Stochastic simulation of droughts. Part 1: point droughts

HYDROLOGICAL PROCESSESHydrol. Process. 22, 854–862 (2008)Published online 11 June 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.6654

Stochastic simulation of droughts. Part 1: point droughts

S. Mohan* and P. K. SahooDepartment of Civil Engineering, Indian Institute of Technology Madras, Chennai 600036, India

Abstract:

The number of drought events derived from the historic streamflow or rainfall series will be limited and produce results that arenot very reliable. This study proposes a drought simulation methodology that uses a long sequence of synthetically generatedmonthly streamflow/rainfall series, from which it is possible to drive a large sample of drought events and the predictionof drought characteristics will be reliable. The modified Herbst method has been used to identify droughts in the generatedstreamflow and rainfall series. The drought simulation procedure is illustrated with a case study of the Bhadra reservoircatchment in Karnataka State, India. Monthly droughts were derived from both historic and generated monthly streamflowand rainfall series. The important drought characteristics were determined and the suitable probability distribution for eachparameter was arrived at after studying seven different probability models. The use of the probability curves thus derivedhas been illustrated with examples (referred to in Part 1 as ‘point droughts’). Similarly, the development and application ofstochastic models for the prediction of regional drought parameters have been illustrated with examples in the accompanyingpaper (Part 2: regional droughts). Copyright 2007 John Wiley & Sons, Ltd.

KEY WORDS streamflow; rainfall; drought simulation; drought prediction

Received 10 July 2006; Accepted 27 November 2006

INTRODUCTION

The term ‘drought’ means abnormal water shortage ofany kind, and has different connotations in different partsof the world. In some parts of the world a period of1 week without rain is termed a drought; in parts ofnorthern African countries, drought becomes effectiveonly after 2 years without rainfall; in Egypt any year thatthe River Nile does not achieve a certain high level is adrought, regardless of rainfall. However, droughts in thetropics produce perhaps more disastrous situations thanfloods.

There are various definitions of drought available. Oneobjective definition, in the sense that it is an unambiguousdefinition of drought, is given by Yevjevich (1967) onthe basis of a time-series Xt of a hydrological variableand a truncation level Xo�t�, as shown in Figure 1.The negative difference between the time-series and thetruncation level leads to drought, i.e. �Xt � Xo�t�� < 0.Each streamflow or rainfall drought event is composedof two parameters, namely its duration D (the numberof consecutive months for which the streamflow orrainfall is below the truncation level also called referencedischarge) and S (the cumulative deficit of streamflowor rainfall for that duration). A third useful measure,the magnitude of drought M (average water deficit), isrelated to drought duration and severity by the equationS D MD. The truncation level may be either a constantvalue or a function of time. In practice, the selection ofthe truncation level is not arbitrary, but is dependent on

* Correspondence to: S. Mohan, Department of Civil Engineering, IndianInstitute of Technology Madras, Chennai 600036, India.E-mail: [email protected]

the water demand and the type of water deficit beingstudied.

Various drought properties may be derived from thehistorical records alone; however, when a historicalrecord, say for 30 years, is analysed, only a smallnumber of droughts can be obtained. Thus, the estimatedstatistical properties for drought parameters derived fromthe historic series are usually associated with large stan-dard errors that would lead to unrealistic conclusions.One way of minimizing this problem is to derive a largepopulation of drought events from a longer generatedseries beyond the historical record. In this study, droughtsimulation has been carried out to obtain reliable resultson the statistical properties of drought parameters thatcan be used for prediction of droughts.

Analytical and data-generation methods have been sug-gested by different workers for analysing drought prop-erties. Analytical expressions were developed by Saldar-riaga and Yevjevich (1970) by which the probability ofsequences of wet and dry years of specified lengths canbe calculated when the basic hydrologic time-series iseither an independent or a dependant stationary series ofa variable that follows the first-order linear autoregres-sive model. Herbst et al. (1966) developed a techniquethat considers a variable truncation level for the eval-uation of monthly droughts from rainfall data. Askewet al. (1971) observed that preservation of historical crit-ical droughts in the generated sequence, obtained byavailable generating techniques, was influenced by therelative aridity of the watershed and by the size ofthe reservoir considered. Millan and Yevjevich (1971)derived the approximate distribution of the longest runlength for an autoregressive AR(1) model. Sen (1980)

Copyright 2007 John Wiley & Sons, Ltd.

STOCHASTIC SIMULATION OF DROUGHTS. PART I: POINT DROUGHTS 855

Xt

D

τb(v)

τc(v)

TIME (MONTHS)

TRUNCATION LEVEL

Md

FLOW

DROUGHT EVENTS

X0

RAINFALL

OR

σ

Figure 1. Definition of point-drought parameters

used the theory of runs to develop analytical solutionsof drought properties. Frequency analysis of multiyeardrought durations was studied by Lee et al. (1986). Wangand Salas (1989) compared the autoregressive movingaverage model and a threshold non-linear model based onwhether the historical droughts were reproduced. Mohanand Rangacharya (1991) proposed a methodology to iden-tify drought parameters from historic data on streamflowand rainfall having a seasonal pattern.

Studies on point and regional droughts are oftenlimited to the analysis of historic flow or rainfall seriesin a specific location or region. A major problem inthese studies is the unavailability of extensive flow orrainfall records, from which one may obtain samples ofmany droughts. The small number of drought events froma particular flow or rainfall series, when subjected tostatistical analysis in order to predict future occurrences,produces results that are not very reliable in predictingfuture events. To overcome this difficulty, this study hasused synthetically generated flow or rainfall series tosupplement the historic flow or rainfall series. Stochasticmodelling and simulation becomes necessary when theunderlying process is so complex that it hinders thederivation of drought properties in closed form. Themain objective was to derive a large population ofdrought events from the generated flow or rainfall seriesto study the statistical and stochastic behaviour of thedrought parameters and to demonstrate their usefulnessfor specific problems in water resources planning andanalysis.

Stochastic models are proposed and tested to representthe underlying flow (or rainfall) data, and then used asthe basis for generation of synthetic inflows (or rainfall).These generated data are, in turn, used to derive thedistribution of drought parameters both for point andfor regional droughts. This paper mainly presents thestochastic model for the analysis of point droughts; theresults of stochastic modelling of regional droughts arereported in Mohan and Sahoo (2007).

DROUGHT SIMULATION

The methodology for drought simulation adopted in thisstudy consists of three phases. In the first phase, the syn-thetically generated data on streamflow and rainfall serieswere obtained and subjected to drought analysis. Theparameters of the droughts thus obtained are subjectedto probabilistic analysis to fit a suitable distribution foreach of the drought parameters in the second phase. Thethird phase involves predicting the drought characteris-tics using the model developed in phase two. These stepsare explained below in detail.

Drought identification

In this study the drought events were identifiedusing the method suggested by Mohan and Rangacharya(1991). The important assumptions adopted in thismethod are as follows:

1. Streamflow/rainfall deficits exceeding the averagedeficit of any month have been included in the evalu-ation of critical drought.

2. The recovery after a drought is not immediate; rather,the residual effect persists for some time until adequatestreamflow passes through (or rainfall falls).

3. A single month could constitute a drought only if noor very little quantity of streamflow passes (or rainfallfalls) in the month of highest streamflow (or rainfall)of an average year.

4. Either a continuous period of slightly above-averagestreamflow (or rainfall) or a few successive monthsof abnormally high streamflow (or rainfall) couldconstitute termination.

5. The drought events are assumed to be independent.It can happen that some of the generated droughtswould be very close to each other and, therefore,mutually dependent. To cope with the problem ofdependence between these drought events, withoutsignificant influence on the accuracy of the final results,

Copyright 2007 John Wiley & Sons, Ltd. Hydrol. Process. 22, 854–862 (2008)DOI: 10.1002/hyp

856 S. MOHAN AND P. K. SAHOO

all droughts with deficits Si, i D 1, 2, . . ., satisfyingthe condition Si < 0Ð01 times the maximum deficit areneglected and are not considered in this study.

6. If the time period separating droughts (interveningtime) is less than or equal to 2 months and the surplusvolume during this intervening period is very small(<0Ð01 times the maximum deficit of either of thedroughts), then these droughts are assumed to becontinued.

MODEL FOR GENERATION OF DATA

An observed flow series contains only limited informa-tion, since it represents only one realization of manypossible realizations that could have occurred. Syntheticgenerated flow (or rainfall) series which differs from theobserved series, but retails many of its properties is con-sidered in this study to take into account all possiblesequences. AR models and the Thomas–Fiering model(Fiering and Jackson, 1971) were studied for their suit-ability in generating streamflow/rainfall series. It wasfound that the Thomas–Fiering model preserved thestatistics of historic data in the generated values togetherwith the drought characteristics. The Thomas–Fieringmodel is given by

xi,j D xj C bj�xi,j�1 � xj�1� C εt�j

√�1 � r2

j� �1�

where xj and xj�1 are the mean flows (or rainfall) inseasons j and j � 1 respectively. �j is the standarddeviation of flow (or rainfall) in the season j. rj is thecorrelation between flows (or rainfall) in the jth and(j � 1)th seasons. bj is the regression coefficient of flow(or rainfall) in the jth season with flow (or rainfall) in the(j � 1)th season. εt is the normal random variate. xi,j isthe flow (or rainfall) generated in the jth season of the ithyear, with j D 1, 2, . . . ., m and i D 1, 2, . . .. If a seasonis a month, then m D 12; for a weekly model, m D 52;for a daily model, m D 365.

It may be noted that in the Thomas–Fiering modelthe rainfall (or flow) in any season j is a sum of threeterms. The first term is the mean rainfall (or flow) inthat season. The second term is the regressed componenton the rainfall (or flow) in the previous season. Thethird term is a random component to reflect the variance.This model accounts for the persistence only up to lag-1.Hence, the Thomas–Fiering model may be viewed asa non-stationary first-order AR model. This model canbe applied directly in the form given in Equation (1),drawing the random variate εt from a normal distributionwith zero mean and unit variance [N(0,1)] provided therainfall (or the flow) in all seasons is normally distributed.

PROBABILITY MODELS AND PARAMETERESTIMATION METHODS

One of the objectives of this study is to apply most ofthe popular and currently advocated probability models to

the derived drought parameters and to reach conclusionsabout the best-fit models based on various goodness-of-fit tests. Eight probability distribution functions that havebeen suggested by various workers for extreme-valuefitting are considered. The probability density functionsof the selected probability models which were used inthis study are as follows.

1. Two-parameter log-normal distribution (LN2):

f�x� D 1

xap

2�exp

{�[ln�x� � b]2

2a2

}�2�

2. Three-parameter log-normal distribution (LN3):

f�x� D 1

��x � c�p

2�exp

{�[ln�x � c� � b]2

2a2

}

�3�

3. Two-parameter Gamma distribution (G2):

f�x� D ba

axa�1e�bx �4�

4. Three-parameter Pearson distribution (P3):

f�x� D(

x � c

b

)a�1 exp[��x � c�

b

]jbja

�5�

5. Three-parameter log-Pearson distribution (LP3):when x � ln�x�, the probability density function (PDF)of the Pearson distribution becomes the PDF of the log-Pearson distribution.6. Log-logistic distribution (LL):

f�x� D

(x � c

b

)�1/a

a�x � c�

[1 C

(x � c

b

)�1/a]2 �6�

7. Exponential distribution (exp)

f�x� D b e�bx x > 0, b > 0 �7�

8. Extreme-value type I distribution (EVI):

f�x� D a eexp[�a�x�b�]�a�x�b� �8�

In the above distributions, a, b and c are the parametersof the distribution, x is the hydrologic variable, and is the gamma function. For the estimation of theparameters of the above distributions, three models ofparameter estimation, namely the method of moments(MOM), the method of maximum likelihood (ML), andthe probability weighed moments (PWM), are considered.The probability models and the corresponding parameterestimation procedures have been discussed in detail byHaktanir (1991).

The proposed methodology for drought simulation isillustrated by means of a case study of the River Bhadra



catchment in Karnataka State, India. The total area of thecatchment is 1968 km2. The mean annual rainfall andmean annual runoff of the catchment are 2482Ð8 mm and2962Ð4 Mm3 respectively. Monthly streamflow recordsfor the period June 1937 to May 1988 pertaining to thesite have been collected. Monthly rainfall records for theperiod June 1956 to May 1983 are available at nine rain-gauges over the catchment. The mean area rainfall overthe catchment was calculated using the Thiessen polygonmethod and used in the study. The number of droughtsin the historical streamflow and rainfall series identifiedby them are 10 and 4 respectively.

To generate a longer sequence of monthly streamflowand rainfall series, the Thomas–Fiering model was used.Several series (up to a period of 1200 years) weregenerated, so that the length of each series was an integermultiple of the historic series length. Basic statisticalparameters, such as the mean, variance, skew and thelag-1 autocorrelation, were estimated for each generatedseries and compared with the corresponding propertiesof the historical series. The comparison of the mean,standard deviation, and skewness of the historic andgenerated sequences of the streamflow and rainfall seriesare shown in Figures 2 and 3 respectively. It can be seenfrom these figures that the mean and standard deviation ofboth the streamflow and rainfall series are well preserved.There were respectively 260 and 270 droughts identifiedin the generated streamflow and rainfall series when theseries were subjected to drought analysis.

Before fitting any distribution for drought parameters,it is necessary to check whether these series are indepen-dent. To this end, autocorrelation coefficients at variouslags were computed for the drought parameters as foundin the generated streamflow and rainfall series, and thetest of significance at the 5% probability level was car-ried out to see whether the sample correlograms belongto the independent cases. The correlograms of droughtparameters were found to be within the 95% confidencelimits. Thus, it can be concluded that the drought param-eters derived from the generated series, namely droughtduration and drought severity, are independent randomvariables.

The comparison of the probability distributions ofstreamflow drought parameters derived from the historicand generated flow series are shown in Figure 4. Sim-ilar comparisons of probability distributions of rainfalldrought parameters derived from the historic and gener-ated rainfall series are shown in Figure 5. It can be seenfrom Figure 4 that the probability distributions of stream-flow drought parameters derived from the historic andgenerated series closely match each other, but the returnperiod of more extreme droughts can only be obtainedfrom the probability distributions of drought parametersderived from the generated series. On the other hand,for rainfall series, the probability distributions of droughtparameters derived from the generated series do notmatch each other due to the limited sample of droughtevents obtained from the historic series (Figure 5). How-ever, for estimating the reliable return period of droughts

Figure 2. Comparison of statistics between historic and generated stream-flow series: (a) mean; (b) standard deviation; (c) skewness

events, the probability distribution of drought parametersderived from the generated series needs to be used.

Table I shows a comparison between the longestdrought durations and greatest drought severities derivedfrom the historic and generated streamflow and rain-fall series. As expected, it was found that the generatedlongest drought duration is longer than that of the his-toric longest drought, and the generated greatest droughtseverity is larger than that of the historic greatest drought.Table II provides a comparison between the mean valuesof the drought magnitudes computed from historic andgenerated flow and rainfall series. When examining thesevalues, it was found that the drought magnitudes resultingfrom generated flow or rainfall series are greater than thecorresponding magnitudes resulting from historic flow orrainfall series. These drought magnitudes, together withthe fitted distribution functions, will be a valuable toolfor prediction of drought parameters.



Figure 3. Comparison of statistics between historic and generated rainfallseries: (a) mean; (b) standard deviation; (c) skewness

DEPENDENCE BETWEEN DROUGHTPARAMETERS

Cross-correlation and regression analysis were carriedout to find the relationship between drought param-eters, namely drought duration and drought sever-ity, derived from the generated series. The correla-tion coefficient between streamflow drought severityand its duration is 0Ð75, whereas the cross-correlationbetween rainfall drought severity and its duration is0Ð76. The relationship between streamflow drought sever-ity and streamflow drought duration is linear and theregression equation is S D 57Ð4D C 228Ð3, where S isthe drought severity and D is the drought duration.The explained variance of the fitted function to thestreamflow drought severity and duration is 0Ð58. The

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

Drought Duration (Months)

No

n-E

xcee

dan

ce P

rob

abili

ty

SimulatedHistoric

(a)

0.0

0.4

0.8

0 2

Drought Severity (Mm3)

No

n-E

xcee

dan

ce P

rob

abili

ty SimulatedHistoric

(b)

1 3 4

Figure 4. Comparison of historic and simulated distribution functions:(a) streamflow drought duration; (b) streamflow drought severity

relationship between rainfall drought severity and rain-fall drought duration is also linear, with the result-ing equation being S D 46Ð1D C 188Ð6. The explainedvariance in this case was also 0Ð58. The relationbetween drought duration and drought severity is cor-related well for less severe droughts than the severedroughts.

DISTRIBUTION FUNCTIONS FOR DROUGHTPARAMETERS

Drought severity

The eight theoretical distribution functions discussedearlier were tried to fit the probability distributions ofstreamflow and rainfall drought severities derived fromthe generated series. The best-fit distribution functionswere selected based on the Kolmogorov–Smirnov (K–S)goodness-of-fit test and chi-square tests at the 5% signif-icance level. Table III indicates the results of the test forstreamflow drought severity. Of the several theoreticaldistribution functions, the log-logistic distribution withthe PWM method of parameter estimation (with Cun-nane plotting position formula) was selected because itcan be expressed in a closed form as well as being flex-ible. The fitted distribution function to the streamflow



0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30Drought Duration (Months)

No

n-E

xcee

dan

ce P

rob

abili

ty

SimulatedHistoric

(a)

0.00

0.40

0.80

0 500 1000 1500 2000

Drought Severity (mm)

No

n-E

xcee

dan

ce P

rob

abili

ty

SimulatedHistoric

(b)

Figure 5. Comparison of historic and simulated distribution functions:(a) rainfall drought duration; (b) rainfall drought severity

Table I. Comparison of extreme droughts derived from the his-toric and generated series

Drought parameter Historic Simulated

Longest streamflow droughtduration (months)

17 47

Greatest streamflow droughtseverity (Mm3)

2672 3547

Longest rainfall drought duration(months)

17 27

Longest rainfall drought severity(mm)

954 2300

Table II. Comparison of the historic and generated droughtmagnitudes

Event Historic Generated

Streamflow (Mm3 month�1) 80Ð44 101Ð11Rainfall (mm month�1) 57Ð04 105Ð33

drought severity is given by

F�S� D 1

1 C(

S C 689Ð52

1401Ð88

)�1/0Ð205 �9�

0.0

0.2

0.4

0.6

0.8

1.0

0 500 1000 1500 2000 2500

Drought Severity (mm)

No

n-E

xcee

dan

ce P

rob

abili

ty

SimulatedTheoretical (LL-PWM(CP))

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4

Drought Severity (103 Mm3)

No

n-E

xcee

dan

ce P

rob

abili

ty

SimulatedTheoretical (LL-PWM(CP))

(a)

1 3

Figure 6. Comparison of stimulated and theoretical probability distribu-tion: (a) streamflow drought severity; (b) rainfall drought severity

where S is the drought severity. The distribution functionfitted to the rainfall drought severity is given by

F�S� D 1

1 C(

S C 384Ð53

796Ð61

)�1/0Ð2150 �10�

The comparisons between the theoretical probabilitydistributions and probability distributions derived fromthe generated streamflow or rainfall series for the droughtseverity are shown in Figure 6, which depicts the close-ness between the simulated and theoretical probabilitydistributions.

Drought duration

In a similar way to that of drought severity, eighttheoretical distribution functions were tried to fit the prob-ability distributions of streamflow and rainfall droughtdurations derived from the generated streamflow and rain-fall series. The best-fit distribution function was selectedbased on the K–S and chi-square tests at the 5% signif-icance level. Of the several distributions, it was foundthat a type I extreme value distribution (with parameterestimation by MOM) fits well to the probability distri-bution of streamflow drought duration derived from thegenerated series. The fitted distribution is given by

F�D� D e� exp[�0Ð16661�d�6Ð709�] �11�



Table III. Results of goodness-of-fit test for various distributions to streamflow drought density

Distributionfunction

Method of parameterestimation

K–S test Chi-squaretest

Remark

Teststatistic (r)

Criticalvalue (r0)

�2 �20

Two-parameter log-normal Method of moments 0Ð138 0Ð084 33Ð57 14Ð07 FailTwo-parameter log-normal Maximum likelihood 0Ð138 0Ð084 33Ð54 14Ð07 FailTwo-parameter log-normal Probability weighted moment 0Ð128 0Ð084 25Ð34 14Ð07 FailThree-parameter log-normal Maximum likelihood 0Ð057 0Ð084 3Ð49 14Ð07 PassTwo-parameter Gamma Method of moment 0Ð058 0Ð084 5Ð16 12Ð59 PassTwo-parameter Gamma Maximum likelihood 0Ð083 0Ð084 10Ð46 14Ð07 PassExponential — 0Ð177 0Ð084 50Ð00 14Ð07 FailExtreme value type I Method of moments 0Ð036 0Ð084 6Ð07 15Ð51 PassExtreme value type I Maximum likelihood 0Ð034 0Ð084 7Ð72 14Ð07 PassExtreme value type I Probability weighted moment 0Ð033 0Ð084 5Ð75 14Ð07 PassThree-parameter Pearson Method of moments 0Ð055 0Ð084 5Ð21 12Ð59 PassThree-parameter Pearson Maximum likelihood 0Ð059 0Ð084 6Ð70 12Ð59 PassLog-logistic Method of moments 0Ð059 0Ð084 6Ð63 12Ð59 PassLog-logistic Probability weighted moment (Weibull

plotting position)0Ð036 0Ð084 3Ð28 12Ð59 Pass

Log-logistic Probability weighted moment(Cunnane plotting position)

0Ð037 0Ð084 2Ð99 12Ð59 Pass

where D is the drought duration series and d is anelement of the drought duration series. It was found thatthe exponential distribution fits well to the probabilitydistribution of rainfall drought duration derived from thegenerated series, and the resulting distribution function isgiven by

F�D� D 1 � e�0Ð16033d �12�

Figure 7 shows the comparison between the fittedprobability distributions and probability distributionsobtained from the generated streamflow series, fromwhich it can be seen that the fitted probability distributionfits well to the simulated probability distribution.

JOINT DISTRIBUTION FUNCTION OF DROUGHTPARAMETERS

Since drought is a two-dimensional process, the jointdistribution of both severity and duration is very impor-tant. Figure 8 shows that the correlation between droughtseverity and drought duration is greater for less severedroughts than for extreme droughts. To substantiate this,ranking the drought severity of the 20 greatest droughtsand the corresponding rank numbers of drought durationhave been worked out, from which it was found thatthe rank numbers of drought severity for the 20 greatestdroughts are not in conformity with the rank numbersof drought durations. Thus, drought duration and droughtseverity for extreme droughts can be logically assumedto be independent of each other. Hence, the joint dis-tribution function of streamflow drought severity and itsduration can be worked out and is given by

F�S, D� D 1

1 C(

S C 689Ð52

1401Ð88

)�1/0Ð205 e� exp[�0Ð166�d�6Ð71�]

�13�

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50


No

n-E

xcee

dan

ce P

rob

abili

ty SimulatedTheoretical (EVI-MOM)

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0 10 15 20 25 30


No

n-E

xcee

dan

ce P

rob

abili

ty

SimulatedTheoretical (exp)

(b)

5

Figure 7. Comparison of stimulated and theoretical probability distribu-tion: (a) streamflow drought duration; (b) rainfall drought duration



0

500

1000

1500

2000

2500

3000

3500

4000

0 10 20 30 40 50


Dro

ug

ht

Sev

erit

y (M

m3)

(a)

0

500

1000

1500

2000

2500

0 10 15 20 25 30


Dro

ug

ht

Sev

erit

y (m

m)

(b)

5

Figure 8. Correlation between drought parameters: (a) streamflowdrought severity and its duration; (b) rainfall drought severity and its

duration

The joint distribution function of rainfall droughtseverity and its duration is given by

F�S, D� D 1

1 C(

S C 384Ð53

796Ð61

)�1/0Ð216 �1 � e�0Ð160d�

�14�It is to be noted here that these joint distribution

functions are applicable only to extreme droughts.

APPLICATION OF THE RESULTS

The following two examples illustrate the applications ofthe results obtained from the point drought analysis.

Example 1

Determine (a) the longest streamflow drought durationand its severity and (b) the largest streamflow droughtseverity and its duration expected within a return periodof 5 years (60 months) in the Bhadra catchment.

Solution. As the return period (60 months) is includedin both the historic flow record and the generated

streamflow series length, the behaviour of the droughtparameters established either from the historic flow seriesor from the generated flow series can be used to determinethe required estimates. If both are used, then a comparisonbetween the estimates resulting from two types of flowseries can be made. The steps to be followed areexplained below.

1. Determination of non-exceedance probability. Theperiodicity of the hydrologic cycle is 12 months andthe expected number of hydrologic cycles within aperiod of 60 months is five. The non-exceedance prob-ability corresponding to this return period p D 1 �1/5 D 0Ð8.

2. Longest drought duration and its severity. From Figure4, corresponding to a non-exceedance probability 0Ð8,the longest drought duration resulting from the historicflow series is 14 months. The longest duration from thegenerated flow series is also 14 months. From Table II,the expected drought magnitude resulting from his-toric flow series is equal to 80Ð44 Mm3 month�1. Theexpected drought magnitude resulting from the gen-erated flow series is equal to 101Ð11 Mm3 month�1.Thus, the expected drought severity during the longestdrought duration from historic series is 14 ð 80Ð44 D1126Ð44 Mm3. The expected drought severity dur-ing the longest drought duration from generatedseries is 14 ð 101Ð11 D 1415Ð54 Mm3. Compared with1124Ð44 Mm3, the value of 1415Ð54 Mm3 must bemore reliable as this estimate resulted from a largepopulation of drought events.

3. Greatest drought severity and its duration. FromFigure 4, corresponding to a non-exceedance proba-bility of 0Ð8, the greatest drought severity resultingfrom historic flow series is 1000 Mm3. The greatestdrought severity resulting from generated flow seriesis 1000 Mm3. Therefore, the expected drought dura-tion corresponding to a severity of 1000 Mm3 resultingfrom historic flow series is 1000Ð0/80Ð44, which isequal to 12Ð4 months. Similarly, the expected droughtduration corresponding to a severity of 1000 Mm3

from generated flow series is 1000/101Ð11, which isequal to 10 months. The latter result may be consid-ered more reliable, as it results from a large populationof drought events.

Example 2

Determine the probable longest streamflow droughtduration and the greatest streamflow drought severitywithin a period of 100 years for the Bhadra catchment.

Solution. These estimates can be made using thedrought simulation results derived in the present study.As a first step, it is necessary to determine the non-exceedance probability corresponding to the given returnperiod. The required estimates can be made by makinguse of the simulated probabilistic relationships or thefitted theoretical probability distribution function for the



drought parameters of interest derived from generatedflow series. As the return period is 1200 months, whichexceeds the historical flow records (624 months), therequired estimates can be obtained by using the simulatedresults.

1. Determination of non-exceedance probability. Theexpected number of yearly cycles is 100. The non-exceedance probability is 1 � 1/100 D 0Ð99.

2. Determination of longest drought duration. FromEquation (11), corresponding to a non-exceedanceprobability of 0Ð99, the probable longest drought dura-tion is approximately 34 months.

3. Determination of the greatest drought severity. Theanticipated greatest severity corresponding to a non-exceedance probability 0Ð99 is obtained for Equation(9) and is given by 2906Ð4 Mm3.

CONCLUSIONS

Based on the simulation study on point droughts, thefollowing conclusions are arrived at.

1. The Thomas–Fiering model for generation of stream-flow and rainfall series was found to be efficient inpreserving the drought characteristics and preservingthe statistics of the historical series.

2. A reliable distribution of drought parameters can beobtained from samples of synthetically generated data.

3. The drought parameters, namely drought duration anddrought severity, are independent random variables,but dependent on each other. However, for extremedrought, the dependence between the drought param-eters is comparatively much less and they can beassumed to be independent of each other.

4. A log-logistic distribution with probability weightedmoment method of parameter estimation fits the best,both to the distribution functions of simulated stream-flow drought severity and rainfall drought severityseries.

5. An extreme-value type I distribution with momentmethod of parameters estimation fits well to thedistribution function of simulated streamflow drought

duration series, whereas an exponential distribution fitswell to the distribution function of simulated rainfalldrought duration series.

6. The derived distribution of drought parameters are use-ful in predicting the longest drought duration with itsseverity and largest severity with its drought durationfor the given period of analysis.

7. The drought simulation methodology proposed in thisstudy is found to be an efficient tool in identifying thedrought parameters and could easily be adapted to anyother similar case studies.

The results presented in this paper need to be modifiedfor catchments in other regions or catchments in otherclimatic conditions. However, the general methodologyproposed in this paper can be easily applied to othercatchments, both for prediction of point droughts and forlong-term prognoses

REFERENCES

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Documents

Stochastic simulation of droughts. Part 1: point droughts