the far-field diffraction pattern of a point-source target observed through a circular aperture yield an exact solution called the Airy pattern 4

This should not be confused with another (perhaps more common) definition of FOV, in which FOV refers to the maximum size of an image observable at a time with a camera.

3.2 Scanning Geometry

47

Fig. 3.12 The Airy Pattern (solid) and a fitted Gaussian having the same FWHM (dashed) as a function of θ D/λ in a linear scale and b logarithmic scale

(5.99). Figure 3.12 shows the Airy pattern as theoretically expected (see Sect. 5.2.4 for mathematical details). A majority of energy resides within a bell-shaped pattern in the middle (known as the Airy disk) bound between ±θd,A ≈ ±1.22λ/D [see (5.100)]. Radiation even from a point source is spread to this extent as the result of diffraction. The Airy disk is surrounded by a series of rings as more evident when plotted in logarithmic scale (Fig. 3.12b), so the diffracted wave energy does not entirely vanish outside θd,A . The full width at half maximum (FWHM) (or −3-dB width) of the Airy pattern is found to be θFWHM ≈ 1.03λ/D .

(3.37)

A crude estimate of θd given by (3.36) turns out be a rather reasonable proxy for more mathematically robust estimates of θd,A and θFWHM . The Airy pattern offers an accurate model of beam pattern or antenna pattern for a diffraction-limited optical system. The beam pattern consists of a mainlobe (the Airy disk) and a sequence of sidelobes (the Airy rings) (Fig. 3.12). Sidelobes are so weak in magnitude relative to the mainlobe that they would not be an issue unless strong signals happen to fall within the sidelobes. In reality, antenna sidelobe clutter often contaminates weather radar measurements and various techniques exist in radar instrument design to suppress sidelobe echoes. The beam pattern is often simplified by a fitted Gaussian (dashed curve in Fig. 3.12), which serves as an excellent approximation as far as sidelobes are negligible. The width of the mainlobe or θFWHM , which nominally defines IFOV, determines the sharpness of a beam. For practical ease, IFOV usually refers to its ground projection, e.g., hθFWHM at nadir where h is the satellite altitude. For example, microwave at a frequency of 10 GHz (λ = 3 cm) observed by a reflector with the effective diameter of 1 m yields θFWHM ≈ 0.03 rad (or about 1.7◦ ), which corresponds to a 21-km-wide

48

3 Satellite Orbit and Scan

nadir footprint for a satellite flying at an altitude of 700 km.5 Any structure with a horizontal scale smaller than the footprint size cannot be clearly resolved, so the IFOV of 21 km serves as a measure of spatial resolution. Given that the minimum observable scale is limited by the breadth of diffraction pattern, spaceborne microwave sensors are a diffraction-limited system. It is evident from (3.37) that the beam is sharper when observed at a shorter wavelength with an identical optical system. This explains why IFOV is in general inversely proportional to frequency for microwave radiometers (Table 2.3). This rule does not apply to high-frequency channels of GMI and microwave sounders (Table 2.4), for which IFOV stays constant across different frequencies. This adjustment is done by a partial illumination of the reflector antenna, which means that only a portion of the reflector is seen to effectively lower the denominator of (3.37), for the purpose of retaining a uniform spatial resolution across frequencies and minimizing the gap between adjacent scans. Visible and infrared instruments have the great advantage over microwave sensors that wavelengths are so short that a meter-size aperture is not necessary to achieve a desired spatial resolution. An IFOV of 1 km from an altitude of 700 km for a thermalinfrared wavelength of 10 µm only requires D = 0.7 cm. Since an aperture larger than this size can easily fit in a compact optical system, the spatial resolution of visible and infrared imagers is not diffraction-limited but detector-limited, that is, controlled by the pixel size of the detector. This is similar to regular digital cameras such as those installed in smartphones unless the f-number (the focal length divided by D) is chosen to be very large. The actual beam pattern can be somewhat distorted from what is expected from IFOV because each satellite footprint is recorded with a finite integration time over which the footprint moves during the scan. This distorted beam width is called the effective FOV or EFOV. EFOV is by design slightly enlarged from IFOV along the scanning direction, while nearly equivalent to IFOV along the satellite track because the movement of spacecraft during the integration time is generally negligible.

3.2.2 GEO Imager Scan GEO satellite imagery as shown in Fig. 3.1 might give the impression that an full-disk image is taken with a single tap as we do on our smartphones. In reality, Fig. 3.1 is a composite of many consecutive scans that take some time to complete a cycle of fulldisk imaging. Old generations of the GEO visible/infrared imagers used to require half an hour for full-disk imagery, but the sampling frequency has been improved to 5–15 min for GOES-R ABI, Himawari 8/9 AHI, and MSG SEVIRI. In addition to the full-disk scans, GEO imagers have a rapid-scan mode in which selected domains are observed more frequently than the full-disk scanning time. 5

The effective diameter is smaller than the physical size of an antenna for various reasons including non-uniform aperture illumination and imperfect reflectivity.

3.2 Scanning Geometry

49

Fig. 3.13 Schematic illustration of the cross-track scan

Rapid scans benefit different applications such as capturing quickly developing cloud systems and deriving the atmospheric motion vector (AMV) with enhanced accuracy. AHI is designed to visit two fixed regions and a third target area to be flexibly chosen to meet operational needs (tracking tropical cyclones etc.) every 2.5 min as well as two landmark areas every 30 s (Bessho et al. 2016). All these rapid scans and a cycle of full-disk scans together are scheduled to accomplish for 10 min. ABI has a 15min scheduling strategy similar to AHI’s with an additional scan mode dedicated exclusively to full-disk scans so the whole disk is covered within 5 min (Schmit et al. 2017). SEVIRI performs rapid scans that sweep a northern third of the MSG-covered domain, latitudes higher than about 15 ◦ N, every 5 min.

3.2.3 LEO Sensor Scan The LEO instruments described in Sect. 2.2 are grouped into three categories in light of scanning geometry: cross-track and conically scanning sensors and non-scanning (nadir-only) instruments. This section outlines the two scanning geometries typical of LEO satellite observations.

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3 Satellite Orbit and Scan

Fig. 3.14 Schematic illustration of the conical scan

3.2.3.1

Cross-Track Scan

Many of scanning LEO sensors, with the notable exception of microwave radiometers, adopt a simple scanning strategy that sensor footprints are scanned in the direction perpendicular to the satellite track (Fig. 3.13). This manner of scanning is called the cross-track scan. Visible/infrared imagers such as MODIS and AVHRR, microwave and infrared sounders (e.g., AMSU and AIRS), and radars such as PR and DPR (except CloudSat, whose line of sight is fixed at nadir) are all cross-track scanning instruments. The swath width of polar-orbiting cross-track scanners is typically about 2300 km or larger so that the longitudinal coverage achieves nearly, if not perfectly, 100% during half a day (that is, either ascending or descending section of 14–15 overpasses; see Fig. 3.3). PR and DPR have a narrower swath of 200–250 km, which takes the instrument almost one week or longer to achieve a complete coverage of the observable domain (note that the time required for complete coverage depends also on the altitude and inclination; see Fig. 3.10). IFOV varies with the scan angle θsc , being smallest at nadir and largest at the swath edges. A footprint is elongated in the cross-track direction nearly proportionally to 1/ cos(θsc ) when θsc is small, but more than proportionally as θsc increases further because the earth’s surface curves. The earth’s sphericity also slightly enlarges the along-track IFOV near the swath edges.

3.2 Scanning Geometry

3.2.3.2

51

Conical Scan

The conical scan, unlike the cross-track scan, makes observations at a constant viewing zenith angle (Fig. 3.14). Microwave radiometers and some other spaceborne sensors (e.g., SSMIS and QuikSCAT) are conically scanning instruments. A key property of the conical scan is that observed surface emissions are relatively stable and predictable over water surfaces, because the reflectivity and emissivity of a smooth surface with a given refractive index are a function only of the incident angle according to the Fresnel equations (see Sect. 5.2.3). This is a great advantage for microwave radiometers, since the ability to accurately separate atmospheric emission/absorption from surface radiation is crucial for the microwave radiometry of the atmosphere and ocean. The viewing zenith angle of most microwave radiometers is 50◦ –55◦ at the ground. IFOV stays unchanged regardless of the scan angle but is heavily elongated in the along-track direction (Table 2.3) as the result of a large viewing angle. A drawback of the conical scan is that the swath width is geometrically restricted by its curved scan lines. In fact, conical scanners generally have a narrower swath than cross-track scanners flying at the same altitude. The swath width of polar-orbiting conical scanners (e.g., SSMI, SSMIS, and AMSR-E) is 1400–1700 km. TMI and GMI have an even narrower swath of 758.5 km (before boost) and 885 km, respectively, since the TRMM and GPM altitudes are lower than the altitudes of polar orbiters.

3.2.3.3

Other Scanning Geometries

Unlike cross-track or conically scanning sensors looking down on the earth, there are other ways to measure the atmosphere from space with a line of sight that never meets the earth’s surface. Among such scanning strategies is the radio occultation (RO) measurements of temperature and water vapor profiles, realized by a constellation of RO satellites carrying receivers of radio signals transmitted by GNSS satellites. Temperature and humidity variations give rise to a perturbation to the radio refractivity of the atmosphere and are therefore the source of uncertainties in the accuracy of GNSS positioning. Reversing this logic, RO satellites, with high-precision atomic clocks onboard, retrieve the temperature and humidity structures from occulted GNSS signals propagating through the atmosphere. RO observations are excellent in vertical resolution (0.5–1 km) but are inferior to cross-track sounders in horizontal resolution (about 200 km). Limb sounding is a methodology to measure the vertical structure of temperature, water vapor, and trace gases in and above the upper troposphere. The measurement principles of limb sounding resemble those of RO in that the atmosphere is observed by a tangential line of sight, except that limb sounders do not use GNSS transmissions but thermal emissions from the atmosphere itself. A broad spectral range from thermal infrared to microwave, including terahertz frequencies in between, is utilized for limb sounding to survey the middle to upper atmospheres.

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RO and limb sounding are not discussed further in this book because cloud and precipitation measurements are not among their major observational targets.

References Bessho K, Date K, Hayashi M, Ikeda A, Imai T, Inoue H, Kumagai Y, Miyakawa T, Murata H, Ohno T, Okuyama A, Oyama R, Sasaki Y, Shimazu Y, Shimoji K, Sumida Y, Suzuki M, Taniguchi H, Tsuchiyama H, Uesawa D, Yokota H, Yoshida R (2016) An introduction to Himawari-8/9 –Japan’s new-generation geostationary meteorological satellites. J Meteor Soc Jpn 94:151–183. https:// doi.org/10.2151/jmsj.2016-009 Prussing JE, Conway BA (1993) Orbital mechanics. Oxford University Press Schmit TJ, Griffith P, Gunshor MM, Daniels JM, Goodman SJ, Lebair WJ (2017) A closer look at the ABI on the GOES-R series. Bull Am Meteor Soc 98(4):681–698. https://doi.org/10.1175/ BAMS-D-15-00230.1

Part II

Basic Physics

Satellite retrieval algorithms are built on different disciplines of physics including statistical mechanics and electrodynamics. Terrestrial thermal radiation cannot be explained without invoking the Planck function (or the blackbody spectrum). The scattering of radiation by cloud and precipitation particles is predicted in terms of Maxwell’s theory of electromagnetic waves. Elementary quantum mechanics is central to the interactions between radiation and atmospheric gases. Three chapters in Part II are devoted to an introduction to the principles of such basic physics.

Chapter 4

Principles of Statistical Mechanics

The interactions between radiation and matter are essential for understanding the physical foundations of remote sensing. This chapter is devoted to a brief review of the statistical mechanical principles underlying the radiative processes.

4.1 Particle Distribution Functions in Phase Space 4.1.1 Phase Space and Density of States From the classical mechanical point of view, the motion of any point object is uniquely determined when its position and momentum are specified at the same time. The 6dimensional phase space constituted of 3 dimensions for position (x, y, and z) and another three for momentum ( px , p y , and pz ) offers a useful system of coordinates in which the behavior of moving objects, such as atmospheric molecules and photons, is described. The particle energy ε is written as  1/2 p2 , ε = mc2 1 + 2 2 m c

(4.1)

where m is the particle mass, c = 3.0 × 108 m s−1 is the speed of light, and p ≡  px2 + p 2y + pz2 . In the Newtonian limit (v  c or p  mc), (4.1) is reduced to be the sum of the rest mass energy mc2 and the kinetic energy of a non-relativistic particle, p2 . ε ≈ mc2 + 2m

© Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_4

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4 Principles of Statistical Mechanics

For a massless (m = 0) particle such as a photon, (4.1) takes a different form, εγ = cp . The energy of a photon εγ is known to be proportional to the electromagnetic frequency ν, (4.2) εγ = hν , where h = 6.63 × 10−34 J s is the Planck constant. The momentum of a photon is thus hν . (4.3) p= c From the quantum mechanical perspectives, microscopic particles may be described in terms of de Broglie wave, just as radiation can be viewed as a stream of particles (photons) as well as electromagnetic waves. The de Broglie wavelength λ is inversely proportional to momentum, λ=

h , p

(4.4)

which is apparently the generalization of (4.3). For a de Broglie wave propagating along the normalized direction vector n the wavenumber vector k is ⎛ ⎞ kx 2π 2π n= p, k ≡ ⎝k y ⎠ = λ h kz

(4.5)

where (4.4) has been used for replacing λ with p. The classical mechanical representation of a “point particle” no longer applies to spatial scales smaller than the de Broglie wavelength, so we instead consider a minimum reference volume xyz that corresponds to (the cubic of) the de Broglie wavelength. Higher modes of de Broglie wave are expressed as kx =

2π 2π 2π l , ky = m , kz = n, x y z

(4.6)

where l, m, and n are positive integers. Equations (4.5) and (4.6) are combined into px = lpx , where px ≡

p y = mp y ,

pz = npz ,

(4.7)

h h h , p y ≡ , pz ≡ . x y z

(4.8)

4.1 Particle Distribution Functions in Phase Space

57

As such, the momentum is discretized by the interval of px , p y , and pz , just as the spatial dimensions are measured in multiples of x, y, and z. In other words, different quantum states are defined in terms of the discrete building blocks of phase space defined by 3 x3 p ≡ xyzpx p y pz . With (4.8), this reference volume turns out to be equal to the Planck constant cubed, 3 x3 p = h 3 , which is known as the Heisenberg uncertainty principle. The uncertainty principle asserts that a quantum state is represented by a multiple of the minuscule building block having the volume of h 3 in phase space. The argument above may be generalized as Number of quantum states in d3 xd3 p = g

d3 xd3 p d3 xd3 p = g , 3 x3 p h3

(4.9)

where d3 xd3 p is a phase-space volume element and an additional factor g is the number of non-translational energy states such as spin and molecular vibration and rotation. Photons, for example, have two states of spin angular momentum (±1), corresponding to two rotationally polarized electromagnetic waves spiraling in opposite directions, and hence g = 2. For later convenience, the density of states is defined as the number of quantum states in a unit phase-space volume element. It is often convenient that d3 p is expressed in the spherical coordinates with respect to p, d3 p = p 2 d p dΩ ,

(4.10)

where the element of solid angle dΩ is a combined measure of zenith and azimuth angles, θ and φ, respectively, as dΩ ≡ sin θ dθ dφ

(4.11)

(see Fig. 4.1 for schematic illustration).1 Solid angle, although dimensionless by nature, is given a unit called steradian (abbreviated as sr) for convenience. The density of states is defined with (4.9), (4.10), and (4.11) by ργ d3 x d3 p =

1

g 2 3 p d x d p dΩ . h3

(4.12)

It would be more appropriate, for notational consistency with other multidimensional displacement elements such as d3 x, to express a displacement of solid angle as d2 Ω rather than dΩ. In this book, however, the latter is adopted in accordance with convention in the literature.

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4 Principles of Statistical Mechanics

Fig. 4.1 Schematic of a solid-angle element dΩ in terms of zenith angle θ and azimuth angle φ defined on a unit sphere. The unit vector Ω defines the direction toward this solid-angle element

The density of states for photons ργ is obtained from (4.3) and (4.12) with g = 2 as ργ d3 x d3 p =

2ν 2 3 d x dν dΩ . c3

(4.13)

4.1.2 Canonical Ensemble Let us consider a system of many particles in equilibrium with the heat reservoir of temperature T . The canonical distribution gives the probability that such a system finds itself in a microscopic state having a certain energy. As will be proved below, the canonical distribution is proportional to   El , exp − kT

(4.14)

where El is the energy of the l-th quantum state, and k = 1.38 × 10−23 J K−1 is the Boltzmann constant. The following argument, although mathematically incomplete, is aimed at giving a glimpse of the theoretical essence of (4.14). For brevity, let us first consider an idealized case where the system of interest consists of N quantum-scale oscillators that each has an energy in multiples of

4.1 Particle Distribution Functions in Phase Space

59

ε0 ≡ hν. If the total energy of the whole system E t is Mε0 , the number of all possible combinations of partitioning this total energy into N oscillators, or equivalently the combinations of M ping-pong balls and N − 1 partitions lined up in a row, is W N (M) =

(M + N − 1)! . (N − 1)!M!

(4.15)

Now we focus on a certain oscillator having the energy of El = Lε0 . This oscillator is assumed to be weakly coupled with the rest of the system, which may be considered as the heat reservoir if its energy Er ≡ E t − El is substantially larger than El . Since the number of possible combinations of partitioning Er to all other N − 1 oscillators in the system is W N −1 (M − L), the chance of finding an oscillator have the energy El is W N −1 (M − L) W N (M) (N − 1) · M(M − 1) · · · (M − L + 1) . = (M + N − 1)(M + N − 2) · · · (M + N − L − 1)

p(El ) =

(4.16)

An asymptotic form of this equation for N  1 and M  L (i.e., E t  El ) is p(El ) ≈

N ML N = L+1 (M + N ) M+N



M M+N

 El /ε0

  El , ∝ exp − Θ

where Θ = ε0 ln[(M + N )/M]. This example, although vastly simplified, suggests that the probability distribution should be a decreasing exponential function of energy as seen in (4.14). Equation (4.16) may be extended to a more general context by replacing W N −1 (M − L) by W (E t − El ) and W N (M) by W (E t ), p(El ) =

W (E t − El ) , W (E t )

(4.17)

where W (E) denotes the number of the microscopic states whose energy is E. The translation of W (E) into a thermodynamic context is facilitated by the microscopic definition of entropy, S(E) ≡ k ln W (E) . (4.18) Combining (4.17) and (4.18), one finds 

 1 [S(E t − El ) − S(E t )] k

  ∂S 1 − , = exp El + O(El2 ) k ∂E

p(El ) = exp

(4.19)

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4 Principles of Statistical Mechanics

The high-order terms with respect to El are negligibly small compared to the leading term because we have assumed E t ≈ Er  El . Using the thermodynamic definition of entropy, 1 ∂S = , ∂E T the canonical distribution (4.14) is derived from (4.19). A statistical ensemble of microscopic states distributed according to (4.14) is called the canonical ensemble. The canonical ensemble founds the basis of many applications to statistical mechanical problems. Among those of great utility is the partition function,    El , (4.20) exp − Z= kT l where applies to all possible states. The partition function appears in the normalization factor of the canonical distribution (4.14),   El 1 exp − . Z kT

p(El ) =

(4.21)

The partition function is useful for calculating macroscopic thermodynamic parameters from an ensemble of microscopic states. The simplest examples include the internal energy,   ∂ 1  El = kT 2 ln Z . U≡ El exp − Z l kT ∂T

(4.22)

Another example is the Helmholtz free energy, F = −kT ln Z .

(4.23)

Equation (4.23) may be derived as follows. A generalized formulation of the entropy, S = −kln p(El )  p(El ) ln p(El ) , ≡ −k

(4.24)

l

which is readily found to be equivalent to (4.18) for a statistical ensemble of a given energy El , where ln p(El ) = ln(1/W ) according to the principle of equal a priori probabilities. Equation (4.24) is rewritten using (4.21) and (4.22) into

4.1 Particle Distribution Functions in Phase Space

S = −k

 l

61

  El − ln Z p(El ) − kT

U + k ln Z . = T Eliminating U and S using the thermodynamic description of the Helmholtz free energy, F = U − T S, one obtains (4.23) above. Some fundamental thermodynamic variables may be derived from the Helmholtz free energy by use of the relationship of dF = dU − d(T S) + μdN = −SdT − pdV + μdN ,

(4.25)

where the first law of thermodynamics, dU + pdV = T dS, has been used for eliminating dU . The last term on the rhs of (4.25) is introduced to take into account changes in the total number of particles N under a given chemical potential μ. Equation (4.25) leads to,   ∂F = −S , (4.26) ∂ T V,N   ∂F = −p , (4.27) ∂ V T,N   ∂F =μ. (4.28) ∂ N T,V

4.1.3 Fermi-Dirac and Bose-Einstein Statistics: Derivation In this section, the canonical distribution (4.14) is applied to a system of many particles having different energies of their own, or an ensemble of different oneparticle quantum states. The number and energy of particles in each one-particle quantum state, denoted by n i and εi (i = 1, 2, · · · ) respectively, satisfy El =



n i εi

(4.29)

ni .

(4.30)

i

and N=

 i

The aim of this subsection is to obtain the probability distribution function of n i .

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4 Principles of Statistical Mechanics

Basic principles of quantum physics impose different constraints on n i depending on whether the particles are fermions or bosons. Fermions are particles that have a half-integer spin such as electrons and protons. Two or more fermions are not allowed to occupy a single one-particle quantum state, known as the Pauli exclusion principle. Bosons, in contrast, are those with an integer spin (photons, for instance) and an unlimited number of bosons can share the same quantum state.

0 or 1 for fermions ni = 0, 1, 2, 3, . . . for bosons

(4.31)

The probability of finding a fermion and a boson at a given energy obey the FermiDirac and Bose-Einstein distributions, respectively, each named after their theoretical founders. The derivation of these distribution functions is provided below. The partition function (4.20) combined with (4.29) becomes Z (N ) =

 n1

n2

···

 ni



1  · · · exp − n i εi kT i

 ,

(4.32)

where the sequence of represents the summation over all possible combinations of (n 1 , n 2 , · · · ) under the constraint of (4.30). Now a certain ( j-th) state is extracted from (4.32) as  n ε  j j Z i= j (N − n j ) , (4.33) ζ j (N ) = exp − kT where Z i= j (N − n j ) ≡



···

n1

and the sequence of



 n j−1 n j+1

⎛

⎞  1 · · · exp ⎝− n i εi ⎠ , kT i= j

(4.34)

applies to all combinations under the constraint of N − nj =



ni .

i= j

Note that (4.34) is the partition function for the subsystem with the j-th state set aside. Assuming that n j  N , a small displacement in Z by subtracting the j-th state can be related to the chemical potential, μ, as μ ≈ −kT which is a discretized form of

ln Z (N ) − ln Z i= j (N − n j ) , nj

(4.35)

4.1 Particle Distribution Functions in Phase Space

μ = −kT

63

∂ ln Z (N ) , ∂N

(4.36)

derived from (4.23) and (4.28). Equation (4.35) yields Z i= j (N − n j ) = exp

n μ j Z (N ) . kT

(4.37)

Equations (4.33) and (4.37) together lead to   n j (ε j − μ) ζ j (N ) p j (n j ) ≡ = exp − , Z (N ) kT

(4.38)

where p j (n j ) denotes the probability of finding n j particles in the j-th state. The expected number of particles for the j-th state is n j ≡



n j p j (n j )/

nj



p j (n j )

nj

    n j (ε j − μ) n j (ε j − μ) = n j exp − exp − kT kT nj nj ⎡ ⎤    n j (ε j − μ) ⎦ ∂ ln ⎣ exp − = kT . ∂μ kT n 

(4.39)

j

Bearing (4.31) in mind, one can expand the summation in (4.39) as     εj − μ n j (ε j − μ) = 1 + exp − exp − kT kT =0

1  nj

for fermions, and ∞  nj



n j (ε j − μ) exp − kT =0

 =

1   εj − μ 1 − exp − kT

for bosons. The Fermi-Dirac distribution is hence derived from (4.39) to be n FD (ε) =

and the Bose-Einstein distribution is

1  , ε−μ +1 exp kT 

(4.40)

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4 Principles of Statistical Mechanics

Fig. 4.2 a Fermi-Dirac distribution for low temperatures as a function of energy normalized with the Fermi energy μ F . Different lines are for different temperatures from absolute zero (solid) to higher values. b As (a) but with temperature increasing further, where the temperature dependence of μ is taken into account

n BE (ε) =

1  . ε−μ −1 exp kT 

(4.41)

From now on, angle brackets and the subscript j are omitted.

4.1.4 Fermi-Dirac Statistics: Basic Properties Unique aspects of the Fermi-Dirac distribution (4.40) best appear when absolute temperature is very low. For a near-zero T , the exponential term in the denominator of (4.40) changes drastically in magnitude as ε − μ changes its sign, so that (4.40) has the asymptotic behaviors of

n (ε) → FD

1 for ε  μ , 0 for ε  μ

and n FD (ε = μ) equals 1/2. The transition at ε = μ becomes discontinuous in the limit of T → 0 (solid curve in Fig. 4.2a). The chemical potential at T = 0 K is called the Fermi energy, μ F , or the energy of the highest occupied quantum state. The Pauli exclusion rule prohibits fermions from condensating into a zero energy state even at absolute zero temperature, so that an additional particle put into the system is forcedly directed to the lowest available empty state immediately above the already occupied states. This is why there is a plateau at unity in Fig. 4.2. Particles are called degenerate when crammed into the domain below the Fermi energy. Degenerate fermion gases have a finite energy (μ F ) even at zero absolute temperature, at odds with the intuition of classical mechanics.

4.1 Particle Distribution Functions in Phase Space

65

As temperature rises from absolute zero (but not too much), the thermal energy excites fermions with an energy slightly short of μ F to an energy level slightly above μ F . This is clearly seen in Fig. 4.2a, where the population jump at μ F is rounded off within an energy range of roughly μ F ± kT as temperature becomes higher. This picture, however, is not exactly correct as temperature increases further because chemical potential itself is also a function of temperature. We next consider how μ changes with T . The density of states for a non-relativistic free fermion gas is written from (4.12) as 4π (4.42) ρ(ε) d3 x dε = 3 (2 m 3 ε)1/2 d3 x dε , h where dΩ has been integrated out into 4π assuming isotropy and p has been replaced by the kinetic energy of fermions, ε = p 2 /2m. The number density integrated over all energy levels,  ∞

N=

ρ(ε)n(ε) dε ,

(4.43)

0

is obtained for fermions by using (4.40) and (4.42) as N=

4π (2 m 3 )1/2 h3

 0

∞

ε1/2  dε . ε−μ exp +1 kT 

(4.44)

This equation yields the chemical potential of fermion gas once N is given. The integral equation (4.44) cannot be solved analytically, but is reduced to a tractable level in the Boltzmann limit, where temperature is so high (and μ is negative) that the exponential term in the denominator is much greater than unity for any ε, i.e.,  μ 1, exp − kT with which (4.44) results in  μ  ∞  ε  4π 3 1/2 N ≈ 3 (2 m ) exp dε ε1/2 exp − h kT kT 0 μ (2π mkT )3/2 = . exp h3 kT The Boltzmann limit thus requires N to meet the following condition,  μ  (2π mkT )3/2 = N  N exp − ≡ N B (T ) , kT h3 or for a given N ,

(4.45)

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4 Principles of Statistical Mechanics

T 

h 2 N 2/3 ≡ TB . 2π mk

(4.46)

Chemical potential is expressed as a function of T by rearranging (4.45):  μ = kT ln

N N B (T )





 3 2π mkT . = kT ln N − ln 2 h2

(4.47)

It is easily shown from (4.47) that μ decreases with increasing temperature when (4.46) holds. Figure 4.2b illustrates the Fermi-Dirac statistics plotted with the temperature dependence of μ taken into account. As temperature increases, particles migrate out of low energy levels and spread over the vastly empty zone at higher energy levels. The distribution function eventually becomes a mere exponential curve, known as the Boltzmann distribution, with a y-intercept much smaller than unity. In the Boltzmann limit, therefore, phase space is only sparsely populated over the whole domain (i.e., n(ε)  1), and fermions have virtually infinite choice of where in phase space to find themselves. In such circumstances, the Pauli exclusion rule no longer matters and classical statistical mechanics holds. While each additional fermion must gain the Fermi energy to join a degenerate system (T < T B ), adding a particle costs little effort at higher temperatures by virtue of the increasing emptiness of phase space. This intuitively explains why chemical potential declines as temperature increases. The terrestrial atmosphere is so thin (or so warm for its density) that the energy distribution of atmospheric gases obeys the Boltzmann distribution. Degeneracy plays a critical role in extremely dense astronomical objects such as white dwarfs and neutron stars. A white dwarf is the ruin of a low mass star such as the Sun that remains after hydrogen and helium cease to burn in the stellar core. The internal temperature gradient supports the star against its own gravity as long as nuclear fusion is at work, while in a white dwarf it is degenerate electron gas that keeps the star from collapsing. Interestingly, when the electrons are relativistic, i.e., their density is so high the Fermi energy outweighs the rest mass energy, degenerate pressure cannot be balanced against gravity in a stable manner. A white dwarf heavier than a certain critical mass lets gravity outrun pressure and would end up in catastrophic collapse.2 This strange property of degenerate stars was first theoretically predicted by Indianborn physicist Subrahmanyan Chandrasekhar, and the critical upper mass for white dwarfs to exist at all (∼1.44 times the solar mass) is called the Chandrasekhar limit.

4.1.5 Bose-Einstein Statistics: Basic Properties As we have seen in Sect. 4.1.3 above, the Bose-Einstein distribution (4.41),

2

When the mass is smaller than the critical mass, the white dwarf would expand until the relativistic limit gives way to the non-relativistic regime where the equilibrium solution is dynamically stable.

4.1 Particle Distribution Functions in Phase Space

67

Fig. 4.3 a Bose-Einstein distribution for low temperatures as a function of energy where μ = 0. Different lines are for different temperatures from absolute zero (solid) to higher values. b As (a) but with temperature increased further, with μ being negative and reducing as temperature increases

n BE (ε) =

1  , ε−μ −1 exp kT 

is derived when an unlimited number of particles are allowed to share any given quantum state. Chemical potential increases with decreasing temperature for bosons just as in the case of fermions described above. A distinct difference from the FermiDirac distribution is that the chemical potential never exceeds zero so that unphysical solutions with a negative n BE are avoided. Unlike the Fermi-Dirac distribution, the Bose-Einstein distribution has a certain critical temperature below which μ stays constant at zero. For any temperature below this criticality, (4.41) diverges to infinity at ε = 0. As a result, (4.41) loses the ability to fully account for the number of bosons in the system, and all those unaccounted for are found to occupy the ground state (ε = 0). This property causes a macroscopic quantum effect known as the BoseEinstein condensation, which gives rise to a strange behavior in a boson gas or liquid cooled down to near absolute zero. A notable phenomenon where the Bose-Einstein condensation plays a critical role is superfluidity, observed for liquid helium with temperatures lower than 2.17 K. Another example is superconductivity, where a pair of electrons acting as if there were a boson are in a state equivalent to the BoseEinstein condensation. The Bose-Einstein distribution is highly skewed toward ε = 0 for low temperatures, while higher energy levels are more and more populated as temperature increases (Fig. 4.3). In the Boltzmann limit (4.46), the Bose-Einstein distribution ends up in the Boltzmann statistics as is the case for the Fermi-Dirac distribution. The most important application of the Bose-Einstein distribution in the context of atmospheric sciences is the energy spectrum of photons known as the blackbody spectrum or the Planck function Bν (T ). The blackbody spectrum is obtained from (4.2), (4.13), and (4.41) with μ = 0 as

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4 Principles of Statistical Mechanics

Bν (T ) d3 x dν dΩ ≡ ργ n BE (εγ ) d3 x dεγ dΩ =

2 hν 2 c3

1  d3 x dν dΩ . hν −1 exp kT 

(4.48)

Chemical potential vanishes for photons since the number of photons does not conserve by nature.

4.1.6 Boltzmann Statistics As we have seen in the previous two Sects. (4.1.4 and 4.1.5), the Fermi-Dirac and Bose-Einstein distributions each converge into the classical mechanical formula called the Boltzmann distribution,   ε−μ , (4.49) n B (ε) = exp − kT in the Boltzmann limit (4.46), T  TB ≡

h 2 N 2/3 . 2π mk

The quantum mechanical constraint (4.31) loses its importance in the Boltzmann limit, since phase space is so sparcely populated that there is virtually no chance that two or more particles happen to occupy the same quantum state. Given that the density of states with no internal degree of freedom (g = 1) is 4π p 2 d p/ h 3 , the probability distribution of non-relativistic free particles in the Boltzmann limit n MB is obtained from (4.49) as 

 2 1 dp p n MB ( p)d p = exp − −μ 4π p 2 3 2m kT h 3/2    2 2 dp h p 1 4π p 2 3 =N exp − 2π mkT 2m kT h   4π N p2 p2 d p . = exp − (2π mkT )3/2 2mkT

(4.50)

It is noted that μ has been replaced with the total number density N so n MB gives N when integrated over p [see (4.47)]. Equation (4.50) is called the Maxwell-Boltzmann distribution, a fundamental formula in classical statistical mechanics. The MaxwellBoltzmann distribution is often expressed in terms of the particle velocity, U = p/m, as

4.1 Particle Distribution Functions in Phase Space

n

MB

69

   m 3/2 mU 2 U 2 dU , (U )dU = 4π N exp − 2π kT 2kT

or n MB (u)d3 u = N

   m 3/2 m|u|2 d3 u . exp − 2π kT 2kT

(4.51)

(4.52)

The expectation of the thermal velocity squared U 2 is U 2 =

1 N



∞

U 2 n MB (U )dU    m 3/2  ∞ mU 2 dU . U 4 exp − = 4π 2π kT 2kT 0 0

Using the integral formula of 

∞

x 4 e−ax dx =

0

3 8



π , a5

the equation above leads to U 2 =

3kT . m

(4.53)

4.2 Boltzmann Equation 4.2.1 Boltzmann Equation for Non-relativistic Particles We have considered so far a particle system in statistical equilibrium with the heat source of a given temperature, T , in which case the probability distribution of particle velocity, U , is found to obey the Maxwell-Boltzmann distribution (4.51) in the Boltzmann limit. This is generally an excellent representation of the tropospheric atmosphere once U is separated from the bulk or macroscopic component of particle velocity unaccounted for by thermal equilibrium: u =U+v ,

(4.54)

where U (|U| = U by definition) is the microscopic velocity due to the thermal motion of particles relative to the fluid motion and v is the macroscopic velocity of the fluid representing the wind in the meteorological context.3 While U is uniquely determined with temperature in local thermodynamic equilibrium, v is not. In the

3

Wind is always subsonic in the troposphere, so that |U|  |v|.

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4 Principles of Statistical Mechanics

remainder of this chapter, we formulate the equation to determine the evolution of u. The number of particles within a control volume in phase space, f (x, p, t)d3 xd3 p ,

(4.55)

may be rewritten as a function of x and velocity, u = p/m, for non-relativistic particles, (4.56) f (x, u, t)d3 xd3 u , where f (x, u, t) has been rescaled with m 3 omitted. The contributions of molecular rotation and vibration are not considered at this point. The time evolution of f (x, u, t) is governed by the momentum exchanges due to particle collisions, ∂f ∂f +u·∇ f +F· = C( f ) , ∂t ∂p

(4.57)

where the second term accounts for the advection of particles, the third term explains the effect of an external force exerted on the particles F, and C( f ), often called collisional integral, summarizes the effects of particle collisions on the probability distribution within a control volume. This equation governing the behavior of f is called the Boltzmann transport equation or simply the Boltzmann equation. For the discussion that follows, F is assumed to be absent, so that ∂f + u · ∇ f = C( f ) , ∂t

(4.58)

4.2.2 Conservation Laws for Non-relativistic Particles The Boltzmann equation offers the most general form of the theoretical framework to predict the space-time evolution of any given particle system. It is, however, highly impractical to explicitly solve the Boltzmann equation in the full 6-dimensional phase space (x and u). A convenient way to circumvent this mathematical challenge is to solve the first few “moments” of f , instead of f itself, by collapsing the momentum dimensions under an appropriate closure assumption. As will become clear later, the first three moments of the Boltzmann equation embody the macroscopic conservations of mass, momentum, and energy. The zeroth moment of the Boltzmann equation is obtained simply by multiplying (4.58) with the particle mass m and integrating over u,  m

∂f 3 d u+m ∂t

  j

uj

∂f 3 d u=m ∂x j

 C( f ) d3 u ,

(4.59)

4.2 Boltzmann Equation

71

The summation over j applies to the three spatial dimensions (x, y, and z). The first term in lhs is the rate of change in the number density of particles N (x, t),  m

∂N ∂f 3 d u=m , ∂t ∂t

or may be interpreted in terms of the mass density ρ,  m

∂f 3 ∂ρ d u= , ∂t ∂t 

where N (x, t) ≡

f (x, u, t) d3 u

(4.60)

and ρ = m N . Bearing in mind that u j is an independent variable in phase space and not a function of t or x j , the second term can be reduced to m

  j

 ∂ ∂f 3 uj d u=m ∂x j ∂x j j =m =

 u j f d3 u

 ∂ (N u j ) ∂x j j

 ∂ (ρu j ) , ∂x j j

(4.61)

where brackets denote the ensemble mean over the velocity space, q (x, t) ≡ N −1

 q(x, u, t) f (x, u, t) d3 u ,

(4.62)

for an arbitrary quantity q(x, u, t). Now we integrate u = U + v in (4.54) over velocity and find that U vanishes everywhere and thus u = v owing to the random and isotropic nature of thermal motion. Equations (4.59)–(4.61) therefore lead to ∂ρ + ∇ · (ρv) = 0 . ∂t

(4.63)

Note that the zero-th moment of the collisional integral, or the rhs of (4.59), is always zero because collisions do not change N . Equation (4.63) is the continuity (or mass conservation) equation of fluid. The first moment of the Boltzmann equation is derived next with the particle momentum mu multiplied before integrating over velocity,

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4 Principles of Statistical Mechanics

 m

ui

∂f 3 d u+m ∂t

or

 ui



uj

j

∂f 3 d u=m ∂x j

 u i C( f ) d3 u , (4.64)

 ∂ ∂ ρu i + ρu i u j = 0 . ∂t ∂x j j

(4.65)

The first moment of the collisional integral vanishes because the particle momentum is conserved during collision. One can rewrite the non-linear term in (4.65) with (4.54) as u i u j = (Ui + vi )(U j + v j ) = Ui U j + vi U j + v j Ui + vi v j = Ui U j + vi v j ,

(4.66)

recalling that U = 0. The isotropic nature of U also allows to reduce the first term of (4.66) into 1 Ui U j = U 2 δi j , (4.67) 3 where U 2 = Ux2 + U y2 + Uz2 . Equation (4.67) with (4.53) and the equation of state for an ideal gas having pressure P, P = N kT , leads to ρUi U j = ρ

kT δi j = Pδi j . m

(4.68)

(4.69)

Letting the momentum flux tensor πi j be πi j ≡ Pδi j + ρvi v j , (4.65) becomes

or in the vector notation,

 ∂πi j ∂ =0, ρu i + ∂t ∂x j j ∂ (ρv) + ∇ · π = 0 , ∂t

with π = PI + ρvv ,

(4.70)

4.2 Boltzmann Equation

73

where I denotes the unit tensor. Equation (4.70) shows the conservation of fluid momentum. Noting that the divergence of the momentum flux tensor may be rewritten as ∇ · π = ∇ P + [∇ · (ρv) + ρv · ∇]v , (4.70) results in

1 ∂v + v · ∇v + ∇ P = 0 ∂t ρ

(4.71)

with the help of (4.63). Equation (4.71) is the momentum conservation equation for an ideal gas in the absence of external force. Finally, the second moment of the Boltzmann equation is obtained by multiplying (4.58) by the particle energy m|u|2 /2, m 2 or



2∂f

m d u+ |u| ∂t 2 3

 |u|

2

 j

∂f 3 m uj d u= ∂x j 2

 |u|2 C( f ) d3 u ,

1 ∂ 1 ∂ ρ|u|2 + ρ|u|2 u j = 0 , 2 ∂t 2 j ∂x j

(4.72)

where the collisional term in rhs again vanishes due to the energy conservation during collisions. The particle velocity is decomposed into the thermal and macroscopic components, 1 1 ρ|u|2 = ρ(U + v) · (U + v) 2 2 1 = ρe + ρ|v|2 , 2

(4.73)

where e ≡ |U|2 /2 is the internal energy of fluid per unit mass, and 1 1 ρ|u|2 u j = ρ(U + v) · (U + v)(U j + v j ) 2 2   1 2 = ρe + P + ρ|v| v j , 2

(4.74)

where (4.69) and the isotropy of U (i.e., U j = 0 and U · v = 0) have been used. Equations (4.72) through (4.74) yield

 

1 1 ∂ ρe + ρ|v|2 + ∇ · ρe + P + ρ|v|2 v = 0 . ∂t 2 2

(4.75)

This is the energy conservation equation for the thermal energy (ρe) and the kinetic energy of fluid (ρ|v|2 /2) summed together.

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4 Principles of Statistical Mechanics

4.2.3 Boltzmann Equation for Photons The Boltzmann equation for photons may be written as ∂ fγ + cΩ · ∇ f γ = Cγ ( f γ ) . ∂t

(4.76)

Equation (4.76) resembles its non-relativistic counterpart, (4.58), except that u has been replaced by cΩ, where Ω is unit direction vector of photons. The distribution function of photons f γ within a control volume in phase space, f γ (x, p, t) d3 x d3 p = f γ (x, p, t) cdt d A p 2 d p dΩ ,

(4.77)

where cdt is the distance over which photons travel for an infinitesimal time of dt through an areal element of d A, and the momentum volume element is replaced by (4.10). As Cartesian coordinates are converted to spherical polar coordinates in momentum space, the number density of photons is now defined within a beam that has a width of dΩ when projected onto unit sphere (see Fig. 4.1). The direction vector of photons is denoted intentionally by a symbol closely linked with the solid angle element, signifying the fact that Ω, identical to unit radial vector in momentum space, is always normal to dΩ. Equation (4.77) is rewritten using (4.3) into f γ (x, p, t) d3 xd3 p = f γ (x, Ω, ν, t)

h3ν2 dt d A dν dΩ . c2

One can see that the three momentum dimensions for photons are decomposed into its scalar magnitude proportional to ν and a two-dimensional unit direction vector Ω. The photon energy per unit phase-space volume, or the energy of photons transported through unit area during unit time into the direction Ω within unit solid angle, is Iν (x, Ω, t) d3 xd3 p = hν f γ (x, Ω, ν, t) d3 xd3 p = f γ (x, Ω, ν, t)

h4ν3 dt d A dν dΩ . c2

The quantity Iν is called specific intensity or radiance, which has the units of [W m−2 sr−1 Hz−1 ]. An alternative form of the Boltzmann equation (4.76) expressed with Iν in lieu of f γ is 1 ∂ Iν + Ω · ∇ Iν = Cˆ γ (Iν ) , (4.78) c ∂t where

h4ν3 Cˆ γ (Iν ) ≡ 3 Cγ ( f γ ) . c

The argument Iν is omitted from now on. The collisional term Cˆ γ (Iν ) is broken down into source and sink terms.

4.2 Boltzmann Equation

75

Fig. 4.4 Schematic of the source and sink in the Boltzmann equation for photons. Diverging arrows with an arrowhead designate photons scattered away, while those with feathers denote photons scattered from different directions into the line of sight

 dσs,ν (Ω → Ω  )dΩ  , Cˆ ν,snk = −ρσa,ν Iν − ρ Iν dΩ   dσs,ν  ˆ Cν,src = ρ jν + ρ Iν (Ω → Ω)dΩ  , dΩ 

(4.79a) (4.79b)

where ρ refers to the mass density (kg m−3 ) of the medium under consideration (atmospheric gases, rain drops, and so on), Iν refers to Iν (x, Ω  , t), and the integral applies to the whole range of solid angle, 



π

dΩ ≡ 0

 sin θ dθ

2π

dφ .

(4.80)

0

The sink term (4.79a) consists of the extinction of radiation due to absorption (first term) and that owing to scattering (second term). The absorption is proportional to the incident radiance by a factor of σa,ν (m2 kg−1 ) or the absorption cross section per unit mass of the medium. While the absorbed radiative energy is converted immediately into the thermal energy of the medium, the scattered energy is simply diverted into other directions, designated by Ω  , without being thermalized (Fig. 4.4). The differential scattering cross section dσs,ν /dΩ  in (4.79a) is by construction

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4 Principles of Statistical Mechanics

merely integrated to the total scattering cross section σs,ν , as Iν is separated out of the integral over Ω  . The first term in the source (4.79b) is the emission of radiation independent of the surrounding radiation field, defined by the emission coefficient jν per unit mass, unit time, unit solid angle, and unit frequency (W kg−1 sr−1 Hz−1 ). The emission of radiation is considered to be isotropic, which is a valid assumption for thermal emissions, throughout this book unless otherwise indicated. The second component of the source is the inverse process to the extinction due to scattering, that is, a bundle of radiances from different directions confluent into Ω (Fig. 4.4). Equation (4.79) is slightly modified for later convenience into Cˆ ν,snk = −ρ(σa,ν + σs,ν )Iν ,  Cˆ ν,src = ρ jν + ρσs,ν Iν pν (Ω  , Ω)dΩ  , where pν (Ω  , Ω) ≡

1 dσs,ν  (Ω → Ω) σs,ν dΩ

(4.81a) (4.81b)

(4.82)

is the normalized differential scattering cross section known as scattering phase function or simply phase function. Scattering phase function by construction satisfies 

pν (Ω  , Ω)dΩ  = 1 .

(4.83)

Note that scattering is here assumed to be elastic, that is, any scattering process that alters ν is not considered. This assumption is not generally valid (e.g., Raman and Compton scatterings) but is broadly applicable to the radiative processes dominant in the earth’s atmosphere. Equations (4.78) and (4.81) are combined into 1 ∂ Iν + Ω · ∇ Iν = −ρ(σa,ν + σs,ν )Iν + ρ jν + ρσs,ν c ∂t



Iν pν (Ω  , Ω)dΩ  ,

which is rewritten for later convenience as

 1 ∂ Iν + Ω · ∇ Iν = ρσe,ν −Iν + (1 − ων )Sν + ων Iν pν (Ω  , Ω)dΩ  , (4.84) c ∂t where σe,ν ≡ σa,ν + σs,ν is the extinction cross section per unit mass (m2 kg−1 ), representing the total radiative energy loss along the direction Ω due to both absorption and scattering,

4.2 Boltzmann Equation

77

ων ≡

σs,ν σe,ν

is a non-dimensional parameter called the single scattering albedo ranging from 0 (i.e., no scattering) to 1 (i.e., no absorption), and Sν ≡

jν σa,ν

(4.85)

is the source function, shown later to be equivalent to the Planck function when local thermodynamic equilibrium is attained (see Sect. 6.1.2). Equation (4.84) is the Boltzmann equation for photons, known widely as the radiative transfer equation.

4.2.4 The Conservation Laws for Photons 4.2.4.1

Moment Equations

Moments of the radiative transfer equation lead to the conservation laws for photons, similar to those for non-relativistic particles derived in Sect. 4.2.2. It is again assumed that scattering is always elastic, so that the radiative transfer equation may be thought of as independent for each ν. The first three moments of monochromatic radiance are given as follows. Radiative energy density: E ν =

1 c 

 Iν dΩ

Radiative flux: Fν,i = Radiative pressure: Pν,i j

Iν Ωi dΩ  1 = Iν Ωi Ω j dΩ c

(4.86a) (4.86b) (4.86c)

The subscripts (i and j) refer to each component of the radiative flux vector and the radiative pressure tensor with Ωi being the projection of unit direction vector Ω onto the i-th axis. For instance, the vertical component of radiative flux Fν,z in a plane-parallel atmosphere is obtained by 

π

Fν,z = 2π 0

 Iν cos θ sin θ dθ = 2π

1

−1

Iν cos θ d(cos θ ) ,

(4.87)

where θ is the zenith angle. It is extremely important not to confuse radiative flux, or often called irradiance, with radiance itself. Radiance is a narrow beam of photons per unit solid angle, while irradiance is the net flux of radiative energy measured as a weighted integral of radiance over solid angle.

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4 Principles of Statistical Mechanics

It is reminded that the contributions of scattering to the source and sink terms are opposite sides of the same coin (compare the two scattering processes illustrated in Fig. 4.4).   Iν

dσs,ν (Ω → Ω  )dΩ  dΩ = dΩ 

 

Iν

dσs,ν  (Ω → Ω)dΩ  dΩ dΩ 

(4.88)

This reciprocity relationship ensures by construction that  σs,ν

  Iν dΩ = σs,ν

Iν pν (Ω  , Ω)dΩ  dΩ .

(4.89)

Applying solid-angle integral to (4.84), one obtains the energy conservation law for photons as ∂ Eν + ∇ · Fν = ρσa,ν (4π Sν − cE ν ) , (4.90) ∂t where (4.89) has been used to eliminate the scattering term. Equation (4.90) claims that the rate of change in E ν is accounted for by the radiative flux divergence and the local exchange of radiative energy with the medium through the emission and absorption. Scattering plays no role in the energy balance because, under the current assumption, scattering only diverts photons from one direction to another without altering the energy density of radiation. The first-order moment of the radiative transfer equation leads to the momentum conservation law for photons.

  ρσe,ν 1 ∂Fν    + ∇ · Pν = −F + ων Iν pν (Ω , Ω)dΩ ΩdΩ c2 ∂t c

(4.91)

Individual elements of the radiative flux vector F and the radiative pressure tensor P are as defined in (4.86). Equation (4.91) is the radiative-transfer equivalent of the momentum conservation law for non-relativistic fluid (4.70).

4.2.4.2

Eddington Approximation

The conservation laws for an ideal gas constitute a closed system of equations once the equation of state (4.68) is given as a closure assumption. For a photon gas, in contrast, such a closure does not exist in general and each moment equation always involves a high-order moment yet to be determined. A notable exception for this occurs when the radiation field is nearly isotropic just like the thermalized atomic motion in an ideal gas. A nearly isotropic radiance may be expressed as a Legendre expansion over the direction cosine with only the zeroth and first terms retained: Iν ≈ Iν,0 + Iν,1 μ ,

(4.92)

4.2 Boltzmann Equation

79

where μ ≡ cos θ is the direction cosine. In this case, the first three moments of radiance (4.86) become 2π Eν = c Fν,r Pν,rr



1

4π Iν,0 , c −1  1 4π Iν,1 , = 2π Iν μdμ = 3 −1  2π 1 4π 1 Iν,0 = E ν , = Iν μ2 dμ = c −1 3c 3 Iν dμ =

(4.93a) (4.93b) (4.93c)

where the subscript r refers to the reference axis (θ = 0) and all other elements of F and P vanish. Scattering phase function is approximated similarly as pν (Δμ) ≈ 1 + gν Δμ

(4.94)

where Δμ ≡ Ω  · Ω and gν is the first-order Legendre moment of the phase function called asymmetry parameter. Asymmetry parameter is a non-dimensional variable representing the bulk angular dependence of scattering, varying from −1 (backward peaking) to 1 (forward peaking). Higher-order coefficients would be integrated out owing to orthogonality and are thus unnecessary to specify under the approximation of (4.92). The energy and momentum conservation laws are found to be dFν,r ∂ Eν + = ρσa,ν (4π Sν − cE ν ) , ∂t ds   1 dE ν ρσe,ν 1 ∂ Fν,r 2 + =− 1 − ων gν Fν,r , c2 ∂t 3 ds c 3

(4.95a) (4.95b)

where s represents the spatial coordinate along the reference axis. It is evident that (4.95) is now solvable for E ν and Fν because (4.93c) acts as a closure, which is a photon-gas counterpart of the ideal-gas equation of state, when the angular dependence of radiance is weak. This near-isotropy assumption given by (4.92) is called the Eddington approximation. If it were not for the time-derivative term in the second equation, (4.95) is mathematically equivalent to the diffusion equation. A flux of photons therefore carries the radiative energy in such a way as heat flux carries the thermal energy by diffusion within a conductive medium, unless the radiation field is highly anisotropic. This diffusive transfer of radiation is primarily owing to successive occurrences of the absorption and emission along a ray of radiation. Scattering has a secondary effect that radiative flux is effectively enhanced or reduced in the momentum equation as seen in the parentheses on the rhs of (4.95b).

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4.2.4.3

4 Principles of Statistical Mechanics

Two-stream Approximation

Another widely-used technique similar to the Eddington approximation is the twostream approximation, in which radiation field is represented by a pair of radiances pointed at opposite directions, Iν↑ ≡ Iν (μ = μ0 ) and Iν↓ ≡ Iν (μ = −μ0 ). The twostream approximation is in a sense a truncated series expansion based on the delta function with respect to direction cosine, just like the Eddington approximation is a truncated Legendre expansion. In both approximations, the first two terms are retained with all higher-order terms omitted. Inserting Iν↑ = Iν δ(μ − μ0 ) and Iν↓ = Iν δ(μ + μ0 ) into (4.93), one finds 2π Eν = c Fν,r = 2π Pν,rr =

2π c

 

1

−1 1

−1  1 −1

Iν dμ =

2π ↑ (I + Iν↓ ) , c ν

Iν μdμ = 2π μ0 (Iν↑ − Iν↓ ) , Iν μ2 dμ =

2π 2 ↑ μ (I + Iν↓ ) . c 0 ν

(4.96a) (4.96b) (4.96c)

As such, the first three moments of radiance can be expressed by a simple combination of Iν↑ and Iν↓ in the two-stream approximation. √ The special case with μ0 = 1/ 3 bears a particular resemblance to the Eddington approximation in that Pν,rr is equal to 1/3E ν (Rybicki and Lightman 1985). Combining the first two equations in (4.96), one obtains √ 3 c Eν + Fν,r , = 4π 4π √ 3 c Eν − Fν,r . Iν↓ = 4π 4π

Iν↑

(4.97a) (4.97b)

Equation (4.97) conveniently serves as boundary conditions to the Eddington approximation (4.95). For instance, let us consider a plane-parallel atmosphere bound between τν = 0 at BOA and τν,a at TOA. Given that Iν↑ is upwelling radiance (μ > 0) and Iν↓ is downwelling (μ < 0), radiances incident on the lower and upper boundaries can be specified by Iν↑ (τν = 0) and Iν↓ (τν = τν,a ), respectively. An example for such applications will be presented in Sect. 8.1.

4.2.5 Radiative Transfer Equation and Optical Depth The time-derivative term in the radiative transfer equation (4.84) is negligibly small under most circumstances (except for some astrophysical applications) because photons move around so fast that the radiation field immediately adjusts itself on a time

4.2 Boltzmann Equation

81

scale of interest. With the spatial derivative replaced with a length element ds along the direction Ω, (4.84) becomes

 dIν = ρσe,ν −Iν + (1 − ων )Sν + ων Iν pν (Ω  , Ω)dΩ  . ds

(4.98)

Equation (4.98) is the time-independent radiative transfer equation, which suffices for applications in satellite remote sensing. The radiative transfer equation determines the radiation field, given a set of radiative properties of the medium: σe,ν , ων , Sν , and pν (Ω  , Ω). For many applications including remote sensing of clouds and precipitation, Sν can be safely replaced by the Planck function (4.48) thanks to Kirchhoff’s law (see Sect. 6.1.2.3). A radically simplified but illustrative example of radiative transfer problems occurs when the extinction term dominates the others (i.e., Sν ∼ 0 and ων ∼ 0) in (4.98). dIν ∼ −ρσe,ν Iν ds A real-life example for this case is a lead apron for X-ray protection, which efficiently absorbs radiation without producing any appreciable thermal emission or scattering. The differential equation above yields an obvious solution,    Iν ∝ exp − ρσe,ν ds .

(4.99)

Radiance exponentially decays over distance at the local rate of ρσe,ν in this particular example. This can be described also in terms of the extinction mean free path of photons, lν,e (m), as 1 lν,e = , (4.100) ρσe,ν which offers a rough measure of how far photons can travel before being absorbed or scattered away. A nondimensionalized length normalized by lν,e over a certain distance (L),  L ds , (4.101) τν (L) = 0 l ν,e is known as optical depth, or alternatively called optical thickness. The radiative transfer equation (4.98) is rewritten using optical depth in lieu of the geometrical length scale as dIν = −Iν + (1 − ων )Sν + ων dτν



Iν pν (Ω  , Ω)dΩ  .

(4.102)

Optical depth, a measure of opaqueness, is physically more intuitive than geometrical depth in the context of radiative transfer. For instance, the cloud effects on the

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4 Principles of Statistical Mechanics

ambient radiation field depend on how optically thick the cloud layer is rather than on its geometrical depth. Clouds could be nearly transparent or opaque, depending not only on the physical thickness (L) but on the cloud properties such as the number density of cloud droplets and the droplet radius, which have control on ρ and σe,ν . Clouds with τ  1 are called optically thick, and those with τ < 1 are considered as optically thin. A practical form of the energy and momentum equations under the Eddington approximation are immediately derived from (4.95) as dFν,r = (1 − ων )(4π Sν − cE ν ) , dτν   1 dE ν 2 1 1 − ων gν Fν,r , =− 3 dτν c 3

(4.103a) (4.103b)

where the time-derivative terms have been omitted.

Reference Rybicki GB, Lightman AP (1985) Radiative processes in astrophysics. Wiley, New York, NY. https://doi.org/10.1002/9783527618170

Chapter 5

Principles of Electrodynamics and Geometrical Optics

We are surrounded in our daily lives by a variety of electromagnetic waves, including visible light delivered from the sun, infrared rays emitted from everything around us, and microwave radiation carrying the data we exchange with smartphones. Just as we exploit electromagnetic waves to see something with our eyes and feel the heat without touching its source, satellite remote sensing is built around the technologies to detect and analyze the electromagnetic waves emitted from and scattered around by target objects. In this chapter, a brief introduction to electrodynamics is first provided to review the theoretical basis how electromagnetic waves are modified in magnitude and direction as they propagate through different media. It is then demonstrated that Maxwell’s equations are of great utility for quantifying the radiative properties of particle scattering and absorption. The second half of the chapter is devoted to a short summary of geometrical optics. Geometrical optics, although a simple, empirical framework of ray tracing predating Maxwell’s theory, is also useful in satellite observations for characterizing the emission and reflection of radiation at the earth’s surface.

5.1 Electrodynamics 5.1.1 Maxwell’s Equations Maxwell’s equations are a set of four elegant equations governing the evolution of electromagnetic field in space and time. ∇ · D = ρe

(5.1)

∇ ·B=0

(5.2)

© Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_5

83

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5 Principles of Electrodynamics and Geometrical Optics

∂B ∂t

(5.3)

∂D +j ∂t

(5.4)

∇ ×E=− ∇ ×H =

The electric displacement D and the electric field E are related to each other in vacuum as D = ε0 E . (5.5) Similarly, the magnetic field H is proportional to the magnetic induction B in vacuum. B = μ0 H

(5.6)

Here ε0 and μ0 are the permittivity and permeability, respectively. The first Eq. (5.1) is Gauss’s law, which claims that electric charge (having a charge density ρe ) serves as the source of an electric field. The second one (5.2) is its magnetic counterpart but without any source, implying the non-existence of magnetic charge or magnetic monopole. The third Eq. (5.3) indicates that a temporal fluctuation of magnetic field creates a circumnavigating electric field, a phenomenon known as Faraday’s law of induction. The last formula (5.4) is a generalization of Ampère’s law, stating that a magnetic field is generated around the electric current having a current density j (original Ampère’s law) and/or by a temporal variation of the electric field (an addition by Maxwell). Maxwell’s equations have non-trivial solutions even in the absence of any electric charge and current. Maxwell’s equations with ρe = 0 and j = 0 are reduced to ∇ ·E=0

(5.7)

∇ ·B=0

(5.8)

∇ ×E=−

∂B ∂t

∇ × B = μ0 ε0

∂E ∂t

(5.9) (5.10)

Applying the curl to (5.9), one finds ∇(∇ · E) − ∇ 2 E = −

∂ (∇ × B) , ∂t

(5.11)

where the vector identity of ∇ × (∇ × E) ≡ ∇(∇ · E) − ∇ 2 E has been applied. Using (5.7) and (5.10), (5.11) is found to be

5.1 Electrodynamics

85

∂ 2E = c2 ∇ 2 E , ∂t 2 where c≡ √

1 . μ0 ε0

(5.12)

(5.13)

Repeating the same operation to eliminate E, one obtains ∂ 2B = c2 ∇ 2 B . ∂t 2

(5.14)

Equations (5.12) and (5.14) are wave equations yielding waves propagating at the phase speed of c, which is found from (5.13) to be 3.0 ×108 m s−1 , equal to the speed of light. Electromagnetic waves are thus the theoretical consequence of Maxwell’s equations with ρe = 0 and j = 0.

5.1.2 Refractive Index and Dielectric Function Now, let us consider the electromagnetic waves propagating through a certain medium. A simple model to formulate this is the electric polarization P representing a bulk effect of electric dipole moment in the medium. An electromagnetic wave incident on an insulating medium induces an electric polarization in an approximate sense as (5.15) P = ε0 χ E , where χ , the electric susceptibility, defines the dielectric properties of the medium. The electric displacement of media is written as D = ε0 E + P = εE

(5.16)

ε ≡ (1 + χ )ε0 .

(5.17)

with Similarly, the effect of magnetization can be incorporated by adjusting μ0 to a proper value specific to the medium, μ. B = μH = (1 + χm )μ0 H

(5.18)

Here χm is called the magnetic susceptibility. Maxwell’s equations are, to the extent that (5.16) and (5.18) are a valid approximation, applicable to the electromagnetic waves propagating through a medium just in the same manner as to those in vacuum. The speed of light in a medium c is modified from c as

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5 Principles of Electrodynamics and Geometrical Optics

1 . c = √ εμ

(5.19)

Note that ε and μ, and hence c as well, vary generally with the electromagnetic frequency unlike ε0 and μ0 , depending on the spectral properties of the medium. This causes a dispersion of electromagnetic waves as the waves propagate through a medium. Among the simplest solutions of (5.12) is a plane wave, E(x, t) = E0 exp(ik · x − i2π νt) .

(5.20)

where ν is frequency and k is wavenumber vector. It is immediately shown from (5.7) that k · E = 0, so that the electric field plane is normal to the direction of propagation. For a wave propagating along the x-axis,   x  , E(x, t) = E0 exp −i2π ν t − c where wavenumber has been replaced by 2π ν/c. As the wave enters from vacuum into an arbitrary medium, the plane-wave solution is slightly modified into     x  nx  E(x, t) = E0 exp −i2π ν t −  = E0 exp −i2π ν t − , c c where n(ν) =

c c (ν)

(5.21)

(5.22)

is refractive index or the ratio of the speed of light in vacuum to that in the medium. Refractive index is almost always larger than unity, implying that electromagnetic waves propagate more slowly, in a phenomenological sense, inside a medium than in vacuum. A reduction in the speed of light may be understood at a microscopic level as the interference of the incident wave with secondary waves induced by the forced oscillation of electric dipoles in the medium. As will become clear later in Sect. 5.2, refractive index is the critical parameter characterizing the optical properties of media for geometrical optics. In general, refractive index is a complex number, n = n r + in i ,

(5.23)

   n r x  ni x  exp −i2π ν t − . E(x, t) = E0 exp −2π νi c c

(5.24)

so that (5.21) is rewritten into

The imaginary part of complex refractive index defines how quickly the wave decays over distance as it proceeds through the medium, while its real part retains the orig-

5.1 Electrodynamics

87

inal, geometrical-optical definition of n. The real and imaginary parts of n together characterize the optical properties intrinsic to the medium. The origin of the imaginary part of n is interpreted in light of Maxwell’s theory as the dissipation of electromagnetic waves through Joule heating. To confirm this, (5.4) is rewritten with the temporal dependence in H and E, proportional to exp(−i2π ν), separated out.    σc σc  ε0 E E = −i2π ν (1 + χ ) + i ∇ × H = −i2π ν ε + i 2π ν 2π νε0

(5.25)

Here σc is the conductivity defined as j = σc E .

(5.26)

Equation (5.25) suggests that the susceptibility χ may be conveniently redefined as a complex number with its imaginary part being σc /2π νε0 . Given a complex susceptibility, the relative permittivity, εν ≡

ε =1+χ , ε0

(5.27)

can be also generalized into a complex number, where its imaginary part is responsible for the energy dissipation proportional to σc /ν to the extent that (5.26) holds to a good approximation. This relative permittivity is better known as complex dielectric function, which is called “function” because it varies with ν in general. In this book, dielectric function εν is distinguished notationally from the permittivity ε by adding a subscript ν. For any medium with negligible magnetic susceptibility (μ = μ0 ), (5.13), (5.19), and (5.22) together lead to  εν =

c  c (ν)

2 = n2 .

(5.28)

The real and imaginary components of n and εν are interrelated as εν,r = n r2 − n i2 and εν,i = 2n r n i . As such, refractive index and dielectric function are simply different measures of the same physical entity. In practice, refractive index is traditionally adopted for visible and infrared optics, while dielectric function is generally preferred for microwave radiometry.

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5 Principles of Electrodynamics and Geometrical Optics

5.1.3 Poynting Vector One of the important and practical consequences of Maxwell’s equations is the electromagnetic energy flux known as the Poynting vector. The Poynting vector is derived from an energetic consideration how the rate of work done by the Lorentz force is balanced against the change of electromagnetic energy density. The rate of work by the Lorentz force per unit volume j · E is written out as   ∂D j · E = E · (∇ × H) − E · ∂t by multiplying E on both sides of (5.4). Applying a vector identity formula, E · (∇ × H) = H · (∇ × E) − ∇ · (E × H) , 

one finds

∂D j · E = H · (∇ × E) − ∇ · (E × H) − E · ∂t

 ,

or equivalently,   ∂D ∂B − ∇ · (E × H) − E · , j · E = −H · ∂t ∂t using (5.3). This equation may be rearranged into so-called Poynting’s theorem, ∂ ∂t



1 2 1 2 εE + μH + j · E = −∇ · S , 2 2

(5.29)

where S≡E×H

(5.30)

is the Poynting vector. Poynting’s theorem states that the energy imbalance between the rate of change in the electromagnetic energy (the term in parentheses on lhs) and the rate of work done by the Lorentz force (the second term) within a control volume is accounted for by the convergence of the Poynting vector S. It follows that S represents the electromagnetic flux bringing the energy in and out of the volume.

5.1.3.1

Radiative Energy Carried by a Plane Wave

Inserting a plane-wave solution (5.20) into (5.9) and (5.18), the electric and magnetic components of an electromagnetic wave and the direction of its propagation are found to be perpendicular to one another.

5.1 Electrodynamics

89

H=

1 k×E 2π νμ

(5.31)

For later convenience, E and H are redefined with the time-dependent component, exp(−i2π νt), taken out. The Poynting vector, when averaged over a period of time much longer than ν −1 , is now expressed as S=

1 Re(E × H∗ ) , 2

(5.32)

where H∗ is the complex conjugate of H. From now on, we consider a plane wave propagating in parallel to the x axis, so that k = kex . Using the vector identity, E × (ex × E∗ ) = ex (E · E∗ ) − E∗ (ex · E), one finds S=

k k 1 Re[E × (ex × E∗ )] = |E0 |2 ex =  |E0 |2 ex . 4π νμ 4π νμ 2c μ

(5.33)

Note that ex · E constantly vanishes. In (5.33), S represents the transport of radiative energy by a plane electromagnetic wave, which is the electromagnetic counterpart of radiance Iν introduced in Sect. 4.2.3.

5.1.3.2

Rate of Electromagnetic Energy Change

Applying a volume integral and Gauss’ theorem to (5.29), one finds the net rate of change in electromagnetic energy, W , as 



∇ · S d3 x = −

W =− V

S · A dA , A

where A is unit vector normal to an areal element d A, which is integrated over the whole surface encompassing a finite volume, V . In spherical polar coordinates, W within a sphere having a radius r is  W (r ) = −

1 S · er r dΩ = − 2 2



Re(E × H∗ ) · er r 2 dΩ ,

(5.34)

where dΩ is a solid-angle element (4.11) and er is unit radial vector.

5.1.4 Mie’s Solution The goal of this section is to solve Maxwell’s equations to theoretically obtain the absorption and scattering coefficients along with the phase function of scattering for a small particle suspended in the air. It is generally a mathematically demanding, if not

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5 Principles of Electrodynamics and Geometrical Optics

impossible, task to obtain such solutions for different types of atmospheric particles such as raindrops and snowflakes. A simplest example is the absorption and scattering by a spherical, homogeneous particle, for which mathematically exact solutions of Maxwell’s equations, widely known as Mie’s solution, exist. Mie’s solution is accurate when applied to cloud droplets, and are also reasonable for raindrops to the extent that they are well approximated as a sphere. The practical utility of Mie’s solution is not necessarily limited to liquid hydrometeors. Although the spherical assumption apparently breaks down for frozen particulates such as snowflakes and ice crystals, Mie’s solution has been often adopted as a useful proxy for the optical properties of non-spherical solid particles in satellite remote sensing. A brief outline of the derivation of Mie’s solution is provided in this section. The readers interested in technical details omitted here are encouraged to consult Bohren and Huffman (1998).

5.1.4.1

Overview

The procedure to derive Mie’s solution involves complicated mathematical operations, but the underlying concept is rather simple. When a plane wave is incident on a spherical particle with a given refractive index, an electric polarization arises and oscillates within the sphere, according to (5.15), in response to the undulating electromagnetic field induced by the incident wave. This oscillating polarization then emits a set of electromagnetic waves propagating away to all directions from the sphere. Mie’s solution is a series of the analytic solutions for these emanating waves (scattered radiation) obtained with the boundary conditions imposed so that the electric fields inside and outside of the particle are continuous at the surface of the sphere. We have seen that Maxwell’s equations are reduced to a pair of wave equations for E and B [(5.12) and (5.14)] when no electric charge nor current exists. The timedependent component of the wave equations may be separated out by assuming the solutions to be proportional to exp(−i2π νt). ∇ 2E + k2E = 0

(5.35a)

∇ B+k B = 0

(5.35b)

2

2

Here k = ±2π ν/c is the wavenumber. A convenient strategy to find the solutions to (5.35) is to first solve a scalar wave equation of the same form to obtain a scalar function Ψ , (5.36) ∇ 2 Ψ + k 2 Ψ = 0, and then compute the two vectors, M and N, defined as M = ∇ × (cΨ )

(5.37)

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91

and N=

1 ∇ ×M, k

(5.38)

where c is an arbitrary constant vector. It can be shown that M and N satisfy their own vector wave equations, ∇2M + k2M = 0 , ∇ 2N + k2N = 0 , and M=

1 ∇ ×N, k

(5.40)

whenever (5.36) holds. In addition, since ∇ · (∇ × a) vanishes for an arbitrary vector a, (5.37) and (5.38) immediately imply that M and N are divergence-free vectors, ∇ ·M = 0, ∇ ·N = 0, just as E and B are in the absence of electric charges [(5.7) and (5.8)]. As such, E and B may be expressed as a linear superposition of the independent solutions of M and N.

5.1.4.2

Scalar Wave Equation

The first step is to solve the scalar wave equation (5.36) and find Ψ . It is most practical to expand the solutions of Ψ into a series of spherical harmonics for the current problem. The scalar wave equation (5.36) is written down in spherical polar coordinates as   1 ∂ 1 ∂ ∂Ψ 1 ∂ 2Ψ 2 ∂Ψ r + sin θ + 2 + k 2 Ψ = 0 , (5.42) 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ 2 where r , θ , and φ are the radial, zenithal, and azimuthal coordinates, respectively. Following the convention, Ψ is decomposed into Ψ (r, θ, φ) = R(r )Θ(θ )Φ(φ) so that (5.42) is broken down into separate ordinary differential equations.

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5 Principles of Electrodynamics and Geometrical Optics

 dR r2 + [k 2 r 2 − n(n + 1)]R = 0 dr    1 d dΘ m2 sin θ + n(n + 1) − Θ=0 sin θ dθ dθ sin2 θ d 2Φ + m2Φ = 0 dφ 2 d dr

(5.43a) (5.43b) (5.43c)

It is evident that the third Eq. (5.43c) has general solutions of cos mφ and sin mφ, where m must be zero or a positive integer under a cyclic boundary condition with respect to φ. The zenithal component of the wave equation, (5.43b), can be transformed with η ≡ cos θ into the associated Legendre differential equation. (1 − η2 )

  d 2Θ dΘ m2 Θ=0 − 2η + n(n + 1) − dη2 dη 1 − η2

(5.44)

The associated Legendre functions Pnm (η) are an orthogonal system of functions known to satisfy (5.44) for −1 ≤ η ≤ 1 and n ≥ m ≥ 0, where n is an integer. Another set of functions called the associated Legendre functions of the second kind are also solutions to the same differential equation but are not considered here because they diverge to infinity at η = ±1. The radial component of the solutions are obtained from the Bessel differential equation,  2  dZ 1 d ρ + ρ2 − n + Z =0, (5.45) ρ dρ dρ 2 √ which is equivalent to (5.43a) for ρ ≡ kr and Z ≡ R ρ. The solutions to (5.45), denoted by z n (ρ), are expressed as a linear combination of any pair from the four kinds of the spherical Bessel functions, two of which are called the spherical Hankel functions. In summary, the scalar wave equation (5.36) yields solutions expressed as Ψe,mn (r, θ, φ) = z n (kr ) Pnm (cos θ ) cos mφ

(5.46a)

Ψo,mn (r, θ, φ) = z n (kr ) Pnm (cos θ ) sin mφ ,

(5.46b)

where the subscripts e and o stands for even and odd components, respectively.

5.1.4.3

Vector Solutions

The vector solutions for M and N are obtained by substituting the explicit form of the scalar solution, (5.46), into (5.37) and (5.38). The components of each vector projected on spherical polar coordinates (er , eθ , eφ ) are found as follows.

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93

m z n (kr )Pnm (cos θ ) sin mφ eθ sin θ d P m (cos θ ) −z n (kr ) n cos mφ eφ dθ m z n (kr )Pnm (cos θ ) cos mφ eθ = sin θ d P m (cos θ ) sin mφ eφ −z n (kr ) n dθ

Me,mn = −

Mo,mn

n(n + 1) z n (kr )Pnm (cos θ ) cos mφ er kr 1 d d P m (cos θ ) + [r z n (kr )] n cos mφ eθ kr dr dθ d m [r z n (kr )]Pnm (cos θ ) sin mφ eφ − kr sin θ dr n(n + 1) = z n (kr )Pnm (cos θ ) sin mφ er kr 1 d d P m (cos θ ) [r z n (kr )] n sin mφ eθ + kr dr dθ d m [r z n (kr )]Pnm (cos θ ) cos mφ eφ + kr sin θ dr

(5.47a)

(5.47b)

Ne,mn =

No,mn

(5.48a)

(5.48b)

It is noted that c in (5.37) has been replaced by r er and hence the radial component does not appear in (5.47). Certain combinations of the vector solutions from (5.47) to (5.48) constitute an orthogonal system. The orthogonality of sin mφ and cos mφ immediately leads to 

 Me,m  n  · Mo,mn dΩ =

Ne,m  n  · No,mn dΩ = 0

for all m, m  , n, n  and   Me,m  n  · Ne,mn dΩ = Mo,m  n  · No,mn dΩ = 0 for all cases but m = m  . The integral over solid angle applies to the whole unit sphere as defined by (4.80). It can be also shown, with some more algebra, that 

 Me,mn  · No,mn dΩ =

Mo,mn  · Ne,mn dΩ = 0 , 

 Me,mn  · Me,mn dΩ =

Mo,mn  · Mo,mn dΩ = 0 ,

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5 Principles of Electrodynamics and Geometrical Optics



 Ne,mn  · Ne,mn dΩ =

No,mn  · No,mn dΩ = 0 ,

except when n = n  .

5.1.4.4

Incident Plane Wave in Spherical Coordinates

Mie’s solution is meant to provide the link between an incident plane wave and outgoing scattered waves. The incident plane wave, although conveniently expressed in Cartesian coordinates, needs to be expanded into spherical harmonics so that it is readily incorporated into a boundary condition at the spherical surface along with the scattered waves. Consider an incident plane electromagnetic wave propagating along the polar axis, (5.49) Ei = E i,0 exp(ik · x) ex = E i,0 exp(ikr cos θ ) ex , where the temporal component exp(−i2π νt) here again has been separated out. A unit vector in the electric field direction ex is ex = sin θ cos φ er + cos θ cos φ eθ − sin φ eφ .

(5.50)

The goal for the moment is to determine the expansion coefficients ai∗ and bi∗ as defined by Ei =

∞  ∞ 

(bie,mn Me,mn + bio,mn Mo,mn + aie,mn Ne,mn + aio,mn No,mn ) . (5.51)

m=0 n=m

As a standard procedure, the coefficients can be found by applying dot product with one of the vectors on the rhs (e.g., Me,mn to find bie,mn ) and integrating it over solid angle. The orthogonality discussed earlier then helps single out each coefficient. It is easily derived from (5.47a), (5.48b), and (5.50) along with the orthogonality of sinusoidal functions that aio,mn = bie,mn = 0 for all m and n and that aie,mn and bio,mn do not vanish only when m = 1. Equation (5.51) is therefore reduced to Ei =

∞ 

( j)

( j)

(bio,1n Mo,1n + aie,1n Ne,1n ) ,

(5.52)

n=1

where the superscript “( j)” refers to the spherical Bessel function of the first kind because its second kind diverges to infinity at the origin and hence is physically irrelevant to plane waves. After lengthy calculations, the incident wave is found to be written down as

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95

Ei = E i,0

∞  n=1

in

2n + 1 ( j) ( j) (M − iNe,1n ) . n(n + 1) o,1n

(5.53)

The magnetic component of the incident wave field is obtained directly from (5.53) using a detemporalized version of (5.9), ∇ × E = i2π νμH ,

(5.54)

with the help of (5.38) and (5.40) to be Hi = −

5.1.4.5

∞  2n + 1 k ( j) ( j) E i,0 (Me,1n + iNo,1n ) . in 2π νμ n(n + 1) n=1

(5.55)

Scattered Wave Solutions

The scattered waves propagating away from the sphere may be expanded into an infinite series similar to (5.53) and (5.55). Es = −E i,0

∞  n=1

Hs =

in

2n + 1 (h) (h) (bn Mo,1n − ian Ne,1n ) n(n + 1)

∞  2n + 1 k (h) (h) E i,0 (an Me,1n in + ibn No,1n ) 2π νμ n(n + 1) n=1

(5.56a) (5.56b)

The superscript “(h)” denotes the spherical Hankel function of the first kind, which has a far-field asymptotic form representing an outgoing spherical wave and is thus relevant to the scattered waves. The spherical Hankel function of the second kind corresponds to an incoming spherical wave and is ruled out in the physical context. The electric and magnetic fields inside the spherical particle are given by E p = E i,0

∞  n=1

in

2n + 1 ( j) ( j) (cn Mo,1n − idn Ne,1n ) , n(n + 1)

∞  2n + 1 k˜ ( j) ( j) E i,0 (dn Me,1n + icn No,1n ) , in Hp = − 2π ν μ˜ n(n + 1) n=1

(5.57a) (5.57b)

where k˜ and μ˜ are the wavenumber and the permeability, respectively, inside the scattering sphere. The four coefficients of an , bn , cn , and dn are to be determined by the boundary condition.

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5 Principles of Electrodynamics and Geometrical Optics

(Ei + Es − E p )r =a · eθ = 0

(5.58a)

(Ei + Es − E p )r =a · eφ = 0 (Hi + Hs − H p )r =a · eθ = 0 (Hi + Hs − H p )r =a · eφ = 0

(5.58b) (5.58c) (5.58d)

These constraints claim that the external (incident and scattered) electromagnetic field must be continuous with the internal field at the surface of the sphere having a radius of a. Combining (5.53), (5.55)–(5.57) with orthogonality relationships into (5.58), one eventually finds an = and bn =

  ˜ ˜ nψ ˜ n (nx)ψ n (x) − ψn (x)ψn (nx) nψ ˜ n (nx)ξ ˜ n (x) − ξn (x)ψn (nx) ˜

(5.59)

 ˜ ˜ n (x)ψn (nx) ˜ ψn (nx)ψ n (x) − nψ ,   ψn (nx)ξ ˜ n (x) − nξ ˜ n (x)ψn (nx) ˜

(5.60)

where the prime designates derivative with respect to the argument in parentheses, ψn (ρ) = ρ jn (ρ), ξn (ρ) = ρh (1) n (ρ) are the Riccati-Bessel functions used in place of the spherical Bessel function of the first kind jn (ρ) and the spherical Hankel function of the first kind h (1) n (ρ), and n˜ =

k˜ . k

Since frequency, ν = ck/2π , should remain unchanged as an electromagnetic wave propagates into or out of the particle, n˜ is equivalent to the relative refractive index of the particle compared to that of the surrounding medium, according to (5.22). For satellite remote-sensing applications, the refractive index of the surrounding medium can be safely assumed to be that of vacuum, so that n˜ can be replaced in practice by the refractive index of the particle for itself. The permeability has been assumed to be identical inside and outside of the sphere. The other coefficients, cn and dn , are obtained as well but unnecessary for the purpose of describing the scattered wave properties. A key parameter characteristic of Mie’s solution is size parameter, x ≡ ka =

2πa 2π νa = , c λ

(5.61)

which is the size, or more precisely the periphery, of the scattering sphere normalized by the wavelength of electromagnetic waves. One of the most notable implications of (5.59) and (5.60) is that the particle-size dependence of Mie’s solution is described

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97

in a sense relative to wavelength. As will be seen later, the asymptotic behaviors of Mie’s solution in the small- or large-particle limit are derived depending on whether the particle is sufficiently tinier or greater than the electromagnetic wavelength.

5.1.4.6

Cross Sections

We are now ready to derive the absorption cross section and scattering cross section of photons by a spherical particle. The rate at which the total electromagnetic energy changes within a spherical domain having a radius of r , containing a scattering particle smaller than the domain (i.e., a < r ) at its center, is ascribed to the energy absorbed into the scattering particle (it is reminded that the surrounding medium is assumed to be vacuum). The rate of energy absorption Wa is evaluated from (5.34) as

1 Re[(Ei + Es ) × (Hi∗ + Hs∗ )] · er r 2 dΩ Wa (r ) = − 2 (5.62) = Wi (r ) + Ws (r ) + We (r ) , where the contributions of the incident and scattered waves and of extinction are

1 Wi (r ) ≡ − (5.63a) Re(Ei × Hi∗ ) · er r 2 dΩ , 2

1 Ws (r ) ≡ − (5.63b) Re(Es × Hs∗ ) · er r 2 dΩ , 2

1 We (r ) ≡ − (5.63c) Re(Ei × Hs∗ + Es × Hi∗ ) · er r 2 dΩ , 2 respectively. The first term (5.63a) is integrated out since the incident plane wave merely goes in and out of the domain, by design, without changing its amplitude. The second term (5.63b) represents the energy brought out of the domain by the scattered spherical waves given by (5.56) and is hence always negative. The last one (5.63c) is the cross term accounting for the interaction between the incident and scattered waves. The physical interpretation of We becomes clear when (5.62) is rearranged into (5.64) We = Wa − Ws = Wa + |Ws | , The rhs is the sum of the electromagnetic energy loss due to absorption and scattering, which is called the extinction of radiation. In spherical polar coordinates, (5.63) is written down as follows.

1 ∗ ∗ Re(E s,θ Hs,φ − E s,φ Hs,θ )r 2 dΩ (5.65a) 2

1 ∗ ∗ ∗ ∗ Re(E i,θ Hs,φ We (r ) = − − E i,φ Hs,θ + E s,θ Hi,φ − E s,φ Hi,θ )r 2 dΩ (5.65b) 2 Ws (r ) = −

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5 Principles of Electrodynamics and Geometrical Optics

Each component of the incident and scattered electromagnetic fields in (5.65) is obtained from (5.53), (5.55), (5.56), (5.59), and (5.60). Since all azimuthal modes but m = 1 are absent from these equations, (5.47) and (5.48) may be simplified with m = 1 as Me,1n = −z n (kr )πn (cos θ ) sin φ eθ − z n (kr )τn (cos θ ) cos φ eφ , Mo,1n = z n (kr )πn (cos θ ) cos φ eθ − z n (kr )τn (cos θ ) sin φ eφ ,

n(n + 1) z n (kr )πn (cos θ ) sin θ cos φ er kr 1 d + [r z n (kr )]τn (cos θ ) cos φ eθ kr dr 1 d [r z n (kr )]πn (cos θ ) sin φ eφ , − kr dr n(n + 1) z n (kr )πn (cos θ ) sin θ sin φ er = kr 1 d [r z n (kr )]τn (cos θ ) sin φ eθ + kr dr 1 d [r z n (kr )]πn (cos θ ) cos φ eφ , + kr dr

(5.66a) (5.66b)

Ne,1n =

No,1n

where πn ≡

(5.67a)

(5.67b)

Pn1 d Pn1 , τn ≡ . sin θ dθ

For the incident wave, z n (kr ) refers to jn (kr ) or ψn (kr )/kr in M( j) and N( j) . Equations (5.53), (5.55), (5.66), and (5.67) are combined into E i,θ

∞ cos φ  n 2n + 1 (ψn πn − iψn τn ) , = E i,0 i kr n=1 n(n + 1)

E i,φ = −E i,0 Hi,θ

∞ sin φ  n 2n + 1 (ψn τn − iψn πn ) , i kr n=1 n(n + 1)

(5.68a) (5.68b)

∞ sin φ  n 2n + 1 k E i,0 (ψn πn − iψn τn ) , = i 2π νμ kr n=1 n(n + 1)

(5.68c)

∞ cos φ  n 2n + 1 k E i,0 (ψn τn − iψn πn ) . i 2π νμ kr n=1 n(n + 1)

(5.68d)

Hi,φ =

(h) For the scattered waves, z n (kr ) is replaced by h (1) and N(h) . n or ξn (kr )/kr in M Equations (5.56), (5.66), and (5.67) together lead to

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99

E s,θ = −E i,0 E s,φ = E i,0

∞ cos φ  n 2n + 1 (bn ξn πn − ian ξn τn ) , i kr n=1 n(n + 1)

∞ sin φ  n 2n + 1 i (bn ξn τn − ian ξn πn ) , kr n=1 n(n + 1)

(5.69a) (5.69b)

Hs,θ = −

∞ E i,0 sin φ  n 2n + 1 (an ξn πn − ibn ξn τn ) , i 2π νμ r n=1 n(n + 1)

(5.69c)

Hs,φ = −

∞ E i,0 cos φ  n 2n + 1 (an ξn τn − ibn ξn πn ) . i 2π νμ r n=1 n(n + 1)

(5.69d)

Inserting (5.69) into (5.65a) and using orthogonality formulae for πn and τn , one obtains ∞ |E 0,i |2  Ws = (2n + 1)Re(−iξ ∗ ξ  )(|an |2 + |bn |2 ) . 2νk n=1 It can be shown that Re(−iξ ∗ ξ  ) is unity. The scattering cross section from Mie’s M is found to be solution σs,ν M = σs,ν

Ws 4π νμ = Ws , Wi k|E 0,i |2

where Wi is the energy flux carried by the incident plane wave as given by (5.33), is found to be ∞ 2π  M σs,ν = 2 (2n + 1)(|an |2 + |bn |2 ) . (5.70) k n=1 The extinction cross section is obtained similarly as M σe,ν =

∞ We 2π  = 2 (2n + 1)Re(an + bn ) . Wi k n=1

(5.71)

M M M M and σe,ν are defined per scattering particle, so σs,ν and σe,ν should be Note that σs,ν multiplied by the number of particles per unit mass to match σs,ν and σe,ν discussed in other chapters. We are now ready to calculate the scattering and extinction coefficients for the Mie problem, given an and bn from (5.59) and (5.60). The frequency dependence of the cross sections, although not explicit in the rhs of (5.70) and (5.71) except in k(= 2π ν/c), arises from complex refractive index and size parameter in (5.59) and M M − σs,ν . (5.60). The absorption cross section is derived immediately as σe,ν Figure 5.1 shows Mie’s solution of the cross sections in terms of the scattering/extinction/absorption efficiency,

100

5 Principles of Electrodynamics and Geometrical Optics (a) liquid water (visible) n=1.33+0i

ext/sca/abs efficiency

5

(b) liquid water (85 GHz) n=3.49+2.09i 5

Q ,e Q ,s Q ,a

4

4

3

3

2

2

1

1

0

0 0.1

1 10 x (size parameter)

100

0.1

1 10 x (size parameter)

(c) liquid water (visible) n=1.33+0i

(d) liquid water (85 GHz) n=3.49+2.09i

1

10

0

10

1

log ext/sca/abs efficiency

10

~x

0

10

-1

10

-1

10

~ x4

-2

-2

10

10

-3

10-3

-4

-4

10

10

10-5

100

~ x4

10 0.1

1

10

100

10-5

0.1

1

x (size parameter)

10

100

x (size parameter)

Fig. 5.1 a Mie’s solution of the scattering efficiency (Q ν,s ), the extinction efficiency (Q ν,e ), and the absorption efficiency (Q ν,a ) as a function of size parameter for the complex refractive index of 1.33 + 0i (the value for liquid water at visible wavelengths). b As a but for n = 3.49 + 2.09i (that for liquid water at a microwave frequency of 85 GHz). c and d As a and b, respectively, but on a double logarithmic scale

Q ν,s =

M σs,ν , πa 2

Q ν,e =

M σe,ν , πa 2

Q ν,a =

M σa,ν , πa 2

(5.72)

or the optical cross sections normalized each by the geometric cross section of the spherical particle. Two representative values of complex refractive index have been chosen: one for liquid water at visible wavelengths (n = 1.33 + 0i, left column) and the other for liquid water at a microwave frequency of 85 GHz (n = 3.49 + 2.09i, right column). Size parameter x can be interpreted either by particle radius or electromagnetic frequency (or the inverse of wavelength) but here let us think of x as a proxy of particle size under a given ν because otherwise a fixed refractive index would be a somewhat unrealistic assumption. The extinction efficiency is identical by design to the scattering efficiency with absorption being absent when the imaginary part of refractive index is zero. In Fig. 5.1a, Q ν,e remains almost zero for x 1 but rapidly increases with x to the maximum around x ∼ 6. A wavy structure apparent beyond the maximum of x arises from an interference between the incident and forward-scattered waves. This interference structure is heavily damped in the presence of absorption (Fig. 5.1b), leaving behind a single, broad peak around x ∼ 1.

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Overall, the behaviors in the two opposite limits (i.e., for particles very small or very large compared to the wavelength of the incident radiation) are quite contrasting against each other. Each limit is investigated next in some depth.

5.1.4.7

Geometrical Optical Limit and Rayleigh Limit

Regardless of n, Q ν,e gradually approaches 2 for x → ∞, where a scattering particle is huge relative to the wavelength. In this limit in x, or the geometrical optics limit, one may expect that only the physical size of the particle would matter for extinction, just as is the case for the silhouette of an object standing in the line of sight. The limiting value of 2, however, suggests that the “silhouette” should be twice as large as the particle. This counterintuitive result, often called the extinction paradox, occurs because an incident wave passing near the edge of the particle, even when not being physically blocked, could be diverted in direction and end up in a value of Q ν,e larger than expected from the principles of geometrical optics. To shed light on key aspects of the other limit (x 1), the same curves are redrawn in double logarithmic plots in Fig. 5.1c–d. For small values of x, a powerlaw dependence of the scattering efficiency (Q ν,s ) on x is apparent with a power-law exponent of 4. This is confirmed by expanding Q ν,s into a polynomial series in x, Q ν,s = or M σs,ν ≈

 2 8  n˜ 2 − 1  4 x + O(x 6 ) , 3  n˜ 2 + 2 

  2  n˜ − 1 2 a 6 4  (2π )5  2 for a λ . 3 n˜ + 2  λ4

(5.73)

It follows that the scattering cross section is proportional to the fourth power of 1/λ and to the sixth power of particle radius when the particle is substantially smaller than the wavelength of the incident wave. This asymptotic behavior holds unless n˜ varies with wavelength (or frequency) more rapidly than does with x. These are features specific to the Rayleigh scattering, and hence (5.73) is called the Rayleigh approximation. The Rayleigh scattering has broad applications beyond the Mie problem because the exact morphological structure of the scattering object (a homogeneous sphere in the present case) is no longer a dominant factor in determining σs,ν when the scatterer is much smaller than wavelength. For instance, the inverse fourth-power law in the Rayleigh regime offers a useful model for the molecular scattering of solar radiation in the earth’s atmosphere. The strong dependence on wavelength accounts for the blueness of daytime sky and the redness of the sun at dawn and dusk, because human eyes see blue (red) color at the shortest (longest) wavelengths of light in the visible spectrum. Cloud droplets, in contrast, are much larger than visible wavelengths and thus the cloud scattering of solar radiation does not have as strong a preference on wavelength as the Rayleigh scattering. This is why clouds are white.

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5 Principles of Electrodynamics and Geometrical Optics

The transition from the Rayleigh to geometrical optical limits is not monotonic. A maximum of Q ν,e occurs when x reaches an order of unity, accompanied by an oscillatory structure particularly when absorption is negligible (Fig. 5.1a). The downhill (large x) side of the maxima could cause a “bluing” effect instead of the reddening due to the Rayleigh scattering. A bluing is far less common than a reddish sunset to occur in the terrestrial atmosphere, while a blue sunset is often observed in the thin but dusty atmosphere of Mars. The absorption efficiency is expanded as n˜ 2 − 1 x + O(x 3 ) , = 4Im n˜ 2 + 2 

Q ν,a



or M ≈ 2(2π )2 Im σa,ν

n˜ 2 − 1 n˜ 2 + 2



a3 for a λ , λ

(5.74)

suggesting that the absorption cross section is inversely proportional to λ and proportional to the volume of the scattering medium (∝ a 3 ) for small particles. The extinction efficiency, or Q ν,s + Q ν,a , is largely dominated by Q ν,a for x 1 since Q ν,a more slowly diminishes to zero than Q ν,s for x → 0. The asymptotic proportionality of Q ν,e with respect to x in the Rayleigh limit is evident in Fig. 5.1d.

5.1.5 Approaches to Non-spherical Particle Scattering Mie’s solution is of great utility for calculating the optical properties of cloud and precipitation particles, but not without limitations. An obvious shortcoming is the assumption of sphericity. Solid hydrometeors such as ice crystals and snow flakes are highly non-spherical, so Mie’s solution is not quantitatively justifiable for frozen scatterers in the atmosphere. A variety of numerical methods have been proposed to deal with non-spherical particle scattering. Among the oldest is the T-matrix method, in which the scattered and incident electromagnetic fields are related to each other in terms of the series expansion coefficients (Waterman 1965, 1971). The T-matrix method is in essence a direct extension of Mie’s solution to non-spherical scatterers. Other widely used approaches include the discrete dipole approximation (DDA) (Purcell and Pennypacker 1973; Draine 1988) and the finite-difference time domain (FDTD) method (Yang and Liou 1996a). In the DDA, the scattering medium is assumed to be an ensemble of electric dipoles forced by the incident wave and all other dipoles in the medium. The FDTD is a scheme to numerically solve Maxwell’s equations on a discretized grid. All the approaches above, although applicable in theory to a scatterer of arbitrary size and shape, become increasingly computationally demanding as size parameter

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103

increases and are impractical for particles much larger than wavelength. A strategy useful for very large particles is a ray-tracing method under the assumptions of geometrical optics. Versions of the geometrical optical method have been developed in attempt to reconcile theoretical gaps between electrodynamics and geometrical optical principles (e.g., Yang and Liou 1996b). Yang et al. (2013) presents an effort to integrate different approaches for constructing a comprehensive dataset of the icecloud optical properties applicable to a broad spectral range from visible to infrared wavelengths. In the microwave range, frozen hydrometeors are never so large as to be in the geometrical optics regime. Liu (2008) conducted DDA simulations to construct a scattering database of ice particles of different crystal habits for microwave frequencies.

5.2 Geometrical Optics Geometrical optics offers a collection of equations relating the incident, transmitted, and reflected rays at an interface of two optically distinct media. Geometrical optics, predating electrodynamics initiated by James Clerk Maxwell, was built originally upon the empirical notion that light propagates in the form of waves. Now that visible light is confirmed by Maxwell’s theory to be a member of the electromagnetic wave family, geometrical optics may be viewed as a branch of electrodynamics rather than constitute its own discipline of physics. Nonetheless, some classic formulae from geometrical optics remain practical for many modern applications including satellite remote sensing. This section provides a brief introduction to some fundamental laws of geometrical optics. The present section is placed intentionally after electrodynamics against the chronological order because, as mentioned above, it is theoretically more straightforward to reintroduce geometrical optics from the perspectives of electrodynamics.

5.2.1 Angles of Reflection and Refraction First introduced are the most fundamental elements of geometrical optics: incident, reflected, and transmitted rays as depicted in Fig. 5.2. Let us consider two optically homogeneous media, having refractive indices of n 1 and n 2 , in contact with each other at the z = 0 plane. The plane of incidence is defined, without losing generality, to accord with the x-z plane. Let us ignore the distinction in polarization (left and right illustrations in Fig. 5.2) for the moment. Electric fields associated with each ray are

104

5 Principles of Electrodynamics and Geometrical Optics (a) Horizontal Polarization (S-Pol)

H H

Ei

ki

Hr i

z

EVi HVi

H

Incident

n1 n2

(b) Vertical Polarization (P-Pol)

z

H i

r

k

r H Reflected r

E

0 y

V

ki

Incident

x

n1 n2

Er i

r

E

H

Reflected

x

0 y

H

t H t

kr

V r

V

Ht

t V

Ht

kt

Et kt

Transmitted

Transmitted

Fig. 5.2 Schematic illustration of incident, reflected. and transmitted rays and the associated wavenumber vectors k and angles θ for a the horizontal (S) polarization and b the vertical (P) polarization. The x and z axes are defined as indicated, with the y axis being normal to the x-z plane. Refractive index is homogeneous in each domain above the x axis (n 1 ) and below (n 2 )

Incident : Reflected :

Ei exp(iki · x − i2π νt) , Er exp(ikr · x − i2π νt) ,

(5.75a) (5.75b)

Transmitted :

Et exp(ikt · x − i2π νt) ,

(5.75c)

where each wavenumber vector is as illustrated in Fig. 5.2. The phase continuity at the interface requires ki · x = kr · x = kt · x at z = 0 . Since k · x = k x x + k z z, the boundary conditions above immediately lead to ki,x = kr,x = kt,x at z = 0 .

(5.76)

The projection of each wavenumber vector onto the x axis is 2π ν n 1 sin θi , c 2π ν n 1 sin θr , = c 2π ν n 2 sin θt , = c

ki,x =

(5.77a)

kr,x

(5.77b)

kt,x

(5.77c)

5.2 Geometrical Optics

105

where k = 2π ν/c and (5.22) have been used. Here θi , θr , and θt are the angles of incidence, reflection, and of transmission (or of refraction), respectively. Two important consequences result from (5.76) and (5.77). The first two equations in (5.77) with ki,x = kr,x imply (5.78) θi = θr , that is, the angle of reflection is identical to that of incidence. The first and third equations with ki,x = kt,x result in n 1 sin θi = n 2 sin θt ,

(5.79)

which, known as Snell’s law, explains that an electromagnetic wave is refracted as propagating across optically distinct (n 1 = n 2 ) media. From the viewpoint of the classical geometrical optics, Snell’s law is explained in terms of the Huygens-Fresnel principle, where a wavefront, or the envelope of an ensemble of expanding wavelets, changes its direction as the wavelets first reaching beyond the interface propagate more slowly (or faster) than those yet to arrive. Snell’s law remains mathematically valid for a complex refractive index, although its physical interpretation is not as straightforward. Wavenumber must be also complex for a complex refractive index, where its imaginary part modulates the amplitude. As we have seen in (5.24), the amplitude of an electromagnetic wave decays over distance as it propagates when refractive index has a finite value in the imaginary part. In such cases, the meaning of “wavefront” becomes less clear in light of the Huygens-Fresnel principle because first-reaching and yet-to-arrive wavelets would be no longer uniform in amplitude even where identical in phase.

5.2.2 Amplitudes of Reflected and Transmitted Rays The orientation of E for a plane electromagnetic wave must be perpendicular to the direction of propagation, while otherwise yet to be determined. The remaining degree of freedom for choosing the orientation of E, e.g., either y or z plane if the wave proceeds in the x direction, is called polarization.1 Polarization does not matter in the relationship among θi , θr , and θt as discussed above but needs to be specified when it comes to the amplitude of waves. The case where E is normal to the plane of incidence is called S-polarization (Fig. 5.2a), and the other case with E confined within the plane of incidence is Ppolarization (Fig. 5.2b). In this book, S- and P-polarization are termed horizontal polarization and vertical polarization, respectively, in accordance with the convention of satellite remote sensing. For the horizontally polarized incident wave, (5.31) is written into

1 Do not confuse with electric/magnetic polarization or a collective effect of dipole moments (5.15).

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5 Principles of Electrodynamics and Geometrical Optics

HiH =

1 n 1 ki ki × EiH = × EiH . 2π νμ cμ |ki |

Using this equation, the x and y elements of EiH and HiH are each related to the complex amplitude E iH as H = E iH exp(iki · x) , E i,y n1 H = E iH Hi,x cos θi exp(iki · x) , cμ

(5.80a) (5.80b)

where the time dependence has been omitted. Similarly, the reflected and transmitted waves in horizontal polarization are H Er,y = ErH exp(ikr · x) , n1 H Hr,x cos θr exp(ikr · x), = −ErH cμ

(5.81a) (5.81b)

and H E t,y = E tH exp(ikt · x) , n2 H Ht,x cos θt exp(ikt · x), = E tH cμ

(5.82a) (5.82b)

respectively. The continuity of electric and magnetic fields at the interface requires that H H H H H H + Er,y = E t,y and Hi,x + Hr,x = Ht,x at z = 0 . E i,y The phase component of these constraints yields (5.78) and (5.79) as already discussed. The amplitude component results in E iH + ErH = E tH , n 1 (E iH − ErH ) cos θi = n 2 E tH cos θt , where θr has been eliminated using (5.78). The amplitudes of the reflected and transmitted waves relative to that of the incident wave are derived as n 1 cos θi − n 2 cos θt ErH = , H n 1 cos θi + n 2 cos θt Ei

(5.84a)

E tH 2n 1 cos θi = . n 1 cos θi + n 2 cos θt E iH

(5.84b)

For vertical polarization, (5.80)–(5.82) are replaced by

5.2 Geometrical Optics

107 V E i,x = E iV cos θi exp(iki · x) , n1 V exp(iki · x) , Hi,y = −E iV cμ V Er,x = −ErV cos θr exp(ikr · x) , n1 V Hr,y = −ErV exp(ikr · x) , cμ

V E t,x = E tV cos θt exp(ikt · x) , n2 V exp(ikt · x) . = −E tV Ht,y cμ

(5.85a) (5.85b) (5.86a) (5.86b)

(5.87a) (5.87b)

It follows that (E iV − ErV ) cos θi = E tV cos θt , n 1 (E iV + ErV ) = n 2 E tV , which lead to n 2 cos θi − n 1 cos θt ErV = , V n 2 cos θi + n 1 cos θt Ei

(5.89a)

E tV 2n 1 cos θi = . n 2 cos θi + n 1 cos θt E iV

(5.89b)

Equations (5.84) and (5.89) are together known as the Fresnel equations.

5.2.3 Reflectivity and Transmittivity The Fresnel equations are more practical if the complex amplitude is replaced by radiative flux for each of the incident, reflected, and transmitted waves. Radiance for the incident ray is obtained with (5.33) to be I (θ ) =

n1 1 |E i |2 δ(θ − θi ) = |E i |2 δ(θ − θi ) , 2c μ 2cμ

where δ is the Dirac delta function. Inserting this into (4.87), the z component of the incident radiative flux is found to be Fi =

π n1 cos θi |E i |2 , cμ

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5 Principles of Electrodynamics and Geometrical Optics

Similarly, radiative fluxes for the reflected and transmitted rays are π n1 cos θr |Er |2 , cμ π n2 Ft = cos θt |E t |2 , cμ

Fr =

respectively. Reflectivity is defined as the normalized magnitude of Fr relative to Fi individually for horizontal and vertical polarizations.    n 1 cos θi − n 2 cos θt 2 FrH |ErH |2   R ≡ H = = n 1 cos θi + n 2 cos θt  Fi |E iH |2    n 2 cos θi − n 1 cos θt 2 FrV |FrV |2 V   R ≡ V = V 2 = n cos θ + n cos θ  F |E | H

i

i

2

i

1

(5.91a) (5.91b)

t

Similarly, transmittivity refers to the relative magnitude of Ft . T

H

 2   2n 1 cos θi    n cos θ + n cos θ  (5.92a) 1 i 2 t  2  2n 1 cos θi n 2 cos θt |E tV |2 n 2 cos θt   (5.92b) = =  V 2 n 1 cos θi |E i | n 1 cos θi n 2 cos θi + n 1 cos θt 

FH n 2 cos θt |E tH |2 n 2 cos θt ≡ tH = = n 1 cos θi |E iH |2 n 1 cos θi Fi

TV ≡

FtV FiV

It is readily shown from (5.91) and (5.92) that RH + T H = 1 ,

RV + T V = 1 .

This is equivalent to the continuity of radiative flux or Fi − Fr = Ft at z = 0, which guarantees that a fictitious generation or loss of radiative energy does not occur at the interface (note that the negative sign of Fr above arises because Fr is upward positive while Fi and Ft are downward positive). Reflectivity and transmittivity can be calculated as a function solely of θi for given n 1 and n 2 once θt is eliminated with Snell’s law, 21  n2 cos θt = 1 − 12 sin2 θi . n2 Figure 5.3 presents R H and R V as a function of the angle of incidence for n 2 = 1.33 + i0 and n 2 = 2.0 + i0 with n 1 fixed at 1. The first case (Fig. 5.3a), intended for a visible light incident upon liquid water, shows that reflectivity stays low for angles of incidence smaller than, say, 60◦ but rapidly increases as θi approaches 90◦ . This matches our daily experiences: pure liquid water is nearly transparent when looked down upon from just above, while the setting Sun above the horizon leaves a clear trace of sunlight reflected on the ocean surface. If a higher value of n 2 is chosen

5.2 Geometrical Optics

109

Fresnel Reflectivity

(a) n2 = 1.33 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Horizontal Vertical

0

(b) n2 = 2.0

10 20 30 40 50 60 70 80 90

0

10 20 30 40 50 60 70 80 90

Angle of incidence [ o ]

Angle of incidence [ o ]

Fig. 5.3 Reflectivity from the Fresnel equations as a function of the angle of incidence for a n 2 = 1.33 + i0 and b n 2 = 2.0 + i0. For both cases, n 1 = 1

(Fig. 5.3b), reflectivity is overall enhanced without any qualitative change over the whole range of θi . Both polarizations yield an identical reflectivity at each end of θi , because the distinction between vertical and horizontal polarizations disappears at θi = 0◦ and an incident ray is obviously unable to be transmitted through the interface for θi = 90◦ regardless of polarizations. Otherwise, horizontal polarization has a consistently higher reflectivity than vertical polarization for any given θi . This fact is conveniently exploited by polarized lenses, designed to selectively cut off the horizontally polarized light for mitigating excessive reflections from surfaces without much disturbing unpolarized lights from elsewhere. A notable property is that R V falls down to zero at a certain angle of incidence before rising back again. This peculiar angle, known as Brewster’s angle, is given as n 2 cos θ B − n 1 cos θt = 0 from (5.91b). Combined with Snell’s law (5.79), this condition is satisfied when sin(2θi ) = sin(2θt ) . Given that 0◦ < θi , θt < 90◦ and θi = θt , the relation above is translated into θi (or θr ) = 90◦ − θt . It follows that the reflected ray is perpendicular to the transmitted ray when the angle of incidence is equal to Brewster’s angle. In this particular case, EtV turns out to be parallel to kr (or normal to ErV ) and hence the dipole oscillations induced by EtV are unable to generate any electric field within the reflected-wave plane. This is why reflectivity for vertical polarization vanishes at Brewster’s angle.

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5 Principles of Electrodynamics and Geometrical Optics

5.2.4 Fraunhofer Diffraction and Airy Pattern Diffraction is a phenomenon that waves traveling through a slit or around an obstacle are deviated in direction and reach outside the area directly behind the slit or inside the shade of the obstacle. As a result, diffraction makes an image “blurred” to some extent even if it is acquired by an ideal optical system (with an aberration-free lens, for instance). Given that remote sensing technologies utilize electromagnetic “waves”, any remote-sensing instrument has a theoretical limit in spatial resolution called diffraction limit. The goal of this subsection is to provide a concise derivation of the Airy pattern, which determines the spatial resolution and beam pattern of satellite instruments as the consequence of the diffraction limit (see Sect. 3.2.1). Let us consider a simple optical system constituted of a narrow aperture and a screen (Fig. 5.4). Electromagnetic waves are incident through the aperture from the source located so far that the incident radiation can be approximated by plane waves. The screen on which diffraction patterns appear is also assumed to be placed at a great distance from the aperture compared to the aperture size. As such, we consider a problem of far-field diffraction known as Fraunhofer diffraction. The Huygens-Fresnel principle claims that the optical field at a certain location should be expressed as the superposition of spherical wavelets from every point on

r

y ,y) x A(

u

x

R

Y

U X

Fig. 5.4 Coordinate systems at the aperture plane (left, with the gray disk indicating an aperture) and at the screen (right)

5.2 Geometrical Optics

111

an incoming wave front. This is translated into an equation to describe the electric field at a given point on the screen (X, Y ) as

E(X, Y ) = E 0

∞

A(x, y) −∞ ∞



E0 ≈ R

−∞

exp(−ikr ) dxdy r

A(x, y) exp(−ikr )dxdy

(5.93)

where x and y are the coordinates in the aperture plane, A(x, y) is an aperture function that is set to zero outside the aperture, k is the wavenumber of spherical wavelets, and the far-field assumption allows to approximate 1/r ≈ 1/R (see Fig. 5.4 for the definitions of r and R). Inside the exponential term, r is expanded around r = R to the first order for obtaining a diffraction pattern. By definition, r = [(X − x)2 + (Y − y)2 + Z 2 ]1/2 = [R 2 − 2(X x + Y y) + x 2 + y 2 ]1/2 ,

(5.94)

where R 2 ≡ X 2 + Y 2 + Z 2 . Again with the far-field assumption (|x|, |y| R), (5.94) is approximated by  Xx + Yy . r ≈ R 1− R2

(5.95)

Inserting (5.95) into (5.93), one finds E(X, Y ) =

E0 exp(−ik R) R



∞ −∞

  k A(x, y) exp i (X x + Y y) dxdy . R

(5.96)

It is noted that the electric field projected onto the screen is in essence the Fourier transform of the aperture function. For a circular aperture symmetric about the z axis, the two-dimensional Cartesian coordinates are transformed into polar coordinates as x = u cos φ,

y = u sin φ,

X = U cos Φ, Y = U sin Φ,

with which (5.96) is rewritten into E(U ) =

E0 exp(−ik R) R



2π 0



∞ 0

  k A(u) exp i uU cos(φ − Φ) ududφ . (5.97) R

Because of the axial symmetry, Φ may be set to zero without losing generality. Replacing the integral over φ by the Bessel function,

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5 Principles of Electrodynamics and Geometrical Optics

i −n Jn (z) = 2π



2π

exp[i(z cos φ + nφ)]dφ ,

0

with n = 0, one obtains E0 E(U ) = 2π exp(−ik R) R





a

J0 0

k uU udu . R

(5.98)

Here the aperture function has been eliminated using  A(x, y) =

1 for u ≤ a , 0 for u > a

where a is the aperture radius. A property of the Bessel functions, d n z Jn (z) = z n Jn−1 (z) , dz

leads to

z J0 (z)dz = z J1 (z) , so that





a

J0 0

  k R kaU 2 J1 . uU udu = a R kU a R

The series expansion of the n = 1 Bessel function, J1 (z) =

1 z + O(z 3 ), 2

indicates that J1 (z)/z = 1/2 for z = 0. Combining these all, the magnitude of electromagnetic energy (or radiance), proportional to |E|2 (see Sect. 5.1.3.1), on the screen is derived from (5.98) as     kaU 2 R J1 PA (U ) = 2 kaU R     πθ D 2 λ J1 = 2 , πθ D λ

(5.99)

where PA (U ) ≡ |E(U )|2 /|E(0)|2 , θ = U/R is the diffraction angle, D = 2a is the aperture diameter, and λ = 2π/k is wavelength. Equation (5.99) illustrates a spatial pattern theoretically expected for the Fraunhofer diffraction of light incident through a circular aperture.

5.2 Geometrical Optics

113

Fig. 5.5 Airy disk and rings on the X Y plane (see Fig. 5.4). Each axis is scaled by θ D/λ

Airy Disk/Rings 3

1

2

D/

1

0.1

0

-1

0.01

-2

-3

0.001 -1

-2

-3

1

0

2

3

D/

Figure 5.5 shows the diffraction pattern given by (5.99) on the X Y plane, which is called the Airy pattern. Given that the smallest zero of J1 (z)/z is z ≈ 3.83, the radius of the central disk, or the Airy disk, is estimated from (5.99) as 

θd,A

λ ≈ 3.83 πD



 ≈ 1.22

λ D

.

(5.100)

Since a series of rings surrounding the Airy disk are substantially weaker in intensity than the Airy disk (note that the colors are in logarithmic scale in Fig. 5.5), θd,A offers a practical measure of the diffraction limit. Actual optical systems are equipped with a lens or a parabolic reflector in place of an aperture so that a diffraction pattern is projected on the focal plane instead of the far-field screen. It can be shown that the Airy pattern is formed on the focal plane just as described above. Equation (5.99) therefore serves as a useful mathematical model of beam pattern for satellite measurements (see Sect. 3.2.1).

References Bohren CF, Huffman DR (1998) Absorption and scattering of light by small particles. WileyInternational Publication Draine BT (1988) The discrete-dipole approximation and its application to interstellar graphite grain. Astrophys J 333:848–872 Liu G (2008) A database of microwave single-scattering properties for nonspherical ice particles. Bull Am Meteor Soc 89:1563–1570. https://doi.org/10.1175/2008BAMS2486.1

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Purcell EM, Pennypacker CR (1973) Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 186:705–714 Waterman P (1965) Matrix formulation of electromagnetic scattering. Proc IEEE 53:805–812. https://doi.org/10.1109/PROC.1965.4058 Waterman PC (1971) Symmetry, unitarity, and geometry in electromagnetic scattering. Phys Rev D 3:825–839. https://doi.org/10.1103/PhysRevD.3.825 Yang P, Liou KN (1996) Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space. J Opt Soc Am A 13:2072–2085. https://doi.org/10.1364/ JOSAA.13.002072 Yang P, Liou KN (1996) Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals. Appl Opt 35:6568–6584. https://doi.org/10.1364/AO.35.006568 Yang P, Bi L, Baum BA, Liou KN, Kattawar GW, Mishchenko MI, Cole B (2013) Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm. J Atmos Sci 70:330–347. https://doi.org/10.1175/JAS-D-12-039.1

Chapter 6

General Theory of Radiative Processes

This chapter focuses on the microscopic physical processes responsible for the emission and absorption of radiation. The opening section describes some basic properties of the Planck function and related physical principles, followed by detailed discussions of the physical background behind the atmospheric spectrum.

6.1 Absorption and Emission of Radiation 6.1.1 Planck Function In this section, we discuss basic properties of radiation in thermal equilibrium known as blackbody radiation. It may seem self-contradictory to call an object “black” when it somehow emits radiation. Human eyes exploit reflected solar radiation to estimate the brightness and colors of surrounding objects while, with no sensitivity to infrared radiation, incapable of detecting thermal emissions from the objects. We therefore tend to think intuitively of a black target as radiatively silent, but we will see in this section that any object absorbing radiation has the ability to emit thermal radiation regardless of colors. The explicit form of the blackbody spectrum Bν (T ) was already derived in Sect. 4.1.5 as the energy spectrum for the Bose-Einstein distribution with a zero chemical potential. 1 2 hν 3   (6.1) Bν (T ) = 2 hν c −1 exp kB T The blackbody spectrum, also named as the Planck function after German physicist Max Planck, has the physical dimensions same as radiance (W m−2 sr−1 Hz−1 ): the © Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_6

115

116

6 General Theory of Radiative Processes a) Blackbody Spectrum B

b) Blackbody Spectrum B

-6

10

8

-8

10

10-10

10

10-12 10-14 -16

10

-18

10

-20

10

100 -4

10

10-8 10-12 -16

-22

10

10

-24

10

T = 2.7 K T = 288 K T = 5777 K

4

B [W/m2/str/ m]

B [W/m2/str/Hz]

10

-20

8

10

10

10

12

10

14

10

16

10

10

-2

10

[Hz]

0

10

2

10 [ m]

4

10

6

10

Fig. 6.1 a Blackbody spectrum Bν (T ) [W m−2 sr−1 Hz−1 ] as a function of frequency ν [Hz] and b Blackbody spectrum Bλ (T ) [W m−2 sr−1 µm−1 ] as a function of wavelength λ [µm] for temperatures of 2.7, 288, and 5777 K

transport of energy in a beam of radiation passing through unit area during unit time within unit solid angle for unit frequency (see Sect. 4.2.3). The blackbody spectrum can be alternatively expressed as a function of wavelength λ in place of frequency ν. Given ν = c/λ, Bν (T ) is converted through the equality (6.2) Bλ (T )dλ = −Bν (T )dν , where the negative sign reflect that λ increases as ν decreases, into Bλ (T ) = −Bν (T )

2 hc2 dν = dλ λ5

1  . hc −1 exp λk B T 

(6.3)

As expected from (6.2), Bλ (T ) has a slightly different physical unit of W m−2 sr−1 µm−1 , where “per frequency” (Hz−1 ) for Bν (T ) has been replaced by “per wavelength” (µm−1 ). The Planck function is plotted as a function of frequency (left) and of wavelength (right) in Fig. 6.1. Three different temperatures are given for illustrative purposes: 2.7 K as the cosmic background radiation, 288 K as a typical earth’s surface temperature, and 5777 K as the solar effective temperature. The Planck function has several notable properties including the Wien displacement law, Rayleigh-Jeans law, Wien law, and the Stefan-Boltzmann law as described below.

6.1.1.1

Wien Displacement Law

Figure 6.1 shows that the Planck function reaches a maximum at a certain frequency that shifts higher as temperature increases. The frequency yielding the spectral peak νm is obtained as the solution of

6.1 Absorption and Emission of Radiation

117

 ∂ Bν (T )  =0, ∂ν ν=νm which is equivalent, by differentiating (6.1) and precluding the trivial solutions of x = 0 and x = ∞, to 3(1 − e−x ) − x = 0 , (6.4) where x ≡ hνm /k B T . The numerical solution of (6.4) is x ≈ 2.82, leading to the relation known as the Wien displacement law, νm = 5.88 × 1010 Hz K−1 . T

(6.5)

German physicist Wilhelm Wien first discussed in 1893 the proportionality of νm to T on the basis not of the precise form of the blackbody spectrum (because the Planck function had not been known yet at that time) but of a generic thermodynamic argument. The Wien displacement law in the wavelength form turns out from (6.3) to be (6.6) λm T = 2.90 × 103 µm K , where λm is the wavelength at which Bλ (T ) reaches its peak. Note that νm λm = c because Bν (T ) and Bλ (T ) are not mathematically identical functions owing to the non-linearity between λ and ν. The Wien displacement law is a convenient formula to estimate, for a given temperature, a typical value of frequency or wavelength around which radiation carries a major portion of thermal energy. According to (6.6), λm is 10 µm for the earth’s thermal emission (T = 288 K) and 0.50 µm for solar radiation (T = 5777 K). These wavelengths both happen to lie in a relatively transparent spectral domain of the terrestrial atmosphere, allowing satellite remote sensors to probe the earth’s surface and lower troposphere by fully exploiting the local thermal emissions and reflected solar radiation.

6.1.1.2

Rayleigh-Jeans Law

The asymptotic form of the blackbody spectrum in the low-frequency limit (hν  k B T ) is obtained by the first-order expansion of the exponential term in (6.1). Bν (T ) ≈

2ν 2 kB T c2

(6.7)

This formula is known as the Rayleigh-Jeans law. Blackbody radiation is proportional to temperature for any given frequency in the Rayleigh-Jeans limit. The RayleighJeans law per unit wavelength is obtained from (6.3) as

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6 General Theory of Radiative Processes

Bλ (T ) ≈

2c kB T . λ4

(6.8)

The Rayleigh-Jeans law holds for the ascending branch in Bν (T ) [the descending branch in Bλ (T )] away from the spectral peak in Fig. 6.1. For T = 288 K, the Rayleigh-Jeans law safely applies for ν < 1012 Hz or 1000 GHz, so that microwave radiation thermally emitted by the earth is well approximated by (6.7). This fact is conveniently exploited by the microwave radiometry of the earth’s atmosphere. The Rayleigh-Jeans law does not involve the Planck constant h, or a “trademark” of quantum physics. Equation (6.8) can be derived from a classical mechanical argument based on the equipartition theorem, where each electromagnetic oscillator is assigned an equal amount of thermal energy, k B T . Integrating (6.7) over all frequencies in attempt to calculate the total radiative energy, however, would diverge to infinity, which is an obviously unphysical outcome known as the ultraviolet catastrophe. The ultraviolet catastrophe had been bothering physicists until (in fact some years later than) Max Planck derived the Planck function with the quantum effects properly taken into account.

6.1.1.3

Wien Law

The high-frequency end of the Planck function, as opposed to the Rayleigh-Jeans limit, has its own asymptotic formula. Equations (6.1) and (6.3) are simplified into Bν (T ) ≈ and Bλ (T ) ≈

  2 hν 3 hν exp − c2 kB T

  2 hc2 hc , exp − λ5 λk B T

respectively, for hν  k B T . These formulae, originally proposed by Wilhelm Wien, are called the Wien law. The Wien law applies to the rapidly dropping segment of the Planck function on the other side of the spectral peak from the Rayleigh-Jeans branch in Fig. 6.1. This rapid decay on the higher frequency side explains in part why thermal emissions from the earth have no contribution to visible radiance while they remain the major source of radiation at microwave frequencies far away from thermal infrared bands. The Wien law had been known before Planck derived the blackbody spectrum but failed to account for measurements at low frequencies.

6.1.1.4

Stefan-Boltzmann Law

The total radiative flux emitted by a blackbody of temperature T may be obtained by integrating the Planck function over frequency,

6.1 Absorption and Emission of Radiation

119

(b) Absorption

(a) Spontaneous emission

n2

n2

A21

h

h

n1

(c) Stimulated emission n2 h

B12

h

B21

n1

n1

Fig. 6.2 Schematic of a two-level model for a spontaneous emission, b absorption, and c stimulated emission. The populations of atoms in the lower (ground) and upper (excited) states are denoted by n 1 and n 2 , respectively. The Einstein coefficients relevant to each process (A21 , B12 , and B21 ) are as indicated in each corresponding schematic



∞

FBB (T ) = π

 Bν (T )dν = π

0

∞

Bλ (T )dλ = σ T 4 ,

(6.9)

0

where the Stefan-Boltzmann constant σ is σ ≡

2π 5 k 4B = 5.67 × 10−8 W m−2 K−4 . 15c2 h 3

(6.10)

The factor of π in (6.9) arises from the solid-angle integral over a half sphere. Equation (6.9), known as the Stefan-Boltzmann law, states that the energy flux carried by blackbody radiation is proportional to the fourth power of temperature.

6.1.2 Einstein Coefficients and Kirchhoff’s Law The derivation of the Planck function in Sect. 4.1.5 did not invoke any microscopic characterizations of emission and absorption processes, relying instead on generic statistical mechanical arguments. In this section, light is cast on the Planck function from a different angle, with focus on the physical principles underlying the emission and absorption of radiation using a simple two-level model describing the microscopic interactions of photons with matter. The overall strategy proceeds as follows. First introduced are the Einstein coefficients, or the transition rates between two energy levels for three individual processes of spontaneous emission, absorption, and stimulated emission. It is proved next from a detailed balance consideration that the three Einstein coefficients are interconnected. The mutual relation of the Einstein coefficients is then shown to lead to the Kirchhoff law, a general rule relating the absorption and emission coefficients for thermal radiation.

120

6.1.2.1

6 General Theory of Radiative Processes

Einstein Coefficients

Let us first consider the spontaneous emission of radiation. Photons with the energy hν can be emanated from atoms at the expense of a loss of the same amount of atomic energy, released when the excited state spontaneously decays to the ground state (Fig. 6.2a). Spontaneous emission increases the number of ground-state atoms per unit volume, denoted by n 1 , at the rate of 

dn 1 dt

 = n 2 A21 ,

(6.11)

A21

where n 2 is the population of atoms in the excited state per unit volume and A21 is the transition rate per atom due to spontaneous emission. Considered next is the absorption of radiation, which decreases n 1 by exciting the atom with the energy received from an incoming photon (Fig. 6.2b): 

dn 1 dt

 = −n 1 B12 Jν ,

(6.12)

B12

where B12 is the transition rate due to absorption and Jν is the radiative energy density (or hν multiplied by the photon number density). A third process known as the stimulated (or induced) emission of radiation occurs when a photon arriving at the atom triggers, rather than being absorbed, the emission of another photon having exactly the same energy. The rate at which the population of ground-state atoms changes due to the stimulated emission may be written as 

dn 1 dt

 = n 2 B21 Jν .

(6.13)

B21

The stimulated emission may be physically less intuitive than the other two processes but is indispensable to explain the Planck function as will become clear later. The Einstein A-coefficient provides the microscopic physical basis of the emission coefficient jν that appears in (4.79): hν ρ jν = 4π



dn 1 dt

 = A21

hν n 2 A21 . 4π

(6.14)

Similarly, the absorption cross section σa,ν is determined by the B-coefficients as ρσa,ν Jν = −

hν 4π

and therefore ρσa,ν =



dn 1 dt



 + B12

dn 1 dt





hν (n 1 B12 − n 2 B21 ) . 4π

, B21

(6.15)

6.1 Absorption and Emission of Radiation

121

An intriguing implication of (6.15) is that the absorption coefficient can be negative when n 1 B12 < n 2 B21 , in which case the incident radiation would be amplified exponentially as it travels through the medium. This phenomenon establishes a physical principle critical to laser technologies.1 When the atomic population obeys the Boltzmann distribution, n 1 B12 is always larger than n 2 B21 and the absorption coefficient is guaranteed to be positive all the time as we will see later.

6.1.2.2

Detailed Balance Relations

A “detailed balance” refers to the state of equilibrium where transitions due to the three competing processes mentioned above statistically balance each other. 

dn 1 dt



 + A21

dn 1 dt



 + B12

dn 1 dt

 =0 B21

When this condition is met, (6.11)–(6.13) together lead to Jν =

n 2 A21 . n 1 B12 − n 2 B21

(6.16)

For the moment, let us assume that atoms are in thermodynamic equilibrium under a given temperature T , that is, the atomic population obeys the Boltzmann distribution:   n2 g2 hν , = exp − n1 g1 kB T

(6.17)

where hν is the energy gap between the two levels (see Fig. 6.2) and g1 and g2 denote the degeneracies of the ground and excited states, respectively. Combining (6.16) and (6.17), one obtains A21 Jν = B12



   g1 B12 hν −1 . exp g2 B21 kB T

(6.18)

It is not surprising that (6.18) has the same form as the Planck function (6.1) because radiation must, by design, obey the blackbody spectrum when under detailed balance

1

Laser stands for light amplification by stimulated emission of radiation.

122

6 General Theory of Radiative Processes

with the atoms in thermodynamic equilibrium. Comparing the coefficients in (6.1) and (6.18), one finds the detailed balance relations,

and

B12 g2 = B21 g1

(6.19)

A21 2 hν 3 = B21 c2

(6.20)

so that Jν = Bν (T ) =

2 hν 3 c2

1  . hν −1 exp kB T 

(6.21)

It is important to note that (6.19) and (6.20) hold whether or not thermodynamic equilibrium is achieved. Although thermodynamic equilibrium was assumed at the beginning of the derivation for ease of discussion, (6.19) and (6.20) per se are valid in more general contexts as well. The detailed balance relations, in theory, could be obtained from quantum physical principles alone without invoking the Boltzmann distribution.

6.1.2.3

Kirchhoff’s Law of Thermal Radiation

The ratio of jν to σa,ν , or the source function as defined in (4.85), is found from (6.14) and (6.15) to be jν n 2 A21 = , (6.22) Sν = σa,ν n 1 B12 − n 2 B21 which is rewritten with (6.17), (6.19), and (6.20) as Sν =

jν = Bν (T ) . σa,ν

(6.23)

Equation (6.23) is known as Kirchhoff’s law of thermal radiation, claiming that the ratio of the emission coefficient to the absorption coefficient accords with the Planck function. Kirchhoff’s law holds whenever the local thermodynamic equilibrium (LTE) holds, that is, the atomic population is governed by the Boltzmann distribution. Kirchhoff’s law is valid whenever the emitting object is in thermal equilibrium with itself (i.e., LTE), while the object needs not necessarily be in equilibrium with ambient radiation (or in radiative equilibrium). The LTE is achieved when the microscopic energy exchange through atomic collisions takes place so frequently that internal energy states are well thermalized independently of the radiative envi-

6.1 Absorption and Emission of Radiation

123

ronment. This fact is of practical significance because the earth’s lower atmosphere is generally not in pure radiative equilibrium. Radiative energy imbalance is compensated in the troposphere by non-radiative processes such as convective heating. The LTE assumption, on the other hand, is widely justified for the terrestrial environments except in the middle and upper atmospheres. The Kirchhoff’s law is therefore safely adopted for interpreting satellite measurements of tropospheric constituents including clouds and precipitation.

6.2 Gas Spectrum The mass of the earth’s atmosphere consists mostly of gaseous components. Algorithms to retrieve the cloud and precipitation properties from space require an effort to carefully evaluate the gaseous emissions and absorption of radiation for isolating out the cloud and precipitation contributions from observed radiance. This section is intended to offer a short course of the background theory behind the spectrum of atmospheric gases from microwave frequencies to infrared wavelengths.

6.2.1 Overview Figure 6.3 shows the spectral transmittance of the earth’s atmosphere for its gaseous components (i.e., in the absence of hydrometeors and aerosols) at sea level. The transmittance is defined by an exponentially decaying function with optical depth [see (4.99)] as    ps

Tν = exp(−τν ) = exp −

ρσe,ν d p

,

(6.24)

0

where ρ is the air density, σe,ν is the extinction cross section, and ps is the sea level pressure. This transmittance by construction represents the fraction of radiative intensity that is lost along a vertical path over the whole thickness of the atmosphere. The extinction due to atmospheric gases arises from molecular scattering, generally called Rayleigh scattering, and molecular absorption. The efficiency of Rayleigh scattering obeys the λ−4 law as we found in the Rayleigh limit of the Mie solutions (Sect. 5.1.4.7). This is seen as a smoothly declining shoulder from visible to ultraviolet wavelengths near the left end of Fig. 6.3. Incoming radiation at even shorter wavelengths is entirely absorbed by electronic transitions in oxygen and ozone molecules. The effect of Rayleigh scattering on the transmittance rapidly tapers off as wavelength increases, creating a spectral “window”, which is a common term for a nearly transparent spectral domain, over the entire range of visible wavelengths. The transmittance is highly variable in the near- and thermal-infrared ranges because vibrational molecular transitions, primarily owing to H2 O and CO2 , produces very strong absorption at certain wavelengths. There are nonetheless narrow,

124

6 General Theory of Radiative Processes

U

1x106

1 Transmittance T

V

IR IR al b i r m s ea er Vi N Th le

R

I ar

F

b

Su

Frequency [GHz] 10000 1000

100000

m

-m

e

av

w

ic

M

300

e

av

w ro

100

50 30 20

10

5

3

Molecular (Rayleigh) 0.8 scattering 0.6 0.4 0.2 0 0.1

0.2 0.3 0.5

O2, O3

Electronic

1

2

3

5

10

20

O3 H2O

H2O CO2

Vibrational

CO2

50 100 200 Wavelength [ m]

H2O

10000 (1 cm)

500 1000 (1 mm)

H2O

O2

100000

H2O

Rotational (H2O) / Fine-structure (O2)

Fig. 6.3 Transmittance of clear-sky atmosphere at sea level as a function of wavelength (µm, labeled at bottom) or of frequency (GHz, labeled at top). The atmosphere is highly opaque at spectral domains shaded entirely in gray. See Sect. 13.2 for the atmospheric model profiles assumed to compute the transmittance

sporadic spectral windows through which warm infrared radiation is allowed to escape from the surface to space. Of particular importance for satellite remote sensing is the one located around 10 µm. This wavelength, lying very close to the peak of the blackbody spectrum for a typical terrestrial temperature (see Sect. 6.1.1.1), provides useful information for measuring the temperature of the earth’s surface and atmospheric constituents. The earth’s atmosphere is extremely absorptive at far-infrared wavelengths (beyond 15 µm) due to H2 O rotational transitions. The inability to detect anything embedded in the fully opaque air from space makes the far-infrared range a vast wasteland for earth’s surface and lower-atmospheric remote sensing. The atmosphere regains transparency at microwave frequencies except within a limited number of H2 O rotational lines and O2 fine-structure lines. The 183 GHz water-vapor line and and 60 GHz molecular-oxygen lines, in particular, are of great utility for humidity and temperature soundings. Spectral windows in between are widely used for satellite measurements of clouds and precipitation as well as the surface properties (e.g., sea surface temperature). The specific wavelength or frequency for each spectral line is determined by the quantum-mechanical laws governing molecular and atomic energy states. Since these energy states are quantized, the resulting spectral lines are also discrete rather than continuous. Individual spectral lines, however, are by no means like a delta function but have a finite width for three reasons: collisional broadening, natural

6.2 Gas Spectrum

125

broadening, and Doppler broadening. The first two arise from the fact that atoms and molecules stay in a certain energy state only for a limited duration of time, owing either to atomic/molecular collisions (collisional broadening) or to the spontaneous emission of radiation (natural broadening). No wave can be strictly monochromatic but somewhat smeared when Fourier-transformed into frequency space unless the wave persists indefinitely without decaying or being truncated. The spectral line profile in this case is known to have a Lorentz profile, as predicted by the Lorentz oscillator model (Sect. 6.3.3). The Doppler broadening arises from the Doppler shift of emission/absorption signals due to thermal motions, having a Gaussian profile in accordance with the Maxwell-Boltzmann distribution (4.52). The collisional broadening is predominant in the troposphere and lower stratosphere in which air density is so high that molecular collisions occur frequently. The Doppler broadening takes its place in the extremely thin atmosphere with a relatively high temperature at higher altitudes. The quantum mechanical basis of major spectral lines inherent in the earth’s atmosphere is outlined briefly in the following subsections. A more profound description of the physical processes underlying microwave gaseous spectra is found in Townes and Schawlow (1975).

6.2.2 Microwave Water-Vapor Bands The electromagnetic spectrum of the terrestrial atmosphere at microwave frequencies is much “cleaner” than the infrared spectrum (Fig. 6.3). Only two species of gases, oxygen and water vapor, give rise to a limited number of microwave absorption lines. Figure 6.4 shows the absorption coefficient of oxygen and water vapor molecules for two different temperature and pressure conditions representing the bottom of the atmosphere (290 K and 1013 hPa) and the tropopause (220 K and 100 hPa). The absorption coefficient (km−1 ) is equivalent to ρσe,ν that appears in (6.24). Microwave water vapor lines arise from pure rotational bands. Water vapor molecules emit or absorb radiation when transitioning between different rotational energy states through a torque exerted on molecular rotation by an electromagnetic wave field. Such molecular interaction with radiation is enabled by the fact that a H2 O molecule has a finite electric dipole moment resulting from its asymmetric structure, while symmetric molecules have no permanent electric dipole moment and do not allow a pure rotational band to emerge. This largely explains why the atmospheric microwave spectrum is relatively deserted, given that major atmospheric gases except water vapor, such as N2 and CO2 , are all symmetric molecules and have no microwave band. A notable exception is oxygen molecules (O2 ) as discussed later in Sect. 6.2.3. The energy of molecular rotation may be written in the classical mechanical analogy as L 2y L2 L 2x + + z , 2Ix 2I y 2Iz

126

6 General Theory of Radiative Processes (a) T=290 K and p=1013 hPa 2

Absorption coefficient [km-1]

10

1

10

0

10

H2O 22 GHz

-1

10

-2

O2 60 GHz

10

-3

H2O 325 GHz

H2O 183 GHz O2 119 GHz

10

-4

10

0

20

40

60

80

100 120 140 160 180 200 220 240 260 280 300 320 340 Frequency [GHz] (b) T=220 K and p=100 hPa

2

Absorption coefficient [km-1]

10

O2 only O2 + H2O (RH=30%) O2 + H2O (RH=100%)

1

10

2

100

1.5 1

-1

10

0.5 0

10-2

50

60

70

10-3 -4

10

0

20

40

80

60

100 120 140 160 180 200 220 240 260 280 300 320 340 Frequency [GHz]

Fig. 6.4 The atmospheric gaseous absorption coefficient (ρσa,ν ) (km−1 ) at microwave frequencies (GHz) for a T = 290 K and p = 1013 hPa and b T = 220 K and p = 100 hPa. Solid, dashed, and dot-dashed lines show three different relative humidities of 0, 30, and 100%, respectively. A zoom-in plot of the 60 GHz O2 lines is shown in the inset (in linear scale)

where L i and Ii are the angular momentum and the moment of inertia, respectively, about each principal axis of inertia (x, y, and z). Following the quantum mechanical convention, the moments of inertia are more conveniently expressed in terms of the rotational constants A, B, and C, as A=

h2 , 8π 2 Ix

B=

h2 h2 , C= , 2 8π I y 8π 2 Iz

(6.25)

where L i has been replaced by h/2π . The rotational constants as defined in (6.25) have the dimensions of frequency (Hz). The rotational constants can be alternatively defined in a slightly different form as A=

h , 8π 2 cIx

B=

h h , C= , 2 8π cI y 8π 2 cIz

6.2 Gas Spectrum

127

so that A, B, and C have the dimensions of wavenumber (typically in cm−1 ). In this book, we consistently use (6.25). From now on, the principal axes of inertia are labelled so that Ix , I y , and Iz are in the increasing order (i.e., A  B  C). The three rotational constants are generally all different as is the case for water vapor molecules, while two or all of them can be identical when the molecular structure holds some symmetry. The degree of asymmetry may be measured by Ray’s asymmetry parameter: κ=

2B − A − C , A−C

(6.26)

which ranges from −1 for A > B = C (a prolate symmetric top) to 1 for A = B > C (an oblate symmetric top). The simplest model of a rotating molecule is a spherical top ( A = B = C). The angular part of the Schrödinger equation in the absence of potential is, − BΛ2 ψrot (θ, φ) = E rot ψrot (θ, φ) ,

(6.27)

where ψrot (θ, φ) is the angular part of wave function and Legendrian Λ2 denotes the angular components of Laplacian, 1 ∂ Λ = sin θ ∂θ 2

  ∂ 1 ∂2 sin θ + . ∂θ sin2 θ ∂ 2 φ

(6.28)

Equation (6.27) yields a series of energy eigenvalues E rot , E rot = B J (J + 1) .

(6.29)

Here J (= 0, 1, 2, . . .) is the quantum number representing the total angular momentum. For a spherical top, a second quantum number K (= 0, ±1, ±2, . . ., and ±J ) is all degenerate, so that E rot is uniquely determined for a given J regardless of K . The degeneracy is lifted partially for symmetric (but non-spherical) tops. A prolate symmetric-top molecule has rotational energy states given by E rot = B J (J + 1) + (A − B)K 2 ,

(6.30)

where K indexes the angular momentum projected onto the axis of symmetry (the x-axis for a prolate top). Equation (6.30) is replaced for an oblate molecule by E rot = B J (J + 1) + (C − B)K 2 .

(6.31)

In each case, E rot is split into J + 1 different states (|K | = 0, 1, 2, . . . , J ) for each J . The rotational energy increases as |K | increases for a prolate symmetric top molecule, while it decreases for an oblate molecule under a fixed J .

128

6 General Theory of Radiative Processes

Fig. 6.5 Schematic of rotational energy levels for an asymmetric top molecule as a function of Ray’s asymmetry parameter, κ

Molecules having three rotational constants all different from one another (e.g., H2 O) are called asymmetric tops. The rotational energy is known to be expressed for asymmetric top molecules as E rot =

A−C A+C J (J + 1) + E K −1 K 1 (κ) , 2 2

(6.32)

where κ is as defined by (6.26). The non-dimensional parameter E K −1 K 1 , introduced to characterize the dependence of E rot on κ, is as schematically delineated in Fig. 6.5. On the left ordinate (κ = −1), E rot is as expected for a prolate symmetric top (6.30) while E rot obeys the solution for an oblate top (6.31) on the right ordinate (κ = 1). For any value of κ in between, each rotational state is specified as JK −1 K 1 , in which 2J + 1 separable states are described in terms of a pair of K −1 and K 1 (= 0, 1, 2, . . . , J ) for a given number of J . Figure 6.5 illustrates, for instance, that the lowest possible state, E rot = C J (J + 1), is J0 J for κ = −1 and the highest level, E rot = A J (J + 1), is J J 0 for κ = 1. The first term in (6.32) accords with the mean of these lowest and

6.2 Gas Spectrum

129

Rotational energy levels for para H2O 20

18

18

542

624

717

Rotational energy levels for ortho H2O 20

541 440

524 515

431

325 GHz

422

413 404

7

6

5

4 K1

3

2

14

10

220

211 111 000

8

14

331

183 GHz

202

16

12

322 313

16

1

0

Erot /h [THz]

533 606

707

625

615 441 532 523 432

12 10 8

6

6

4

4

2

2

0

0

616

505

423

330

8

22 GHz

514

414

321 312 303

221 212 110 0

101 1

2

3

4 K1

5

6

7

8

Fig. 6.6 Rotational energy levels (E rot / h [103 GHz]) of para H2 O (left) and ortho H2 O (right). Individual states are denoted by JK −1 K 1 and a group of those having the same J are indicated by a gray oval. The microwave water vapor bands at 22 and 183 GHz arise from transitions 616 ← 523 and 313 ← 220 , respectively

highest energy states, a deviation from which is described by the second term. The quantity E K −1 K 1 satisfies a permutation relation, E K −1 K 1 (κ) = −E K 1 K −1 (−κ) , so that the diagram of Fig. 6.5 has point symmetry. The rotational energy levels are obtained for H2 O from (6.32) with the rotational constants of A = 835 GHz, B = 435 GHz, C = 279 GHz, and resultantly κ = −0.44. The calculated rotational energy states are plotted in Fig. 6.6. The selection rule, which is a quantum mechanical constraint to allow radiative transitions to occur, is J = 0 and ±1. Any transition is hence forbidden beyond directly neighboring ovals drawn in Fig. 6.6. An additional restriction applies to the parity of wave function. When the axis of symmetry (the axis parallel to the permanent dipole moment) is associated with the intermediate rotational constant B as is the case for H2 O, radiative transitions are permitted between two K −1 K 1 states of the same parity: a transition within (K −1 ,K 1 ) = (even, even) and (odd, odd) or within (even, odd) and (odd, even). A water vapor molecule in the former state is called

130

6 General Theory of Radiative Processes

para H2 O while the latter is ortho H2 O. Figure 6.6 illustrates para and ortho H2 Os separately, because transitions are allowed only within each section. Figure 6.6 shows that although most rotational transitions of H2 O lie in the teraHertz range (submillimeter and far-infrared bands), there are two notable transitions that fall in the microwave frequency range: the 22.235 GHz line arising from the 616 ← 523 transition and the 183.31 GHz line from 313 ← 220 . These two water vapor lines are indeed prominent in the microwave spectrum of the earth’s atmosphere (Fig. 6.4), and are widely utilized for water vapor retrieval by microwave radiometry and sounding. All other radiatively allowed transitions, including a prominent 325.15 GHz line of 515 ← 422 (Fig. 6.4), account for a thick forest of water-vapor absorption lines from sub-millimeter to far-infrared spectral ranges.

6.2.3 Microwave Oxygen Bands Molecular oxygen is a homonuclear diatomic molecule and, because of the symmetry, does not have a permanent electric dipole moment. Oxygen molecules instead have a magnetic dipole moment arising from unpaired electron spins in the ground state. Rotational magnetic-dipole transitions of molecular oxygen, however, fall in a submillimeter range as will be seen later, and hence a microwave spectrum lacks rotational O2 bands. Molecular oxygen instead possesses fine structure lines at microwave frequencies, resulting from the split of each rotational state into sublevels due to the reorientation of electron spins. Two unpaired electrons together give rise to a spin triplet in an oxygen molecule, allowing three angular momentum states of J = N and N ± 1 where N represents the orbital angular momentum including molecular rotation. The spin angular momentum interacts with the other angular momenta associated with the molecular rotation and electron orbit in different ways. Of particular importance for microwave spectroscopy are the electron-spin interaction due to molecular rotation and the spin-spin interaction. The former refers to the interaction of spin magnetic moment with the magnetic field produced by molecular rotation through changes of the electronic charge distribution in the molecule, while the latter arises from the two parallel spins interacting with each other. These effects, called the fine structure effects, cause a split of each rotational energy state into three sublevels as mentioned above. Differences among the sublevels fall in a microwave frequency range. Note that electrons are all paired in the ground state for most other molecules such as N2 and their spin angular momenta are canceled out, leaving behind no molecular magnetic dipole moment. This is why nitrogen molecules, also lacking electric dipole moment, have no microwave spectral line despite the predominant abundance in the earth’s atmosphere. The fine-structure energy levels for J = N , N + 1, and N − 1 are

6.2 Gas Spectrum

131

E fs (N ) = B N (N + 1) , E fs (N + 1) = B N (N + 1) − hν+ (N ) , E fs (N − 1) = B N (N + 1) − hν− (N ) , where the fine structure separations for oxygen molecules, or ν± (N ) for the transitions from N to N ± 1, have been theoretically derived by Miller and Townes (1953) as γ ν+ (N ) = λ + − B(2N + 3) + 2   γ 2 1/2 γ

λ2 − 2λ B − + (2N + 3)2 B − , 2 2

(6.33)

and γ ν− (N ) = λ + + B(2N − 1) − 2   γ 2 1/2 γ

2 2 λ − 2λ B − + (2N − 1) B − , 2 2

(6.34)

except for N = 1 where (6.34) is replaced by ν− (1) = 2λ + γ .

(6.35)

Here γ and λ are a measure of the interaction of spins with molecular rotation and of the spin-spin interaction, respectively. Miller and Townes (1953) fit the parameters B, γ , and λ with experimental data to be B = 43.1016 − 1.4 × 10−4 N (N + 1) GHz , h γ = −0.25272 GHz , and

λ = 59.5016 + 5.75 × 10−5 N (N + 1) GHz ,

where the N (N + 1) terms arise from the centrifugal distortion of molecules. Transitions within each gray oval depicted in Fig. 6.7a are the fine structure transitions of molecular oxygen. Only odd values of N are allowed to exist for the following reasons. Oxygen molecule nuclei (16 O) have zero nuclear spin and obeys the Bose-Einstein statistics. A quantum mechanical law requires that the total wave function should be symmetric with respect to an interchange of identical boson nuclei. Individual wave functions to be considered here are nuclear spin, electronic, and rotational states. The nuclear spin state must be symmetric for zero-spin nuclei, while the ground electronic state of O2

132

6 General Theory of Radiative Processes (b) O2 fine structure separations

(a) O2 energy levels

60.3

17

715

834

600

62.5

58.4

N=3

400

425 369

487

200

Total orbital angular momentum (N )

774

1000 Frequency [GHz]

59.6

N=5

1200

800

N->N+1 N->N-1

19

1400

15 13 11 9 7 5 3

119

56.3

1

N=1

0 J=N-1

J=N

J=N+1

0

10 20 30 40 50 60 70 80 90 100 110 120 130 Frequency [GHz]

Fig. 6.7 a O2 fine structure transitions (solid) and rotational magnetic-dipole transitions (dashed) for N = 1–3. Transition frequencies are labeled in [GHz]. Three states sharing the same N is grouped together by a gray oval. b Frequencies of the fine structure lines

is anti-symmetric. It follows that the rotational wave function is anti-symmetric so as to meet the requirement that the overall wave function of O2 should be symmetric. Even values of N , associated with symmetric rotational states, thus do not exist for (16 O)2 . Dipole rotational transitions with N = ±2 are nominally forbidden but are made possible by a rotational energy splitting due to the spin-spin coupling. A rotational transition between neighboring odd states is hence allowed, although all beyond the microwave frequency range, as shown by dashed lines (for transitions that meet the selection rule J = 0, ±1) in Fig. 6.7a. It is fine-structure transitions (J = ±1 and N = 0) that account for microwave absorption lines of O2 . Figure 6.7b shows that all the fine-structure lines gather within a narrow range around 60 GHz except one at 119 GHz (N = 1 and J = 0 ← 1). In the lower troposphere, the O2 60 GHz band appears as if it were a single broad line as the result of the Lorentz broadening of different lines overlapped together (Fig. 6.4a). Microwave temperature sounders are designed typically to target this 60 GHz band. These overlapped lines are separated into individual lines at high altitudes where pressure broadening is less efficient, as evidenced by an inset plot in Fig. 6.4b. The 119 GHz band remains an isolated line regardless of pressure because it is raised by a single transition in the first place.

6.2 Gas Spectrum

133

6.2.4 Infrared Molecular Bands The nuclear distance of a molecule vibrates around an equilibrium state if energy sufficient to excite a certain vibrational state is provided through molecular collisions and/or the absorption of radiation. The simplest model of molecular vibration is a diatomic molecule bound in a spherically symmetric potential expanded around an equilibrium nuclear distance, re , as   2  dV  1 2 d V + (r − re ) + O(r 3 ) , V (r ) = V0 (re ) + (r − re ) dr r =re 2 dr 2 r =re

(6.36)

where the zeroth-order constant can be offset to zero without losing generality and the first-order coefficient also vanishes because an equilibrium is achieved (dV /dr = 0) at r = re . The potential in the vicinity of r = re is hence approximated to be V (r ) ≈

1 k(r − re )2 , 2

(6.37)

which is a harmonic oscillator potential with k being a constant analogous to the spring constant in classical mechanics. The radial part of the Schrödinger equation, complementary to its angular counterpart (6.27), is     B ∂ 1 2 ∂ 2 − 2 r + k(r − re ) ψvib (r ) = E vib ψvib (r ) , r ∂r ∂r 2

(6.38)

where ψvib (r ) is the radial part of wave function. The energy eigenvalues for molecular vibration are found to be      1 h 1 k = n+ , (6.39) E vib = hν0 n + 2 2π μ 2 where n = 0, 1, 2, . . . and μ is the reduced mass of the nuclei. The total energy of a rotational-vibrational state is   1 (6.40) E = E rot + E vib = B J (J + 1) + hν 0 n + 2 for a spherical top molecule (6.29). The selection rule is ΔJ = ±1 and Δn = ±1. While pure rotational spectral lines lie generally in the microwave frequency range as shown in the preceding sections, vibrational transitions involve much greater energy intervals and hence are found in infrared spectrum. A transition between neighboring numbers of n gives rise to an infrared band around a wavelength of c/ν 0 , comprised of a series of lines (i.e., rotational fine structure) attributed to transitions between J and J ± 1. Transitions accompanying an decrease in J as n increases by one (ΔJ = −1) are called P branches and those with ΔJ = +1 are R branches.

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6 General Theory of Radiative Processes

H2O Symmetric stretching ~2.73 m

Asymmetric stretching ~2.66 m

Symmetric stretching ~7.50 m (Infrared inactive)

Asymmetric stretching ~4.26 m

Bending ~6.27 m

CO2 Bending ~15.0 m

Fig. 6.8 Schematic picture of different vibration modes for H2 O (top) and CO2 (bottom)

The molecules especially salient in the infrared spectrum of the earth’s atmosphere are water vapor and carbon dioxide. It is more complicated than for a diatomic molecule to characterize the vibrational modes of polyatomic molecules such as H2 O and CO2 . Polyatomic molecules have multiple vibration modes each having a distinct value of k, or equivalently, of ν 0 . Figure 6.8 presents a schematic illustration of different vibration modes for H2 O and CO2 along with the wavelength of each transition (c/ν 0 ). Molecular vibration occurs, with the center of mass being stationary, in three distinct manners of symmetric stretching, asymmetric stretching, and bending (or scissoring). All the three vibrational modes appear in infrared spectrum (i.e., infrared active) for H2 O owing to its permanent electric dipole moment. A CO2 molecule, on the other hand, does not have any permanent dipole moment because of its symmetry, and rotational-vibrational transitions are allowed to interact with radiation only through a temporary dipole moment that occurs for the asymmetric stretch and bending modes. The symmetric stretch mode of CO2 does not induce a temporary dipole moment and is thus infrared inactive, but instead plays a key role in Raman scattering at visible wavelengths (Raman active). The infrared spectra of H2 O and CO2 are shown in Fig. 6.9, in which the individual vibrational modes presented in Fig. 6.8 appear as indicated. The symmetric and asymmetric stretching modes of H2 O are so close in wavelength to each other that they are not visually separable in the spectrum. Each mode accompanies rotational fine structure on both sides of the line center as seen in the zoom-in plots shown on the right (b

6.2 Gas Spectrum

135 (a) H2 O broadband

(b) H 2 O (~6. 3 m) 1

Sym./Asym. Stretch

Overtones/ Combination

-18

Bend

Pure rotation

0.8 0.6

2

10

cm /molecule]

-16

10

-20

-20

10

a [10

2 a [cm /molecule]

10-14

10-22

0.4 0.2

P branch R branch

10-24 0 2

1

3

5

7

10

15

20

30

5 5.5

Wavelength [ m]

6 6.5 7 7.5

8

Wavelength [ m]

(c) CO 2 broadband

(d) CO 2 (~4.3 m) 1 cm /molecule]

Asym. Stretch

Overtones/ Combination

10-16

Bend

-18

0.8

-20

-18

10

-22

10

P branch

0.6

R branch

2

10

a [10

2 a [cm /molecule]

10-14

0.4 0.2

-24

10

0 1

2

3

5

7

Wavelength [ m]

10

15

20

30

4.2

4.24 4.28 4.32

Wavelength [ m]

Fig. 6.9 b Broadband spectrum of the H2 O absorption cross section (cm2 ) per molecule for T = 296 K and p =1 atm. b Zoom-in plot of a selected vibration mode as indicated by dotted box in a. c–d As in a–b but for CO2 . The original data are provided by the HITRAN database (Rothman et al. 1987) via HITRAN on the Web (https://hitran.iao.ru/)

and d). A cluster of spectral lines on the lower-energy (larger wavelength) side correspond to the P branch, while those on the higher-energy (smaller wavelength) side belong to the R branch. The gradual slope extending to wavelengths larger than 10 µm for H2 O consist of pure rotational transitions (see Sect. 6.2.2). The other groups of near-infrared lines unaccounted for by the aforementioned vibrational modes are overtones and combination bands. Overtones arise because molecular vibration is not exactly harmonic as assumed in (6.38). A deviation in energy eigenvalues from the harmonic oscillator model expands as n becomes larger, and accordingly the selection rule is no longer limited to Δn = ±1. A combination band occurs when multiple fundamental vibrations are excited at the same time, giving rise to a spectral line at a certain combined frequency. The overtones and combination bands are considerably weaker in intensity compared to the individual fundamental modes (note that the ordinate is in logarithmic scale in Fig. 6.9a, c). The ability to absorb infrared radiation makes H2 and CO2 greenhouse gases. Water vapor is the most dominant greenhouse gas in the earth’s atmosphere, owing mainly to its 6.3-µm band and a sequence of pure rotational lines in far infrared. Carbon dioxide is efficient in absorbing the earth’s thermal emissions particularly at its 15-µm band. Another important greenhouse gas is ozone (O3 ), which is a triatomic molecule having a bent structure just like H2 O. Ozone exhibits a striking absorption

136

6 General Theory of Radiative Processes

feature due to the asymmetric stretching mode around 9.6 µm. The O3 bending line is located at 14.3 µm, which is masked out by the strong CO2 absorption mentioned above.

6.3 Condensate Spectrum 6.3.1 Overview Radiation interacts not only with gaseous H2 O but also with water in liquid or ice phase, enabling us to utilize remote sensing techniques for detecting clouds and precipitation. We have seen in Sects. 6.2.2 and 6.2.4 that an H2 O molecule has a permanent electric dipole moment, allowing efficient interactions of water vapor with electromagnetic waves. This principle applies also to water condensates, although liquid water and ice have its own mechanism to interact with radiation in a way different from water vapor. Intermolecular interactions, negligible for atmospheric gases, play key roles in the condensate spectra. The dielectric functions εν (5.27) of liquid water and ice are plotted in Fig. 6.10. The real part of εν stays nearly at a constant around 1.75–1.8 for liquid water and 1.7– 1.75 for ice across visible and UV wavelengths, which are translated by (5.28) into the well known values of 1.33 and 1.31, respectively, in refractive index. The imaginary part is extremely small at those wavelengths, indicative of the non-absorptive nature of liquid water and ice for visible and UV lights. The imaginary part of εν rapidly grows in magnitude as wavelength increases into the infrared range. The primary source of near-infrared absorption by water condensate is the molecular vibrational transitions discussed for the gaseous H2 O spectrum in Sect. 6.2.4. The behavior of εν becomes increasingly complex at longer wavelengths as hydrogen-bond (H-bond) vibrational modes and librational modes come into play. Both the real and imaginary parts of εν for liquid water rise dramatically in the microwave domain (note that the graph is scaled separately for microwave in Fig. 6.10 to accommodate this rapid increase). The orientational polarization of water molecules or so-called Debye relaxation, responsible for the microwave dielectric function of liquid water, explains why liquid water efficiently absorbs microwave radiation. We observe this effect on a daily basis when heating food in a microwave oven. In the following subsections, we discuss the individual processes underlying the condensate spectrum.

6.3 Condensate Spectrum

U

IR le r IR al b i m a s er Vi Ne Th

V

6

R

I ar

F

m

m

b-

Su

Frequency [GHz] 10000 1000

100000

1x10

5

137

e

av

w

300

e

av

w ro

ic

M 100

50 30 20

10

5

3 100 80

3

60

2

40

1

20 0 100000

0 0.1

0.2 0.3 0.5

1

6

Log Im( ) (300 m)

4

10

5

3

2

10

1

10

0

10

-1

10

-2

10

-3

Log Im( ) (>300 m)

Re( ) ( 300 µm) is on the right axis. The water temperature is assumed to be 0 ◦ C for microwave frequencies (the temperature dependence is otherwise minimal)

6.3.2 Microwave Properties of Water Condensate 6.3.2.1

Orientational Polarization and Debye Relaxation

Let us consider a viscous fluid comprised of molecules having a permanent electric dipole moment. This fluid would be polarized if exposed to a static electric field, Es , as the orientation of molecules align with the imposed field in the manner described by (5.15). (6.41) P = ε0 χs Es

138

6 General Theory of Radiative Processes

The subscript s refers to a static field. We next consider a case in which some finite time, or relaxation time τ R , is required for polarization to settle on an equilibrium state since fluid viscosity prevents molecular motion from responding instantaneously to an exerted electric field. Such a relaxation process may be modeled as P − ε0 χs Es dP =− , dt τR

(6.42)

   t , P = ε0 χs Es 1 − exp − τR

(6.43)

which yields the solution of

in case that polarization is absent at t = 0 and Es is constant over time. As readily expected, (6.43) eventually agrees with (6.41) but not until the e-folding time of τ R has elapsed. In another situation where the external field E imposed for a sufficiently long time prior to t = 0 is suddenly removed at t = 0, the governing equation and its solution for the subsequent evolution of P are

and

P dP =− dt τR

(6.44)

  t , P = ε0 χs Es exp − τR

(6.45)

respectively. Polarization exponentially decays with time in this latter case as the molecular orientation randomizes and loses the memory of the electric field initially imposed. When the imposed electric field varies over time as is the case for electromagnetic waves, the two contrasting behaviors above would repeat one after another. Such electric polarization that arises from changes in the molecular orientation is called orientational polarization in distinction to ionic and electronic polarizations. A radiation incident on the fluid is exemplified by making E sinusoidally vary with time, dP + P = ε0 χs E0 exp(−i2π νt) . τR (6.46) dt Assuming the solutions of P in the form of P0 exp(−i2π νt), the differential equation (6.46) is reduced to an algebraic equation, −iν τˆR P0 + P0 = ε0 χs E0 , which is satisfied by the solution P0 =

1 + iν τˆR ε0 χs E0 , 1 + ν 2 τˆR2

(6.47)

6.3 Condensate Spectrum

139

Fig. 6.11 The real (solid) and imaginary (dashed) parts of dielectric function for the orientational polarization as a function of ν τˆR . The electric susceptibility χ is assumed to be 80, a value close to the liquid water susceptibility at a room temperature

where τˆR ≡ 2π τ R . The real and imaginary parts of dielectric function (5.27) are found by equating the rhs of (6.47) and ε0 χ E0 to be χs 1 + ν 2 τˆR2

(6.48)

ν τˆR χs , 1 + ν 2 τˆR2

(6.49)

OP εν,r =1+

and OP = εν,i

respectively. Figure 6.11 shows (6.48) and (6.49) as a function of ν τˆR . When ν is much smaller OP OP approaches to 1 + χs and εν,i vanishes. In this limit, the than 1/τˆR (i.e., ν τˆR  1), εν,r incident electromagnetic wave oscillates so slowly relative to τˆR that an electric dipole is given a sufficient time to change its orientation and closely follows the electric field vector at any moment. In the other limiting case or ν τˆR  1, ν is so high that an electric dipole is no longer able to keep up with the quickly varying electric field, shutting off any interaction between radiation and polarized molecules. Resultantly, OP OP reduces to unity (or the dielectric constant of vacuum) while εν,i declines to zero εν,r OP again. Meanwhile, the imaginary part of εν becomes largest for an intermediate frequency around ν ≈ 1/τˆR . It is implied that the incident radiation decays most quickly as it travels through a cloud of polarized molecules when the relaxation time is comparable to the period of electromagnetic oscillation. This reflects the fact that radiative energy dissipates through the work done by the electric torque as it rotates polarized molecules against the fluid viscosity. The energy dissipation is inefficient for ν τˆR  1 because molecules are almost static with respect to electric orientation, and so is it for ν τˆR  1, in which case molecules are virtually immobilized against rotation. As a result, the radiative energy is most effectively transferred into the fluid when ν τˆR ∼ 1. The relaxation time τ R is determined by a dynamic balance between the incident radiation forcing molecules to align parallel to the electric field and the thermal

140

6 General Theory of Radiative Processes

motion acting to randomize the molecular orientation. The fluid viscosity counteracting the electric torque is of critical importance for the radiative energy dissipation into the fluid thermal energy. Dutch scientist Peter Debye first formulated this problem (Debye 1929), and the relaxation of the orientational polarization in a viscous fluid is called the Debye relaxation. A simple theory to determine τ R is illustrated next. The original formulation of the Debye relaxation process is given by Chap. 5 of Debye (1929). Viscous Torque We first estimate the viscous torque, T η , exerted on a rotating water molecule. The molecule is approximated by a solid sphere with the radius a, immersed in a homogeneous fluid having the viscosity η. The fluid motion around the sphere rotating around the polar (z) axis is assumed to be in the form of ⎞ ⎞ ⎛ 0 ur ⎟ ⎜ u = ⎝u θ ⎠ = ⎝ 0 ⎠ ,  uφ r φ sin θ ⎛

(6.50)

or in Cartesian coordinates ⎞ ⎛  ⎛ ⎞ ux −y φ ⎟ ⎜  ⎟ u = ⎝u y ⎠ = ⎜ ⎝ xφ ⎠ , uz 0 



where φ = φ(r ) is the fluid angular velocity dependent only on r ≡ Let us consider a scalar quantity defined as 

r

M(r ) =

(6.51)



x 2 + y2 + z2.



r  φdr  ,

0

with which (6.51) is rewritten into ∂M , ∂y ∂M uy = , ∂x uz = 0 . ux = −

(6.52a) (6.52b) (6.52c)

The Navier-Stokes equation is reduced to the Stokes equation, ∇ p = η∇ 2 u ,

(6.53)

6.3 Condensate Spectrum

141

for a steady flow with a small Reynolds number. If we assume that pressure gradient is negligibly small for the spatial scale of interest, (6.52c) and (6.53) are combined into   dM 1 d r2 = constant , ∇2 M = 2 r dr dr or equivalently,

1 d 3 (r φ) = constant . r 2 dr 





Under the boundary conditions φ(r = a) = φ 0 , where φ 0 is the angular velocity of 

the solid sphere, and φ(r → ∞) = 0, the solution is found to be 

φ=

a 3 r



φ0 .

Given the tangential viscous stress at the rotating sphere surface,

pφr

⎛ ⎞   ∂ uφ

∂φ ⎠ =η r = η sin θ ⎝r ∂r r ∂r r =a



= −3ηφ 0 sin θ ,

(6.54)

r =a

the net viscous torque exerted on the entire sphere surface is 

π

Tη =

pφr a sin θ · 2πa 2 sin θ dθ  π  3 sin3 θ dθ = −6π ηa φ 0 0

0 

= −8π ηa φ 0 . 3

(6.55)

Probability Distribution of Sphere Orientation Let f θ be the probability distribution function of θ , or the angle at which the molecule’s electric dipole moment is pointed with respect to the external electricfield vector (Fig. 6.12). The temporal evolution of f θ is considered to be determined by the thermally induced Brownian motion altering the sphere orientation and the sphere rotation brought about by the electric and viscous torques. These processes may be expressed by an equation as  ∂ fθ = κ∇ ˆ θ2 f θ − ∇θ · ( f θ a θ ) , ∂t

(6.56)

142

6 General Theory of Radiative Processes

Fig. 6.12 Schematic of a solid sphere representing an H2 O molecule, having an electric dipole moment μd and subject to an external electric field E

E0

Brownian motion

Electric torque

Viscous torque d

where the subscript θ of differential operators denotes their zenithal component. The first term on the rhs represents the Brownian motion taking the form of a diffusion equation, while the second term accounts for the rate of change in f θ ascribed to the 

sphere rotation having the angular velocity θ. Equation (6.56) is rewritten into    1 ∂ fθ TE ∂ fθ ∂ fθ = sin θ κˆ θ − , ∂t sin θ ∂θ ∂θ 8πa 3 η

(6.57) 

where κˆ θ = κ/a ˆ 2 . Equation (6.55) has been used for eliminating θ , interchangeable 

with φ 0 , assuming an equilibrium state in which the electric torque T E is balanced by Tη . The solution of f θ under a thermodynamic equilibrium, ∂ fθ =0, ∂t

(6.58)

should satisfy the Boltzmann distribution (4.49),   Uθ f θ ∝ exp − , kB T

(6.59)

where the potential energy Uθ equals the work done to rotate the sphere against the electric torque it feels, that is,

6.3 Condensate Spectrum

143

 Uθ = −

T E dθ .

(6.60)

Equations (6.59) and (6.60) together imply ∂ fθ dUθ f θ TE =− = fθ . ∂θ dθ k B T kB T

(6.61)

Combining (6.57), (6.58), and (6.61), one finds    κˆ θ 1 1 ∂ sin θ T E f θ − . 0= sin θ ∂θ kB T 8πa 3 η For this equation to hold regardless of θ , it is required that κˆ θ =

kB T . 8πa 3 η

(6.62)

Substituting (6.62) back in (6.57), one obtains 1 ∂ fθ 8πa η = ∂t sin θ 1 = sin θ 3

   ∂ ∂ fθ sin θ k B T − fθ TE ∂θ ∂θ    ∂ ∂ fθ sin θ k B T + f θ μd E sin θ , ∂θ ∂θ

(6.63)

where μd is the electric dipole moment of a water molecule. Relaxation Time Now we are ready to derive a formula to evaluate τ R . As in Sect. 6.3.2, considered here again is the case where the electric field initially imposed is suddenly removed at t = 0. Equation (6.63) for t > 0 is 8πa 3 η

  kB T ∂ ∂ fθ ∂ fθ = sin θ . ∂t sin θ ∂θ ∂θ

(6.64)

The solution f θ turns out to be in the form of   t , f θ = f 0 + f 1 cos θ exp − τR

(6.65)

where the decay time τ R is taken after (6.45). Substituting (6.65) into (6.64), one finds 4πa 3 η τR = . (6.66) kB T

144

6 General Theory of Radiative Processes

Given the liquid water viscosity for 20 ◦ C, η ∼ 1 × 10−3 kg m−1 s−2 and an atomicscale sphere radius a ∼ 1 Å= 10−10 m, (6.66) yields 3 × 10−12 s. This approximate estimate is in reasonable agreement with an experimentally derived value of 8 × 10−12 s (Bohren and Huffman 1998). The corresponding electromagnetic frequency, or the relaxation frequency, is 1/(2π τ R ) ∼ 20 GHz. Recalling that radiative energy is dissipated at a greatest rate for 2π ντ R ∼ 1, one can see that liquid water is an efficient absorber of microwave radiation.

6.3.2.2

Microwave Properties of Liquid/Ice Water

Figure 6.13 shows the dielectric functions of liquid water and ice at microwave frequencies. The liquid water dielectric function (Fig. 6.13a) is almost precisely as expected from the orientational polarization theory (Fig. 6.11), with a relaxation frequency around 20 GHz. Frozen water has a very different dielectric function from liquid water. The imaginary part of ice dielectric function is smaller by orders of magnitude than the liquid water counterpart over the whole microwave frequency range. This results from the immobility of H2 O molecules in solid water, largely disabling the Debye relaxation mechanism. The smallness of Im(εν ) of ice water implies that, in contrast to rain and cloud liquid water, solid hydrometeors hardly absorb microwave radiation. Ice water is dominated by liquid water also in Re(εν ) but the difference (liquid to ice ratio) is not as large as a factor of three once microwave frequency goes beyond ∼70 GHz. Although frozen hydrometeors are invisible in emission/absorption signals, microwave instruments have reasonable chance to detect snow and graupel particles by exploiting scattering signals. These basic properties of dielectric functions establish the fundamental principles of microwave remote sensing of clouds and precipitation. Further details in the application of the Debye relaxation to microwave dielectric properties is discussed in Chap. 9.5 of Bohren and Huffman (1998).

(a) Dielectric function of liquid water

(b) Dielectric function of ice water

80

0.08

8 Real (left) Imag (right)

40

20

0

0.06

4

0.04

2

0.02

)

6

0

0 1

100 10 Frequency [GHz]

Im(

60 Re( )

Re( ) and Im( )

Real Imag

1

100 10 Frequency [GHz]

Fig. 6.13 Microwave dielectric function as a function of frequency [GHz] for a liquid water at 20 ◦ C and b ice water at 0 ◦ C. The real and imaginary parts are drawn by solid and dashed lines, respectively. The imaginary part is labeled on the right axis in b

6.3 Condensate Spectrum

145

6.3.3 Infrared Properties of Water Condensate 6.3.3.1

Lorentz Oscillator Model

The Lorentz oscillator model, or simply the Lorentz model, is a simplistic but illustrative model offering a classical-mechanical analogue of the molecular vibration theory. The model is built upon the equation of motion for a damped harmonic oscillator driven by an external forcing mimicking an incident electromagnetic wave. m

dy d2 y + ky = eE0 exp(−i2π νt) . + mγ dt 2 dt

(6.67)

Here y, e, and m are the displacement, charge, and mass of the oscillator, respectively, γ represents the damping strength, and k is the spring constant as introduced for the molecular vibrational potential (6.37). An ensemble of oscillators induce a finite electric polarization, P = N ey, where N is the number of oscillators per unit volume. The equation governing P is derived from (6.67) as k dP d2 P + P = 4π 2 ν 2p ε0 E0 exp(−i2π νt) . +γ 2 dt dt m where νp ≡

1 2π



N e2 mε0

(6.68)

1/2 (6.69)

is called the plasma frequency. The solution to (6.68) is found to be P0 =

ν 2p (ν02 − ν 2 ) − iγˆ ν

ε0 E0 .

(6.70)

where P is assumed to be P0 exp(−i2π νt) and 1 ν0 ≡ 2π



k m

1/2 , γˆ ≡

γ . 2π

Comparing (6.70) with (5.15) and (5.27), one obtains dielectric function for the Lorentz model. LM εν,r = 1+ LM = εν,i

ν 2p (ν02 − ν 2 ) ν 2p (ν02 − ν 2 )2 + γˆ 2 ν 2 γˆ ν 2p ν

ν 2p (ν02 − ν 2 )2 + γˆ 2 ν 2

(6.71a) (6.71b)

146

6 General Theory of Radiative Processes Dielectric function (Lorentz model) 4.5 Real Imag

4 Re( ) and Im( )

Fig. 6.14 The real and imaginary parts of dielectric function from the Lorentz model as a function of ν/ν0 for γˆ = ν0 and ν p = 2ν0

3.5 3 2.5 2 1.5 1 0.5 0 0

0.5

1 / 0

1.5

2

LM LM The typical properties of εν,r and εν,i are as shown in Fig. 6.14. The imaginary part of dielectric function exhibits a bell-shaped absorption feature centered roughly at ν ∼ ν0 . This is an effect mathematically equivalent to the resonance of a sinusoidally driven harmonic oscillator. The particular shape of the absorption profile given by (6.71b) is called the Lorentz profile, providing a model of the spectral line shape for natural and pressure broadenings (Sect. 6.2.1).

6.3.3.2

Infrared Properties of Liquid/Ice Water

Figure 6.15 shows the infrared spectrum of the dielectric function for liquid water and ice. Despite its simplicity, the Lorentz model excellently explains the overall spectral structure of near-infrared variational modes (Fig. 6.15a). The stretching (∼2.7 µm) and bending (∼6.3 µm) modes of H2 O vibration (Fig. 6.8) are evident, although the central wavelengths are slightly shifted from those for gaseous phase. The qualitative characteristics of these vibration modes in εν,r and εν,i well resemble the theoretical prediction by (6.71b) as delineated in Fig. 6.14. The far-infrared spectrum, on the other hand, contains unique features absent from the gas spectrum, attributed to the libration and H-bond stretching modes (Fig. 6.15b). Libration refers to a partial rotational motion of molecules repeated back and forth around a certain orientation, driven under the strong influence of intermolecular forces preventing free molecular rotation. The librational absorption by liquid water has a broad mid- to far-infrared signature in Im(εν ) extending to wavelengths beyond ∼14 µm. Ice librational absorption, in contrast, is marked by a sharp, narrow peak of Im(εν ) at ∼12 µm, a feature instrumental for ice-cloud retrieval using split-window thermal infrared channels. The narrowness of the ice librational profile is due to the highly restricted mobility of H2 O molecules against rotation in solid water. Ice absorption peaks at ∼45 µm and ∼65 µm correspond to the intermolecular H-bond stretching, although these wavelengths are not of use for atmospheric remote sensing because of the heavy absorption by water vapor.

References

147 (a) Intra-molecule vibrational modes

(b) Inter-molecule vibrational/librational modes 5

2

Stretch

1

)

liq Real ice Real liq Imag ice Imag

Re( ) and Im(

Re( ) and Im( )

3

Bend

4

Libration

3 2 1

H-bond Stretch 0

0 2

3

4 5 6 Wavelength [ m]

7

8

10

20

30

40 50 60 70 Wavelength [ m]

80

90

100

Fig. 6.15 a The dielectric function of liquid water (thick) and ice (thin) as a function of wavelength in the near-infrared range. The real and imaginary parts are drawn in solid and dashed curves, respectively. b As a but in the far-infrared range

References Bohren CF, Huffman DR (1998) Absorption and scattering of light by small particles. WileyInternational Publication Debye P (1929) Polar molecules. Dover Publications Miller SL, Townes CH (1953) The microwave absorption spectrum of (O16 )2 and O16 O17 . Phys Rev 90:537–542 Rothman LS, Gamache RR, Goldman A, Brown LR, Toth RA, Pickett HM, Poynter RL, Flaud JM, Camy-Peyret C, Barbe A, Husson N, Rinsland CP, Smith MAH (1987) The HITRAN database: 1986 edition. Appl Opt 26:4058–4097. https://doi.org/10.1364/AO.26.004058 Townes CH, Schawlow AL (1975) Microwave spectroscopy. Dover Publications

Part III

Measurement Principles

Part III is constituted of key chapters of this book, delineating the physical and mathematical principles behind the satellite measurements of clouds and precipitation. The first three chapters are devoted to passive remote sensing utilizing thermal infrared radiation, solar radiation, and microwave radiation. Each chapter begins with the theory of radiative transfer at different levels of complexity, followed by simulated spectra to demonstrate the impacts of clouds and/or precipitation on satellite measurements. The measurement principles of active sensors are detailed in a separate chapter. The last chapter is dedicated to an overview of the mathematical basis of satellite retrieval algorithms.

Chapter 7

Infrared Sensing

Thermal infrared radiation is a convenient tool for cloud remote sensing. Infrared emissions from cloud tops are often considered to be a proxy of the temperature there (and hence of cloud top height) regardless of day or night. Infrared brightness temperature, however, is not always a reasonable substitute for the physical temperature of clouds and could be largely misinterpreted if analyzed without care. The ultimate goal of this chapter is to present the utility and limitations of the satellite infrared measurements of cloud properties such as cloud top temperature, particle size, and thermodynamic phase.

7.1 Radiative Transfer in Non-scattering Atmospheres 7.1.1 Infrared Properties of Cloud Particles This first section offers an introduction to radiative transfer problems in a nonscattering medium. The omission of the scattering term, achieved by a zero single scattering albedo (ων = 0), vastly simplifies the radiative transfer equation (4.98) and makes its solutions easy to interpret intuitively. Non-scattering radiative transfer problems serve as a robust theoretical basis of thermal infrared observations of clouds. Figure 7.1 shows the single scattering albedo (ων ) of liquid and ice clouds as a function of wavelength. The single scattering albedo stays almost precisely at unity for visible wavelengths while dropping to lower values for wavelengths longer than 1 µm. The complex spectrum of ων at infrared wavelengths arises from the absorption features due to the intra- and inter-molecular vibrational modes and librational modes of water condensates (Sect. 6.3.3.2; see also the imaginary part of the dielectric function for liquid and ice waters in Fig. 6.10). As evident in Fig. 7.1, ων for cloud water and ice in the infrared range remains well above zero, suggesting that a cloudy © Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_7

151

152

7 Infrared Sensing Single Scattering Albedo for Cloud Water/Ice Water Cloud Ice Cloud

1 0.8 0.6 0.4 0.2 0 0.2 0.3

0.5

1

2

10 5 3 Wavelength [ m]

20

50

100

Fig. 7.1 Cloud single scattering albedo at visible and infrared wavelengths for liquid cloud (solid) and ice cloud (dashed). See Sect. 13.2 for the microphysical model assumptions

atmosphere is only partially “non-scattering” even at infrared wavelengths. In effect, nevertheless, the assumption of ων = 0 provides a reasonable theoretical framework of infrared radiative transfer as will be confirmed in Sect. 7.2.2.

7.1.2 Mathematical Formulation and Solution The radiative transfer Eq. (4.98) is reduced to dIν = ρσa,ν (−Iν + Sν ) ds

(7.1)

for a non-scattering medium (ων = 0), where Iν , σa,ν , and Sν are radiance, the absorption cross section per unit mass, and the source function for a given electromagnetic frequency ν, respectively, with ds and ρ being a length element and the mass density of the medium. Note that the absorption cross section is equivalent to the extinction cross section σe,ν when scattering is absent. Radiance is a beam of energy carried by photons per unit area during unit time into unit solid angle within unit frequency, having the units of W m−2 sr−1 Hz−1 [see (4.11) and Fig. 4.1 for the definition of solid angle ]. The reasoning behind these notoriously complex units may be quickly summarized as follows. Radiance is proportional to the energy density of photons in phase space, (7.2) hν f γ d3 xd3 p = hν f γ (cdt d A)( p 2 d p d) ,

7.1 Radiative Transfer in Non-scattering Atmospheres Fig. 7.2 Schematic drawing of radiative transfer in a non-scattering medium. A volume element of the medium is indicated by a cylinder

153

Absorption: –I d Emission: S d I ( +d )

I( )

d

where f γ is the number density of photons per phase-space volume. The volume element d3 x is equal to ds d A = cdt d A (d A is an area element normal to the photon propagation direction). With the momentum of photons replaced by p = hν/c, (7.2) turns out to be (7.3) hν f γ d3 xd3 p = Iν dt d A dν d , where Iν = h 4 ν 3 f γ /c2 (see Sect. 4.2.3 for more details). Equation (7.3) explains the dimensions of radiance. When radiance is expressed per unit wavelength instead of per unit frequency, the units of radiance read W m−2 sr−1 µm −1 . A more practical form of (7.1) is dIν = −Iν + Sν , dτν

(7.4)

where optical depth τν is defined as dτν = ρσa,ν ds

(7.5)

for a scattering-free medium [see (4.101) for the general definition]. For applications to satellite remote sensing, ds is more conveniently replaced by dz/ cos θ , where z denotes the vertical axis in a plane-parallel atmosphere and θ is the satellite zenith angle. Figure 7.2 presents an intuitive interpretation of (7.4). Radiation incident on the medium Iν (τν ) is partially absorbed into a volume element by an amount of −Iν dτν , while an emission of Sν dτν from the volume element is added to the outgoing radiation Iν (τν + dτν ). The source function Sν is formally determined with the Einstein coefficients in the context of the detailed balance relations (see Sect. 6.1.2.2). In a very tenuous medium where collisional transitions do not overwhelm radiative transitions, Sν is dependent on the ambient radiation field and hence the radiative transfer equation needs to be solved iteratively. Under the assumption of local thermodynamic equilibrium (LTE), on the other hand, the source function depends solely on temperature as

154

7 Infrared Sensing (a) Schematic

(b) RT solutions for a non-scattering LTE atmosphere

I(

c,

2

) I / B (T)

1.5

c,

T( ) 0

1

0.5

I (0)=Is,

0 1

0

2

4

3

Cloud Optical Depth

5

c,

Fig. 7.3 a Schematic of the coordinate system for (7.7). b Solutions of the radiative transfer equation for a non-scattering cloud with a uniform temperature. Radiance normalized by the Planck function is plotted as a function of cloud optical depth for different boundary conditions from 0 to 2 by the interval of 0.25

Sν = Bν (T ) =

2 hν 3 c2

1  hν −1 exp kB T 

(7.6)

according to Kichhoff’s law (Sect. 6.1.2.3). Here Bν (T ) is the Planck function or the blackbody spectrum for a given temperature T (Sect. 6.1.1). The LTE approximation, well satisfied in the troposphere and lower stratosphere, is safely applicable to the remote sensing of clouds and precipitation. Throughout the rest of this book, Sν is replaced by Bν (T ). Let us consider a layer of non-scattering cloud having an optical depth of τc,ν , or cloud optical depth, and an in-cloud temperature profile given by T (τν ) (see Fig. 7.3a).1 Assuming that the atmosphere is transparent outside the cloud at the frequency of current interest, the formal solution of (7.4) is found to be 

τc,ν

Iν (τc,ν ) = Is,ν exp(−τc,ν ) +

Bν [T (τν )] exp[−(τc,ν − τν )]dτν .

(7.7)

0

The first term on the rhs arises from the surface emission Is,ν incident on cloud base decaying exponentially as it travels through the cloud. The second term is a collective effect of in-cloud thermal emission at τν (0 ≤ τν ≤ τc,ν ) and the attenuation over an optical distance of τc,ν − τν . The formal solution (7.7) arrives at the exact solution of Iν (τc,ν ) = Is,ν exp(−τc,ν ) + Bν (T )[1 − exp(−τc,ν )] 1

(7.8)

Cloud optical depth, sometimes abbreviated as COD or COT (cloud optical thickness), is defined at an infrared wavelength throughout this chapter. A discussion on the wavelength dependence of cloud optical depth will be given in Sect. 8.3.2.

7.1 Radiative Transfer in Non-scattering Atmospheres

155

if temperature is uniform throughout the cloud layer. Figure 7.3b shows Iν (τc,ν ) from (7.8) for different values of Is,ν . It is clear that Iν increases with τc,ν when initially smaller than Bν (T ) but decreases otherwise so that Iν → Bν (T ) for τ  1 in all cases. As such, radiance is equilibrated with local temperature in a non-scattering, optically thick cloud. The satellite-observed radiance is sensitive particularly to cloud top temperature for a sufficiently opaque cloud, in which a large portion of emissions from deep inside are absorbed before escaping out of the cloud. This is a key property of infrared radiative transfer as discussed later.

7.1.3 Absorption and Emission Lines Non-scattering radiative transfer offers an illustrative example for the formation of absorption and emission lines. Let us consider a hypothetical spectral profile of optical depth as a proxy of a molecular line around the central frequency of ν0 (Fig. 7.4a). This model is meant to emulate a line profile engraved in the absorption cross section as depicted in Fig. 6.4 since τν is proportional to σa,ν for a given airmass ρds [see (7.5)]. Two different spectra are simulated for contrasting values of the background blackbody temperature Tbg as

Background Radiation Tbg

Medium T Observer

(b) I for cold background

(a) Optical depth profile

(c) I for warm background

5

B (Tbg) 4 3

B (T)

B (T)

2 1

B (Tbg)

0 0 ( - 0)/

0

0 ( - 0)/

0

0 ( - 0)/

0

Fig. 7.4 a Model spectral profile of optical depth mimicking a molecular line. b Simulated spectrum for a cold background (Tbg < T ), resulting in an emission line. c Simulated spectrum for a warm background (Tbg > T ), yielding an absorption line. The spectral width is assumed to be so narrow that the Planck function does not substantially vary over the plotted range of ν

156

7 Infrared Sensing

Iν (τν ) = Bν (Tbg ) exp(−τν ) + Bν (T )[1 − exp(−τν )], which is just another application of (7.8). In the first case, the background radiation is assumed to be colder than the medium (i.e., Tbg < T ). This example yields an emission line as plotted in Fig. 7.4b. As expected from Fig. 7.3b, thermal emission from the relatively warm medium dominates the observed radiance near the line center (ν = ν0 ) where optical depth is much larger than unity, while cold radiation from the background directly arrives at the observer at line wings without being disturbed by the medium. The same reasoning applies in the opposite way to the formation of an absorption line in the other case of a warm background or Tbg > T (Fig. 7.4c). It is therefore a temperature gradient along the line of sight that generates a spectral line, and the sign of the gradient differentiates whether it is emission or absorption. Down-looking satellite sensors typically observe absorption lines rather than emission lines because air temperature decreases upward in the troposphere. By contrast, air temperature increases with height above the tropopause up to the stratopause, so stratospheric gases could produce emission lines in cases where warmer radiation from the lower troposphere is somehow masked out (an example for this will be given in Sect. 7.2.2). Limb sounders detect emission lines because they measure radiation from middle- and upper-atmospheric trace gases against a deep-space background.

7.1.4 Infrared Brightness Temperature Radiance Iν is translated into brightness temperature Tb by inverting the Planck function (6.1), 1 hν (7.9) Tb,ν = ,  kB 2hν 3 ln 1 + 2 c Iν or using an alternative definition per unit wavelength (6.3), Tb,λ =

hc λk B

1 .  2hc2 ln 1 + 5 λ Iλ

(7.10)

Brightness temperature is equivalent to the actual temperature of the emitting medium if the medium is a blackbody. Otherwise Tb , varying in general with frequency/wavelength, is not guaranteed to accord with the physical temperature of the medium. Brightness temperature is of practical utility, however, even if the observational target is not a blackbody, as normally is the case. The example given above in Fig. 7.4 demonstrates that Iν is almost equal to Bν (T ) at a frequency where τν  1 (ν ≈ ν0 in this particular case). It follows that temperature is measurable radiatively

7.1 Radiative Transfer in Non-scattering Atmospheres

157

by Tb for an optically thick, non-scattering object under LTE, which is the physical principle employed by infrared thermography. Brightness temperature, on the other hand, may not be as good a metric of real temperature at the spectral line wing (ν = ν0 ), because radiance there is a mixture of emissions from the background and the medium itself. This issue will be revisited later in the context of the infrared observations of cloud top temperature.

7.2 Infrared Spectrum of the Atmosphere The spectrum of atmospheric transmittance is a rich forest of gas-absorption lines in the thermal infrared range (3–15 µm) (Fig. 6.3). Some of the narrow-band imager channels are chosen to be right on absorption lines while others are off the lines, that is, window channels. An example snapshot from Himawari-8 AHI observations is given by Fig. 7.5, showing simultaneous images of a water-vapor band (Band 8 at 6.2 µm) and of a window channel (Band 13 at 10.4 µm). The most striking difference between the two images is that Band-8 tends to observe substantially colder Tb s (more whitish), delineating large-scale uppertropospheric moisture patterns that are absent in Band 13. Band-13 instead clearly captures the areas covered with cold (high) clouds, marking a sharp contrast with warm surfaces outside the clouds. This section is aimed at illustrating the theoretical basis for how to interpret infrared observations as shown in Fig. 7.5. Technical details in radiative transfer simulations of infrared spectra are given in Sect. 13.2.

Fig. 7.5 Regional (Japan-area) observations by Himawari-8 AHI at Band 8 (6.2 µm, left) and Band 13 (10.4 µm, right) acquired at 1800 UTC 19 June 2021. Brightness temperature varies from warm to cold values as the shade changes from black to white. Image credit: JMA Meteorological Satellite Center (https://www.data.jma.go.jp/mscweb/data/himawari/)

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7 Infrared Sensing

7.2.1 Effects of Water Vapor Figure 7.6 shows the simulated clear-sky spectra of infrared Tb for a reference atmospheric profile, along with perturbed profiles having relative humidities (RHs) up to 100% at all heights and down to 10% as plotted in the left panel. Solar radiation is not included in the simulations. The spectral response function (SRF), or also known simply as the response function, of each AHI band (B7–B16) is superimposed on the spectra in Fig. 7.6. SRFs, characterizing the filter properties unique to each band, are generally different in details from one sensor to another even for “identical” bands with an equivalent central wavelength. Brightness temperature observed by a certain band is the SRF-weighted average of monochromatic Tb,λ over wavelength. Infrared Tb is in general warmer (colder) for a drier (moister) atmosphere. This is understood in terms of non-scattering radiative transfer as described in Sect. 7.1 above. A moister atmosphere has a larger infrared optical depth for wavelengths at which water vapor is absorptive of radiation. At these wavelengths, satellite-observed infrared radiance is radiatively saturated at a higher altitude, which is closer to the spacecraft, in a moist atmosphere than in a dry atmosphere. The observed Tb therefore “feels” a colder temperature when the atmosphere is more humid, since air temperature decreases with height in the troposphere, where a vast majority of the atmospheric water vapor resides. The sensitivity of Tb to moisture varies with wavelength. Brightness temperature at the 6.2 µm band (Band 8) can be warmer by almost 20 K in a dry (RH = 10%) air compared to the reference profile. This particular band exploits strong watervapor absorption due to the H2 O bending vibrational mode (Sect. 6.2.4), resulting in Tb s of about 220–240 K at wavelengths where the Band-8 SRF is close to unity.

1 0.8

280

250 300

T(p)

400

RH10%

500 600 700

RH100%

0

2

0.6

260 O3

240 220

0.4

CO 2

0.2

CO 2

H2O

q(p)

850 1000

Tb [K]

Pressure [hPa]

200

Simulated IR Tb and Himawari-8 SRF (B7-B16) B8 B9 B10 B11 B12 B13 B14 B15 B16

B7

300

0

200 4

6

8 10 12

Vapor mixing ratio [g/kg]

SRF

Temperature [K] 180 200 220 240 260 280 300 150

3

4

5

6

7

8

9

10

11

12

13

14

15

Wavelength [ m]

Fig. 7.6 (left) Model atmospheric profiles of temperature (dashed) and vapor mixing ratio (solid and shaded). Water vapor is perturbed around a reference profile up to a relative humidity of 100% at all levels and down to 10%. (right) Nocturnal, clear-sky spectra of simulated infrared Tb for the reference and perturbed profiles, where the moist (dry) profile yields colder (warmer) values of Tb . See Sect. 13.2 for the microphysical model assumptions. The SRFs of AHI Bands 7–16 are plotted together (scaled on the right). The Himawari-8 AHI SRF data were obtained from JMA Meteorological Satellite Center (https://www.data.jma.go.jp/mscweb/en/himawari89/space_ segment/spsg_ahi.html)

7.2 Infrared Spectrum of the Atmosphere

159

This can be interpreted that the vapor absorption occurs at pressure levels of 400 hPa or above where air temperature is colder than 240 K (see the left panel), on the basis of the argument in Sect. 7.1.3 and Fig. 7.4. Band 8 is hence sensitive to the upper-tropospheric moisture. An increase in RH from the reference profile does not introduce as much a difference in Tb , because the reference atmosphere is already nearly radiatively saturated around this wavelength. Bands 9 and 10 at 6.9 and 7.3 µm, respectively, are located on the P-branch side of the same H2 O absorption-line complex. A modest vapor absorption at these bands leads to a relatively warm Tb compared to Band 8, making Bands 9 and 10 useful for sounding the mid-tropospheric water vapor. The Tb sensitivity to water vapor is weak or absent outside the H2 O absorption lines. The window channels at wavelengths of 10–13 µm (Bands 13–15) exhibit a warm Tb of 280–290 K with only a modest dependence on humidity perturbations. These wavelengths are not subject to appreciable water-vapor absorption, located in a narrow spectral gap dividing the vibrational modes of H2 O molecules from their pure-rotational modes dominating longer wavelengths (Fig. 6.9a). The fact that the atmosphere, if clouds are absent, is nearly transparent at the wavelengths of 10– 13 µm is a convenient coincidence for measuring the earth’s surface temperature from space, given that the blackbody spectrum reaches the maximum near these wavelengths for a typical terrestrial temperature range (Sect. 6.1.1.1). Brightness temperature is entirely insensitive to water vapor within the CO2 absorption bands at 4.3 µm and 15 µm and the O3 band at 9.6 µm. At these wavelengths, water-vapor emissions are largely absorbed by layers of carbon dioxide and ozone lying above, leaving little trace in satellite-observed Tb .

7.2.2 Effects of Clouds We next consider the effects of clouds on infrared Tb by inserting a layer of cloud into the reference atmosphere. Two cases are simulated: one with low cloud (0.5–1.5 km in height) and the other with high cloud (10–11 km). LWC and IWC are set to be 0.1 g m−3 homogeneously through each 1-km-thick cloud layer, resulting in the LWP (IWP) of 100 g m−2 (see Sect. 8.3.1 for terminological definitions). With this value of LWP and IWP, the given cloud layer is optically thick at infrared wavelengths. Infrared Tb is only slightly lower outside the gas absorption lines in the presence of a low cloud than for the clear-sky case, while substantially colder when a high cloud takes the place of the low cloud. At the window channels (e.g., Bands 13–15), infrared Tb is found to be a reasonable proxy of cloud top temperature, which is 278 K for the low cloud and 217 K for the high cloud. Warm radiation from the low cloud is overridden entirely by the H2 O emissions within the water-vapor bands of Bands 8–10. High-cloud emissions, to the contrary, are not affected by the watervapor absorption because the high cloud is located above a substantial body of water vapor. The CO2 and O3 spectra emerge as emission lines instead of absorption lines in the high-cloud case. As demonstrated by Fig. 7.4, a spectral line from a warm

160

7 Infrared Sensing Temperature [K] 180 200 220 240 260 280 300 150

T(p)

280

0.8

260

0.6

240

O3

0.4

High cloud

220

850 Low cloud (liquid) 1000 0.1 0.12 0.06 0 0.02

CO 2

SRF

Tb [K]

Pressure [hPa]

250 High cloud (ice) 300

500 600 700

1

Low cloud

200

400

Simulated IR Tb and Himawari-8 SRF (B7-B16) B8 B9 B10 B11 B12 B13 B14 B15 B16

B7

300

0.2

CO 2

H2O

0

200 3

4

5

3

LWC/IWC [g/m ]

6

7

8

9

10

11

12

13

14

15

Wavelength [ m]

Fig. 7.7 As Fig. 7.6 but for cloudy-sky simulations with a layer of low cloud or of high cloud. (left) Model atmospheric profiles of temperature (dashed) and LWC/IWC (shaded). (right) Simulated Tb spectra for the low-cloud and high-cloud cases as well as the reference (clear-sky) case. Zeroscattering (ων = 0) simulations (dotted) are hardly discernible from the original spectra except for wavelengths shorter than 6 µm. Vertical arrows indicate the cloud effect on Tb

foreground (the stratosphere in this case) against a colder background (high cloud in the upper troposphere) is observed as an emission line. Apart from these gaseous emission features, brightness temperature is constant over a broad spectral range, suggesting that the cloud considered here is well approximated by a blackbody. Figure 7.7 also shows “zero-scattering” simulations (dotted) for the low- and highcloud cases by omitting the single scattering albedo (ων = 0). The dotted curves are hardly discernible from the reference simulations with the exception of wavelengths shorter than 6 µm. This fact assures that a non-scattering radiative transfer model, significantly simpler and faster than a full-specification radiative transfer model, is not just of conceptual value but of practical utility for simulating infrared Tb . It is noted that ων does not necessarily have to be near zero for the non-scattering assumption to be valid (Fig. 7.5). This is explained by two reasons. The first is the nonlinearity in the dependence of radiance on ων as discussed later in Sect. 9.1. Another reason is the fact that the infrared radiation field in an optically thick atmosphere, due to strong gas absorption and/or an optically thick cloud, is largely dominated by locally saturated thermal radiation. Under such a situation, the last two terms of the radiative transfer equation (4.102) representing the two source functions, one for local thermal emission and the other from scattered radiation, are combined together into the Planck function regardless of ων , 

Iν pν ( , )d ≈  (1 − ων )Bν (T ) + ων Bν (T ) pν ( , )d = Bν (T ) , (1 − ων )Bν (T ) + ων

(7.11)

when Iν ≈ Bν (T ). Equation (7.11) assures that a cloud layer, even with a non-zero ων , behaves as if it were a fully absorptive medium whenever Iν is nearly equilibrated

7.2 Infrared Spectrum of the Atmosphere

161

with Bν (T ). The effects of scattering may not be negligible at window channels if the cloud is not optically thick enough for (7.11) to hold to a good approximation. Optically thick clouds were considered in this section for ease of demonstration. In the Band-13 snapshot of Fig. 7.5, bright, white portions of the image are likely optical thick cold clouds, but extensive gray (i.e., relatively warm) features surrounding these clouds are less easy to interpret, dependent generally on both the temperature and optical depth of clouds. Infrared observations of clouds including semi-transparent ones are discussed in the next section.

7.3 Infrared Properties of Clouds Among various manners to characterize semi-transparent high clouds from satellite observations, the brightness temperature difference (BTD) between infrared splitwindow channels has been proved to be useful. The split-window channels refer to a pair of adjacent bands, roughly 1 µm apart in band-center wavelength, in the thermal-infrared spectral window of 10–13 µm. Traditional 5-channel imagers such as NOAA AVHRR carrying 11 and 12 µm bands (Table 2.2) have offered a long history of research exploring the utility of split-window channels, ranging from the atmospheric correction in measuring sea/land surface temperature to a separation of volcanic ashes from underlying clouds in infrared imagery. This section describes the physical principles behind methodologies to measure the cirrus-cloud properties using infrared Tb and BTD. Also reviewed are the utility of the 8.5–11 µm BTD for detecting cloud thermodynamic phase and the CO2 slicing method for measuring cloud top temperature.

7.3.1 Split-Window BTD Method Satellite-measured brightness temperature at split-window channels is theoretically expected from (7.8) to be B11 (Tb,11 ) = εs,11 B11 (Ts ) exp(−τc,11 ) + B11 (Tc )[1 − exp(−τc,11 )]

(7.12)

B12 (Tb,12 ) = εs,12 B12 (Ts ) exp(−τc,12 ) + B12 (Tc )[1 − exp(−τc,12 )] ,

(7.13)

and

where the subscripts 11 and 12 refer to a pair of split-window channels (typically at wavelengths of 11 and 12 µm), εs is surface emissivity, Ts is surface temperature, and Tc is cloud top temperature. The clear-sky absorption is neglected in the current formulation.

162

7 Infrared Sensing

Since the split-window channels are close in wavelength to the peak of blackbody spectrum (i.e., ∂ Bλ /∂λ ≈ 0) for tropospheric temperatures, it is assumed from now on that B11 (T ) ≈ B12 (T ) . It is also assumed that εs,11 ≈ εs,12 . Given these simplifications, BTD, ΔTb,11−12 ≡ Tb,11 − Tb,12 , is approximated by ΔTb,11−12 ≈ −Δτc,11−12 exp(−τc,11 )ΔT0 , where

 ΔT0 ≡ [εs,11 B11 (Ts ) − B11 (Tc )]

∂ B11 (Tb,11 ) ∂T

(7.14)

−1 (7.15)

is a radiometric temperature difference between the surface and cloud top. To derive (7.14), the differences in Tb and in τc between the split-window channels have been assumed to be relatively small, so that B11 (Tb,11 ) − B12 (Tb,12 ) ≈

∂ B11 (Tb,11 )ΔTb,11−12 ∂T

and exp(−τc,11 ) − exp(−τc,12 ) ≈ −Δτc,11−12 exp(−τc,11 ) , where Δτc,11−12 ≡ τc,11 − τc,12 . The absorption coefficient (ρσa,ν , where ρ denotes LWC or IWC) of cloud water and ice is plotted as a function of wavelength in Fig. 7.8. Given that the absorption

(b) Absorption Coefficient for Cloud Ice

(a) Absorption Coefficient for Cloud Water 4

Dm = 5 m Dm = 10 m Dm = 20 m

3

D0 = 10 m D0 = 20 m D0 = 40 m

3

12 m 11 m

2

2

12 m 11 m

8.5 m

8.5 m

a,

[km-1]

4

1

1

0

0 7

8

9

10

11

12

Wavelength [ m]

13

14

15

7

8

9

10

11

12

13

14

15

Wavelength [ m]

Fig. 7.8 The absorption coefficient (ρσa,ν ) at thermal infrared wavelengths for (a) cloud water and (b) cloud ice with ρ = 10−2 g m−3 . Different curves in each panel are obtained with different particle sizes: the mode diameter Dm is chosen to be 5, 10, and 20 µm (re = 3.4, 6.8, and 14 µm, respectively) for cloud water, and the median volume diameter D0 is 10, 20, and 40 µm (re = 4.3, 8,6, and 17 µm, respectively) for cloud ice. See Sect. 13.2 for details in the microphysical model assumptions

7.3 Infrared Properties of Clouds Fig. 7.9 Schematic illustration of the ΔTb -Tb diagram for cirrus clouds. See text for details

163

Tb,11-12 c exp(

Ts

Tc exp(

c,11)

c,11)

T0

Tb,11

c

coefficient multiplied by the cloud layer thickness equals τc [see (7.5)], the spectral gradient of ρσa,ν from 11 to 12 µm is proportional to Δτc,11−12 . The spectral gradient is striking particularly for cloud ice, ascribed to the absorption due to the ice libration mode (see Fig. 6.15b). This property is a key ingredient in the BTD-based methods for cirrus cloud measurements. Note that Δτc,11−12 is negative because the absorption coefficient increases with wavelength from 11 to 12 µm. Figure 7.9 shows a schematic diagram of the infrared properties of clouds projected onto the ΔTb -Tb plane. The abscissa is Tb,11 , which is interpreted by (7.12) as a weighted average of Ts and Tc varying with the cloud transmittance exp(−τc,11 ). Brightness temperature would be equivalent to Tc for a zero-transmittance (or blackbody) cloud layer, while approaching Ts (since εs,11 ∼ 1) as the cloud transmittance increases toward unity. In contrast, −Δτc,11−12 , having a finite positive value for an optically thick cloud, diminishes as the cloud becomes transparent (τc,11 , τc,12 → 0). Since ΔTb,11−12 is roughly proportional to the product of exp(−τc,11 ) and −Δτc,11−12 as shown by (7.14), clouds with various optical depths would fall onto an arching curve in the ΔTb -Tb plane, with its both feet landing at Tb,11 = Tc and Ts on the abscissa (ΔTb,11−12 = 0). An arch as predicted Fig. 7.9 has been known in actual infrared observations of cirrus clouds since early studies (Inoue 1985; Wu 1987). It is apparent that the arch would shrink horizontally and vertically as the difference between Tc and Ts diminishes. The height of the arch for a given τc depends also on Δτc . Among the factors accounting for Δτc,11−12 is the mean size of cloud particles as evident in Fig. 7.8, which shows that smaller cloud-ice particles give rise to a larger absorption coefficient for a given IWC. The sensitivity to the particle size is greater at 12 µm than at 11 µm, so that −Δτc,11−12 enlarges with increasing particle size. This fact sets the basis of the split-window BTD method to retrieve the optical depth and mean particle size of cirrus clouds (Prabhakara et al. 1988). The split-window BTD method is in theory applicable also to warm clouds, although not as useful as is for cold clouds. For warm clouds, Tc is so close to Ts that the variability in τc is difficult to differentiate radiometrically in infrared radiation. Radiative transfer simulations are conducted to confirm the simplistic theoretical prediction of an arching curve in the Tb -ΔTb plane. An ice cloud is inserted into

164

7 Infrared Sensing (a) Cloud Optical Depth at 11 m

(b) Cloud Effective Radius [ m] 5

7 6

7

40

6

35

4 5

15

5

Tb,11-12 [K]

30 3

4 3

2 3

2 5

2

20

4

25 30 35 m

3

25 20

1

2

4 1

1

15

1 0

0 210

220

230

240 250 260 Tb,11 [K]

270

280

290

0

10 210

220

230

240 250 260 Tb,11 [K]

270

280

290

Fig. 7.10 The ΔTb -Tb diagram for cirrus clouds based on split-window (11 and 12 µm) channels. (a) cloud optical depth at 11 µm with a contour interval of 0.5. (b) cloud effective radius with a 5-μm interval. The microphysical model setups are summarized in Sect. 13.2

a standard atmosphere between the heights of 10 and 11 km. IWC is varied from 10−3 to 10−1 g m−3 , while the median volume diameter of ice particles are perturbed independently from 2 µm to 100 µm (see Sect. 13.2 for further details), resulting in a broad range of cloud optical depth τc,11 and cloud effective radius re , defined by (13.3). Monochromatic brightness temperatures at 10.99 and 12 µm are calculated for Tb,11 and ΔTb,11−12 under a fixed cloud top temperature without any SRF considered for brevity.2 Figure 7.10 clearly shows arching patterns with a systematic horizontal gradient of τc,11 (left) and with a vertical re gradient (right). It follows that τc,11 largely accounts for Tb,11 while re is primarily responsible for ΔTb,11−12 under a given τc,11 , as expected from (7.12) and (7.14). The left foot of the arch is rooted in the blackbody solution of Tb,11 ∼ Tc = 217 K, which is attained for τc,11 → ∞. In this limit, Δτc,11−12 practically diminishes to zero, so BTD provides useful information of re only for optically thin cirrus. The both ends of arches do not precisely meet zero BTD, because the clear-sky absorptivity, mainly owing to water vapor, is not uniform across the two split-window wavelengths (see Fig. 7.6). The right foot of the arch is more offset upward than the left, since the surface emissions are attenuated by abundant lower-tropospheric moisture while the cloud-top emissions are affected only slightly by upper-tropospheric water vapor. The particle absorption and scattering properties in the present calculations are based on Mie’s solutions assuming that the cloud particles are spherical ice. More realistic assumptions with a variety of ice crystal habits would modify quantitative

The wavelength is slightly offset from 11 µm in order to avoid a weak absorption feature at 11 µm.

2

7.3 Infrared Properties of Clouds

165

aspects of the simulated results, but do not qualitatively alter the overall arching structure (Cooper and Garrett 2010).

7.3.2 Detection of Cloud Thermodynamic Phase The water absorptivity stays relatively low at wavelengths of 8–10 µm (Fig. 7.8), sandwiched between the near-infrared vibrational bands and the mid- to far-infrared intra-molecular bands (Sect. 6.3.3). This fact inspired a methodology to exploit the BTD between 8.5 and 11 µm, ΔTb,8.5−11 ≡ Tb,8.5 − Tb,11 , for separating cloud ice from cloud water (Ackerman et al. 1990; Strabala et al. 1994). The underlying idea is based on the fact that the spectral gradient of the cloud absorption coefficient from 8.5 and 11 µm is substantially steeper for ice water than for liquid water (Fig. 7.8). The 8.5–11-µm version of (7.14) is ΔTb,8.5−11 ≈ −Δτc,8.5−11 exp(−τc,8.5 )ΔT0 ,

(7.16)

where Δτc,8.5−11 ≡ τc,8.5 − τc,11 , so that the aforementioned difference in Δτc,8.5−11 between ice and liquid water implies that ΔTb,8.5−11 should be sensitive to cloud thermodynamic phase. Ice clouds have a greater magnitude of −Δτc,8.5−11 than liquid clouds, leading to a distinctly positive ΔTb,8.5−11 compared to a smaller positive value for liquid clouds. It is noted that this method does not efficiently work for optically thin clouds, for which Δτc,8.5−11 is nearly zero regardless of ice or water. In real atmospheres, warm clouds have a negative ΔTb,8.5−11 when the thermal emissions from low cloud tops undergo a non-negligible water-vapor attenuation. The water-vapor absorption is somewhat stronger at 8.5 µm than at 11 µm (compare B11 and B14 in Fig. 7.6), resulting in a negative value of ΔTb,8.5−11 .

7.3.3 CO2 Slicing Method It is found from (7.12) that infrared brightness temperature at a window channel is a useful proxy of cloud top temperature for an optically thick cloud but is less so as cloud emissivity, (7.17) εc ≡ 1 − exp(−τc ) , decreases away from unity. The CO2 slicing method, proposed originally in early years of satellite infrared sounding (Menzel et al. 1983), is a technique to evaluate εc and Tc at a time from CO2 sounding channels. A conceptual sketch of the algorithm theoretical basis is delineated below to give a physically intuitive outline of the methodology. The CO2 slicing method utilizes multiple channels from the CO2 bending band at 13–15 µm (Figs. 6.9c and 7.6). A CO2 absorption of upwelling thermal radiation

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7 Infrared Sensing

Fig. 7.11 A simplified picture describing the CO2 slicing method. Shaded boxes represent an opaque lower-tropospheric layer, entirely concealing a low cloud inside (dotted) from being detected. See text for details

I

I

all 1

clr 1

I

cld 1

I

I

all 2

cld 2

I

clr

Tc

Tp2 Tp1

1

2

makes the surface and low clouds invisible while mid- to high-level clouds in a foreground remain observable. Choosing a proper set of wavelengths within the CO2 band, one can radiometrically “slice out” the atmosphere at different levels. This is schematically depicted in Fig. 7.11. Let us consider clear- and cloudy-sky infrared radiances, I clr and I cld respectively, at two wavelengths λ1 and λ2 , the latter of which is more absorptive than the former. For the sake of brevity, the clear-sky atmosphere is assumed to be extremely opaque by the CO2 absorption below a specified level while transparent above that level. This is equivalent to replacing the weighting function with a delta function (see Sect. 9.5). Under this simplification, clear-sky emissions at λ1 and λ2 agree with the blackbody radiation for temperatures T p1 and T p2 , respectively. Any cloud lower than the specified level is by construction undetectable, while a cloud at a higher level is observable and considered in the present model. Infrared radiance at each wavelength is expressed using (7.8) as clr = Bλ1 (T p1 ) , Iλ1 cld Iλ1 = Bλ1 (T p1 ) exp(−τc ) + Bλ1 (Tc )[1 − exp(−τc )] ,

(7.18a) (7.18b)

all cld clr Iλ1 = f c Iλ1 + (1 − f c )Iλ1 ,

(7.18c)

7.3 Infrared Properties of Clouds

167

for λ1 and clr = Bλ2 (T p2 ) , Iλ2 cld Iλ2 = Bλ2 (T p2 ) exp(−τc ) + Bλ2 (Tc )[1 − exp(−τc )] ,

(7.19a) (7.19b)

all cld clr Iλ2 = f c Iλ2 + (1 − f c )Iλ2 ,

(7.19c)

for λ2 , where f c is the cloud cover within an FOV and Iall is the all-sky radiance. Cloud optical depth τc is common to (7.18c) and (7.19c) since the two wavelengths are so close to each other that the spectral difference in τc is negligible. Equations (7.18c) and (7.19c) are combined into clr all − Iλ1 = f c εc [Bλ1 (T p1 ) − Bλ1 (Tc )] , Iλ1

(7.20a)

clr all Iλ2 − Iλ2 = f c εc [Bλ2 (T p2 ) − Bλ2 (Tc )] ,

(7.20b)

where εc is as defined by (7.17). Equation (7.20b) provides cloud top temperature Tc and f c εc , known collectively as effective cloud emissivity, for given radiances from a pair of CO2 absorption channels. The clear-sky radiance is not directly available from observations, so in practice synthetic measurements obtained with a radiative transfer model are substituted for I clr . The rhs of (7.20b) represents the cloud emission blended with penetrating radiation from below, which depends on the temperature of the cloud itself (Tc ) and how much portion of the background radiation penetrates it through ( f c εc ). As such, a combination of warm (T = T p1 ) and cold (T = T p2 ) backgrounds enables to determine Tc and f c εc independently. A more precise formulation of (7.20b) is found in Menzel et al. (1983).

7.4 Summary Key points from this chapter are summarized as follows. 1. Radiation incident on a non-scattering LTE medium approaches the blackbody spectrum Bν (T ) as propagating through the medium. Brightness temperature is practically equivalent to the physical temperature of the emitting medium when the medium is optically thick (τν  1). This behavior forms the physical basis of infrared remote sensing. 2. The spatial variability of upper-tropospheric water vapor is well captured by brightness temperature at the H2 O band centered around 6.2 µm, and high clouds are clearly detectable at window channels with wavelengths of 10–13 µm. 3. Infrared brightness temperature is conveniently used as a proxy of cloud top temperature for opaque clouds, while not a good metric of the physical temperature for optically thin clouds. 4. The utility of infrared remote sensing is marginal for low clouds because of the weak temperature contrast against the surface.

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5. A combination of thermal-infrared brightness temperature and the split-window BTD conveys useful information of cirrus cloud properties, especially cloud optical depth and cloud effective radius. The split-window BTD is not suitable for low clouds. 6. The distinct difference in the 11-µm absorptivity between ice and liquid water enables to detect cloud thermodynamic phase using the 8.5–11-µm BTD. The phase determination is not as efficient for optically thin cloud as for opaque clouds. 7. Cloud top temperature (or eventually cloud top height) can be evaluated from multiple CO2 absorption channels for both opaque and semi-transparent clouds. Low clouds are indiscernible at the CO2 band, so the CO2 slicing method is targeted on mid- to upper-tropospheric clouds. The cloud phase determination based on ΔTb,8.5−11 is used for the MODIS products (Platnick et al. 2003). The CO2 slicing technique is also adopted for the MODIS cloud products while not applicable to traditional visible/infrared imagers without CO2 channels such as AVHRR, for which the split-window BTD is still useful to constrain the cloud emissivity (Heidinger and Pavolonis 2009).

References Ackerman SA, Smith WL, Revercomb HE, Spinhirne JD (1990) The 27–28 october 1986 FIRE IFO cirrus case study: spectral properties of cirrus clouds in the 8–12 μm window. Mon Wea Rev 118:2377–2388. https://doi.org/10.1175/1520-0493(1990)1182.0.CO;2 Cooper SJ, Garrett TJ (2010) Identification of small ice cloud particles using passive radiometric observations. J Appl Meteor Climatol 49:2334–2347. https://doi.org/10.1175/2010JAMC2466.1 Heidinger AK, Pavolonis MJ (2009) Gazing at cirrus clouds for 25 years through a split window. Part i: Methodology. J Appl Meteor Climatol 48:1100–1116. https://doi.org/10.1175/ 2008JAMC1882.1 Inoue T (1985) On the temperature and effective emissivity determination of semi-transparent cirrus clouds by bi-spectral measurements in the 10 μm window region. J Meteor Soc Jpn 63:88–99 Menzel WP, Smith WL, Stewart TR (1983) Improved cloud motion wind vector and altitude assignment using VAS. J Climate Appl Meteor 22:377–384. https://doi.org/10.1175/15200450(1983)0222.0.CO;2 Platnick S, King M, Ackerman S, Menzel W, Baum B, Riedi J, Frey R (2003) The MODIS cloud products: algorithms and examples from Terra. IEEE Trans Geosc Remote Sens 41:459–473. https://doi.org/10.1109/TGRS.2002.808301 Prabhakara C, Fraser RS, Dalu G, Wu MC, Curran RJ, Styles T (1988) Thin cirrus clouds: Seasonal distribution over oceans deduced from Nimbus-4 IRIS. J Appl Meteor 27:379–399. https://doi. org/10.1175/1520-0450(1988)0272.0.CO;2

References

169

Strabala KI, Ackerman SA, Menzel WP (1994) Cloud properties inferred from 8–12-μm data. J Appl Meteor 33:212–229. https://doi.org/10.1175/1520-0450(1994)0332.0.CO; 2 Wu MLC (1987) A method for remote sensing the emissivity, fractional cloud cover and cloud top temperature of high-level, thin clouds. J Climate Appl Meteor 26:225–233. https://doi.org/10. 1175/1520-0450(1987)0262.0.CO;2

Chapter 8

Visible/Near-Infrared Imaging

Satellite visible imagery is intuitive in interpretation because visible remote sensing is the way by which human eyes observe the surrounding world. Our eyes are an optical device with a set of overlapping spectral filters that help us make sense out of the world through “colors”. RGB-composite satellite images (e.g., Fig. 3.3) visualize the entire earth that would be seen if we could fly out of the planet. Satellite visible/infrared imagers, however, are not designed just to mimic human vision. In the context of meteorological applications, the primary purposes include the quantitative evaluation of the cloud physical properties such as cloud optical depth and effective radius. This chapter is devoted to a review of the methodologies of visible/near-infrared cloud measurements from satellites with focus on their theoretical background.

8.1 Radiative Transfer in Scattering Atmospheres Figure 7.1 in the previous chapter shows not only that clouds are highly absorptive of infrared radiation, but also that the cloud single scattering albedo stays very close to unity for wavelengths shorter than 1 µm, implying that visible light is scattered by clouds without being absorbed. This section presents the formulation of the scattering radiative transfer equation, complementary to the non-scattering radiative transfer discussed in Sect. 7.1.

© Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_8

171

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8 Visible/Near-Infrared Imaging

8.1.1 Mathematical Formulation and Solution for ων = 1 The radiative transfer equation (4.98) is simplified into    dIν    = ρσs,ν −Iν + Iν pν (Ω , Ω)dΩ ds

(8.1)

for a purely scattering (i.e., non-absorbing) medium (ων = 1), where σs,ν is the scattering cross section per unit mass and pν (Ω  , Ω) is scattering phase function from a certain direction Ω  to the line of sight Ω. The length element ds is defined along Ω. Radiances in the directions of Ω and Ω  are denoted by Iν and Iν , respectively. Similarly to (7.4), (8.1) may be more conveniently rewritten into dIν = −Iν + dτν



Iν pν (Ω  , Ω)dΩ  ,

(8.2)

where optical depth τν is given by dτν = ρσs,ν ds

(8.3)

for a purely scattering medium. The first term on the rhs of (8.2) represents the loss of radiative energy due to the photons diverted away, while the second term accounts for the source from the scattered photons confluent into the line of sight (Fig. 8.1). As noted in Sect. 7.1, the current formulation is directly applicable to a plane-parallel atmosphere by replacing ds by dz/ cos θ in (8.3). The formal solution of (8.2), similar to (7.7) for a non-scattering atmosphere, may be written as  τc,ν  exp[−(τc,ν − τν )] Iν pν (Ω  , Ω)dΩ  dτν . Iν (τc,ν ) = Iν (0) exp(−τc,ν ) + 0

(8.4) A prior knowledge of Iν is required to find Iν , making the purely-scattering radiative transfer equation (8.2) difficult to solve. This contrasts to the non-scattering cases

I ( +d )

I( )

d Fig. 8.1 Schematic drawing of radiative transfer in a purely scattering medium. A volume element of the medium is indicated by a cylinder. Small arrows with an arrowhead designate photons scattered away, while those with feathers denote photons scattered from different directions into the line of sight (cf. Fig. 4.4)

8.1 Radiative Transfer in Scattering Atmospheres

173

(b) Step 2: Find I (

(a) Step 1: Obtain J ( )

↓

I (

I( c,

c,

)=I*,

)

)

c,

J( )

J( ) 0

c,

c,

0

↑

I (0)=Is,

I (0)=Is,

Fig. 8.2 Schematic illustration of the two-fold strategy to derive an analytic solution of the purelyscattering radiative transfer equation. a Step 1 and b Step 2

under LTE, where the source function is independent of the ambient radiation and hence radiance is easily obtained [see (7.8)]. It is nonetheless attempted in this section to derive an analytic expression of (8.4) so that radiance on both sides of the equation is found in a self-consistent manner. To facilitate this, a two-fold approach is employed (Fig. 8.2). The first step is intended to obtain the scattering source function or the integral over solid angle in (8.4) under the Eddington approximation. In the second step, (8.4) is computed to yield the top-of-the-atmosphere (TOA) radiance using the outcome of the step one. The detailed procedures for each step are described next.

8.1.1.1

Step 1: Scattering Source Function

Two major assumptions are made to reduce the problem to a tractable level. The first assumption is that scattering is isotropic, that is, pν (Ω  , Ω) ≡ 1/4π . With isotropic scattering, (8.4) is reduced to  Iν (τc,ν ) = Iν (0) exp(−τc,ν ) +

τc,ν

Jν (τν ) exp[−(τc,ν − τν )]dτν .

(8.5)

0

where

1 Jν ≡ 4π

 Iν dΩ =

c Eν , 4π

(8.6)

is the angular-mean radiance, proportional to the radiative energy density E ν [cf. (4.86a)]. The second assumption, the Eddington approximation, is introduced for the ease of obtaining J (τν ) in (8.5). As discussed in Sect. 4.2.4.2, the Eddington approximation

174

8 Visible/Near-Infrared Imaging

offers a useful closure relationship for the radiative energy and momentum equations when the radiation field is nearly isotropic [i.e., (4.92)]. The energy and momentum conservation equations under the Eddington approximation given by (4.103) are combined into 1 d2 Jν = (1 − ων )[Jν − Bν (T )] (8.7) 3 dτν2 under the assumptions of LTE, Sν = Bν (T ), and isotropic scattering (gν = 0). For a purely scattering atmosphere (ων = 1), (8.7) is simply a diffusion equation with no source term, d2 Jν =0, dτν2 which yields a general solution, Jν (τν ) = J0,ν + J1,ν τν .

(8.8)

Here J0,ν and J1,ν are constants to be determined with boundary conditions. The boundary conditions are specified by invoking the two-stream approximation (Sect. 4.2.4.3). Equation (4.97) along with (4.103b) and (8.6) leads to 1 d Jν , Iν↑ = Jν − √ 3 dτν 1 d Jν Iν↓ = Jν + √ , 3 dτν

(8.9a) (8.9b)

for gν = 0 (i.e., isotropic scattering). We consider a transparent atmosphere except within a cloud layer having an optical depth of τc,ν . The cloud is illuminated at the top by incoming solar radiation I∗,ν and at the bottom by the surface-reflected radiation Is,ν , which could be near zero over open ocean or rather bright when the surface is covered with ice or overcast by lower-level clouds. The boundary conditions are Iν↑ (τν = 0) = Is,ν at cloud base , Iν↓ (τν = τc,ν ) = I∗,ν at cloud top ,

(8.10a) (8.10b)

as illustrated in Fig. 8.2a. Equations (8.8), (8.9), and (8.10) together result in      1 1 2 −1 Is,ν τc,ν + √ , J0,ν = √ I∗,ν + τc,ν + √ 3 3 3   2 −1 J1,ν = (I∗,ν − Is,ν ) τc,ν + √ . 3

(8.11a) (8.11b)

8.1 Radiative Transfer in Scattering Atmospheres

175

Equation (8.8) with (8.11) describes the spatial structure of Jν inside the cloud layer (0 ≤ τν ≤ τc,ν ). It is reminded that (8.11) is only approximately exact because the Eddington approximation is valid when radiation field is quasi isotropic. This is justifiable deep inside an optically thick cloud, while less accurate near the cloud base and top and/or for an optically thin cloud. Another assumption of a uniform phase function is not an optimal substitute for the scattering phase function of clouds, which is typically forward peaking (Sect. 8.4). The solution (8.11) is not valid for quantitative applications but is useful for intuitive interpretations as demonstrated later.

8.1.1.2

Step 2: TOA Radiance

We are now ready to derive the upwelling radiance at TOA, Iν (τc,ν ). Equation (8.5) is rewritten with (8.8) as 

τc,ν

Iν (τc,ν ) = Is,ν exp(−τc,ν ) +

(J0,ν + J1,ν τν ) exp[−(τc,ν − τν )]dτν .

(8.12)

0

Inserting (8.11) into (8.12), one eventually finds      1 2 −1 Iν (τc,ν ) = Is,ν + (I∗,ν − Is,ν ) τc,ν − 1 − √ [1 − exp(−τc,ν )] τc,ν + √ . 3 3

(8.13)

Figure 8.3 shows the normalized solution Iν (τc,ν )/I∗,ν as a function of τc,ν for different values of Is,ν . The TOA radiance grows rapidly with cloud optical depth for a small Is,ν , while the contrast between optically thin and thick clouds weakens as Is,ν increases toward I∗,ν . It follows that scattered solar radiation is sensitive to cloud optical depth over a dark background but clouds are less discernible from the background as the surface becomes brighter. This behavior is qualitatively similar to the non-scattering solution depicted in Fig. 7.3. Note that cases with Is,ν > I∗,ν , although mathematically possible, are not shown in Fig. 8.3 because the surface-reflected radiance is in reality unable to exceed the incoming solar radiation. An important difference is that radiation is entirely saturated at a moderate optical depth (τν ∼ 5, see Fig. 7.3) in a non-scattering atmosphere, in contrast to a purely-scattering atmosphere in which reflectance retains a sensitivity to τc,ν until τc,ν reaches 10 or even higher. This is a notable advantage of visible/near-infrared remote sensing to measure cloud optical depth. The non-scattering equivalent of the purely-scattering solution (8.13) is Iν (τc,ν ) = Is,ν + (Bν (T ) − Is,ν )[1 − exp(−τc,ν )]

(8.14)

[cf. (7.8)]. The factor of [1 − exp(−τc,ν )], or cloud emissivity (7.17), is the reason why radiation is rapidly saturated in a non-scattering atmosphere as τc,ν increases

176

8 Visible/Near-Infrared Imaging RT solutions for a purely scattering atmosphere 1

I / I *,

0.75

0.5

0.25

0 0

5

10

15

20

25

30

Cloud Optical Depth

Fig. 8.3 Analytic solutions of the purely-scattering radiative transfer equation. The TOA radiance Iν (τc,ν ) normalized by the incoming solar radiance I∗,ν is plotted as a function of optical depth for different boundary conditions (specified by Is,ν ) of 0, 0.25, 0.5, 0.75 from bottom to top

beyond one. A similar term appears also in the purely-scattering solution but is √ accompanied by another term proportional to τc,ν /(τc,ν + 2/ 3), which dominates for large τc,ν once the exponential term is saturated out. This second term arises because the scattering source function Jν varies with τc,ν unlike in a non-scattering atmosphere where the source function Bν (T ) is determined only by local temperature. This property is characteristic of scattering atmospheres, allowing clouds to grow brighter beyond the exponential saturation as they optically thicken.

8.1.2 Mathematical Formulation and Solution for ων  = 1 The radiative transfer solution derived above is valid only when ων is strictly unity. Single scattering albedo, on the other hand, does not stay precisely at one for wavelengths beyond 1 µm (Fig. 7.1). The formulation is now expanded to more general cases with ων < 1. It is noted that the radiative transfer equation discussed here, despite the inclusion of absorption, is still not a complete formulation since the local source of thermal emission remains omitted. This treatment is justified by the fact that thermal radiation of terrestrial origin is negligibly small compared to solar radiation in the near-infrared range ( 0). As such, Rλ given by (8.21) is a measure of the fractional upwelling flux of reflected solar radiation relative to the incoming solar flux, S0,λ cos θ0 , with the caveat that the assumption of isotropic Iλ is in general a poor premise. The theoretically predicted normalized TOA radiance, Iν (τc,ν )/I∗,ν , plotted earlier in Figs. 8.3 and 8.4 may be considered to be a practical substitute for Rλ .

8.2 Visible/Near-Infrared Spectrum of the Atmosphere

181

Reflectance is brightened by clouds across the whole visible and near-infrared spectral ranges outside the gas-absorption bands, which are mostly attributed to water vapor with a notable exception of the 0.76-µm O2 band. Very optically thick clouds give rise to a reflectance of 0.8 or even higher, whereas sub-visible clouds (τc,0.65 < 1) are only barely discernible from the clear-sky reflectance. This intuitively predictable property, as expected also from the simplistic analytic solution (Fig. 8.3), is common to low and high clouds. Low clouds, however, are more heavily affected by H2 O absorption than high clouds because a substantial amount of the water vapor lying above low cloud tops is confined to below the high clouds considered here. The reflectance of an opaque cloud gradually decreases from visible to nearinfrared wavelengths. This is understood in terms of the spectral gradient in the cloud ων . Among the physical factors responsible for the near-infrared ων is the size of cloud particles as discussed next.

8.2.2 Clouds with Different Effective Radii Figure 8.7 shows synthetic spectra similar to Fig. 8.6 but for three different cloud effective radii (re , see Sect. 8.3.1 for the definition). LWC and IWC are chosen to be proportional to cloud effective radius so that cloud optical depth is approximately same among all runs [see (8.28)]. The spread of reflectance spectra for different values of re is modest at visible wavelengths, but expands as entering into the near-infrared range. Near-infrared radiance is weaker for a larger re under a given cloud optical depth. This striking sensitivity of near-infrared reflectance to re will be discussed further in Sect. 8.3.3.

8.3 Visible/Near-Infrared Properties of Clouds This section gives an overview of visible/near-infrared measurements of the cloud physical properties. First outlined are some fundamentals such as the relationship among LWP, cloud optical depth, and cloud effective radius (Sect. 8.3.1) and the wavelength dependence of cloud optical depth (Sect. 8.3.2). The physical basis for the satellite retrieval of cloud optical depth and cloud effective radius is reviewed in Sect. 8.3.3.

8.3.1 Liquid/Ice Water Path and Cloud Effective Radius The mass density of condensate is called liquid water content (LWC) or ice water content (IWC), depending on cloud thermodynamic phase. LWC and IWC are expressed in the units of kg m−3 or g m−3 . In the meteorological literature, the density of con-

182

8 Visible/Near-Infrared Imaging (a) Low cloud effective radius 150

400 500 600 700 850 1000

c

≈24, 24, 23 5

0

15 10 re [ m]

0.8

0.8

0.6

0.6

0.4

H2O

O2

0.4 H2O

H2O

0.2 20

0

0 0.2

(c) High cloud effective radius 150

0.4

0.6

0.8

1

1.2 1.4 1.6 1.8 Wavelength [ m]

2

2.2

2.4

2.6

(d) Simulated Reflectance for high clouds and Himawari-8 SRF (B1-B6) B1 B2 B3 B4 B5 B6 1

1

200

0.8

0.8

400

c

Reflectance

250 300

≈23, 20, 20

500 600 700 850 1000

0.6

0.6 O2

0.4

0.4 0.2

0.2 0

5

10 15 re [ m]

20

0.2

SRF

Reflectance

Pressure [hPa]

250 300

SRF

H2O

200

Pressure [hPa]

(b) Simulated Reflectance for low clouds and Himawari-8 SRF (B1-B6) B1 B2 B3 B4 B5 B6 1

1

0

0 0.2

0.4

0.6

0.8

1.2

1

1.4

1.6

1.8

2

2.2

2.4

2.6

Wavelength [ m]

Fig. 8.7 Simulated spectrum from visible to near-infrared wavelengths with a layer of low cloud a–b or of high cloud (c–d). a The vertical profile of low-cloud effective radius for three different setups. LWP is adjusted so that τc,0.65 , indicated in the panel, is roughly same among different runs. b Simulated radiance spectra with low clouds having different effective radii as well as the reference (clear-sky) case. c As a but for high-cloud effective radius. d As b but for high clouds. The Himawari AHI SRFs are superimposed as in Fig. 8.6

densate is often described in terms of mixing ratio, with which LWC and IWC are defined as LWC = ρa qw ,

(8.22a)

IWC = ρa qi ,

(8.22b)

where qw and qi are liquid and ice water mixing ratios, respectively, and ρa denotes the mass density of dry air. LWC may be defined to contain all liquid condensates (cloud water and rain) or can refer to the cloud component only. A similar ambiguity exists also in the definition of IWP. LWC and IWC may be called “cloud LWC” and “cloud IWC” to avoid confusion in case that precipitating condensates are precluded from the definition. This is a tricky problem when it comes to practical applications, because satellite instruments are generally not uniformly sensitive to different hydrometeor species nor able to unambiguously separate one species from another. Liquid water path (LWP) and ice water path (IWP) are the column integrated quantities of LWC and IWC:

8.3 Visible/Near-Infrared Properties of Clouds

183



zt

LWP =

z  bzt

IWP =

ρa qw dz ,

(8.23a)

ρa qi dz ,

(8.23b)

zb

where z b and z t are cloud base and top heights, respectively. In practice, the integral can be applied to the whole thickness of the atmosphere in (8.23) because LWC and IWC vanish outside the cloud layer anyway, with the caveat that, in such a case, different cloud layers would be combined into a single estimate of LWP or IWP in the presence of overlapping clouds. LWP and IWP are typically expressed in the units of kg m−2 or g m−2 . Satellite measurements from visible/infrared imagers give cloud optical depth and effective radius (Sect. 8.3), but do not directly yield the mass of condensate. Instead, a simple formula exists to evaluate LWP from the measurable variables. Given a cloud particle size distribution (PSD) n c (r ) as a function of particle radius r , 

4πρw LWP = 3



zt

∞

r 3 n c (r )dr dz ,

(8.24)

0

zb

where ρw is the mass density of liquid water (103 kg m−3 ). Here cloud droplets are assumed to be spherical, which is a valid approximation for liquid water clouds. Cloud optical depth is  τc,ν =

zt



zb

∞

0

M σe,ν n c (r )dr dz ,

(8.25)

M where σe,ν is the extinction cross section per particle from Mie’s solutions (5.71). M may be Since cloud droplets are substantially larger than visible wavelengths, σe,ν replaced by its limiting value in the geometrical optics regime (see Sect. 5.1.4.7), M ≈ 2πr 2 , σe,ν



so that τc ≈ 2π

zt zb



∞

r 2 n c (r )dr dz ,

(8.26)

(8.27)

0

where τc represents cloud optical depth for visible radiation. If n c (r ) is uniform over height, (8.24) and (8.27) are combined into LWP ≈

2 ρw τc re , 3

where cloud effective radius is defined by

(8.28)

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8 Visible/Near-Infrared Imaging

 re ≡ 

∞ 0

∞

r 3 n c (r )dr .

(8.29)

r n c (r )dr 2

0

Cloud effective radius, sometimes abbreviated as CER, is the ensemble mean of particle radius weighted by the geometric cross-sectional area, or approximately the extinction cross section, of cloud droplets. This particular measure of mean particle size was first introduced by Hansen and Travis (1974) along with its higher-order moments called the effective variance and effective skewness. The effective radius as defined by (8.29) is widely used in satellite meteorology, not only for clouds but also for aerosols. The assumption that n c (r ) is independent of height is generally not valid. The diffusional growth of cloud droplets in an ascending air parcel results in an upward increase of droplet size. A semi-analytic model taking this effect into account leads to the factor of 5/9 instead of 2/3 using the cloud-top value of re in (8.28) (Szczodrak et al. 2001). IWP may be related to τc and re in a similar formula, although a precise formulation is difficult because ice crystals are complicated in shape. The size of non-spherical hydrometeors may be defined in different ways including the radius or diameter of a sphere of equivalent volume or the maximum dimension of a particle.

8.3.2 Visible Versus Infrared Optical Depth of Clouds Cloud optical depth τc depends on wavelength. In general, τc is defined at a visible wavelength unless otherwise noted, but it is worthwhile to briefly discuss how τc varies between visible and infrared wavelengths. Figure 8.8 shows the ratio of cloud optical depths at visible (0.65 µm) to thermal infrared (11 µm) wavelengths, namely τc,0.65 and τc,11 , as a function of cloud effective radius re . For both liquid and ice clouds, τc,0.65 exceeds τc,11 for small values of re but the discrepancy quickly diminishes with increasing re until it virtually disappears for re >∼ 15 µm. This is as expected from Mie’s solution (Sect. 5.1.4). Cloud effective radii well exceeding 11 µm may be interpreted in the framework of the geometrical optics regime (Sect. 5.1.4.7) for both τc,0.65 and τc,11 . In the geometrical optics limit, the extinction cross section depends neither on wavelength nor refractive index and thus τc,0.65 ≈ τc,11 . For smaller values of re , τc,11 reduces with re while τc,0.65 remains in the geometrical optics regime, resulting in an excess of τc,0.65 over τc,11 as shown in Fig. 8.8. The precise shape of the curves plotted in Fig. 8.8 would change if the microphysical assumption is altered. See Sect. 13.2 for detailed descriptions of the microphysical models currently used for defining re .

8.3 Visible/Near-Infrared Properties of Clouds Fig. 8.8 The ratio of visible (0.65 µm) to thermal infrared (11 µm) cloud optical depths as a function of effective radius for liquid (solid) and ice (dashed) particles

185 Visible/infrared optical depth ratio

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Fig. 8.9 The R0.65 -R2.3 diagram of a cloud optical depth with a contour interval of 2 and b cloud effective radius for warm clouds with a 2-µm interval. Section 13.2 provides details on radiativetransfer model setups

8.3.3 Retrieval of Cloud Optical Depth and Effective Radius A methodology was proposed by Nakajima and King (1990) to retrieve the optical depth and effective radius of low clouds from visible and near-infrared observations. The idea is based on the spectral sensitivities of reflectance to τc,ν and re as shown by Figs. 8.6 and 8.7. The overall strategy is applicable to both ice and warm clouds and has been adopted for the MODIS cloud optical property product (Platnick et al. 2017) as well as GEO cloud products (e.g., Letu et al. 2019). Figure 8.9 shows a version of the Nakajima and King (1990) diagram, that is, contours of cloud optical depth and effective radius mapped on the R0.65 -R2.3 plane, where visible reflectance at 0.65 µm (R0.65 ) and near-infrared reflectance at 2.3 µm (R2.3 ) are obtained from radiative transfer simulations (see Sect. 13.2 for details). Cloud optical depth (Fig. 8.9a) increases in gradation from left to right, implying that

186

8 Visible/Near-Infrared Imaging -3

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P3 for a given brightness temperature (top right panel). Repeating this procedure to the whole range of R would result in a continuous distribution of the posterior probability as plotted in the bottom right panel. In short, the prior probability distribution (bottom left) is narrowed into

11.3 Inversion Models

261 R=R1 R=R2 R=R3

P(T b|R)xP(R) /N

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Rain rate

Fig. 11.2 Schematic illustration of the Bayesian estimation. The likelihood P(Tb,ν |R) for selected rain rates of R1 , R2 , and R3 (top left) multiplied by the prior probability P(R) (bottom left) yields rescaled distributions (11.13) (top right). Repeating this procedure for all Rs for a given observation results in a continuous curve of the posterior probability (bottom right)

a posterior probability distribution (bottom right) that satisfies a given observation within uncertainty. This is the concept at the heart of the Bayesian estimation. A satellite data product is, in most cases, desired to be just a number rather than the whole probability distribution itself. Multiple ways exist to pull a representative value out of the probability distribution. Calculating the expected value [see (11.14) below] is one of preferred approaches, while finding the mode is another. The Bayesian estimation is called the maximum a posterior (MAP) estimation when the optimal solution is defined as the mode of the posterior probability distribution, that is, an estimate that is most likely to occur for a given observation. The prior distribution P(x) is often assumed to be Gaussian, just as is the likelihood, for the sake of computational ease. The Gaussian assumption, however, is not always a justifiable model of P(x). The probability distribution of precipitation is highly non-Gaussian, heavily skewed toward small values. A practical approach to flexibly accommodate a non-Gaussian P(x) into Bayesian algorithms, originally proposed by Kummerow et al. (1996), is to use an a priori database of synthetic hydrometeor profiles constituted of a broad spectrum of precipitating clouds. The expected value of x for a given observation y,  x =

x P(x|y)dx ,

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11 Mathematical Basis of Retrieval Algorithms

is rewritten with Bayes’ theorem (11.10) by x =

1 N

 x P(y|x)P(x)dx .

The integral is then replaced by the summation over the whole database ensemble using (11.12) as, 1  xi P(y|xi ) N i   1  1 = xi exp − [y − ys (xi )]T (O + S)−1 [y − ys (xi )] , (11.14) N i 2

x ≈

where i = 1, 2, . . . refers to the ensemble members. The key assumption in (11.14) is that the database ensemble reasonably represents the probability distribution of precipitation occurrence P(x) realized in nature. In the latest version of the operational GPM microwave precipitation algorithm (the Goddard Profiling algorithm or GPROF), an a priori database consisting of observed precipitation profiles from past GPM DPR and GMI measurements is implemented in a Bayesian estimation algorithm (Randel et al. 2020).

11.3.2 Maximum Likelihood Estimation The prior probability distribution P(x), when appropriately specified, assures that the Bayesian estimation is mathematically exact in light of Bayes’ theorem. In practice, however, P(x) is rarely known in advance with confidence. Given that the precise form of P(x) is intrinsically unknown, it is also an acceptable tactic to simply assume that P(x) is constant. In a popular inversion technique known as the maximum likelihood estimation, the solution is obtained by maximizing the posterior probability distribution for a given observation under a fixed prior probability. As such, the maximum likelihood estimation is equivalent to the MAP estimation with a uniform prior probability distribution. Procedures of the maximum likelihood estimation are shown in Fig. 11.3. The resulting distribution of the posterior probability (P2 > P1 > P3 ) differs from the previous case (P1 > P2 > P3 ) shown in Fig. 11.2 because different assumptions of the prior probability distribution affect the way errors propagate. Since heavy rains are less frequent than light rains, the assumption of a uniform prior distribution could lead to an overestimation of rain rate from statistical perspectives (recall the high chance of false arrests in the example anecdote in Sect. 11.2).

11.3 Inversion Models

263

P(T b|R)

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R=R1 R=R2 R=R3

Likelihood

P2 P1

P3 Tb

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Prior Probability

P(R)

P(R|Tb)

P2

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P1

P3

R1 R2 R3

Rain rate

R1 R2 R3

Rain rate

Estimate

Fig. 11.3 As Fig. 11.2 but for the maximum likelihood estimation

11.3.3 Deterministic Estimation An error in the prior probability distribution could propagate into a retrieval bias through the insufficiency in information content. For an extreme example, if your broken instrument ended up in sending you nothing but random signals (i.e., no information from observations), the posterior probability distribution would be identical to the prior probability distribution as is. To the contrary, the estimation would hardly depend on the prior probability distribution when the observation is linked one-to-one, or deterministically, with the geophysical variable of interest. A simplest example for one-to-one relationship between an observable and a geophysical variable is the Z -R relation for radar measurements (Sect. 10.2.1). The Z -R relation is conveniently built upon the fact that radar reflectivity monotonically increases with rain rate under a fixed DSD assumption. In reality, instrument error and forward model uncertainty (e.g., DSD assumption) always exist, challenging the accuracy of deterministic formulae such as the Z -R relation. It is nonetheless often practically useful to assume that the instrument and model uncertainties are negligibly small. Such inversion strategies denying the need of any error model are termed here as the deterministic estimation. Figure 11.4 illustrates the deterministic estimation interpreted in light of Bayes’ theorem. The likelihood is constituted of a set of delta functions for selected values of R, each of which represents a deterministic forward model Tb,ν = Tb,ν (R). By

11 Mathematical Basis of Retrieval Algorithms R=R1 R=R2 R=R3

P(T b|R)

Likelihood

Likelihood x Prior P(T b|R)xP(R) /N

264

Tb

Obs

Tb

Prior probability does not matter.

P(R|Tb)

P(R)

Posterior Probability

Rain rate

R1 R2 R3

Rain rate

Estimate

Fig. 11.4 As Fig. 11.2 but for the deterministic estimation

design, a specific form of the prior probability distribution needs not to be known, which is a practical strength of the deterministic estimation. The posterior probability distribution for a given observation is also a delta function, ruling out all possibilities but the “truth”. Utility of the deterministic estimation ranges from prescribed analytic functions such as the Z -R relation to sophisticated inversion models based on huge multivariate lookup tables (LUTs). An obvious drawback is that the deterministic estimation potentially skews the statistics because of the zero-uncertainty assumption: if the chance of failure in the Alien detection was ignored, the rate of false arrest as high as 93% (11.3) would not be explained at all.

11.4 Summary This chapter may be concluded briefly as follows. 1. Satellite retrieval algorithms are an inverse problem solver to find geophysical variables from electromagnetic signals received by satellite instruments. Bayes’ theorem serves as a mathematical framework in which different inversion models are interpreted.

11.4 Summary

265

2. The Bayesian estimation fully exploits the idea central of Bayes’ theorem, requiring the prior probability distribution to be prescribed in advance. The Bayesian estimation is mathematically robust, while challenged by the intrinsic difficulty that our knowledge on the prior probability distribution is generally limited. 3. The maximum likelihood estimation is a special case of the Bayesian estimation, in which the solution is obtained by maximizing the posterior probability distribution for a given observation under a uniform prior distribution. The Bayesian estimation is reduced to the deterministic estimation when the instrument and forward model uncertainties are neglected. The maximum likelihood and deterministic estimations are of great practical utility in that the prior probability distribution does not bother, while susceptible to statistical bias due to the simplifications. Unbiased estimation with uniform global coverage is crucial for some applications, while low latency and high spatial/temporal resolution are prioritized for others. It is, however, difficult to meet both the requirements at the same time. A Bayesian algorithm with a well-tuned priori probability distribution would in theory offer statistically unbiased estimates, but would likely underestimate the threat of destructive storms because extreme rainfall events are far rarer, and thus less weighted in Bayesian estimates than modest rains. The forward model is typically built with radiative transfer simulations that produce synthetic satellite measurements for a range of input variables. An alternative approach is empirical prediction models trained by machine learning (e.g., neural network and random forest algorithms). Machine learning also has the potential of exploring new strategies to solve the inversion problems. Thanks to recent advances in the computational capabilities of processing big data, satellite retrieval strategy is being diversified rapidly with emerging methodologies. The mathematical concepts outlined in this chapter nonetheless remain useful in the construction of overall algorithmic structure.

References Kummerow C, Olson W, Giglio L (1996) A simplified scheme for obtaining precipitation and vertical hydrometeor profiles from passive microwave sensors. IEEE Trans Geosci Remote Sens 34:1213–1232. https://doi.org/10.1109/36.536538 Masunaga H, Matsui T, Tao W, Hou AY, Kummerow CD, Nakajima T, Bauer P, Olson WS, Sekiguchi M, Nakajima TY (2010) Satellite data simulator unit. Bull Amer Meteor Soc 91:1625–1632. https://doi.org/10.1175/2010BAMS2809.1 Randel DL, Kummerow CD, Ringerud S (2020) The Goddard profiling (GPROF) precipitation retrieval algorithm. In: Levizzani V, Kidd C, Kirschbaum DB, Kummerow CD, Nakamura K, Turk FJ (eds) Satellite precipitation measurement: volume 1. Springer International Publishing, pp 141–152. https://doi.org/10.1007/978-3-030-24568-9_8 Rodgers CD (2000) Inverse methods for atmospheric sounding. World Scientific Stephens GL, Kummerow CD (2007) The remote sensing of clouds and precipitation from space: a review. J Atmos Sci 64:3742–3765. https://doi.org/10.1175/2006JAS2375.1

Part IV

Applications

The previous chapters are dedicated mainly to theoretical aspects of satellite remote sensing. The last part of this book focuses more on the application aspects of cloud and precipitation measurements from space. Efforts have been made worldwide to integrate observations from a constellation of satellites into global datasets of clouds and precipitation. Such datasets are outlined in the first chapter of Part IV. The second chapter offers a brief overview of satellite data simulators, which are software package to carry out forward-model simulations of satellite measurements. Synthetic radiances and spectra presented in this book were obtained through one of those satellite simulators. The model assumptions and setup parameters for the plots throughout the book are summarized at the end of the chapter.

Chapter 12

Global Datasets of Clouds and Precipitation

Satellite measurements play crucial roles in the construction of global observation datasets of clouds and precipitation. Many of such datasets are open to public free of charge wherever you live in the world. In general, one only has to fill in a web-based user registration form (which is sometimes not required) to gain online access to the data. Although it all looks easy, you might feel lost in how to get started with satellite data if you are new to it. The challenge is not that you cannot find what you want but that there are so many choices it is hard to decide which one best suits your needs. This chapter is meant to be a concise guidance for those in need of a little assistance. It is not attempted in this chapter to present a complete list of satellite-based cloud/precipitation products, since otherwise the reader would be inundated with excessive information. Instead, we will put focus on a limited number of datasets that are widely used in the science and commercial/industrial sectors. Following an introduction to the data processing levels, selected data products are summarized without going too much into technical details.

12.1 Data Processing Levels Satellite data are classified by the data processing level (Levels 1–4). The overall definition as summarized in Fig. 12.1 is universal, with the caveat that different dataproducing agencies define subcategories (e.g., Levels 1A and 1B) in their own ways. There is a classifier called Level 0 reserved for raw satellite data, but Level 0 data are normally limited for in-house use only. Level 1 data are observed parameters such as radiance (Iν ), brightness temperature (Tb,ν ), and attenuation-uncorrected effective radar reflectivity (Z e,m ). Level 1 data are stored at sensor resolution (i.e., on a footprint-by-footprint basis) along with

© Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_12

269

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12 Global Datasets of Clouds and Precipitation

Level 3

Level 2

Level 1 Observed quantities

Geophysical Variables

Geophysical Variables

(I , Tb , Z e,m , etc.)

Level 4 Geophysical Variables processed further

mapped onto a

at sensor resolution

at sensor resolution

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longitude-latitude grid

external data

Retrieval

Spatial/temporal

Algorithm

Averaging

Analysis/ Data Assimilation

Fig. 12.1 Brief summary of data processing levels

ancillary information including geolocation and time stamps. Calibration coefficients are provided together or pre-applied to the parameters. Level 1 products are aimed at expert users who run their own retrieval algorithm or in need of observed parameters rather than retrieved variables for their analysis. Level 2 products contain geophysical variables (e.g., cloud optical depth, LWC, and surface rain rate) retrieved from Level 1 data. Earth observing satellite missions typically have their own science team assigned to develop a standard algorithm to produce Level 2 products. Level 2 products consist of instantaneous geophysical variables at instrument resolution in the same manner as Level 1 data do, fully exploiting the sampling capability of the sensor. General users may not necessarily require instantaneous, high-resolution observations as they are. Users may instead expect satellite datasets to be formatted more like reanalysis data in that all variables are neatly sorted on a regular geographical grid. Level 3 products exist to meet such user needs, offering geophysical variables mapped onto a longitudinal-latitudinal grid. The grid averaging generally involves a temporal mean as well (monthly mean, for instance), while daily or subdaily finegridded Level 3 products exist. The latter case is in effect regarded as instantaneous observations and is sometimes classified as Level 2 despite that it is a gridded product. Level 4 refers to an enhanced product containing the satellite-retrieved variables processed further with a numerical model or an analysis model, often blended with a variety of observations from external sources. Level 4 is the category for high-level products not as widely produced as Levels 1–3 but could grow more common in the future.

12.2 Global Cloud Datasets Clouds are a key element of Earth’s energy budget through the interactions with atmospheric radiation, but remain among the largest sources of uncertainty in future climate projections. Global datasets of cloud physical properties are invaluable for understanding the climate system in further depth and for the assessment of climate model performance. A climate data record (CDR) of cloud properties derived

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271

from generations of satellite observations is in particular of critical value for climate change monitoring. First summarized in this section are cloud variables available from satellite observations, followed by a quick overview of existing global cloud datasets.

12.2.1 Cloud Variables Table 12.1 summarizes cloud variables derived from satellite observations. The variable most frequently used is probably cloud amount.1 Cloud amount, however, is difficult to define without ambiguity because cloud amount varies with the detectability of clouds. The total cloud amount over the globe is estimated from a collection of satellite observations as 0.68 ± 0.03 for clouds with optical depths larger than 0.1, while this number could exceed 0.7 when subvisible cirrus is included or, conversely, much smaller for a conservative τc threshold (Stubenrauch et al. 2013). Cloud optical depth is an important variable, closely related to cloud reflectance at visible wavelengths. Cloud top temperature is another key parameter that is readily converted to cloud top height or cloud top pressure if air temperature profile is known. Cloud effective radius (see Sect. 8.3.1 for definition) and cloud thermodynamic phase (liquid, ice, or mixed phase) convey valuable information on the cloud microphysical state. There are ways to estimate the vertically integrated mass of condensates (cloud LWP and IWP) from passive instruments, while the detailed vertical structure of LWC and IWC is obtained only from radars and lidars. The physical principles underlying the derivation of each variable are described in some detail in earlier sections of this book (see the rightmost column of Table 12.1).

12.2.2 Cloud Datasets The International Satellite Cloud Climatology Project (ISCCP) D-series (Rossow and Schiffer 1999) is arguably the most widely used among global cloud datasets. The ISCCP products are constructed with visible and infrared measurements from a number of operational GEO and LEO satellites to maximize the spatial and temporal coverages. Nine cloud types (e.g., cirrus, deep convection, and stratocumulus) are classified in terms of cloud top pressure and optical depths. The ISCCP D-series consists of DX (quasi Level 2) and D1/D2 (Level 3) products. The DX product is a high-resolution (30-km), 3-hourly dataset of selected cloud variables sorted by individual satellites. The D1 product is 3-hourly, 280-km (2.5◦ ) equal-area gridded data with all satellites integrated together, which is accumulated to monthly-mean 3-hourly statistics in the D2 product. The series-D data period 1 Similar

terms such as cloudiness, cloud cover, and cloud fraction are used interchangeably with cloud amount in most (although not all) cases.

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12 Global Datasets of Clouds and Precipitation

Table 12.1 Typical cloud variables available from satellite data products. Vis, NIR, TIR, and MW denote visible, near infrared, thermal infrared, and microwave Cloud variables Observables Instruments/Methods Cloud amount Cloud cover profile Cloud optical depth

Cloud emissivity Cloud LWP/IWP

Vis/NIR Iλ and/or TIR Tb Radar Z e and lidar β Vis Iλ TIR Tb Vis/NIR Iλ TIR Tb TIR Tb TIR and CO2 -band Tb TIR and CO2 -band Tb Vis/NIR Iλ

Cloud LWC/IWC profile

MW Tb Radar Z e and lidar β

Cloud effective radius Cloud top temperature

Cloud thermodynamic phase TIR Tb TIR Tb Polarized Vis/NIR Iλ Lidar depolarization

Vis/IR imager W-band radar and lidar Vis/IR imager (Sect. 8.3.3) Split window (Sect. 7.3.1) Vis/IR imager (Sect. 8.3.3) Split window (Sect. 7.3.1) Single TIR channel or split window CO2 slicing (Sect. 7.3.3) CO2 slicing (Sect. 7.3.3) Vis/IR imager (Sects. 8.3.1 and 8.3.3) Microwave radiometer (Sect. 9.3.2) W-band radar (Sect. 10.2.1) and/or lidar (Sect. 10.3.2) Diagnosed with cloud top temperature TIR spectral difference (Sect. 7.3.2) Polarimeter (Sect. 8.4.4) Polarimetric lidar (Sect. 10.3.3)

spans from mid 1983 to the termination of the NASA ISCCP processing at the end of 2009. As ISCCP data production was transitioned from NASA to NOAA, the ISCCP H-series (Young et al. 2018) has been released with an enhanced spatial resolution (10 km and 1.0◦ instead of 30 km and 2.5◦ ) for an extended period from 1983 to beyond 2009. Figure 12.2 presents the 2015 annual-mean cloud amount from the ISCCP Hseries Level-3 monthly (HGM) product for selected cloud types, including those most important in modulating Earth’s radiation budget (e.g., cirrus and stratocumulus) and others closely related to precipitation (deep convection and nimbostratus). It is noteworthy that the cirrus amount (Fig. 12.2a) appears to be roughly a “mirror image” against the stratocumulus amount (Fig. 12.2e) as if they were mutually exclusive. This may be explained in a climatological context, given that tropical cirrus clouds are associated with a moist upper troposphere typical of convectively active regions (Luo and Rossow 2004), while maritime stratocumulus clouds prefer a stably stratified atmosphere over cold ocean (Klein and Hartmann 1993). At the same time, a practical limitation of visible/infrared imaging might be partly involved as well that low-level clouds could be underrepresented when frequently overcast by higher-level clouds as is the case in the deep Tropics. The likely underestimation of low-cloud amount is more evident in Fig. 12.2e, showing that shallow cumulus, prevailing across the global ocean, is falsely suppressed in cloud amount within the ITCZs. Overlapped

12.2 Global Cloud Datasets

273

Fig. 12.2 2015 annual-mean cloud amount from the ISCCP HGM product (Young et al. 2018) for selected cloud types of a cirrus, b deep convection, c altostratus, d nimbostratus, e stratocumulus, and f cumulus. An area encompassing the Himalayas and Tibetan Plateau has no data for low-level clouds (e and f) because of altitude

clouds are better captured by active sensors such as CloudSat CPR and CALIPSO CALIOP as will be detailed later. Early generations of the GEO satellites did not have an instrument with nearinfrared channels indispensable for retrieving cloud effective radius (Sect. 8.3.3). For this reason, cloud effective radius is fixed at a prescribed value in the ISCCP algorithm. Polar-orbiting meteorological satellites, on the other hand, typically carry an imager, notably AVHRR given its availability over decades, that is equipped with a minimum set of channels necessary for deriving essential cloud variables including droplet radius. The Pathfinder Atmospheres-Extended (PATMOS-x) dataset (Heidinger et al. 2014) exploits this advantage of a long-term record of AVHRR cloud observations. A similar approach has been explored in recent data products such as the Climate Monitoring Satellite Application Facility (CM SAF) Cloud, Albedo And Surface Radiation dataset from AVHRR data second edition (CLARAA2) (Karlsson et al. 2017) and ESA’s Cloud_cci (Cloud Climate Change Initiative) (Stengel et al. 2017). Infrared sounders such as TOVS and HIRS and modern multi-spectral imagers, MODIS in particular, benefit from additional wavelength bands including the 15-µm CO2 band. The CO2 band enables a reliable estimation of cloud top temperature and effective cloud emissivity through the CO2 slicing method (Sect. 7.3.3). Detailed descriptions of different satellite cloud datasets may be found in a thorough review by Stubenrauch et al. (2013).

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Although ISCCP clouds are sorted into nine cloud types differentiated by cloud top pressure and optical depth, visible/infrared imagers are not an optimal tool for observing vertical cloud profiles. As noted above, low- and mid-level clouds are undetectable when concealed by overlying upper-level clouds. Cloud base height is difficult to constrain by passive sensors alone, which could be an additional source of observational uncertainty in Earth’s radiation budget (Stephens et al. 2012). In contrast, W-band radar and lidar, namely CloudSat CPR and CALIPSO CALIOP, offer a detailed picture of vertical cloud structure at the expense of a vastly limited sampling frequency in comparison with the rich collection of visible/infrared imagers. The CALIPSO lidar is able to detect tenuous clouds without the ability of seeing through deep inside a thick cloud layer, while the CloudSat radar is less susceptible to the attenuation of echo signals but is less sensitive to thin cirrus than CALIOP (Sect. 10.3). Microwave radiometry from space offers a complementary estimation of LWP over ocean to the visible/near-infrared retrieval of LWP. The Multisensor Advanced Climatology of Liquid Water Path (MAC-LWP) dataset (Elsaesser et al. 2017) provides long-term, bias-corrected estimates of cloud LWP and total (cloud plus rain) LWP from all available microwave radiometers.

12.3 Global Precipitation Datasets Precipitation is not only a potential cause of devastating natural hazards such as flush floods but also a critical component of Earth’s energy budget. Various independent efforts have been made to construct global precipitation datasets integrating observations from multiple satellites and rain gauge networks. The retrieval algorithms are different in technical details but share basic strategies as reviewed next. The global and zonal-mean distributions of surface precipitation from selected products will be also presented. Note that all information provided is latest at the time of this writing (January 2022) but may be outdated in the future.

12.3.1 Multi-satellite Precipitation Datasets Many, if not all, precipitation products provide a single variable of surface precipitation rate, marking a contrast to the cloud products storing a variety of variables (Table 12.1). This does not necessarily assure that satellite precipitation algorithms are built simpler than cloud algorithms. Microwave radiometry is instrumental for measuring precipitation from space, with the caveat that more assumptions are required for microwave precipitation retrieval over land than over ocean (Sect. 9.4). The satellite network of microwave observations is not as dense as that of visible and infrared imagery, mainly due to the fact that no GEO satellite carries microwave instruments.

12.3 Global Precipitation Datasets

275

Table 12.2 Selected list of multi-satellite (with or without gauge data) precipitation products. “Global” means the full 90◦ S–90◦ N coverage, “Long term” refers to a period dating back to the 1980s or earlier, and “TRMM/GPM” stands for a period within the TRMM and GPM era (1998–). References were chosen from recent articles if multiple sources exist in the literature so that the reader is given access to the latest information (as of January 2022) on each product Product name Resolution Period Coverage References Long term Long term

50◦ S/N land Global

CMORPH GPCP monthly

0.05◦ , daily etc. 2.5◦ , monthly/pentad 8 km, half-hourly 0.5◦ (v3)

TRMM/GPM Long term

60◦ S/N Global

GPCP daily

0.5◦ (v3)

TRMM/GPM

Global

GSMaP

0.1◦ , hourly

TRMM/GPM

60◦ S/N

HOAPS IMERG

0.5◦ , Long term 6-hourly/monthly 0.1◦ , half-hourly TRMM/GPM

Global

MSWEP PERSIANN PERSIANN CCS PERSIANN CDR

0.1◦ , 3-hourly 0.25◦ , hourly 0.04◦ , hourly 0.25◦ , daily

Global 60◦ S/N 60◦ S/N 60◦ S/N

CHIRPS CMAP

Long term TRMM/GPM TRMM/GPM Long term

80◦ S/N ocean

Funk et al. (2015) Xie and Arkin (1997) Xie et al. (2017) Huffman et al. (2019) Huffman et al. (2001) Kubota et al. (2020) Andersson et al. (2010) Huffman et al. (2020) Beck et al. (2019) Hsu et al. (1997) Hong et al. (2004) Ashouri et al. (2015)

Table 12.2 is a selected list of multi-satellite precipitation products combined with or without ancillary sources of observation data. The retrieval strategy adopted for CMAP, CMORPH, GPCP, GSMaP, IMERG, and TMPA2 consists of three distinct components. 1. Surface precipitation rate is retrieved from multi-platform microwave radiometer and sounder measurements. 2. Infrared measurements are then employed to fill in the temporal and spatial gaps where microwave observations are unavailable. 3. The satellite precipitation estimates over land are adjusted to ground-based measurements from rain gauge networks. The first component provides the baseline estimation of precipitation. Passive microwave instruments available for this module are carried by polar orbiters (e.g., Aqua AMSR-E, DMSP SSM/I and SSMIS, GCOM-W AMSR2, and NOAA and MetOp AMSU/MHS) as well as by two sun-asynchronous spacecrafts (TRMM TMI and GPM GMI). Sun-asynchronous orbits have the advantage of crossing all polar 2

The TMPA (TRMM 3B42) product (Huffman et al. 2007) now has been replaced by IMERG.

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orbits, offering an opportunity for the inter-calibration of all participating instruments (Skofronick-Jackson et al. 2017). Another merit of the TRMM and GPMcore satellites is high-quality radar or combined radar-radiometer measurements of precipitation. The GSMaP passive microwave algorithm uses a LUT constructed off-line with TRMM PR and GPM DPR observations (Kubota et al. 2020). An a priori database constituted of precipitation profiles from the GPM combined radarradiometer product (Grecu et al. 2016) is implemented in the current version of the GPROF microwave algorithm (Randel et al. 2020). The GPROF algorithm has been adopted for the microwave algorithm of GPCP (Huffman et al. 2019), CMORPH (Xie et al. 2017), and IMERG (Huffman et al. 2020). The second component is somewhat tricky, since infrared radiance only detects cloud tops with little sensitivity to precipitation deep inside or beneath the cloud layer. The GOES Precipitation Index or GPI (Arkin and Meisner 1987) and its variants (e.g., OLR precipitation index or OPI) are an empirical infrared-based proxy of precipitation. CMAP, one of the oldest products of this kind, is a blend of rainfall estimates from different sources including GPI (Xie and Arkin 1997). PERSIANN is built on a neural network technique with which a statistical relationship is established between microwave and infrared observations (Hsu et al. 1997) (note that microwave measurements are used only for training purposes and are not incorporated in the end product of PERSIANN). The infrared module of GPCP was based on GPI/OPI in early versions but has been evolved to more sophisticated approaches (PERSIANN CDR is incorporated for processing GEO infrared data in GPCP version 3). CMORPH, GSMaP, and IMERG pursue a different strategy based on Lagrangian cloud tracking, which enables a high-resolution space-time extrapolation from sporadic microwave measurements. A methodology to derive motion vectors from GEO infrared imagery was originally developed for CMORPH (Joyce et al. 2004) and was later refined with a Kalman filtering technique for GSMaP (Ushio et al. 2009). IMERG combines PERSIANN CCS (Hong et al. 2004) with microwave precipitation estimates through a Kalman filtering approach. The third component is intended to correct potential bias in satellite precipitation estimates over land exploiting rain gauge data as the reference. GPCC (Becker et al. 2013) and CPC (Xie et al. 2007) each produce global gridded analysis products by integrating existing rain gauge networks. CMORPH and GSMaP use the CPC gauge analysis, while GPCP and IMERG rely on the GPCC gauge data. There are other products that employ a different approach. HOAPS precipitation is derived along with other water-budget parameters from microwave (SSM/I and SSMIS) observations over ocean without the help of infrared information. In contrast, the CHIRPS product is a blend of infrared and gauge estimates of precipitation over land. MSWEP precipitation is obtained as a weighted ensemble of various sources of data including gauge analysis, infrared data, reanalysis data, and existing precipitation products such as CMORPH and GSMaP. Among the frequently asked questions from colleagues searching for a precipitation dataset for use in their work is “which product is the best?”. If “best” is meant to be most accurate, my answer would be “I can’t tell”, partly for diplomatic reasons but mostly because I honestly do not know. Satellite retrieval depends on many assump-

12.3 Global Precipitation Datasets

277

tions that are not always valid, as discussed throughout Part III of this book. Rain gauge networks are limited in spatial representativeness where deployed sparsely (or not deployed at all). Precipitation as we can measure hinges upon such an elusive “truth”. It is hard to tell which product is more accurate than another if no reliable absolute reference exists in the first place. It is nevertheless possible to give some practical recommendation regarding which product is more suited than others for a certain application. As evident from Table 12.2, different products have different spatial/temporal resolutions and coverages. The data periods may be grouped into two categories of “long term” (dating back to the 1980s or earlier) and “TRMM/GPM era” (beginning in 1998 or later). Products with a very high temporal resolution (hourly or half-hourly) all limit themselves to the TRMM/GPM era, while the long-term products are 3-hourly (MSWEP), 6-hourly (HOAPS), or targeted on daily or even longer time scales (CMAP, CHIRPS, GPCP, and PERSIANN CDR). The long term products are crucial for climatological studies analyzing the precipitation CDR, and the high-resolution datasets are optimal for meteorological and hydrological applications focused on the last two decades. CMORPH, GSMaP, IMERG, and PERSIANN (CCS) each have a near-real-time product distributed with a latency within 4 h, offering invaluable observation data for operational weather forecasts.

12.3.2 Global Distribution of Precipitation Given that the precipitation “truth” is difficult to define as discussed above, it is beneficial to investigate to what extent different precipitation products agree with or differ from one another. In this short subsection, selected products are compared in terms of the global distribution and zonal mean of precipitation. More thorough reports on the current status of global precipitation datasets may be found in Sun et al. (2018) and Roca et al. (2021). Figure 12.3 shows the global map of 2015 annual-mean precipitation from selected products. All the datasets share a similar geographical pattern punctuated by the intertropical convergence zones (ITCZs) and mid-latitude storms over the northern Pacific and Atlantic. The magnitude of ITCZ precipitation, however, is evidently larger in some products than others, while the inter-product bias in the tropics can be reversed in sign at southern mid-latitudes (60◦ S-45◦ S). Global precipitation products, even though based on a similar set of input observations (LEO microwave sensors, GEO infrared imagers, and gauge networks), are subject to systematic errors that do not easily average out in annual climatology. It is noted that some of the datasets are not mutually independent since certain products have been calibrated against another (e.g., CMORPH over ocean and PERSIANN CDR are adjusted on a monthly basis to an earlier version of GPCP). To facilitate a quantitative comparison, zonal-mean precipitation is plotted in Fig. 12.4. The ITCZ precipitation maximum and subtropical minima on its both sides emerge in a coherent manner, except that the peak value varies among the products.

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Fig. 12.3 2015 annual-mean surface precipitation from selected datasets of a CMORPH (biascorrected or CRT) version 1.0, b GPCP version 3.1, c GSMaP version 7, d HOAPS version 4 (ocean) and CHIRPS version 2.0 (land, 50◦ S–50◦ N), e IMERG version 6, and f PERSIANN CDR. All products have been adjusted in resolution to a common 1◦ × 1◦ grid. The remapped data are provided by courtesy of Fumie A. Furuzawa (b) Zonal mean rainfall (Land)

(a) Zonal mean rainfall (Ocean) 10

10 CMORPH GPCP GSMaP HOAPS IMERG PERSIANN

Rainfall [mm/d]

8 6

6

4

4

2

2

0 -60

-45

-30

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30

CMORPH GPCP GSMaP CHIRPS IMERG PERSIANN CPC (gauge) GPCC (gauge)

8

45

60

0 -60

-45

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Fig. 12.4 Zonal-mean precipitation for 2015 calculated for the products shown in Fig. 12.3 for a ocean and b land. Two gauge datasets (CPC version 1.0 and GPCC Full Data version 2020) are plotted together in b for comparison

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Oceanic precipitation begins to exhibit a significant inter-product spread as latitude increases beyond 30◦ S/N, potentially attributable to various reasons including the ad hoc separation of rainfall from cloud LWP in microwave radiometry (e.g. Stephens and Kummerow 2007) and intrinsic difficulties in the treatment of frozen precipitation. The spread is smaller over land than over ocean, likely as the result of the gauge adjustment. The two gauge products (CPC and GPCC) do not agree well with each other at low latitudes, where satellite-based products are split into two groups as well depending partly on which gauge products they are adjusted to.

References Andersson A, Fennig K, Klepp C, Bakan S, Graßl H, Schulz J (2010) The hamburg ocean atmosphere parameters and fluxes from satellite data - HOAPS-3. Earth Sys Sci Data 2:215–234. https://doi. org/10.5194/essd-2-215-2010 Arkin PA, Meisner BN (1987) The relationship between large-scale convective rainfall and cold cloud over the western hemisphere during 1982–84. Mon Wea Rev 115:51–74. https://doi.org/ 10.1175/1520-0493(1987)1152.0.CO;2 Ashouri H, Hsu KL, Sorooshian S, Braithwaite DK, Knapp KR, Cecil LD, Nelson BR, Prat OP (2015) PERSIANN-CDR: daily precipitation climate data record from multisatellite observations for hydrological and climate studies. Bull Amer Meteor Soc 96:69–83. https://doi.org/10.1175/ BAMS-D-13-00068.1 Beck HE, Wood EF, Pan M, Fisher CK, Miralles DG, van Dijk AIJM, McVicar TR, Adler RF (2019) MSWEP v2 global 3-hourly 0.1◦ precipitation: methodology and quantitative assessment. Bull Amer Meteor Soc 100:473–500. https://doi.org/10.1175/BAMS-D-17-0138.1 Becker A, Finger P, Meyer-Christoffer A, Rudolf B, Schamm K, Schneider U, Ziese M (2013) A description of the global land-surface precipitation data products of the global precipitation climatology centre with sample applications including centennial (trend) analysis from 1901present. Earth Sys Sci Data 5:71–99. https://doi.org/10.5194/essd-5-71-2013 Elsaesser GS, O’Dell CW, Lebsock MD, Bennartz R, Greenwald TJ, Wentz FJ (2017) The multisensor advanced climatology of liquid water path (MAC-LWP). J Climate 30:10,193-10,210. https://doi.org/10.1175/JCLI-D-16-0902.1 Funk C, Peterson P, Landsfeld M, Pedreros D, Verdin J, Shukla S, Husak G, Rowland J, Harrison L, Hoell A, Michaelsen J (2015) The climate hazards infrared precipitation with stations - a new environmental record for monitoring extremes. Sci Data 2(150):066. https://doi.org/10.1038/ sdata.2015.66 Grecu M, Olson WS, Munchak SJ, Ringerud S, Liao L, Haddad Z, Kelley BL, McLaughlin SF (2016) The gpm combined algorithm. J Atmos Oceanic Technol 33:2225–2245. https://doi.org/ 10.1175/JTECH-D-16-0019.1 Heidinger AK, Foster MJ, Walther A, Zhao X (2014) The pathfinder atmospheres - extended AVHRR climate dataset. Bull Amer Meteor Soc 95:909–922. https://doi.org/10.1175/BAMSD-12-00246.1 Hong Y, Hsu KL, Sorooshian S, Gao X (2004) Precipitation estimation from remotely sensed imagery using an artificial neural network cloud classification system. J Appl Meteor 43:1834– 1853. https://doi.org/10.1175/JAM2173.1 Hsu K, Gao X, Sorooshian S, Gupta HV (1997) Precipitation estimation from remotely sensed information using artificial neural networks. J Appl Meteor 36:1176–1190. https://doi.org/10. 1175/1520-0450(1997)0362.0.CO;2

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Huffman GJ, Adler RF, Morrissey MM, Bolvin DT, Curtis S, Joyce R, McGavock B, Susskind J (2001) Global precipitation at one-degree daily resolution from multisatellite observations. J Hydrometeor 2:36–50. https://doi.org/10.1175/1525-7541(2001)0022.0.CO; 2 Huffman GJ, Adler RF, Bolvin DT, Gu G, Nelkin EJ, Bowman KP, Hong Y, Stocker EF, Wolff DB (2007) The TRMM multi-satellite precipitation analysis: quasi-global, multi-year, combinedsensor precipitation estimates at fine scale. J Hydrometeor 8:38–55 Huffman GJ, Adler RF, Bolvin DT, Hsu K, Kidd C, Nelkin EJ, J T, Xie P (2019) Algorithm theoretical basis document (ATBD) for global precipitation climatology project version 3.0 precipitation data. NASA/Goddard Space Flight Center, Greenbelt, MD, USA Huffman GJ, Bolvin DT, Braithwaite D, Hsu KL, Joyce RJ, Kidd C, Nelkin EJ, Sorooshian S, Stocker EF, Tan J, Wolff DB, Xie P (2020) Integrated multi-satellite retrievals for the global precipitation measurement (GPM) mission (IMERG). In: Levizzani V, Kidd C, Kirschbaum DB, Kummerow CD, Nakamura K, Turk FJ (eds) Satellite precipitation measurement: volume 1. Springer International Publishing, pp 343–353. https://doi.org/10.1007/978-3-030-24568-9_19 Joyce RJ, Janowiak JE, Arkin PA, Xie P (2004) CMORPH: a method that produces global precipitation estimates from passive microwave and infrared data at high spatial and temporal resolution. J Hydrometeor 5:487–503. https://doi.org/10.1175/1525-7541(2004)0052.0. CO;2 Karlsson KG, Anttila K, Trentmann J, Stengel M, Fokke Meirink J, Devasthale A, Hanschmann T, Kothe S, Jääskeläinen E, Sedlar J, Benas N, van Zadelhoff GJ, Schlundt C, Stein D, Finkensieper S, Håkansson N, Hollmann R (2017) CLARA-A2: the second edition of the CM SAF cloud and radiation data record from 34 years of global AVHRR data. Atmos Chem Phys 17:5809–5828. https://doi.org/10.5194/acp-17-5809-2017 Klein SA, Hartmann DL (1993) The seasonal cycle of low stratiform clouds. J Climate 6:1587–1606 Kubota T, Aonashi K, Ushio T, Shige S, Takayabu YN, Kachi M, Arai Y, Tashima T, Masaki T, Kawamoto N, Mega T, Yamamoto MK, Hamada A, Yamaji M, Liu G, Oki R (2020) Global satellite mapping of precipitation (GSMaP) products in the GPM era. In: Levizzani V, Kidd C, Kirschbaum DB, Kummerow CD, Nakamura K, Turk FJ (eds) Satellite precipitation measurement: volume 1. Springer International Publishing, pp 355–373. https://doi.org/10.1007/978-3-030-24568-9_20 Luo Z, Rossow WB (2004) Characterizing tropical cirrus life cycle, evolution, and interaction with upper-tropospheric water vapor using Lagrangian trajectory analysis of satellite observations. J Climate 17:4541–4563 Randel DL, Kummerow CD, Ringerud S (2020) The Goddard profiling (GPROF) precipitation retrieval algorithm. In: Levizzani V, Kidd C, Kirschbaum DB, Kummerow CD, Nakamura K, Turk FJ (eds) Satellite Precipitation measurement: volume 1. Springer International Publishing, pp 141–152. https://doi.org/10.1007/978-3-030-24568-9_8 Roca R, Haddad ZS, Akimoto FF, Alexander L, Behrangi A, Huffman G, Kato S, Kidd C, Kirstetter PE, Kubota T, Kummerow C, L’Ecuyer TS, Levizzani V, Maggioni V, Massari C, Masunaga H, Schröder M, Tapiador FJ, Turk FJ, Utsumi N (2021) The joint IPWG/GEWEX precipitation assessment. WCRP report 2/2021. World Climate Research Programme (WCRP), Geneva, Switzerland. https://doi.org/10.13021/gewex.precip Rossow WB, Schiffer RA (1999) Advances in understanding clouds from ISCCP. Bull Amer Meteor Soc 80:2261–2288 Skofronick-Jackson G, Petersen WA, Berg W, Kidd C, Stocker EF, Kirschbaum DB, Kakar R, Braun SA, Huffman GJ, Iguchi T, Kirstetter PE, Kummerow C, Meneghini R, Oki R, Olson WS, Takayabu YN, Furukawa K, Wilheit T (2017) The global precipitation measurement (GPM) mission for science and society. Bull Amer Meteor Soc 98(8):1679–1695. https://doi.org/10. 1175/BAMS-D-15-00306.1 Stengel M, Stapelberg S, Sus O, Schlundt C, Poulsen C, Thomas G, Christensen M, Carbajal Henken C, Preusker R, Fischer J, Devasthale A, Willén U, Karlsson KG, McGarragh GR, Proud S, Povey AC, Grainger RG, Meirink JF, Feofilov A, Bennartz R, Bojanowski JS, Hollmann R (2017) Cloud

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property datasets retrieved from AVHRR, MODIS, AATSR and MERIS in the framework of the cloud_cci project. Earth Syst Sci Data 9:881–904. https://doi.org/10.5194/essd-9-881-2017 Stephens GL, Kummerow CD (2007) The remote sensing of clouds and precipitation from space: a review. J Atmos Sci 64:3742–3765. https://doi.org/10.1175/2006JAS2375.1 Stephens GL, Li J, Wild M, Clayson CA, Loeb N, Kato S, L’Ecuyer T, Stackhouse PW, Lebsock M, Andrews T (2012) An update on Earth’s energy balance in light of the latest global observations. Nat Geosci 5:691–696. https://doi.org/10.1038/ngeo1580 Stubenrauch CJ, Rossow WB, Kinne S, Ackerman S, Cesana G, Chepfer H, Girolamo LD, Getzewich B, Guignard A, Heidinger A, Maddux BC, Menzel WP, Minnis P, Pearl C, Platnick S, Poulsen C, Riedi J, Sun-Mack S, Walther A, Winker D, Zeng S, Zhao G (2013) Assessment of global cloud datasets from satellites: project and database initiated by the GEWEX radiation panel. Bull Amer Meteor Soc 94:1031–1049. https://doi.org/10.1175/BAMS-D-12-00117.1 Sun Q, Miao C, Duan Q, Ashouri H, Sorooshian S, Hsu KL (2018) A review of global precipitation data sets: data sources, estimation, and intercomparisons. Rev Geophys 56:79–107. https://doi. org/10.1002/2017RG000574 Ushio T, Sasashige K, Kubota T, Shige S, Okamoto K, Aanashi K, Inuoe T, Takahashi N, Iguchi T, Kachi M, Oki R, Morimoto T, Kawasaki ZI (2009) A Kalman filter approach to the global satellite mapping of precipitation (GSMaP) from combined passive microwave and infrared radiometric data. J Meteor Soc Japan 87A:137–151. https://doi.org/10.2151/jmsj.87A.137 Xie P, Arkin PA (1997) Global precipitation: a 17-year monthly analysis based on gauge observations, satellite estimates, and numerical model outputs. Bull Amer Meteor Soc 78:2539–2558. https://doi.org/10.1175/1520-0477(1997)0782.0.CO;2 Xie P, Chen M, Yang S, Yatagai A, Hayasaka T, Fukushima Y, Liu C (2007) A gauge-based analysis of daily precipitation over east asia. J Hydrometeorol 8:607–626. https://doi.org/10. 1175/JHM583.1 Xie P, Joyce R, Wu S, Yoo SH, Yarosh Y, Sun F, Lin R (2017) Reprocessed, bias-corrected CMORPH global high-resolution precipitation estimates from 1998. J Hydrometeor 18:1617–1641. https:// doi.org/10.1175/JHM-D-16-0168.1 Young AH, Knapp KR, Inamdar A, Hankins W, Rossow WB (2018) The international satellite cloud climatology project H-series climate data record product. Earth Syst Sci Data 10:583–593. https://doi.org/10.5194/essd-10-583-2018

Chapter 13

Satellite Data Simulators

Satellite data simulators, a general term for a software package containing a set of atmospheric radiative transfer codes and a user interface, provide synthetic satellite measurements (observables such as radiance and/or retrieved variables) for geophysical variables input by users. In other words, a satellite data simulator generates virtual observations from hypothetical satellite instruments. This chapter offers a concise introduction to satellite data simulators with a selected list of simulator packages. The second section is devoted to detailed descriptions of model setups in the radiative transfer simulations carried out for this book.

13.1 Overview There are a range of applications in which satellite data simulators play a crucial role. Satellite data simulators are, for example, an indispensable tool for the development of retrieval algorithms for future satellite programs. This is especially so when the satellite program involves new instrument technology that does not currently exist. Satellite data simulators provide synthetic measurements of hypothetical satellite instruments “flying” over a simulated atmosphere, serving as a testbed for the algorithm being developed. Satellite data simulators are also instrumental for the assessment of numerical atmospheric models in comparison with satellite observations. The satellite-based diagnosis of model performance has been often carried out in terms of geophysical variables such as surface rain rate. This is a useful approach but could be misleading for two reasons. First, observation data have its own source of uncertainty arising from a variety of assumptions in the retrieval algorithm. Indeed, different precipitation datasets do not precisely agree with one another even after being averaged to an annual mean (Sect. 12.3.2). Second, surface precipitation is more of the effect of convective dynamics than the cause for it, so tuning the model just to match observed © Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5_13

283

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13 Satellite Data Simulators

Level 1 Observation

Satellite Retrieval

Compare

(Inversion model) Synthetic Observation

Agree? Level 2/3/4 Data product

Satellite Simulator (Forward model) Cloud model

Fig. 13.1 Schematic illustration of satellite simulator as a model evaluation tool. Adapted from Masunaga et al. (2010) with modifications. ©American Meteorological Society

precipitation has a risk of distorting the model physics to get the right answer for wrong reasons. Satellite data simulators offer an alternative way to assess the model performance against satellite observations without the need for geophysical variables (Level 2 or higher-level products, see Fig. 12.1 for data processing levels). As illustrated by Fig. 13.1, satellite data simulators are a forward model (Sect. 11.1) generating the synthetic observables from model outputs directly comparable with Level 1 data, circumventing the uncertainties originating from retrieval algorithms. Another advantage is that satellite data simulators have the potential of unraveling hidden bias embedded deep in the model physics, for instance in cloud microphysical assumptions (e.g., Masunaga et al. 2008), that may be difficult to track down by a limited collection of the variables stored in Level 2–4 products. Among the popular simulators useful for model assessment is the Cloud Feedback Model Intercomparison Project (CFMIP) Observation Simulator Package (COSP) (Bodas-Salcedo et al. 2011; Swales et al. 2018). The primary target of COSP is the evaluation of climate model simulations with focus on the reproducibility of clouds and their radiative effects. A unique feature of COSP is the Subgrid Cloud Overlap Profile Sampler (SCOPS), a preprocessing module to interface large-scale model grid profiles with subgrid-scale cloudy columns. SCOPS was originally developed

13.1 Overview

285

for the ISCCP simulator, which is also a part of COSP. In addition to the ISCCP simulator, COSP contains a collection of existing simulators specialized for individual instruments (e.g., CloudSat CPR, CALIPSO CALIOP, Terra MISR, and Terra/Aqua MODIS). Unlike other simulators, main COSP outputs are “pseudo retrieval” of geophysical variables in a format compatible with Level 3 satellite products (e.g., ISCCP). Satellite Data Simulator Unit (SDSU) (Masunaga et al. 2010) is aimed at the evaluation of cloud resolving models against radar, microwave radiometer, and visible/infrared imager observations. SDSU supports a particle size distribution (PSD) library, allowing the user to flexibly customize the cloud microphysical assumptions for radiative transfer simulations. Goddard SDSU (G-SDSU) (Matsui et al. 2013, 2014) is built upon an early version of SDSU but significantly expanded with additional capabilities including lidar and broadband simulators and a satellite orbit and sensor scan module. Joint Simulator for Satellite Sensors (Joint-Simulator) (Hashino et al. 2013) was developed upon G-SDSU with new modules such as a Doppler radar simulator. In addition to these simulator packages widely used in the research community, there is a constant demand from operational users for efficient passive-sensor simulators fast enough to work with an operational data assimilation system. Such simulators include Radiative Transfer for TOVS (RTTOV) (Saunders et al. 2018 and references therein) and Community Radiative Transfer Model (CRTM) (Chen et al. 2008; Ding et al. 2011).

13.2 Satellite Data Simulations for This Book Synthetic satellite measurements throughout this book were computed using SDSU (Masunaga et al. 2010). The visible and infrared module is based on a plane-parallel discrete-ordinate method (Nakajima and Tanaka 1986, 1988; Stamnes et al. 1988) with a k-distribution scheme for the gas absorption table (Sekiguchi and Nakajima 2008). The number of Gaussian quadrature points is chosen to be 12 for the solarradiation regime (wavelengths shorter than 3 µm) and 4 for the thermal-infrared regime (3 µm or longer). The microwave transfer module is an Eddington scheme (see Sect. 4.2.4.2) developed by Kummerow (1993) and the radar simulator is as described in Masunaga and Kummerow (2005). This section summarizes the atmospheric and microphysical model setups for simulations conducted in earlier chapters.

13.2.1 Model Atmospheres Figure 13.2 shows the vertical structure of geopotential height, air temperature, and water-vapor mixing ratio assumed in the simulations. The reference atmospheric model is the global-mean profile adopted from the U.S. Standard Atmosphere 1976. Also used are the low-latitude and winter high-latitude profiles. Only the tropospheric

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13 Satellite Data Simulators

Pressure [hPa]

(c) Water vapor mixing ratio

(b) Air temperature

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Fig. 13.2 The reference (global-mean) profiles (solid) and selected regional profiles for low latitudes (dashed) and winter high-latitudes (dotted). The plots are focused on pressure levels below 100 hPa, while the whole troposphere and stratosphere (up to 47 km in height) are included in the calculations. a Geopotential height (km), b air temperature (K), and c water vapor mixing ratio (g kg−1 )

portion is shown in Fig. 13.2 so that the water vapor profiles in the lower troposphere are clearly legible. The top of the atmosphere is set to be 47 km in the calculations to include the stratospheric ozone and carbon dioxide necessary for the infrared gas-absorption bands (4.3, 9.6, and 15 µm).

13.2.2 Cloud Microphysics Next described are the cloud microphysical models applied to the radiative transfer simulations. Four hydrometeor species are considered: warm (liquid-water) cloud, ice cloud, liquid precipitation (rain), and frozen precipitation (snow). For brevity, other solid precipitation types (graupel and hail) are not considered throughout this book. An analytic function is assumed for the PSD of each hydrometeor type. Cloud PSD is uniquely determined once LWC or IWC is specified along with a given value of mean particle size. The specific definition of “mean particle size” varies with the PSD functions as will become clear below. Rain and snow PSDs are a single-moment scheme constrained only by given LWC or IWC. Warm cloud droplets are assumed to obey a log-normal PSD,   (ln D − ln Dm,c )2 dD N0,c , exp − N (D)dD = √ 2σ 2 D 2π σ

(13.1)

where D is the diameter of droplets, Dm,c is the mode diameter, and σ is the dispersion. The dispersion is fixed at 0.35 (Nakajima and King 1990). Here the scaling

13.2 Satellite Data Simulations for This Book

factor is obtained as N0,c =

287

  6W 9 2 , σ exp − 3 πρc Dm,c 2

(13.2)

where W is LWC and ρc is the cloud water particle density (1.0 ×103 kg m−3 ). In satellite remote sensing, the effective radius, 

∞

r e ≡ 0 ∞

r 3 n(r )dr ,

(13.3)

r 2 n(r )dr

0

is a preferred measure of mean particle size (here r is the particle radius, equivalent to D/2) (see Sect. 8.3.1 for details). Inserting (13.1) into (13.3), the mode diameter is found to be translated into re as re =

  5 2 Dm,c exp σ 2 2

(13.4)

for a log-normal PSD. A gamma distribution is assumed for cloud ice,  N (D)dD = N0,i

D D0,i

μ

  D dD , exp −(3.67 + μ) D0,i

(13.5)

with μ = 1. The first-order gamma function, known to be a reasonable substitute for the PSD of ice crystals smaller than 100 µm (McFarquhar and Heymsfield 1997), is adopted for visible and infrared radiative transfer calculations. The intercept parameter N0,i is given by 6W (3.67 + μ)μ+4 , (13.6) N0,i = 4 Γ (μ + 4) πρi D0,i for given values of IWC (W ) and the median volume diameter (D0,i ). The cloud-ice particle density ρi is 0.917 × 103 kg m−3 . The effective radius and median volume diameter are related to each other as re =

D0,i 3 + μ , 2 3.67 + μ

(13.7)

for a gamma distribution. The rain drop size distribution is assumed to be an exponential PSD, N (D)dD = N0,r exp(−Λr D)dD, where the slope parameter Λr is determined by

(13.8)

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 Λr =

π N0,r ρr W

1/4 ,

(13.9)

when W , N0,r , and ρr are specified. The snow size distribution is also an exponential PSD, (13.10) N (D)dD = N0,s exp(−Λs D)dD , with

 Λs =

π N0,s ρs W

1/4 .

It is assumed that N0,r = 2.2 × 107 m−4 with ρr = 1.0 × 103 kg m−3 and N0,s = 1.0 × 108 m−4 with ρs = 0.1 × 103 kg m−3 . As an exception, the snowflake density ρs is allowed to vary with D for the simulations of DFR and the Ku-band reflectivity shown in Fig. 10.9. The absorption and scattering cross sections of hydrometeors are computed first for each particle size from Mie’s solutions (Sect. 5.1.4) and are then integrated together as a weighted average with the PSDs described above. Mie’s solutions are exact for optically homogeneous spheres, and hence offer an accurate scattering model of cloud droplets and (to a somewhat lesser degree) of rain drops. Ice crystals and snow are far from spherical in shape nor homogeneous in crystal structure. Mie’s solutions are not a good quantitative approximation for solid hydrometeors but are still of utility as a qualitative measure. The Maxwell-Garnett model (Garnett 1904) with an ice matrix and air inclusions is adopted for the dielectric function of snow, for which the volume fraction of the inclusions is determined so the resulting particle density agrees with ρs .

13.2.3 Simulation Setups Table 13.1 presents the model setups for visible and infrared simulations of cloud variables in (Chaps. 6–8). For all simulations listed in Table 13.1, satellite zenith angle is set to be zero (i.e., nadir looking) over ocean background. Solar zenith angle is fixed at 30◦ for visible and near-infrared simulations. In Table 13.2, the model setups for microwave radiometer and radar simulations are summarized (Chaps. 9–10). Satellite zenith angle is 53◦ to compute microwave brightness temperature (Chap. 9), while fixed at zero for radar simulations (Chap. 10). Rain LWC is assumed to be constant over height from 0 to 5 km at a value specified in Table 13.2, topped by upward decreasing snow IWC as IWC = IWC0

15 − z for z > 5 km 10

with IWC0 is defined to be continuous with LWC at 5 km.

(13.11)

Transmittance Cloud ων TIR Tb TIR Tb

TIR Tb TIR Tb Cloud ρσe,ν TIR Tb Vis/NIR Iλ Vis/NIR Iλ Vis/NIR Iλ Vis/NIR Iλ τc,0.65 /τc,11 Vis/NIR Iλ

6.3 7.1 7.6

7.7

8.8 8.9

8.7

7.8 7.10 8.6

Parameter

Figure

Global T = 273.15 K Global RH = 10%-100% Global Global T = 273.15 K Global Global Global Global Global – Global

Atmosphere

0.5–1.5 – – – 0.5–1.5 – 0.5–1.5 – – –

– – – –

Warm cloud z (km)

0.1 – 0.01 – 0.003, 0.03, 0.3 – 0.05, 0.1, 0.2 – 0.01 0.01–0.2

– – – –

LWC (g m−3 )

10 – 5, 10, 20 – 10 – 5, 10, 20 – 5–50 2–30

– 10 – –

Dm,c (µm)

– 10–11 – 10–11 – 10–11 – 10–11 – –

– – – –

Ice cloud z (km)

– 0.1 0.01 0.001–0.1 – 0.003, 0.03, 0.3 – 0.05, 0.1, 0.2 0.01 –

– – – –

IWC (g m−3 )

– 40 10, 20, 40 10–100 – 40 – 10, 20, 40 5–100 –

– 20 – –

D0,i (µm)

Table 13.1 Radiative transfer model setups for simulations shown in Chaps. 6–8. Vis, NIR, and TIR denote visible, near infrared, and thermal infrared, respectively. Precipitating hydrometeors are absent in all these runs

13.2 Satellite Data Simulations for This Book 289

290

13 Satellite Data Simulators

Table 13.2 Radiative transfer model setups for simulations shown in Chaps. 9–10. “Low Lat.” and “W. High Lat.” refers to the low-latitude and winter high-latitude atmospheric profiles, respectively, shown in Fig. 13.2. The sounding weighting function is denoted by Wν (z) (9.28). Cloud water and ice are absent in all these runs Figure

Parameter

Atmosphere

Surface

Rain

Snow

z

LWC

z

IWC

(km)

(g m−3 )

(km)

(g m−3 )

9.1

Rain/snow ρσe,ν , ων

T = 273.15 K

–

–

0.03, 0.3, 3

–

0.03, 0.3, 3

9.8

MW Tb

Low Lat.

Ts =300.4 K u 10 = 5 m s−1

–

–

–

–

MW Tb

×0.5, RH=100%

As above

–

–

–

–

MW Tb

Low Lat.

As above

0–5

0.03, 0.3, 3

–

–

MW Tb

Low Lat.

As above

0–5

0.03, 0.3, 3

5–15

Eq. (13.11)

MW Tb

Low Lat.

Ts =300.4 K ν,s = 0.9

0–5

0.03, 0.3, 3

–

–

MW Tb

Low Lat.

As above

0–5

0.03, 0.3, 3

5–15

Eq. (13.11)

MW Tb

Low Lat.

Ts =300.4 K u 10 = 5 m s−1

0–5

0.03–3

–

–

MW Tb

Low Lat.

As above

0–5

0.03–3

5–15

Eq. (13.11)

MW Tb

Low Lat.

As above

0–5

0.03–3

–

–

MW Tb

Low Lat.

as above

0–5

0.03–3

5–15

Eq. (13.11)

9.14a

T Wν (z)

Global

–

–

–

–

–

9.14b

WV Wν (z)

Low Lat.

–

–

–

–

–

9.14c

WV Wν (z)

W. High Lat.

–

–

–

–

–

10.3

σˆ b,ν , Z

T =300.4 K

–

–

10−3 –10

–

–

10.4

k

T = 300.4 K

–

–

10−3 –10

–

–

10.5

LWC, Z m

Low Lat.

–

0–5

10−3 –10

5–15

Eq. (13.11)

10.7

PIA

Low Lat.

–

0–5

10−3 –10

5–15

Eq. (13.11)

10.8

|K ν |2

T = −10, 0, 10, 20 ◦ C

–

–

–

–

–

10.9

DFR, Z eKu

T = 273.15 K

–

–

10−2 –10

–

10−2 –10

9.9

9.10

9.11

9.12

References

291

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Index

A ABI, 11, 19 Absorption, 120 Absorption cross section, 75, 120 Mie’s solution of, 99 Absorption efficiency, 99 Absorption line, 155–156 Acronyms, list of, xiii Active sensors, 16 AGRI, 12, 19 AHI, 12, 19, 157 AIRS, 13, 23 Airy pattern, 47, 113, 215 AMSR2, 20, 205 AMSR3, 20 AMSU, 10, 11, 13, 21 Antenna gain, 226 Antenna pattern, see beam pattern Aperture efficiency, 227 Aqua, 13, 19, 20, 23, 41 AMSR-E, 12, 20, 51 Asymmetry parameter, 79 ATMS, 11, 21, 220 A-Train, 13 Attenuation of radar echo, 232, 233 attenuation coefficient, 236 attenuation correction, 242, 244 PIA, 243, 244 AVHRR, 10, 11, 18, 50, 168

B Back-scattering coefficient (lidar), 248 Back-scattering cross section (radar), 227 Bayesian estimation, 260–262 Bayes’ theorem, 257–259 Beam pattern, 47, 113, 215

Blackbody spectrum, 67, 115 Boltzmann constant, 58 Boltzmann distribution, 68 Boltzmann equation for non-relativistic particles, 69–73 for photons, 74–77 Bose-Einstein distribution, 63, 66, 115 Boson, 62 Brewster’s angle, 109 Bright band, see radar bright band Brightness temperature infrared, 156 microwave, 198 Brightness Temperature Difference (BTD), 161

C CALIOP, 12, 25, 249, 251, 274, 285 CALIPSO, 12, 25, 42, 225, 249, 251, 274 Canonical distribution, 58 Canonical ensemble, 58–61 Carbon dioxide infrared bands, 134–136 vibrational transitions of, 134 CATS, 25 CERES, 13 Chemical potential, 61 CHIRPS, 276 CLARA-A2, 273 Cloud amount, 271 Cloud cci, 273 Cloud effective radius, 164, 183, 185–187, 287 Cloud optical depth, 154, 184–187 CloudSat, 12, 25, 42, 225, 249, 251, 274, 285 Cloud thermodynamic phase, 165, 190

© Springer Nature Singapore Pte Ltd. 2022 H. Masunaga, Satellite Measurements of Clouds and Precipitation, Springer Remote Sensing/Photogrammetry, https://doi.org/10.1007/978-981-19-2243-5

293

294 Cloud top temperature, 155, 161, 167 CMORPH, 276 Collisional broadening, 125 Conductivity, 87 Conical scan, 19, 21, 51 Convective rain, 235, 241 CO2 slicing method, 165–167, 273 COSP, 284 CPC, 276 CPR, 12, 25, 274 CrIS, 11, 23 Cross-track scan, 21, 50 CRTM, 285

D Data processing level, 269–270 DDA, 102 Debye relaxation, 140 Density of states, 57 Depolarization ratio, 250, 251 Detailed balance, 121 Deterministic estimation, 263–264 DFR, see dual frequency ratio Dielectric factor, 228 Dielectric function, 87 liquid/ice water, 136, 144, 146 Diffraction limit, 46, 110, 206 DMSP, 10, 20, 21 Doppler broadening, 125 DPR, 15, 24, 50, 225 Dual frequency ratio, 245

E Eddington approximation, 78–79, 173 Effective cloud emissivity, 167 Effective radius, see cloud effective radius EFOV, 48 Einstein coefficients, 120–121 Electric displacement, 84, 85 Electric field, 84 Elektro-2, 12 Emission coefficient, 76, 120 Emission line, 155–156 Emissivity, 200 cloud, 165, 167 surface, see surface emissivity EOS, 13 Extinction cross section, 76 Mie’s solution of, 99 Extinction efficiency, 99

Index F FDTD method, 102 Fengyun (FY), 11, 12, 19, 23 Fermi-Dirac distribution, 63 Fermion, 62 Field of view, see FOV Forward model, 255 Forward problem, 255–256 FOV, 46–48 Fraunhofer diffraction, 110 Free energy (Helmholtz), 60 Fresnel equations, 107, 201

G GCOM-W, 20 GEO, 9, 29–31 operational satellites, 11–12 GEO-KOMPSAT, 12 Geometrical optical limit, 101 Geometrical optics, 103 Geostationary orbit, see GEO GIIRS, 12, 24 Glory, 189 GMI, 15, 20, 51 GMS, 11 GOES, 11, 19 GOSAT-GW, 20 GPCC, 276 GPCP, 276 GPI, 276 GPM, 15, 20, 24, 42, 225 GPROF, 276 Ground clutter, 241 G-SDSU, 285 GSMaP, 276

H Halo, 190 Himawari, 11, 19, 30, 157, 178 HIRAS, 23 HIRS, 23 Hitchfeld-Bordan equation, 242–243 HOAPS, 276 HSB, 23

I IASI, 11, 23 Ice water content, 181 Ice water path, 182 IFOV, 46 IMERG, 276

Index INSAT, 12 Inverse problem, 255–256 Inversion model, 256, 259 ISCCP, 271

J Joint-Simulator, 285 J2 perturbation, 33 JPSS, 10, 19, 21, 23

K Kirchhoff’s law, 122 k-Z relation, 234–236

L Landsat, 41 LEO, 9, 31–33 operational satellites, 10–11 Lidar, 25 Lidar ratio, 248 Likelihood, 258 Liquid water content, 181 Liquid water path, 182 Local thermodynamic equilibrium (LTE), 122 Lorentz oscillator model, 125, 145–146

M MAC-LWP, 274 MADRAS, 16, 20 Magnetic field, 84 Magnetic induction, 84 MAP estimation, 261 Marshall-Palmer distribution, 234 Maximum likelihood estimation, 262 Maxwell-Boltzmann distribution, 68 Maxwell’s equations, 83–85 Megha-Tropiques, 15, 20, 23 Meteor-M, 11 Meteosat, 12, 19 MetOp, 11, 18, 21, 23 MHS, 21 Microwave radiometer, 19–21 Mie’s solution, 89–102 MISR, 13, 285 MODIS, 12, 19, 32, 50, 168, 285 MSWEP, 276 MTSAT, 11

295 N Natural broadening, 125 NOAA satellite, 10, 18, 21, 23 Non-uniform beam filling, see NUBF NPOESS, 10 NUBF, 214–216 O Operational satellite missions, 8–12 OPI, 276 Optical depth, 81, 153, 172, 195 Optical depth of clouds, see cloud optical depth Optical thickness, see optical depth Oxygen molecule fine-structure transitions of, 124, 130, 132 microwave bands, 130–132 Ozone infrared bands, 136 P PARASOL, 14 Passive sensors, 16 Path-integrated attenuation, see attenuation of radar echo, PIA PATMOS-x, 273 PCT, 216–218 Permeability in free space, 84 Permittivity in free space, 84 relative, 87 PERSIANN, 276 Phase function, see scattering phase function Phase space, 55 PIA, see attenuation of radar echo, PIA Planck constant, 56 Planck function, see blackbody spectrum POES, 10 Polarization electric, 85, 137 of electromagnetic waves, 105, 201 orientational, 137–140 Polarization corrected temperature, see PCT Polar orbit, see sun-synchronous orbit Polar(-orbiting) satellite, see sunsynchronous satellite POLDER, 14, 191 Posterior probability, 258 Poynting’s theorem, 88 Poynting vector, 88

296 PR, 12, 14, 24, 50, 225 Precession, 35–37 Prior probability, 258 PSD, 228

Q QuikSCAT, 51

R Radar, 24–25 Radar back-scattering cross section, see back-scattering cross section (radar) Radar bright band, 239 Radar constant, 230 Radar reflectivity, 228 Radar reflectivity factor, 231 effective, 232 Radar equation, 229 Radar reflectivity factor, 230 Radiance, 74, 152 Radiative energy density, 77 Radiative equilibrium, 122 Radiative flux, 77 Radiative pressure, 77 Radiative transfer equation, 77, 80–82 non-scattering, 152–155 partially scattering, 176–178 scattering, 172–176 scattering and emitting, 198 scattering and Emitting, 195 Radio occultation, 51 Rainbow, 188 RainCube, 24 Rapid scan, 48 Rayleigh-Jeans law, 117 Rayleigh limit, 101–102, 209 Rayleigh scattering, 101, 123, 188 Reflectance, 179 Reflectivity in geometrical optics, 108 radar, see radar reflectivity Reflectivity factor radar, see radar reflectivity factor Refractive index, 86, 87 complex, 86 Research satellite missions, 8, 12–16 Response function, see spectral response function Revisit cycle, 39–42 RTTOV, 285

Index S SAPHIR, 16, 23 Satellite data simulator, 283–285 ScaRaB, 16 Scattering angle, 187 Scattering cross section, 76 Mie’s solution of, 99 Scattering efficiency, 99 Scattering phase function, 76, 187–191 SCOPS, 284 SDSU, 285 SEVIRI, 12, 19 Single scattering albedo, 77 Size parameter, 96 Snell’s law, 105 Solid angle, 57 Sounder infrared, 23–24 microwave, 21–23 Sounding, 218–220 Source function, 77 Specific intensity, see radiance Spectral response function (SRF), 158 Speed of light, 85 Split-window BTD method, 161–165 channels, 18 Spontaneous emission, 120 SSM/I, 10, 20 SSMI, 51 SSMIS, 10, 21, 51 Stefan-Boltzmann constant, 119 Stefan-Boltzmann law, 118 Steradian, 57 Stimulated emission, 120 Stratiform rain, 235, 241 Sun-asynchronous orbit, 9, 42–44 satellites, 14–16 Sun-synchronous orbit, 9, 37–39 satellites, 13–14 Suomi NPP, 10, 19, 21, 23 Surface clutter, see ground clutter Surface emissivity, 199, 205 land, 202 ocean, 201 Surface reference technique, 243 Susceptibility complex, 87 electric, 85 magnetic, 85

Index

297 V VIIRS, 11, 19 VIRS, 14 Visible/infrared imager, 17–19

T TEMPEST-D, 20 Terra, 13, 19, 32, 41 TIROS-1, 7 TIROS-N, 10 T-matrix method, 102 TMI, 12, 14, 20, 51 TOVS, 23 Transmittance (of the earth’s atmosphere), 123 Transmittivity in geometrical optics, 108 TRMM, 12, 14, 20, 24, 42, 225 TROPICS, 20 Two-stream approximation, 80

W Water vapor infrared bands, 134–136, 158–159 microwave bands, 125–130, 207, 209 rotational transitions of, 124, 125, 129 vibrational transitions of, 134 Weighting function, 219 Wien displacement law, 116 Wien law, 118 Window channel, 157

U Ultraviolet catastrophe, 118

Z Z-R relation, 234–236 Z-W relation, 234–236