Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability: Volume 5 Weather Modification Experiments [Reprint 2020 ed.] 9780520313903

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PROCEEDINGS OF FIFTH

BERKELEY

THE

SYMPOSIUM

VOLUME V

PROCEEDINGS of the F I F T H BERKELEY SYMPOSIUM ON MATHEMATICAL STATISTICS AND PROBABILITY Held, at the Statistical Laboratory University of California June 21-July 18,1965 and December 27,1965-January 7,1966 with the support of University of California National Science Foundation National Institutes of Health Air Force Office of Scientific Research Army Research Office Office of Naval Research VOLUME V

WEATHER MODIFICATION EDITED BY L U C I E N M . L E C A M AND J E R Z Y N E Y M A N U N I V E R S I T Y O F C A L I F O R N I A PRESS BERKELEY AND LOS ANGELES

1967

UNIVERSITY OF CALIFORNIA

PRESS

B E R K E L E Y AND LOS A N G E L E S CALIFORNIA

CAMBRIDGE UNIVERSITY LONDON,

COPYRIGHT ©

PRESS

ENGLAND

1967,

BY

THE REGENTS OF THE UNIVERSITY OF CALIFORNIA

The United States Government and its offices, agents, and employees, acting within the scope of their duties, may reproduce, publish, and use this material in whole or in part for governmental purposes without payment of royalties thereon or therefor. The publication or republication by the government either separately or in a public document of any material in which copyright subsists shall not be taken to cause any abridgment or annulment of the copyright or to authorize any use or appropriation of such copyright material without the consent of the copyright proprietor. L I B R A R Y OF CONGRESS CATALOG CARD N U M B E R :

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CONTENTS OF PROCEEDINGS VOLUMES I, II, III, IV, AND V Volume I—Theory of Statistics General Theory T. W. ANDERSON and S. M. SAMUELS, Some inequalities among binomial and Poisson probabilities. R. R. BAHADUR, An optimal property of the likelihood ratio statistic. GEORGE A. BARNARD, The use of the likelihood function in statistical practice. D. BASU, Problems relating to the existence of maximal and minimal elements in some families of statistics (subfields). YU. K. BELYAEV, On confidence intervals and sets for various statistical models. FRIEDHELM EICKER, Limit theorems for regressions with unequal and dependent errors. R. H. FARRELL, Weak limits of sequences of Bayes procedures in estimation theory. R. H. FARRELL, J. KIEFER, and A. WALBRAN, Optimum multivariate designs. JAROSLAV HAjEK, On basic concepts of statistics. J. L. HODGES, JR., Efficiency in normal samples and tolerance of extreme values for some estimates of location. J. L. HODGES, JR. and E. L. LEHMANN, Moments of chi and power of t . WASSILY HOEFFDING, On probabilities of large deviations. PETER J. HUBER, The behavior of maximum likelihood estimates under nonstandard conditions. OSCAR KEMPTHORNE, The classical problem of inference—goodness of fit. HIROKICHI KUDO, On partial prior information and the property of parametric sufficiency. YU. V. LINNIK, On the elimination of nuisance parameters in statistical problems. J. MACQUEEN, Some methods for classification and analysis of multivariate observations. KAMEO MATUSITA, Classification based on distance in multivariate Gaussian cases. EMANUEL PARZEN, On empirical multiple time series analysis. YU. V. PROHOROV, Some characterization problems in statistics. ROY RADNER, A note on maximal points of convex sets in I C . R. RAO, Least squares theory using an estimated dispersion matrix and its application to measurement of signals. KAROLY SARKADI, On testing for normality. R. A. WIJSMAN, Cross-sections of orbits and their application to densities of maximal invariants.

Sequential Procedures PETER J. BICKEL and JOSEPH A. YAHAV, Asymptotically pointwise optimal procedures in sequential analysis. DAVID BLACKWELL, Positive dynamic programming. Y. S. CHOW and H. ROBBINS, A class of optimal stopping problems. Y. S. CHOW and H. ROBBINS, On values associated with a stochastic sequence. ARYEH DVORETZKY, Existence and properties of certain optimal stopping rules. THOMAS S. FERGUSON, On discrete evasion games with a two-move information lag. M. V. JOHNS, JR., Two-action compound decision problems. JERZY LOS, Horizon in dynamic programs.

Information Theory TOSIO KITAGAWA, Information science and its connection with statistics. ALFRED RENYI, On some basic problems of statistics from the point of view of information theory. MILLU ROSENBLATT-ROTH, Approximations in information theory. J. WOLFOWITZ, Approximation with a fidelity criterion.

Nonparametric Procedures PETER J. BICKEL, Some contributions to the theory of order statistics. RALPH A. BRADLEY, Topics in rank-order statistics. Z. GOVINDARAJULU, L. LE CAM, and M. v

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RAGHAVACHARI, Generalizations of theorems of Chernoff and Savage on the asymptotic normality of test statistics. PRANAB K U M A R SEN, On a class of two-sample bivariate nonparametric tests. I. VINCZE, On some questions connected with two-sample tests of Smimov type.

Volume II—Part I—Theory of Probability Probability on Algebraic Structures SIMEON M. BERMAN, Sign-invariant random elements in topological groups. R. GANGOLLI, Abstract harmonic analysis and Levy's Brownian motion of several parameters. LEONARD GROSS, Abstract Wiener spaces. E D I T H MOURIER, Random elements in linear spaces. CZESLAW RYLL-NARDZEWSKI, On fixed points of semigroups of endomorphisms of linear spaces. A. and C. IONESCU TULCEA, On the existence of a lifting commuting with the left translations of an arbitrary locally compact group.

Distributions in Functional Spaces H E R M A N N DINGES, Random shifts of stationary processes. TAKEYUKI HIDA and NOBUYUKI I K E DA, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral. K I Y O S I I T O , Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral. A. V. SKOROHOD, On the densities of probability measures in functional spaces. L E S T E R E. DUBINS and DAVID A. F R E E D MAN, Random distribution functions.

Stochastic Processes and Prediction HARALD CRAMER, A contribution to the multiplicity theory of stochastic processes. R. M. DUDLEY, On prediction theory for nonstationary sequences. K. URBANIK, Some prediction problems for strictly stationary processes. A. M. YAGLOM, Outline of some topics in linear extrapolation of stationary random processes.

Martingales M. BRELOT, Capacity and balayage for decreasing sets. LESTER E. DUBINS and GIDEON SCHWARZ, On extremal martingale distributions. STEVEN OREY, F-processes. VOLKER STRASSEN, Almost sure behavior of sums of independent random variables and martingales.

Special Problems D. A. DARLING, Some limit theorems associated with multinomial trials. G E R A R D DEBREU, Integration of correspondences. WILLIAM FELLER, On regular variation and local limit theorems. MILOSLAV JIRINA, General branching processes with continuous time parameter. E U G E N E LUKACS, On the arithmetical properties of certain entire characteristic functions. H E R M A N RUBIN, Supports of convolutions of identical distributions. E. SPARRE ANDERSEN, An algebraic treatment of fluctuations of sums of random variables. LAJOS TAKACS, On combinatorial methods in the theory of stochastic processes.

Volume II, Part II—Theory of Probability Markov Processes R. M. B L U M E N T H A L and R. K. GETOOR, Accessible terminal times. LEO BREIMAN, First exit times from a square root boundary. E . B. D Y N K I N , General lateral conditions for some diffusion processes. H. K E S T E N , The Martin boundary of recurrent random walks on countable groups. MINORU MOTOO, Application of additive functionals to the boundary problem of Markov processes (Levy's system of (/-processes). TADASHI UENO,

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A survey on the Markov process on the boundary of multidimensional diffusion. HIROSHI KUNITA and TAKESI WA TAN ABE, Some theorems concerning resolvents over locally compact spaces. DAVID G. KENDALL, On Markov groups. J A N E M. O. SPEAKMAN, Some problems relating to Markov groups. DAVID WILLIAMS, Uniform ergodicity in Markov chains. J. G. BASTERFIELD, On quasi-compact pseudo-resolvents. J A N E M. O. SPEAKMAN, A note on Markov semigroups which are compact for some but not all t > 0. D A N I E L RAY, Some local properties of Markov processes. DONALD ORNSTEIN, A limit theorem for independent random variables. CHARLES STONE, On local and ratio limit theorems. J O H N L A M P E R T I , Limiting distributions for branching processes. SAMUEL K A R L I N and J A M E S McGREGOR, Uniqueness of stationary measures for branching processes and applications. WALTER L. SMITH, Some peculiar semi-Markov processes. WALTER L. SMITH, A theorem on functions of characteristic functions and its application to some renewal theoretic random walk problems. F. SPITZER, Renewal theorems for Markov chains.

Ergodic Theory R O B E R T J. AUMANN, Random measure preserving transformations. J. R. BLUM, H. D. BRUNK, and D. L. HANSON, Roots of the one-sided TV-shift. R. V. CHACON, A geometric construction of measure preserving transformations. ARSHAG B. HAJIAN and Y U J I I T O , Conservative positive contractions in L1. KONRAD JACOBS, On Poincar6's recurrence theorem. SHIZUO KAKUTANI, Ergodic theory of shift transformations. U L R I C H K R E N G E L , Classification of states for operators. KLAUS K R I C K E B E R G , Strong mixing properties of Markov chains with infinite invariant measure. CALVIN C. MOORE, Invariant measures on product spaces. JACQUES NEVEU, Existence of bounded invariant measures in ergodic theory. M. ROSENBLATT, Transition probability operators.

Volume III—Physical Sciences Astronomy E . M. B U R B I D G E and G. R . BURBIDGE, Evolution of galaxies. W. H. McCREA, Age distribution of galaxies. T H O R N T O N PAGE, Masses of galaxies: singles and members of multiple systems. BEVERLY T. LYNDS, Space distribution of small dark nebulae. W. C. LIVINGSTON, On correlations between brightness, velocity, and magnetic fields in the solar photosphere.

Physics R . L. DOBRUSHIN, Existence of phase transitions in models of a lattice gas. J. M. HAMMERSLEY, Harnesses. H E R B E R T SOLOMON, Random packing density.

Spectral Analysis M. S. BARTLETT, The spectral analysis of line processes. B E N O I T MANDELBROT, Sporadic random functions and conditional spectral analysis: self-similar examples and limits.

Control Processes J O H N B A T H E R and H E R M A N CHERNOFF, Sequential decisions in the control of a spaceship. R I C H A R D BELLMAN, On the construction of a mathematical theory of the identification of systems. P. W H I T T L E , The deterministic stochastic transition in control processes and the use of maximum and integral transforms.

Reliability R. E. BARLOW and A. W. MARSHALL, Bounds on interval probabilities for restricted families of distributions. YU. K. BELYAEV, B. V. G N E D E N K O , and A. D. SOLOVIEV, On some stochastic problems of reliability theory. Z. W. BIRNBAUM and J. D. ESARY, Some

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inequalities for reliability functions. B. V. G N E D E N K O , Some theorems on standbys. FRANK PROSCHAN and RONALD PYKE, Tests for monotone failure rate. A. D. SOLOVIEV, Theory of aging elements.

Volume IV—Biology and Problems of Health Information, Processing, and Cognition MARY A. B. BRAZIER, The challenge of biological organization to mathematical description. R I C H A R D BELLMAN, Adaptive processes and intelligent machines. H. J. B R E M E R M A N N , Quantum noise and information. VIOLET R. CANE, Mathematical models for neural networks. E D W A R D A. FEIGENBAUM, Information processing and memory. JULIAN FELDMAN, Recognition of pattern in periodic binary sequences. WALTER R E I T M A N , Modeling the formation and use of concepts, percepts, and rules.

Demography NATHAN KEYFITZ, Estimating the trajectory of a population. M I N D E L C. SHEPS, Uses of stochastic models in the evaluation of population policies. I. Theory and approaches to data analysis. E D W A R D B. P E R R I N , Uses of stochastic models in the evaluation of population policies. II. Extension of the results by computer simulation.

Ecology DOUGLAS G. CHAPMAN, Stochastic models in animal population ecology. E. C. PIELOU, The use of information theory in the study of the diversity of biological populations. J. G. SKELLAM, Seasonal periodicity in theoretical population ecology.

Epidemiology C. C. SPICER, Some empirical studies in epidemiology. D. E. BARTON, F. N. DAVID, EVELYN F I X , M A X I N E M E R R I N G T O N , and P I E R O MUSTACCHI, Tests for spacetime interaction and a power function. P I E R O MUSTACCHI, F. N. DAVID, and EVELYN F I X , Three tests for space-time interaction: a comparative evaluation. NORMAN T. J. BAILEY, The simulation of stochastic epidemics in two dimensions. R O B E R T BARTOSZYfrSKI, Branching processes and the theory of epidemics. J. GANI, On the general stochastic epidemic. H. E. DANIELS, The distribution of the total size of an epidemic.

Genetics THEODOSIUS DOBZHANSKY, Genetic diversity and diversity of environments. HOWARD LEVENE, Genetic diversity and diversity of environment: mathematical aspects. G. MALECOT, Identical loci and relationship. OSCAR K E M P T H O R N E , The concept of identity of genes by descent. D. E. BARTON, F. N. DAVID, EVELYN FIX, and M A X I N E M E R R I N G T O N , A review of analysis of karyographs of the human cell in mitosis. J. O. IRWIN, A theory of the association of chromosomes in karyotypes, illustrated by Dr. Patricia Jacobs' data. WALTER F. BODMER, Models for DNA mediated bacterial transformation. SAMUEL KARLIN, J A M E S McGREGOR, and WALTER BODMER, The rate of production of recombinants between linked genes in finite populations. SAMUEL KARLIN and J A M E S McGREGOR, The number of mutant forms maintained in a population. R. C. LEWONTIN, The genetics of complex systems. P. A. P. MORAN, Unsolved problems in evolutionary theory.

Chance Mechanisms in Living Organisms S. R. BERNARD, L. R. SHENTON, and V. R. RAO UPPULURI, Stochastic models for the distribution of radioactive material in a connected system of compartments. P R E M

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S. P U R I , A class of stochastic models of response after infection in the absence of defense mechanism. J . GANI, Models for antibody attachment to virus and bacteriophage.

Cellular Phenomena H E R B E R T E. K U B I T S C H E K , Cell generation times: ancestral and internal controls. H. R U B I N , Cell growth as a function of cell density. W A L T E R R. STAHL, Measures of organization in a model of cellular self-reproduction based on Turing machines. D. G. B U R N E T T - H A L L and W. A. O'N. WAUGH, Sensitivity of a birth process to changes in the generation time distribution.

Carcinogenesis D A V I D L I N D E R and STANLEY M. G A R T L E R , Problem of single cell versus multicell origin of a tumor. WOLFGANG J. B t l H L E R , Single cell against multicell hypotheses of tumor formation. T. T I M O T H Y C R O C K E R and B E R Y L J . N I E L S E N , Chemical carcinogens and respiratory epithelium. K. B. DEOME, The mouse mammary tumor system. D A V I D W. WEISS, Immunology of spontaneous tumors. M. B. S H I M K I N , R. W I E D E R , D. MARZI, N. GUBAREFF, and V. S U N T Z E F F , Lung tumors in mice receiving different schedules of urethane. M. W H I T E , A. G R E N D O N , and H. B. JONES, Effects of urethane dose and time pattern on tumor formation. J E R Z Y N E Y M A N and E L I Z A B E T H L. SCOTT, Statistical aspect of the problem of carcinogenesis.

Experimentation F. YATES, A fresh look a t the basic principles of the design and analysis of experiments. P. A R M I T A G E , Some developments in the theory and practice of sequential medical trials. H E R M A N C H E R N O F F , Sequential models for clinical trials. J E R O M E C O R N F I E L D and SAMUEL W. G R E E N H O U S E , On certain aspects of sequential clinical trials. B R A D L E Y E F R O N , The two sample problem with censored data. M A R V I N A. S C H N E I D E R M A N , Mouse to m a n : statistical problems in bringing a drug to clinical trial.

Decision Theory in Medical Diagnosis L E O N A R D R U B I N , M O R R I S F. COLLEN, and G E O R G E E. GOLDMAN, Frequency decision theoretical approach to automated medical diagnosis. C H A R L E S D. FLAGLE, A decision theoretical comparison of three procedures of screening for a single disease. L E E B. LUSTED, Logical analysis in medical diagnosis. J. T. CHU, Some decision making methods applicable to the medical sciences.

Volume V—Weather Modification Physical Background M. N E I B U R G E R , Physical factors in precipitation processes and their influence on the effectiveness of cloud seeding.

Large Randomized Experiments LOUIS J. B A T T A N and A. R I C H A R D KASSANDER, JR., Summary of results of a randomized cloud seeding project in Arizona. J . B E R N I E R , On the design and evaluation of cloud seeding experiments performed by Electricité de France. W A Y N E L. D E C K E R and PAUL T. S C H I C K E D A N Z , The evaluation of rainfall records from a five year cloud seeding experiment in Missouri. DONALD L. E B E R L Y and L E W I S H. ROBINSON, Design and evaluation of randomized wintertime cloud seeding at high elevation. K . R. GABRIEL, The Israeli artificial rainfall stimulation experiment. Statistical evaluation for the period 196165. L E W I S 0 . G R A N T and PAUL W. M I E L K E , JR., A randomized cloud seeding experiment

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at Climax, Colorado, 1960-65. E. P É R E Z SILICEO, A brief description of an experiment on artificial stimulation of rain in the Necaxa watershed, México. PAUL SCHMID, On "Grossversuch I I I , " a randomized hail suppression experiment in Switzerland. E. J. SMITH, Cloud seeding experiments in Australia.

Relevant Climatological Studies S. A. CHANGNON, JR. and F. A. HUFF, The effect of natural rainfall variability in verification of rain modification experiments. THOMAS J. HENDERSON, Tracking silver iodide nuclei under orographic influence.

Nonrandomized Operations G L E N N W. B R I E R , THOMAS H. C A R P E N T E R , and D W I G H T B. KLINE, Some problems in evaluating cloud seeding effects over extensive areas. HANS G E R H A R D MÜLLER, Weather modification experiments in Bavaria.

Methodological Discussion A R N O L D COURT, Randomized cloud seeding in the United States. L. G. DAVIS and C. L. HOSLER, The design, execution and evaluation of a weather modification experiment. A R C H I E M. KAHAN, The Bureau of Reclamation's Atmospheric Water Resources Research Program. VUJICA M. YEVDJEVICH, Evaluation of weather modification as expressed in stream flow response. J E R Z Y N E Y M A N and ELIZABETH L. SCOTT, Some outstanding problems relating to rain modification. J E R Z Y N E Y M A N and ELIZABETH L. SCOTT, Appendix. Planning an experiment with cloud seeding. JERZY NEYMAN and ELIZABETH L. SCOTT, Note on the Weather Bureau ACN Project. J. M. WELLS and M. A. WELLS, Note on Project SCUD. J E R Z Y N E Y M A N and ELIZABETH L. SCOTT, Note on techniques of evaluation of single rain stimulation experiments. R O B E R T B. DAVIES and P R E M S. P U R I , Some techniques of summary evaluations of several independent experiments. BARRY R. JAMES, On Pitman efficiency of some tests of scale for the Gamma distribution. F R A N K YATES, Discussion of reports on cloud seeding experiments.

Observational Data A collection of data from cloud seeding experiments in five countries.

PREFACE Berkeley Statistical Symposia, held every five years, is to stimulate research through lectures by carefully selected speakers and through prolonged personal contacts of scholars brought together from distant centers. Accordingly, particular Symposia last from four to seven weeks. On occasion, and this was the case with the Fifth Symposium, they are conducted in two parts, one in June-July, emphasizing theory, and the other in December-January, emphasizing applications. The winter part of the Fifth Symposium was held in conjunction with the 132nd Annual Meeting of the American Association for the Advancement of Science. The Proceedings of the Symposia are intended to present a comprehensive cross-section of contemporary thinking on problems of probability and statistics and on selected fields of application. The rapid growth of research in statistics and especially in probability makes it increasingly difficult to achieve a complete coverage of the field, but sincere efforts are made to invite to the Symposia representatives of all the existing schools of thought, each individual having complete freedom of expression. The organization of the theoretical part of the Fifth Berkeley Symposium was carried out, and the contributors were selected, with the participation of an Advisory Committee composed of Professors J. L. Doob, S. Karlin, and H. Robbins, delegated for this purpose by the American Mathematical Society and by the Institute of Mathematical Statistics. In addition, we had the assistance of Professor D. L. Burkholder, the Editor of the Annals of Mathematical Statistics. The interest of the American Mathematical Society and of the Institute of Mathematical Statistics and their help are deeply appreciated. While a broad coverage of contemporary work in the theory of probability and statistics is difficult, the field of applications of these disciplines is currently so wide that the program of a single symposium can include no more than a few particular domains. The domains covered at the Fifth Symposium were selected on two principles. First, some applied problems appeared as subjects of studies by outstanding probabilists and statisticians invited to the Symposium on account of their work in theory. Second, an effort was made to delineate a few fields of substantive studies that appear particularly promising for probabilistic and statistical treatment. One of the most fruitful fields of this category is undoubtedly biology and problems of health. Here we profited greatly by the advice of Drs. LaMont Cole, Jerome Cornfield, F. N. David, Louis Hellman, Samuel Greenhouse, Hardin Jones, Samuel Karlin, David Krech, Lincoln Moses, Curt Stern, Michael B. Shimkin, and Cornelius Tobias. Quite a few of these colleagues are connected with the broad research activity of the National Institutes of Health and helped to bring to our attention many novel and important subfields of research. T H E P U R P O S E OF T H E

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In the field of astronomy we are deeply indebted to Drs. N. U. Mayall, Rudolph Minkowski, and Thornton L. Page. For advice in the field of meteorology we are grateful to Drs. Earl Droessler, James Hughes, Dwight B. Kline, Morris Neiburger, Jerome Spar, Edward P. Todd, and P. H. Wyckoff. Special thanks are due to Dr. Kenneth B. Spengler, Secretary of the American Meteorological Society. Following the established tradition, Volume I of the present Proceedings is given to the theory of statistics. Volume II is devoted to the theory of probability. Because of the large amount of material, about 1000 pages in print, this volume had to be divided into two parts formed through a somewhat arbitrary classification of papers. Volume III includes papers related to physical sciences: astronomy, theory of control, physics, and the theory of reliability. Volume IV, on biology and problems of health, includes papers on information and brain phenomena, on chance mechanisms in live organisms, on epidemiology, on genetics, on medical diagnosis, on clinical trials, on carcinogenesis and cellular phenomena, on demography, and on ecology. Some of these subdomains are already subjects of well developed statistical treatment. Others appear to offer interesting and important possibilities. Compared to the Proceedings of the earlier Symposia, Volume V, being entirely given to the problem of artificial weather modification, is an innovation. With the classification adopted for the first four volumes, weather modification would fit Volume III. I t is assigned a special volume because of the specificity of the domain and because of its separateness from all the other fields dealt with in Volume III. Also, the novelty of the problem of weather modification, considered by itself and as a field for statistical research, indicated the desirability of producing a comprehensive coverage of the more extensive experiments. Finally, it appears probable that the readership of the material being published in Volume V will be essentially different from that expected to be interested in Volume III. The fifth Symposium would not have been possible without very substantial financial support from various sources. Hearty thanks are due to Dr. Clark Kerr, President of the University of California, for a special grant made several years in advance of the Symposium. Without this grant, no planning and no initial steps for the organization of the Symposium would have been possible. This initial triggering grant of the University of California was later supplemented by the subsidy of the University Editorial Committee, without which the publication of the Proceedings, to be sold at a reasonable price, would have been a very difficult problem. To a very considerable extent, the theoretical part of the Symposium and the part concerned with physical sciences, were financed by The Program in Mathematics of the National Science Foundation, by the Air Force Office of Scientific Research, by the Army Research Office, and by the Office of Naval Research. The large program on biology and problems of health was made possible by a grant of the National Institutes of Health. The

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program on weather modification was organized using a grant of the Atmospheric Sciences Section of the National Science Foundation. Finally, we wish to record special help from the Office of Naval Research, in the form of air transportation for a number of foreign participants in the Symposium. It is our pleasure to acknowledge gratefully the generosity of the governmental institutions enumerated. The vitality of our Symposia and the growth of the Proceedings, from 500 pages in 1945 to about 3,000 in 1965, seem to indicate that the funds provided are being spent to fill a real need. The problems connected with the publication of such an amount of technical material are very substantial, especially since some of the material was originally written in languages other than English and required translation. All efforts were made toward speedy publication at a reasonable price, and we are pleased to acknowledge the excellent cooperation and assistance we received from the University of California Press. For the translation of manuscripts, we are indebted to Drs. Amiel Feinstein> Morris Friedman, and Mrs. C. Stein. We are also indebted to several of our colleagues in the Department for work connected with the preparation of manuscripts for the printer. Special thanks are due to Professors E. L. Scott, M. Loeve, E. W. Barankin, to Drs. Carlos-Barbosa Dantas, W. Biihler, Nora Smiriga, Grace Yang, to Mr. Steve Stigler, and Mrs. M. Darland. We are pleased to acknowledge the technical help of Mrs. Sharlee Guise and MrsCarol Rule Roth. For taking care of the many complexities of editing technical manuscripts we are deeply indebted to Miss Susan Jenkins whose patience and skill deserve superlative praise. Thanks are also due to Mrs. Virginia Thompson for her greatly appreciated assistance in the same process. To Mr. August Fruge, the Director of the University of California Press, we extend heartfelt thanks for financial, technical, and moral support in publishing so much difficult material. Special thanks are due also to Joel Walters, Editor of the University of California Press. In spite of all our efforts, we found ourselves unable to keep up with the schedule of publication proposed by the Press, but we must thank them for helping us to keep the delays at a minimum and for producing a publication in accordance with the usual excellent standards of the University of California Press. Many thanks are due to our Administrative Assistant, Miss M. Genelly for taking care of many financial and organizational difficulties. For transportation, housing, and other logistic problems connected with the organization of the meeting itself, very valuable assistance was received from the staff of the Laboratory and in particular from Miss June Haynes and Mrs. J. Lovasich. As was the case on many earlier similar occasions, for supervising and taking care of the innumerable intricacies of local organization we are deeply indebted to our colleague Professor Elizabeth L. Scott. It is a pleasure to express here our deepest appreciation.

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Last but not least we wish to thank the Department of Statistics of the University of California, Berkeley, and all our colleagues therein, for their sympathetic attitude and help. Particular thanks are due to David Blackwell. During the winter part of the Fifth Symposium, the Statistical Laboratory lost one of its organizers as well as one of its most active members. Our colleague and cordial friend, Professor Evelyn Fix died of a heart attack on December 30, only a few hours after she acted as one of the hostesses at the banquet of the Symposium. Sit ei terra levis! LUCIEN L E CAM

JERZY NEYMAN

Director, Statistical

Laboratory

July, 1967

PREFACE TO VOLUME V The idea of organizing a special Weather Modification Section of the Fifth Symposium, and of publishing a separate volume of the Proceedings, specifically given to weather control problems, occurred to us in the fall of 1964 as a result of studying the seven voluminous annual reports on the Swiss experiment Grossversuch III. Even though the primary purpose of this experiment was hail prevention, the reports contain a wealth of data on rainfall and on other collateral factors. The analysis we performed indicated not only the presence of real effects of seeding on rainfall, but also the puzzling circumstance that in some not clearly defined conditions these effects are positive and in some other conditions they are likely to be negative. Since over the last two decades a number of randomized cloud seeding experiments have been carried out, it appeared timely to organize a systematic review of as many of them as practicable. Jointly they may reveal an intelligible pattern of the effects of cloud seeding, even though in separate experiments these effects seem inconclusive or contradictory. Weather control is a new field of experimentation with peculiarities not encountered in other fields which stimulated the development of experimental designs and of statistical techniques that are now available. Our hope is that the Weather Modification Section of the Symposium and the present volume will serve as a stimulus for novel statistical research, theoretical as well as empirical, which will be beneficial both to statistics and to applied meteorology. I t was a pleasure to find that the above ideas met with the approval of the Weather Modification Program of the National Science Foundation whose financial help is gratefully acknowledged. As we see it, the inspiration of fresh statistical research, and also an advance in the understanding of the complex reactions of the weather to external stimuli,

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depend very much on ready availability of observational data which could serve for verification of various tentative hypotheses that may be formulated. Thus, an effort was made to assemble a substantial collection of experimental results on rainfall and on some collateral factors. This collection is published at the end of the volume. To our knowledge, the present review of weather control studies is the fifth to be published. The first review, which was brought to our attention by Dr. George Steck of Sandia Corporation, appeared more than 70 years ago. The relevant article, written by M. W. Harrington under the title "Weather making, ancient and modern," is contained in the Smithsonian Report, 1894, pp. 249-270. The paper makes a distinction between scientific methods of rain making and those based on superstition. It appears that already at that time some of the "scientific" methods had been patented. The paper is interesting and at times amusing reading. The second published summary of weather modification studies seems to be the controversial Final Report of the Advisory Committee on Weather Control, Vol. 1 and 2, Washington, D. C., 1957. This was followed by two reports of the Panel on Weather and Climate Modification of the National Academy of Sciences—National Research Council, the Preliminary Report of 1964 (NASNRC Publication 1236), and the two volume Final Report of 1966 (NAS-NRC Publication 1350). Even though contemporaneous and concerned with the same general domain, the two volumes of the Final Report of the Panel have little in common with the present volume of the Proceedings. There is no duplication. Lucien M. LeCam

Jerzy Neyman

Elizabeth L. Scott

CONTENTS Physical Background M. NEIBURGER—Physical Factors in Precipitation Processes and Their Influence on the Effectiveness of Cloud Seeding Large Randomized

Experiments

Louis J . B A T T A N and A. R I C H A R D K A S S A N D E R , JR.—Summary of Results of a Randomized Cloud Seeding Project in Arizona J.

BERNIER—On the Design and Evaluation of Cloud

Seeding Experiments Performed by Electricité de France

WAYNE

L.

1

DECKER

and

PAUL

T.

29 35

SCHICKEDANZ—The

Evaluation of Rainfall Records from a Five Year Cloud Seeding Experiment in Missouri

DONALD L . EBERLY a n d LEWIS H . R O B I N S O N — D e s i g n

55

and

Evaluation of Randomized Wintertime Cloud Seeding at High Elevation

65

K. R. GABRIEL—The Israeli Artificial Rainfall Stimulation Experiment. Statistical Evaluation for the Period 1961-65

91

and P A U L W . M I E L K E , J R . — A Randomized Cloud Seeding Experiment at Climax, Colorado, 1960-65

115

P É R E Z SILICEO—A Brief Description of an Experiment on Artificial Stimulation of Rain in the Necaxa Watershed, Mexico

133

SCHMID—On "Grossversuch I I I , " a Randomized Hail Suppression Experiment in Switzerland . .

141

LEWIS O. GRANT

E.

PAUL

E. J. SMITH—Cloud Seeding Experiments in Australia Relevant Climatological Studies S. A.

CHANGNON,

JR. and F. A. H U F F — T h e Effect of xvii

161

xviii

CONTENTS

Natural Rainfall Variability in Verification of Rain Modification Experiments THOMAS J .

HENDERSON—Tracking Silver Iodide Nuclei

Under Orographic Influence

177 199

Nonrandomized Operations GLENN W . BRIER, THOMAS H . CARPENTER, a n d

DWIGHT

B. KLINE—Some Problems in Evaluating Cloud Seeding Effects Over Extensive Areas 209 H A N S G E R H A R D MÙLLER—Weather Modification Experiments in Bavaria 223 Methodological Discussion A R N O L D COURT—Randomized

States

Cloud Seeding in the United

237

and C. L . HOSLER—The Design, Execution and Evaluation of a Weather Modification Experiment 253 A R C H I E M . KAHAN—The Bureau of Reclamation's Atmospheric Water Resources Research Program . . . .271 V U J I C A M. YEVDJEVICH—Evaluation of Weather Modification as Expressed in Streamflow R e s p o n s e . . . . 283 J E R Z Y N E Y M A N and E L I Z A B E T H L. SCOTT—Some Outstanding Problems Relating to Rain Modification . . . . 293 J E R Z Y N E Y M A N and E L I Z A B E T H L . SCOTT—Appendix. Planning an Experiment with Cloud Seeding 327 L. G . DAVIS

JERZY NEYMAN a n d

ELIZABETH L . S C O T T — N o t e o n

the

Weather Bureau ACN Project 351 J . M . W E L L S and M . A . WELLS—Note on Project SCUD 357 J E R Z Y N E Y M A N and E L I Z A B E T H L . SCOTT—Note on Techniques of Evaluation of Single Rain Stimulation Experiments 371 R O B E R T B . D A V I E S and P R E M S. PURI—Some Techniques of Summary Evaluations of Several Independent Experiments 385

CONTENTS

XIX

JAMES—On Pitman Efficiency of Some Tests of Scale for the Gamma Distribution 389 F R A N K YATES—Discussion of Reports on Cloud Seeding Experiments 395 BARRY R .

Observational

Data

A Collection of Data from Cloud Seeding Experiments in Five Countries Introduction I. Arizona Experiments II. Australian Experiment (with Explanatory Note by E. J. Smith) III. An Experiment in France, 1963-64 IV. Israeli Experiment, 1961-65 V. Swiss Hail Prevention Experiment " Grossversuch III," 1957-63

399 400 407 424 426 449

PHYSICAL FACTORS IN PRECIPITATION PROCESSES AND THEIR INFLUENCE ON THE EFFECTIVENESS OF CLOUD SEEDING M. NEIBURGER D E P A R T M E N T OF M E T E O R O L O G Y ,

U N I V E R S I T Y OF C A L I F O R N I A ,

LOS

ANGELES

"An old wisdom from the Far East says: you cannot tell how the flower looks if you know only the seed. You have to know first how the bud looks like. It appears that in our age of modern technology of the conquest of space and time, we sometimes tend to forget this simple wisdom. So, for instance, when man tries to make or prevent rain or hail artificially without even knoioing all the intermediate steps that Nature has provided in the evolutionary process that leads from a cloud droplet or ice crystal to a rain drop or hailstone." —from the foreword to the Proceedings of the International Conference on Cloud Physics, May 2June 1, 1965, byHelmut Weickmann 1. Introduction The above quotation focuses on the crux of the situation with which I shall deal. From the physical standpoint, the problem with respect to the effectiveness of attempts to modify the precipitation process is that while we have a fairly clear understanding of the process in qualitative terms, adequate theories and observational data for quantitative evaluation of process rates are not available. Consequently, we cannot tell whether the natural process in a given case will proceed at optimum efficiency, or whether a particular change in the conditions will lead to an increase or a decrease in the rate. Under these circumstances it is likely, if not inevitable, that insofar as they have any effect at all the attempts to affect precipitation by cloud seeding will in some cases increase it and in other cases decrease it, with results which are impossible to predict and difficult to detect. In the following I shall review the status of our knowledge of the process of formation of precipitation and point out the implications of our present knowledge and the requirements for future research to enable the intelligent selection of cases and procedures to seed to produce a particular effect in optimum fashion. 1

2

FIFTH BERKELEY SYMPOSIUM: NEIBURGER

2. Qualitative description of precipitation processes Broadly, the processes of formation of precipitation may be divided into the dynamic processes, concerned with the motions of air currents which give rise to the general conditions for the formation of clouds and precipitation, and the microphysical processes, concerned with the growth of the individual precipitation particles from gas phase by condensation and from smaller cloud particles by collision and coalescence. There is, of course, a strong interaction between the two kinds of processes. The upward motions determine the rate of cooling due to expansion and thus control the rate at which the microphysical processes go on. The release of latent heat in condensation and the drag of the particles formed affect the buoyant forces which determine the upward motion. While the dynamic processes are prerequisite to the microphysical ones, it is convenient to discuss the processes of particle growth first, and subsequently to turn to the larger scale setting in which it occurs. I t is a fact of common experience that clouds can remain in the sky for long periods without precipitating. Since clouds consist of water particles, liquid or solid, which are heavier than air, this phenomenon requires explanation; the usual explanation is, of course, that the particles are being sustained in a current of air moving upward as fast as or faster than they are falling. Alternatively, the particles may evaporate as they fall from the base of cloud and vanish into vapor a short distance below. Measurements show that the radii of drops in nonprecipitating liquid clouds are in the range 2 to 20 microns, with the modal radius usually between 5 and 10 microns. These drops have terminal velocities ranging from 0.05 to 5 cm/sec, so that very slight upward flow of air would be required to offset their falling. Further, it has been shown that if drops of these sizes fall out of a cloud into air with 90 per cent relative humidity they would evaporate before they go as much as one meter. Rain drops, on the other hand, range in radii from 0.1 mm to 3 mm, with terminal velocities from 70 cm/sec to 9 m/sec. They thus can fall relative to ascending air and reach the ground before evaporating, even when the vertical air velocity is considerable and low humidities prevail below the clouds. The key difference between cloud and precipitation is thus the particle size, and the central question in precipitation physics concerns the conditions under which the particles can grow, roughly one to ten million times in mass, from cloud size to precipitation size. The process of condensation by itself can be shown to be much too slow to explain the rates at which precipitation forms. For instance, the development from clear air to showers in the course of a summer day may occur in a matter of an hour or less. While condensation results in very rapid growth of drops to the size of average cloud drops, say 10 microns, continued growth is progressively slower, and with the number of drops which form, there is not enough water vapor available for millimeter drops to be produced by condensation alone.

PHYSICAL FACTORS IN PRECIPITATION

3

The two ways that cloud particles can grow rapidly to precipitation are (1) by collision and coalescence, and (2) by the three phase, or Bergeron process. The nature of the first process is obvious: if the cloud drops are not of uniform size the larger ones will fall relative to the smaller, and tend to overtake and capture them. After collecting one small drop the large drop becomes larger, falls faster, and is more effective in collecting others. But as we shall see, because of the tendency for the air to carry drops around each other, there are limitations on the initiation of this process. The three phase process is based on the fact that drops remain liquid at temperatures below 0° C, and ice crystals, if they form, are much fewer in number than the liquid drops. Since the equilibrium vapor pressure over ice is lower than that over water at the same (subzero) temperature, there is a strong gradient of vapor density away from the liquid drops toward the ice crystals, so that rapid transfer of water occurs from the drops, which evaporate, to the crystals, which quickly grow large compared to the preexisting supercooled drops. The crystals fall relative to the remaining small drops and collect them. Process (2) thus may initiate process (1), and the two acting together can readily lead to the formation of precipitation sized particles in subfreezing clouds. In warm clouds which precipitate, collision and coalescence alone must be the activating process. As a drop falls, the air ahead of it is pushed out of its way, and if a smaller drop is contained in that air it likewise will tend to be carried out of its way by the moving air. Because of its inertia the smaller drop may be struck by the large one if it is not too far from the axis of fall. The fraction of the small drops in its way which would be collected by a large drop is called the collision efficiency E. It is a function of the radii A and a of the large and small drops, as well as their density and the density and viscosity of the medium (air) through which they fall. If Yc is the limiting distance from the axis of fall of the large drop, within which the small drop must lie to be collected, the collision efficiency is (1)

E = n / { a + A) 2 = yV{ 1 +

p)\

where yc = YJA and p = a/A are nondimensional values of the limiting distance and the small radius, in units of the large radius. It is convenient to show yc as a function of p for various values of A. Evaluations of yc have been carried out by computing drop trajectories for various conditions [11]. It turns out, as shown in figure 1, that for large A (greater than 60 n) the geometric value corresponding to E = 1 is approximated for most values of p; but for relatively small A, say 20 E is at most about 0.25, and is zero for half of the range of p (for p < 0.3; >0.8); and for A ^ 18, E is zero for all values of p. (It should be mentioned that these computed values of yc and E involve approximations, and a recent investigation suggests that, rather than a complete cutoff for A 5= 18, the values for small A are not much smaller than for A = 20.

4

FIFTH B E R K E L E Y SYMPOSIUM:

NEIBURGER

P = °5/at FIGURE 1

Linear collision efficiencies yc of water drops of radius a L falling through air, as a function of the ratio p of the radius of the small droplet as to the large. If strong electric fields are present they would modify the collision efficiency, but present indications suggest that strong fields arise only after some cloud particles have become large or ice has formed in the cloud.) This result, together with the data presented earlier on drop sizes in clouds, indicates that precipitation by the warm process should not be expected to take place from most clouds. If, as the data shows, most clouds do not contain drops of radius as large as 20 n, the likelihood of drop growth by coalescence is very slight. I t is only in those cases where the size spectrum is broadened to include a sizable number of drops up to 30 or 40 microns in radius that the coalescence process is likely to be initiated. To understand why clouds sometimes have broad drop size spectra which lead to warm rain, but frequently do not, we must look to the details of the condensation process. 3. Condensation and the formation of clouds Saturation vapor pressure or one hundred per cent relative humidity is defined in terms of the vapor in equilibrium with a plane surface of pure water. In the

PHYSICAL FACTORS IN PRECIPITATION

5

absence of surfaces, however, the vapor pressure can exceed the saturation value several fold before condensation will begin. In the atmosphere surfaces are always present, in the form of particles of haze or dust. These particles are predominantly in the size range 0.005 m to 5 y.. The lower limit is due to the tendency for smaller particles to agglomerate rapidly because of Brownian motion. Particles larger than one micron will tend to settle out even though the effect of turbulence is to diffuse them upward. Typical observed size distributions of nuclei are shown [7] in figure 2. It will be seen that the most frequent sizes are the smallest. While individual cases vary considerably Junge found that on the average the frequency of sizes varies as the inverse fourth power of the equivalent radius. The reason why nuclei are needed for condensation to occur at reasonable humidities is that the equilibrium vapor pressure of a spherical drop is inversely proportional to its radius. For this reason condensation will occur more readily on large than on small nuclei. Condensation on the few large nuclei will keep the vapor pressure from rising to the higher value required for condensation on the small ones, and the number of cloud drops will thus be much smaller than the total number of nuclei. Among the various particles in the atmosphere some are hygroscopic, that is, they tend to attract water even at relative humidities less than 100 per cent. These nuclei will grow into droplets before the nonhygroscopic nuclei of the same size. Figure 3 shows the equilibrium vapor pressure of nuclei composed of various quantities of hygroscopic material as a function of their size. Each curve shows a maximum, and if this maximum vapor pressure is exceeded the drop will grow continuously, for the larger it becomes the smaller its equilibrium vapor pressure. T h e f a v o r e d nuclei are large hygroscopic

nuclei.

In general the number of nuclei which are effective is in the range 50 to 1000 per cubic centimeter, and this correspondingly is the number of cloud drops formed. The exact number and size distribution depends on the rate of cooling (updraft velocity) during the time when saturation is being attained and exceeded and the spectrum of nucleus sizes. The rate of growth of a single drop of radius r growing from a nucleus of equivalent radius r0, density pn, and hygroscopicity r , in an environment with temperature T and vapor pressure e is (2)

dr dt

D rPsRvT

Here D is the coefficient of molecular diffusion, Rv the gas constant, and es(T) the saturation pressure of water vapor, L is the latent heat of condensation, ps the density and a the surface tension of the liquid, and S = (psLr dr/dt)/kT, the fractional difference between the temperature of the drop and the ambient temperature, where k is the thermal conductivity. That the rate of growth decreases with increasing size is readily demonstrated

rV

•

X

\

0

f

ro

E

/ r

\

\

3 (12)

+—+

4 (20)

A—A

5

>

\

I05

(4)

\ \

w X

t

10'

o

o

10 -i

\) \

* X + \

\

.-2 10'

10"'

\

1.

1 \

•

\ A

\

r2 10

.-I 10 ^

radius

I0 L

10'

->-

FIGURE 2

Average size distributions of natural aerosols occurring in a city (curves 1, 2 and 5) and at a mountain peak (3 and 4). The curves are composites of measurements made for the various ranges of sizes by different techniques at different times.

co

>

c o

t-< S ï**

Z )

C\J

E5 < cr

S M P O

m

I

co

O

'G 03 >

D

Sx

00

tß a> m»

a s

CO £ ' 3 O1 H

8

FIFTH BERKELEY SYMPOSIUM:

NEIBURGER

10"' cm 1 0 0 0 microns

Nucleus Radius FIGURE 4

Distributions of nucleus sizes used in drop growth computations. Solid curves: cumulative distributions, referred to left hand scale of ordinates. Dashed curves, differential concentrations, referred to right hand scale.

9

PHYSICAL FACTORS IN PRECIPITATION

by this equation. Table I shows values of dr/dt for various r0 and r, assuming T = 278K and e = 1.01e„. The rates in table I cannot be applied to estimate TABLE

I

RATE OF GROWTH OP DROPS ON A 0 . 1 MICRON "RADIUS" SALT NUCLEUS, IN A ONE PER CENT SUPERSATURATED ENVIRONMENT

r (microns) 0.13 0.25 0.46

1.00

ÔT — (microns/sec) at

r (microns)

197.22 40.17 3.27 0.77

2.51 4.64 10.00 100.00

at

(microns/sec) 0.34 0.16 0.075 0.007

the rate of growth of drops in a cloud, for as the drops grow the condensed vapor reduces the environmental vapor pressure at a rate which depends on the number of activated nuclei. To see the size distribution which develops in a cloud, the growth equation must be applied to specific nucleus distributions. C. W. Chien and I [9] computed some cases of cloud growth for certain typical rates of cooling and nuclei spectra. The distributions of nucleus sizes assumed are shown in figure 4. In the type A distribution there are more than 1000 nuclei per cm3 larger than 0.01 n, there are 115 per cm3 larger than 0.1 ju, and 1.5 per liter larger than 1 /¿. In the type B distribution, which may be typical of air near the ocean, the first two numbers are almost unchanged, but there are 50 per liter larger than one micron, and 14 per m 3 larger than 10 microns. As a simulation of an active convective cloud, we used a distribution of vertical velocities based on the measurements of the Thunderstorm Project [3] as shown in figure 5. The air is assumed to start near the ground with a relative humidity of 75 per cent and type A nucleus distribution. The growth curves for nuclei of various sizes are shown in figure 6. Once the relative humidity slightly exceeds 100 per cent the drops formed on nuclei of 0.1 n radius or larger grow rapidly, while those formed on smaller nuclei do not continue to grow. The size distributions after various time intervals are shown in figure 7. Shortly after the cloud forms the separation between the cloud drops and the inactivated nuclei show up, with the drop size mode at a radius of about 7 microns. At the end of the computation, corresponding to a rise of the air parcel to 9 km, the mode is at 20 n, with about 70 per cm3 greater than 16 n but only one per liter greater than about 22 ¡i. This very narrow drop size spectrum would not favor collision and coalescence even though the critical size for nonzero collision efficiency is slightly exceeded. The influence of the rate of cooling was tested by computing the growth of the same nuclei at a much smaller cooling rate, corresponding to a constant vertical velocity of 17 cm/sec. The growth of the cloud drops took longer, of course, and the computation was continued only until the modal cloud drop

^ O

o° —

o°

(qui)d

> » s « P Oí "t? O £ £ '3 a S •o —i oT «i C >D Sí "S ^ o g T3 O a) o«(I S a> oS o «tj o3 Ö ° 3 o 8 a s a a? oa -, «3 03 -do

£^

U)

i

O T

e> 0 o * o T

o

W W \ w eocococccocococococococococococo

I H H H H H H H H H H N N N N N N N N N O M M C O W e O C O M I I I I I I I I I I I I I I I I I I I I I I I I I I I I I i I I I I I cococococococococococococococoeococococococococococoeocoeocoeocococococo CO cO cO CO CO CO CO ® tì t O ( Û c o o c o t O Ö î O ( Û c o c o c o , fa

= & + 722(2).

The average unseeded precipitation in the control z is defined by (6.6), (6.21)

2 = (Ji + J ^ i J i Z i +

W

The average unseeded precipitation in the target is then, (6.22)

fa(z)

= ft + fts.

The effect of seeding is taken to be (6.23)

A?/2 = fa{z) -

H2).

The per cent increase is then, (6.24)

? W , R = 100Ay 2 /Hz)-

It is clear that many estimates similar to equation (6.24) can be defined by replacing z with any observed value of z. An alternate approach that avoids this difficulty and in fact, one of the earliest used in cloud seeding evaluations, is to compute (6.25)

dk = y?> -

m(42)),

for the K seeded events k = 1, • • • , K. These are the differences between the seeded value of y and its unseeded value as predicted by the control line. Let

84

FIFTH B E R K E L E Y SYMPOSIUM: E B E R L Y A N D

(6.26)

d = K-1

ROBINSON

jt dk. k=i

The estimate becomes (6.27)

K,r = lOOd/Y,

where Y is some estimate of the average unseeded target precipitation, possibly from equation (6.22) or historical records. 7. Transformations No square root, logarithmic, or other transformations of the basic data have been made. Other investigations have employed such transformations, mainly to stabilize the residual variance as required by the theory, although this desirable end has the bad side effect of complicating the estimation procedures. In particular, biased estimates must be corrected [11]. Estimation of the per cent increase is a central issue in any evaluation of cloud seeding. It is so important that the analysis very likely should be handled as an estimation problem with confidence intervals developed for what are essentially, in all procedures, ratio estimates. 8. Computer processing The analyses discussed in the next sections are based on about 150,000 gage hours of accumulated precipitation data, of which about 25,000 gage hours were nonzero. Readings for each individual gage were punched on cards, each card containing 24 hourly values at least one of which was nonzero. (No cards were punched for rainless or snowless days.) This file of about 2500 cards was then checked, matched, merged, and processed with a deck of 76 event cards containing all the meteorological and operational data. Total precipitation for each 12-hour event was calculated for each gage and output as a gage detail file for each gage and event processed. These 12hour totals were combined for all gages within a particular target or control area and divided by the total gages in the target or control. This information was output as an area average file for each area and event processed. The precipitation data listed in table II were obtained from this output. Thus the numbers represent area wide 12-hour totals, divided by the number of gages in the area. The gage detail file also includes, for each event, 12-hourly averages, standard deviations, and coefficients of variations, sorts of each of these, and a frequency distribution of the total gage record for each 0.01 inch increment of precipitation. Since areas (and events) could be redefined with ease, it was quite simple to investigate modified target and control areas. In fact, a modern digital computer with large memory is almost indispensable to relieve the tedium and

CLOUD SEEDING IN THE SIERRA

85

remove the errors in these computations. The work was done on an IBM 7094 in the Fortran language. Once the precipitation data had been boiled down to the events of interest, standard and modified regression programs were employed to calculate the results in tables IVb and Vb. 9. Summary of analysis of westerly wind cases As pointed out earlier, each seeded period is stratified into categories of wind direction according to the standard 16 point rose. This section describes the results of the analysis of the cases that were determined to be associated with a west wind over the Lake Almanor watershed. Twenty three usable 12-hour periods with winds from WSW through NW occurred during the season. Of these 23 cases, the east burner group was used in 11 cases and the west burners used in 12 cases. This resulted in target precipitation gages being seeded 12 times and not seeded 11 times. With the standard gage groupings, the ratio, regression, and covariance analyses showed an increase in precipitation when the clouds were seeded prior to moving over the target gages; that is, when the west burners were on. This increase is largest for the westerly cases when the — 5° C level was at or below 7500 feet. The chance of erroneous identification of distinct regression lines in this cold situation is, for the regression analysis, less than five per cent. This result is the main significant effect to be reported (see table IVb.) The per cent increases for the westerly cold cases are by the ratio method, 62.2 per cent; by the regression method, 52.8 per cent; and by the covariance method, 68.1 per cent, so that in round numbers the increase is 60 per cent. With warm and cold combined into one sample, the ratio, regression, and covariance percentage estimates are respectively 41.6, 41.7, and 42.4 per cent, or 40 per cent in round numbers. Regression analysis showed a difference in seeded and unseeded lines significant at the 10 per cent level. The covariance analysis was significant at the 15 per cent level, the bivariate analysis of variance at the 25 per cent level. As described earlier, several different groupings of gages have been used to investigate the distribution and magnitude of any increase. Table IV shows the results of the analysis of the gage grouping which was thought to represent gages most likely to be seeded; that is, arrangement BT. Increases of 54.5 per cent using the ratio method, 53.9 per cent using the regression method, and 53.5 per cent using the covariance method were observed for combined westerly cases. The increases for the cold cases were, for the same analyses, respectively, 79.6, 75.1, and 88.8 per cent. Significance of these results is about the same as for the standard gage group. Data from the other gage groupings are not included in this summary. The most interesting result of the other grouping analyses was the fact that the

86

FIFTH BERKELEY SYMPOSIUM: EBERLY AND ROBINSON

target gages closest to the burners showed more of an increase than the gages farthest from the burners. 10. Summary of analysis of southerly wind cases For 28 usable events associated with southerly winds, 15 were identified as warm and 13 were identified as cold. The east burners seeded the east target 16 times and the west burners seeded the west target 12 times. Estimates of per cent changes due to seeding along with the elaborating statistics are shown in table V. No effects of interest comparable to the westerly cold situation were detected. In particular, no positive changes were observed that could be identified with significantly different seeded and unseeded regression lines. Covariance analysis and bivariate analysis of variance were both used; regression analysis was not performed because of time limitations and the unpromising prior results. (The basic data for analysis is, of course, available in table II.) Regression analysis is not likely to detect any significance either, except perhaps for the cold cases. Unfortunately this small sample is highly unbalanced. On the whole, an excess of negative signs occurs in the estimated effects. There is little doubt, however, that this is a chance occurrence, although the temptation might exist to postulate a minus effect in warm cases. No improvement was detected in the level of significance for the different groupings of the target gages. This is to say, no changes were observed which could be confidently attributed to seeding when investigations of just the gages closest to the burners, gages farthest from the burners, and so forth, were made. 11. Discussion of results Statistical analysis of the meteorological data appears to have isolated the general weather conditions where seeding has been effective. This has led to an investigation of the physical mechanisms which might be operating to cause an increase in precipitation. Reviewing briefly the results of the statistical analysis, a positive change was observed in the standard target area during seeding when the winds were from the west. In round numbers, the increase was 40 per cent for all westerly cases and 60 per cent when only the cold cases were analyzed. When the peripheral target gages were eliminated from the analysis and the buffer zone increased, the increases became about 54 per cent for all cases and about 80 per cent for the cold cases. The target gages were further divided into close in targets and targets farthest from the burners. This analysis showed a bigger increase close in than was observed in the farthest target. The implication in the aboye result was that a distance effect relationship was occurring. To investigate this further, individual gages were analyzed down-

87

CLOUD SEEDING IN THE SIERRA

wind, under westerly flow, of two different burners, Stover Mountain and Butt Mountain. Results of this study are shown in figure 5. Average gage catch was normalized to the catch of the control gages and plotted for the seeded periods and for the unseeded periods. The solid line is

4

6

8

10

12

14

16

IS

GAGES E A S T OF STOVER M T N .

x— KEY: SEEDED UNSEEDED .CAGES 1. FR 2 . MH 3.

CH

4 . RC 5. 5 M 3.

2

4

6

8

4.

10

5.

12

6 . SM

14

16

18

20

22

DISTANCE FROM BURNER - M I L E S FIGURE 5

Analysis of the effects of seeding under westerly flow downwind of two individual silver iodide burners, 1963 data.

24

88

FIFTH B E R K E L E Y SYMPOSIUM: E B E R L Y A N D

ROBINSON

the seeded catch. The broken line is the unseeded catch. In both cases the precipitation was observed to decrease with distance when no seeding was carried out. A rise in terrain at about ten miles from the Stover burner accounts for the increase at that distance. Seeded cases showed the opposite pattern; that is, precipitation was observed to increase out to about seven miles and then decrease. In every case, the seeded precipitation catch was more than would have been expected. Under westerly flow the Almanor watershed is in the lee of the Sierra; that is, the mountains are highest to the west of the watershed. The watershed drains to the Pacific through the deep Feather River Canyon which is south and southwest of the basin. Air moving from the west to the east passes the crest of the Sierra and begins a gradual descent which averages at least 1000 feet in 10 to 20 miles. Compression of the air increases both the pressure and temperature of the parcel, causing evaporation of the cloud particles. In the mountains the flow becomes complicated and successive waves are often produced in the lee of the mountain. However, at Almanor the overall tendencies will be for the air to be sinking over the watershed with superimposed waves downwind of the crest of the Sierra under westerly flow. Discussed below are three physical mechanisms which are probably contributing to increasing the precipitation under westerly flow when the clouds are seeded. (a) Freezing the subcooled (temperatures less than 0° C) drops in a cloud will allow the individual drops to grow larger than a water drop could grow from condensation alone under the same atmospheric conditions. This is possible because the vapor pressure over ice is smaller than over water; thus, ice can attract more vapor from the air and can cause the adjacent water droplets to evaporate with this vapor also being attracted to the ice. By growing larger, the ice crystals can begin to fall through the cloud (having overcome the buoyant effect of the cloud), and still grow by sweeping up other cloud particles. (b) Freezing of the cloud particles increases the expected life of a cloud particle. More energy is required to evaporate an ice particle than is required to evaporate a liquid drop. Also, because the vapor pressure is lower over ice than over water, the gradient of available vapor below a cloud would permit ice particles to fall farther before they would start to evaporate. The saturation pressure over water at — 5° C is 4.22 millibars while over ice at the same temperature the saturation pressure is 4.03 millibars. (c) Latent heat is released to the atmosphere when water freezes. From calculations, the available heat from this source would appear to be large enough to greatly increase the buoyancy of the seeded parcel of air. For example, if the assumption is made that seeding will convert a cloud downwind of a burner that is 3.4763 X 108 meters 3 in volume (the approximate seeded volume of air with temperature less than — 5° C out to about 1.5 miles from the burner) and that the average water content of the cloud is 1 gram per meter 3 , then approx-

89

CLOUD SEEDING IN THE SIERRA

imately 1.16398 X 1011 joules of heat would be released. This is enough heat to raise the temperature of such a volume of dry air by 3° C. An increase of temperature of a volume of air over the temperature of the surrounding air would have the effect of displacing the warmed air in an upward direction until the surrounding air mixes sufficiently with the warmed air to bring it into equilibrium. Thus, the seeded parcel of air, being more buoyant than the surrounding air, resists the downward motion in the lee of the crest preventing evaporation of the cloud particles and maintaining the cloud for a longer period of time. The analysis of individual gages downwind of burners referred to earlier suggests that the effect of these physical mechanisms is to increase and displace the precipitation to the east with a maximum increase occurring at about seven miles from the burner. This appears to be a physically reasonable possibility. No statistically significant result was detected from seeding clouds associated with southerly flow over the Almanor watershed. One factor which is thought to be an active suppressant of any effect is the thermally stable air often associated with southerly flow. This is in contrast with westerly flow where the atmospheric conditions usually are measured to be unstable. Stable flow could prevent or retard the silver iodide from reaching the clouds at the levels and in the concentrations required for effective seeding. Also, the release of latent heat under stable conditions is less likely to be sufficient to induce vertical motion. The possibility has been suggested that the increase under southerly flow might be occurring farther downstream. There is no support for this case. The diffusion of smoke from a point source would reduce the cloud concentration of silver iodide to a level below one particle per liter in less than ten miles. The same analysis was completed for the southerly cases as was described above under westerly flow; that is, only close in gages were analyzed, only gages farthest from the burners were analyzed, and so forth. This comprehensive work gave no clue to any effects. REFERENCES "Weather and climate modification," Report of the Special Commission of Weather Modification, December 1965.

[ 1 ] NATIONAL SCIENCE FOUNDATION,

[ 2 ] N A T I O N A L ACADEMY OF S C I E N C E S — N A T I O N A L RESEARCH COUNCIL, Weather

and

Climate

Modification, Problems and Prospects, Vol. 1 and 2, Final Report of the Panel on Weather and Climate Modification, Washington, 1966. [ 3 ] N . H . FLETCHER, The Physics of Rainclouds, Cambridge, Cambridge University Press, 1962. [ 4 ] T . A . J E E V E S , L . L E C A M , J . N E Y M A N , and E . L . SCOTT, "On the methodology of evaluating cloud seeding operations," Bulletin 16, California State Water Resources Board, 1955. [5] P. J. MEADE, "Meteorological aspects of the peaceful uses of atomic energy," Technical Note 33, World Meteorological Organization, WMO-No. 97, TP 41, Geneva, WMO Secretariat, 1960.

90

FIFTH BERKELEY SYMPOSIUM: EBERLY AND ROBINSON

[6] F. PASQUILL, Atmospheric Diffusion, New York, Van Nostrand, 1962. [7] H. E. CRAMER et al., "Meteorological prediction techniques and data system," Final Re-port, Contract No. DA-42-007, CML-552, Bedford, Geophysics Corporation of America, 1964. [8] H. SCHEFFÉ, The Analysis of Variance, New York, Wiley, 1959. [9] T. W. ANDERSON, An Introduction to Multivariate Analysis, New York, Wiley, 1959. [10] P. A. P. MORAN, "The power of a cross-over test for the artificial stimulation of rain," Austral.

J. Statist.,

V o l . 1 (1959), p p . 4 7 - 5 2 .

[11] J. NEYMAN and E. L. SCOTT, "Further comments on the 'Final Report of the Advisory Committee on Weather Control'," J. Amer. Statist. Assoc., Vol. 5 (1961), pp. 580-600.

THE ISRAELI ARTIFICIAL RAINFALL STIMULATION EXPERIMENT STATISTICAL EVALUATION FOR THE PERIOD 1961-65 K. R. GABRIEL H E B R E W UNIVERSITY,

JERUSALEM

1. Introduction A rainfall stimulation experiment is being carried out in Israel by silver iodide seeding from an aircraft in a randomized crossover design. The operations are directed by Electrical and Mechanical Services (Mekorot, Ltd.), Mr. M. Cohen, Director, and are financed by the Israeli Ministry of Agriculture. The experiment is conducted under the guidance of the Rainfall Committee whose chairman is Professor E. D. Bergmann, and the related research work is performed at the Hebrew University, under the direction of Professor J. Neumann. The author is responsible for the statistical design and evaluation. Daily rainfall data are provided by the Israeli Meteorological Service from its regular network of raingage stations. The present statistical design of the experiment [9] was adopted when an earlier design based on weekly units [8] was abandoned after a few weeks because those units were considered unsuitable for detailed analysis. Earlier analyses excluded a small number of days, twelve, on which the aircraft could not be operated. Since the decision to ground the aircraft was not independent of atmospheric conditions, this exclusion might have introduced a slight bias. Therefore, the present analysis includes these few days and the results differ very slightly from those published earlier [10], [11], [12]. The experiment is based on comparison of amounts of precipitation in two areas of Israel: the North, and the Center, as shown in figure 1. These are separated by a buffer zone to avoid contamination of the atmosphere in one area when the other is being seeded. (The southern, more arid, part of Israel has been excluded from the experiment because its rainfall regime is different.) The interarea comparison reduces day to day variability of observations on precipitation, as rainfall in the two areas is highly correlated. The correlation between daily amounts of precipitation in the two areas was found to be r = 0.81, when means of eight stations were taken in each area and a square root transformation used to reduce heteroscedasticity and nonnormality. The amount of precipitation in each area is estimated by a simple average of daily precipitation re91

92

FIFTH BERKELEY SYMPOSIUM: GABRIEL

Map of Israel showing both experimental areas and the interior areas (shaded). Dots indicate raingages used in analysis as of 1964-65. corded at different stations of the area. This method was considered to be simple and objective, and in view of high between station correlations it is probably as accurate as the intricate weighting procedure employed in Australia [1]. These amounts will be denoted N and C for the North and Center, respectively, with

ISRAELI EXPERIMENT

93

subscripts n and c indicating the area which was designated to be seeded, for example, Nc would be the amount in the North on a Center seeded day. The experimental variable is defined as N — C, the difference between the two area averages. In some studies (see, for example, [25]) a ratio was used instead of a difference, so that periods with different amounts of precipitation received the same weights. Use of differences, on the other hand, ensures that an increase on a very rainy day be weighted more heavily than one on a day with little rain. It is not really known which method is more sensitive to potential seeding effects (see sections 3 and 5 below). One would suspect that the use of ratios might greatly increase variability and hence reduce the efficiency of the experiment. Each experimental unit-—24 hours—is designated ahead of time, that is, before the season starts, to be seeded in one or the other of the two areas. This designation is random and independent from day to day so that about one half of the days are designated to be North seeded and the rest to be Center seeded. Random designation permits probabilistic evaluation, that is, testing for statistical significance of the differences between the two sets of days. The results of the experiment are evaluated by comparing North seeded days with Center seeded days, using the difference N — C as the comparison variable. Such a crossover design with a double comparison of days and areas was first proposed by Adderley and used in Australia. It was used also in Canada [16]. Its statistical properties were discussed by Moran [20]. The advantages of this crossover design are readily demonstrated under the simplifying assumptions of normal distribution and equal variances of precipitation in both areas. Let r denote the correlation between N and C. A crossover experiment requires only (1 — r)/2 of the number of observations (days) needed by a single area design. Also, a crossover design requires only 1/2(1 + r) of the number of observations needed if a control area is added with a view to predicting target rainfall from it. With r = 0.81 as in Israel, these proportions are 8.5 per cent and 27.6 per cent, showing that much shorter periods of experimentation are required with the crossover design. It should be noted that even in the absence of correlation the crossover design still reduces the length of an experiment by one half. Recent remarks by Bowen [4], [5] have raised the fear that these designs may be inefficient because the effect of silver iodide seeding may persist beyond the seeded day (or other experimental unit). Nominally unseeded days might show residual effects of earlier seeding, and the difference between seeded and unseeded days might be less than the real effect of seeding. All designs with short experimental units would therefore underestimate the true increase in rainfall due to seeding. Correct estimates could be obtained only from experiments with sufficiently long intervals between units, which would ensure that the effects of silver iodide seeding in one unit could not carry over to the next unit. As of now, there is no compelling evidence that such persistence exists, and

94

FIFTH B E R K E L E Y SYMPOSIUM:

GABRIEL

we feel that the use of the present type of short unit design is justified. However, the possibility that persistence exists should be investigated carefully, and data from current experiments may be tested for the existence of such effects (see sections 2, 5.4, 5.5). Seeding techniques and equipment are similar to those used by the CSIRO group in Australia. Silver iodide is seeded in acetone solution by means of double burners fixed under the wing of a DC-3 plane. The solution is burned at the rate of 13 liters per hour, which corresponds to vaporization of 800 to 900 grams of Agl. The aircraft takes off whenever cloud conditions appear favorable, but seeding is carried out only after the cloud seeding oificer has ascertained that cloud tops reach or exceed the — 5° C level. The extent of seeding in each season is shown [24] in table I. Seeding takes place just below cloud bases in an area TABLE

I

S E E D I N G : N U M B E R OF FLIGHTS, D A Y S AND

Number of seeded days Number of seeding flights Hours of seeding

HOURS

1961

1961-62

1962-63

1963-64

1964-65

20 26 91

34 61 117

36 63 130

39 65 179

43 70 172

displaced upwind from the target by half the wind speed per hour, according to the prevailing direction and speed of the wind. The experiment runs throughout the rainy season and includes each day as North seeded or Center seeded, according to the random designation and irrespective of whether seeding is actually carried out (see table II). It would be TABLE

II

U N I T S EMPLOYED IN THE E X P E R I M E N T

Season 1961 half 1961-62 1962-63 1963-64a 1963-64b 1964-65

Date

Period

Unit of Time

19. 2.61-15. 4.61 15.10.61- 5.11.61 7.11.61-15. 4.62 16.10.62-15. 4.63 1.11.63- 8. 1.64 9. 1.64-30. 4.64 16.10.64-15. 4.65

weekly

0800 to 0800 hrs

daily daily daily daily daily

2000 2000 2000 0800 0800

to to to to to

2000 2000 2000 0800 0800

hrs hrs hrs hrs hrs

advantageous to restrict the evaluation of the experiment to those days on which seeding is feasible and to exclude the large number of days without rain clouds which cannot add to the sensitivity of the experiment. However, this is not permissible. Since seeding is attempted only when suitable clouds appear in the area designated to be seeded, the choice of days for seeding is biased in favor

95

ISRAELI EXPERIMENT

of days with good rainfall conditions in the designated area. The present evaluation relates, strictly speaking, to the designation to be seeded, rather than to actual seeding. 2. Overall evaluation The main object of the experiment is to examine the possibility of stimulating rainfall by cloud seeding, and the secondary object is to identify conditions favorable or unfavorable to such stimulation. Various meteorological measurements and observations are being made. The importance of such measurements has often been stressed [22], It is hoped to obtain interesting detailed analyses by classifying days according to synoptic conditions. The present statistical evaluation includes some first attempts at such analyses. Detailed analyses of counts of freezing nuclei and other observations taken from the seeding aircraft and on the ground have been published separately [15]. Overall results in terms of average daily precipitation for each season and all seasons together are given in table III. Results for each area are presented sepaTABLE M E A N D A I L Y PRECIPITATION

(mm)

III PER STATION OF E A C H

AREA,

IN S E E D E D AND U N S E E D E D A R E A S WITH S / N S R A T I O

Seeded means all days designated to be seeded, whether actually seeded or not. Unseeded refers to all unseeded days. Note the slight difference between this table and all the following tables from which "dry" days are excluded. Station

Designation Total 1961 1961-62 1962-63 1963-64a 1963-64b 1964-65

North Precipitation Center North Seeded Seeded Nn Nc

Center Ratio S/NS Kn/Nc

Precipitation North Center Seeded Seeded üc Cn

Ratio S/NS Cn

Average Ratio S/NS

3.232

3.038

1.064

2.351

2.934

1.248

1.152

2.495 3.209 2.044 2.957 4.318 4.262

0.744 4.350 2.911 2.471 3.254 3.185

3.353 0.738 0.702 1.197 1.327 1.338

1.050 1.827 1.157 3.991 2.406 4.010

0.853 3.391 1.991 3.417 3.393 3.882

0.812 1.856 1.722 0.856 1.410 0.968

1.650 1.170 1.099 1.012 1.368 1.138

rately for days designated to be seeded in the North and for days designated to be seeded in the Center. Beside the mean daily precipitation figures, the table shows the S/NS ratio, defined as the ratio of mean amounts N„ to Nc in the North and ~Cc to C„ in the Center. As designation of seeded area was random, the days of both kinds should have had similar amounts of natural precipitation, except for random variability.

96

FIFTH BERKELEY SYMPOSIUM: GABRIEL

Thus the amounts Nn and C„ on days designated to be North seeded should have been similar to the amounts Nc and Cc, on days designated to be Center seeded. In the absence of random variability the S/NS (seeded/nonseeded) ratio of each area would have been equal to one if seeding had had no effect, whereas if seeding had been effective the S/NS ratio of each area would have exceeded one. In fact, neither of these explanations fits the results in table III which show that for each season, except 1963-64b, the S/NS ratios for the two areas deviated from one in opposite senses. In other words, Nn — Nc has had the same sign as Vn — Vc. Apparently, in any one season, the random differences in country wide precipitation between North seeded days and Center seeded days were much larger than any possible seeding effect. The results of table III must not be taken to indicate differential seeding effects in the two areas. They merely show that rainfall stimulation experiments based on randomized seeding in a single area are subject to enormous variability. To give any hope of revealing seeding effects, single area experiments must be continued for a large number of years. For the initial design of the Santa Barbara experiment it was calculated that well over ten years of experimentation in one area are required to give over 75 per cent chance of finding a 5 per cent significant result if seeding increases precipitation by 20 per cent ([22], p. V-32). The Israeli experiment requires only four to five years for the same chance of finding a significant result [8]. Part of the difference may be due to different rainfall regimes, but most of it results from the advantages of the crossover design which allows interarea comparisons. The extent of this advantage has been discussed in section 1 above. As the difficulties with single area experiments were foreseen, the Israeli experiment has been based from the outset on interarea comparisons by means oi N — C differences or N/C ratios. Random variability for such comparisons is considerably reduced, since most day to day variation affects N and C similarly, and thus does not affect either N — C or N/C. Using these comparisons one may expect conclusive results with a relatively short experiment [20]. The product of the S/NS ratios for the North Nn/Nc and for the Center (Jc/(Jn gives the double ratio ,

.

seeded amounts in N X seeded amounts in C _ NjJc unseeded amounts in N X unseeded amounts in C Nat CENTER NORTH w CENTER $ -

•• • ' •'"'•• •• • .

.

NORTH k> -

.... . . . . .

! ••

-

1

-

CM

CENTER | -

•

NORTH 5 CENTER Iom - •

•

•'•••

• •• •

. . . . . . .

NORTH J CENTER S NORTH CENTER & -30

-20 -10 0 10 20 NORTH-CENTER DIFFERENCE IN PRECIPITATION (mm) FIGURE 2

Plot of daily North-Center difference in precipitation, per season and seeding region. If this is not a random effect (a matter which it is difficult to determine since the number of days on which such large effects have been noticed is very small), it raises interesting speculations about the possible effect of cloud seeding. Could it be that the effect of seeding is negligible on most days, but is very considerable on some particular days? I t would be most helpful if one could identify some synoptic peculiarities of the few days concerned, and research should continue in this direction. Schleusener [26] has pointed out that similar observations have been made in the Mexican experiment. Pérez Siliceo et al. [27] write ". . . suggests that seeding is not always effective but that, whenever the right meteorological conditions occur, it is highly efficient . . . and ". . . increases in the rainfall at the target area are due to a relatively small number of cases in which seeding is effective."

106

FIFTH BERKELEY SYMPOSIUM: GABRIEL

5.3. Normal tests. It is usually convenient to use the statistical techniques that have been derived for normally distributed variables. The present variable, however, is markedly nonnormal (see figure 3) with an excess of values near zero

NORTH-CENTER

DIFFERENCE

IN PRECIPITATION

(mm)

FIGURE 3

North-Center difference in daily amounts (mm) of precipitation. Histogram of observed distribution and normal curve with same mean and variance. and of very large deviations, both positive and negative. Moreover, there is a slight dependence between the values for successive days (see section 3). Hence, tests based on normal theory are at best approximate. Normal theory methods use means and hence are more sensitive to individual large observations than are ranking methods such as the Wilcoxon-MannWhitney test of section 3. Indeed, the normal test calculated for the entire experiment reaches the critical point for 5 per cent significance even without breakdown or adjustment for a concomitant variable. A slightly better approximation to normality could be obtained by using the difference between the square roots of North and Center precipitation figures [8]. 5.4. Season to season differences. I t might be that real effects within each season of experimentation were swamped by analyzing the days of all seasons together. This could happen if season to season variation were appreciably larger than day to day Variation within each season. Means and variances for each season are shown in table I X along with the corresponding analysis of variance. The F ratios are very close to one, so that there is no evidence of extra season to season variation, and no indication that the days of all seasons should not be analyzed together. To test whether seeding effects might have been the same in all seasons, the variance of percentages of positive comparisons in different seasons is computed and checked against the chi square distribution with (k — 1) degrees of freedom,

ISRAELI EXPERIMENT TABLE

107

IX

D A T A ON N O R T H - C E N T E R D I F F E R E N C E S FOR E A C H

SEASON

Table excludes six rainy days on which the plane was not operational.

Total 1961 1961-62 1962-63 1963-64a 1963-65b 1964-65

Between years Within years F ratio

Days 143 14 30 26 10 30 33

North Seeded Days Mean Variance 105.781 2.229 3.061 3.707 3.186 -2.408 3.137 0.358

Days 132

Center Seeded Days Mean Variance 107.472 0.210

60.634 125.826 74.077 100.396 94.474 144.823

12 27 22 9 24 38

-0.669 2.283 3.241 -3.666 -0.065 -1.651

6.462 243.197 48.968 195.935 24.317 106.312

d.f.

Sum of Squares

Mean Square

5 126

595.985 13482.847

119.197 107.007 1.114

d.f.

Sum of Squares

Mean Square

5 137

454.135 14566.767

90.827 106.327 0.854

where k is the number of seasons compared (see appendix). The variance, as noted at the bottom of table V, is far from significant, so that there is no evidence that seeding effects have not been the same in all seasons. The homogeneity of seasons as regards rainfall amounts and seeding effects is also evident from the fact that the percentage of positive comparisons and the significance levels as estimated from all days together—first line of table V—and as obtained by pooling the seasons—bottom of table V—do not differ much. Clearly, there is no point in keeping the seasons apart in this analysis. 5.5. Monthly differences. Analyses of the data taking each month separately are of interest for two reasons. First, there might be meteorological differences resulting in different effectiveness of seeding in different months. Second, if Bowen's [5] suggestion of persistence of seeding effects is true, any possible seeding effects in later months would be obscured by contamination, and seeding effects should show most clearly in the earlier months of each season. Data for each month are presented in table X, merging the same months of different years. Some differences between months appear but are not significant, the probability of random variation as large as this being 17 per cent. If anything, the effects of seeding would seem most evident in March and April, contrary to what one would expect if persistence existed. Indeed, neither the season to season variations nor the within season variations of the results of this experiment lend support to Bowen's persistence hypothesis. 5.6. Differences according to amounts of natural rainfall. The effect of seeding is unlikely to be the same under all conditions. A variety of synoptic factors probably has a role in furthering or inhibiting the artificial stimulation of

108

FIFTH BERKELEY SYMPOSIUM: GABRIEL TABLE ANALYSIS BY

X MONTH

AVERAGE S / N S R A T I O AND PERCENTAGE OF COMPARISONS WITH W I L C O X O N - M A N N - W H I T N E Y

POSITIVE

T E S T OF SIGNIFICANCE

"Dry" days have been excluded. Positive means that N — C rainfall difference is greater on the N seeded day than on the C seeded day.

Month Oct.-Nov. Dec. Jan. Feb. Mar.-Apr. Pooled average Between month variation

Number of Days

Average S/NS Ratio

Positive Comparisons Per Cent

Significance Level

34 50 67 62 68

1.176 1.093 1.145 1.021 1.663

51.43 46.56 61.57 45.41 66.83

0.444 0.662 0.053 0.730 0.009

55.35

0.064

6.366

4 d.f.

0.174

precipitation. Analyses by prevailing types of clouds, their water content, presence of natural freezing nuclei, and so forth, might indicate which factors make seeding effective. However, in the present experiment it took a certain time until the collection of data on relevant variables was systematized; hitherto few measurements have been available for a period long enough to warrant being included in this statistical analysis. A rough idea of differential seeding effects according to amounts of natural rainfall could be obtained by classifying days according to the amounts of precipitation in the buffer zone, and testing for seeding effects separately in each class. Such an analysis is presented in table X I for all seasons together, and seems of interest in view of suggestions that seeding has different effects under conditions of heavy precipitation than under conditions of little or no natural precipitation. Positive seeding effects appear for all amounts of natural rainfall (as measured in the buffer zone) except the relatively small group of days with 2.51 to 5.00 mm of rain. Again, in all groups except that one, more than 50 per cent of the comparisons are positive. However, the test of differences between groups is not significant even at the 20 per cent level, so that these differences may well be due to chance. 5.7. Differences according to the concomitant variable. The breakdown according to the difference between rainfall in the buffer zone and the South was used above (section 3) for a more sensitive pooled test of the whole experiment. In addition to this we may consider the individual rows of table VI and see if seeding may have been differentially effective in different groups. The test for differences between groups is far from significant, so that one would tend to conclude that there is no difference in effectiveness of seeding in

109

ISRAELI EXPERIMENT TABLE

XI

ANALYSIS ACCORDING TO PRECIPITATION IN B U F F E R AVERAGE S/NS

RATIO AND PERCENTAGE OF POSITIVE

ZONE

COMPARISONS

WITH W I L C O X O N - M A N N - W H I T N E Y T E S T OF SIGNIFICANCE

" D r y " days have been excluded. Positive means that N — C rainfall difference is greater on the N seeded day than on the C seeded day.

Precipitation in Buffer Zone None Some Up to 1 mm 1.01-2.50 mm 2.51-5.00 mm 5.01-12.50 mm Over 12.50 mm

Number of Days

Average S/NS Ratio

Positive Comparisons Per Cent

Significance Levels

502 281

0.970 1.155

53.03

0.190

88 42 37 56 58

1.376 1.142 0.925 1.327 1.145

51.33 61.10 37.94 66.03 54.04

0.415 0.123 0.894 0.020 0.303

54.43

0.105

Pooled Between group variation

5.863

—

4 d.f.

0.210

different groups. However, the rows of table VI form a pattern which cannot be ignored (but to which the test of significance is not sensitive). Positive seeding effects appear in all groups except those with a small positive buffer-South difference. But this is the most frequent size of difference. If this pattern is not random, seeding is more effective when buffer-South differences are unusuallylarge or small, that is, either very large or negative. Could it be that when rainfall conditions deviate from the mode in the sense of an unusual distribution over different parts of the country, there is some atmospheric instability which furthers seeding effectiveness? 5.8. Differences according to cloud temperatures. An interesting additional breakdown would be according to cloud top temperatures which are supposed to be a crucial factor in determining silver iodide seeding effectiveness. However, cloud levels are difficult to measure precisely and vary a good deal within any one day. Hence, the present analysis is limited to temperatures at the 700 mb altitude, as recorded by a radiosonde, in the hope that this may give some indication of cloud temperature effect. The data are presented in table X I I with days grouped by single degrees centigrade. It is not easy to interpret this table. The S/NS ratios jump around pretty erratically and the percentages of positive comparisons do not vary at all significantly. And yet, the largest ratios and the most nearly significant percentages are almost all at temperatures —5° C, —6° C and perhaps also —4° C and —7° C. Could this be merely a chance pattern?

110

FIFTH BERKELEY SYMPOSIUM: GABRIEL TABLE XII ANALYSIS ACCORDING TO TEMPERATURES AT 7 0 0 m b AVERAGE S/NS

R A T I O AND PERCENTAGE OP POSITIVE

WITH W I L C O X O N - M A N N - W H I T N E Y

ALTITUDE COMPARISONS

T E S T OF SIGNIFICANCE

"Dry" days have been excluded. Positive means that N — C rainfall difference is greater on the N seeded day than on the C seeded day. Temperature (centigrade)

Number of Days

Average S/NS Ratio

Positive Comparisons Per Cent

Significance Level

up to - 1 0 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 2 and above

9 12 20 19 26 26 43 33 20 23 10 18 22

0.942 0.931 0.894 1.454 1.263 1.367 0.936 1.104 0.818 1.292 0.581 1.022 1.139

75.00 47.22 50.00 50.00 66.67 66.86 58.26 45.56 34.07 60.61 16.67 39.38 38.33

0.110 0.564 0.500 0.500 0.077 0.077 0.178 0.668 0.875 0.194 0.956 0.775 0.822

51.71%

0.316

Pooled average Between temperature variation

12.909 12 d.f.

0.376

Data for the 1965-66 season have become available after the presentation of this report. The root double ratio is 1.473 for 1965-66 and 1.184 for the entire 1961-66 period (table III). The significance level, as computed in table IV, is 0.027 for the 1961-66 period. For the interior parts of the areas (table VII) the root double ratio is 1.273 for 1961-66 (1.642 for 1965-66) and the level of significance 0.009. It is my pleasant duty to acknowledge the help and advice obtained from the Australian CSIRO weather modification team, especially in the early stages of planning this experiment. Many people have been helpful to me in writing this report, including all my colleagues in this project and other workers in this field with whom I have discussed earlier versions. I am especially indebted to Professor J. Neumann's guidance and help and also to the suggestions of Mr. A. Gagin and Mr. K. S. Rosner. I am indebted to Professor G. Morlat of the Institut de Statistique, Université de Paris, for a discussion which clarified the choice of a concomitant variable (section 3).

111

ISRAELI EXPERIMENT

Mrs. Raya Steinberg and Mr. Y. Avichai have very conscientiously carried out the calculations and helped in the preparation of this report. 0

0

0

0

0

APPENDIX On pooling tests and testing homogeneity. A comparison of a North seeded day

with a Center seeded day is counted positive if the variable on the former day exceeds that on the latter. Let n, and rrii be the sample numbers of North seeded and Center seeded days, respectively, in the ith of k groups. Let U< denote the number of positive comparisons out of the niirii comparisons in the ¿th group, so that Ui/n{nii is an unbiased estimate of the probability p, that a comparison in group i be positive. It is well known that (A.l)

Yar

=

+

Ylnimi

and that for samples which are not too small ( a

U i

2)

r

i

2

n

m

i

~ r

nam \jii + mi + 1J

has a standard normal distribution if p, = 1/2. This can be used to test p, = 1/2, that is, the hypothesis of stochastic equality of the variables on both types of day, in what is known as the Wilcoxon-Mann-Whitney test ([17], section 9.6). The present form of the test is equivalent to the rank sum form as given in many textbooks. Further, if it is assumed that pi = p2 = • • • = Pk = p, say, a weighted average estimate of p is £ l2Ut/{m (A 3)

'

whose variance is

^

=

£ 12mm,/(m i

+ mt + 1) + mi + 1)'

(A.4)

Var f = f~£ L i (fti + mi + 1)J if p = 1/2. This allows the testing of hypothesis pi = p2 = • • • = pk = 1/2 by means of the statistic (A.5)

( f - Ì ) (Var p) 1/2

which is normally distributed under the hypothesis (if the samples are not very small). This pooled test is due to van Elteren [7]. Another test of the hypothesis p\ = p i = • • • = Pk = 1/2 makes use of the variance estimate

112

FIFTH BERKELEY SYMPOSIUM: GABRIEL

(A.6)

s2 =

Z i nim,i(ni

12 M

+ ra, +

1)

- p i

12Ui i ìli + mi + 1

which, under t h e h y p o t h e s i s , h a s a chi square distribution w i t h (k — 1) degrees of freedom. T h i s statistic is equal to t h e s u m of squared d e v i a t i o n s of t h e U i / n i W i i e s t i m a t e s f r o m their m e a n p, i n v e r s e l y w e i g h t e d b y V a r (t/ 1 :/n i m l ). If either t h e p t e s t or t h e s 2 t e s t is significant, p\ = pi = • • • = Pt = 1 / 2 m u s t b e rejected. If o n l y t h e former is significant, w e m a y h a v e p\ = pi = • • • = Pk = p 1 / 2 , t h e sign of t h e statistic indicating w h e t h e r p > 1 / 2 or p < 1 / 2 . If o n l y t h e latter is significant w e m a y conclude t h a t t h e different p/s differ f r o m o n e a n o t h e r b u t on the average t h e y are near 1 / 2 . Care m u s t b e t a k e n in t h e use of t h e s e t e s t s w h e n pi = p 2 = • • • = Pk = is n o t true, b e c a u s e t h e n t h e variance is n o t as described above.

1/2

REFERENCES

[1] E. E. ADDERLEY and S. TWOMEY, "An experiment on artificial stimulation of rainfall in the Snowy Mountains of Australia," Tellus, Vol. 10 (1958), pp. 275-280. [2] J. BERNIER, "Sur le contrôle statistique des opérations de 'Pluie Provoquée'," Centre de Recherches et d'Essais de Chatou, Electricité de France, 1962. [3] V. P. BHAPKAR, "Some non-parametric median procedures," Ann. Math. Statist., Vol. 32 (1961), p p . 8 4 6 - 8 6 3 .

[4] E. G. BOWEN, "Lessons learned from long-term cloud seeding experiments," Proceedings of the International Conference on Cloud Physics, Tokyo and Sapporo, Tokyo, International Union of Geodesy and Geophysics; Meteorological Society of Japan, 1965, pp. 429-433. [5] , "The effect of persistence in cloud seeding experiments," J. Appl. Meteor., Vol. 5 (1966), pp. 156-159. [6] D. R. Cox, Planning of Experiments, New York, Wiley, 1958. [7] P. VAN ELTEREN, "On the combination of independent two sample tests of Wilcoxon," Bull. Inst. Internat. Statist., Vol. 37 (1960), pp. 351-361. [8] K. R. GABRIEL, "Statistical design of an artificial rainfall stimulation experiment in Israel," Le Plan d'Expériences (edited by D. Dugué), Paris, Centre National de la Recherche Scientifique, 1963, pp. 147-163. [9] , Statistical Considerations in Planning a Rainfall Stimulation Experiment—Second Design, Jerusalem, Hebrew University, 1963 (in Hebrew). [10] , "Preliminary results of the Israeli rainfall stimulation experiment. Statistical evaluation of the first three seasons," mimeographed report submitted to Electrical and Mechanical Services (Mekorot Ltd.), 1963. [11] , "Additional results of the Israeli rainfall stimulation experiment. Statistical evaluation of the first four seasons," mimeographed report submitted to Electrical and Mechanical Services (Mekorot Ltd.), 1964. [12] , "L'expérience de pluie provoquée en Israel. Quelques résultats partiels," Journal de Recherches Atmosphériques, Vol. 2 (1965), pp. 1-6. [13] , "Calculation of confidence bounds by means of non-parametric tests," First Israel Conference on the Uses of High Speed Computers, Rehovot, 1964 (in Hebrew). [14] K. R. GABRIEL and J. NEUMANN, "A Markov chain model for rainfall occurrence," Quart. J. Roy. Meteor. Soc., Vol. 88 (1962), pp. 90-95. [15] A. GAGIN, "Ice nuclei, their physical characteristics and possible effect on precipitation initiation," Proceedings of the International Conference on Cloud Physics, Tokyo and

ISRAELI EXPERIMENT

[16]

[17] [18] [19] [20]

113

Sapporo, Tokyo, International Union of Geodesy and Geophysics; Meteorological Society of Japan, 1965, pp. 155-162. W. L. GODSON, C. L. CROZIER, and J. D. HOLLAND, "Silver iodide cloud seeding by aircraft in Western Quebec, Canada, 1959-63," Ottawa, Canadian Department of Transport, Meteorological Branch, 1965. N. L. JOHNSON and F. C. LEONE, Statistics and Experimental Design, Vol. 1, New York, Wiley, 1964. W. H. KRUSKAL, Personal communication, 1965. M. E. LOPEZ and W. E. HOWELL, "Cloud seeding at Medellin, Columbia, during the 1962-64 dry seasons," J. Appl. Meteor., Vol. 4 (1965), pp. 54-60. P. A. P. MORAN, "The power of a cross-over test for the artificial stimulation of rain," Austral.

J. Statist.,

V o l . 1 (1959), p p . 4 7 - 5 2 .

[21] J. NEYMAN, E. L. SCOTT, and M. VASILEVSKIS, "Statistical evaluation of the Santa Barbara randomized cloud-seeding experiment," Bull. Amer. Meteor. Soc., Vol. 41 (1960), pp. 531-547. [22] , "Evaluation of seeding operations in Santa Barbara and Ventura counties in 1957, 1958, and 1959," Santa Barbara Weather Modification Project, Interim Report of the Board of Directors, Sacramento, State of California, Department of Water Resources, 1960. [23] B. V. R. MURTHY, Private communication, 1964. [24] K. S. ROSNER, "Annual reports on cloud seeding by aircraft in Israel, 1960/61, 1961/62, 1 9 6 2 / 6 3 , 1 9 6 3 / 6 4 , 1 9 6 4 / 6 5 , " m i m e o g r a p h e d (in H e b r e w ) .

[25] A. K. ROY, B. V. R. MURTHY, and R. C. SRIVASTAVA, "Cloud seeding trials at Delhi during monsoon months, July to September (1957-1959)," Indian J. Meteor. Geophys., V o l . 12 (1961), p p . 4 0 1 - 4 1 2 .

[26] R. A. SCHLEUSENER, Personal communication, 1965. [27] E. PÉREZ SILÍCEO, A. AHUMADA A., and P. A. MosiSo, "Twelve years of cloud seeding in the Necaxa Watershed, Mexico," J. Appl. Meteor., Vol. 2 (1963), pp. 311-323. [ 2 8 ] E . J . SMITH, E . E . ADDERLEY, a n d D . T . WALSH, " A c l o u d - s e e d i n g e x p e r i m e n t in t h e

Snowy Mountains, Australia," J. Appl. Meteor., Vol. 2 (1963), pp. 324-332.

A RANDOMIZED CLOUD SEEDING EXPERIMENT AT CLIMAX, COLORADO, 1960-65 LEWIS 0 . GRANT and PAUL W. MIELKE, JR. COLORADO STATE UNIVERSITY

1. Introduction Investigations of snowfall and snowfall modification were initiated in the Central Colorado Rockies by Colorado State University during the latter part of the 1959-60 winter season. These studies have continued and expanded since that time. The broad objectives are to obtain an increasingly complete understanding of mountain clouds, their precipitation processes, and changes in their behavior when artificial ice nuclei are supplied. The evaluation of the effects of seeding on precipitation, consequently, forms a basic but single phase of the program. The specific objective of the program with respect to weather modification is to determine if changes in precipitation result from the ground releases of silver iodide on the upwind mountain slopes. If the observed results appear negative, an understanding of the reasons for this is essential; and, if results are positive, it is considered important to determine how this occurred and how even more precipitation might be obtained with operational changes. This paper deals primarily with the experimental design and statistical analyses used in the detection of actual changes in precipitation. The analyses of physical factors intermittent between the actual seeding and resulting precipitation are also integral parts of this experimental design. These physical factors include the material transport mechanisms, ice nuclei concentrations, in-cloud processes, and so forth. 2. Design of experiment Atmospheric variability severely complicates studies of precipitation processes and their modification. This variability extends to all phases of the process including the general atmospheric circulation in which the cloud system forms, the thermodynamics of the cloud itself, the physical characteristics of the cloud, and to the precipitation which is highly variable in intensity, form, and amount. Consequently, statistical evaluations based on a sound experimental design are essential to this project. The basic features of the experimental design are the following. 115

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FIFTH BERKELEY SYMPOSIUM: GRANT AND MIELKE

(a) Randomization is employed in obtaining the seeded and nonseeded samples. (b) The randomization is unrestricted (see Neyman and Scott [4]). (c) The experimental time unit is 24 hours. This is a compromise that minimizes variations in the physical parameters during an event and is still long enough to lower the "noise" level to reasonable values when establishing correlations with upwind precipitation controls. (d) The observations of meteorological variables, in both seeded and nonseeded cases, are made as intensively as feasible during all stages of the precipitation process. The following is a description of the operational procedures. (a) Randomization is accomplished in a manner established by Dr. Donald Bentley, formerly of the CSU Mathematics and Statistics Department. This involves drawing 100 paired slips from a container at the start of each season. A chronological ordering of decisions is prepared. (b) The suitability of each day is determined by the Denver Branch of the U. S. Weather Bureau. The criteria of a suitable day is that Leadville, Colorado, which transmits airway data every two hours, is expected to have at least 0.01 inches of precipitation in the 24 hour period. The forecast is made some six to eight hours prior to the start of each experimental period. The Weather Bureau forecasters have no access to the seeding decisions or specifically how they will be used. (c) Two changes have been made in the start of an experimental period since the project started. The time unit, however, has remained constant at 24 hours. During the spring of 1960, the period extended from 1600 MST (D — 1) until 1600 MST the following day (D). This procedure, however, exposed snow which fell during the night, followed by daytime clearing, to day melting and evaporation. The interval was changed to 0800 MST (D - 1) to 0800 MST (D) for the 1960-61 winter season. This largely eliminated the daytime melting problem. It also coincides with the low point in a very pronounced cycle in diurnal variation in the intensity of precipitation. The experimental interval was changed slightly to the 0900-0900 interval for the 1961-62 and subsequent winter seasons to accommodate snow observers. (d) The "yes" or "no" for suitable experimental conditions from the Weather Bureau is relayed daily by teletype to Colorado State University. The relay has been through the High Altitude Observatory in Boulder, the Climax Molybdenum Co., and directly to the CSU Weather Station at different intervals during the experiment. (e) A phone relay operator at the Colorado State University Weather Station processes the forecast and takes appropriate action to initiate seeding operations if the next consecutive listing of a random decision is for an operational day. (f) Generator operators at Minturn, Red Cliff, south of Tennessee Pass, and south of Leadville turn on silver iodide ground generators at 30 minutes prior to the start of the experimental period, run them continuously and terminate

EXPERIMENT AT CLIMAX, COLORADO

117

operations 30 minutes prior to the end of the period. The procedure is the same for operators at Aspen and Reudi but a one hour lead time is used. Specific units used are changed during the interval as specified by the Weather Bureau wind forecast. A CSU modified Skyfire, needle type ground generator is used for seeding at the rate of about 20 gm of Agl per hour. These generators have been extensively calibrated in the CSU cloud chamber to establish the temperature activation characteristics of the particles produced. Figure 1 shows the output of effective ice nuclei from these generators (CSU modified Skyfire) with respect to other highly efficient Agl generators and with respect to Fletcher's theoretical curve. These units produce about 1014 particles/gm Agl effective at —12° C and 4 X 1016 particles/gm Agl effective at - 2 0 ° C. (g) Snowboard observers hired locally—Leadville, Minturn, Frisco, and Breckenridge—read daily some 70 snowboards located at about one mile intervals over Fremont, Hoosier, and Vail Passes. Climax is located near the summit of Fremont Pass. The summit area of Fremont Pass is considered the primary target for the seeding. The Hoosier Pass network forms a backup for the Fremont Pass network further downwind from the seeding sites. Four observations are made at each site: (1) snow depth; (2) the weight of a core of snow taken at an average depth location as established from the depth measurements; (3) a collected water sample for future analysis; and (4) observer's description of drifting or melting conditions, if any. The "water depth" of new snow is calculated from the weight of the sample. The snowboards over each Pass are read in a systematic order each day. The observers have no information as to seeding decisions, and in general, know very little about the project. A number of changes in observers have been necessary due to the part time nature of the job, moves, sickness and so on. Mr. Robert Rinker, a local resident in the area and an employee, directly supervises the local observation program. (h) As many other physical observations as possible are made on seeded and nonseeded days. Ice nuclei observations, two groups of at least three readings, have been made on almost all days. A detailed weather observation is taken at the time of each ice nuclei observation. Various other physical observations have been possible on at least a portion of the experimental days. These include ice crystal replications, cloud photography, radar photography, and atmospheric electricity, and have been made with increasing intensity as the program progresses, but still on a minority of the experimental days. A second ice nuclei counter was installed at a similar but completely unseeded upwind site in the fall of 1962. Data from this site were intermittent during the first year but have been continuous since the fall of 1963. Ice nuclei counters at these two sites have been compared with each other and a similar unit at NCAR in Boulder. These units have also been interchanged. (i) A program of systematic in-cloud sampling on seeded and nonseeded days is just getting underway.

FIFTH BERKELEY SYMPOSIUM: GRANT AND MIELKE

TEMPERATURE (°C) FIGURE 1

Seeding output, as a function of temperature, for generator used in the Climax experiment (effective nuclei/gm Agi).

EXPERIMENT AT CLIMAX, COLORADO

119

3. Statistical evaluation 3.1. General -procedures. The evaluation procedures were established at the outset of this project after numerous discussions with members of the statistics staff at Colorado State University. During the discussions, the following specific decisions were reached. (a) The basic evaluation procedure would be the general procedure developed by Thom [6] utilizing normalized target and control relationship established for the control and for seeded cases. (b) Terrain upwind of the target is unsuitable for the establishment of special control stations; therefore, standard Weather Bureau Stations southwest, west, and northwest of the target were chosen for control stations. (c) Other evaluation procedures would also be used to provide maximum understanding of the processes involved. The following criteria were considered to be important in formulating specific analyses: storm type; upper air temperature and lapse rate; direction and velocity of air flow; concentrations of ice nuclei; variation among experimental years; variation among various calendar months; time of day; use of various generator combinations; variation of nonseeded cases with historical data. Statistical investigation to establish the control stations was started in the first year of the project and completed during the third year. The historical data from the Weather Bureau gage at Climax were used in determining control stations. These investigations were for the sole purpose of establishing stations that correlated well with the target region. 3.2. Statistical procedures. In this study, two different approaches are used to evaluate possible differences in precipitation amounts during seeded and nonseeded periods. The first approach is based on a parametric technique introduced by Thom [6]. The second approach involves an application of distribution free procedures. If differences in precipitation amounts are assumed to be scale changes, then either approach will yield point estimates of these scale changes. Both of these approaches will be outlined in this section. 3.2.1. Parametric approach. Since a complete description of the technique devised by Thom [6] is available in the literature, only a few highlights of this procedure will be given here. Under the assumption that precipitation data may be approximated by a gamma distribution, this technique begins by transforming the raw data into normalized data which is suitable for the application of a simple regression analysis. Since the basic information for both seeded and nonseeded periods consists of paired observations (that is, target and control), the only acceptable information is the set of all paired observations in which neither the target observation nor the control observation is zero. With this in mind, the remainder of the technique is quite straightforward. If the expectation of the resulting normalized test statistic is taken in terms of the assumed underlying gamma distributed

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FIFTH BERKELEY SYMPOSIUM: GRANT AND MIELKE

variables, then a point estimate of a scale change during the seeded period with respect to the nonseeded period is easily obtained. 3.2.2. Distribution free approach. The two techniques discussed in this section employ simple ranking procedures. I t is the desire of the authors that modifications within the framework of these distribution free techniques will provide satisfactory analyses for a variety of investigations. Both techniques utilize the same basic information for seeded and nonseeded periods. This information consists of all pairs of target and control observations. Also, these techniques avoid elimination of data resulting from transformations used in certain parametric techniques when either target or control observation of a pair is zero. Let [(Ti, Ci), • • • , (Tn, C„)] represent n pairs of target (T) and control (C) observations for the nonseeded period. Similarly, let [(T(, Ci), • • • , (T'm, C'm)] represent m pairs of target and control observations for the seeded period. Also, let TV = m + n be the total number of pairs of target and control observations in the pooled seeded and nonseeded periods. The design of this study requires that the number of observations of the seeded and nonseeded periods are approximately the same. A description of each technique (that is, Technique A and Technique B) follows. Technique A. The underlying assumption in employing this technique, which is associated with the Wilcoxon method, is that if seeding has no effect on the amount of precipitation, then Ti — Ci and Tj — C) represent observations from identical distributions. The first step is to rank the N pooled values (1) T\ — Ci, • • • , Tn Cn, T'i C'i, T'm Cm from smallest to largest according to their values on the real line. When dealing with the type of meteorological data under consideration here, many tied observations will occur if the total number of observations is reasonably large. This leads to a partitioning of the ranked N pooled values into r disjoint classes of tied observations. These r classes are, in turn, ranked in the same order as their respective tied observations were ordered relative to the initial ranking of the N pooled observations. Let i* be the number of tied observations in the Mh smallest class according to its rank, that is, —

—

• • •

,

—

(2)

t h = N. k= 1 Also, let Sk denote the partial sums of the number of tied observations, (3)

Sh = Z k, k= 1

h = 1, 2, • • • , r,

and So = 0, Sr = N. Then define the mean rank of the /cth class of tied observations by (4)

Rk = f [ ( S w + 1) + tk

+ 2) + • • • + &].

If j is used as an index over the observations in each class of ties, then the

E X P E R I M E N T AT CLIMAX,

COLORADO

121

expression W, corresponding to the Wilcoxon statistic in the case of ties, is defined by (5)

W = t £ Rkb]k k=\j=1 where Sjt = 1 if the j t h observation in the fcth class of ties was a seeded observation, 0 otherwise. The mean of W is given by (6)

vw = ±m(N + 1).

The variance of W (Hemelrijk [1]) is given by

*

=

If tk = 1 for k = 1, • • • , r — N (that is, there are no ties), then the variance of W reduces simply to (8)

aw = ^ mn{N + 1).

For a reasonably large number of observations, as in the present study, in the case seeding has no effect on precipitation, the statistic (9)

U =

aw

is distributed approximately as the normal distribution with mean zero and variance one. A point estimate of the scale change using the present technique is obtained in the following simple manner. Essentially, this estimation procedure is accomplished by replacing the seeded observations (T[ — C[, • • • , T'm — C'm) in the N pooled values by specifically adjusted nonseeded observations (ATi — Ci, • • • , AT m — Cm). This adjustment of the nonseeded observations consists of two steps. The first step, if necessary, is either to add or subtract observations from the n nonseeded observations in order to match the sample size m of the seeded observations. This is done in such a manner that alteration of the empirical nonseeded distribution will be minimized. To accomplish this, consider the n nonseeded observations (7\ — C\, • • • , Tn — Cn) ranked from smallest to largest according to their values on the real line. If n = m, this first step is unnecessary. If n > m, the n — m ranked nonseeded observations which most closely correspond to the n — m cumulative distribution values i/(n — m + 1) for i — 1, • • • , n — m, of the empirical nonseeded distribution are eliminated (for example, if n — m = 1, then the median value of the empirical nonseeded distribution is eliminated). If, however, n < m, then the m — n ranked nonseeded observations which most closely correspond to the m — n cumulative distribution values i/(m — n + 1) for i = 1, • • • , m — n, of the empirical nonseeded distribution are added to the n existing nonseeded obser-

122

FIFTH BERKELEY SYMPOSIUM: GRANT AND MIELKE

vations (for example, if m — n = 1, then the median value of the empirical nonseeded distribution is added to the existing n nonseeded observations). The second step is to modify the size adjusted nonseeded observations with appropriate target scale changes (that is, different values of A) yielding adjusted nonseeded observations of the form (ATi — Ci, • • • , AT m — Cm). Next, for each replacement of (Ti - C'u • • • , T'm - C'm) by (ATi — Ci, • • • , ATm — Cm), the associated values of Wa and f/A are determined for selected values of A (for example, A = 0.80, 0.85, • • • , 1.95, 2.00). Then by using interpolation, the value of Ua which corresponds to the observed value of U obtained from the m seeded observations yields the estimate of the scale change A. It should be noted that this scale change estimate is based strictly on the empirical nonseeded data. Also, for a reasonably large sample, approximate A alternative power curves may be obtained for an arbitrary set of significance levels. Technique B. Again, the underlying assumption when applying this technique, which follows Mood, is that if seeding has no effect on the amount of precipitation, then Ti — Ci and T'} — Cj represent observations from identical distributions. However, the first step now is to rank the N pooled values Ti — Ci, • • • , T„ — Cn, T[ — C'i, • • • , T'm — C'm from smallest to largest according to their absolute values. Again, the encounter with ties results in the partitioning of the ranked N pooled values into r disjoint classes of tied observations. Also these r classes are again ranked in the same order as their respective tied observations were ordered relative to the N pooled observations. Again, let tk be the number of tied observations in the Mh smallest class according to its rank, Sh = U = i tk, Sr = N and S0 = 0. This time, define the mean value assigned to the Mh class of tied observations to be (10)

Qk = jk [(&_! + l) 2 + (St-t + 2)' + • • • + Sll

If j is used once again as an index over the observations in each class of ties, then the expression X, which is motivated by a distribution free test given by Mood [3], is defined by (11)

x = ± £ QkSjk k=l¿=1

where S,-* was previously defined in Technique A above. The mean and variance of X are given respectively by (12)

nx = l m ( N + 1)(2AT+ 1)

and

+ 3 (n* - ± tl) - 12\ ± iSl-i \ k=X / Lfc = l

+ £ tl(tk + l)(Sfc_i~|)4=1 JJ

EXPERIMENT AT CLIMAX, COLORADO

123

If tk = 1 for k = 1, • • • ,r = N (that is, there are no ties), then the variance of X simplifies to (14)

= ^

mn(N + l)(2tf + 1)(82V + 11).

Again, in the case where seeding has no effect on the precipitation and the number of observations is quite large, the statistic (15)

V =

ox

is distributed approximately as the normal distribution with mean zero and variance one. Finally, a point estimator of the scale change based strictly on the empirical nonseeded distribution is obtained in a manner analogous to that described in Technique A above. Also, for a reasonably large sample, approximate alternative power curves may be determined for an arbitrary set of significance levels. Preliminary numerical results of empirical precipitation data show that a power comparison of these techniques depends specifically on the relationship between target and control.

4. Summary of preliminary analysis of data 4.1. Ice nuclei observations. Large concentrations of seeding material are arriving in the target area on seeded days. Figure 2 shows the type of increases that are being observed consistently in seeding sequences. This sample was selected since ice nuclei observations were made intermittently throughout the 24 hour period. April 14th and 15th show background concentrations observed prior to the start of the seeded sequence. April 16th through 24th are omitted since they involve a mixing of seeded and nonseeded cases with only the two ice nuclei observations sequences each day. As can be seen, concentrations were low on the 25th but increased by a factor of ten shortly after seeding started. It can be noted that peak values observed during the day were nearly one hundred times greater than on previous or following days. Considerable variations occurred during the period but the values were consistently higher than during adjacent unseeded periods. It can also be noted that values lowered during the night. Seeding operations terminated at 0830 on the 27th. Concentrations were considerably lower on the 27th and at least near original background levels. Figure 3 shows the percentage of low concentrations of ice nuclei observed at Climax during the respective operation seasons through 1964. The following are considered to be interesting aspects of these data. (a) Concentrations of ice nuclei were, in general, less than 1 /liter at —20° C at Climax before the experiment started and also at the upwind station near Steamboat Springs during 1963-64. The pre-experiment and the upwind Steam-

se

E CL