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Some Analysis on The Diurnal Var;

of Rainfall Over The Atlantic 0

Tepper Gill, Shien Perng and Arthu

Department of Mathematics

Howard University

Washington, D.C. 20059

(NASA-CH-164937) SUMS ANALYSIS ON THE

DIURNAL VARIATION OF RAINFALL OVE6 THEATLANTIC OCEAN (Howard U"iv.) 55 pHC A04/MF AO1 CSC

t» ^

August, 1981

C' NASA STi FACIUIY. tiAC' ESS DEPT.

2

This research was carried out under a grant from the NASA, Goddard Space

4

Ac)uiowledgement

'Me authors wish to thank Mr. Charles Laughlin and Dr. Raym ml Wexler

of National Aeronautics mid Space Administra Lion, Goddard Space Flight Cer,

for their assistance -md helpful discussions throughout the period of rest

and Mr. Vernon Paterson of National Oceanic and Atmospheric Administratic

for his assistance in data preparation.

Summary

Diurnal variation of rainfalls over the oceans has long been a ,treat concern

of members _in the meteorological communit •In this paper we examined such

Variation using part of data collected from the CARP Atlantic Tropical Experiment

(GATE), which was conducted in 1974 in the North Atlantic ocean. The data were

collected from 10,000 grid points arranged as a 100 x 100 array; each grid covered a

4 square km area. The amount of rainfall was meauured every 15 minutes during tale

experiment periods using c-band radars. We analyzed data collected in Phases I

and II of CATE which were conducted between 1790h and 197th days and between 209th

and 227th days of the year , respectively.

Two types of analyses were performed oil the data: A unlysis of diurnal varia-

tiun wits done on each of grid point y: bared oil rainfall averages at noon and

at midnight, and time series analysis an selected grid points based oil hourly

averages of rainfall.

Since there are no known distribution model which hesc describes the rainfall

amount, nonparnmetric methods were used to examine the diuratil variation. This

kind of methods was selected because of its model free nature. Kolmogorov-

Smirnov test was ised to test if the rainfalls at noon and at midnight have the

same statistical distribution. Wilc.oxon signed-rank test wits used to test If the

noon rainfall is heavier than, equal to, or lighter than the midnight rainfall.

These tests were done oil each of the 10,000 grid points at which the data are

available.

Among 10,000 grid points, data are not available at 1,872; these grid points

are around the boundary of tie square covered by CATE. In addition, there are

1,743 grid points where no conclusion was drawn due to insufficient frequency of

rain at noon or at midnight during the experiment periods. Both Kolmogorov-

Smirnov and Wilcoxon signed-rank tests were conducted at the rest of 6,385 grid

points.

F

11

With Kolmogorov;S...Irnuv test, it was found that at one out of 10 chance of

error, the rainfall distributions at noon and at midnight are same at 5,425 grid

points and different at 960 grid points. In term of percentage, they are 15.0 and

85.02, respectively. This is in contrast to 10 and 90%, respectively, if the as-

sumption of no difference between noon and midnight rainfall distributions had been

true. A chi-tielua re test showed that this split of percentage was not followed.

There are muA more grid points where the assertion of difference was made than

expected. Thus, overall the temporal rainfall distribution at noon and at midnight

are different,

With Wilcoxon signed-rank test it was found that at one out of 10 chance of

error, the numbers of grid points at which the noon rainfall is less than, equal

to, and more than tha midnight rainfall are, respectively, 430, 4,890, and 1,065.

In term of percentage, they are 6.7, 76.6, and 16.7%, respectively. This is in

contrast to 10, 80, and 10%, respectively, if the assumption of no difference had

been true. A chi-square test concluded that the midnight rainfall is not equal

to the noon rainfall; the noon rainfall is convincingly higher than the midnight

rainfall.

Time series analysis was conducted on some selected grid points in both

Phases I and II. This analysis is designated to detect if there is a short term

cycle in the rainfall pattern during the experiment periods and to examine the

temporal correlational behavior of the rainfall.

For Phase I, 20 grid points are randomly selected from 8,128 grid points at

which data are available and-for Phase II, 16 grid points are strategically selec-

ted. For these selected grid points the hourly averages of rainfall are obtained.

For cacti of these grid points, we thus obtained a time series consisting of the

hourly averages of rainfall. There are 36 time series, 20 for Phase I and 16 for

Phase II.

a

Although there were no uniform shape of the autocorrelation functions of

these time series, 9 out of 20 in Phase I and 9 out of 16 in Phase II are basically

negative exponential curves. Tito values for the auto-correlation decrease rapidly

as the values for lag increase. The medians of the auto-correlations for these

series at lags 1, 2, 3 and 4 are 0.46, 0.17, 0.15, and 0.10 for Phase I and 0.41,

0.12, 0.08, and 0.04 for Phase II, respectively. The medians at lags 5 or greater

are not significantly different from 0 for series in both Phases.

There are short term cycles detected in some of the time series. These cycles

are sparae and irregular. For Phase T, a 10-hour cycle is found in one series,

15-hour cycle in another series and 25-hour cycle in another one. A 12.5-hour cycle

is found in two series. For Phase IT, the cycles found are usually longer; 13-hour

cycle in two series, 50-hour cycle in 3 series and u 1.00-hour cycle in one series.

Thus, short term cycles exist in the oreanic rainfall, but they are not prevail.

It is interesting to note that the rainfall distribution in Phase II is very

even among all locations. 'They vary little from location to location. But in

Phase 1, the story is different. The value for the mean ranges from 0.0014 cm/hr

to 0.1320 cm/hr in Phase I; the latter is 94 times of the former. The variation

in the variance is also dramat{c in Phase I. The largest variance is 9,349 times

of the smallest. For Phase IT, it is only 25 times. This phenomenon probably as-

sociates with the heavy thunder storm activities usually happen in June or July

during which Phase I of GA'I'T, was conducted.

Chapter 1. Objectives and Background

1.1 Introduction

1.2 Background

1.3 Approach

Chapter 2. Statistical Methodologies

- 2.1 Kolmogorov-Smirnov

I 2.2 Wilcoxon Signed Rank Test

2.3 Chi-square Test

Chapter 3. Temperal Analysis on :iourly Rainfall Averages

3.1 Objectives of The study

3.2 Sample Design and- Data

3.3 Auti Correlation

3.4 Power Spectrum

3.5 Distribution of Total Power

3.6 Rainfall Average

Chapter 4. Results on niurnal Analysis

4.1 Introduction

4,2 Correlation

4.3 Kolmogorov-Smirnov 'rest

4.4 Wilcoxon Signed Rank Test

4.5 Chi-square Test

4.6 Summary on Diurnal Analyses

Chapter I: 1bjectives and llackgrouna

1.1 Introd•acti.on

Many members of the meteorological community have long been concerned with the

possibility of diurnal rainfall variation over the oceans. A comprehensive study

of the observational evidence for the acceptance of the hypothesis that such a

variation exists was completed by Jacobson (1976). Cray assumes that this is the

case and proceeds to attempt an explanotion based on a radiational cooling pro M c

theory. Much of the observational data used by Jacobson was based on small islane

data in th ,e Pacific ►

It should be noted that the work of Jacobson points to the existence of a

maximum in the morning and a minimum at night; this is in contrast to a recent

paper of Weick ►nan, long and Hoxit (1977), where a maximum appears at night and a

minimum appears in the morning. The Wei.ckman tet al) paper is based on CATE ship

measurements over the mid-Atlantic ocean. Assuming that Cray's theory explains

the variation over the Pacific.; the Weickman (et al) study makes it clear that we

cannot expect the same theory to apply to the Atlantic.

Questions immediately arise, both of P. theoretical and a methodological nature.

On the theoretical level, one i.ants to know if the variation is seasonally depen-

dent and how; is it latitudinally or long1tuLinally dependent and how; are there any

significant fluctuations in the variational distribution from year to year in a

given season; how doL, this affect our current overall estimates of world rainfall

rate; how does this affect current sampling plans to estimh;e the oceanic rainfall

budgrt?

On the methodological level, there have been grumblings concerning the possible

belief in any of the reports on rainfall variations over the oceans. This is due

to many factors: the lack of quality control over data collected from both small

islands and ships; the lack of any reasonable distril,utior, of reliable data col-

lection source; over a given region; the use of unreliable and/or untested data

-z-

manipulation methods and statistical procedures.

ThR_purpose of this report is to present a comprehensive unbiased analysis

of the question of the existence or non-existence of a diurnal rainfall variation

over the Eastern Atlantic,_ Our study is based on use of validated radar data

from a dense network of ships in the Eastern Atlantic during phases I and II of

GATE. In order Lo clearly define the scope of our study, the srecific research

objectives were:

I. To determine if ,iiurnal rainfall variations exist ami determine when

maxima and minima occ ar.

II. To identify and analyze periodic behavior in oceanic rainfall.

III. To develope a map showing those areas where rainfall variations exist.

Because of criticisms of previous efforts, we attempted to constrain our

approach methodologically in such a manner as to minimize tt•chnic:al objections

to our conclusions. This lead us to the following, specific methodological ob-

irctives.

1. To determine if there was any mathemat'.cal difference in the empirical

distributions of rainfall at mo rning verses evening.

II. To determine if there was any statistical difference between the mean

hourly rainfall rates in the mornings verses evenings.

III. To determine if there was any temporal. correlation between rainfall in

morning and evening.

1.2 Background

The GARP Atlantic Tropical Experiment (GATT:) was conducted in 1974 using a

total of twelve ships arranged in two hexagonal arrays which were exact for the

first two phases and distorted during the last phase. It is clear from Figure 1,

Acknowledgement

The author wish to thank Mr. Charles Laughlin and Dr. Raymond Wexler

of National Aeronautics and Space Administration, Goddard Space Flight

Center for their assistances and helpful discussions throughout the period

of researe}, and Mr. Vernon Paterson of National Oceanic and Atmospheric

Administration for his assistance in data preparation.

the extent and nature of the distortion. The ships in the inner hexagon were 165

Km apart while those in the outer hexagon •.acre .445 Km apart. This report Is

based only on data from phases I and II.

The ships in the array made standard surfacr, synoptic observations on an

hourly basis and upper air measurements every six hours, such measurements were

increased in frequency during periods of precipitation. Many ships made ocean-

ographic and radiation measurements, collected boundary layer profile data, and

recorded surface meterological data; however, the- data used in this study come

from the use of twos-band radars used to obtain special distributions of rain-

fall not available from ocher sources.

The c-band radars were choosen to minimize attenuation problems and maximize

spatial resolution. The radars were equipped with automatic digital processing

and recording equipment and the anteinian were stabilized to compensate for possible

roll and pitch.

Data was collected at 15 min. intervalP on a 24 hour basis. Antenna tilt

sequences of 360° scans at a series of 12 increasing tilt angles were collected

out to a range of 250 km. The reader is referred to Hft+dl.ow (1975) for details

and further discussion on the radar systems along with further references.

1.3 Approach.

The distribution of rainfall has been studied extensively from many different

vantage points and a number of probability models have been developed. Thom (1968),

Simpson (1972), Johnson and Mielke and Mielke (1973) are among the major workers

In this field. There is no single distribution which best describes rainfall and

any proposed model may be criticized at a number of levels. In particular, the

very nature of the diurnal rainfall variation makes the use of any model problemaric.

i

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-3&-

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Fig. 1

-4-

Ilia approach takop '.n this study is based on the use of three powerful methods

in mathematical statistics. The first two methods are used to accomplish the first

research objective and pare f the others. We use two non-parametric statistics,

the Kolmogorov-Smirnov and the Wilcoxon signed rank. Non-parametric statistics

are definition, those statistics that are independent of any particular model

for the distribution which generates the data, and thus, all conclusions drawn

are model indgendent.

This approach only depends on the assumption that the underlying distribution

is of continuous type; that is that measurements take values in a continuous in-

terval or union of intervals and does not take any specific value with positive

probability. It is clear that rainfall measurement is not of continuous type,

and in fact takes the value zero with large probability. For our ntudy, this is

not a major problem since we are only interested in comparisons conditioned on

the event of rainfall, thus our approach uses Coo method of conditional probabi-

lity. The measurement of comparisons conditioned on the event of nonzero rain-

fall is a continuous distribution. For related applications, see McAllister

(1969) and Gringarten (1970).

The third method used in this report is based on the assumption that the

hourly rainfall averages may be viewed as a time series. This approach allows

us to examine the serial correlation of the hourly rainfall, to detect short

term cycles, to measure and compare rainfall at different locations and to

es"ablish confidence intervals for the mean value of hourly rainfall at different

locations. We are thus able to accomplish our last two research objectives with-

in imposed methodological constraints.

In the second chapter we disco~s the Kolmogorov-Smirnov and Wilcoxon signed

Rank statistics and analyze their application to the problem of rainfall variation

between noon and midnight. In the third chapter, we apply time series methods to

the hourly rainfall data and provide a detailed analysis of results. In Chapter 4,

we report results of the diurnal analyses.

Chapter 2: Statistical Methodologies

In this section tw discuss the statistical foundations which ut;derpin our

approach. Since the Wilcoxon test is well documented in the literat-ire and well

known to applied researchers, we denote a major portion of this section to the

explication of the Kolomogorov-Smirnov statistic which is less used and less

known in applied research circles.

2.1 Kolmogorov-Smifrnov.

We would like to compare the distribution -f rainfall and rainfall rate at

noon with that occuring at midnight. We expect that if there is a diurnal varia-

tion then chose two Distributions may be different. It is possible that there

could be a diurnal variation and yet the distributions are the same, so that this

assumption is biased in favor of no variation.

Let (x 11 y l )J... (X no Yn) represent n-observations of rainfall over some

region C. The x's represent the measurement at midnight and the y's at noon. In

order to satisfy our assumption of continuity, we only cons±der those pairs

(x, y) for which either x > 0 or y > 0. Let F 1 (x) by the cumulative distribution

function for midnight and F 2 (y) be the cumulative distribution for noon. These

are the unknown true conditional distribution;. We can now state that if there is

no variation, then we expect that FI (x) - F2 (y); that is, it is desired to test

the statistical hypothesis:

H: F 1 ()') - F2 (Y) (2.1)

verses the al.terrate hypothesis:

A: F 1 (x) f F2 (Y) (2.2)

Under this framework, the basic hypothesis (11) is that the distribution of midnight

rainfall measure is the same as the distribution of noon rainfall measure. Based

on the data, we would like to confirm or reject this hypothesis. In rejecting 11

we would be making the conclusion that the alternate hypothesis (A) is true.

-6-

Since we don't hpve direct access to F 1 (x) and F2 (y) we construct and analyze

the empirical distributions F W and F 2 (y), where n denotes the number of sample

points. Define CM by;

1 if X Z 0

E(A) -

(2.3)

0 if a < 0

and set:n

Fn(x) a - E C(x - x3)1

(2.0

Fn(Y) - E C (Y - Y^)-1

Since the superscript identifies the distribution, we shall let a denote the

independent variable so that equation (2.4) now becomes:

Fn(^) n n C(X - X^) i . 1,2 (2.5)-1

where A= . x^ if i - 1 and a J Q y^ it i - 2. The following theorems s'.iow how the

empirical distributions are related to the true distributions. We interpret

F M to be the proportion of X which are less than X.

Theorem 2.1. The expectation and covariance of F M satisfy:

1) E(Fn(X)) - F i M

2) Cov {F1(X), F J) ) - n {rein (F i (X), Fi (T)) •• Fi ( X ) Fi(Z))

Theorem 2.2

1) (Strong law of large numbers)

Fi (X) -► Fi (with probability 1)

(2.6)

(2.7)

^A

I-s

1^

2) limlog log t

Fi(J1) (1 - Fi 01) an,*m

t^3) sup IF M - x i MI 4 0 (with prubability 1) (Cantilli-Glivenko lemma).

—oo< A<w

Proofs of the above results require very delicate analysis and are presented

in order to provide us with the relationship between true and empirical distributions,

the relevent literature incluues: Githman 195:3, 1954, Glivenko 1933, Gnedenko and

Kolmogory 1933, 1941, and Smirnov 1936, 1937, 1939.

The next two theorei.is are fundamental to our testing procedure, define E

andn

D12 by: (i ® 1, 2)

El a V n sup IF iM - F I M 1 (2.8)

r -CQ<A<«0 n

Dn2z sup IF M- F2 M1

(2.9)D21 - n12

n n

Theorem 2.3

1) Di1 2 is independent of F M

i Q 1, 2

22) 11.", PnI 2 1)n 2 < YJ ^ 1 - e-2r ^ ¢(Y) for 0 < Y <

n-►-

Proof: We shall obtain the proof of theorem 2.3 as a special case of theorem 2.4.

Theorem 2.4. Set c - Q2nYl (greatest integer) then:

( 2n )1 _ n -1 < <

Y 11( 2n) 2

aa

Proof: 1,et us arrange our 2n-observations (xl' y d ... (X no yn ) in increasing

order of magnitude X1 , ... X2n . Consider the r ►ew set of random variable s Y1' ••• Y2n'

defined by:

1 if Xk is a midnight observation

Y a (2.11)

-1 if Xk is a noon observation

kSet SO n 0 and Sk - r Xk . It is easy to see that nD12 a cup S ;.et us note

J u l n 0<k<2i k

that Stn = 0; if we plot the points (k, S k ) for k - 0,1,.-.2n in the (u,v) plane

and connect these points by straight-line segments, the v c-)mponent will increase

by one unit at n points (corresponding to rainfall observation at noon) and will

decrease by one unit at,the cemainir.g n points.

S

(01 0) i

2n k

rig. 2

Figure 2 represents a typical trajectory of the process, the dotted lines

indicate that the curve has been broken. Since there are n increases and n de-

creases, the total number of possible trajectories is (n2n ); furthermore as our

null hypothesis is that both distributions are the same, this means that all

trajectories are equally likely. (Another bias in favor of the null hypothesis)

Hence, the probability of each trajectory is (2nd . In our geometric interpre-

n

tation, the required probability satisfying Q^ n (Y) Pr( Dn2 < c] may be stated

L

J _ ( 2n )/(2n)n-c n

(2.12)

-4-

as the probability tha — lie entire trajectory will be below the line u - C. Now

the total number of trajectories which do not cross the line u - c is the same as

2n( n ) - (the number of trajectories which reach the line u - c), as there are n - c

possible ways to reach the line u - c, there are ( 2n ) possible distinct trajectoriesn-c

which reach the line u - c; thus the number of trajectories which do not reach the

line u - c `--o ( 2nl ) (n c) so that:

( 2n) - ( 2n )

Q^(Y) —nn_c- (2n M2n11 2n n )

( n)

We may now provide a proof of theorem 2.3:

Proof: Let us prove part 2) first; set

If we use Stir]ings formula:

k! a kk+1/2 0- (1+ p (k))( 21i) (2.13)

where p(k) -► 0, k -► -, and note that:

J

2

(n + c) (n - c)

we have:

In n+1/2 n 2

+ 0n+r.+l/2 (n - 0nl-c.+1/2 e-(n- c+1/1.)

2n+1 -2n

nn+c^Fl /2(1 + c/n)n+c:+ l%2 a-(n+c) nn-c+l 2U - c/n)n-c -4-l/2 a-(n-c)

(1 + ,t/n)-(n+c+1 /2) (1 - c/n)- (n-c+1/2) ( 1 + O(n)) (2.14)

-10-

Using the Maclaurin evp-nsion for log J, we get to the firs! order:

2log J - n + 0(n)

2Recall that c - Q2nyJ so that n 2y2 , equation (2.15) becomes:

log J Z -21 2 + 0(n) so that:

J - exp {-2y 2 ) • (1 + 0(n))

(2.15)

(2.16)

If we now use equation (2.10) we got

Q'11 (Y) '° 1 - exp {-2y 2 ) (1 + 061)) 'C' t(Y) - lim 0 1 (Y) - 1 - exp {-2y2}IrKu

Let us now prove part 1, we will show that 1) 12 is independent of F i (a) byn

showing that Fi is independent of F i (X); it suffices to show this for rn since the

same method applies to En

Proof:

Define v(t) as the inverse function of the event F 1 (s) < t this means that

Is _< v(t)) it; the inverse event, and this event has probability F t ( (t)) - t,

hence if we set t - F 1 (s) then Lip remains unchanged if F 1 (s)is repinced by the

uniform distribution. Our proof is complete if we recall that given any probability

distribution, there exists a transformation which transforms it to the uniform

distribution. Hence any statistic which is invariant under the uniform distribution

is invariant under all distributions;.

Returning to our test procedures, two types of possible error may be commit-

ted in confirming or rejecting the basic hypothesis: type I and type II. Type I

error occurs if the basis hypothesis H is rejected while it is true and type II

' error occurs if H is confirmed while iL is not true. Type II error is usually

used to determine the testing a,,proach. Since our approach has been determined by

other constraints, we shall only concern ourselves with type I error.

-11-

Let a be the probability of type I error, that is:

a - Pr [11 is true and H is rejected]

This number a is referred to as the significance level of the test. In this study,

we choose a be .10. This implies that there is a 1/10 chance of rejecting H while

H is actually true. This level was determined by the number of data _points avail-

Jlug.

In order to implement our procedure, we must determine a constant Y such that,

assuming It is true:

Pr [1) 12 > Y] . .10

pr [D12 < Y] - .90

This gives us a 90% chance of accepting H when 11 is true. The results of this test

are reported in Chapter 4.

2.2 Wilcoxon Signed hank Test.

Instead of the alternative hypothesis A in the Kolomogorov-Smirnov test, we

consider two alternative hypotheses Al and A2 defined as

A1: The midnight rainfall is (stochastically) greater than the noon rainfall,

A2: The midnight rainfall is (stochastically) less than the noon rainfall.

Using the notations in sub-section 2.1, A l and A2 can be denoted symbolically as

Al : F1 > F2 and A2 : Fl < F2.

Thus, the two testing problems considered in the Wilcoxon signed rank test one.

H: F1 = F 2vs A1 : F1 > F2

and H: F1 . F2 vs A2 : F1 < F2

-12-

However, these two to^r+ ,il, problems are statistically equivalent, respectively,

to If F2 < F2 vs A1 : F 1 > F2 and N2 : F1 > F2 vs A2 : F1F2. If Al is not rejected in

the first problem and A2 is not rejected in the-second problem then we conclude

that II: F1 = 1?2 holds.

As in the K - S test, there are two types of possible error in each of these

two problems. Again, we consider only type I errors. To test H vs Al , type I

error is committed if H (or Ii 1 ) is rejected while H: F F` is actually true, i.e.

confirming A 1 : ,'1 < F2 while H: F 1 a F2 is actually true. The significance level 011

is given by

a1 I' 1{ [rejecting III - P 11 [accepting A11.

To test H vs A2 , type I error is committed if A 2 : F 1 > F2 is accepted while

11: F 1 = F2 in actually true. The significance level is given by

a2 = 1 . 11 f rejecting 111 - P H [accepting A 2 1 .

Note that rejecting; H implies accepting A l in the first testing problem If vs Al

and accepting; A2 in the second testing problem H vs A2 . In this study, we adopted

N= a2 = .10.

Let (x1,y1),..., (Xns yn ) be as defined in subsection 2.1. The testing pro-

cedure of Wilcoxon signed test consists of deriving a statistic V based on

(xl'yl)'...' (xn'yn) and choosing constants C 1 and C2 (C2 • C1 ) corresponding to

al and a2 , respectively, such that for testing 11 vs Al

rejecting; 11 (or accepting A1 ) if and only if V > C1

and for testing If A2

rejecting H (or accepting A2 ) if and only if V S C2.

e detail, the reader is referred to Lehmann (1975).

►c above discussion that when If is true, PH [Vn < C2 1 - . 10,

-13-

PH [C

2< VN < C 1 ) _ .LC ind P

IlIV '1 Z C 1 ) - .10. Thus, when H is true, the chance

of concluding that midnight rainfall is (stochastically) greater than, equal to

and less than the noon rainfall are, respectively 10%, 80% and 10%.

The result of this test is reported in Chapter 4.

2.3. Chi-square Test.

The tests describe in subsection 2.1 and 2.2 are applied to the midnight and

noon rainfall measures at each grid of the GATE data. It was seen that when H is

true, the chances of drawing e conclusion that F Q F 2 1 2and F 0 F , are, respectively,

90% and 10% and that F < F 2 , 1- 1 - F2 and F > F2 are, respectively 102, 80% and 10%. We

use X 2-test to test whether these percentages are actually obtained in the result.

Let 1) 1 be the probability of making the jth conclusion, j - 1,2,..., k, where

k is the number of different conclusions (k - 2 in K-S test and k - 3 in Wilcoxon

test). Let: n' be the number grids at which jth conclusion is made, j - 1,2,..., k

and n = n + n2 +...+ nk , the number of grids in the study. Define

2 k ( _ J:_Xk = j°1 np j

If the given percentages of different conclusions are followed, Xk should be

relatively small. Thus, if Xk is large, we may conclude that the percentages do

not hold true and should have been otherwise. For instance, in the Wilcoxon test,

P 1 . .10, P2 - .80 and P 3.10. If X3 is too large, then the percentages of 10,

80 and 10% do not hold and thus the hypothesis H: F 1 - F2 is not true. To determine

"how large" is "too large", a cutoff constant can be ound corresponding to k and

the desired significance level of the X 2-test. With a - .05 (i.e. 5% chance of

making wrong conclusion), C - 3.84 for k - 2 and C - 5.99 for k = 3.

Results of the X2-test are reported in Chapter 4.

►ugh for practical purpose.

Chapter 3

Teml•iral Analysin on Hourly Rainfall Averages

The hourly rainfall averages were considered as observations in a time series.

Analyses on the time series were undertaken for both Phase I and II data in

experiment. Properties of such time series at selected locations in the GATE

experiment were investigated. In the following sections, the objective of the study,

the data and sampling design as well as the analyses and their results are reported.

3.1. Objective of The Study

The objectives of this study are:

• To examine the serial correlation of the hourly rainfall.

• To detect the short term cycles in the rainfall activities.

• To measure and compare the power of rainfall activities at different localities.

• To establiHii a confidence interval for the mean of hourly rainfall averages.

The serial correlation of hourly rainfall averages reveal the linear dependence

of the rainfall on the rainfall of the preceeding hours. These statistics are

closely related to the duration of rainfall activities.

High autocorrelation with large lags implies long duration and vice versa.

The serial correlation is also related to an assumption in the studies in Chapter 2:

The diurnal analysis of rainfall. It was implicitly assumed in thc: study that the

noontime and midnight rainfall rates are independent. Here we consider the corelation

between rainfalls several hours apart. While no correlation between rainfalls in

different hours do not imply independence between them, they may be considered close

-15-

It has been suspected that a two-day cycle exists in oceanic rainfall. One

objective of the study is either to confirm or disconfirm this conjecture.

The third objective is to establish a quantity to measure the "ample" of rainfall

activities. Power is a term in time series analysis designated to measure the amount

of variation it the series. Since for most of the time there is no rain. A large

value of this quantity may be a good indication of frequent rainfall or even thunder-

storm activity.

3.2. Sampling Design and Data

3.2.1. Sampling Design

In order to have unbiased results as well as to account for the variation of

rainfall in different locations, both randomly selected sample and strategically

selected sample were used in the study.

The region of the experiment was put into a coordinate system as shown in

Figure 3.2.1. Each grid is assigned a pair of integers between 0 and 100 as longti-

tude and latitude coordinates. The grid represents a square of 4 km x 4km, which is

the spatial resolution for the experiment.

There are 10,000 (100 x 100) grids in the coordinate system. Rainfall measures

were consistently observed in 8,128 of these grids, scattering around the center of

the big square. For this study, a sample of grids is taken for each phase from those

grids in which the rainfall measure is available.

For the Phase I data a randomly selected sample of 20 grids were used. For

each grid in the sample, we have a time series consisting of hourly rainfall averages.

Thus, from Phase I data of the experiment, we obtain 20 time series, each represents

the hourly rainfall averages in one of the selected grids. These 20 selected grids

are shown in Figure 3.2.1.

L

-16-

• Phase I

x Phase II

x(2,50)

(5,43)10

` x x20 (18034) (18,66)

^ (21,47)

30 (33,64) xx x (34,82)

i (34,18) (34,50)(35,55) (37,94)

40 (38,7). (44,23) (40,89)

(46,18) • (49,97)

50 (50,18)(50 ,2) x (50,34)

(50,66) (50. 8)I 0

(53,35)n^

60 (57,23)•4 (67,9) x x (64 90)

(65,2)x (66.18) (66,50) (66 82)a

I 70 (70,13)

(74 91)4

80 x x(82,34) (82,66)

i (86,28)

90

Ia

(98,50) (92,70)100 x

U 10 20 30 40 50 60 70 80 90 100

Longtitude Coordinate

Figure 3.2.1. Grids Selected for Study in Temporal Analysis.

-17-

For Phase II data. a strategically selected sample of 15 grids were used. The

sampled grids are also shown in Figure 3.2.1.

3.2.2. Data and Missing Values.

Hourly rainfall averages are computed by adding the rainfall measured in the

hour and then dividing by the number of rainfall measurements within the hour. This

Is done for each hour in the duration of Phases I and II in the CATE experiment.

The nature of the study calla for an uninterrupted sequence of hourly rainfall

averages. It was found that there are hours in both Phases I and II which do not

have rainfall measurements. This situation usually occured in the earlier stage of

both pharces. We decided to cut short the time series to accomodate our needs. For

Phase I, the actual duration usCd for this study is from the 12th hour of 182nd day

to the 17th hour of the 197th day. (The Phase I experiment lasts from the 0th hour

of the 179th day to the 17th hour of the 197th day.) (All Julian hours and days).

For Phase II the actual duration used is front 20th hour of the 214th day to the

21st hour of the 227th day. (The Phase II experiment lasts from the 0th hour of the

209th day to the 21st hour of the 227 day.)

In both Phases I and II there still exists an hour without measurement. This is

filled with method of interpolation. It is expected that the effect of this is

negligiM e.

We could have used the original rainfall measure of every quarter hour as the

observati^n (term) of the time series, instead of the hourly rainfall average. However,

there are so many missing values in the quarter-hourly measures throughout the experi-

ment in both Phase I and II that it is not advisable to use them directly.

-18-

3.3. Auto Correlation

It is well known that the hourly rainfall averages of adjacent hours are

positively correlated. That is given that there was an above average rainfall. in

a specific hour, it is very likely that there would be an above average rainfall

in the next hour, and possibly in the hour thereafter, etc. Auto-correlation

measures the correlation of the hourly rainfall of hours apart. The computation

formula are given in subsection 3.3.1 and refults reported in the subsequent sections.

3.3.1. Computation Formulas.

Let xI ' x2,...,xn denote the hourly rainfall averages at a given grid of the study.

Note that n is the number of hourly rainfall averages in the grid. Define the

sample mean and sample variance as

_n nx- 1 Ex S2-1 E (x -x) 2 .nt-1t nt-1 t

Uen the rth auto-correlation of the series is defined to be

n-rRr

n- R(r) - E (xt X) (X- x)/s 2 , r

t-1

The number r in the formula is called 1^ of the auto-correlation, and k is the

maximum number of lags for the auto-correlation to L computed. The auto-correlation

R - R(r) considered as a function of r is called an auto-corre lation func tion.

It should he pointed out that, unlike in a random sample (fro g a population of

rainfalls), xl,x2,...,xn are not independent and identically distributed. The x

denote the t-th hourly rainfall average starting at sc.,ic specific hour. The length

n of time series is 366 for series in Phase I data and 328 for series in Phase II

data.

-19-

3.3.2. Correlograms.

The correlogram of a time series is the plot of its auto-correlation function

R(r) against the lag r. Just as the histogram is studying sampling problems, the

correlogram is descriptive and informat i ve in studying the time series. It is a

visual device which is useful to perceive the linear relationship of rainfalls

several hours apart and to Identify the mechanism generating the rainfall series.

However, it in not an objective of this study for the latter, i.e. to identify the

model of the series.

In the following paragraphs, we make some observations of the correlograms for

time series in both phases I and II.

As said earlier, the length of the time series is 366 for phase I and 328

for phase II. Corresponding to these sample sizes, the approximate 95% confidence

interval for the correlation coefficient of 0 is (- .12, .12). That is, if it is

hypothesized that the correlation coefficient is zero, the chance of having the

computed coefficient to be out of interval (- .12, .12) is 5% or 1 out of 20. The

5% chance of error is a generally accepted level in practice. Thus, if the auto-

correlation of a time series at any given lag is in the interval (- .12, .1.2), we

may safely assume the value 0 (with 5% chance of being wrong.)

Examining the correlogram, it is found that there are no

uniform and specific shape for the correlograms for all time series. The shape

varies greatly from series to series. The length of these time series may be blamed

for the non-uniformity in the correlogram. In general, a timc series of length 366

or 328 should be long enough to obtain a meaningful result. But for the time series

-z0-

on hand, this may not be true, because there are too many zeros (more than 4 out of

5) for most series) for the hourly rainfall average. It is very likely that just a

few hours of heavy or even moderate amount of rainfall at a different time would

change the bhape of the correlogram dramatically.

first consider the correlograms for rainfall series in phase I.

Even though there are no uniform shape for the correlogram, nine (9) of them

share the same baric (negative) exponential shape. These are for time series at

grids (5,43), (33,64), (38,7), (40,89), (44,23), (46,18), (49,97), (50,18), and

(92,70). For most of these time series, the auto-correlations of lag 5 or higher are

small and within the interval (••.12,.12), hence may be considered zero with 95%

confidence.

Time series at grids (64,90) and (74,91) have very similar lobed exponential

shape of correlogram. This may not come as a surprise because these two gridF are

only about 40 km apart. The correlogram for the tim^ series at (21,47) have moder-

ately high auto-correlation at lags 1.0,30, and 42. This may be an indication 10

hours cycle and will be examined in next section (see 4.2).

Time series at grid (40,89) have consistently large auto-correlation i;t small to

moderate lag numbers. This is largely due to a long string of hours with persistent

rainfall on 183rd and .184th Julian dayn at the locricion.

'rile magnitudes of the auto-correlation for time series at grids (35,55), (37,94),

(38,7), (57,23), (65,62) and (67,9) do nct seem to depend on the lag numbex; the magni-

tudes do not: decrease as the lag number increase. In particular, those at (37,94),

(38,7) and (65,62) are almost all negligible at 95% confidence level.

Next, consider the correlogram in Phase II.

The correlogram forms essentially a negative exponential curve for the rainfall

averages at grids (18,34), (50,2), (50,24), (50,66), (50,98), (66,18), (66,50), (66,82)

and (82,34). Thus 9 out of 16 correlograms assume this shape. Tile auto-correlations

of first order are positive number of mrderate ma7nitude and decrease exponentially

-21-

as the order increa»es.

For son ►e of the tln ►e series, auto-correlations are significantly non-zero for

large lagm. Vor th(- series rt grid (2,50), R ( 38) is .25, and for tilt, serteti at (18,66),

R(45) is .18 while R(t) it; only .24. At ('14,18), the auto-correl tit ton-.4 raised for lairs

between 38 and 46 to tit; high as .24.

At (98,50), the correlogram behaves like it wave with peak at ings IS, 28

and 40. The power spectral density will tit , examined closely if cycles exist.

3.3.3. Median and confidence interval.

nit- motlian and its Confidence Interval of the auto-correlations are oi-italned and

reported in this section. Since the correlation does not seem to li t, sYmmett ict ► 11y

distribctted and the number of correlations avallabte for the study is limited (20 for

phc ► sv 1 and lb for pl ► r ► se 1T) we use non-parametric approach instead of the more conven-

t.ional one wbert , normal (1, all s11ape) distribution are assumed.

I.et y l * y 2 < ... vn be the auto-correlation (of tiny given lair) ordered In

increasing m-der. In 1'11aso T, it 20 where , tilt- 25111 :ind 75th percent Iles are y, and

Y16 reslivet ively and the tnedi ►tm m ° I (Y 10 i Yll)'

in p ltnse IT, it 16, the 25th and

175111 percentile:; are y

4 and y 13 1

respect ivt-ly and tit(- med inn to 2

(y 8 + Yy) .

It it; we l knourn (see e. g. 133 1, p. 181 ) that for 0 .<_ i _< 1 ^- n

PIy i < m < y l l G . F. 1 (11 1

) ( ) x ( I)'t-x r.x'i

Thus (y i , v i ) Is it % confidence for n► , the Illectiall. BY using; .^ binomial table

(e. g, ( 331) 1 t is fou ► .kt t hat for it 20 (phase 1)

Ply < m < YlbI

o .9041-.00`)9 - .9882 s .99

and for n - 16 (Phase 11)

P1 Y4 < tit < y 13 1 - .9788 a .98.

-22-

Hence (y 5 , y16) is a 99% confidence interval for the medium of the auto -correlation

with any given lag in Phase I and N' y13 ) a 98% confidence interval in Phase H.

The chances of error are 1% and 22 in Phases I and II, respectively, in saying that

the medium is in the interval.

Table 3.3.1 and Table 3.3.2 summarize the means, medians, 25th and 75th percent•

iles, ranges, lengths of confidence intervals of the auto-correlations for lags 1

through 12 in Phases I and II data, respectively. The range is defined to be the

maximum of correlations minus minimum of the same. Figure 3.3.1 and Figure 3.3.2

show the medium, 25th and 75th percentile along with confidence interval for the

medium in Phases I and 11 data respectively.

It is seen that in.hoth Figure 3.3.1 and Figure 3.3.2, the median with lag 1 is

moderate (in 40's) and there is a considerable drop between lags 1 and 2. The mediums

with lags 2 and 3 are about the same magnitude. Those with lag 5 or higher are so

small that they are negligible. In fact, their confidence intervals contain 0 for

most of them. Thus, the hypothesis that meal ian is 0 ,!ould not have been rejected.

3.3.4. Concluding Remarks.

The auto-correlation function for each time series in both Phases 1 and II

were studied. The length of series is 366 in Phase I and 328 in Phase II. Since

most (more than 80%) of the rainfall are 0, the auto-correlation function seem un-

stable in sense that the shape of the function are uite different from series to

series. However, 9 out of 20 in Phase I and 9 out of 16 in Phase ?I are essentially

negative exponential curves. For these series, the auto-correlation with lag 1

ranges from .46 to .82 in Phase I and from .34 to .68 in Phase 11. The value decrease

exponentially as the lag increase. Three of the correlation functions show scme re-

peating peaks and valleys. These series r:ay have short term cycles and will be

examined for such in the next section. At lag 1, the median of the auto-correlation

is .46 for phase I series and .41 for phase II series. At lags 2 and 3, the values

-23-

for the auto-correItit ion drop considerably for both phases I and H. The values

are not significant for lag larger than or equal to S. Thus we may say with caution

that rainfalls of S or more hours apart are uncorrelated. This statement is not

always true. There are occasions that the auto-correlation with larger lag is signs--

ficantly non-zero.

3.4. Power Spectrum

TEe power spectrum is useful tool to detect a cycle in the time series. If the

power at it frequency is large comparing to the powers of neighboring frequencies,

then we may conclude that there Iv it cycle In the time series corresponding to the

frcquvacy.

In this section, we use power spectrum estimate of hourly rainfall averages to

detect short term cycles. We start with a brief review of the methology in subsection

3.4.1. The analysh-, and results are reported i.n subsection 3.4.2.

3.4.1. Power spectrum of the t iaie series.

To farilltate the intei-pretatton of the results of s pectral analysis an over-

simplitied version of time series representation and its analysis; is presented here.

This version is intended for readers who have no background in time series analysis

and want to get hold of some conceptccal meaning of the result to be reported in the

following subsection. Readers with background in spectral analysis of time series

may skip th.Is subsection and proceed to subsection 3.4.2.

We assume that a time series x(t) may be written as

X(t) _ {{

CO

E z eixjt

J'-p0 S

whprc- i _ I a0 0 0, X -1 _ - a1

and z z^, the z's are random variables (with

0 for J # 0). The restrictions on the a 0 and z's imply that x(t) is a real-

random variable for each t, the time. The representation of x(t) says that

v'►N

1Kn

;I

k I N f- O% .-1 M O% O M to 4N Un N

1^Q+ aC ^O u1 n n M N N N. . . . . . . . . . .

n

r-1

v

CI

M

c^SiU O 10 N tlD O 00 11 ON 00 ►n N r--1u N .-r r-4 O p O O r-1 r4 r-A

C!axa^^n

G%.v n Ln O N N N N N N r-1 r-111 r-1 r i r i O O O O O O O O

CI

r-1 r--1 r--1 M r-1 r--4 r-4 r-1^Y M .-r r-1 O O O O O O O Or-1 O O O

• i i 1 i i i i Ii

H^ 0 r-1 r-A d M r-d N N M N M M

O O

1

O

1

O

I

O

1

O

1

O

1

O

1

O

1

O

1

O

1

O

1

tq i

e-1 N M -1 in %O r- co m O r-1 N(ti .-1 .-1 .-d

nrnu

CI

Ma^ANuNofaxuN

Z NCi

ri

dl

Eid+I

i-25-

HHdW

^dd•iii

^p

O•rlRIr-1}GlNOU

dWOMuaNu^or3

HNNOb

CR

6

K

11 N Cl N N O 0 0 0 0 0 0

n

1

en M 1- N 1- 1- N in rl in ^C •-1^7 ^7 cn M N N M N N r4 r-4 •"

•I f^

cn M N N N N r-1 r4 r-1 O

nM

r-i

01 HM M N Q OO m^ 1^ M O vy.i n M N N O O V C O ^ . 1

q IduNOlax^nr

t++ ' r N co 17 H . -1 .-1 N H .-i N NrA O O O O O O O O O O

^^ 1 i 1 1 1 1 I

Jr-1v

d'H !•1 en N N N M M H 00^ M O O O O O O O O O O Oy 1 1 I I 1 1 I I

uwa,auinN

u1 r-i N N N ^t

;

tn

q ! N O O O O O O O O O O O

1 1 1 1 1 1 1 I 1 1

^I

T-A N c9 It 00 O% OH9-4 C4

Fi

MM

0oMW

a

0

SO

N

dw0

w

Mr-1Y'IuuNC!aa^^n

b

Au.nNb

wt"`I

uparbMW

guu

w

ld

b

du

u^^b

u W

eaa

d

-26-

O O O O O O O O O O O O%D Ln -1 Cn N rl O •-1 N M -4

1 1 1 1 1

suoT3t?jajjoo-oany

-LZ-

GI CI.-1 •-1•d 44u u

u u

YI .0 A

Ln tnr` N

O 0 0 O

.••-1

N

ud

OvOa►

WO

041

.-1

u.

uHNa.cu

n

,Cu

N

b

w.-1

01u1^1

0)

uqNbAw

u

u,rw

N

A

M

dH

vlW

a►u

u^bbwq HuwM000

-26-

the series can be decomposed into contribution of harmonic at frequency a^ or that

the series is the superimposed sum of harmonics at frequency AJ.

It can be shown then that the power of the time series can be written as

power of x(t) _ (power of x(t) at frequency a^J.

The power of x(t) at frequency X

measure the amount of variation, or intensity, of

the series at frequency X J . In particular if z a c i , a fixed constant (i.e. not it

chance variable), then the power at X i is c 2 J . Note that lei I is the magnitude of

the term c I e ixjt in.the representation. (i.( , . le I = Ic- i IC i/jtj ). When z is a

chance variable, the power of x(t) at frequency a^ is the variance of z i , E1z21, and

the power of x(t) is the total variance at all frequencies, which is also the variance

of the ensemble x(t), t = 1,2.....N.

The power spectrum of the time series describes the distribution of the total

power of x(t) at different frequencies. Define the bower of x(t) at frequency Xi

as a function of a J . '1'lav function so defined is the power spectrum of the series

x(t). At frequency Xj , the value of this aunction measures the contribution of theis t

harmonic e i to the total power of x(t). if the contribution at the frequency a3

Is large, coanparing to thc• neighboring; frequency, then there is a cycle imbedded in

the time series x(t) at frequency X J . It is understood that the power spectrum, or

spectral analysis in general, is useful in other respects such as model building;,

prediction, filtering; and control simulation and optimization, etc. We shall restrict

our study to explore the short term cycles of the series. For detail discussion,

the Interested readers are referred to Jenkins and Watts 1371 and Koopmans 142).

3.4.2. Power spectrum estimate.

The power spectrum estimate is computed using FT-FREQ subroutine of the Inter-

national Mathematical and Statistical. libraries (IMSI.). Due to the limited accessi-

bility of the 1MSL to the author,;, only preliminary exploratory study of the analysis

-29-

is undertaken. The study is restricted to explore the short term cycles in the

data. it is noted that the length of the time series is short for detecting cycles:

366 for series in Phase I and 328 in Phase II.

In Phase I, the series at grid (21, 41) show strongly a 10 hour-cycle, at grid

(53,35) a 25 hour cycle, at grid (64,90) and (65,62) a 12.5 hour cycle, and at

(74,91) a 15 hour cycle.

In Phase I1, the series at (98,50) shows a 13 hour cycle, and a 5 hour cycle,

at (82,34) a slight 50 hour cycle, at (66,82) a 100 hour cycle, at (66,50) a slight

33 hour cycle,at (50,98) it hour cycle, at (50,2), 14, 7 and 5 hour cycles, at

(34.38) a 50 hour cycle and at (18,34) a 50 hour cycle.

From the obt;ervation in the last two paragraphs, it does not seem to have

dominant cycle prevail to all time scrics. A cycle of 12-13 hours is observed in

two of Phase I Series and one of Phase l.I series. A cycle of 50 hours (or more

likely 2 days) is observed in three series in Phase 1I.

3.5. Distribution of Totnl Power

It was observed in the last section that the power varies wildly among; series.

This is true for bath total power and power at all frequencies. To some extent,

the power measure the "amount" of rainfall activities at a specific frequency. The

total power is in fact the variance of the hourly rainfall averages (the ensemble).

Since most of the hourly rainfall averages are zero, large variance would indicate

a large of rainfall from time to time or maybe frequent thunderstorm activities.

Figures 3.5.1 and 3.5.2 show, the plot of the total powers of times series at

selected grids in Phase I and Phase II, respectively. Tables 3.5.1 and 3.5.2 list

L• he values.

It is observed that the values of the total power of the series in Phase I

vary from .002 to 1.589. This variation is dramatical considering that there are

U.

-30-

366 observations involved. The value of 1.589 occured at grid (65.62). It is

found that at this grid there was e..ceptional.ly largo: amount (23.71 cm/hour) of

rainfall at one time. At grids (37,97) and (53,35) the total powers are .187

and Al, respectively. These values are also considerably large comparing to

the rest.

The total power of series in Phase II ranges from .002 to .050. The largest

value is 25 times of the smallest values. The ratio is moderate if one notes that

the corresponding ratio in Phase I is 9,349. Thus the rainfall activities were

somewhat similar among all localities during Phase II of the experiment while they

were dramaticnlly different during Phase I.

-31-.187

a

r 60vw00

70av0

80.a

1.000

90

40

20

50

10

30

0

117

10 20 30 40 50 60 - 70

Longtitude Coordinate

80

90 100

Figure 3.5.1. Total Power of Hourly Rainfall Averages in Phase I at Selected Grids.

0

10

20

30

40

70 80 90 1001.0 20 30

90

100

0

80

a^

. 50

Hg 60

0^v0

70

40 50 60

Longtitude Coordinate

.050

-32—

Figure 3.5.2. Total Power of Hourly Rainfall Averages in Phase II atSelected Grids.

1u

.04 r.N o3

aui . 02NF .01

.00

10 20 30 40 50 60 70 80 90 100

Longtitude Coordinate

1. The Mean of Hourly Rainfall and its 95% Confidence Interval inPhase I at Selected Grids.

0

10

20

70

80

90

100 --

0

irc

-33-

30

ub

40b00

50a^v

60I

1w

u

0%

-34-

.04

,-^ .03H

.0211

. 01,4a .00

I

1(

2(

3(

4(

0 50uCO.a

N 60

00Ub 70

NM

a 80

90

100 --+— '0 10 20 30 40 50 60 70 80 90 100

Longtitude Coordinate

Figure 3.6 . 2. The Mean of Hourly Rainfall and its 95% Confidence Intervalin Phase II at Selected Grids.

-39-

V3.6. Rainfall Average

The mean of the hourly rainfall averages and its 952 confidence interval were

listed in Tables 3.5.1 and 3.5.2. Although the mean of averages should be closer

to normal distribution than the mean of raw rainfall, the distribution of the mean

of averages is far from being normal. In addition, the hourly rainfall averages

are not independent either, as was observed in section 3 of this chapter. There-

fore Tables 3.5.1 and 3.5.2 should be viewed with cautions.

Figures 3.6.1 and 3.6.2 display the mean, along with its 95% confidence

interval, of the hourly nverageK, according to the location of obKervations. It

Is interesting to note that the rainfall dit.tribut.ion in Phase 11 is very even among

all locations. The in.igqitude of the mein and the length of its confidence interval

are comparable. They vary vary little from location to location. But in Phase l•

the Mary is completely different. The value of the mean ranges from .0014 cm/hr.

to .1320 em/hr; the latter iK 94 times of the former. The length of the confidence

interval varies dramatically from location 1,-) location ale:o.

I.

-36-

N

O/

F1+

NQOcoN

r-1td

W

^^1py

PR

D+rl

wO

Olu

N

bcoo

M 00 O ^T ^D ^D ^D M N O %D V1 QT r l% V1 1- 00 %T 00 %nF-1 N C% 0 N rl .-1 1-1 r-1 1- O M9 O r-1 O N m r- 0% %.Dr4 O M M O O 0 r-i r4 .t .-1 00 rl O% 0 O O 00 r-4 NO 0 O O -4 O O O O O O O O O N O O O O O

•w w w w w w w w w w w w w w w w w w w wON in %D 0 -T N N m O -T N QT in m u1 N C\ ^ 00 r-N ^ -4 %D -7 rl r-1 r-1 N O M rl M O M -4 O %D00 ON

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-38-

Chapter 4: Results on Diurnal Analysis

4.1 Introduction.

In this part of our report, the results of the Kolmogorov-Smirnov, Wileoxon

Signed Rank and the Chi-square Goodness of fit tests are reported as they were

applied to the noon and midnight average rainfall rates at each grid (4 square

Km) of the 160,000 square Km array of GATE.

Special programs were written to perform the analysis utilizing standard

subprograms from S.S.P. One program prints Lite results of both tests in a tabular

form while two other programs were designed to present the output of the results

pictorially as a 100 x 100 cartesian array of alpha-numeric symbols, each repre-

senting the outcome of the statistical procedures indiented above. The graphical.

approach makes it easy to detect clustering; and/or perioetic behavior in any region

of the array.

Data condi oned on the event of rain for noon or midnight were combined

for the first two phases of GATE to produce a temporal resolution of 35 days at

each point of the 100 x 100 array. Noon and midnight rate vectors of length 35

were generated.

Fifteen minute instantaneous radar precipitation data for phases one and two

were used in the study. To minimize the effects of missing data, there was 107

records used from phase one and 188 used from phase two. The noon (12th hour)

and midnight (o th hour) rainfall rates were obtained by taking averages of all

data Moth one hour before and one hour after noon and midnight. Thus the neighbor-

hood about both noon and midnight was a one hour 'r.l.dius.

ri-39-

4.2 Correlation

In Chapter 3. a detailed analysis of Lite time-series approach to the question

of temporal correlations was presented. Recall that a major ( implicit) assumption

of the Kolmogorov-Smirnov and the Wilcoxon Signed Rank tchts is that the noon and

midnight data are independent. A fundamental conclusion from the temporal anal sus

jll. 3.3.1 and 3.3.^Ij__Spames 26 and 27 is that with a 98% confidence level. rainfall

5 or more a ar^ . are uncorrelated.

It is well known that uncorrelated data need not be independent, however from

a practical point of view, this is all that can be expected. In it more technical

vain, there can be no difference between the two concepts unless we are a priori

given Lite joint distribution function, which is a part of the unknown information

In this study.

4.3 Kolmogo -oy-Smirnov 'rest

A major objective was to determine if there is any mathematical difference

in the empirical. distribution of rainfall at noun verses midnight. This is

equivalent to our test of the null hypothesis that noon and midnight distribution~

are the same verses the alternate hyoothesis that they are different.

In this study we surveyed 10,000 grid points; 1,872 were excluded because

they had too few rainfall events for analysis; 1,742 had noon and midnight data

distributed in such a way that we could not reach any conclusion. I n 5,425 g rid

areas, we found that the nu ll hypothesis could tie accep ted and in 960 grid areas;

we found tint the null hypothesis must be rdected and the alterrinte hypothesis

accepted.

Figure 4.2.1 displays the results pictorially in a 100 x 100 array. The

letter D indicates that there is a significant difference between noon and midnight

data, (reject hull hypothesis) A blank indicates that there is no difference. It

is easy to see that zero rainfall rates in both phases dominate the western border

/law

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-4o-

of the array. Heavy clustering of U's can be detected in the north eastern and

south western region of the array. We expect that thin is duo to the occurance

of heavy rain in these regions throughout the thirty five day period of phase 1

and 2. This also i.mLlies thnt_there t^ hea periodic spatial distribution of

areas where there is a_definit:e diurnal rainfall variation surrounded by regions

where there is no variation.

4.4 The Wilcoxon Signed Rank Test

The Wilcoxon test, allows us to test the null hypothesis that the mean distri-

bution of rainfall rata for noon and midnight are the same verses the alternative

hypothesise that one is greater than the other.

In this study, 1,872 grid areas were excluded because they had too few events for

analysis; for 1,743 we could reach no conclusion. In 4,890 &rid areas we found that

the null IyLc^l'hesis could be accc^Lted while in 1,495 of the gridaic^;ts we found that

the null ji)a^othesis must be rejected and the al ternate hypothesis accepted.

ThIs result it expected since the Wilcoxon test is less conservative than the

Kolmogorov-5mirnov test. It is possible for the distributions of rainfall at noon

and midnight to be the same and yet the means are different.

Figure 4.3.1 displays the results pictorially in a 100 x 100 array. The letter

1. indicates that noon rainfall is significantly Less than midnight rainfall, the

letter G indicates that noon rainfall is significantly greater than midnight and a

blank indicates no difference.

The results of Wilcoxon signed-rank test match those of Kolmogorov-Smirnov

test, as far as whether there is difference in the rainfall distribution at noon

and in the midnight is concerned. from Figure 4.4.1, it is observed that the rain-

fall activities at noon and in the midnight during the experiment period are rare

in the western and north-western parts of the area, as indicated by "?". The noon

rainfall is greater than the midnight rainfall in the southern and northeastern

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•_ ,Fie ^••^ i6•. 7?1777777!7!7 y7 G4 G 4GG1•G" ^1 Gam. C^{jf (^,.___*fit-•ii^••6••••ii.•i66ei••'► 9?7?7 r,7 •7^J/7 4a G 4C.G 4 G G G C.G 4 •6.6••6.•••••.•••x•.••NN N 16.1•••••••••16••71 71717i?77G 777E 4au 44{iG6GGG GGGG •666.6••6666• N1••6 H • ♦••••••••••••• • 6••••••••••97997 ► 9799 GGTG6GGGG6G GGGGGGGG ••••••••••••••••••....•••••••l616.61.66••166^• •6N 66 y797j77G ,yGGGGG.G G.._^ CL4^t GG^j •••6.66•••••••••6••••6.6•••••16 66. •6110^ i•• ♦••6 ^6 •• •6•• ^7717b 3,GaGG44GG44G4 G_Y(.GQ^_ •••^1.11• ►••^6^^661066••61••!._..♦166.1i•1•^i• • •6^.•.••^• •_^ 666.777b79GGG77WyGGG_¢ GGt^yGG •6p•••06666•tt^.•666'06.61600i0••••••1. 16 ••e6^o. 6•a 6. 66••7r•••.^. 67 Gu •..0 4Gfti^j§GGG4GG •••••6•••••6•••6.60•••6.66••66•••6 ^,•6NHH:11^••►•s•^; •1.6 N616M •••1 H 6•I^G6YY bGfr4h GG4G G •66.66.6 ►6•.6.61 N•1N H161•N61611.6•

• - •

NUUN < 416T47 70 NIDNI4M ( NJUN(6) NO MTOTAL

•to Lou ^e10 I/1s keys 10000

Fig. 4.4.1

•t!

G

••••••••777777777 {. Guu ? 44u^•as► •aT•f7lJ7i?7 T1 ►7• ^ 4G 744 G• 6i•6i1•it 1 7771111711 •. 77 J71 '1 ot•GG GG••61.16••1 t`i)79rT?7?iG 9 7 77 7GG07M.4 G Ltn;.•••61.6116 T17777T9/171711/9? 7117 ?GGGGGGGG GGGG♦11111.1N• 6777717T17T ??94177777 G G_r-3 q 00^.G....^♦••^.•••.••7i 777J?? 77 77?7717717TG 6GoG4G000aGG6^••••ii••i••!•7J 117 7177?9777TJr7j ^: U7 ua:74 y{rObY:^•i6^Ti•6T6•^^

.iJ 71777?G771'11 )/)711 ( rG V44GyvG^u4,

0600

GGGG l 9 •.•••••.•••

-41-

parts, as indicated by "C". The places where midnight rainfall is greater ere

scattering..

4.5 Chi-Square Test

The c.hi-dquare test is used to provide a further check on the results of

the Kolmogorov-Smirnov and Wilcoxon signed rank tests. This test allows us to

determine whether or not the percentages used for analysis using the other two

statistics were actually preserved in the results of the analysis. In this case

the X2-test is defined by: (see Chapter 2)

2 k (it_nP^ )X k t' nP3-1 J

k

where it F n n is the numlirr of grids for which the J-th conclusion was made;n= l j

. Pi is the probability of making the n-th conclusion.

We use an a - .05 for the level of significance for each experiment.

The X2 test, applied to the K-S experiment.

In the K-S test we have the following, information:

k-2 j-1,2.nl - 5425 P1 - .90 P2 - .10

n3 C 960

n = n + n2 - 6385

c .05,2 - 3.84

So2 - (5425 - 6385 (.90 ) 2 + (960 - 6385 ( .10))2

X2 6385 (.90) 6385 (.10)

- (321.5 ) 2 + (321.5)2 = 17.99 + 161.88 - 179.87 > 3.845746.5 638.5

hence, the percentages of 80 and 20 do not hold at the 5% level of significance

and thus the null hypothesis 11 0 : F1 F2 is not true end must be rejected.

f

-42-

The X 2Teet applied to the Wilcoxon lixreriment.

In the Wilcoxon experiment we have the following information:

k -3 j-1,2,3.3

n 430 n2 - 4890 n3 - 1065 n - E ni - 6385

1

P 1.10 P 2 - .80 P3 - .10

c .05,3 - 5.99

2 - X 430 - 6385).10 2 + 4890 - ( 63851 .80)) 2 + ( 1065 - 6485 (.10)]2X 3 6385 (.10) 6385 (.80) 6385 (.10)

- 208.5 2 + ( - 218jj + ( 426.5 2L - 68.03 + 9.30 + 284.67 - 362.00 > 5.99639 5108 639

Thus, the percentages of 10, 80; and 1.02 do not hold at the 5% level of significance

and hence the null hypothesis H0: F - F2 is not true and must be rejected.

4.6. Summary on Diurnal Analyses.

Two non-pa rametric tests were used to detect if there is a diurnal variation

in oceanic rainfall using CATE data. Nan-parametric methods are choson because of

their model free characteristic.

The test was undertaken at grid points where data are available. Of the 10,000

grid points in the study area, 1872 were excluded because data are not available;

these points are around the boundary of the square. In addition, there are 1743

grid points where no conclusion was obtained due to insufficient frequency of rain

in the midnight and at noon during the experiment period. The analysis was done

at the rest of 6385 grid points.

The Kolmogorov-Smirnov test was used to test whether there is a difference in

the rainfall distributions between noon and midnight. At one out of 10 chance of

error, it was found that out of 6385 grid points, the rainfall distributions at

-43-

noon and at midnight are different in 960 grid points and are same at 5,4125 grid

points, which are 15.0 and 85.0%, respectively. If there were no difference in

the rainfall distributions, these percentages would have been 10 and 90% respec-

tively. A uhi-square teat at one out of 20 chance of error shows that this split of

percentage was not followed. There are more grid points where the assertion of

difference is made than expected. Thus, overall the rainfall distributions at

noon and at midnight are different. This concludion could have been made at

practically any significant level.

The Wilcoxon signed-rank test was used to test if the noon rainfall is less

than, equal to, or greater than the midnight rainfall. It was found that, at one

out of 10 chance of error, the numbers of grid points fall in these three categories

are, respectively, 430, 4,890, and 1,065. In terms of percentage, they are 6.73,

76.5, and 16.68%, as compared to 10, 80, and 1.0%, respectively, which are expected

if the assumption of no difference had been true. At one out of 20 chance of error,

the chi-square test concluded that the midnight rainfall is not equal to the noon

rainfall; the noon rainfall. is convincingly greater than the midnight rainfall.

Thin; conclusion could have been made at practically any significantly level.

From the analysis it was also found that the Wilcoxon signed-rank test is more

sensitive than the Kolmogorov-Smirnov test in detecting the difference in the mid-

night and noon rainfalls. This is to be expected by the nature of these two tests.

s

r_tl

x

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