THE USE OF SIMULATION TECHNIQUES FOR SEQUENTAL GENERATION OF SHORT-SIZED RAINFALL DATA AND ITS APPLICATION IN THE ESTIMATION OF DESIGN FLOOD

H.D.Sharma*, Dr.A.P.Bhattacharya** and S.R.Jindal;t**

ABSTRACT

The studies based on rainfall runoff data are considerably vi- tiated in the event of inadequate data, as the reliability o€ the probabilities of occurrence is reduced, It is, however, possible to get over the lacuna of inadequacy of data by creating bigger-sized artificial series of rainfall. The use of such a series gives grea- ter precision in the estimations or projections based on expected m a ximum rainfall with specified levels of occurrence and also provides better insight into possible patterns of behaviour. This is done by the procedure of sequential generation fo data by the use of simula- tion techniques. Making use of these, the technique has been applied for generating rainfall series of 100 nombers on the basis of recor- ded rainfall data for a period of ten years, The generated rainfall series was compared with the historical data which showed strong co- rrelation.

flood for Yamuna river at Okhla (Delhi). These results have been used for the estimation of design

Les procédés qui consistent à déduire les écoulements des p r 5 cipitations voient leur efficacité considérablement diminuée lorsque les observations concecnant celles-ci sont insuffisantes, par suite de l'imprécision qui regne alors sur l'estimation des probabilités de ces précipitations. On peut essayer de tourner la difficulté en créant artificiellement de longues séries d'observations pluviométri ques. L'utilisation de telles séries conduit a une meilleure préci- sion des estimations ou des prédéterminations basées sur la pluie m a ximale attendue avec une probabilité donnée; elle permet aussi une meilleure vue suc les schémas possibles du comportement des précipi- tations. On procede par génération séquentielle des données, en uti- lisant les techniques de simulation, On donne comme exemple la cons- titution d'une série de 100 ans à partir d'une période de 10 ans d'observations. La séries engendrée , comparée avec la sérìe histori que, met en 'evidence une forte corrélation.

de projet à Okhla, sur le fleuve Yamuna [Delhi). Ces résultats ont étd utilisés pour l'estimation d'une crue

* Director , Irrigation Research Institute , Roorkee, U .P,

$:* Research Officer , Basic Research Division, Irrigation Research Institute, Roorkee, U.P.

;'<*st Statistical Officer , Basic Research Division , Irrigation Research Institute , Roorkee , U ,P.

420

i. INTRODUCTION

1.1 In all iqr-gothetical investigations, particularly in the estimation

of design flood of river basins, it 1s essential to have an idea of the

distribution of rainfall as also the relationship between rainfall a d

runoff. This is, however, not always possible in case of small sized

data, extending over 8ay 10 to 20 years as is usually met vit

oractice, as these may not be representative of the vorst possible

conditions prevaillng in the catchment. On account of such shortcomings,

it is likely that the findings based thereon may not be realistic. This

difficulty may be overcome by resorting to the technique of sequential

generation with the aid of which it is possible to artificially create

n 3

larger sized data series.

2. CONCWT OF W?UENTIAL GEWERI'EION

2.1 sequential generation is a statistical process usiag Monte Carlo

methods to produce a random sequence of hydrologic or any other data on the basis of a stochastic model for the hydrologic process. Monte

Carlo method is an experimental or m e r i c a l probability method used

for the statistical sampling of random variables. The sequence so

generated makes possible detailed study of the performance of various

hydrologic events, thus helping the development of well balanced hydo

rologic designs.

2.2 Unless the record is too meagre to be considered as a represento-

tive sample, the statistical parameters derived from It should enable

the hydrologist to construct a suitable model that wlll generate hydrologic information for as long a period of time as desired. Bnce

the statistical parameters of the population of the generated data

are necessarily the same as those estimated from the bistorical date,

the new information is limited that are inherent in the observed record.

errors of measurement and sampling

4 21

2.3 The procedure of sampling by shuffling; cards which waa among the

srllest techniques can be simplified by the use of random number tables.

naugh random number tables are available as punched cards, with Increase

3g use of digital cornputor, mathematical methods for generating pseu-

F n d o m numbers within the computing machine have been developed'in order

eliminate the need for extensive input of random numbers.

2.4

3 the basis of required statistical levels of errors and confidence,

Lthough the optimal size may be determined more realistically by compar-

the cost of the Increased sample size with the benefits of the corres-

The size of the hydrological data to be generated may be estimated

mding increase in accuracy, provided that the benefit and cost data

:e available.

, ANàLYSIS OF RAINFALL QATA

3.1

?corded at New Delhi for a period of 10 years from 1956 to 1965. They

ive been arranged in such e manner that the storm starts with the first

burly rainfall and ends at the 6th hourly rainfall, although in reality

Le arrangement may be vitiated in some cases by the occurrence of a

Bizzle before the recording of the main _portion of the storm or by beaks within the duration of the storm. The recorded data may be seen

I Table I.

The rainfall data analyse8 herein pertain to 6 hour annual storms

FORMULBTION OF THE MATHEMûTICAL MODEL

:.1 To develop a suitable model to represent the time degendent

ndom process of the hourly rginfalls, the following non-stationary

rkov-chain niodel(l) was found to be consistently satisfactory.

.) Ven Te Chow, Handbook of Applied Hydrology, pp. 8-93, McGraw Hill Book Co.

422

......... (1) where xt x the hourly rainggll of any one of B annual

storms at the t hour,

xt-1 z the hgurly rainfall at the preceding or the (t-i) h hour,

t = time in hour ranging from 1 to m,

r = Markov Chain Coefficient,

6~ = random component due to hourly rainfall xt ,

For the first hour when t = 1, the trend component r Xt,l become

zero and X1 may be taken to be equal to €1 . The Markov Chah Coeffi- cient r and the random component €G may be determined from the give1

rainfall data by the method of least squares by fitting a straight

line between Xt and Xt-1.

4.2 For the rainfall data recorded at New Delhi Station, the storm

duration m = 6 hours and number of annual storms, Ns10. The distribi

tion parameters, mean and standard deviation of the historical rain.

fall data were determined for each hour and are given in column 2 ai

3 of Table II. The values of the random component et and the Markov Chain Coefficient r were worked out by the method of least squares

and are shown in columns 4 end 5 in Table II.

4.3 In the present analysis based on sequential generation, the

oractice followed has been to generate 100 pseudo-random numbers fo:

uniform distribution of the first hourly rainfall by I.B.M. Compute:

1401, whose programme is given in igpendix I. These 100 generated

random mmbers of a uniform distribution have been taken as first

hourly rainfalls of 100 storms and have been utilized for computing

100 second hourly rainfalls by the Markov-chain model given in

equation (1).

423

4.4 The rainfall data have been generated for each successive hour on

the basis of the rainfall in tlx? previous hour according to the Markov

chain model formulated. Knowing the Markov chain coefficient r and

random component 6,for the second hour derived from the historical data

(vide Table II) a random series of 100 second hourly rainfalls can be comouted by means of equation (i). These 100 generated second hourly

rainfall were then utilized to compute 100 third hourly rainfalls with

the help of Markov chain coefficient r and random component (vide

Table II) by using equation (i). This procedure has been repeated for

successive hourly rainfalls until serles of 100 hourly rainfalls for

all the six hours were generated. The involved operations were carried

out on IBM computer, 1620 as per programe given in Appendix II. The

sequentially generated data has been shown in Table III.

3

4.5 The cumulative probability function P(x) of the variate X may be

obtained by the following equation;

.........o (2)

where ,.ho 5 Y & ,h~ fiois the lower limit of the variate X which may be assumed to be zero

an8 is the upper limit of variaue X.

4.5.1 In the present analysis, the total hourly rainfall of annual

storms have been worked out by adding all the six hourly rainfalls for

each storm of historical data as well as generated data as per column 8

of Tables I and III respectively. The cumulative probability per cents have been evaluated by the use of equation (a for ten storms of the historical data as ?er column (9) of Table I as also for 100 storms of

the generated data as per column (9) of Table III.

424

5. EsTIYVìTION OF DESIGN FLOOD WITH THE AID OF GENERATED RAINFALL S ~ I I

It is possible to derive a series of runoffs from the generated 5.1

rainfall series provided that the relationship between rainfall and

off for a particular basin is known. In the present Case, in which

sequential generation techniques have been applied for only on rainfall

station in the Yamuna catchment, vie. New Delhi and for w N c h 110 rain-

fall-runoff relationship was available, an assum3tion has been made tha

surface runoff from rain storm is 80 per cent of rainfall during the

period of high floods when most of the catchment is saturated and in-

filtration losses are of low order. Based on this preamble, a series

of runoffs may be assumed to be generated. The abovezentioned series

can be utilised to compute the peak floods with the help of unit

hydrograph developed at the gauge site and other methods.

1

5.2 The series of 100 peak floods comguted for the river Yamuna at

Okhla (catchment area = 6811 sq. Kms.) shown in column 10 of Table

III has been used to derive the following stochastic model on the

Dasis of princinles of stochastic hydrology reported earlier for

the estimation of design

wnere yo is the design flood and Tk is the recurrence interval.

5.3 From Mg. 4 based on above, the design flood with a recurrence

interval of 500 years works out to 7794.5 cumec for the Yamuna river

at Okhla (Delhi). It may however be 2ointed out that this should be

talen to be more as an illustration of the application of the techn-

ique of sequential generation for the estimation of the design flood

in view of the limitations of the rainfall data for the entire catch-

ent and - ilitv of a rainfall ru ela t i o =hi D (2) ,,ttZharya>A8P., Jindal, S.R. and RamJ%ff :Estimation of Design--

Flood of the Ganga Fiver by processes of Stochastic hydrology", U. 2. Annual Besearch rieport, 1967 (Technical Memorandum No. 37) .

425

5. DISCUSSION OF RLiSULTS

6.1 Figure 1 gives a comparison between the worst possible raiaall

;tarm of the historical data and the generated series an the basis of

I gra3hical plot between time in hours and hourly rainfall. It is

.ndicated that there is close Conformity for the entire storm dura-

,ion comorising six hours.

6.2

Iata with respect to cumulative probability distributian of rainfall at

he third hour, at which the peak rainfall was rècorded in the observa-

ional as well as seqtientially generated data as per Figure 2. Close

Gra^hical comparison has been made bbtwecn historical and generated

ionformity is indicated between the two distributions.

6.3 similar comlarison has also been made for the two series for total

ix-hourly rainfall for the annual storms as shown in Figure 3. Close

ionformitg is observed in this case as well, both for ehird hourly rain-

'all and total six-hourly rainfall, which provides added evidence regard-

ng the representativeness of the sequentially generated series.

6.4 m o m the generated rainfall series, it has been ,possible to derive cm?

runoff serles which has been utilised toda series of 100 peak dischar-

es. The latter orovide the background for the derivation of a stochastic

ode1 wherefrom a hypothetical 500-year design flood for the Yamuna

iver at okhla (Delhi) may be estimated.

COI;CLU~IOMS

7.1

he size of the historical data, particularly in such investigations

herein this may be a limiting factor for analytical studies.

7.2

y El finite duration discrete non-stationary process that is ameneble to

The technique of sequentiaï generation may be adopted for increasing

storm rainfall is a time dependent raridom series and may be treated

athematical formlation and analysis. For rainfall at New Delhi, the istorical data of hourly rainfall in the annual storm has been regresented

y nan-stationary Markov-chain model, the data consisting of ten .six-

426

hourly storms.

7.3 A compari n of the historical and generat LI 100 y ar data, both

for third hourly rainfall and total six hourly rainfall, shows that the

sequentially generated series is fairly representative of the charactei

istics of the historical data.

7.4 The generated hydrologic series of rainf'all has been utilised to

estimate the design flood of the Yamuna river at Okhla(De1hi) with a

recurrence, interval of 500 years.

The authors wish to acknowledge the useful help extended by

Messrs Ramjeet and D.C.Mltta1 in the analysis and computational work.

APPENDIX I

Fortran program for the generation of PseudÕrandom numbers in Uniform Mctribution. 4

SE Q spm FORTUN STATENE2iT

C GEN-RATION OF 100 ?SEUDORANDUM Nuz.[BERS IN UNIFORM DI SJXtBUTION

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10 IALFA- 10**17 -C 3 IRN1 = 10*(10**19-1) -b 7 8% 0.0 N = l

91 READ 95,B 95 FC-WAT (F4.2)

Do 2 I = 1,100 IRN IRNl*IALFA R S N = IRN RSJ!N= RUN* 10.0**(-20) SN = (B-A)* RSTN .) A R = N + 1 IRNI= Im 2RIhT loo, SW

100 FORMAT (2E 16 e 8) 2 COICTIhUE

3 GO TO 91 IF (SENSE ShkTCS O) 92,3

92 STOP 555 END

10 100

42 7

APPENDIX II

Fortran program for the generation of i00 slx hourly rainfall storms for New Delhi Station by Markov-chain Model.

DIMEESIONS X(iOO),A(lOO) ,B(100) ,C(iOO) ,D(iOO) ,G(100), DIEIEKSI ONS Y ( 100 ,V ( 100 )

READ 100, (X(I), I = 1,100) FORMAT (~oF7.4) cupi = 0.0 SUMA = 0.0 SUMB = 0.0 SUlC = 0.0 m!D = 0.0 SUMG = 0.0 smfl = 0.0 DO 200 I = 1,100 b(1)' 0.973 -k 1*551*X(I) B(1) 2 15.023 46.694*A(I) C(I) = 12.297- 0*036*B(I) D(1) z L.871 + O. 106*C(I) G(1) = 0.138 4 0.400*D(I) Y(1) = X(1) + A(1) t B U ) t C(l) -k D(I)S G(1) V (I ) = 664.9 *Y (I ) suMx= s w + X(1) SUMA = SUMA t A(1) SüMl3 = SUMB f B(I) swc SUMC -t C(I) SUMD = SUMD + D(1) SUMG = SUIVIG + G(1) SUMV = smn +V(I) PUNCH300, X(I ,A (I 1 , B(I 1 , C (I ) , D( I , G (I ,Y (1 PUNCH350 ,V (I )

350 FORMi1T (FS0.4) 300 FORMAT (7F10.4) 200 CGEJTI NUE

400 FORMAT (6F12.4) PUNCH 400, SUMX, S W , SUME , SUMC, SUMD , SUMG ,"UI\+CH 500,SUMV

STO? ENI)

500 FORMAT (F 25.4)

428

TABLE I

Historical hourly rainfall data for annual storms for New Delhi Station

1 2 3 4 5 6 7

9 10

a

20; 7.56 13.9.57 29-90 58 6.9.59 5.10.69 24.9.61 20.9062 8.8.63 14.7.64 2.9.65

0.25 1. 80 2.00 ‘O. 10 O. # o. 10 0.30 1.50 O. 40 1.90

O. 50 2.10 3.30 4.60 O. 80 0.40 O. 50 l e 80 1.50 7.80

19.30 22. so 42.00 54.20 19.10 8.50 21.50 new 30.00 61.20

14.75 8.10 13. so 3.50 9000 5.50 9.10 22.00 17.20 9.30

4.06 1.02

11.90 5.60 0.50 0.20 2.00 0.40 2.40 0.08

1.80 1.50 1.80 0.90 0.70 0.20

5030 3060

0.10 0.10

39.88 43.40 78.30 63.10 31.70 16.98 31.60 56. So 51.80 81.10

49.2 53.5 96.5 770 8 39.1 20.9 39.0 69.7 63.9 100 o

TABLE II

fa Parameters of the Markov-Chain Model(hour1y rainfall of annual storms

of New mihi ‘Sation.

Time í t 1 Mean (mm/hour) Stendard Random Markov-Chai n in hours devi at i on component coefficient

(mmhour 1 et r II

1 2 3 4 b

o. 875 O 8122 ..I - 1

2 2.330 2.3522 0.973 l o 551

3 30.620 16o7600 15.023 6.694

4

5

11.195 5.8190 12.297 -0 036

3.056 3.4928 1.871 0.106

6 l b 360 1.8311 0. 138 0.400

429

i

I Co

431

4 3 2

FIG 1 - DISTRIBUTION O F WORST R A I N F A L L S T O R M FOR N E W DELHI

10 30 50 80 90 95 99 99.8 999! CU M U L A T IV E PR OB AB ILtTY PERCE NT

F IG.2 - CUMLILATIVE PROBABILITY DISTRIBUTION OF THIFiC 4 0 U R L Y R A I N F A L L IN A N N U A L S T O R M S

4 3 3

E E

O I -I 4 I-- O 40 60 80 90 95 98 99 99.8 99.99 k- 20

CU M U LAT IVE PROBABIL I TY PERCE NT

RECURRENCE INTERVAL (Tk) IN Y E A R S

FIG.4- STOCHASTIC M O D E L F O R ESTIMATION O F DESIGN F L O O D (Yo1 F R O M R E C U R R E N C E INTERVAL ( TK)