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Hydrological Sciences-Journal-des Sciences Hydrologiques, 45(2) April 2000

285

Use of a two-component exponential distribution in partial duration modelling of hydrological droughts in Zimbabwean rivers THOMAS R. KJELDSEN, ALLAN LUNDORF & DAN ROSBJERG Department of Hydrodynamics and Water Resources, Technical University of Denmark, Building 115, DK-2800 Lyngby, Denmark e-mail: [email protected] Abstract An investigation of hydrological droughts has been conducted, based on the truncation level approach: each drought event is characterized by its duration and deficit volume. The truncation level is defined to reflect the expected natural water availability and, therefore, is evaluated monthly as a fixed percentile of the monthly flow-duration curve. Thus, the problem of expected zero flow of ephemeral rivers during the dry season is taken care of. The data material consists of daily discharge data from ten Zimbabwean rivers, and both ephemeral and perennial rivers are included in the analysis. The partial duration series approach is used to predict the severity of future droughts, i.e. the T-year events. The two-component exponential distribution is adopted as exceedence distribution for both duration and deficit volume. The parameters of the two-component exponential distribution are estimated using the maximum likelihood method. A method for calculating the T-year event and an approximate expression of the uncertainty of the T-year events have been developed. An observed problem of underestimation of observed deficit volumes is reduced by the introduction of censoring in the partial duration series. A better description of the observed events has been obtained by censoring the duration and deficit volume series. A relationship between optimal censoring and the coefficient of variation of the drought series has been indicated.

Utilisation d'une distribution exponentielle à deux composantes pour la modélisation de la durée des étiages de rivières du Zimbabwe Résumé Nous avons réalisé une étude des étiages basée sur un modèle de dépassement de seuil. Un seuil reflétant la disponibilité naturelle de l'eau (exprimé pour chaque mois comme un certain quantité de la courbe débit-durée, permettant ainsi de tenir compte des débits nuls des rivières intermittentes) ayant été fixé, chaque étiage est défini par sa durée et par le volume du déficit. Les données utilisées sont les débits journaliers de dix rivières du Zimbabwe. L'analyse inclut à la fois des rivières intermittentes et des rivières pérennes. Le modèle de dépassement a été utilisé pour estimer la sévérité des futurs étiages. La loi exponentielle à deux composantes a été adoptée pour modéliser la durée des étiages ainsi que les volumes des déficits. Les paramètres de la loi exponentielle à deux composantes ont été estimés par la méthode du maximum de vraisemblance. Une méthode de calcul de l'événement de période de retour T, ainsi qu'une expression approximative de l'incertitude qui lui est associée ont été développées. Nous avons pu constater un problème de sous-estimation des volumes des déficits qui a pu être corrigé en tronquant certaines données des séries de durées partielles, ce qui a permis une meilleure description des événements observés. Une relation entre le seuil de troncature optimal et le coefficient de variation des séries a été établie.

INTRODUCTION No exact definition of the term drought is found in the hydrological literature, but a drought implies a period of time where the supply of water cannot meet the water

Open for discussion until I October 2000

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demand. Such a situation can occur because of hydrological, technical, pollution or management problems. Therefore, it is important to distinguish between drought and non-drought water shortage. A drought is related to deficit in the water availability in the hydrological cycle. The water demand from a specific source must reflect the expected water availability of the considered source and, therefore, a drought is also a period of less water than expected. As water availability and demand vary in both time and space, a drought is a relative term that has to be analysed according to the actual situation. By intuition, a short period of supply not being able to fulfil demand without any significant consequences is not a "real drought", thus making a censoring of the drought series necessary.

HYDROLOGICAL DROUGHTS An objective definition of surface water droughts was presented by Yevjevich (1967) based on the truncation level approach. The basis for the drought analysis is a time series of river discharge (annual, monthly or daily discharge). The truncation level is an analytical interpretation of the water demand or the expected availability of water. When the flow in the river falls below the truncation level, a drought has occurred. The drought events are characterized by the three parameters duration, deficit volume and time of occurrence, denoted D, S, and T, respectively. In this study only the properties of D and S are investigated. Substantial amounts of literature exist on the modelling of droughts using annual runoff, e.g. Sen (1980). Zelenhasic & Salvai (1987) applied the truncation level approach with daily discharge data in an investigation of hydrological droughts using partial duration series (PDS) data. They used discharge data from two Yugoslavian rivers and concluded that both duration and deficit volume could be described by the one parameter exponential distribution. As daily data were used, the duration was modelled as a continuous random variable, even though it is in fact discrete. Madsen & Rosbjerg (1998) adopted the same approach in an investigation of droughts from two Danish streams. They concluded that the generalized Pareto distribution was able to give a satisfactory description of the duration, but unable to model the deficit volume properly. The lack of fit for the deficit volume was due to the presence of many small and few very big events. Woo & Tarhule (1994) analysed droughts from four Nigerian rivers using the same approach (daily discharge, steady truncation level and PDS). They identified two major clusters of drought events (both duration and deficit volume), a big population of small events and a small population of big events. As a solution to the problem, they removed the biggest events and divided the remaining events into two sub-populations, which were modelled separately. The small droughts were modelled according to a two-parameter Weibull distribution and the long droughts were modelled by the two-parameter normal distribution. Common for all the above mentioned investigations is that the applied truncation level is calculated as a certain percentile in the flow-duration curve for each river and assumed steady during the year, e.g. the truncation level is equal to the flow exceeded 75% of the time on an annual basis. Mathier et al. (1992) extended the truncation level model. They applied monthly runoff data and a truncation level, which varied from month to month according to the flow in the river.

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ADOPTED DEFINITIONS Zimbabwe has a semiarid climate with a single rainy season followed by a dry period. Therefore, the expected flow of Zimbabwean rivers varies significantly from season to season. As a result of the prevailing climatic conditions, the variable truncation level method has been adopted in this investigation. A truncation level, which is calculated as a fixed percentile of the flow duration curve for each month, will reflect the expected flow of the river and therefore give a more realistic drought pattern than a steady annual truncation level. It is the experience of the authors that the problem of many small and a few big events remains the same for all truncation levels defined between 50 and 90%. In this study, a flow-duration percentile of 75% has been chosen. As ephemeral rivers have been included in the investigation, a problem of defining the term drought has occurred. When considering an ephemeral river, the expected flow during the dry season is zero, which gives a truncation level equal to zero in that period. The problem occurs when a drought starts in the rainy season and continues into the following dry period. When the river runs dry in the period where it is expected to run dry, does the drought from the rainy season continue into this dry period? In this study, the following approach has been adopted: if a drought is present in the rainy season and continues into the dry season with zero truncation level, the duration of the drought will increase while the deficit volume remains constant, as illustrated in Fig. 1. When the rainy season is expected to occur once again, the truncation level will be greater than zero. If the flow in the river is still below the truncation level, the drought continues. If, on the other hand, the flow exceeds the truncation level, the drought has ended. Perennial rivers yield no problem regarding the drought definition. A very important aspect to consider in drought modelling is the problem of dependency between successive drought events as recognized by Zelenhasic & Salvai (1987). The problem occurs when a prolonged drought period is interrupted by short

Discharge Truncation level

L Sep

Oct

Nov

Dec

Jan

Feb

Mar Apr Time

May

Jun

Jul

Aug

Fig. 1 Abstraction of drought events using a variable truncation level.

Sep

Oct

Nov

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insignificant wet periods, which divide the prolonged drought into two or more smaller droughts. The presence of minor droughts instead of one major drought will distort the estimation of the parameters in modelling the drought events. Three methods of solving the problem were compared by Tallaksen et al. (1997): the inter-event time and volume criterion method, the moving average (MA) method, and the sequential peak algorithm (SPA). They concluded that the MA and the SPA methods were preferable in combination with PDS modelling. Furthermore, the SPA method was found to be superior in terms of abstracting large events, which are considered to be the most important in this study. Therefore, the SPA method has been adopted in this study. An introduction to the SPA method in drought modelling is given by Tallaksen etal. (1997).

DATA MATERIAL The data material analysed in this study originates from Zimbabwe. Ten gauging stations categorized as measuring natural flow were selected (see Fig. 2). All ten flow series contained minor periods (maximum one year) of missing measurements.

Fig. 2 Map of Zimbabwe indicating location of gauging stations.

Use of a two-component exponential distribution in partial duration modelling

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Applying the conceptual NAM rainfall-runoff model filled holes in the flow series. Historical records of runoff, precipitation and evaporation from each site were used to calibrate and validate the NAM model before modelling the missing periods of runoff, The climate of Zimbabwe varies from a very dry area in the southwestern part bordering the Kalahari Desert (mean annual precipitation, MAP of 500 mm) and a more humid area in the Eastern Highlands (MAP > 1000 mm). The different climatic conditions give rise to two different types of river: perennial and ephemeral, both of which are included in this investigation. Table 1 is a schematic presentation of the ten rivers. Table 1 Information on the ten gauging stations. Station

A61 B78 C6 C12 C22 C24 D28 E2 E49 E114

Area (km )

622 49 1036 5180

231 189 70 541 1010

197

River type

Start year

Number of years

Eph Eph Eph Eph Per Per Per Eph Per Per

1972 1969 1948 1950 1952 1953 1927 1928 1959 1973

24 28 47 47 44 43 69 66 35 22

PARTIAL DURATION SERIES (PDS) MODELLING The PDS consists of all the drought events abstracted from the initial flow series that exceed a certain threshold value XQ. The statistical modelling of the PDS adopted in this study is based on the following two assumptions: (a) The occurrence of droughts follows a time homogeneous Poisson process with constant intensity X (expected number of drought events per time unit): P{N(t) =n} = ^—pexp(-to)

(1)

where N(t) is the number of droughts in the time interval [0;t]. (b) The magnitude X of the threshold exceedence follows the probability density function (pdf) f(x), also denoted the exceedence distribution. As described in the introduction, previous studies of hydrological droughts show that generally the extracted drought events belong to two sub-populations, where the upper population, i.e. the severe droughts, are found difficult to model with the proposed exceedence distributions. Rossi et al. (1984) successfully applied a twocomponent exponential (TCE) distribution to model series of annual maximum flood data from Italy consisting of two sub-populations. The flood series consisted of a lower population of normal events and an upper population of flood events from special Mediterranean storms. In the following sections, the TCE distribution is introduced in PDS modelling and procedures for estimating the T-year event and the corresponding uncertainty are presented. First, a brief introduction to the TCE distribution is given.

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Consider a PDS with a total number of events N in time t. The series consists of a contribution from a lower population N\ and from an upper population A^: N = N]+ N2

(2)

Both N\ and N2 occur according to independent Poisson processes. The Poisson intensity is therefore calculated as: X = Xi+X2

(3)

The pdf of the TCE-distribution is derived as a mixture of two exponential distributions: f(x) = fl(x) +

(l-p)f2(x)

(4)

where both fi(x) and f2(x) are one parameter exponential distributions: j =1,2

f ; 0 ) = — exp

(5)

CI;

Therefore f \ (l_p) _ e x p - i f (x) = p — exp —x (6) a2 \a2 j V"i J if x > 0, otherwise zero. The corresponding cumulative distribution function (CDF) is given as: f

F(x) = p\ 1-exp - x

\

r

+ (l-p)

1-exp

\

(7)

—X

if x > 0, otherwise zero. The probability of a drought event belonging to the lower population is: (8)

p = -X

The three parameters in the TCE distribution are estimated simultaneously using the maximum likelihood (ML) method as proposed by Hasselblad (1969). The natural logarithm of the likelihood function is given as: ( •'

(9) nf(*,-> =i>(f(*,.)) v ;=i The ML estimators of a\, a2 map are found by setting the partial derivatives of ln(Z) with respect to a\, aj and/» equal to zero: ln(Z,) = In

dln(L) da,

^

I

-1

~L777^P — w f(*;) \[a.

ex

'-O

X,

"1 J

"1 a,

PV

f

&z2

,=1 f(x ( )

[a

f-x^

-x ; a2 ) a

(10)

=0

Texp

-exp

'-O

=0

(11)

Use of a two-component exponential distribution in partial duration modelling

dp

h

291

(12)

f(*,-)

From equations (10), (11) and (12), the estimators of the three parameters are calculated iteratively by means of: f.(x,.|9)x.

I f O; 6) f.(x,.9) I f(x,.|9) />"' =

P

i

(13)

J = 1. 2

f,(x,.|e)

(14)

f(x,.|9)

As the equations are implicit functions of the parameter estimators, the estimation of the parameters has to be carried out by iterative solution of the three equations. The parameter vector 9 = (a\v, a2v,pv) contains the parameter values after the vth iteration. According to Hasselblad (1969), between 100 and 1000 iterations should enable a solution to the equations.

J-YEAR EVENT In PDS modelling, the T-year event is calculated as the 1 1 - -— quantile in the exceedence distribution. F(x7.) = l -

1

(15)

XT

When the TCE distribution is used as exceedence distribution, the J-year event xT is found by combining equations (7) and (15) and solving the equation: f

—x. pexp

exp —x.

pexp *,

j

\

1 XT

0

(16)

A determination of the variance of the estimator of the T-year event xT is complicated by the fact that xT is not given as an explicit function. Still xT is a function of a\, ai,p, X and T: XT = g(ci], a2, p,X,T)=

g(9)

(17)

where 9 is the parameter vector 9 = {ci\, a2,p, X, T). The estimation uncertainty of xT is governed by sampling variability of the estimators of a\, a2, p and X. Since the return period T is a fixed quantity, it does not contribute to the uncertainty. The approximate variance of xT is calculated using a Taylor-approximation to equation (17):

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Thomas R. Kjeldsen et al.

varix,

varifl +

%

van{PY p\ +

'' U-

da,

(18)

dg] ( dg^

okJ...

var{À}+2 - ^ \oa.j \da2j

cov\a,,a2

where m means that the derivatives must be evaluated at the mean values of the parameters. Independence between X and the other parameters has been assumed. As the parameter p is a function of X, independence between p and the other parameters has been assumed as well, i.e.: covja,,/)} = cov|a 2 ,/?j = cov|a,,A,| = cov|a,,A,} = 0

(19)

The covariance between p and X, cov{p, X} is approximately zero, which is further elaborated in the Appendix. The partial derivatives of the function xT = g(Q) are calculated following the approach presented by Rasmussen & Rosbjerg (1991): f

\

jcexp -x.

pexp

exp \

U

2J

1 =0 XT

(20)

dg The partial derivatives —— are calculated from:

dh

oh

(21)

• =0 59,. -+- dxT dQ; Analytical expressions of —— are calculated through equations (20) and (21). (9)

var{a,}

cov{a,,a 2 }

cov{ai,p}

cov{a,,a 2 }

var{a,}

cov{a2,p}

cov {p,a^}

cov {p,a2}

var{p}

(22)

where i(9) is the information matrix which contains the second order partial derivatives of the natural logarithm of the likelihood function. f

d2 da

i(9) =

•j-(lnZ)

d2 daxda2 (lnX) d2 (InZ) daxdp

•(In I )

d2 (InZ) daxdp

d2 da„—fQnL)

d2 (InL) da2dp

datda2

d2

——-(In I ) oa2op

d2

-r(lnZ)

op

(23)

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293

Analytical expressions of

-(lnZ), ——-—(InZ), 7 (In L) and —r- (In L) are oa{ aaxoa2 oa{ op rather cumbersome and therefore omitted, but can be calculated through equations (10), (11) and (12). The variance of the Poisson parameter X is estimated as: varlA,} = —t where both X and t are defined in connection with equations (2) and (3).

APPLICATION In the following, the proposed framework is applied to PDS of drought events (both duration and deficit volume) extracted from the Zimbabwean rivers using the SPA method. Deficit volume is normalized using the mean runoff (m3 day 4 ) from the considered river to be able to compare results from different rivers. Therefore, both duration and deficit volume are given in days. Figure 3(a) and (b) shows the two drought characteristics, deficit volume and duration, plotted against the non-exceedence probability P{X