An analytical formulation of return periodof drought severity

B. Bonaccorso, A. Cancelliere, G. Rossi

Abstract. Planning and management of water resources systems under droughtconditions often require the estimation of return periods of drought eventscharacterized by high severities. Among the several methods proposed fordescribing droughts, the run method is the most suitable to provide an objectiveidentification and characterization of drought events. According to such a method,droughts are identified as consecutive intervals where the investigatedhydrological variable is continuously below a fixed threshold, and may bedescribed by means of two characteristics, namely, drought duration and droughtseverity. Since both characteristics are necessary to estimate water deficit risks,frequency analysis of drought events cannot be based on the same approachgenerally used for flood analysis, such as maximum annual series or partialduration series of a single characteristic. In particular, the evaluation of returnperiod for drought events needs to consider both duration and severity in order totake into account the pluriannual duration of several droughts. Very often areliable analysis of the probabilistic structure of droughts based on the observedsamples, using an inferential approach, cannot be properly carried out due to thelimited number of drought events which can be identified even on quite longhistorical series. This problem has been faced by Shiau and Shen (2001), who havedetermined the conditional distribution of drought severity given a droughtduration on the basis of generated hydrological series. In this paper their approachis extended by deriving analytically the parameters of the probability distributionof drought severity based on the stochastic process describing the underlyinghydrological variable. More specifically, a gamma distribution is adopted to modeldrought severity and its parameter are theoretically determined as a function of thethreshold level and the coefficient of variation of annual precipitation seriesassumed independent and lognormal distributed. Then, the return period ofdrought events with severity greater than or equal to a fixed value is computed asthe mean interarrival time of drought events with a certain severity or greater.Such procedure has been applied on 88 annual series of data recorded in Sicilianrainfall stations, by computing for each series the return period corresponding to

Stochastic Environmental Research and Risk Assessment 17 (2003) 157–174 � Springer-Verlag 2003

DOI 10.1007/s00477-003-0127-7

B. Bonaccorso, A. Cancelliere (&), G. RossiDepartment of Civil and Environmental Engineering,University of Catania, Viale A. Doria 6, 95125 Catania, Italye-mail: acance@dica.unict.it

The present research has been carried out in partial fulfillment ofthe first author Doctoral Degree. The research has been developedin the framework of both the European Commission INCO-MEDproject WAMME (contract no. ICA3-1999-00014) and researchfunded by CNR, Group for Prevention from HydrogeologicalDisasters (no. 00.00543.42).

157

drought of fixed severity and by plotting the spatial distribution in order toprovide a description of drought spatial variability at a regional level.

Keywords: Drought, Severity, Return period

List of symbols

Cs skewness coefficientCv coefficient of variationDc drought severityD�c dimensionless drought severityDt deficitfDc

probability density function of drought severity Dc

fDtprobability density function of deficit Dt

fn;k probability that an event with Ld � k occurs for the first time at nth trialFDc

cumulative distribution function of drought severity Dc

FDtcumulative distribution function of deficit Dt

l run lengthL drought interarrival timeLd drought durationLn nondrought durationn generic trialN number of trials needed to the first occurrence of a critical eventNDc

number of drought events between two droughts with severity � dc

p0 probability of deficitp1 probability of surplusr parameter of the gamma distributionsx sample standard deviationSn;k probability of Ld � kT return periodTDc

return period of drought severity Dc

x0 threshold levelxx sample meanXt hydrological process

Greek symbols

a parameter for computing thresholdb parameter of the gamma distributionUðxÞ cumulative distribution function of a standard normal variablek drought critical durationlx mean of the hydrological process Xt

ly mean of the process Yt ¼ lnðXtÞrx standard deviation of the hydrological process Xt

ry standard deviation of the process Yt ¼ lnðXtÞ

1IntroductionDrought is usually defined as a significant temporary reduction in water avail-ability below the expected amount for a specified period and for a particularclimatic zone. Since droughts are strictly related to random phenomena, such asprecipitation or streamflow, they are random as well and therefore an appropriate

158

way to investigate them is by means of probability theory and stochastic pro-cesses methods.

Probabilistic characterization of droughts is extremely important, primarily inthose regions where an accurate water resources planning and management re-quires a detailed knowledge of water shortages due to hydrological droughts(Frick et al., 1990; Rossi et al., 1995). Moreover, the estimation of return periodsassociated to severe droughts can provide useful information in order to improvewater systems management under drought condition.

Probabilistic features of drought have been investigated extensively since Ye-vjevich (1967) proposed the use of run concept as a method to identify andcharacterize in an objective way drought events and their statistical properties.According to this concept, a drought event is defined as a succession of con-secutive intervals (negative run) where the hydrological variable under investi-gation remains below a threshold level. Once such a threshold is fixed, the maindrought characteristics are defined as drought duration Ld (negative run length)and as sum of the deficits over the run length Dc (negative run sum or droughtseverity).

A reliable analysis of the probabilistic structure of droughts cannot be properlycarried out based on the observed samples using an inferential approach, due tothe limited number of drought events which can be identified even on quite longhistorical series. In order to overcome such difficulty the probabilistic behaviourof drought characteristics has been generally derived analytically, assuming agiven stochastic structure of the underlying hydrological series (e.g., Downer etal., 1967; Llamas and Siddiqui, 1969; Sen, 1976, 1977, 1980; Guven, 1983; Zele-nhasic and Salvai, 1987; Mathier et al., 1992; Sharma, 1995; Shiau and Shen, 2001).Although such an approach has found large application in literature, exact ana-lytical derivation of probability distribution of drought severity and/or of thebivariate distribution of drought severity and drought duration, remain unsolvedproblems due to the difficulty to obtain a solution in closed form even assuming asimple stochastic structure for the underlying variable.

Further, direct derivation of return period of drought characteristics hasproven also to be a difficult task. Indeed, since drought duration is variableandoften multi-year, it is not possible to identify one or more events for each timeunit (year), as it usually happens in flood analysis, where for example maximumannual floods are investigated. As a consequence, the traditional Bernoulli trialsapproach, which leads to the well known formula for return period as a functionof nonexceedence probability T ¼ 1

1�P½Xt�xt �, cannot be adopted. Derivation of

return period of drought severity must therefore take into account droughtduration in order to properly model the interarrival time of events.

The return period can be defined in different ways for different applications.Some authors (e.g., Lloyd, 1970; Loaiciga, and Marino,1991; Shiau and Shen, 2001)have assumed the return period as the average elapsed time between occurrences ofcritical events (i.e., floods or drought events). An alternative definition of returnperiod is the average number of trials required to the first occurrence of a criticalevent (e.g., Vogel, 1987; Bras, 1990; Douglas et al., 2002).

Following the second definition, Fernandez and Salas (1999) have providedanalytical formulations for estimating return periods of drought events withduration greater than or equal to a critical value for both time independent andMarkov time dependent series by using a recursive algorithm proposed bySchwager (1983). However, such an approach is based on a run concept which isdifferent from the one proposed by Yevjevich. In particular, according to this

159

definition a sequence of n observations has as many runs of length l as there areuninterrupted successions of exactly l events of the same kind.

Using the same definition of run, Chung and Salas (2000) estimated the returnperiod of drought events with given duration for dependent hydrological droughtevents by using DARMA models, which are more adequate than Markovianmodels for modeling hydrological processes with a more persistent time depen-dence.

Shiau and Shen (2001) have derived theoretically the return period of inde-pendent and identically distributed hydrological droughts with a severity greaterthan or equal to a fixed value, as a function of the expected value of droughtinterrival time and the cumulative distribution function (cdf) of drought severity,whose parameters are estimated by an inferential approach applied to generatedseries.

The aim of the present study is to derive an analytical formulation for the returnperiod of drought severity. In particular the parameters of the distribution ofdrought severity, assumed gamma, are expressed as a function of the parameters ofthe distribution of the underlying hydrological series, assumed log-normal, and ofthe generic threshold level (parametrized as a function of the mean and of thestandard deviation of the same hydrological series). The derived distribution is thenused to determine the return period of drought severity. The result is an analyticalexpression which shows that the return period of dimensionless drought severityonly depends on the coefficient of variation of the original series and the thresholdparameter.

The procedure here developed extends the method proposed by Shiau andShen (2001), since it provides an expression of the return period of droughtseverity as a function of the statistical characteristics of the original underlyingvariable, and of the threshold parameter.

Application of the methodology on historical long records of precipitation inSicily shows an excellent agreement between theoretical and observed returnperiods of drought severity, which indicates the validity of the derived analyticalexpressions.

An analysis of the spatial variability of drought severity in Sicily region is alsoillustrated and maps of return period of drought events with a given dimen-sionless severity are presented.

2Derivation of cumulative distribution function of drought severityRecalling the drought definition given by the run method, a drought event can beconsidered as negative run characterized by two main properties: the run-lengthand the run-sum which, in terms of drought, correspond respectively to droughtduration and drought severity. In particular, if xt is the value assumed at time t bythe hydrological variable, the drought duration Ld is the length of negative run, ornumber of consecutive intervals where xt < x0, followed and preceded by at leastone interval where xt � x0; the drought severity Dc is the sum of the single deficitsDt ¼ x0 � xt over the drought duration Ld.

From the above definitions it follows that drought severity Dc is a randomvariable function of the underlying hydrological process Xt and of the thresholdxo. More formally, it can be defined as:

Dc ¼XLd

t¼l

Dt ¼XLd

t¼l

ðx0 � xtÞ for Xt � x0 ð1Þ

160

where Dt and Ld are both random variables. Derivation of the pdf of Dc can bepursued by combining the pdf of severity conditioned on a fixed length Ld withthe distribution of drought length Ld, by means of the total probability theorem(Sen, 1977):

fDcðdcÞ ¼

X1

k¼1

fDcjLd¼kðdcÞ � fLdðkÞ ð2Þ

If the distribution of single deficits Dt is known, it should be possible from atheoretical standpoint, to derive the distribution of DcjLd since DcjLd ¼

PLd

t¼1 Dt.In practice, the above derivation can be carried out in closed form only in veryfew cases, depending on the distribution of Dt which, in turn, depends on thedistribution of the underlying process Xt and on the threshold x0. Therefore atraditional approach consists in assuming a parametric distribution for Dc,which can be fitted to series of observed drought severities (Sen 1980; Guven,1983; Zelenhasic and Salvai, 1987; Mathier et al., 1992; Sharma, 1995; Shiau andShen 2001). On the other hand, following a non parametric approach, themoments of drought severity Dc can be derived as a function of the moments ofDt and Ld. This suggests to estimate the parameters of the distribution of Dc asa function of the moments of Dt and Ld. Indeed, under the hypothesis ofindependent processes the first two moments of Dc are (Sen, 1977):

E½Dc� ¼ E½Dt� � E½Ld� ð3Þ

Var½Dc� ¼ E½Ld� � Var½Dt� þ Var½Ld� � E½Dt�2 ð4Þ

For stationary and time-independent series, the drought duration Ld follows ageometric distribution with parameter p1 ¼ P½xt > x0� (Yevjevich, 1967), that is:

fLdðldÞ ¼ p1 � ð1� p1Þld�1 ð5Þ

and therefore

E½Ld� ¼1

p1; Var½Ld� ¼

1� p1

p21

It can be easily shown that the distribution of single deficits Dt is simply thetruncated distribution of the underlying random variable with sign changed andshifted of x0 namely:

fDtðdtÞ ¼

1

p0� fXtðx0 � dtÞ � IðdtÞð0;1Þ ð6Þ

In particular, if Xt is lognormal with parameters (ly; ry), the probability densityfunction of Dt is:

fDtðdtÞ ¼

1

p0� 1ffiffiffiffiffi

2ppðx0 � dtÞ � ry

� e�1

2�lnðx0�dt Þ�l

yry

h i2

� IðdtÞð0;1Þ ð7Þ

161

with p0 ¼ Uln x0�ly

ry

� �and the first two moments of single deficits Dt can be

computed by using the following equations:

E Dt½ � ¼ x0 �lx

p0� D ð8Þ

and

E D2t

� �¼ x2

0 þl2

xer2y

p0�W� 2lxx0

p0� D ð9Þ

where lx ¼ elyþr2y2 ;D ¼ U

ln x0 � ly

ry� ry

� �and

W ¼ Uln x0 � ly

ry� 2ry

� �:

From Eqs. (8) and (9), the variance of Dt can be derived as:

Var Dt½ � ¼ E D2t

h i� E Dt½ �2¼

l2xe

r2y

p0�W� l2

x

p20

� D2 ð10Þ

Combining Eqs. (8) and (10) with Eqs. (3) and (4), the moments of Dc can beexpressed as a function of the parameters of the distribution of Xt (assumedlognormal) and of the threshold x0. The latter can be parametrized as (Yevjevich,1967):

x0 ¼ lx � arx ¼ lx 1� aCvð Þ

where lx; rx and Cv are respectively mean, standard deviation and coefficient ofvariation of Xt, while a is a parameter. Substituting, the following expressions forp0; p1;D and W are obtained:

p0 ¼ U1

2ry þ lnð1� aCvÞ

� �; p1 ¼ 1� p0 ¼ 1� U

1

2ry þ lnð1� aCvÞ

� �

D ¼ U � 1

2ry þ lnð1� aCvÞ

� �; W ¼ U � 3

2ry þ lnð1� aCvÞ

� �

Using these expressions in Eqs. (3) and (4), the following results are determined:

E Dc½ � ¼lx

p1� p0ð1� aCvÞ � D

p0

� �ð11Þ

Var Dc½ � ¼ l2x � er2

yW

p0p1� D2

p20p1þ p0

p21

1� Dp0

� �� aCv

� 2( )

ð12Þ

162

Generally, gamma and exponential distribution are employed to describe thedrought severity distribution (Zelenhasic and Salvai, 1987; Mathier et al., 1992;Shiau and Shen, 2001). In the present work a gamma distribution is assumed fordrought severity Dc:

FDcðdcÞ ¼

Zdc

0

1

bCðrÞz

b

� �r�1

e�zbdz ¼ P r;

dc

b

� �ð13Þ

where P r; dc

b

� �¼R dc

b

01

CðrÞ tr�1e�t dt is called incomplete gamma function (Abra-

mowitz and Stegun, 1965). For such a distribution, the expected value and thevariance are respectively:

E Dc½ � ¼ r � b; Var Dc½ � ¼ r � b2

Solving for r and b in the above expressions and substituting Eqs. 11 and 12, thefollowing parameters of gamma distribution can be derived as:

rr ¼ p0 1� aCvð Þ � D½ �2

er2yWp0p1 � D2p1 þ p0ðp0 � D� ap0CvÞ2

ð14Þ

and

bb ¼ lx

er2yWp0p1 � D2p1 þ p0ðp0 � D� apoCvÞ2

p1p0 p0 1� aCvð Þ � D½ �

" #ð15Þ

Since ry is related to Cv through the analytical expression Cv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffier2

y � 1p

, in turn,p0; p1;D and W depend on Cv and a, therefore the parameters of the distributionof drought severity Dc are function of a;Cv and lx.

A particular case is shown in what follows. If a threshold equal to the mean isassumed, i.e. x0 ¼ E Xt½ � ¼ lx ¼ e

12r

2yþly , the following expressions for expected

value and variance of Dc are obtained:

E Dc½ � ¼ lx

U 12 ry

�� U � 1

2 ry

�

U � 12 ry

�U 1

2 ry

�" #

ð16Þ

Var½Dc� ¼ l2x

er2yU � 3

2ry

�

U 12ry

�U � 1

2ry

��U � 1

2ry

�

U 12ry

�� �2þU 1

2ry

��U � 1

2ry

�� �2

U 12ry

�U � 1

2ry

�� �2

( )ð17Þ

Therefore, the parameters of the gamma distribution become:

rr¼U 1

2ry

��U �1

2ry

�� �2

er2yU �3

2ry

�U �1

2ry

�U 1

2ry

�� U �1

2ry

�� �3þU 12ry

�U 1

2ry

��U �1

2ry

�� �2 ð18Þ

and

163

bb¼

lx

er2yU �3

2ry

�U 1

2ry

�U �1

2ry

�� U �1

2ry

�� �3þU 12ry

�U 1

2ry

��U �1

2ry

�� �2

U �12ry

�U 1

2ry

�U 1

2ry

��U �1

2ry

�� �( )

ð19Þ

Once again, since ry is univocally related to Cv; for the case when a is equal to 0the parameters of the gamma distribution are completely defined by the coeffi-cient of variation Cv and the mean lx of the underlying variable Xt.

3Estimation of return period of drought severityThe return period of a drought may be considered as the average elapsed time, ormean interrival time, between two drought events with a fixed severity or greater.The interarrival time L can thus be defined as the period of time from the be-ginning of a drought to the beginning of the next one, that is, the sum of droughtduration Ld and nondrought duration Ln. Fernandez and Salas (1999) give aslightly different definition of interarrival time between two droughts, namely theperiod of time between the end of a drought and the end of the next one. It maybe worthwhile to note that both definitions are equivalent for what follows,therefore all the analytical derivations will be developed with reference to the firstdefinition.

Under the assumption of independent and identically distributed droughtevents, Shiau and Shen (2001) have developed a procedure, described below, toderive the return period of a drought with severity Dc greater than a fixed value dc.

The interarrival time between two drought events with severity equal to orgreater than a fixed value can be written as:

TDc¼XNdc

i¼1

Li ð20Þ

where Li is the drought interarrival time between any two successive droughtevents (i.e., Li ¼ Ldi

þ Lni) and Ndc

is the number of drought events until theoccurrence of the next drought with severity greater than dc (refer to Fig. 1 forfurther details).

The return period is defined as the expected value of TDc, which through Eq. 20

can be expressed as:

E TDc½ � ¼ E

XNdc

i¼1

Li

" #; i ¼ 1; . . . ;Ndc

ð21Þ

Making use of a property of conditional expectation (Mood et al., 1974):

EXNdc

i¼1

Li

" #¼ E E

XNdc

i¼1

LijNdc

" #" #ð22Þ

and reminding that for Li identically distributed,

164

EXNdc

i¼1

LijNdc

" #¼ Ndc

E L½ � ð23Þ

Equation 21 becomes:

E TDc½ � ¼ E Ndc

½ �E L½ � ð24Þ

Let FDcðdcÞ be the cumulative distribution function (cdf) of drought severity.

Recalling the definition of Ndc, it can be proven that it has a geometric distri-

bution:

P Ndc¼ nð Þ ¼ FDc

dcð Þn�1 1� FDcdcð Þ½ � ð25Þ

with expected value:

E Ndc½ � ¼ 1

1� FDcðdcÞ

ð26Þ

Combining Eqs. (23) and (26), Shiau and Shen (2001) have found out that thereturn period can be evaluated by the following equation:

E TDc½ � ¼ E L½ �

1� FDcðdcÞ

ð27Þ

Hence, the evaluation of E½TDc� requires the knowledge of the expected value of

interarrival time L and the cdf of drought severity Dc.The drought and non drought durations have the same geometric distribution

with parameters respectively p1 and p0 ¼ 1� p1. Therefore the expected value ofinterarrival time L is equal to the sum of the expected values of two geometricdistributions, namely:

E L½ � ¼ E Ld½ � þ E Ln½ � ¼1

p1þ 1

p0¼ 1

p1p0ð28Þ

Fig. 1. Interarrival time TDcbetween droughts with severity � dc (represented by hatched

area)

165

Combining Eqs. (27) and (28), one gets:

E TDc½ � ¼ 1

p1p0� 1

1� FDcðdcÞ

ð29Þ

The above equation allows to derive the return period E½TDc � once the cdf ofdrought severity is known. Substituting the gamma distribution for Dc introducedin the previous section, Eq. (29) may be written as:

E TDc½ � ¼ 1

p1p0� 1

1�R dc

01

bCðrrÞzb

� �r�1e�

zbdz

ð30Þ

which can be also expressed as:

E TDc½ � ¼ 1

p1p0� 1

1� P rr;d�c lx

bb

� � ð31Þ

where d�c ¼ dc

lxis the dimensionless drought severity and rr and bb are function of

the underlying lognormal distribution of the parameters of the hydrologicalvariable. It is interesting to observe that substituting rr and bb from Eqs. (14) and(15) in Eq. (31), the return period of dimensionless drought severity d�c willdepend only on a and Cv, i.e. on the parameter which fixes the threshold and onthe coefficient of variation of the underlying hydrological series.

The method proposed in the present paper is first applied to generated data.Indeed, a 100,000-years rainfall series lognormal distributed is generated with co-efficient of variation Cv equal to 0.2 and 0.25. Drought events are identified by usingrun analysis and the corresponding durations and severities are determined.

In order to verify the goodness of fit of the gamma distribution to droughtseverities, the sample dimensionless severities identified on the generated seriesand the quantiles obtained for the corresponding empirical frequency by theinverse of the gamma distribution are plotted in Fig. 2 for different a and Cv. Inparticular, since the threshold has to be representative of the water demand, a hasbeen chosen equal to 0 and 0.2, so that x0 is respectively equal to the long-periodprecipitation mean and to lx � 0:2rx, which can represent the level of watersupply recognized as a drought beginning (Yevjevich, 1967).

As it can be noticed, the values are well distributed along the theoretical line,which means the gamma distribution provides a good approximation of droughtseverity behaviour.

Further, the return period for the generated series is evaluated by definition asthe averaged elapsed time between two drought events with severity greater than afixed value of dimensionless drought severity d�c . Then, the return period iscomputed again by using Eq. (31). More specifically, a gamma distribution withparameters derived by means of Eqs. (14) and (15) is applied.

A comparison between the return periods of a set of d�c obtained either nu-merically or by applying the derived model to the generated data is shown inFig. 3 for different values of threshold and Cv. Numerical results are denoted bysymbols, while theoretical results are represented as continuous lines.

Return periods computed by the model fit well to numerical results, except forthe larger dimensionless drought severity d�c because of the smaller number ofdrought events corresponding to such severities in the generated data set.

166

Hence, the reliability of the proposed procedure under the assumption ofrainfall series lognormal distributed and drought severity Dc gamma distributedis verified on generated data.

4Applications

4.1Model testing on observed precipitation dataThe developed model has been tested on four historical precipitation seriescovering a rather long period of observation recorded in four stations in Sicily(Italy). The characteristics of investigated stations are reported in Table 1 fromwhich it can be inferred that coefficients of variations range from 0.24 to 0.28.Also, skewness coefficients Cs, sensibly different than 0, indicate the non-nor-mality of the series under study. Drought events have been identified on theobserved data by run method using two different thresholds corresponding to thevalues a ¼ 0 and a ¼ 0:20 respectively.

Then, the return periods of drought events corresponding to different di-mensionless severities d�c have been estimated by computing the mean interarrivaltime between successive observed droughts characterized by dimensionless se-verities dc=lx greater than several values d�c . In particular, d�c ranging from 0.05 to0.6 have been considered. Larger values of d�c would have led to unreliable returnperiod estimates due to the limited number of identified droughts on the series.Observed return periods have been compared with the theoretical values

Fig. 2. Comparison between numerical sample quantiles and theoretical quantiles ofdimensionless drought severity d�c

167

computed by means of the procedure outlined in Sect. 3. More specifically, foreach series and for each threshold, the parameters of the distribution of droughtseverity (assumed gamma), have been estimated on the basis of the coefficient ofvariation of annual precipitation (assumed log-normally distributed) and of thethreshold parameter a, through Eqs. (14) and (15). Then theoretical return pe-riods have been computed through Eq. (31). The log-normality of annual pre-cipitation has been preliminarly tested for all four series through the chi-squaretest, which did not reject the null hypothesis of log-normality at a 5% significancelevel.

The comparison of observed return periods and theoretical ones is reported inFig. 4, for each series and for each threshold parameter a. It can be inferred fromthe plots a general close agreement between observed and theoretical values. Thisseems to confirm the adequacy of the developed theoretical approach to modelreturn periods of annual precipitation drought severity.

From the figure, it can be inferred that observed values are closer to thetheoretical ones when the threshold level is equal to the mean, that is a ¼ 0. Apossible reason for this might be because of the larger number of identifieddrought events compared to the case when the threshold x0 is equal to xx� 0:20sx.Even in such a case however, the model looks suitable for computing returnperiods of drought events.

Fig. 3. Return period of drought events derived on the basis of generated data Xt log-normal distributed either numerically (�) or by theoretical model (�) for various Cv and a

Table 1. Characteristics of the stations used for testing the proposed model

Stations Period ofobservation

No. of yearsof observation

Cv Cs

Agrigento 1886–1998 113 0.27 0.51Trapani 1881–1998 118 0.27 0.34Petralia 1881–1998 118 0.24 0.67Caltanissetta 1879–1998 120 0.28 0.69

168

4.2Spatial analysis of return period of drought events in SicilyAn analysis of drought in Sicily has been carried out by applying the proposedmethodology for estimating the spatial variability of drought return periods overthe whole sicilian territory. For this purpose, annual precipitation series recordedbetween 1921 and 1998 at 171 rainfall stations located over the region have beeninvestigated. A preliminary trend analysis was carried out in order to identifyseries presenting significant linear trend, to be excluded from further investiga-tions. On the basis of Student’s t test, for 83 out of the original 171 annualprecipitation series the no-trend hypothesis was rejected at a significance level5%, thus leaving only 88 stations. Then, chi-square test has been applied in orderto test the goodness of fit of lognormal distribution to such precipitation series.The null hypothesis was rejected at 5% significance level only for 4 out of 88series, which indicate the adequacy of the lognormal distribution to model annualprecipitation in Sicily. Statistics of annual precipitation series recorded in theselected 88 stations are reported in Table 2.

For each rainfall series the coefficient of variation was estimated as the ratiobetween the sample standard deviation and the sample mean. This allowed tocompute the parameters rr and bb through Eqs. (14) and (15) for different a values

Fig. 4. Return period of drought events observed on annual precipitation series (�) andtheoretical values (�) vs. dimensionless severity for different thresholds

169

and finally to estimate the return period E½TDc� for fixed dimensionless severity d�c

by Eq. (31). In the present application, a values 0 and 0.20 and dimensionlessseverity d�c ¼ 1 have been considered. The return periods computed at each

Table 2. Statistics of annual precipitation in the selected stations

No Stations Mean(mm)

Cv Cs No Stations Mean(mm)

Cv Cs

1 Calvaruso 955.6 0.27 0.57 45 San Cataldo 607.2 0.25 0.402 San Saba 758.6 0.28 2.55 46 Enna 519.1 0.25 0.633 Santa Lucia

del Mela825.2 0.23 0.58 47 Granci 543.2 0.27 0.87

4 Castroreale 734.6 0.24 0.42 48 Caltanissetta 466.7 0.29 0.685 Barcellona 774.4 0.22 0.45 49 Licata 501.3 0.31 0.446 Montalbano

Elicona964.9 0.24 1.19 50 Butera 541.2 0.30 0.79

7 Tindari 693.0 0.17 )0.08 51 Gela 408.1 0.30 0.698 Tortrici 1084.8 0.18 0.17 52 Acate 768.5 0.25 0.249 Alcara Li Fusi 981.7 0.19 0.35 53 Cozzo Spadaro 478.4 0.27 1.28

10 San Fratello 931.7 0.18 )0.30 54 Noto 588.4 0.34 0.6211 Castel di Lucio 846.4 0.20 1.15 55 Canicattini B. 684.0 0.30 0.9012 Tusa 816.9 0.24 0.04 56 Palazzolo A. 891.9 0.30 0.5713 San Mauro C. 846.2 0.18 )0.02 57 Floridia 904.3 0.34 0.4914 Castelbuono 793.0 0.20 )0.15 58 Siracusa 753.4 0.37 0.6515 Cefalu 681.4 0.21 0.21 59 Francofone 554.6 0.32 0.5816 Termini I. 688.9 0.19 0.45 60 Lentini 711.8 0.32 0.4417 Vicari 780.2 0.20 )0.14 61 Cesaro 649.9 0.38 0.3718 Campofelice 670.2 0.30 0.19 62 Maletto 631.4 0.28 0.8019 Marineo 737.9 0.20 0.16 63 Bronte 651.7 0.22 0.7520 San Giuseppe J. 685.4 0.17 0.37 64 Capizzi 718.8 0.27 0.9421 Alcamo 689.3 0.18 0.39 65 Cerami 859.3 0.22 0.5422 Specchia 473.8 0.25 0.23 66 Ragalna 507.7 0.28 1.4423 Lentina 491.8 0.23 0.65 67 Centuripe 664.1 0.31 1.0024 Fastaia 506.3 0.22 0.13 68 Motta 494.7 0.30 1.1325 Ciavolo 481.7 0.30 1.11 69 Simeto 474.0 0.34 1.9226 Marsala 523.6 0.27 1.72 70 Leonforte 467.4 0.38 1.8027 Castelvetrano 981.9 0.21 )0.19 71 Valguarnera 454.1 0.33 1.0928 Roccamena 534.8 0.19 )0.18 72 Catenanuova 643.0 0.27 1.0729 Gibellina 681.8 0.23 0.31 73 Mirabella I. 465.6 0.31 1.0030 Sambuca 668.5 0.20 0.37 74 Caltagirone 583.8 0.34 0.9131 Prizzi 743.6 0.22 1.12 75 Ramacca 590.0 0.30 0.6032 Piano del L. 830.9 0.21 0.40 76 Castel di J. 543.2 0.28 0.4433 Ribera 576.7 0.20 0.67 77 Nicolosi 617.8 0.29 0.5134 Cattolica E. 822.9 0.20 0.01 78 Linguaglossa 1131.9 0.42 2.1135 Casteltermini 608.9 0.25 0.58 79 Fieri 1226.2 0.37 1.1036 S. Caterina V. 585.8 0.24 0.45 80 Viagrande 1345.5 0.37 1.2537 Vallelunga 626.0 0.24 0.63 81 Piedimonte E. 1123.7 0.35 1.5738 Mussomeli 614.9 0.25 0.38 82 Catania 1068.5 0.32 0.6339 Racalmuto 644.9 0.21 0.17 83 Floresta 813.7 0.34 0.7540 Bompensiere 604.2 0.21 0.65 84 Roccella V. 682.1 0.43 1.2341 Raffadali 714.3 0.22 0.62 85 Antillo 1072.2 0.31 1.0642 Agrigento 501.3 0.27 0.79 86 Casalvecchio S. 765.8 0.29 0.5343 Canicattı 627.7 0.24 0.18 87 Camaro 1356.9 0.31 1.4744 Palma di M. 804.1 0.22 0.73 88 Santo Stefano B. 954.1 0.28 0.69

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station have been spatially interpolated. Since the interest of the study is to obtaingeneral information on the spatial distribution of return period of drought eventsin Sicily and to detect significant differences in drought behaviour rather than tofind accurate point estimates of drought return period, the inverse square dis-tance has been chosen as interpolation method for the sake of simplicity. Inparticular, such interpolator assumes that each point has a local influence thatdecreases with distance and it weights the points closer to the processing cellgreater than those farther away.

Once that the interpolation method has been selected, maps of the spatialvariability of return period for drought with dimensionless severity d�c ¼ 1 inSicily have been plotted by means of Arcview-GIS software for the two consideredthreshold parameter values a. The two maps, depicted in Figs. 5 and 6 provideuseful insights on the spatial distribution of droughts in Sicily. Obviously, areascharacterized by lower return periods are prone to more frequent drought eventsof the same severity or greater.

From both maps it can be observed that the South-Eastern side and the farWestern side are more frequently affected by severe droughts. In particular, thereturn period of drought occurrence for the areas around Catania (No. 82) andSiracusa (No. 58) is less than 100 years for both of the two values of threshold. Onthe other hand, for the central part of the region the variability of return periodchanges considerably depending on the threshold used for drought identification.This last result underlines the importance in selecting an appropriate threshold inrelation to the water demand which has to be satisfied.

5ConclusionsAnalytical expressions for the return period of both dimensional and dimen-sionless drought severity have been derived assuming the underlying hydrological

Fig. 5. Spatial distribution of theoretical return period of drought events with severityequal to the mean annual rainfall and threshold x0 ¼ xxða ¼ 0Þ

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variable to be independently and identically log-normally distributed. In order toevaluate the return period, the probability distribution of drought severity isrequired. The latter can be derived analytically by combining the conditionaldistribution of drought severity given a drought duration and the distribution ofdrought duration. Since a direct analytical derivation for the distribution of Dc isnot possible, here a gamma distribution has been assumed. The parameters ofsuch distribution have been expressed as a function of the mean and the coeffi-cient of variation of the hydrological variable and of the adopted threshold.Although from a theoretical standpoint the gamma assumption for drought se-verity is not rigorous, simulation studies based on generated log-normal serieshave shown that the gamma distribution is appropriate to model the probabilisticbehaviour of drought severity.

Substitution of such cdf in the formulation of return period developed byShiau and Shen (2001), allows to express the return period of droughtseverity as a function of the adopted threshold and of the statistics of theunderlying hydrological variable. In particular, assuming a threshold functionof the mean and of the standard deviation of the original process, it has beenshown that the return period of dimensionless severity depends only on thecoefficient of variation of the hydrological series and on the parameterthat defines such a threshold. The latter results allows to characterize therecurrence interval of drought severity on the basis of few parameters, whichcan be easily computed on an observed sample of the hydrological variableunder investigation.

The developed expressions have been tested on fairly long series (almost 120years) of annual precipitation observed in Sicily. The comparison between the-oretical and observed return periods for different dimensionless drought sever-ities indicates a remarkable agreement, which points out the validity of theproposed methodology, and seems to indicate the possibility to extrapolate,

Fig. 6. Spatial distribution of theoretical return period of drought events with severityequal to the mean annual rainfall and threshold x0 ¼ xx� 0:2sx ða ¼ 0:2Þ

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within reasonable limits, information about return period of drought severity,beyond the length of the observed series.

The potential of the methodology for the assessment of drought risk withina region has been investigated through the evaluation of the spatial variabilityof return period of droughts with dimensionless severity greater than afixed value over the Sicily region. In particular, maps showing variabilityof return period of drought severities have been created by interpolation oflocal values computed in 88 rainfall stations. The maps indicate that theSouth-Eastern side and the far Western side are more frequently affectedby severe droughts.

Forthcoming research is being carried out in order to develop a bivariateformulation of return period that takes into account both duration and severity ofdrought events.

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