On the definition and modelling of streamflow ... - Semantic Scholar

Hydrological Sciences-]ournal-des Sciences Hydrologiques, 42(1) February 1997

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On the definition and modelling of streamflow drought duration and deficit volume LENA M. TALLAKSEN Department of Geophysics, University of Oslo, P. O. Box 1022 Blindent, N-0315 Oslo, Norway

HENRIK MADSEN Department of Hydrodynamics and Water Resources, Technical University of Denmark, Building 115, DK-2800 Lyngby, Denmark

BENTE CLAUSEN Department of Earth Sciences, Aarhus University, Ny Munkegade, Building 520, DK-8000 Aarhus C, Denmark

Abstract The threshold level approach is used to define drought characteristics, i.e. drought duration and deficit volume from time series of daily streamflow. Three different procedures for pooling dependent droughts are compared: a method based on an inter-event time and volume criterion (IC), a moving average procedure (MA), and a method based on the sequent peak algorithm (SPA). The extreme values of drought duration and deficit volume are analysed using both an annual maximum series (AMS) and a partial duration series (PDS) approach. Two Danish catchments with very different flow regimes were used in the study. The IC and MA methods provided virtually the same sample statistics of the AMS of drought duration and deficit volume for all thresholds considered. The results of the SPA method differed significantly from the other two methods for high thresholds due to the presence of multi-year droughts. For analysis of seasonal droughts the SPA method is restricted to low thresholds. The occurrence of a large number of zerodrought years for low thresholds may significantly reduce the information content of the AMS, and in this case the PDS model is superior. The problem of minor droughts in the PDS was implicitly reduced by using the MA and SPA methods, and in this respect these methods have an important advantage as compared to the IC method.

Définition et modélisation de la durée et du volume déficitaire des étiages Résumé Un seuil de débit a été utilisé pour définir des caractéristiques des étiages, à savoir leur durée et leur volume déficitaire, à partir de séries chronologiques de débits journaliers. Trois procédures différentes permettant de regrouper des étiages dépendants ont été comparées: une méthode fondée sur un critère de durée entre événements et de volume (IC), une procédure de moyenne mobile (MA) ainsi qu'une méthode fondée sur l'algorithme du pic sequent (SPA). Les valeurs extrêmes de la durée d'étiage et du volume déficitaire ont été analysées en utilisant aussi bien une approche de type durée partielle (PDS) qu'une approche de type maximum annuel (AMS). Deux bassins versants danois présentant des régimes d'écoulements très différents ont été utilisés dans cette étude. Les méthodes IC et MA fournissent pratiquement les mêmes statistiques de maximum annuel de durée d'étiage et de volume déficitaire pour tous les seuils considérés. Les résultats de la méthode SPA diffèrent significativement de ceux des deux autres méthodes pour les seuils élevés, en raison de la présence d'étiages dont la durée atteint plusieurs années. L'analyse des étiages saisonniers par la méthode SPA est limitée aux seuils faibles. La présence d'un grand nombre d'années sans étiage pour des seuils faibles peut réduire significativement le volume d'information traité par l'approche AMS, et dans ce cas l'approche PDS est préférable. Le problème posé par les sécheresses mineures, dans le cas de l'approche PDS, a été implicitement réduit par l'utilisation des méthodes MA et SPA qui, dans ce cas, présentent un avantage significatif par rapport à la méthode IC.

Open for discussion until 1 June 1997

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L. M. Tallaksen et al.

INTRODUCTION Prolonged droughts are critical for industrial, agricultural and domestic water resources and may seriously affect the natural environment. Droughts are regional in nature and critical drought conditions occur when there is an extreme shortage of water for long durations over large areas. The properties of severe regional droughts are not well understood, and presently prediction of these droughts is not possible. Drought studies have suffered from the lack of sufficient regional data sets and consistent methods for at-site drought analysis. In this study the focus is on hydrological droughts in terms of streamflow deficits. Low flow studies traditionally characterize droughts in terms of the minimum annual n-day average discharge. In the United States and in the United Kingdom, n equal to seven days is the definition most used (TCLFE, 1980; Gustard et al., 1992). This method, however, considers only one measure of the drought, the drought magnitude, which may be insufficient in some applications. A method that simultaneously characterizes streamflow droughts in terms of duration and deficit volume is the threshold level method presented by Yevjevich (1967), which defines droughts as periods during which the flow is below a certain threshold level. The definition by Yevjevich (1967) was originally based on the statistical theory of runs for analysing a sequential time series with a time resolution of one month or longer. Statistical properties of the distributions of water deficits, run-length (drought duration) and run-sum (deficit volume) were recommended as parameters for at-site drought definition. The method provides an objective definition of droughts which forms the basis for characterizations of continental or large area droughts. The relationship and time lag between the synoptic situation of a given meteorological condition and the corresponding development of a continental drought need to be determined to be able to predict large continental droughts. To achieve this goal, the regional extent of historical droughts need to be investigated, and this requires an objective and consistent tool for describing both the time of occurrence, duration and severity of site-specific droughts. The threshold level approach has also been used in the analysis of streamflow droughts from a daily recorded hydrograph (Zelenhasic & Salvai, 1987). The present paper discusses the simplifying assumptions of drought definition made by Zelenhasic & Salvai (1987) and suggests alternative procedures. In addition, the paper discusses methods for the estimation of the extreme value properties of drought duration and deficit volume. The main objective of this investigation is to provide guidelines for at-site drought definition which is applicable at the regional scale, i.e. over larger areas covering different physiographic and climate conditions.

DEFINITION OF DROUGHT EVENTS A sequence of drought events is obtained from a streamflow hydrograph by considering flow situations where the discharge is below a certain threshold level, q0 (Fig. 1). Each drought event is characterized by its duration, 1) indicate that seasonal droughts dominate. EXTREME VALUE MODELLING Two different procedures were applied for modelling extreme drought duration and deficit volume: the annual maximum series (AMS) model, and the partial duration series (PDS) model. Annual maximum series In the AMS approach the largest event within a hydrometric year is extracted for the extreme value analysis. The hydrometric year in this study is defined as the calendar year since low flows mainly occur in the summer. However, it might happen that a drought begins in one year and ends in the following year. In that case the drought is not split but assigned to the year in which the time of drought occurrence belongs. Generally, the drought with the annual maximum duration will also be the one with the maximum deficit volume, but this is not always the case, especially in years with only minor droughts. Consequently, the AMS of both duration and deficit volume are analysed. The AMS model experiences a problem when a relatively low threshold level is used. In some years drought conditions do not occur, i.e. the flow never becomes less than the threshold level, implying that the AMS includes zero values. Estimation of the drought properties should in this case be based on the sample of non-zero values. However, too many zero-drought years will reduce the sample size and seriously affect the extreme value modelling. A conditional probability procedure for the frequency analysis is appropriate when the number of observations recorded as zero is small (Stedinger et al., 1993).

Definition and modelling streamflow drought duration

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In the case of high threshold levels a prolonged drought might include low flow periods from two or more consecutive years (multi-year droughts). The multi-year drought is assigned to the year that corresponds to the date of occurrence as defined above. If the year of onset (or termination) of the multi-year drought does not coincide with the year of occurrence, the maximum of any drought in the year of onset (or termination) is defined as the annual maximum. The occurrence of multiyear droughts may imply that in some years, annual maxima are not specified. Analytically, this situation is treated as if the AMS included zero-drought years. Clausen & Pearson (1995) applied the AMS approach in a regional drought study in New Zealand and found that the three-parameter log-normal distribution best fitted both drought duration and deficit volume. Partial duration series In the PDS model all drought events are taken into account, and hence intuitively it provides a more consistent definition of the extreme value region than the AMS approach. However, a potential problem of the PDS approach is the large number of minor droughts which may distort the extreme value modelling. Following the exclusion of minor droughts, Zelenhasic & Salvai (1987) applied the PDS model assuming a Poisson distributed number of drought events and exponentially distributed drought duration and deficit volume. This model was extended by Madsen & Rosbjerg (1995b) who assumed a generalized Pareto distribution for duration and deficit volume.

APPLICATION Daily flow data from two Danish catchments with contrasting geology were used in the study. The data covered a common period of 68 years (1 January 192631 December 1993). The flow regime of catchment 14.01 is persistent with a relatively flat flow duration curve, whereas catchment 59.01 has a more flashy regime

of time flow exceeded 10

14.01

0.1 59.01 0,01 -2,5

1.5 -0,5 0,5 Standardized normal variate Fig. 3 Flow duration curves for catchment 14.01 and 59.01. -1,5

2,5

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Table 1 Catchment characteristics and flow properties. Catchment characteristics Area Length of main stream Main stream slope Average soil type index ' (weighted by area) Mean annual precipitation (1971-1990) Flow properties Mean annual flow 50 percentile of the flow duration curve 70 percentile of the flow duration curve 90 percentile of the flow duration curve 1 Soil types are defined by Madsen & Platou (1983); 4: Sandy clay, 5: Clay, 6: Heavy clay or silt.

Catchment 14.01

Catchment 59.01

213.8 km2 31.3 km 1.4%o 1.4 834 mm

130.2 km2 23.4 km 2.0%0 3.1 721 mm

2342 1 s 4 895 1 s"1 2200 1 s"1 4191s-' 170 1 s» 1963 1 s"1 64 1 s 4 1670 1 s'1 1: Coarse sand, 2: Fine sand, 3: Clayey sand,

with a steeper flow duration curve (Fig. 3). Catchment 14.01 has predominantly sandy soils that produces a high groundwater contribution as opposed to the soils in catchment 59.01 which have a high content of clay (see Table 1). Although catchment 59.01 covered a wider range in flow values, larger fluctuations during drought periods were found for catchment 14.01. Neither of the catchments experiences zero discharge values. The AMS and PDS of drought duration and deficit volume were derived for both catchments using three threshold levels, viz. the 50, 70 and 90 percentiles (Q50, Q70 and Q90) of the flow duration curve. Basic catchment characteristics and flow properties for the two catchments are given in Table 1. The AMS were analysed with respect to the sensitivity of statistical characteristics of drought duration and deficit volume for different values of the parameters tc and pc (IC method) and averaging interval, td (MA method). The drought characteristics obtained using the SPA method were a function only of the threshold level, and the method does not include any parameters for pooling droughts. Sensitivity analysis of the IC and MA methods The definition of the pooled drought characteristics using the IC method, (equation (2)), implies that the mean value of the drought duration is strictly an increasing function of the critical duration, tc, and critical ratio, pc. The mean value of the deficit volume, however, does not possess this strict behaviour and will in practice tend to an upper limit as the parameters tc and pc increase (or even reach a distinct global maximum for particular values of tc and pc). This feature may be used to determine the "optimal" values of tc and pc, i.e. the values that from a water resources engineering viewpoint yield the most critical drought conditions. Five values of tc (0, 2, 5, 10 and oo) and eleven values of pc (0, 0.025, ..., 0.25) were used in the sensitivity analysis. In the case of the MA method the larger the averaging interval, td, the more droughts from the original series will be pooled, and the mean values of drought


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duration and deficit volume will increase. The mean values, however, are not strictly increasing functions of td. As td becomes large, the smoothing of the time series will reduce the duration and deficit volume of the droughts. An "optimal" value of the parameter td may then be found when the mean values reach a constant or maximum level. Six different values of td (1, 5, 10, 15, 20 and 30 days) were applied in the sensitivity analysis. From the AMS of non-zero values the mean values of drought duration, E{D}, and deficit volume, E{S}, were calculated. The relative mean values for the Q90 threshold level are shown in Fig. 4 as a function of tc and pc, and in Fig. 5 as a function of td. The mean values are shown relative to the values obtained from the original series which did not contain any pooling of drought events. High sensitivity means that the pooled mean values show large deviations from the values obtained from the original series, whereas in the case of low sensitivity the relative mean values are close to one. High sensitivity due to changes of the parameters tc and pc, or td, reflects the fact that a large number of droughts or a few very large droughts are pooled. Catchment 59.01 showed in general much lower sensitivity compared to catchment 14.01 due to its less fluctuating hydrograph during low flow. The difference, however, reduced with a lower threshold level, and the two catchments showed similar relative values at Q90 for both the IC and MA method. The properties of D were more sensitive than the properties of S mainly due to the different methods for calculating the pooled drought characteristics (equation (2)). The drought characteristics using the IC method were very sensitive to even small changes of tc and/?c (Fig. 4). In general, E{S} reached a constant level about pc = 0.1. In addition, the change of tc from 5 days to infinity had only a minor

Catchment 14.01

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Pi^rrs^s 0,05

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Fig. 4 Sensitivity in mean drought duration and deficit volume with critical ratio, pc and critical duration, tc (days) at threshold level Q90 (IC method).

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impact on £{5}. This general picture was observed for all threshold levels, with the exception of catchment 14.01 using Q50 as a threshold level where a major break in the curves was observed for tc = oo and pc = 0.1. This is due to the presence of multi-year droughts, which had a pronounced effect on the shape of the distributions (see discussion below). The relative drought characteristics for the moving average series showed a general increase up to td = 10 or 15 days for all threshold levels, after which the curves flattened or even decreased for higher averaging intervals (Fig. 5). A deviation from this general trend was found for catchment 14.01, which showed an increase up to td = 20 days for the highest threshold level Q50. Catchment 14.01

-Catchment 14.01

5

10

15

20

25

Averaging interval (td)

30

5

- Catchment 59.01

10 15 20 25 Averaging interval (tà)

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Fig. 5 Sensitivity in mean drought duration and deficit volume with averaging interval, td (days) at threshold level Q90 (MA method).

Based on the sensitivity plots, pc = 0.1 and tc = 5 days (IC method) and td = 10 days (MA method) were chosen as consistent measures for pooling dependent droughts. These parameter values were used in the following analyses.

Comparative analysis of the different pooling methods using AMS Mean values The mean values of drought duration and deficit volume are given for different threshold levels and pooling methods in Table 2. To facilitate a comparison between the two catchments, discharge was standardized by the mean flow, and both duration and deficit volume have the unit of time (days). The IC and MA methods yielded similar results in most cases, whereas the SPA method generally resulted in higher values for E{D}. This was particularly noticeable for catchment 14.01 using the Q50 and Q70 threshold levels and is due to the presence of multi-year droughts. At the Q90 threshold level, neither methods experienced multi-year droughts for the two catchments and similar values of E{D} were found. The calculation of deficit volume in the SPA method was not significantly affected by the presence of multi-year droughts as it was determined as the maximum accumulated deficit recorded during the drought period. For E{S} all three methods provided virtually the same results, especially at the Q70 and Q90 threshold levels.


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Table 2 Estimated mean values of the AMS of drought duration (£>) and deficit volume (5). E{D} (days) Q50 Q70 Catchment 14.01 lC:pe == 0.1; tc =5 days MA: td = 1 0 days SPA Catchment 59.01 \C:Pc == 0.1; tc =5 days MA: td = 10 days SPA

Q90

E{S} (days) Q50 Q70

Q90

129.5 118.8 318.9

78.1 79.0 111.3

41.3 44.4 46.8

20.4 19.1 27.1

9.0 9.0 8.8

2.8 2.8 2.9

152.2 146.5 166.0

107.7 102.9 115.0

38.7 41.0 44.3

47.1 45.9 48.0

10.5 10.1 10.7

1.1 1.2 1.2

The two catchments showed almost identical values of E{D] at the Q90 threshold level, whereas significantly higher values were found for catchment 59.01 at the Q70 and Q50 threshold levels. This might be explained by the different low flow regimes of the two catchments. Catchment 14.01 had a high clustering of droughts. Although many of these were pooled, the number of droughts per year X, was still significantly higher in this catchment. This implies that the annual maximum drought duration on average was smaller since both catchments had the same number of days with flow below the threshold level. Consequently, E{D} would be lower in catchment 14.01. The difference in X between the catchments decreased for decreasing threshold level, and similar values of X were found at the Q90 threshold level. For E{S} the difference between the two catchments was ambiguous. For the Q90 threshold level E{S} was significantly larger at catchment 14.01, whereas the reverse relationship was observed for Q50. Thus, for instance, if a positive correlation between E{S} and an explanatory variable (e.g. a soil index) existed for Q90, then the same two variables would be negatively correlated for Q50. This suggests that either different explanatory variables are important for different threshold levels or that the standardization of S (by division with the mean flow) introduces spurious correlation. L-moment diagrams To depict contrasts between different samples the Lmoment diagram is a powerful tool (Hosking & Wallis, 1993; Vogel & Fennessey, 1993). Sample estimates of L-skewness and L-kurtosis for the different pooling methods are compared to the theoretical relationships for a number of different distributions in Fig. 6. The L-moments were estimated using unbiased estimates of the probability weighted moments (Landwehr et al., 1979). All three pooling methods showed very similar relationships for catchment 59.01 for both duration and deficit volume. The IC and MA methods also agreed well for catchment 14.01, whereas only the SPA method was consistent with the other two methods at the Q90 threshold level and large deviations were found for Q70 and Q50. All major deviations in the diagrams can be explained by the presence of multi-year droughts. Disregarding series dominated with multi-year droughts, the diagrams show that the distribution of S was more long-tailed than the distribution of D, and the distributions of both D and S became more long-tailed when the threshold level was lowered. For

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Catchment 14.01: AMS of deficit volume

Catchment 14.01: AMS of duration

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Catchment 59.01 : AMS of deficit volume

Catchment 59.01: AMS of duration

L-skewness

L-skewness

Fig. 6 L-moment diagrams for the AMS of drought duration and deficit volume for the IC, MA and SPA methods, plotted together with the theoretical relationships for the Generalized Extreme Value (GEV), Generalized Pareto (GP), three-parameter lognormal (LN) and Pearson type 3 (P3) distributions.

all threshold levels catchment 14.01 had a more long-tailed distribution of both D and 5 than catchment 59.01. Zero-drought years The number of zero-drought years is given in Table 3 for different threshold levels and averaging intervals in the MA method. For the IC method the number of zero-drought years is independent of the parameters tc and pc and equals the values given for td = 1 day (original series). This is also the case for the SPA method. However, additional zero-drought years might occur due to the presence of multi-year droughts. The number of zero-drought years increases primarily with a lowering of the threshold level, but also with an increasing number

Table 3 Number of zero-drought years for the 1, 10 and 30-days moving average series using different threshold levels. Catchment 14.01 Q50 Q70 0 3 tc = 1 day 5 td = 10 days 0 13 td = 30 days 1

Q90 24 28 34

Catchment Q50 0 0 1

59.01 Q70 5 5 6

Q90 18 22 32


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of averaging days in the MA method. The increase as td changes from 1 to 10 days, however, was low for all threshold levels. At Q90 it amounted to four years in both catchments, corresponding to 6% of the total length of the series. Thus, the difference between the IC and MA method was small in this respect, and the presence of zero-drought years was first of all related to the choice of threshold level. The number of zero-drought years using Q50, Q70 or Q90 as the threshold level showed consistent results for the two catchments. Multi-year droughts The values of the index m in equation (4) for evaluating the presence of multi-year droughts are given in Table 4. The only indication of multiyear droughts given by the index (m < 1) was for catchment 14.01 and threshold level Q50, which agreed well with the results obtained for the IC and MA methods. Although no multi-year droughts were present for any of the chosen threshold levels (provided that the "optimal" parameter values were chosen), multi-year droughts were found for catchment 14.01 based on the IC method using only a slightly higher threshold level than Q50 (e.g. the mean annual flow), or by a small increase in tc (from 5 to 10 days). In the case of the MA method it was necessary to increase the moving average interval td from 10 to 30 days in combination with using the mean annual flow as the threshold level to obtain multi-year droughts. In general, the number of multi-year droughts increased with increasing threshold level and increasing values of the parameters pc, tc or td. The SPA method on the other hand gave multi-year droughts for both catchments. One multi-year drought (at Q50) was found for catchment 59.01, whereas the number of multi-year droughts were 23, 5 and 0 for catchment 14.01 at the Q50, Q70 and Q90 threshold level, respectively.

Table 4 Multi-year drought index (ni).

a m

Catchment 14.01 Ç, = 0.121; n = 2.342 Q50 Q70 Q90 0.939 0.838 0.713 0.50 1.34 2.37

Catchment 59.01 Cv = 0.322; n = 0.895 Q50 Q70 0.468 0.190 1.65 2.52

Q90 0.072 2.88

Ranking of the largest drought events The three methods for pooling dependent droughts were also compared by looking at particular extreme droughts. The ten largest deficit volumes in the 68 year annual maximum series are ranked for both catchments in Table 5. The comparison was made between the drought series obtained by using the original flow series with no pooling of droughts, and the IC, MA and SPA methods for pooling droughts. Although included in Table 5, direct comparison cannot be made with series containing multi-years droughts. If one compares the more extreme droughts using Q90 as the threshold level, there was a good agreement between the ranking of the most severe droughts. Overall, the same 9 years were included in the ranking lists, although some of the lower ranked droughts shifted order. A similar picture was found for Q70, whereas larger

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differences are seen using Q50 as the threshold level. Table 5 reveals an important difference between the two catchments. For catchment 14.01 the dry years in the mid 1970s clustered together and were the most severe drought years of the record, whereas consecutive years did not tend to cluster for catchment 59.01. This suggests that there may have been persistence in the annual maximum drought series for catchment 14.01 due to the larger groundwater reservoirs in this catchment. A run analysis based on annual flow series also revealed persistence for catchment 14.01 (Madsen & Rosbjerg, 1995a). If the correlation in the drought series is significant, this should be taken into account, for instance by using a Markov model to describe the persistence (e.g. Rosbjerg, 1977).

Comparative analysis of the different pooling methods using PDS L-moment diagrams The L-moment diagrams of D and S obtained by using all drought events (for the same pooling criteria as used for the AMS model) are shown in Fig. 7. Contradictory to the AMS L-moment diagrams, significant deviations between the pooling methods can be observed. The large skewness found in the series for catchment 14.01 at the Q70 and Q50 threshold leveis using the SPA

Table 5 Ranking of the 10 largest annual droughts (year) in each catchment for the Q50, Q70 and Q90 threshold levels (Org = Original series; IC = Inter-event time and volume based criterion; MA = Moving average procedure; SPA = Sequent peak algorithm). SPA*

Q70 Org

75 48 33 91 42 69 59 26 65 55

76 77 75 74 49 41 43 78 90 93

Q50 Org

IC

MA

Catchment 14.01 75 76 76 76 75 75 77 74 78 74 77 77 41 74 41 49 49 41 47 78 73 43 47 78 93 49 43 34 90 43 Catchment 59.01 38 38 38 76 76 76 59 59 59 75 75 75 53 47 53 47 53 69 64 48 47 92 48 48 92 64 64 77 57 92

76 59 38 75 59 92 53 38 69 76 47 41 64 39 48 47 92 69 77 48 * series containing multi-year droughts

MA

SPA*

Q90 Org

IC

MA

SPA

76 75 77 74 41 49 78 47 43 33

76 75 77 74 49 41 78 43 48 90

76 49 41 34 73 47 43 48 93 90

76 75 77 74 49 78 42 47 90 73

76 75 74 77 78 49 42 90 73 47

76 75 74 77 49 78 42 90 47 73

76 75 74 77 78 49 47 41 42 90

59 38 76 75 47 92 53 39 41 69

59 38 75 76 47 92 53 41 39 69

59 38 75 76 47 92 53 39 41 69

59 38 34 47 39 44 92 41 76 40

59 38 39 34 47 76 92 40 69 44

59 38 39 47 34 76 92 40 44 69

59 38 39 34 47 76 92 40 41 44

IC


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L-skewness L-skewness Fig. 7 L-moment diagrams for the PDS of drought duration and deficit volume for the IC, MA and SPA methods, plotted together with the theoretical relationships for the Generalized Extreme Value (GEV), Generalized Pareto (GP), three-parameter lognormal (LN) and Pearson type 3 (P3) distributions.

method were, as was the case for the AMS approach, due to the presence of multiyear droughts. Disregarding series with multi-year droughts, the distributions obtained by using the IC method exhibited in general a more long-tailed behaviour than those obtained using the MA and SPA methods (which showed very similar values). The points for the different threshold levels are located very closely in the diagram for catchment 14.01, whereas a large range in values is seen for catchment 59.01. Minor droughts The skewness of the distributions of D and S was very pronounced using the IC method. Thus there is a strong indication that the very minor droughts in the sample distorted the extreme value modelling in this case. The sensitivity of deficit volume with respect to the exclusion of minor droughts is shown in Fig. 8 for the Q90 threshold level. Droughts with D < rE{D) or S < rE{S} were excluded prior to the PDS modelling (but after pooling of dependent droughts). As expected, the distributions of D and S became less skewed when the minor droughts were excluded (the points moved in a SW direction in the L-moment diagram as the exclusion factor r increased). The two catchments, however, behaved differently for increasing r. Whereas the drought characteristics of catchment 14.01 seemed to stabilize about r=0.3, catchment 59.01 shows large variability in the whole range of

0,

L. M. Tallaksen et al,

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Catchment 14.01

0,0!

A

PDS

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•

AMS

, '

-GP -P3

I 0,4

*P'

.0,1 A 0,05 SPA

_ •' '

t

- ""

¥ 0,3

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—J

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_^'— MA^ AAiA *

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L-skewness

Fig. 8 Sensitivity in L-moments with exclusion of minor droughts for the PDS deficit volume at the Q90 threshold (IC method). Included in the L-moment diagrams are corresponding values for the MA and SPA methods, and the theoretical relationships for the Generalized Pareto (GP) and Pearson type 3 (P3) distributions.

r-values from 0 to 1, Because of the large sensitivity of D and 5, the choice of exclusion factor r may be rather crucial for the extreme value modelling. One may argue that the choice of r is related to the choice of a parent distribution. In some cases a proper fit to a specified parent may only be achieved by excluding a significant part of the droughts. Thus, the choice of r should be based on a compromise between the increase in sampling variability due to the abstraction of a small sample and the bias due to the lack of fit of the parent distribution (Madsen & Rosbjerg, 1995b). The L-moment statistics obtained from the PDS modelling using the MA and SPA methods are also included in Fig. 8. In the MA method minor droughts were implicitly excluded by using smoothed series. Figure 9 shows the number of droughts with duration less than six days as a function of averaging interval td and threshold level. There was substantial decrease in the number of minor droughts for increasing td. Catchment 14.01 had the largest number of minor droughts, although the difference between the two catchments reduced for increasing averaging interval. Generally, the use of the MA method reduced the problem of minor droughts to such an extent that for most applications it was not required to include a separate exclusion criterion. The number of minor droughts using the SPA method was nearly twice as high as for the MA method (td = 10 days), but it was still sufficiently low not to impose serious problems in the PDS modelling.


Catchment 14.01

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Catchment 59.01 160 -i 140 120 -

A ^ ^

1008060-

—•— Q90 -m-

s*

\

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^&~Q50

-is*--—— * — — . —i 5

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20

25

Averaging interval t d (days)

Fig. 9 Number of minor droughts (duration