SO-260

Research Nat@

1960

Rainfall Data Simulation

T. L. Rogerson

SUMMARY

A simple simulation model to predict rainfall

for individual storms in central Arkansas is

described. Output includes frequency distribution tables for days between storms and for

storm size classes; a storm summary by day

number (January 1 = 1 and December 31 =

365) and rainfall amount; and an annual storm

summary that includes monthly values for rainfall and number of storms. The model should

be adaptable to other areas where 10 or more

years of rainfall records are available.

Additional keywords: Computer models,

model studies, simulation analysis.

INTRODUCTION

In the past few years hydrologic models have

been developed to simulate various components

of the water balance system. Some have simulated rainfall, but most have used actual rainfall

as input to their hydrologic models. A few have

simulated the complete water balance system.

The objectives of this study were to (1) develop

a model to simulate rainfall for individual storms,

(2) develop the model so that it could be used

as a submodel in a water balance system,

and (3) compare the model output with actual

rainfall records.

The model developed herein uses rainfall probabilities and parametric functions to simulate

rainfall. It was classified as stochastic-empirical

according to Clarke’s (1973) definitions: it is

stochastic in that the rainfall variable is random

and has distributions in probability, and it is

empirical in that it is based on observed data. It

should be useful as a submodel for systems

(such as soil water, runoff, tree growth, sediment

and water balance) that need rainfall as input,

and should be adaptable to any area where 10

or more years of rainfall records are available.

MODEL DEVELOPMENT

The model was written in GPSS, an IBM

discrete simulation language (IBM 1971, IBM

1973) available on IBM and Univac computers.

The language offers great flexibility and allows

model modification without extensive reprograming. The model was programed and used

on the IBM 370-l 55 computer at the University

of Arkansas.

Rainfall models can be developed for individual storms, daily amounts or amounts over

some other selected time interval. The model

reported here was developed from individual

storms. An individual storm was defined as a

period of rain followed by at least 12 hours

without rain. The time unit selected for this

Southern Forest Experiment Station/T-l0210 U.S. Postal Services Bldg., 701 Loyola Avenue, New Orleans, La. 70113

Forest Service, U.S. Department of Agriculture.

Serving Alabama, Arkansas, Louisiana, Mississippi, Eastern Oklahoma, Tennessee, Eastern Texas.
simulation was 1 day, so only one storm per

day could occur. For this model, it was assumed

that whatever happens one day (storm or no

storm) was independent of the previous day. The

model was developed from 13 years of rainfall

data from the Alum Creek Experimental Forest

in central Arkansas. Because the model only

allows one storm per day, if two observed

storms occurred on the same day the storm

nearest to another day was said to occur on that

day. Likewise, an observed storm that occurred

on two or more days was said to occur on

the day on which most of the rain occurred.

The model relies mainiy on four functions that

determine (1) days to next storm and (2) amount

of rainfall. The model can be run for 1 year

or as many years as desired. A program listing

of the model is given in the Appendix.

On the first day of simulation the model calculates days to the first storm (fig. 1). The number

of days to the first storm is the integer of

the equation: DBSD x ADBSM + 0.5 where

days between storms (DBSD) and average

number of days between storms by months

(ADBSM) are continuous and discrete functions’

respectively. The DBSD function was developed

from a cumulative frequency distribution for

days to next storm (fig. 2). During the 1961-

1973 period, days between storms averaged

4.8203 days and ranged from 1 to 30 days.

The monthly means for number of days to next

storm varied by such a degree (fig. 3) that the

DBSD function alone could not adequately

define days between storms throughout the

year. Therefore, to obtain monthly distributions

of days to next storm, I developed the ADBSM

function from day number at the end of each

month (Januav= 31 and December = 365) and

average number of days between storms for

that month (fig. 3). Then I adjusted the DBSD

function by dividing the upper limit of each

days to next storm class by the average days

between storms for the 13 years of data: for

example, for day one, 1.499/4.8203 = 0.3110.

The lower limit of the DBSD function was set

1 With a continuous function, when an argument value lies

between two successive Xi points the program interpolates

linearly to obtain a dependent funciton value between the two

associated Yi points. With a discrete function, all argument

values between Xi and Xi-1 have the same function value.

Figure 1 .--Rainfall simulation flow chart.

so the month with the fewest average days

between storms (April) could produce a storm

on the following day. In determining days to

next storm, the program generates a uniformly

distributed six-digit fractional random number

between 0 and 1. This number determines the

value from the DBSD function. The day number

determines the value from the ADBSM function.

Days to next storm, which is distributed about

the monthly mean, is the integer of the product

of these two values and a rounding factor of 0.5.

Then, if days to next storm = 7? is not

one, it is reduced by one (fig. 1). The simulation then proceeds to day end, day + 7,

and days to next storm = 7?. When the number

of days to next storm = I? equals one, the

rainfall amount is calculated from the equation:

RAIN x ASSM. where storm size (RAIN) and

average storm size by months (ASSM) are

continuous and discrete functions respectively.

The RAIN function was developed from a cumulative frequency distribution of O.lO-inch storm

size classes (fig. 4). Rainfall ranged from 0.01

to 8.30 inches and averaged 0.726 inches during

the 1961-1973 period. Because of the large

variance in mean monthly storm size (fig. 5)

T. L. Rogerson is Research Forester, Southern Forest Experiment Station, Forest Service-USDA Fayetteville, Arkansas.

2
the RAIN function could not adequately define

storm size in months that were much different

than the mean storm size of 0.726 inches.

Therefore, to obtain monthly distributions of

storm size, I developed the ASSM function

from day number at the end of each month

and average storm size for that month (fig. 5).

Then I adjusted the RAIN function by dividing

the upper limit of each storm size class by the

average storm size for the 13 years of data: for

example, for the 0.10 size class, 0.10/0.726

= 0.1377. The lower limit of the RAIN function

was set so the month with the lowest average

storm size (February) would produce a storm

of at least 0.01 inch. In determining rainfall

amount, the program generates a uniformly distributed random number between 0 and 1, which

determines the value from the RAIN function.

The day number determines the value from the

ASSM function. The product of these two values

is rainfall amount which is distributed about the

monthly mean.

The simulation then calculates a new value

for days to next storm. Then proceeds to day

end, day + 1, and days to next storm = l?.

The simulation continues in this way for the run

time selected.

Output from this rainfall simulation model consists of (1) annual frequency distribution table for

days between storms, (2) annual frequency distribution table for storm size classes, (3) storm

summary by day number and rainfall amount

(table l), and (4) annual storm summary that

includes monthly and annual values for rainfall

and number of storms (table 2).

Table 1 .-Example of individual storm summary

Table 2.-Example of annual storm summary

- ___-

Storm Summary

Rainfall (inches) No. of storms

January 2.91 6

February 1.11 4

March 3.81 7

April 7 72 1 0

May ’ :66 9

June 7.46 9

July 2.68 6

August 2.41 6

September 1.39 3

October 0.91 3

November 8.74 9

December 3.14 5

Annual 46.93 7 7

The model can be easily altered to provide

different tables, graphs, and forms of output.

Any variable in the simulation can be in the

statistical output on an annual basis. A TABULATE statement allows a user to produce frequency distribution tables with the following

output: sum of the variable, number of table

entries, mean variable value, standard deviation,

and the number of times the variable fell within

each frequency class. Graphs can be plotted

from any of the variables in the simulation.

Additional output can be obtained by access

to user-written FORTRAN subroutines with the

HELPA statement.

To use the model in areas other than central Arkansas, users must develop the DBSD, .li

ADBSM, RAIN, and ASSM functions from at

least 10 years of local rainfall data.

Storm Summary/For Year 5

Day number Rainfall (inches) MODEL ACCURACY

6 0.47

8 0.42

9 0.05

1 3 0.81

1 5 1.13

2 1 0.02

350 0.04

351 0.38

352 0.11

361 2.04

363 0.56

I compared simulated rainfall for 20 years

with the 13 years of record used to develop

the model. Comparing simulated data against

the data used to develop the model is not

comprehensive validation; however, in this case

I chose to use all the data available to develop

the model. Chi-square tests were conducted on

actual and simulated data for days to next storm,

storm size classes, storms per month, and rainfall

per month. The tests showed no differences

between actual and simulated data. The simulated data had one less storm per year, and

3
Table 3.-Compaf’son of 20 years of simulated rainfall data and actual rainfall data for 1961-7973

- ___-

Number of storms annually Average days Annual precipitation (inches) Average storm

Average Range between wet days Average Range size (inches)

Actual 75.77 57-94 4.82 54.98 39.93-82.79 0.726

Simulated 74.75 61-89 4.89 53.82 37.51-74.45 0.720

- -

simulated annual rainfall was an inch less than

actual (table 3). The range of simulated storms

per year was less than recorded during the study

period. The range of simulated annual rainfall

was somewhat smaller than that of actual, but the

actual rainfall of 82.79 inches in 1973 was

nearly 30 inches greater than normal and would

probably only occur once in every 100 years.

To determine if the model was simulating the

extreme values, I compared largest simulated

rainfall to largest actual and largest calculated for

a loo-year return period for storm, month, and

annual values. The simulated storm and month

values compared very favorably with the actual

and loo-year return period values (table 4). The

simulated annual value of 74.45 inches was

lower than both the actual and lOO-year return

period values, but the actual value of 82.79

‘inches, as mentioned above, is probably the

maximum that occurs once in every 100 years.

From these comparisons it can be said that

the model accurately simulated rainfall events

with the frequency and size as occurred between 1961-1973. This does not mean the

model will accurately simulate rainfall events

as they occur in the future as rainfall patterns

may change. The model should be useful as a

submodel for soil water, runoff, tree growth,

sediment, water balance, and other systems

needing rainfall as input.

Table cl--Largest Rainfall values

Storm Month Annual

--------------inches--------------

Actual a.25 14.22 82.79

Simulated 8.89 19.49 74.46

1 OO-year’ 12.19 20.12 111.63

‘Calculated from 13 years of actual data.

LITERATURE CITED

Clarke, R. T. user’s manual. Ed. SH 20-0851-l. IBM,

1973. A review of some mathematical models White Plains, N.Y.

used in hydrology, with observations on International Business Machines Corporation.

their calibration and use. J. Hydrol. 19:1-20. 1973. General purpose simulation system V-OS

International Business Machines Corporation. operations manual. Ed. SH 20-0867-3. IBM,

1971. General purpose simulation system V White Plains, N.Y.

4
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TIME BETWEEN STORMS (DAYS)

Figure 2.-Cumulative frequency distribution of days between storms.

JAN FEB MAR APR MAY JUN JUL AUQ NOV DEC

Figure 3.-Mean number of days between storms by-!r?nths.

_ :’

5
STORM SIZE (INCHES)

Figure 4.-Cumulative frequency distribution of storm size.

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Figure S-Mean storm size by months.

6
APPENDIX

BLOCK

NlMBER **T-Ix OPERATION A,B,CP ,E,F,G,H ,I mm

SIMUL?il'E

*******************************************************

* PRESIM MmEL *

* RAINFALL SIMUL?fl'IU!I FOR ALCM CREEK EXPT FORST *

* BASED ON DATA FROM 1961-1973 *

******************************************************

*

* AVERAGE NWBER O;lD;Y$ BETWEEN STOR!fS BY WNTHS FUNCTION

ADBSM FUNCTICN

31,5.676/59,4.893/90,i.429/120,4.062/151,4.429/181,4.535/212,4.112

~43,4.528/273,5.200/304,6.106/334,5.652/365,5.167

* tUiYS BEWEENS'IDEMS DISTRIBVTI~ FUNCTICN

DBSDFUNCl'IW RN2,C31

0,.1232/.2802,.3110/.4213,.5184/.5188,.7259/.6234,. 9333/.6995,1.1408

.7492,1.3482/.7970,1.5557/.8345,1.7632/.8579,1.97~6/.8893,2.1781

.9036,2.3855/.9228,2.5930/.9360,2.8004/.9452,3.0079/.9543,3.~54

.9665,3.4228/.9746,3.6303/.9807,3.8377/.%58,4.0452/.9888,4.2526

.9909,4.4601/.9919,4.6676/.9929,4.8750/.9939,5.0825/.9949,5.2899

.9960,5.4974/.9970,5.7048/.9980,5.9123/,,6.3272 *

* AVERAGE STCRM SIZE BY MONTHS FUNCTI(w

ASSM FUNCTION Cl,Ql2

31,639/59,610/90,814/120,692/151,749/18~,623/212,650/243,694/273,829

304,802/334,865/365,795 *

* PREcIPITATIm EVENT SIZE CLASS DISTRIBUTI@J

RAIN FUNCTICM RN3,C48

0,.0164/.1706,.1377/.3036,.2755/.4142,,487

.6284,.8264/.6711,.%42/.7107,1.102/.7421,,1.240/.7695,1.377/.7970,1.515

.8213,1.653/.8447,1.791/.8619,1.928/.8792,2.066/.8843,2.204/.8964,2.342

.9015,2.479/.9147,2.617/.9228,2.755/.9299,2.893/.9350,3.030/.9421,3.168

.9482,3.306/.9553,3.444/.9584,3.581/.9624,3.719/.9645,3.857/.9665,3.994

.9685,4.132/.9736,4.270/.9787,4.821/.9807,5.096/.9817,5.234/.9838,5.372

.9888,5.510/.%98,5.64~/.9909,6.061/.9929,6.336/.9939,6.612/.9949,6.887

.9959,7.025/.9970,8.540/.9980,8.815/.9990,9~901/.99%,11.433~~0,13.499

*

MOED FUNZl?IO!U XH3,Dll

31,59/59,90/90,120/120,151/151,181/181,212/212,243

243,273/273,304/304,334/334,365

*

STINT FVARIABLE FN$ADBSM*FN$CBSDt.5

PRCIP FVARIABLE FN$ASSM*FN$RAIN

RMULT ,743,451

INITIAL

INITIAL ii?!1

SIm Mmax MX,i2,2 *

* FULIwoRDSAVEVALUES HALFWRDSAVEVALUES * Xl DAYS TO NEXT SWRM XHl TEST VALUE FOR FIRST STORM EVENT * X2 DAILY PRECIP XH2HEADING TESTVALUE * x3 l?ma!HLYPREIcIP XH3 M(MH END DE * X4 ANNtAL PRECIP * x5 rJsmrHLYPR~IP * X6 bKNTHLYN0. OF STORM!?

7
* X7 NO. OF SmIW EVENTS * X8 MATRIX TNPUI'CCUNTER * x9 STORMS ANNUALLY * X10 MATRIX OUTPUT CQUNTER *

**** mE WDm, ****:

*

hi,;6Ll

1,Ki,iH

1,VSSTTNT

Xl,l,C@JT

*

GENERATE

TEST E

SAVEVAUJE

SAVEVALUE

NOTlTESTE

SAVEVALUE

TABULATE

SAVEVALUE ** ovppm **

TABULATE

SAWVALUE

SAVEVALUE

SAVEVALUE

S4VEVALUE

HELPA

TRANSFFR

CCNT SAVEVALUE

MONTTESTE

SAVEVALUE

MSAVEVAWE

MSAVEVAWE

SAVEVALUE

SAVEVALUE

SAVEVALUE

SAVEVALUE

TEST E

ASSIGN

REPT SAVEVALUE

SAVEVALUE

SAVEVALUE

HELPA

LOOP

HELPA

NXDAY TERMINATE *

l,V$STMT

1

3t,X2

4+,X2

7+,Kl

2t,Kl,XH

PREiCUT,Cl,X2,XH2

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10,12

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6,MX$STOM(Xl0,2)

PRESuM,X5,X6,XlO

10,REPT

ANLJ?RE,X4 ,X9

1

SIMULATE DAY

TEST FOR INITIAL DAY OF SIM

TESTVAUJE

DETERMINE INITIAL STORM DATE

STOF@4EVENT'IWAY

SIZE OF STORM

TABULATE STORY SIZE

DETERMINE DAYS To NEXT .S'IGRJY

TABULATEDAYSBEZWEENSTOmE

MDNTHLYPRFCIP

ANNUAL PRECIP

Sl'ORMEVENTCOUNIER

HEADINGCOUNTER

PRIN'IWT DAY AND PRECIP AWJW

TRWSFER 'IQ MWFH END TEST

REDUCEDAYS 'N3NFXT S'IDFWBY 1

TEST FORDAY MONTH ENDS

CUJNTER FOR STORAGE VALUES

STORAGE OF PRFCIP / M3N'l-H

STORAGE.OF SToRlvLs / M3NTH

INITIAL M3lTHLY FROCTP TO 0

STORMS ANNUALLY

INITIAL STORMS TO 0

NEW MONTH END D4TE

TEST FOR END OF YEAR

IWNTHLY PRIN'IWT m

COUNTER

RECALLMQ-JTHLYPRECIP

RECALLMdJTHLYNO.CFSTORMS

MWTHLYPRECIP& No. OFSTOR!!

MONTHLY PRINTOUT LOOP

ANNUAL PRECIPANDNO.OFSl'ORMS

* TABLE DEFINITIRJ CARDS

1 TABLE lA,o,1,35

2 TABLE x2,0,100 *

* CCNTROLCAPDG

Er CLEAR

INITIAL

START

CLEAR

INITIAL

START

PREOtPT,PRESUM,ANLPRE

365

Xl,XHl

XH3,31

365

Xl,XHl

XH3,31

365

CLEAR Xl,XHl

a