Hydrological Sciences–Journal–des Sciences Hydrologiques, 49(5) October 2004
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Choice of reliability, resilience and vulnerability estimators for risk assessments of water resources systems THOMAS RODDING KJELDSEN & DAN ROSBJERG Environment & Resources DTU, Technical University of Denmark, Building 115, DK-2800 Kongens Lyngby, Denmark
[email protected]
Abstract Definitions and estimators of water resources system reliability (the probability that the system will remain in a non-failure state), resilience (the ability of the system to return to non-failure state after a failure has occurred) and vulnerability (the likely damage of a failure event) have been thoroughly investigated. A behaviour analysis addressing monotonic behaviour, overlap and correlation between the estimators was carried out by routing time series of monthly runoff through a reservoir with a specified storage volume that is operated according to a fixed operation policy. Estimation based on historical time series is shown to be problematic and a procedure encompassing generation of synthetic time series with a length of at least 1000 years is recommended in order to stabilize the estimates. Moreover, the strong correlation between resilience and vulnerability may suggest that resilience should not be explicitly accounted for. Key words reliability; resilience; vulnerability; reservoirs
Choix d’estimateurs de fiabilité, de résilience et de vulnérabilité pour les analyses de risque de systèmes de ressources en eau Résumé Les définitions et les estimateurs de fiabilité (probabilité que le système reste dans un état de non-défaillance), de résilience (aptitude du système à revenir à un état de non-défaillance après une défaillance) et de vulnérabilité (dommage vraisemblable d’une défaillance) d’un système de ressources en eau ont été examinés précisément. Une analyse de comportement appliquée au comportement monotone, au recouvrement et à la corrélation des estimateurs a été menée, en imposant des séries temporelles d’écoulement mensuel à un réservoir dont le volume de stockage est spécifié et qui est géré selon des règles opérationnelles établies. L’estimation basée sur des séries temporelles historiques se révèle être problématique, et une procédure de synthèse de séries temporelles d’une longueur minimale de 1000 ans est recommandée afin de stabiliser les estimations. En outre, la forte corrélation entre la résilience et la vulnérabilité suggère que la résilience pourrait ne pas être explicitement prise en compte. Mots clefs fiabilité; résilience; vulnérabilité; réservoirs
INTRODUCTION The objective of this paper is to review the estimators of reliability, resilience and vulnerability (R-R-V) proposed in the literature and to examine which combination of these would be the most appropriate for use in connection with a multi-objective risk assessment of a water resources system, such as suggested by, for example, Loucks (1997), where the sustainability of a scenario is expressed as the product of R-R-V. This sustainability criterion has been applied to existing water resources systems by Kay (2000) and Kjeldsen & Rosbjerg (2001). The use of reliability has been an integrated part of analysis and design of water resources systems for more than a century. Since the publications by Hashimoto et al. Open for discussion until 1 April 2005
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(1982) and Fiering (1982), the use of the additional risk criteria of resilience and vulnerability has been widely discussed; see for example, Moy et al. (1986), Jinno et al. (1995), Kundzewicz & Laski (1995), Vogel & Bolognese (1995), Kundzewicz & Kindler (1995), Srinivasan et al. (1999) and Vogel et al. (1999). It is common for all estimators to rely on the statistical characteristics of failure periods for estimation. Several estimators of R-R-V have been proposed, but few of the studies mentioned above discuss which is the most appropriate combination of estimators to use. To investigate the most appropriate combination, the estimators were investigated with regards to monotonic behaviour and degree of overlap. The exact nature of these characteristics is discussed in the later sections. FAILURE The following description of R-R-V is based on the assumption that the system under consideration at a given time t can be in either a satisfactory (i.e. non-failure) state NF or an unsatisfactory (i.e. failure) state F. In this study the focus is on water supply systems, and, therefore, the NF state occurs when water supply is able to meet water demand and, hence, the F state is when supply cannot meet demand. Moving from time step t to t + 1, the system can either remain in the same state or migrate to the other state. The duration of the jth excursion into a failure period is denoted d(j) and the corresponding deficit volume is denoted v(j), j = 1, …, M, where M is the total number of failure events. The definitions of d(j) and v(j) are illustrated for a single failure event in Fig. 1. The deficit volume of the failure event is calculated as the cumulative difference between demand and availability as: d ( j)
v( j ) = å [D(t ) − Y (t )]
(1)
t =1
where d(j) is the duration of the failure, D(t) and Y(t) are the water demand and the water actually supplied, respectively. The following sections describe how to estimate R-R-V from the extracted series of failure duration and deficit volume. Reliability
The oldest and most widely used performance criterion for water resources systems is reliability, which is defined by Hashimoto et al. (1982) as:
Rel = P{S ∈ NF }
(2)
where S is the system state variable under consideration. The most widely accepted and applied definition is occurrence reliability, which can be estimated as: M
Rel = 1 −
å d ( j) j =1
(3) T where d(j) is the duration of the jth failure event, M is the number of failure events, and T is the total number of time intervals.
3 -1 Water / demand Watersupply supply [m3s -1(m ] s )
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6.0
Supplied Supply
5.5
Threshold Demand level
757
5.0
v(j)
4.5 4.0 3.5
d(j)
3.0
Date
Fig. 1 Characteristics of duration and deficit volume of a failure event. Dots represent water supplied, Y(t), and the solid line water demanded, D(t).
Resilience
Resilience is a measure of how fast a system is likely to return to a satisfactory state once the system has entered an unsatisfactory state. Hashimoto et al. (1982) define resilience as a conditional probability:
Res = P{S (t + 1) ∈ NF | S (t ) ∈ F}
(4)
where S(t) is the system state variable under consideration. This definition of resilience is equal to the inverse of the mean value of the time the system spends in an unsatisfactory state, i.e.: ì1 Res1 = í îM
ü d ( j )ý å j =1 þ M
−1
(5)
where again d(j) is the duration of the jth failure event and M is the total number of failure events. Moy et al. (1986) defined resilience as the maximum consecutive duration the system spends in an unsatisfactory state. To make this definition comparable with the definition of Res1 in equation (5), resilience is expressed as the inverse of the maximum duration as:
{
}
Res 2 = max{d ( j )} j
−1
(6)
Kundzewicz & Kindler (1995) argued that the definition based on maximum value is better than that based on mean value, as the presence of small insignificant events may lower the mean value compared to the same situation, but without the small events as illustrated by Kundzewicz & Kindler (1995), which might lead to non-monotonic behaviour (see section on monotonic behaviour). The extent of non-monotonic behaviour observed in estimates of resilience and vulnerability obtained from short samples will be further discussed later. Srinivasan et al. (1999) highlighted the same problem but further argued that using the maximum duration might mask resilient
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behaviour in the rest of the series compared to other series with shorter maximum duration but more non-resilient behaviour in the remaining series. Vulnerability
Vulnerability is a measure of the likely damage of a failure event and was defined by Hashimoto et al. (1982) as: Vul = å e( j ) h( j )
(7)
j∈F
where h(j) is the most severe outcome of the jth sojourn in unsatisfactory state and e(j) is the probability of h(j) being the most severe outcome of a sojourn into the unsatisfactory state. Hashimoto et al. (1982) and Jinno et al. (1995) based their vulnerability measure on the total water deficit experienced during the entire jth sojourn into F, i.e. deficit volume. This definition is suited for reservoirs as the most severe outcome of a reservoir state is often empty, h(j) = 0. As a further simplification of equation (7), both studies considered the probability of each event to be equal, i.e. e(1) = ... = e(M) = 1/M, where M is the number of failure events. Therefore, they estimated vulnerability as the mean value of the deficit events v(j) as: Vul1 =
1 M
M
å v( j )
(8)
j =1
Again, Kundzewicz & Kindler (1995) argued that the maximum event as proposed by Moy et al. (1986) might be a better estimator than the event-based mean value, i.e.:
Vul 2 = max{v( j )} j
(9)
where v(j) is the deficit volume of the jth failure event. As an alternative to the maximum observed value, estimates of resilience and vulnerability can be based on the pth fractile in the empirical cdf or a standard cdf fitted to either the duration or the deficit volume of the observed failure events as:
{
}
Res3 = Fd−1 ( p) Vul3 = Fv−1 ( p)
−1
(10)
where Fd and Fv are the cdf of duration and deficit volume, respectively. Difficulties associated with the use of standard cdf for modelling of duration and deficit volume of failure events from southern African rivers have been reported by Kjeldsen et al. (2000), and in this study only the use of the empirical cdf was attempted with p = 0.9. In summary, two different sets of estimators of resilience and vulnerability are available from the literature. One set of estimators focuses on the average of observed failure characteristics, while the other set focuses on the properties of the observed maximum values of the failure properties. If an analyst is left with the task of simply characterizing the R-R-V of a water resources system without further specification, a choice between these different types of estimators has to be made. The remainder of this paper attempts to investigate reasons why certain estimators, and combinations of estimators, should be preferred.
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SYSTEM EXPERIMENTS Behaviour analysis
The investigations of the statistical properties of R-R-V, with regards to monotonic behaviour and overlap, are based on series of failure events, d(j) and v(j), extracted from time series using behaviour analysis. Time series of monthly runoff are routed through a reservoir with a specified storage volume S(t) and demand D(t) as: S (t + 1) = S (t ) + Q (t ) − D (t ) S (t + 1) < 0 Þ S (t + 1) = 0
(11)
S (t + 1) > S max Þ S (t + 1) = S max
where S(t) is reservoir storage at the beginning of time step t, Q(t) is inflow to the reservoir in time step t, D(t) is demand from the reservoir in time step t and Smax is the reservoir storage capacity. Surplus water is spilled, and the reservoir is assumed to be full at the beginning of each simulation. In this study the reservoir is operated according to a standard operating policy as defined in Fig. 2.
Spill to downstream
D 1:1
D
Smax + D
Water availability (storage plus inflow)
Fig. 2 Standard reservoir operating policy.
To investigate the behaviour of different estimators of R-R-V, time series of monthly discharge from four rivers—two in South Africa and two in Zimbabwe—were collected. All gauging stations were reported recording natural flow, i.e. with a minimum of anthropogenic influence. The data series were chosen based on record length and data quality, especially concerning missing data. Annual and monthly flow statistics of the historical series are shown in Tables 1 and 2, respectively. The monotonic behaviour of the estimators was investigated using time series of historical monthly runoff. For investigating the correlation between estimators and the record length required to obtain stable estimates, it was necessary to apply time series of monthly runoff generated from stochastic models estimated from the historical data. A Box-Cox transformation of the time series of annual runoff was applied in order to make the series approximately Gaussian, followed by fitting of appropriate ARMA(p,q) models to the series. Generated annual series of runoff were converted into time series of monthly runoff using the method of fragments for disaggregation.
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Table 1 Annual runoff characteristics of selected southern African gauging stations. River
Gauging Country station
Length of record
Palala A5H004 South Africa 1956–1995 Quencwe R2H008 South Africa 1947–1997 Mazoe D28 Zimbabwe 1927–1995 Umshagashi E2 Zimbabwe 1929–1994 AR: annual runoff; MAR: mean annual runoff.
Catchment area (km2) 638 62 223 541
MAR (106 m3) 71.5 0.72 19.8 43.3
CV of AR 1.19 0.91 0.87 1.10
Years with zero runoff (%) 0 0 0 3
Table 2 Monthly runoff statistics for the selected gauging stations. River Jan. Feb. Mar. Apr. May Mean (106 m3): A5H004 12.05 13.94 11.10 5.51 4.78 R2H008 1.04 1.01 1.13 0.64 0.43 D28 3.75 5.13 4.14 1.81 0.93 E2 9.65 12.08 7.78 2.89 1.40 6 3 Standard deviation (10 m ): A5H004 16.22 18.39 15.43 6.25 8.40 R2H008 1.79 1.59 1.79 1.29 1.56 D28 3.53 4.95 4.27 2.11 1.06 E2 14.93 16.79 10.80 3.90 2.04 Correlation coefficient with flow of previous month: A5H004 0.54 0.64 0.77 0.85 0.74 R2H008 0.33 0.39 0.57 0.52 0.09 D28 0.49 0.65 0.75 0.90 0.93 E2 0.43 0.69 0.69 0.81 0.92 Percentage of zero flows: A5H004 0 0 2 0 2 R2H008 16 10 10 10 22 D28 0 0 0 1 3 E2 11 12 14 18 26
Jun.
Jul.
Aug.
Sep.
2.69 0.08 0.65 0.68
1.77 0.09 0.57 0.41
1.31 0.36 0.46 0.21
0.90 0.16 0.31 0.10
4.34 0.09 0.70 1.01
2.17 0.15 0.57 0.61
1.59 1.36 0.46 0.36
0.95 0.10 0.98 0.95
0.87 0.15 0.99 0.97
2 22 3 38
5 18 3 42
Oct.
Nov.
Dec.
0.89 0.46 0.26 0.07
2.35 0.89 0.48 0.48
9.22 0.58 1.62 5.48
1.13 0.31 0.33 0.21
1.03 1.02 0.28 0.19
2.44 1.81 0.83 1.01
17.61 0.77 2.02 9.72
0.91 0.36 0.99 0.96
0.96 0.12 0.98 0.90
0.79 –0.04 0.85 0.35
0.41 0.51 0.36 0.33
0.63 0.32 0.19 0.06
5 30 3 46
10 22 3 54
15 12 4 58
0 4 6 38
0 6 1 20
This method was found by Srikathan & McMahon (1980) to deal effectively with months of zero runoff as encountered in the adopted flow series. MONOTONIC BEHAVIOUR
To introduce the concept of a monotonic estimator, as used in this study, consider a storage–yield–reliability relationship. If storage increases or yield decreases the reliability will increase, hence reliability is considered a monotonic estimator with respect to storage volume and draft. To investigate the degree of monotonic behaviour of the estimators of resilience and vulnerability, the following combinations of active reservoir storage capacity (S) and draft (a), both normalized with mean annual runoff (MAR), were used: storage: S = 0.5, 1.0, 2.0 and draft: a = 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95. The water demand is specified on an annual basis and afterwards disaggregated uniformly on each month. Each historical time series of monthly runoff is routed
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through the reservoir and the number of failure months, the duration of each failure and the water deficit accumulated during each failure are recorded. Based on the statistics of the failure events sample estimates of resilience and vulnerability are Res1, Res2, Res3, Vul1, Vul2 and Vul3. A similar behaviour of the resilience and vulnerability estimators was observed across the four rivers. Thus, Fig. 3(a)–(f) shows typical examples of the estimated resilience and vulnerability as a function of draft and active storage volume, illustrated by data from gauging station D28 in the Mazoe River in Zimbabwe. When estimation of resilience and vulnerability is based on mean values of duration and deficit volume (Res1 and Vul1), the sample estimates generally exhibit a non-monotonic behaviour as, for a specified storage volume, the estimates do not increase monotonously as the draft increases (see Fig. 3(a) and (b)). Also, when con(b)
0.14
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0 0.5
0.6
0.7 0.8 Draft/MAR
0.9
1.0
0.5
0.6
0.7 0.8 Draft/MAR
0.9
1.0
Fig. 3 Monotonic behaviour of (a) Res1 using mean value of failure duration; (b) Vul1, using mean value of failure deficit volume; (c) Res2 using maximum failure duration; (d) Vul2 using maximum deficit volume; (e) Res3 using F-1 (0.9) in the cdf of duration; and (f) Vul3 using F-1 (0.9) in the cdf of deficit volume. The unit of vulnerability is m3 s-1, corresponding to the runoff in one month.
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sidering resilience and vulnerability as a function of storage volume with a fixed draft, a similar non-monotonic behaviour can be observed. A similar tendency can be observed when estimating resilience and vulnerability as Res3 and Vul3, i.e. as the 0.9th fractile in the empirical cdf of failure duration and deficit volume respectively, as evident from Fig. 3(c) and (e). The use of maximum observed values of duration and deficit volume for estimation of resilience and vulnerability, Res2 and Vul2, give sample estimates that are monotonic with respect to both draft and storage volume (see Fig. 3(c) and (d)). The results from this investigation support the observations reported by Kundzewicz & Kindler (1995) that obtaining sample estimates of resilience and vulnerability based on mean values of failure duration and deficit volume using time series of historical length is not appropriate as the sample estimates are nonmonotonic. Likewise, system experiments show that sample estimates obtained using the 0.9th fractile of the empirical cdf of failure duration and deficit volume exhibit the same non-monotonic behaviour. Maximum values appear to be more applicable in terms of monotonic behaviour. However, the largest event observed in a n-year period has a very large sampling variance (Vogel & Stedinger, 1988) and is, therefore, a less appropriate estimator. No recommendation can be made based on an investigation of the degree of monotonic behaviour alone. The non-monotonic behaviour observed when using mean values or the 0.9th fractile in the empirical cdf is related to the limited number of failure events experienced when using time series of historical extent. Many studies have used time series of historical length for evaluation of resilience and vulnerability. However, the estimation precision of both resilience and vulnerability can be enhanced by the introduction of stochastic streamflow models. By generating long series of synthetic streamflow, the number of failure events can be increased, leading to more robust estimates of resilience and vulnerability, concerning both mean and maximum values. Hashimoto et al. (1982) used 10 000 years of synthetic data for evaluation of R-R-V and Vogel & Bolognese (1995) used 100 million years of data. The record length required for robust estimation of resilience and vulnerability is discussed later.
OVERLAP
Overlap, or dependence, between different criteria in multi-objective decision making may lead to inaccurate ranking of the considered scenarios. To illustrate the effect of overlap, consider the comparison between two policy options with respect to two criteria, a and b. Assume that the final ranking of the policy options is based on the product of the two criteria. If a and b are significantly correlated, then b can be expressed as a linear function of a as b = αa + β, where α and β are considered constants. The product of a and b is given as ab = a(αa + β) = αa2 + βa, and hence, the ranking of the two policy options is based on the outcome of criterion a (or b) only. Rogers et al. (1997) investigated overlap between selected environmental indicators using principal component analysis. In this study, the overlap between different estimates of reliability, resilience and vulnerability is investigated by generating many synthetic time series of monthly runoff, estimating R-R-V for each time series and then quantifying the degree of overlap in terms of correlation coefficients between series of
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sample estimates. The procedure for each of the four rivers is as follows: 1. Identify, estimate and validate stochastic streamflow model. 2. Generate a synthetic time series of monthly runoff of historical length. 3. Route time series through the reservoir with specified storage volume and water demand. 4. Estimate Rel, Res1-3 and Vul1-3. 5. Repeat steps 2–5 a total of 100 times. The following combinations of storage volume and water demand were used (both normalized with MAR): storage: S = 0.5, 1.0, 2.0 and draft: a = 0.5, 0.70, 0.90. The generated time series of monthly runoff are routed through the reservoir using behaviour analysis. For each combination of storage volume and water demand, the correlation coefficient between the 100 coherent estimates of R-R-V is calculated as: cov{•,•} (12) ρ= (var{•}var{•}) expressing the linear relationship between the estimates of R-R-V. The higher the value of the correlation coefficient, the more the two considered estimators overlap, thereby reducing the amount of information gained by using both criteria compared to using two independent criteria. Combinations of estimators shown in Table 3 were investigated and the results of each combination of storage volume and water demand plotted. The inverse of the three resilience estimators, rather than the estimators themselves, were used in order to obtain a linear relationship between the variables, thus making the correlation coefficient a more useful measure of overlap. As with the monotonic behaviour, the observed behaviour of the R-R-V estimators did not differ significantly from river to river and, thus, the figures illustrate typical behaviour exemplified by data from gauging station A5H004 in the Palala River in northern South Africa. Table 3 Combinations of R-R-V for investigation of overlap. Res1 Res2 Res3 Rel
Vul1 ü ü ü ü
Vul2 ü ü ü ü
Vul3 ü ü ü ü
Rel ü ü ü
Resilience–vulnerability
Nine different combinations of resilience and vulnerability were investigated. Figure 4(a)– (c) shows the correlation coefficient between estimators of the inverse of resilience and vulnerability, i.e. each part of the figure contains results from one of the rows in Table 3. In general the correlation coefficients between vulnerability and the inverse of resilience are positive and between 0.5 and 1.0. This shows that a system with high sample resilience also has a low sample vulnerability, which agrees with the findings of Kundzewicz & Laski (1995). It should be noted that the correlation coefficients between pairs of estimators based on the same summary statistics, such as mean value, maximum value or the 0.9th fractile, are close to one, indicating a total overlap between resilience and vulnerability in these cases.
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Vul2
Vul3
Vul1
(b)
1.0
1.0
0.8
0.8
Correlation
Correlation
(a)
0.6 0.4
S = 0.5 S = 1.0 S = 2.0
0.2 0.0
Vul2
0.6 0.4
S = 0.5 S = 1.0
0.2
S = 2.0
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0.9
0.5
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(d)
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Res1 0.0
0.8
-0.2
0.6 0.4
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0.2
S = 2.0
S = 1.0
0.0 0.7
0.9
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0.5
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1.0
0.5
0.5
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Vul3
Res2
Res3 S = 0.5 S = 1.0 S = 2.0
-0.4 -0.6 -0.8 -1.0 0.5
0.7
0.9
0.5
0.7
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Draft/MAR
(e)
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Vul2
Vul3 S = 0.5
0.0
S = 1.0
Correlation
-0.2
S = 2.0
-0.4 -0.6 -0.8 -1.0 0.5
0.7
0.9
0.5
0.7
0.9
0.5
0.7
0.9
Draft/MAR
Fig. 4 Correlation between (a) inv{Res1} and Vul1, Vul2 and Vul3 (mean, max, fractile); (b) inv{Res2} and Vul1, Vul2 and Vul3; (c) inv{Res3} and Vul1, Vul2 and Vul3; (d) Rel and inv{Res1}, inv{Res2} and inv{Res3} (mean, max, fractile); and (e) Rel and Vul1, Vul2 and Vul3.
Reliability–resilience
As may be seen from Fig. 4(d), the correlation between reliability and the inverse of the resilience expressed through either mean values, fractiles or maximum values is of the same order of magnitude between –0.3 and –0.8. Considering the mean value of failure duration, there appears to be a tendency of increasing numerical correlation with decreasing storage capacity and increasing water demand. The opposite mechanism is observed when considering the maximum value of observed failure duration, i.e. increasing storage volume and decreasing water demand. No such tentative conclusions can be reached when considering the 0.9th fractile in the empirical cdf of the duration of failure duration.
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The high negative correlation indicates that reliable systems tend to have a high degree of resilience (keeping in mind that the experiment considers the correlation between reliability and the inverse of resilience), which corresponds to the findings of others, for example, Kundzewicz & Kindler (1995) and Srinivasan et al. (1999). All three methods of estimating resilience gave correlation coefficients of the same order of magnitude. Reliability–vulnerability
Figure 4(e) shows the correlation coefficient between reliability and vulnerability expressed through mean values and maximum values, respectively. The order of magnitude of the correlation coefficient is generally between –0.3 to –0.8, i.e. the same as for the correlation between reliability and resilience. The negative correlation indicates that systems with high reliability tend also to be less vulnerable, which corresponds to the findings of Kundzewicz & Laski (1995). The correlation coefficients behave in a similar way to the pattern observed in the correlation between reliability and resilience. For vulnerability expressed through mean values, the correlation coefficient decreases with increasing storage volume and increases with increasing water demand. For vulnerability expressed as the maximum value, the correlation increases with increasing storage volume and decrease with increasing water demand. REQUIRED RECORD LENGTH
As is evident from the investigations concerning monotonic behaviour, the use of time series of historical length to estimate resilience and vulnerability leads to nonmonotonic behaviour when using mean values, due to the limited number of failure events. To enhance the estimation of resilience and vulnerability, the stochastic streamflow models were used to generate longer series and thereby more failure events. The objective of this investigation was to obtain an indication of the record length required to give robust estimates of resilience and vulnerability. The procedure for estimating the required record length is as follows: 1. Identify, estimate and validate a stochastic streamflow model. 2. Select the required record length (T years). 3. Generate synthetic time series of monthly runoff with a record length of T. 4. Route time series through the reservoir with specified storage capacity and water demand. 5. Estimate Res1, Res3, Vul1 and Vul3. 6. Repeat steps 2–5 for each chosen value of T. The investigation was made considering a reservoir with a storage volume equal to MAR and water demand a = 0.5, 0.7 and 0.9. The following record lengths were considered T = historic, 100, 200, 500, 1000, 5000, 10 000 and 100 000 years. The required record length was assumed reached when further extension of the time series did not significantly change the estimate of the resilience or vulnerability. The reliability and vulnerability estimated as the maximum duration and deficit volume of observed failure periods (Res2 and Vul2) will increase as the record length increases; hence, no required record length can be determined and these estimators were not
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20
8
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15 10 5 0 10
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100000
Fig. 5 Required record length for (a) Res1; (b) Res3; (c) Vul1; and (d) Vul3. The unit of vulnerability is m3 s-1, corresponding to the runoff in one month.
considered. The procedure outlined above was repeated for each of the four rivers and the general tendency was that the same pattern could be observed in most rivers for all estimators; therefore, tentative guidelines for required record length can be given. Typical results are illustrated in Fig. 5(a)–(d) obtained using data from gauging station E2 in the Umshagashi River in Zimbabwe. For all rivers, the estimates of resilience and vulnerability obtained using mean values and the 0.9th fractile reached a constant level when using record lengths of 1000 years. CONCLUSIONS
Based on the results of the investigation of R-R-V, the following tentative conclusions can be made. Using time series of historical length is problematic, especially when estimating resilience and vulnerability through the use of mean values of failure events characteristics, as it leads to a non-monotonic behaviour of the estimates. Using maximum values of duration and deficit volume proved to give more monotonic estimates; however, masking of resilient behaviour in the remainder of the series and large estimation uncertainty made maximum values estimated from historical time series less appealing. Considering the overlap between R-R-V, it became clear that combinations of resilience and vulnerability based on the same summary statistics should not be used, to avoid total overlap. It might be beneficial to abandon either resilience or vulnerability from a multicriteria analysis; alternatively, they could be
Choice of reliability, resilience and vulnerability estimators for risk assessments
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used together while abandoning reliability instead. Abandoning resilience might appear as a step away from the issues related to sustainability, such as discussed by Loucks (1997). However, the strong correlation between vulnerability and resilience guarantees that systems with low vulnerability (which are potentially the most sustainable) also have a high degree of resilience. Finally, the non-monotonic behaviour of the estimators led to the investigation of the required record length. From the investigation it is clear that the length of the time series to be used for estimating resilience and vulnerability should be at least 1000 years. Acknowledgements The authors wish to express their gratitude to the Department of Water and Forestry, South Africa, and the Ministry of Lands and Water Resources, Zimbabwe, for providing the data material. The suggestions and comments made by two anonymous reviewers are also gratefully acknowledged. REFERENCES Fiering, M. B. (1982) Alternative indices of resilience. Water Resour. Res. 18(1), 33–39. Hashimoto, T., Loucks, D. P. & Stedinger, J. (1982) Reliability, resilience and vulnerability for water resources system performance evaluation. Water Resour. Res. 18(1), 14–20. Jinno, K., Zongxue, X. Kawamura, A. & Tajiri, K. (1995) Risk assessment of a water supply system during drought. Water Resour. Devel. 11(2), 185–204. Kay, P. A. (2000) Measuring sustainability in Israel’s water system. Water Int. 25(4), 617–623. Kjeldsen, T. R. & Rosbjerg, D. (2001) A framework for assessing water resources system sustainability. In: Regional Management of Water Resources (ed. by A. H. Schumann, M. C. Acreman, R. Davis, M. A. Marino, D. Rosbjerg & Xia Jun) (Proc. Maastricht Symp., July 2001), 107–113. IAHS Publ. 268. IAHS Press, Wallingford, UK. Kjeldsen, T. R., Lundorf, A. & Rosbjerg, D. (2000) Use of a two components exponential distribution in partial duration modelling of hydrological droughts in Zimbabwean rivers. Hydrol. Sci. J. 45(2), 285–298. Kundzewicz, Z. W. & Kindler, J. (1995) Multiple criteria for evaluation of reliability aspects of water resources systems. In: Modelling and Management of Sustainable Basin-scale Water Resources Systems (ed. by S. P. Simonovic, Z. Kundzewicz, D. Rosbjerg & K. Takeuchi) (Proc. Boulder Symp., July 1995), 217–224. IAHS Publ. 231. IAHS Press, Wallingford, UK. Kundzewicz, Z. W. & Laski, A. (1995) Reliability-related criteria in water supply studies. In: New Uncertainties Concepts in Hydrology and Water Resources (ed. by Z. W. Kundzewicz), 299–305. Cambridge University Press, Cambridge, UK. Loucks, D. P. (1997) Quantifying trends in system sustainability. Hydrol. Sci. J. 42(4), 513–530. Moy, W.-S., Cohon, J. L. & ReVelle, C. S. (1986) A programming model for analysis of the reliability, resilience, and vulnerability of a water supply reservoir. Water Resour. Res. 22(4), 489–498. Rogers, P., Jalal, K. F., Lohani, B. N., Owens, G. M., Yu, Chang-Ching, Dufournaud, C. M. & Bi, J. (1997) Measuring Environmental Quality in Asia. Harvard University Press, Boston, Massachusetts, USA. Srikathan, R. & McMahon, T. A. (1980) Statistical generation of monthly flow for ephemeral streams. J. Hydrol. 47, 19–40. Srinivasan, K., Neelakantan, T. R., Shyam Narayan, P. & Nagarajukumar, C. (1999) Mixed-integer programming model for reservoir performance optimization. J. Water Resour. Plan. Manage. ASCE 125(5), 298–301. Vogel, R. M., Lane, M., Ravindiran, R. S. & Kirshen, P. (1999) Storage reservoir behaviour in the United States. J. Water Resour. Plan. Manage. ASCE 125(5), 245–254. Vogel, R. M. & Stedinger, J. R. (1988) The value of stochastic streamflow models in overyear reservoir design applications. Water Resour. Res. 24(9), 1483–1490. Vogel, R. M. & Bolognese, R. A. (1995) Storage–reliability–resilience–yield relations for over-year water supply systems. Water Resour. Res. 31(3), 645–654.
Received 11 November 2003; accepted 20 May 2004
