Methods in Climatology: Second Edition, Revised and Enlarged, including Some Methods in General Geophysics [2nd ed., rev. and enl. 2nd printing 1962. Reprint 2014] 9780674187856, 9780674187153

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METHODS IN

CLIMATOLOGY

METHODS IN CLIMATOLOGY Second Edition, Revised and

Enlarged

INCLUDING

Some Methods in General

Geophysics

V. CONRAD HARVARD UNIVERSITY

and L. W. P O L L A K DUBLIN INSTITUTE FOR ADVANCED STUDIES

HARVARD UNIVERSITY

Cambridge, Massachusetts 19 6 2

PRESS

COPYRIGHT 1944, 1950 BY T H E P R E S I D E N T AND F E L L O W S OF HARVARD COLLEGE

Second

Distributed

Printing

in Great Britain

OXFORD UNIVERSITY

by

PRESS

LONDON

P R I N T E D IN T H E U N I T E D STATES OF AMERICA

Preface The difficulty, in principle, with which a book on methods has to contend is the selection of the methods to be presented. Methods may be invented for and applied to the solution of an individual problem, or of a group of problems. Methods of the first class are here left out of account, since they are usually so intimately connected with the special problem that they are not of general interest, and because the multifariousness of problems with their peculiar methods is unlimited. Thus, of the many constructive criticisms and comments contained in reviews of the first edition of this book only those could be considered that pertain to methods of the second class—those applicable to groups of problems. In accordance with one of the suggestions, the use of the punched-card system and of other mechanical aids to computation in climatology, and generally in geophysics as well, is now discussed, although in a condensed way. Complying with another broad demand of a reviewer, a periodography containing the fundamentals of a general theory of cycles was incorporated in this edition. The original plan of the book was to serve students in climatology, and also in geography, geology, botany, agriculture, and kindred sciences with which climatology plays an important role, as a convenient guide for dealing "with the original data of observation." It is hoped that this aim is maintained also in the second edition by a new arrangement which permits the reader to find his way easily through the book according to his problems and knowledge. Part I corresponds essentially to the entire book in its first edition. It is concerned with methods that are useful primarily in representing climatological phenomena and solving climatological problems. Naturally, these methods and concepts can partly be transferred to other geophysical problems of different nature, either directly or in a modified form. This first part has been fully revised, and enlarged by the addition of a number of new items. The most important insertions are V

vi

PREFACE

as follows. A section on "Computation of the arithmetic mean" has been added to Chapter II. The new Chapter III, "Aids to computation," gives various useful methods of representing variables, as well as the method of nomograms. Chapter IV, "Curve fitting," is much expanded, especially by the addition of the sections on "The Gaussian curve" and "Arithmetic probability paper." Chapter V now includes discussions of "Lamont's correction" and "The harmonic dial." The sections on "Cumulated temperature" and "Degree days" in Chapter VI are expanded. In Sec. 7.2 the anthropoclimatic drying power and climatic classifications according to drying and cooling power are inserted. The "Frequency distribution of daily amounts of precipitation" is an important element, though rather neglected, and is of special interest for the physics of the atmosphere ; the method of representing and calculating this distribution is given in Sec. 7.4.7. "Cumulative summation" and "Hythergraphs," the last sections of Chapter VII, are new. The same is true of Sec. 9.2, "The construction of a map of isolines. The gradient," which should at least demonstrate that a good deal of thought—both geophysical and geometric—is necessary to make reliable maps. Section 9.6, "Hyetographic curve," is a new addition and Sec. 9.8 is greatly expanded, especially the paragraphs on "Boundaries between desert, steppe, and tree climates" and "Precipitation effectiveness index." The third section of Chapter X, "Dynamic climatology," was rather thoroughly changed to lay greater stress on this fundamental portion of modern climatology. Part II is entirely new. Chapter XI describes mechanical and electrical devices for handling large numbers of observations, without regard, of course, to the sources of the material, whether climatological, geophysical, biological, or any other. In this chapter descriptions of punched-card machines are included, and their special advantages for climatology and meteorology are demonstrated. In connection with the discussion of extensive numerical materials, the remainder of Part II deals with "Cycles," treated in a form adapted to general geophysical problems, including, therefore, climatological and meteorological ones. Theory and methods are given for revealing hidden periodicities and for testing the

PREFACE

vii

reality of the results obtained. The last chapter presents Fuhrich's autocorrelation method, which seems to be superior to earlier attempts. Grateful thanks of the authors are due to Professor Charles F. Brooks, of Harvard University, for his encouragement and interest in this second edition also. We are indebted to the Milton Fund of Harvard University for a grant to facilitate typing and other necessary routine work. Warmest thanks are extended to Mr. J . D. Elder, Science Editor of the Harvard University Press, for his indefatigable interest in the manuscript of this book and for all his efforts to bring out the book in its best form. V.C. L.W.P. Cambridge, Massachusetts Dublin, Ireland September 1949

Preface to the First

Edition

TO IDA CONRAD, M Y COMRADE I N L I F E AND W O R K , T H I S BOOK IS DEDICATED

Climate influences the surface of the earth, and this conversely, in its varieties, determines the climate under otherwise equal conditions. This intimate mutual connection makes climatology and climatography appear as parts of geography, because they are essentially necessary to describe the surface of the earth and its changes. These ideas find their expression in the fact that generally in colleges and universities, climatology as a whole is treated in the geographical departments. Perhaps the dependent role of climatology may be attributable also to the fact that geographers have so greatly furthered this science. For the most part, climatographies have been written by geographers. Therefore, geographical methods are kept in the forefront, and specifically climatological methods are not so much used. It would be satisfying if this book offered a bridge connecting the two realms. To judge from the author's many years of experience in Europe, followed by a few years in the United States, the student in geography has perhaps sufficient knowledge of climatology, but he does not know how to deal with the original data of observations. He is, I should say, familiar with the results but not with the ways of getting them. This holds, particularly, for the different methods of mathematical statistics which can be adapted to an individual representation of the climates; only thus can too schematic descriptions be avoided. This book should make the student acquainted with a number of methods of mathematical statistics and theory of probability applicable at once to climatological problems. Approximations are used as much as possible in order to facilitate arithmetic. The student who is interested in the mathematical proofs and derivations as well as in their philosophical background must be referred to the special technical literature. ¡X

χ

PREFACE

The general introduction presents climatology as a world science, and its international organization. T h e number of observations in the meteorological register makes the necessity of statistical methods evident. Their special application to climatological series is discussed. An instruction, easy to understand, is given for computing periodic phenomena by means of harmonic analysis. In the foregoing, the overwhelming role of the frequency distribution has been emphasized. This idea led L. W. Pollak to introduce a new system of treating climatological observations by means of automatic statistical machines. This system covers the entire body of the observational d a t a and may represent the future of rational climatological statistics. Pollak's method is not discussed here, however, because only with the resources of great institutions is it practicable. The general representation of climatological elements, factors, etc., is followed by discussions of the special characteristics of the single elements. The radiation elements are not dealt with. As far as frequencies and different correlations with other elements are concerned, the methods are identical with those applied to other climatological elements, on the one hand. On the other, such a discussion would have had to include the physical and astronomical nature of the subject. This exceeds the range of this book. The first two parts of the present book are concerned with the variations of the elements in the course of time a t one fixed place. The third part presents the comparison of the elements which are observed synchronously at different places, and arrives a t their geographical distribution. Critical scrutiny of the observational d a t a leads to the examination of the homogeneity of climatological series and t o the reduction to an identical period. These problems remind me of H a n n ' s quotation of Francis Bacon, the great English philosopher: Si quis hujusmodi rebus ut nimium exilibus et minutis vacare nolit, imperium in naturarti nec obtinere nec regere poterit.1 '"A scholar who is unwilling to take pains over such investigations because they seem too insignificant and microscopic will be able neither to gain nor to maintain mastery over nature."

xi

PREFACE

These efforts are of fundamental importance not only for the climatologist but also for everybody who is interested in climatological series, as are the long-range forecaster and those dealing with periodicities and tree-growth analysis. Comparisons of data at different places reveal the concept of coherent and noncoherent climatic regions and of the climatic divide. These ideas show also clearly that the theory of correlation in the realm of climatology should play a significant role. An intermediate chapter deals with this subject. Climatological examples demonstrate the computations so that everyone with high-school training should be able to understand the mathematical procedure. Linear correlation, simplifications of the calculation, and regression equations are discussed. Finally, the formulas of partial correlation of three variants are added. Rather much space is devoted to graphical methods of representation, especially in connection with mountainous regions. The great importance of anomalies and isanomals is shown and supported by examples. At the end of this chapter, air-mass climatology, continentality, etc., are mentioned. The fourth section gives suggestions for the arrangement of a more or less complete climatography. In the appendix, models for climatic tables are presented, a table for easy calculation of the probable error, an auxiliary table for computing the equivalent temperature, and a table giving the numbers of the consecutive days of the year. The content and order of treatment in this book greater part, the experience the author acquired in climatology and in supervising the theses of his University of Vienna, Austria. A number of the taken from the author's papers.

reflect, for the his lectures in pupils at the examples are

V. c. Cambridge, Mass. October 1, 19 M

Contents

P A R T I.

M E T H O D S IN

CLIMATOLOGY

INTRODUCTION I. G E N E R A L

1

REMARKS

3

1 . 1 . CLIMATIC ELEMENTS. CLIMATOLOGICAL ORGANIZATION

. . .

3

1.1.1. Climatic elements 1.1.2. Climatological organization 1.1.8. The meteorological register 1 . 2 . SOME REMARKS UPON OBSERVATION OF CLIMATIC ELEMENTS

3 5 8 .

1.2.1. Discontinuities in climatological series 1.2.2. Observation of air temperature. Errors caused by radiation and by stratification of air close to the ground 1.2.8. Measurement of precipitation II. S T A T I S T I C A L A N A L Y S I S O F C L I M A T I C E L E M E N T S

.

2 . 1 . G E N E R A L STATISTICAL CHARACTERISTICS

2.1.1. 2.1.2. 2.1.3. 2.1.4.

System of characteristics Primitive characteristics Variate. Absolute extremes. Range of variation Class intervals. Frequency distribution. Absolute and relative frequency 2.1.5. Histogram and frequency curve. Mode. Modal class 2.1.6. Elementary characteristics. Median, quartiles, deciles 2 . 2 . ARITHMETIC MEAN

12

12 13 14 17 17

17 17 17 19 21 24 28

2.2.1. 2.2.2. 2.2.3. 2.2.4·

Deviations. Average variability Computation of the arithmetic mean Standard deviation. Variance. Coefficient of variation Cornu's theorem. Unilaterally and bilaterally limited elements 2.2.5. A practical application of standard deviation in climatological and kindred series 2.2.6. Higher characteristics. Skewness

28 30 36 41 46 48

2 . 3 . SPECIAL VARIABILITIES. M E T H O D OF RANDOM SAMPLES. A P P L I CATIONS TO CLIMATOLOGY AND METEOROLOGY

2.3.1. Different kinds of variability xüi

50

50

xìv

CONTENTS 2.3.2. Absolute and relative variability and, other measures of variation 2.3.3. Method of random samples applied to precipitation observations 2.3.4· Method of random samples applied to cloudiness and duration of sunshine

III. A I D S T O

COMPUTATION

54 56 58 61

3 . 1 . G R A P H S AND MODELS

61

3.1.1. Graphical representation of two variables 3.1.2. Elimination of an independent variable 3.1.3. Representation of three or more variables. The model

61 64 65

3 . 2 . NOMOGRAMS AND COMPUTATION CHARTS

72

3.2.1. Charts of functions of one variable 3.2.2. Cartesian computation charts 3.2.3. Method of alignment

73 76 77

3.3. TABLES IV. S O M E

80

PROBLEMS

SMOOTHING

OF CURVE

OF NUMERICAL

FITTING,

AND

SERIES

84

4 . 1 . T H E STRAIGHT L I N E

84

4 . 2 . T H E PARABOLA

88

4-2.1. The level of the turning point of the curve 4-2.2. The standard distribution

91 91

4 . 3 . T H E STRAIGHTENING OF CURVES 4 . 4 . T H E CURVE OF GROWTH : Y = 4 . 5 . T H E CURVE OF DECAY: Y = 4 . 6 . T H E CURVE Y =

AEHX OR Y =

92 A 1061

94

AX~B

97

AXB

101

4.7. T H E CURVES y — q = abx~p AND y — q = a(x — p)m

. . .

4 . 8 . T H E G A U S S I A N CURVE

105

4-8.1. Arithmetic probability paper

111

4 . 9 . T H E SMOOTHING OF NUMERICAL S E R I E S V. H A R M O N I C

114

ANALYSIS

119

5 . 1 . T H E ANALYSIS

119

5 . 2 . T H E SYNTHESIS (EVALUATION OF T H E S E R I E S ) 5 . 3 . T H E SIMPLIFIED STRAIGHTFORWARD METHOD OF

103

127 EVALUATING

A F O U R I E R SERIES

129

5 . 4 . C H E C K I N G OF HARMONIC ANALYSIS. H I G H E R TERMS

130

5 . 5 . R E L A T I V E AMPLITUDES

132

5 . 6 . T I M E S OF T H E E X T R E M E S

132

5 . 7 . LAMONT'S CORRECTION

133

XV

CONTENTS 5.8. SUMMARY OF SECTION 5.7. DEFINITIONS AND FORMULAS FOR THE COMPUTATION OF NONCYCLIC CHANGE

139

5.9. T H E HARMONIC DIAL

142

5.9.1. The diurnal variation of temperature for different months of the year 6.9.2. Discussion of the harmonic dial 5.9.8. Annual course of duration of sunshine at different levels in the Eastern Alps VI.

CHARACTERISTICS OF SOME SELECTED E L E M E N T S (I) 6.1. TEMPERATURE

155 155

6.2. ATMOSPHERIC PRESSURE AND WATER-VAPOR PRESSURE

. . .

7.2.1. 7.2.2. 7.2.8. 7.2.4·

Relative humidity Equivalent Drying power Cooling power

155 164 167 168 168 170 175 176

CLIMATIC

7.1.1. Resultant wind velocity calculated from records of direction and velocity 7.1.2. Resultant wind direction calculated from frequencies of the direction alone 7.1.8. The resultant run of the wind 7.1.4· The steadiness of the wind 7.2. SOME COMBINED ELEMENTS

151

CLIMATIC

6.1.1. Hours of observation. The "true mean" 6.1.2. Dates at which the average temperature crosses certain thresholds 6.1.3. Duration of temperatures above and below certain thresholds 6.1-4· Other characteristics of temperature conditions. Spells of cold and hot days 6.1.5. Cumulated temperatures 6.1.6. Degree days 6.1.7. Relative temperatures

VII. C H A R A C T E R I S T I C S O F S O M E S E L E C T E D E L E M E N T S (II) 7.1. W I N D

145 149

178 178

178 181 182 184 186

temperature

7.3. CLOUDINESS

7.8.1. Clear and cloudy days 7.8.2. Fog 7.4. PRECIPITATION

7.4-1· Probability of a day with precipitation

186 187 191 194 198

198 200 201

201

xvi

CONTENTS 7.4.2. 7.4.3. 7.4.4. 7.4-5.

Probability that a day with precipitation is a day with snowfall Amounts of precipitation within certain periods The annual course of the amounts of precipitation Absolute and relative variability. Quotient of variation. Method of random samples applied to records of rainfall 7.4-6. Rain intensity 7.4.7. Frequency distribution of daily amounts of precipitation 7.4.8. Wet and dry spells 7.4-9. Characterization of the hydrometeoric type of a period 7.4.10. Snow 7.4.11. Index of aridity 7-4-12. Rain histogram 7-4-1S. Cumulative summation 7.4.I4. Hythergraphs

VIII. S P A T I A L C O M P A R I S O N O F C L I M A T I C E L E M E N T S 8.1.

UNIFORM GIONS.

8.2.

TENDENCIES

RELATIVE

CRITERIA

OF AVERAGE

WEATHER

OVER LARGE

(I) 2 2 2 222

HOMOGENEITY

226

8.2.1. Helmert's criterion 8.2.2. Abbe's criterion 8.3.

REDUCTION

OF

CLIMATOLOGICAL

227 227 AVERAGES

TO

A

CERTAIN

PERIOD

8.8.1. 8.3.2. 8.8.3. 8.8.4. 8.3.5.

Method of differences The reduction of precipitation series. Method of ratios Physical explanation of the method of ratios The reduction of other elements to a given period Limits of the method of reducing climatological series to a normal period 8.3.6. The length of a normal period 8.3.7. Interpolation of missing observations 8.3.8. Coherent and incoherent climatic regions and their statistical evidence

8.4.

CORRELATION

8.4.1. 8.4.2. 8.4.3. 8.4-48.4.5.

205 205 205 212 215 215 217 218 218 220

RE-

HOMOGENEITY

OF RELATIVE

202 202 203

Linear correlation Example of calculating a correlation Simplifications in computing correlations Regression equation Partial correlation

232

232 235 236 237 238 239 241 241 242

242 245 247 249 254

xvii

CONTENTS IX. S P A T I A L C O M P A R I S O N O F C L I M A T I C

ELEMENTS

(II)

9 . 1 . T H E ISOGRAM

258 258

9 . 2 . T H E CONSTRUCTION OF A MAP OF ISOLINES. T H E GRADIENT

.

.

261

9.3.1. Temperature distributions in mountainous regions 9.3.2. Examination of lapse rate 9.3.3. Mapping temperature conditions of a mountainous country 0reduction to a given level) 9.3.4. Actual temperatures 9.3.5. Standard curves. Anomalies. Isanomals

272 272

9 . 3 . METHODS OF ANOMALIES

272

9 . 4 . APPLICATIONS OF THE METHOD OF ANOMALIES AND ISANOMALS

9.4.}. 9.4-2. 9.4.3. 9.4-4· 9.4-5. 9-4-6. 9.4.7.

Vegetative period and altitude Temperature and geographic latitude Pleions, meions Snow cover and altitude Duration of sunshine versus elevation Anomalies of precipitation Isanomals of variability of precipitation

273 274 275 276

276 278 280 280 282 282 285

9 . 5 . PRECIPITATION PROFILES

285

9 . 6 . HYETOGRAPHIC CURVE. PERCENTILE AREAL DISTRIBUTION OF RAINFALL ON THE HEMISPHERES

287

9.7. WIND

291

9.7.1. 9.7.2. 9.7.3. 9.7.4.

291 293 294 295

Wind roses Wind roses for different elements Maps of streamlines Streamlines based only upon frequencies of direction

9 . 8 . NUMERICAL CHARACTERIZATION OF DIFFERENT CLIMATIC FEATURES

296

9.8.1. Continentality 9.8.2. Boundaries between desert, steppe, and tree climates 9.8.3. Precipitation effectiveness index X.

ESSENTIAL ELEMENTS

OF T H E

CLIMATOGRAPHY

1 0 . 1 . ARRANGEMENT OF THE CLIMATOGRAPHY

10.1.1. The introduction 1 0 . 2 . STATIC CLIMATOLOGY

10.2.1. 10.2.2. 10.2.3. 10.2.4.

296 300 304 .

307 307

308 309

Climatic tables 309 Records of self-registering instruments 310 Variation of temperature and precipitation with elevation 311 Graphical representation 312

1 0 . 3 . DYNAMIC CLIMATOLOGY

10.3.1. General remarks

316

316

xviii

CONTENTS 10.3.2. Models of dynamic clima, tographies 10.8.3. Statistical air-mass climatology IO.34. Historical note

10.4.

THE

10.5.

DESCRIPTION

BIOCLIMATOLOGICAL

PART

11.1.

SLIDE

PART

OF CLIMATIC

328

PHENOMENA

328

II. COMPUTING PERIODOGRAPHY

X I . COMPUTATION TRICAL

317 320 323

WITH

MECHANICAL

DEVICES. AND

ELEC-

DEVICES

332

RULES

332

11.1.1. Slide rules for reduction of pressure 11.1.2. Slide rules for hygrométrie calculations 11.2.

CALCULATING

AND COMPUTING

11.2.1. Accounting 11.3.

ELECTRIC

11.4.

PUNCHED-CARD

333 334

MACHINES

335

machines

CALCULATING

336 MACHINES

MACHINES

337

AND METHODS

339

11.4-1· Introduction 11.4-2. Historical note 11.4.3. Punched-card methods in climatology 11.4-4· Punching methods and forms of cards 11.4-5. Verification of punched cards 11.4.6. The sorting machine 11.4.7. The tabulator 11.4-8. Auxiliary equipment 11.4-9. The mark-sensing system II.4.IO. Methodolpgical hints 11.4-11. The punch card 11-4-12. Coding and punching 11.4-13. Organization II.4.I4· Uses of punched-card systems in meteorology and climatology

X I I . PERIODOGRAPHY. 12.1.

INTRODUCTION

12.2.

PERIOD

KNOWN:

HIDDEN

PERIODICITIES

. . . .

339 340 342 343 343 345 345 346 346 346 349 350 350 351

354 354

BUYS-BALLOT

SCHEDULE

12.2.1. Computation of amplitude and phase angle without folding 12.2.2. Analysis interval not an integral multiple of period; nonharmonic wave

355

358 360

CONTENTS

xix

12.2.3. Period not an integral multiple of the time interval between two successive observations. Darwin's method 12.2.4. Influence of other periods on the columnar means of the average row in the Buys-Ballot schedule 12.3.

LENGTH

AND NUMBER

OF PERIODS UNKNOWN

MUTUALLY

TEST

OF

DEPENDENT

371 373 375 377

SECONDARY 381 .

381

13.1.1. Fourier series for random values, distribution of the amplitudes, expectancy 13.1.2. The K-test of significance 13.1.3. Example of the application of the theory of random series 13.1.4. Analysis of frequencies

383 389 392 394

13.2.

TEST

370

SIGNIFICANCE.

VARIABLES.

ANALYSIS 13.1.

368 369

12.3.1. Schuster's periodogram 12.3.2. Trial periods. Rearrangement of the data into a Buys-Ballot schedule 12.3.3. Practical method of rearrangement. Harmonic analysis of the average row 12.3.4. Periodogram for one simple period. Width of maximum 12.3.5. Periodogram for two simple periods (sine waves), κ-curve, selection and density of trial periods X I I I . PERIODOGRAPHY.

364

OF SIGNIFICANCE

INVESTIGATION

(REALITY)

OF MUTUALLY

OF A SUPPOSED

DEPENDENT

PERIOD

.

VALUES

396

13.2.1. Schuster's first method 13.2.2. Schuster's second method 13.2.3. Schuster's quantitative test of significance of any {persistent) trial period 13.2.4. Example 13.2.5. Adjustment of expectancy for quasi persistence. Effective expectancy 13.2.6. Summary 13.2.7. Application to Set Ζ 13.3.

SECONDARY PERIOD.

13.4.

ANALYSIS.

PHASE

RESULTS

DETERMINATION

OF

WITH

DEFICIENCIES

AND CRITICISM OF

PERIODOSCHUSTER'S

14.1.

411

PERIODOGRAPHY. WITH

THE

FUHRICH'S

403 406 407 410

THE AID OF SCHUSTER'S

METHOD XIV.

399 401

TRUE

DIAGRAM

OBTAINED

GRAM ANALYSIS.

ACCURATE

396 398

AID

DETERMINATION OF

METHOD

AUTOCORRELATION

OF

PERIODS 413 413

XX

CONTENTS

1 4 . 2 . FUHRICH'S CRITERION FOR REALITY AND THE CHARACTERISTICS OF HIS METHOD

421

14-2.1.

CRITERION FOR REALITY

14.2.2.

14.2.3. 14-2.4· 14-2.5. 14-2.6. I4.2.7. 14-2.8.

HYPOTHESES OF APPLICATION

422

421

Independence of the length of time subjected to analysis 422 Length of the interval analyzed 423 Number of transformations required 423 Conclusions from the intermediate results 423 Sifting the periodic components according to their importance 423 Equal amplitudes of the periodic components 423

14.2.9.

DAMPED OSCILLATIONS

424

14.2.10.

APERIODIC TERM (SECULAR COMPONENT, TREND)

14-2.11.

REPRESENTATION OF ARTIFICIAL FUNCTIONS

14-2.12.

INFLUENCE OF DISTURBANCES

424 424 424

14.2.13.

ARTIFICES AND MECHANICAL METHODS

14.2.14.

APPLICATIONS

425 426

1 4 . 3 . EXAMPLE OF FUHRICH'S METHOD

426

1 4 . 4 . APPLICATION OF THE AUTOCORRELATION METHOD TO PERIODIC FUNCTIONS DISTURBED BY ERRATIC SHOCKS

429

1 4 . 5 . MOST RECENT DEVELOPMENTS. T H E SPECIFICATION OF DISTURBED PERIODIC TIME SERIES

433

1 4 . 6 . COMPUTATION OF CORRELOGRAMS FROM HARMONIC CONSTANTS APPENDIX I . FORMS OF CLIMATIC TABLES

435 439

APPENDIX I I . VALUES OF Y / X / N FOR SELECTED INTEGRAL VALUES OF Η FROM 1 ΤΟ 1 , 0 0 0 , 0 0 0

442

APPENDIX I I I . VALUES OF K(T,B) FOR USE IN CALCULATING THE EQUIVALENT TEMPERATURE Θ FROM AIR TEMPERATURE T ( ° C ) , AIR PRESSURE B (MMHG), AND RELATIVE HUMIDITY

443

APPENDIX I V . T H E DAYS OF AN ORDINARY YEAR NUMBERED CONSECUTIVELY, BEGINNING JANUARY 1 APPENDIX V . VALUES OF Φ0(Χ)

=

APPENDIX V I . VALUES OF F ( Z ) =

APPENDIX V I I . VALUES OF Q(X)

444 1

V2X Ι — - = V2T =

E~IX'

AND Ξ =

R A / E~IX' ÀX

I F 1 — - = / V 2?R

« E~IX' DX

E~IX'.

.

.

445

446

448

Figures 1. Structure of a climatological network 2. Blank form of the international climatological register 3(a). Old form of register of cooperative stations of the U.S. Weather Bureau 3(b). New form of register of cooperative stations of the U.S. Weather Bureau 4. Histogram of relative frequencies with superposed smoothed frequency curve 5. Normal distribution showing the areas (percent) between consecutive ordinates for abscissas of one, two, three standard deviations left and right from the center (frequency of the arithmetic mean) 6. Frequency curve of March rainfall for 21 years at Helwan, Egypt . . 7. Frequency distribution of cloudiness 8. Frequency distribution (percent) of the deviations from the mean daily minimum of the coldest months of 53 years at State College, Pa. based on the probable error 9. Skewed frequency distributions, showing the theoretical position of mode and mean 10. Relative variability and rainfall 11. T h e relation between cloudiness and duration of sunshine 12. Average temperatures of the parallels of latitude and their variation with latitude 13. Vector ruler 14. "Temperature surface" of Berlin 15. Stereoscopic photograph of "wind surface" model, Donnersberg Observatory, Bohemia (858 m) 16. Normal frequency surface 17. Frequency surface of interdiurnal variability of cloudiness and average cloudiness of the previous day, Donnersberg Observatory (Bohemia) 18. Anemo isopleths, Donnersberg Observatory (Bohemia) 19. Drawn stereogram of the track of a pilot balloon 20. Quality of rainfall stations in County Cork, Ireland 21. Nomogram for reducing vapor pressure to sea level 22. Léon Lalanne's nomogram for multiplication 23. Construction of an alignment chart 24. Nomogram for y = ·\/αβ 25. Nomogram for the density of dry air 26. Yearly amplitude of air pressure and altitude 27. Observations represented by the quadratic equation χ = A + By -f- Cy1 28. Range of annual rainfall as a function of length of the observational period, using a logarithmic scale in one direction 29. Range of annual rainfall as a function of the length of the observational period, using semilogarithmic paper 30. Observations represented by the formula y = ax~b 31. Mean wind velocity a t various heights above the ground a t Nauen, plotted on ordinary cross-section paper xxi

5 9 10 11 22

41 43 45

48 49 55 59 62 63 65 66 67 68 69 70 71 76 77 78 79 81 84 88 95 96 98 101

xxii

FIGURES

32. The data of Fig. 31 plotted with logarithmic scales on both axes . . . 33. Mean wind velocity at various heights above the ground at Nauen, represented by the equation υ = a log (h + q) + 6 34. Frequency distribution (normal coordinates) of annual rainfall at Padua, Italy (1725-1924) 1 r' , 35. Graphs of the function Q(x) = J e~i' dx before and after straightening 36. Test of annual rainfall at Padua, Italy (1725-1924) on arithmetic probability paper 37. Graphical smoothing using the binomial coefficients as weights . . . . 38. Example of superposition of two waves 39. Diagram for determining the phase angle 40. (a) The annual course of the relative duration of sunshine at a given average level of a mountainous region. (b) The first two constituents of the calculated curve 41. Representation of amplitude a*, its components pk and its phase angle At on the harmonic dial 42. Representation of 0.5 sin [(360°/24)i + 207°] (a) on a vector diagram, (6) on a harmonic dial 43. Harmonic dial for the first term of the 1871-1915 average diurnal variation of temperature at Cahirciveen (Valentia Observatory), Ireland 44. Harmonic dial for the second term (semidiurnal wave) of the average diurnal variation of temperature at Cahirciveen 45. Harmonic dial for the third term (8-hr wave) of the average diurnal variation of temperature at Cahirciveen 46. Harmonic dial for the first (whole-year) term of the yearly variation of relative duration of sunshine at different levels in the Eastern Alps . . 47. Harmonic dial for the second (half-year) term of the annual variation of relative duration of sunshine at different levels 48. Shifting of the angle x m a x of the first term represented in Cartesian coordinates 49. Average daily course of temperature at a place in Central Europe . . 50. Correction to give the "true mean" if ( M + m)/2 is known. Isolines for the United States, January 51. Correction to give the "true mean" if ( M + m)/2 is known. Isolines for the United States, July 52. Diagram explaining the interpolation of the dates at which temperature crosses a given threshold, upward and downward 53. Relative temperatures at Bismarck, N.D 54. Correlation between the frequency of wind directions and their average velocity 55. Climogram representing the correlation of monthly averages of relative humidity and temperature for Boston, Mass 56. Graphical solution of the transcendental equation log (1 + d) = 0.12d 57. Cumulative monthly rainfall amounts, expressed as percentages of annual total (cumulative percentages) 58. Examples of G. Taylor's hythergraphs 59. Variations of January temperature at Moscow in the years 1860-1870 60. Increase of range of the temperature extremes with length of period of observation

102 104 107

113 114 118 120 126 127 143 144 147 147 148 151 152 153 155 162 163 165 176 182 186 210 220 221 223 240

FIGURES 61. Dot chart and regression lines of the deviations from the mean annual level of Victoria Nyanza and from the mean number of sunspots . . 62. Isopleths representing the annual course of air temperature in the high northern latitudes 63. Construction of a rainfall map; first phase 64. Construction of a rainfall map; second phase 65. Construction of a rainfall map; third phase . . . 66. Construction of a rainfall map; final phase . . . 67. Linear and nonlinear interpolation 68. Graphical determination of gradient from three scalar values . . . . 69. Geometric representation of d//ds 70. Example of construction of gradient from three scalar values . . . . 71. Isanomals of the duration of the vegetative period in Switzerland . . 72. Isanomals of temperature 73. Isanomals of relative duration of bright sunshine in the Eastern Alps in winter 74. Precipitation profile across the Bernese Alps 75. Profile of snowfall across New England 76. A set of profiles exhibiting the geographic distribution of the mean depth of snow over Finland in March 77. Hyetographic curve for the whole surface of the earth 78. Areas of the two hemispheres receiving specified amounts of average annual rainfall 79. The wind "star" . 80. Streamlines over the Balkans 81. Streamlines of New England in January 82. Streamlines of New England in July 83. Graphical representation of the boundaries between desert and steppe climates 84. Graphical representation of the boundaries between steppe and tree climates 85. Boundary lines between the climates Aw, Am, Af 86. Nomogram for determination of the (P£)index 87. Isolines of the dates on which the curve of the annual course of temperature rises to 70°F in the Mediterranean region 88. Isolines of the dates on which the curve of the annual course of temperature drops below 70°F in the Mediterranean region 89. Isolines of the duration in weeks of an average temperature 3: 70°F (duration of the hot season) in the Mediterranean region 90. Isolines of the number of months with an amount of rain g 0.2 in. (duration of the dry season) in the Mediterranean region 91. The SW type of weather over Finland 92. Distribution of pressure, wind, and temperature during cold, normal, and warm seasons in the United States: Winter 93. Distribution of pressure, wind, and temperature during cold, normal, and warm seasons in the United States: Spring 94. Distribution of pressure, wind, and temperature during cold, normal, and warm seasons in the United States: Summer 95. Distribution of pressure, wind, and temperature during cold, normal, and warm seasons in the United States: Fall 96. Eroded mountain slopes with westerly exposure above Puka, Albania .

xxiii 253 258 261 261 262 262 266 267 268 270 278 281 283 286 287 288 289 290 292 295 296 297 302 303 304 306 314 314 315 315 319 324 325 326 327 329

xxiv 97. 98. 99. 100. 101. 102. 103. 104. 105.

106. 107. 108. 109. 110. 111. 112.

113. 114. 115.

TABLES Hay barn at Saas Grund, Switzerland, standing on high stone pillars . Humidity slide rule Hollerith punched card Tabulating cards used for in-station punching at observing weather stations in Czechoslovakia Samples of punch cards used by the U.S. Weather Bureau Daily variation of air temperature, Donnersberg (Bohemia) Mean daily variation of air temperature, Donnersberg (Bohemia) . . Nonharmonic wave y = a sin [(2χ/τ)ί + A] Buys-Ballot schedule for the trial period 29 of the International Magnetic Character Figures produced by the tabulator from manually rearranged punched cards Periodogram for one simple period, y = a sin [(2π/r)t + A] Periodogram of a function for various lengths of the analysis interval . Frequencies of amplitudes for any trial period of a population with Gaussian distribution · Periodogram of 342 random numbers Harmonic dial for each of the 25 rows in Table 96. Sine waves of trial period 6 Quasi persistence in the International Magnetic Character Figures, 1906-1933 Graphs of y, = 8 sin 36.09634% - 5 sin 9.74028% + random fluctuations ί„ and the first and second transformed series (according to Fuhrich's method) Graphs of simple harmonic functions of unit amplitude with superposed random fluctuations Graph of a disturbed harmonic function, experimental series Correlogram of y, = 100 sin [(2ττ/7> + 9 0 ° ]

330 335 341 344 349 357 358 363

374 377 378 386 394 402 406

428 429 431 437

Tables 1. Comparison of areas and heights above ground level of rain gages in various countries 2. January daily minimum temperatures a t Mount Washington, N.H. for four years 3. Frequency table grouping the data of Table 2 into class intervals . . 4. Distribution of relative frequencies of temperature a t Vienna and Scutari in summer 5. Two methods of arranging data: (a) in an array; (b) grouped . . . . 6. Frequency distribution of minimum temperatures for 25 months of July 7. Dates of the last killing frost in Vancouver, Wash, and Bismarck, N.D., and deviations from the mean date 8. Annual rainfall a t Padua, Italy (1725-1924) 9. Computation of arithmetic mean of the annual rainfall at Padua, using a provisional mean 10. Computation of arithmetic mean and standard deviation of annual rainfall a t Padua (1725-1924), using grouped data 11. March rainfall for 21 yr (1904-1924), a t Helwan, Egypt

15 18 20 20 24 25 29 31 32 34 44

TABLES

XXV

12. Frequency distribution of cloudiness for 9 P.M. at Prague, Czechoslovakia 44 13. Classification of deviations from the mean daily minimum of the coldest months of 53 yr at State College, Penna 47 14. Comparison between average variability and intersequential variability 51 15. Various statistics regarding interdiurnal variability of temperature . . 53 16. Method of random samples applied to rainfall at Batavia, Java . . . 58 17. Observed duration of sunshine 5 (percent of possible duration), observed cloudiness C (percent of visible sky), and 5 + C for various climates 59 18. Relationship between yearly amplitude of atmospheric pressure and altitude above sea level 85 19. Variation of temperature with altitude in an Alpine valley in winter . 89 20. Average temperature at various heights above the bottom of an Alpine valley in winter 92 21. Range of annual rainfall and length of the observational period . . . 95 22. Representation of observations by an exponential curve y = ax~b . . 98 23. Mean wind velocity at various heights above the ground at Nauen (near Berlin, Germany) 101 24. Mean wind velocity at various heights above the ground at Nauen, represented by a semilogarithmic system of coordinates 103 25. Annual rainfall at Padua, Italy (1725-1924) (grouped data) 108 26. Fitting a normal frequency distribution to the distribution of annual rainfall at Padua, Italy . . . . 109 27. Scale of arithmetic probability paper 112 28. Smoothing of data, using the binomial coefficients as weights 116 29. Coefficients cos kiz and sin kiz for 12 equidistant values 122 30. Coefficients cos kiz and sin kiz for 24 equidistant values . . . 123 31. Harmonic analysis of the annual course of the variable 5 124 32. Evaluation of equation for S (computed in Table 31) 128 33. Harmonic synthesis of first periodic term of Fourier series 130 34. Daily variation of mean atmospheric pressure at Cahirciveen (Valentia Observatory), Ireland in March 1932 137 35. Harmonic constants of the average diurnal variation of temperature for the period 1871-1915 at Cahirciveen (Valentia Observatory), Ireland. 145 36. Ratios of amplitude of the first terms of the Fourier series, from Table 35 149 37. Constants of the harmonic analysis of the annual course of the monthly percentages of the effective possible duration of sunshine at different levels in the Eastern Alps 150 38. Angles %max in degrees and calendar dates, calculated from the phase angles A¡ and A2 in Table 37 152 39. Average daily course of temperature 156 40. Observation hours in the United States 157 41. The annual course of c (F deg) for combinations of different hours of observation 158 42. Correction required to reduce the mean of the extremes to the true mean, for different average periodic daily ranges and different elevations . . 162 43. The annual course of temperature at Bismarck, N.D 166 44. Average cumulated daily minimum temperatures g 0°C for Paris, winter 1872-73 to 1911-12 169

xxvi

TABLES

45. Warm and cold winters in Vienna, Austria, characterized by the totals of daily means of temperatures S 0°C, 1873-74 to 1923-24 46. Average number of degree days at Boston, Mass. and Chicago, 111. . . 47. Average number of degree days for different climates of North America 48. Frequency distribution of hourly temperatures at Kew Observatory (Richmond, England), February 1929 49. Total of degree days, based on 35°F standard indoor temperature for February and for the heating season at Kew Observatory in the fiveyear period 1929 to 1933 50. Example of the calculation of wind elements, 24-yr January average for Boston, Mass 51. Direction and velocity of wind at Batavia Observatory, January 1925 52. Equivalents of the numbers of the Beaufort wind scale 53. Average monthly equivalent temperatures at Boston, Mass 54. Equivalent temperatures computed from the equation θ = t + 2e compared with the results from the correct formula 55. Climatic classification according to drying power 56. Anthropoclimatic classification of drying power 57. Climatic classification according to dry cooling power 58. Differences of frequencies of clear days and cloudy days and their relation to cloudiness 59. Probability that a day with precipitation is a day with snowfall . . . 60. Monthly precipitation at a place in Central Europe, represented by different methods 61. Portions of the frequency distribution of daily amounts of rainfall at Batavia, Java 62. Frequency of daily amounts of precipitation at Zurich, Switzerland . 63. Statistics of wet spells (1901-1940) at Lugano, Switzerland 64. Form for tabulating snow conditions for a number of stations . . . . 65. Annual course of precipitation represented by cumulated amounts and percents 66. The quasi constancy of the differences between mean January temperatures at two places 69 mi apart 67. The quasi constancy of ratios of rainfall in July at two places 11 mi apart 68. Fluctuations of rainfall amounts as percentages of the average values, illustrating the quasi constancy of ratios 69. Abbe's criterion applied to a series of January temperatures at New York and New Haven 70. Abbe's criterion applied to the rainfall series of Table 67 71. Mean January temperatures for different periods at St. Paul, Minn. . 72. Mean rainfall at Maiden Island in different periods 73. Example of quasi constancy of January temperature difference, from Table 66 74. Example of quasi constancy of July rainfall ratio, from Table 67 . . . 75. Mean January temperatures, from Table 66 76. Example of July average rainfall, from Table 67 77. Example of calculating a correlation 78. Calculating a correlation by means of the original numbers reduced by a constant value 79. Annual deviations of the level of Victoria Nyanza and of the annual sunspot numbers

170 172 172 174 174 178 179 184 189 190 193 194 198 200 202 203 206 209 214 216 219 224 225 226 229 230 232 233 233 233 234 236 246 249 252

TABLES 80. S t a n d a r d values of t h e duration of the vegetative period a t different altitudes in Switzerland 81. Anomalies of t h e duration of the vegetative period a t two places in Switzerland, north and south of the Alps 82. S t a n d a r d distribution of t e m p e r a t u r e with geographic latitude . . . . 83. Variation of precipitation with height 84. Areas of the e a r t h ' s surface receiving specified a m o u n t s of rainfall . . 85. Areas of t h e northern and southern hemispheres receiving specified a m o u n t s of average annual rainfall 86. Wind roses for different elements observed a t Mossel Bay, Spitsbergen 87. Relative frequencies of different air masses a t Aachen, G e r m a n y . . . 88. Degree of continental i tv derived from frequencies of continental and maritime air masses, for Pennsylvania and for Western Europe . . 89. C o m p u t a t i o n of sums of products by progressive digiting ( s u m m a r y multiplication) 90. Accuracy of representing 24 equally spaced ordinates of the function 100 sin [(2τ/7)ί + 90 o ] by a Fourier series with 12 terms 91. Selection of observations m a d e a t solar hours for computing t h e M-tide according to D a r w i n ' s method 92. W i d t h s of m a x i m u m for different values of Ν 93. Values of W(«cE,) = exp

xxvii 276 277 279 284 290 291 293 321 322 348 363 366 379

Ζ «sj, W{KE2) = exp ( - κ 2 ) ,

W{KE·?) = e~' for various values of Κ . . . * 94. R a n d o m numbers, obtained by throwing two dice 95. T e s t of significance of a 12-yr period in annual rainfall a t Vienna . . 96. T e s t of significance of a trial period with o u t s t a n d i n g ordinate in t h e a m p l i t u d e periodogram 97. Set Z, 150 random n u m b e r s 98. T e s t of conservation of Set Ζ 99. T e s t of significance of a trial period of lengths 6 in t h e numbers of Set Ζ . 100. P a r t s of t h e original series to be correlated according to Fuhrich's method 101. Values of y, = 8 sin 36.09634% - 5 sin 9.74028° κ + random fluctuations, and t h e first and second transformed series (according to F u h r i c h ' s method)

390 393 399 401 403 408 409 415

427

PART I M E T H O D S IN

CLIMATOLOGY

Introduction Climatography describes the climate, climatology explains it. Climatography arranges and prepares the raw material supplied by observation; it is the foundation of climatology. Seemingly there is a vicious circle, because it is hardly possible to make a climatography without a knowledge of climatology. This is the present state in which the two branches of science supplement each other. In its early beginning, climatography was an occasional description in words and pictures by travelers, chroniclers, writers, poets, and painters. The invention of the thermometer and its regular use for observing air temperature were the first steps toward a quantitative climatography. Galileo Galilei invented the thermometer in 1597; it became an accurate physical instrument in 1772, when de Luc reintroduced mercury as a thermometric substance. Rain gages are supposed to have been used in India in the fourth century B.c. The oldest regular measurements of rainfall were made in Palestine in the first century A.D. With these observations at hand, it has been possible to compare the hydrometeoric climate of ancient Palestine with the present one in a more or less exact way. The invention and development of other instruments has furthered regular quantitative climatological observations. Climatic elements that cannot be observed with the aid of instruments are estimated by means of arbitrary scales. A single observation, picked from a series, has usually no climatological significance; it can be compared to a single exposure of a motion-picture film. Only chronologic arrangement and appropriate combination of observations yield a sufficient basis for computing average variations and average states. The nature of our subject determines the necessary analytic and synthetic methods. First of all, the concept of climate must be defined. Climate is the average state of the atmosphere over a particular place or region of the earth's surface, related to a particular epoch and taking into con1

2

METHODS

IN

CLIMATOLOGY

sideration the average and extreme variations to which the atmospheric state is subject Observations are made at isolated points. Only by comparing these data can the climate of the whole region be interpolated from place to place. It is therefore the principal and fundamental aim of climatological methods to make the climatological series comparable. The more closely this goal is approached the more reliable is the climatography. Long-range forecasts or investigations into hidden periodicities depend even more on reliability and exactness, and therefore upon the comparableness of climatological series. Only with comparable series can true averages, true variations, and true probablities be determined and used for making further estimates and inferences. *V. Conrad, Die klimatologischen Elemente und ihre Abhängigkeit von terrestrischen Einflüssen, in Köppen-Geiger, ed., Handbuch der Klimatologie (Berlin, 1936), vol. IB.

Chapter I

GENERAL

REMARKS

1 . 1 . CLIMATIC E L E M E N T S . CLIMATOLOGICAL ORGANIZATION

1.1.1. Climatic Elements Climatic elements are the components of the climate. The combination of all elements occurring at a given moment makes the weather; the average weather—that is, the average state of the atmosphere—is called the climate. The climate depends upon the climatic factors, such as geographic latitude, geographic longitude, altitude, distribution of land and water, which practically do not vary over long periods—for example, the period of recorded history. Thus, the variable elements and the quasi-invariant factors should not be confused with one another. It is not possible to enumerate all the elements, because their number can be increased arbitrarily, but the following may be listed : 1. 2. 3. 4. 5. 6. 7. 8.

Radiation, incoming and outgoing, Temperature of the air and of the surface of the earth, Wind direction and velocity, Humidity and evaporation, Cloudiness and sunshine, Precipitation, Snow cover, Atmospheric pressure.

The last item is included because of its intimate relation to the instantaneous and average states of weather and atmosphere. The difficulty is that these items do not really represent single elements, but groups of elements. The incoming radiation, for example, is divided into two main parts: direct radiation from the sun and radiation from the sky. Each part can be subdivided into any number of elements. Thus, the total radiation, including the energy of the entire spectrum, can be divided into elements characterized by the energy of parts of the spectrum, for example, the infrared, the visible, the ultraviolet portions. Other radiation elements are the transmission coefficients of certain wavelengths, the turbidity factor and coefficient, the blueness of the sky, photometric and photochemical 3

4

METHODS

IN

CLIMATOLOGY

brightness, atmospheric back radiation, range of the ultraviolet spectrum, and so forth. 1 The radiation group is only one example. The same conditions exist with all the other groups. This very great variety is no drawback; on the contrary, a clever climatographer, while utilizing the usual elements, should define new elements which give the best, the shortest, and the most accurate contribution to the description of the climate under consideration. For the present purpose, it is perhaps of greater importance to classify the elements from another point of view : 1. Primitive elements, directly observed or estimated, 2. Combined elements, some of which can be determined also instrumentally, 3. Derived elements. Examples of elements in group 1 are temperature, precipitation, wind velocity, wind direction, cloudiness. Examples of elements in group 2 are cooling power, drying power, continentality, equivalent temperature, potential water power. Cooling and drying power, equivalent temperature and other combined elements can be determined instrumentally, or they can be calculated by means of a combination of two or three primitive elements. The third group is obviously the most extensive and offers much scope to the imagination of the climatologist. A few examples may illustrate the multifariousness of the elements of this group: turbidity factor and coefficient, variabilities, durations of characteristic average states of the atmosphere, lapse rates of temperature, frequencies, probabilities, anomalies. From the classification of the elements as primitive, combined, and derived, their infinite multiplicity is obvious;2 hence the need for some methodical hints on handling these numerous data and for selecting the best in order to get the most concise and exact description of the climate. Observations do not always reproduce natural conditions in the right way. There are many sources of error. An appropriate critique and treatment of the data should eliminate the systematic as well as the random 'For the multiplicity of radiation elements, see the monograph of F. W . P. Götz, Das Strahlungsklima von Arosa (Berlin, 1926). From the same point of view, W . Mörikofer, "Meteorologische Strahlungs Messmethoden," in Emil Abderhalden, Handbuch der biologischen Arbeitsmethoden (Berlin, 1939), Abt. I I , pt. 3, is of interest. 2 One or another of the climatic elements enumerated in the foregoing list may be unknown to the reader. Some of them are explained in the following chapters in so far as they are of méthodologie interest. Further explanations can be found in N . Shaw, Manual of meteorology ( N e w York, 1926-31); V. Conrad, Fundamentals of physical climatology (Harvard University Press, 1942) ; and in V. Conrad, Die klimatologischen Elemente und ihre Abhängigkeit von terrestrischen Einflüssen, in Köppen-Geiger, ed., Handbuch der Klimatologie (Berlin, 1936), vol. IB.

GENERAL

REMARKS

5

errors. These corrections are related to a series of observations made at one place over a period of time, and a comparison of observations made simultaneously at different places. The preparation of original observations for purposes of comparison, whether as single values or as a series, is one of the main tasks of climatological methods. 1.1.2. Climatological Organization The observations of a station A must be comparable with those of station B, say, south of A ; the observations of C, west of A, should be compared with those of either A or B, or both, and so on for all directions of the compass (see Fig. 1). Thus climatology is a world-wide science and we

F

FIG. 1. Structure of a climatological network, understand why Sir Napier Shaw writes: "No country, not even the largest, is self-sufficient in the material for study, because that material must be coextensive with the whole world." 3 In 1781, the meteorological organization, the "Mannheimer Akademie," was founded and this was the first step toward world-wide climatological studies. In its publications, Ephemerides Societatis Meteorologicae Palatinae, even the results of observations for two American places were included.4 3

N. Shaw, op. cit., vol. 1, p. 160. The data were communicated to the Mannheimer Akademie by one of its members, Samuel Williams, Hollis Professor of Mathematics and Natural Philosophy at Harvard. For Williams's scientific activities, see B. M. Varney, "Early meteorology at Harvard College," Monthly Weather Rev. 3 6 , 1 4 0 (1908). Professor F. Steinhauser, Vienna, Austria, kindly identified these two places; they are Cambridge and Bradford, Massachusetts. 4

6

METHODS

IN

CLIMATOLOGY

Nowadays, the governments of civilized countries have organized central offices which administer and extend the network of climatological (meteorological) stations. The expression "network" is taken from the "network of stations" upon which the geodetic survey is based. Figure 1 indicates more fully the meaning of the term. The observations of each station must be comparable with those of all the others. The aim of the study of the reciprocal relations between the climatic elements at different places is to determine the circulation of the atmosphere (physics of the actual state of the atmosphere), and to get full information about average and extreme values of climatic elements—at least close to the earth's surface (physics of the average state of the atmosphere). For these large-scale purposes, there must be a network, internationally administered, that covers the entire globe. Regional meteorological efforts were internationally organized at a relatively early stage. An international conference on maritime meteorology was held at Brussels in 1853, and the first international meteorological congress at Vienna, Austria, in 1873, was attended by official delegates from a large number of countries. On this occasion an "International Meteorological Committee" was elected in order to assure international meteorological and climatological collaboration without interruption during the interval between the congresses. This international organization has the following duties: 1. To regulate and advise concerning comparable observations; 2. To define phenomena such as rime, snow cover, day with thunderstorm, day with precipitation, etc.; 3. To regulate the organization of a network for a country. Such a network should consist of stations of different orders. (а) A Central Office, or Central Institute, is the chief office entrusted by the Government with the management, collection, and publication of the meteorological observations of the country (e.g., the U.S. Weather Bureau); (б) A Central Station (Section Center) is a subordinate center for the management and collection of observations from a certain province or state (e.g., the U.S. Weather Bureau, Boston, Mass.) ; (c) A Station of the First Order is an observatory in which, without the collection of observations from other stations, meteorological observations are conducted on a large scale, i.e., either by hourly readings or by the use of self-recording instruments; (d) Stations of the Second Order are the stations where complete and regular observations on the usual meteorological elements—atmospheric pressure, temperature, and humidity, wind, cloud, rain, and hydrometeors, etc.—are conducted; (e) Stations of the Third Order, finally* are the observing stations where only a greater or smaller portion of these elements are observed. 5 6

N. Shaw, op. cit., vol. 1, p. 163.

GENERAL

REMARKS

7

A knowledge of the regulations and definitions of the international organization is of great importance to anyone who has to use the meteorological records of other countries. The most essential resolutions and recommendations of the International Meteorological Committee during the years from 1872 to 1905 are summarized in the International Meteorological Codex.® In spite of the age of this report (published in English by the British Meteorological Office in 1909), the recommendations are valid and valuable today, with the exception of the sections on aerologie observations and on weather maps and forecasting. A Spanish version was published by the Observatorio Central de Manila in 1913, and includes the resolutions adopted between 1872 and 1910.7 Another function of the International Meteorological Organization is the publication of the Réseau Mondial, which contains the observations of a global network of meteorological stations. This title represents the climatological and meteorological ideal: a network that covers the entire surface of the earth, administered in a uniform manner and serving all mankind. The publication in question is only the beginning of an ideal, as yet unrealized. It was a great scientific advance when the British Meteorological Office initiated it in 1911. Monthly averages of pressure, temperature, and precipitation at about 500 places are published for each year. The stations are grouped in 10° zones of latitude. It is noteworthy that meteorological data are supplied for all the zones from 80°N to about 61°S. South of this latitude no permanent meteorological station is being operated. The data and elements given for each station in the Réseau Mondial, averaged for the months and for the year (with the exception of the three coordinates), are: 1. 2. 3. 4.

Latitude; Longitude; Elevation (meters); Pressure (millibars), (а) Mean at station level, (б) Mean at mean sea level (M.S.L.), (c) Deviation from normal; 5. Temperature (°K), 8 (a) Mean maximum, (b) Absolute maximum with date of occurrence, (c) Mean minimum, (d) Absolute minimum with date of occurrence, •Prepared by G. Hellmann and H. H. Hildebrandsson. 'N. Shaw, op. cit., vol. 1, p. 166. For further information, see the Official Reports of the International Conferences. •The Kelvin, or absolute, temperature scale is the one most widely used in scientific work; its degrees are the same size as those of the centigrade scale, but its zero is (approximately) -273°C.

8

METHODS

IN

CLIMATOLOGY

(e) Arithmetic mean of (a) and (e), ( / ) True mean temperature, (g) Deviation from normal; 6. Precipitation, (а) Monthly total (millimeters), (б) Deviation from normal.

Another document of the international development of climatology and meteorology is the World Weather Records, which was assembled and arranged by H. Helm Clayton in consequence of a resolution of the International Meteorological Conference adopted at the meeting at Utrecht in 1923. This publication9 gives for each year of record the monthly and annual values of atmospheric pressure, temperature, and precipitation, and averages of these elements for about 390 places. The great advantage of this publication is that it offers long and, so far as possible, homogeneous series which overlap one another for the period of observation. Thus, the work is indispensable if variations of the three elements have to be compared. 1.1.8. The Meteorological Register The meteorological register contains the records of the original, regular observations made at the climatological station. All climatological studies are based ultimately upon the registers of the stations. The utmost international uniformity of the registers is essential for investigations that cover a region larger than one national network. For this reason the International Meteorological Organization, at its meeting at Utrecht in 1874, suggested a special form for the register of stations of the second order. Figure 2 shows this international form with a few supplements. As far as the units of pressure and elevation, the temperature scale, and so on, were concerned, uniformity could not yet be achieved. This form, which assumes that the elements mentioned are observed three times a day, is used by most of the central institutes of Europe and the U.S.S.R. In many networks the register is more or less modified, according to the various conditions and requirements. It is to be hoped that these modifications will be restricted more and more and will eventually give way to complete uniformity. As for the organization of the United States Weather Bureau, J . B. Kincer may be quoted :10 9Smithsonian miscellaneous collections (Washington, D.C., 1927, 1934, 1947), vols. 79, 90, 105. '""Climate and weather data for the United States," in Climate and man (Yearbook of Agriculture, U.S. Department of Agriculture, Washington, D.C., 1941), p. 689. R. H. Weightman, Chief, Station Facilities and Meteorological Observations Division, was kind enough to communicate to us the numbers of stations mentioned here. They are valid for the end of 1948 and replace those given by Kincer.

ε 3 1/1 en -—* a a °.OOOS©i-H may be obtained from (8.2 + 3 X 9.6 + 3 X 9.2 + 8.6)/8. T h e method applied in Table 28 has the additional advantage that it can be easily checked. T h e sum of the t{k) is equal to the sum of the t{k~1) in the previous column diminished by half the sum of the first and last terms in this column. For example, Σ / u m = 67.40, Σ ¿ ( i i ) = 76.01, /í iiJ = 9.15, /¿u) = 8.08, and 76.01 which agrees closely with ^

£(9.15 +

8.08) = 67.39;

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0 0 — ι 1 iH — ι 1

C O O I 1 — t M ( 1 1 I

t ^0 o O I o I Nc o 0o Tf 1—1 O Oc • f « t ^t ^Ο O I o o M I c o — t 1

++++

1 1

"

ο ο c o c o f < i o c o • M ,-¡ ( Ν

< — < " b w w

c o C D C O o

1 1

Il

> 0 0 0

D O I 0 0M ( O I • f i • f iC — r i 1 — τ 1 — τ 1 — i < M I

+++++++

I

( Ν C D Ή

1 1 I

+++++++

S o < m

I

o O I i o oc — i 1 Ν

+++++++

σ ιT J H M ( r H

( Ν Ν

M I c o

++++

< 3 5 0 • f0 4

o

++

1 1

I

O O c o H H O O

1

_

1

I

+

1

1

I C O < M O M • f t• f «C

1 3 5 C O > o 1 I

+

+

1

T HO t ^ C D f- 0 5c O I C O S o C a O

++ t ' < N Il

+

• s q 0

+

y—i r Ht—l o

H Ht — 1 r-Ί o r — ι1 r H,

o c o c o

II

^ 1 3 5 n it M I

+

1 1 I

< 3 5 < 3 5 0 5c c o c o C O C O < 3 5 C O C O o C O +

c a -

li

o

t-

++

++

1

++

++

1

1

1

O D o O O O oc oC oC OC oC oC r—i M — ι IM ι 1 — I i< — ι 1 ( I — < M M M I

c o B Q

s

I

0 5

Η

++

++

1 1

I I f iO O-fi O o• O

+++

1

o

r--

1—1

Ί**

1 1 1

0 • < f 0

< M ( • f i• f iM — T< r HT H iH

++++++

^

1 1 1

8?:

1 1 1

C O C C O c C O C O c o t ^C O O o i o o oO I oo o i oo ooi p iH0 o o o o 0 o o M I o o c o o o c o M I M I o o c o M I o o C O

^ S o

++

Ϊ 5 & C

1 I

I Nho t ^M I eT ψo

t - < N I • f i• f iM — ι Iτ — l — I
C O c o O C O M ( c o M I c o < M c o c o M I C O M I < M

»

3 5 3 5 5o < 3 5 3 5 o0 oO S o< o1 o1 < M C O M I C D < M C D M ( C O M ( C O M ( C D

3

++

σ υ

1

++

1 I

1 1

++

1

T 3 e

«

< N

II a , < o O C O C O

< b O I O I O I O I » O l oO I O I Ό Ό> o O l O I O I o* oo O I O I o O O» Ot l b O I O I I IO • f «O O I C D Ό C D

+++

1

rt >

+++

I

1

o M I M I I < M 0 00 0o o ΝM o o 0 0o M I C O M ( M I c o M I < M C O c o c o C O

ω

• Φ o O I O • f• I f «> iH• < f • f

c i c o

1 17

" Ψ • « f

++++++

> o O I O I • f « 7

I

1

o « *

s

_

M Μ Η

> >

VI

ν

e o

III

u J η c H

O O I > o • f» C D O I C D O I

1 1 1

H H

> > H

> >

X

J H

S S

Ά

HARMONIC

ANALYSIS

129

As far as the degree of approximation is concerned, attention should be directed to the convergence of the resulting series. The ratios Α2/Α! = 0.32,

Α3/Α, = 0.16,

ajAT

= 0.05

show that there is a good convergence indeed. For practical purposes, the series in the present problem can be cut off after the second term.8 5.3. T H E

SIMPLIFIED

STRAIGHTFORWARD

METHOD

OF

EVALUATING

A

FOURIER SERIES

The straightforward method of evaluating a Fourier series, used above, can be simplified, if we start with the form of the series given in Sec. 5.1, namely, y = a0 + p¡

p¡

cos 2x +

q2

iz + qx s i n iz + p2

c o s 2 iz +