Modelling and Management of Sustainable Basin-scale Water Resource Systems (Proceedings of a Boulder _ Symposium, July 1995). IAHS Publ. no. 231, 1995. 377

On the modelling of extreme droughts

HENRIK MADSEN & DAN ROSBJERG Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, Building 115, DK-2800 Lyngby, Denmark

Abstract In order to analyse extreme droughts, the truncation level approach is applied. This model considers both the drought duration and the deficit volume as drought characteristics. Definition and modelling of droughts based on discrete and continuous time series, respectively, are discussed, and estimation procedures based on estimated probability distributions are presented. The model is applied to two Danish rivers with contrasting geology. The distribution of the drought duration seems to be unaffected by the flow regime, except in the analysis of annual series where the degree of persistence is important. On the other hand, the distribution of the deficit volume is significantly correlated to the flow properties. The relation between deficit volume and catchment characteristics is, however, not unambiguous, but depends on the trunca­tion level.

INTRODUCTION

In general terms, droughts refer to severe water shortages in all domains of the hydro-logical cycle. This implies that droughts may have very significant impacts on a number of human activities, including, for instance, impacts on the environment, on the perfor­mance of water supply systems and on the hydropower potential. In terms of environ­mental impacts, droughts may seriously affect the vegetation and the natural soil cover and eventually cause desertification. Droughts also affect the dilution capability of rivers, groundwater balances and the deposition of sediments in lakes and reservoirs, just to mention some effects with consequences for water quality and water availability.

Frequency analysis of extreme drought characteristics is an important tool in the evaluation of drought impact. The traditional approach is based on the modelling of annual minima series of droughts of specified duration. This approach, however, involves only one measure of a drought - the drought magnitude. If one is interested in modelling droughts in terms of duration and magnitude (or deficit) simultaneously, the annual minima series approach is not applicable. A method that considers both dura­tion and deficit as drought characteristics is the truncation level approach presented by Yevjevich (1967). Although this definition originally was based on the statistical theory of runs and used for analysing time series with a time resolution not less than one month, it has also been used in the analysis of low streamflows from a daily recorded hydrograph (Zelenhasic & Salvai, 1987).

The main objective of this paper is to evaluate and expand the truncation level approach applied to both discrete and continuous time series. The focus is on analytical modelling, i.e. description of drought characteristics by probability distributions. The

378 Henrik Madsen & Dan Rosbjerg

model is applied to time series of streamflows from two Danish catchments.

DROUGHT DEFINITION

In the truncation level approach droughts are defined as periods during which the hydrological determinant (e.g. streamflow) is below a certain threshold level. The fundamental characteristics are: (a) the drought duration D, which is the distance between a downcross and a following

upcross of the truncation level; and (b) the deficit volume S (often termed the drought severity), which is the sum of deficits

within the dry spell period. A third characteristic, the drought magnitude M, may be defined from the other two as M = SID. The selection of the truncation level is not arbitrary but depends on the type of drought to be studied. For instance, in the analysis of drought impacts on a water supply system, the truncation level should be chosen equal to the water demand. The truncation level can also be chosen on the basis of economic considerations. In Mathier et al. (1992) the acceptable water deficit based on economic requirements of a hydro-power plant was used as the truncation level.

A major problem in the truncation level approach concerns the clustering of dry spell periods. During a prolonged dry period it is often observed that the hydrological determinant exceeds the truncation level in a short period of time, thus dividing a large drought in a number of minor dry spells that are mutually dependent. In practice one will treat the dry period as one drought because short periods of time with insignificant excess volumes will not reduce the impacts of the drought (measured as one drought) significantly. If the system has not recovered after one dry spell, the impacts from a succeeding dry spell is more severe than one would observe if the system was totally recovered prior to the onset of the second dry spell. Hence, a consistent definition of a drought depends on the degree of recovery of the system between two dry spell periods. The degree of recovery may be expressed in terms of the inter-event time and the corres­ponding excess volume, respectively. Two dry spell periods with characteristics {dl,sl) and (d2, s2), respectively, are assumed to be mutually dependent if: (a) the inter-event time is less than a critical value tc; and (b) the ratio between the excess volume and the preceding deficit volume is less than a

critical value pc. The two dry spell periods are then pooled into a single drought event with the

characteristics dpool = dx + d2 and spool = sx + s2. In Zelenhasic & Salvai (1987) the definition of droughts was based only on one criterion, the inter-event time. The above definition, however, is more consistent because recovery is dependent on both the inter-event time and the excess volume.

MODELLING OF SEQUENTIAL TIME SERIES

For drought analysis based on sequential time series with a time resolution of not less than one month, Yevjevich (1967) introduced the theory of runs. It is assumed that the process described by the hydrological determinant Xt, relative to the truncation level x0,

On the modelling of extreme droughts 379

is stationary, i.e. the probabilities p and q of, respectively, a surplus and a deficit, where p + q = 1, are constant at all time steps /. In case of a monthly discretization, the resulting time series is generally non-stationary due to seasonal variations. The series of standardized monthly values obtained by removing periodicity in the mean and the standard deviation can, however, be assumed to be stationary. Alternatively, a monthly varying truncation level equal to a given percentile of the monthly distribution of the hydrological determinant can be used (Mathier et al., 1992).

Due to the large inertia of some processes within the hydrological cycle, persistence may be present. In order to describe this dependence structure Sen (1976) used the two-state lag-one Markov model. The two states are defined by the transition probabilities

r = P{xt < x0\xi_l < x0} v = P{xt > * J * M < x0} (1)

where r + v = 1. Using this approach the probability function of drought duration D becomes (Sen, 1976)

P{D = d} = (l-r)rd-1 d = 1,2,... (2)

which is seen to constitute a geometric distribution. In the case of independent observa­tions D is also geometrically distributed but with the parameter r = P{xt < xQ\xiA <x0} = P{xt < x0} = q. The deficit volume S is equal to the sum of deficits within the dry spell period. Contrary to the distribution of D, the distribution of S has no closed analytical form.

The extreme values of drought characteristics are important when defining esti­mators of design events. Sen (1980) argued that the occurrences of droughts can be described by a Poisson process. Thus, the number N(t) of drought events in the time interval [0,f] is Poisson distributed:

P{N(t) = n} = ^lexp(-Xt) n = 0,1,2,... (3)

where \t is the mean number of drought events in [0,t]. Assuming that droughts are independent, the distribution of the maximum drought duration L(t) and the maximum drought deficit Z(t) in [0,/] are given by (Sen, 1980):

P{L(t) < d} = exp[-Xr(l -P{D < d})]

P{Z(r) < s} = exp[-Xr(l-P{S < s})]

Utilizing!) as being geometrically distributed, i.e. P{D < d} = 1 — r4, an analytical expression of L(t) is obtained from equation (4). The Poisson parameter X can be expressed in terms of q and r

* = pixi+i > xo>xi ^ xo) (5)

= P{xt ^ XQ}P{X1+1 > x0\x( < x0} = q{\ - r)

The estimation of the extreme drought characteristics may be performed in two ways: (a) estimation based on the statistical properties of the hydrological determinant X; or (b) estimation based on the sample properties of the time series of the drought

characteristics D and S.

380 Henrik Madsen & Dan Rosbjerg

Denoting by F,(®) and F2(®, •) the marginal distribution function of Xand the bivariate distribution function of (X,-, XM), respectively, the estimators of q and r based on X become:

q = F,(x0;a) r = W JWOJÔ)

where â = (â1( â2, ...)are the estimated parameters in F,(®), and §_ = (j8,,j82,...) are the estimated parameters in F2(®, • ) . An estimate of X is then found from equation (5). Sen (1976) and Rosbjerg (1977) used the normal distribution function for the marginal distribution and the bivariate normal distribution for the simultaneous distribution of successive events. For normally distributed X, Sen (1977) derived analytical expressions of the mean and the variance of S in case of both independent and dependent processes. Based on these results any two-parameter distribution may be adopted to describe S as exemplified by Giiven (1983) who used the two-parameter gamma distribution.

The sample estimation procedure is based on the samples of, respectively, drought durations dt and drought deficits st, i = 1, 2, ..., N where TV is the number of observed droughts. Maximum likelihood estimators of the Poisson parameter X and the transition probability r read:

X = ^ f = 1- 1 £ { D } = ! 5 X (7) t Ê{D} N& •

The properties of the deficit volume are estimated from the sample by assigning a distribution to S. Compared to the estimation based on the time series of X, the sample estimation procedure is much more simple. However, the reliability of the sample para­meters depends on the sample size of drought events. In the analysis of multi-year droughts (using a yearly discretization) the number of observed droughts may be small even in large records, implying large sampling uncertainties. In this case the estimation based on X may be more efficient. The assumption of Xbeing normally distributed may, however, introduce large model errors (bias). Thus, it is not evident which estimation procedure to apply in practice when analysing multi-year droughts. For time intervals less than one year the sample estimation procedure is preferable in most cases.

MODELLING OF CONTINUOUS TIME SERIES

Daily time series may be assumed to be a sufficiently close approximation to the continuous series. Modelling of continuous time series is closely related to the modelling of discrete series based on the theory of runs. However, two basic differences must be emphasized. First, the theory of runs must be applied to stationary time series, while continuous time series are always non-stationary. Second, the abstraction of droughts from stationary, discrete time series facilitates a consideration of drought durations of more than one year, while abstraction from continuous time series are more suitable for studying droughts related only to the dry season.

The analysis of droughts abstracted from continuous time series may be performed in two ways using either the annual maxima series (AMS) or the partial duration series (PDS) model.

On the modelling of extreme droughts 381

The AMS approach may yield some problems with respect to the definition of the extreme value region. In wet years the abstracted droughts may not belong to the true extreme value region, and, in addition, very wet years may not produce any droughts at all, i.e. the sample of annual maxima may include zero values. Thus, a consistent AMS model should include some kind of censoring. In the PDS model all abstracted droughts are taken into account. However, the abstraction procedure normally produces a bunch of minor droughts that may distort the extreme value modelling. Zelenhasic & Salvai (1987) defined minor droughts as droughts with deficit volumes less than 0.5-1 % of the maximum observed deficit volume, and all these minor droughts were excluded from the analysis. This approach is very sensitive to outliers and a definition based on a given percentage of the mean values of the drought duration and the deficit volume, respectively, may be more consistent.

Having abstracted the relevant droughts the modelling of the extreme drought characteristics using the PDS approach is similar to the model presented by Todorovic & Zelenhasic (1970) in the analysis of floods. It is assumed that the occurrence of droughts can be described by a Poisson process. If the intensity of the process is periodic with a one-year period, which seems reasonable, the number of drought events N(f) in [0,t] is Poisson distributed according to equation (3). Estimation of the maximum drought characteristics is then obtained from equation (4) using estimated probability functions of D and S.

APPLICATION

The drought models are applied to streamflow records from two Danish rivers with contrasting geology and soil type. The catchment of St. 14.01 is predominantly sandy soils, whereas the catchment of St. 59.01 is dominated by soils with a high content of clay. These contrasting catchment characteristics produce different flow regimes cf. the flow duration curves (FDC) shown in Fig. 1. St. 14.01 has a flat FDC reflecting the low variability of the daily mean flow due to a relatively high baseflow contribution. The steep FDC at St. 59.01 indicates a high variability of daily flow values. The records consist of 68 (St. 14.01) and 76 (St. 59.01) years of daily observations, respectively.

% of time flow exceeded 10 3

| 1

I 0,

0.01 -2.5 -l.S 1.5 -0.5 0.5

Standardized normal variate

Fig. 1 Daily flow duration curves for the two catchments.

St. 14.01

St.59.01

2.5

382 Henrik Madsen & Dan Rosbjerg

The model based on the theory of runs was applied to time series with yearly and monthly discretization, respectively. In the case of annual series the truncation level was chosen equal to the median, whereas a monthly varying truncation level equal to the median in the monthly distributions was applied to the monthly series. The PDS modelling approach was applied to the daily series. In this case the 90% quantile of the FDC was chosen as the truncation level since this quantile is often used as a standard low flow index. In the analysis of annual and monthly time series no restrictions were imposed in order to ensure independent drought events. In the analysis of daily series two dry spell periods were pooled if the inter-event time was less than tc = 6 days and the ratio between the excess volume and the preceding deficit volume was less thanpc

= 0.3. To facilitate a comparison between catchments the deficit volume was divided by the mean flow, implying that S has the dimension of time.

The results from the analysis of multi-year droughts are shown in Figs 2 and 3. The observed distributions of D are compared to the estimated geometric distributions in Fig. 2 using the observed sample of D and estimation based on the properties of the basic time series X, respectively. For both stations the annual series are well described by the normal distribution. When the truncation level is chosen equal to the median, then q = Vi and an estimate of the parameter r is then given by (e.g. Sen, 1977) r = Vi + 1/TrArcsin p where p is an estimate of the lag-one serial correlation coefficient. As can be seen from Fig. 2, the two estimation procedures differ only slightly. At St. 59.01 r is close to q indicating that the persistence is relatively modest. At St. 14.01 the observed distribution of D is significantly more skewed than the geometric distribution. In this case the simple two-state Markov model is insufficient to fully describe the persistence. The distributions of 5 are compared to the generalized Pareto distribution (GPD) in Fig. 3. Using the estimation procedure based on X, the mean and the variance of S can be expressed in terms of r and p and the mean and the variance of X (Sen, 1977). The scale end the shape parameter in the GPD is then estimated using the method of moments. As can be seen from Fig. 3, this estimation procedure and the one based on the observed sample of S differ in the tail of the distribution, especially at St. 59.01. The distribution of S at St. 14.01 is long-tailed (negative shape parameter) due to the very extreme observation corresponding to d = 10 years. The distribution of S at

Station 14,01 Station $9,01

Drought duration a' (years) 2 3 4 5 6

Drought duration d (years)

Fig. 2 Histogram of D compared to the estimated geometric distribution using sample estimation (GEOl) and estimation based on the time series (GE02), respectively.

On the modelling of extreme droughts 383

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1.6

1.4

1.2

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0.8

0.6

0.4

0.2

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Station 14.01

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1.4

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0.0 1.0 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Fig. 3 Observed distribution of S compared to the estimated GPD using sample estima­tion (GPl) and estimation based on the time series (GP2), respectively, p denotes the exceedance probability.

St. 59.01 has a positive shape parameter implying an upper bound of the distribution. This feature, however, is not evident from any physical catchment characteristic.

In the analysis of monthly runs only sample estimation has been applied. The observed distribution of D is slightly more skewed than the geometric distribution at both stations (not shown). In addition, the sample estimates of r are almost identical. Hence, the different degree of persistence at the two stations, that was present in the analysis of annual runs, is not observed in the monthly series. In Fig. 4 the observed distributions of S are compared to the estimated GPD. Both stations have long-tailed distributions with almost identical shape parameters. However, the mean values of S differ significantly, the mean value being greater at St. 59.01 than at St. 14.01. This implies that the deficit volume is more severe in the flashy river at St. 59.01.

The abstracted samples of D and S from the daily time series were truncated prior to the analysis in order to exclude minor droughts. For both variables a threshold level equal to 0.3 times the mean value of the total abstracted sample was chosen. The distribution of D is well described by the GPD at both stations (not shown). The scale and the shape parameter are almost identical at the two stations implying that D is not affected by the different flow regimes. The distributions of 5 are shown in Fig. 5. At both stations the GPD implies some lack of fit in the tail of the distribution. This is espe­cially pronounced at St. 14.01 due to very extreme observations. The mean value of S is significantly greater at St. 14.01 than at St. 59.01 which is in contrast to the analysis of monthly series. This difference, however, is mainly due to the use of different truncation levels. For a low truncation level the deficit volume is more severe in persis­tent streams compared to more flashy rivers.

CONCLUSIONS

The truncation level approach is applied for a simultaneous modelling of drought dura­tion D and deficit volume S. Analytical estimation procedures for the evaluation of extreme drought characteristics are presented. A consistent definition of drought events based on the inter-event time and the corresponding excess volume is introduced.

Henrik Madsen & Dan Rosbjerg

Station 14.01 station 59,01

Fig. 4 Observed distribution of S compared to the estimated GPD (monthly series), p denotes the exceedance probability.

Station 1401 Station 59.01

Fig. 5 Observed distribution of 5 compared to the estimated GPD (daily series), p denotes the exceedance probability.

Especially in the analysis of daily time series some restrictions should usually be imposed in order to ensure independent events. The model is applied to annual, monthly and daily time series, respectively, from two Danish rivers. For the very persistent stream the two-state lag-one Markov model seems to be inappropriate to describe the distribution of multi-year drought duration. In the analysis of monthly runs the geometric distribution provides a better fit, although it is not perfect. The distributions of D are almost identical at the two stations in the analysis of monthly and daily series implying that D is not affected by the flow regime. The distributions of S seem to be well described by the GPD, except in cases where very extreme deficits are observed. In the analysis of monthly and daily time series the shape parameter of the distribution of S is not significantly affected by the flow regime. The mean value of S, however, depends on the catchment characteristics. For high truncation levels, S is more severe in flashy rivers compared to persistent rivers, while a low truncation level imply that S is more severe in persistent rivers compared to flashy rivers.

On the modelling of extreme droughts 385

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Rosbjerg.D. (1977) Crossings and extremes in dependent annual series. Nordic Hydrol. 8, 257-266.

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Sen, Z. (1977) Run-sums of annual flow series. J. Hydrol. 35, 311-324.

Sen, Z. (1980) Statistical analysis of hydrologie critical droughts. J. Hydraul. Div. ASCE 106(HY1), 99-114.

Todorovic, P. & Zelenhasic, E. (1970) A stochastic model for flood analysis. Wat. Resour. Res. 6, 1641-1648.

Yevjevich, V. (1967) An objective approach to definitions and investigationsof continental hydrologicdroughts. Colorado State Univ., Hydrol. Paper no. 23, Fort Collins, Colorado, USA.

Zelenhasic, E. &Salvai, A. (1987) A method of streamflow drought analysis. Wat. Resour. Res. 23(1), 156-168.