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
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