Advances in Water Resources 27 (2004) 889–898 www.elsevier.com/locate/advwatres

Treatment of precipitation uncertainty in rainfall-runoff modelling: a fuzzy set approach Shreedhar Maskey a

c

a,*

, Vincent Guinot b, Roland K. Price

c

UNESCO-IHE Institute for Water Education, P.O. Box 3015, 2601 DA Delft, The Netherlands b Universite´ Montpellier 2, Maison des Sciences de l’Eau, 34095 Montpellier Cedex 5, France UNESCO-IHE Institute for Water Education, P.O. Box 3015, 2601 DA Delft, The Netherlands Received 22 December 2003; received in revised form 9 June 2004; accepted 6 July 2004 Available online 12 September 2004

Abstract The uncertainty in forecasted precipitation remains a major source of uncertainty in real time flood forecasting. Precipitation uncertainty consists of uncertainty in (i) the magnitude, (ii) temporal distribution, and (iii) spatial distribution of the precipitation. This paper presents a methodology for propagating the precipitation uncertainty through a deterministic rainfall-runoff-routing model for flood forecasting. It uses fuzzy set theory combined with genetic algorithms. The uncertainty due to the unknown temporal distribution of the precipitation is achieved by disaggregation of the precipitation into subperiods. The methodology based on fuzzy set theory is particularly useful where a probabilistic forecast of precipitation is not available. A catchment model of the Klodzko valley (Poland) built with HEC-1 and HEC-HMS was used for the application. The results showed that the output uncertainty due to the uncertain temporal distribution of precipitation can be significantly dominant over the uncertainty due to the uncertain quantity of precipitation.
2004 Elsevier Ltd. All rights reserved. Keywords: Disaggregation; Flood forecasting; Fuzzy sets; Genetic algorithm; Precipitation; Uncertainty

1. Introduction Since flooding is an inherently uncertain natural process, the reliability and credibility of flood forecasting and warning systems cannot be guaranteed without properly incorporating the sources of uncertainty into the forecasting and warning systems. Estimating uncertainty in forecasts enables an authority to set risk-based criteria for flood warning and offers potential for additional economic benefits of forecasts to every rational decision maker [15]. The present discussion deals with

*

Corresponding author. Tel.: +31 15 2151755; fax: +31 15 2122921. E-mail addresses: [email protected] (S. Maskey), guinot@ msem.univ-montp2.fr (V. Guinot), [email protected] (R.K. Price). 0309-1708/$ - see front matter
2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2004.07.001

uncertainty in a deterministic rainfall-runoff-routing model. The various sources of uncertainty associated with the model outputs can be classified as [18]: (1) Model uncertainty (due to assumptions in the model equations, model building, and other forms of imprecision in conceptualising the real system) (2) Input uncertainty (due to imperfect forecasts of future precipitation, evaporation, etc.) (3) Parameter uncertainty (due to imperfect assessments of model parameters) (4) Natural and operational uncertainty (due to unforeseen causes, e.g. glacier lake overflow, landslides and debris flows, etc., malfunctioning of system components (hardware and software), erroneous and missing data, human errors and mistakes, etc.).

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The latter category is broad and is usually not addressed by uncertainty assessment in operational flood forecasting systems. Model uncertainty may be assessed by comparing results from different models [22]. The parameter uncertainty can reflect part of the uncertainty associated with the model structure [7]. Such assessments of model uncertainty however, remain largely subjective. One of the important developments for the treatment of uncertainty in water systems modelling is the generalised likelihood uncertainty estimation (GLUE) by Beven and Binley [2]. Some examples of the application of the GLUE methodology in rainfallrunoff modelling are [3] and [5]. The present discussion is limited to the treatment of uncertainty in precipitation as an input uncertainty. Real time flood forecasting using a rainfall-runoff-routing model requires the forecasted precipitation for the forecast period [4]. The precipitation uncertainty represents the major part of the input uncertainty [14]. The uncertainty in the forecasted precipitation results from the uncertainty in (i) the magnitude, (ii) the temporal distribution over the forecast period, and (iii) the spatial distribution over the catchment. The present paper proposes a methodology for the propagation of the precipitation uncertainty through a deterministic rainfall-runoff-routing model. The methodology is independent of the structure of the model and can therefore be used with any deterministic, rainfall-runoff-routing model. It uses the temporal disaggregation of the precipitation (over a period) into subperiods and allows the uncertainty in both the temporal and spatial distributions of the rainfall to be accounted for. The methodology is detailed in the framework of fuzzy set theory for uncertainty propagation using the extension principle (see Appendix A). The methodology however can be easily adapted for implementation in the framework of the Monte Carlo technique for a probabilistic approach. The method is applied to a model of the Klodzko (Poland) watershed, based on the HEC-1 [28] and HEC-HMS [29,30] modelling software. 2. Methodology

2.1. Temporal disaggregation of precipitation for uncertainty assessment In catchment modelling, some of the forcings may be observed with a period larger than the typical reaction time of the catchment. A typical example is a rainfall measured or forecasted on an hourly basis when the catchment response time is half an hour. In this case, using the (average) measured forcing directly as an input to the model may lead to underestimation of the amplitude of the model response (because the variations in the signal are smoothed out into the averaged measurement). This is particularly true when peak values in the output are related to peak values in the inputs (e.g. peak rainfall resulting in peak runoff). In order to estimate better the peak values of the model output, the inputs must be reconstructed at a time scale smaller than the typical reaction time of the hydrological system under study. Failing to generate an input at smaller time scale than the catchment response time may introduce error/uncertainty in the model outputs. If a model with multiple inputs is to be used (e.g. a catchment model using records from several rainfall gauges) the spatial distribution of the inputs is also a source of uncertainty. In addition, the average rainfall rate over a given subbasin is not necessarily equal to the point value measured at the corresponding rainfall gauge. Besides the imprecision attached to the measuring device, the point measurement introduces uncertainty in the average input over the subbasin to which the measuring device is attached. The idea presented here consists in dividing the temporal period over which the accumulated rainfall is known into a fixed number of subperiods and to randomly disaggregate the accumulated sum over subperiods. The disaggregated rainfall should aggregate up to the (measured) accumulated sum. The disaggregated precipitation signal is then used as an input to the rainfallrunoff-routing model. Let the accumulated sum of the precipitation for the subbasin i (i = 1, . . ., m) during a given time period of T be Pi. Dividing the period T into n subperiods, Pi is disaggregated over the subperiods as n X pi;j i ¼ 1; . . . ; m ð1Þ Pi ¼ j¼1

The methodology of uncertainty assessment in flood forecasting due to precipitation is presented in this section. The principle of disaggregation is presented in Section 2.1 and the representation of the uncertainty in the magnitude of the precipitation by a fuzzy membership function is presented in Section 2.2. The algorithm for the method is presented in Section 2.3, whereas the algorithm used to determine the minimum and the maximum values of the model outputs and a simplification of the methodology are described in Sections 2.4 and 2.5 respectively.

where pi,j is the disaggregated quantity for the jth subperiod (Fig. 1). For the sake of simplicity coefficients bi,j are introduced, such that pi;j ¼ P i bi;j

ð2Þ

where 0 6 bi;j 6 1 Pn j¼1 bi;j ¼ 1

8ði; jÞ 8i

) ð3Þ

S. Maskey et al. / Advances in Water Resources 27 (2004) 889–898

891

µ (Pi)

Pi

1

Pi /n

p i,1

p i,j

p i,n

j=1

j

j=n

Pi,min Pi,mc

t

T

Pi,max

Pi

Fig. 2. A triangular membership function with three values: Pi,min, Pi,mc and Pi,max.

Fig. 1. Average and disaggregated precipitations.

In the present approach the coefficients bi,j are generated randomly and independently from each other, subject to the constraints (3); the method is described in Section 2.3. Eqs. (2) and (3) allow the quantity pi,j to take any value from 0 to Pi. Allowing the coefficients bi,j to vary over the subperiods accounts for different possible temporal distributions over the various subbasins, whereas varying the coefficients over subbasins accounts for the spatial variations of the rainfall field. Note that this methodology is based on the implicit assumption that the length of a subperiod is larger than the correlation time of the precipitation signal. If it is not the case, the coefficients bi,j should not be generated independently from each other. The length of the subperiod, however should not be too large, in which case the resulting signals will be close to the average signals over the period. Note that methods exist, that allow signals to be generated taking into account the signal correlation [19,24]. In the absence of any information about the correlogram of the rainfall signal, such methods cannot be used and the coefficients bi,j must be generated independently from each other. Therefore, for the present implementation of the methodology the disaggregated signals are generated without considering their correlations. However, it is important to note that this methodology for propagating the precipitation uncertainty can still be used with correlated signals if available. Also note that the number of subperiods has a direct influence on the maximum possible rainfall intensity. This is because the maximum possible intensity is obtained when the rainfall is concentrated over one subperiod, with a rainfall intensity nPi/T. 2.2. Precipitation uncertainty represented by a membership function The representation of uncertainty using fuzzy set theory on the accumulated precipitation, Pi (i = 1, . . ., m) requires the membership function (MF) of the Pi to be known. In the absence of sufficient information about the MFs, it is assumed that only the range of uncertainty given by the minimum (Pmin) and maximum (Pmax) values are available. It is also assumed that the given value of Pi is its most credible value represented by Pi,mc. With

the 3 values Pmc, Pmin and Pmax a triangular MF (Fig. 2) is assumed, given as 8 0 if P i < P i;min > > > > P i P i;min > < P i;mcP if P i;min 6 P i 6 P i;mc i;min ð4Þ lðP i Þ ¼ P i;max P i > if P i;mc 6 P i 6 P i;max > > P i;max P > i;mc > : 0 if P i > P i;max The qualitative meaning of the triangular membership function is the following. The ‘‘true’’ value of precipitation, Pi, is certainly included between Pi,min and Pi,max and is likely to be close to Pi,mc. The key words are ‘‘included’’ and ‘‘close’’. These words constitute the only information that one has about the problem [23]. By doing so the uncertainty or imprecision associated with the single value of the forecast precipitation is represented using a fuzzy number on the basis of the minimum information available about the anticipated precipitation. 2.3. Algorithm for the propagation of uncertainty The precipitation represented by a MF and disaggregated into subperiods is propagated using the extension principle of fuzzy set theory. Some definitions on fuzzy sets are presented in Appendix A. The extension principle [31,32] is performesd by the a-cut method [26]. An example of an a-cut for a MF and corresponding lower bound (Pi,LB) and upper bound (Pi,UB) are shown in Fig. 3. The Pi,LB and Pi,UB are the minimum and the maximum values of the precipitation, respectively for the given a-cut. Let a function f represents the rainfall-runoff-routing model with the precipitation as input and the runoff, Q, as output:

µ (Pi ) 1

(α )

(α )

Pi , LB

Pi , UB

α Pi,min

Pi,mc

α -cut level

Pi,max

Pi

Fig. 3. An a-cut level and corresponding upper and lower bounds. The definitions of Pi,min, Pi,mc and Pi,max are identical to those in Fig. 2.

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Q ¼ f ðpi;j ; i ¼ 1; . . . ; m; j ¼ 1; . . . ; nÞ ¼ f ½ðp1;1 ; . . . ; p1;n Þ; . . . ; ðpm;1 ; . . . ; pm;n Þ

ð5Þ

A general procedure for the propagation of the three forms of precipitation uncertainty through the model (Eq. (5)) can be decomposed into the following steps. (1) Select a value of a 2 [0, 1] (called an a-cut level) for the MF of the input. Use the same value of a for the MFs of all inputs. (2) For the given a-cut, determine for all i = 1,. . .,m the ðaÞ ðaÞ lower bound, P i;LB , and the upper bound, P i;UB (see Fig. 3). ðaÞ ðaÞ (3) Generate randomly a value P i;ðrÞ between P i;LB and ðaÞ P i;UB for all i = 1, . . ., m. This value corresponds to the accumulated sum over the period T, i.e. Pi in Eq. (1). (4) Generate the coefficients bi,j,(s) randomly between 0 and 1. For each i, only n  1 coefficients are generated. If the sum of the first n1 coefficients exceeds 1, reject all the coefficients and generate a new set of n  1 coefficients. Repeat this process until the sum is smaller than or equal to 1. The nth coefficient is then determined as follows: bi;n;ðsÞ ¼ 1 

n1 X

bi;k;ðsÞ

ð6Þ

k¼1 ðaÞ

(5) Use the value P i;ðrÞ generated in step (3) and the coefficients bi,j,(s) generated in step (4) in Eq. (2) to disaggregate the rainfall over the subperiods; i.e. ðaÞ

ðaÞ

pi;j;ðrÞ;ðsÞ ¼ P i;ðrÞ bi;j;ðsÞ

ð7Þ

ðaÞ

(6) Use the pi;j;ðrÞ;ðsÞ ði ¼ 1; . . . ; m; j ¼ 1; . . . ; nÞ as inputs to the model (Eq. (5)). This leads to a model ðaÞ output QðrÞ;ðsÞ . (7) Repeat steps (4)–(6) s times. This means that for every value of the accumulated precipitation obtained from step (3), s number of disaggregated sets of precipitation are generated. This produces s outputs. (8) Repeat steps (3)–(7) r times. This produces r · s outputs. From the r · s outputs determine the minimum and maximum of the outputs. These are assumed to be equal to the lower and upper bounds of the output, that is 9 ðaÞ ðaÞ QLB ¼ minðQðrÞ;ðsÞ Þ > = r;s ð8Þ ðaÞ ðaÞ ; QUB ¼ maxðQðrÞ;ðsÞ Þ > r;s

The values of s and r are primarily governed by the size of the search space for the determination of minimum and maximum values of the outputs. The search space in this example is the function of m and n. Obviously, more number of subbasins and subperiods means more possible combinations

of input sets. Different types of algorithms for the determination of the minimum and maximum values may deal differently for the values of s and r, such as using stopping criteria. The algorithm used in the present study for the determination of maximum and minimum values is presented in Section 2.4. (9) Repeat steps (2)–(8) for as many a-cuts as needed to produce a complete MF for the output. 2.4. Algorithm used for the determination of minimum and maximum An algorithm is needed for the determination of the maximum and minimum values of the model outputs (Eq. (8)). In the present study a genetic algorithm (GA) is used. GAs are search algorithms based on the mechanics of natural selection and natural genetics [10]. Various applications of GAs are reported elsewhere [12,17,20]. A simple GA consists of (i) a random selection of the initial population, (ii) a selection for mating, (iii) crossover, and (iv) mutation. The version of the GA used in the present study consists of tournament selection [9], uniform crossover [21], two children per pair of parents, jump and creep mutations [6], elitism and niching [11]. In the present methodology (Section 2.3) the tasks for steps (3), (4), (7) and (8) are replaced by the GA operations. The number of parameters (N) for the GA is given by N ¼ m þ ðn  1Þm ¼ nm

ð9Þ

where m and n are the number of subbasins and subperiods respectively, as defined in Section 2.1. Instead of the parameters s and r (Section 2.3), GA uses initial population (Np) and maximum number of generation (Ng). In the present study, Np = 50 and Ng = 40 were used for the example with three subperiods (Section 4.1) and Np = 200 and Ng = 50 were used for the example with six subperiods (Section 4.2). 2.5. Simplification of the methodology An obvious drawback of the above methodology is that the search space becomes larger as the number of subbasins and subperiods increases, which makes the determination of the maximum and minimum values very time consuming. In an attempt to reduce the search space, several checks revealed that the minimum and maximum values of the model outputs (Qmin and Qmax) ðaÞ actually correspond to the lower bound ðP i;LB Þ and the ðaÞ upper bound ðP i;UB Þ values (for the given a-cut) respectively, of the precipitation determined in step (2) of the algorithm. Therefore, instead of looking for all possible ðaÞ values of precipitations between LB and UB, P i;LB is ðaÞ used to evaluate Qmin and P i;UB is used to evaluate Qmax.

S. Maskey et al. / Advances in Water Resources 27 (2004) 889–898

This means that the model function is assumed monotonic with respect to the quantity of the accumulated precipitation in the forecast period. This simplification brings the value of r in the algorithm down to 2: one for the upper bound and one for the lower bound. In terms of the number of parameters for the GA (Eq. (9)), N reduces to (n  1)m. This is an important conclusion as it helps to reduce the computational effort, particularly when working on a large catchment with many subbasins.

SB 1

SB 2 SB 4

JN 1

SB 7 RH 5

JN 2 RH 4

RH 3

The methodology was applied to a flood forecasting model of Klodzko valley (Poland) on the river Nysa Klodzka. The river serves as a highland tributary of the upper Odra and the lag time between rainfall and runoff is very short (see [16,27]). The basin under consideration consists of nine subbasins. The basin covers a total area of 1744 km2 with the subbasin areas ranging from 64 to 280 km2. The data observed during the flood of July 1997 were used. The town of Klodzko located on the river Nysa Klodzka was virtually ruined by this flood with several casualties and the destruction of numerous houses [16]. The model was built with HEC-1 and HEC-HMS. In particular, HEC-HMS (its Calibration Module) was used for model calibration and HEC-1 was used for simulation. The options used in the present application to represent the rainfall-runoff processes are the Soil Conservation Service (SCS) Curve Number method for runoff volume computation, ClarkÕs unit hydrograph method for direct runoff computation, the exponential recession method for baseflow computation and the Muskingum method for flow routing. Readers are referred to [29] for the details of these methods. The selection of the methods in the present model was governed particularly by the availability of the data. The analysis of the influence of different methods is beyond the scope of this study. A schematic diagram of the model is shown in Fig. 4. Some of the parameters of the model are estimated by calibration and some are estimated from available data and engineering judgement. The calibration is performed using the calibration module of HEC-HMS. In particular, the SCS Curve Number, the time of concentration, the storage coefficient and the Muskingum K were determined by calibration. Observed precipitation data (cumulative for every 3 h) were available. Due to the lack of forecasted precipitation data, the observed precipitation was assumed as a forecasted precipitation for the application of the present methodology. The MF representing the uncertainty in the quantity of the forecasted precipitation for each subbasin is obtained by taking minimum and maximum

SB 3 RH 1

SB 5

SB 8 JN 4

Legend

RH 2

JN 3 RH 6

SB 6

Subbasin

RH 7

Junction

SB 9 JN 5

Outlet

3. Application example

893

Subbasin outlet River reach

Fig. 4. Schematic diagram of the watershed (SB=subbasin; RH=reach; JN=junction).

values as 0.7 and 1.3 times the given precipitation. Although the minimum and maximum bounds are taken arbitrarily, discussion with experts suggested that the assumption is not too far from reality. The starting date for simulation was 4 July 1997 at 06:00 h, while the fore cast of precipitation started on July 6 1997 at 00:00 h (i.e. 42 h from the point of start of simulation). Since the interval of the available precipitation data is 3 h the size of the forecast period is also taken as 3 h. The very short lag time between the rainfall and runoff justifies the use of the present methodology of uncertainty assessment with disaggregation. Three subperiods of 1 h each and six subperiods of half an hour each are considered for disaggregation. It is to be noted that for each forecast, only the uncertainty in the forecast precipitation during the same forecast period is considered and no uncertainty is assumed in the precipitation previous to the current forecast period. For example, to forecast the flow for 3 h, no uncertainty is assumed in the precipitation up to the time 0 h; similarly, to forecast the flow for 6 h no uncertainty is assumed in the precipitation up to the time 3 h, and so on. A total of 20 forecasts (20 · 3 h=60 h) from 6 July at 00:00 to 8 July at 09:00 were carried out.

4. Results All the results presented and discussed here are based on the forecast downstream of junction 5 (Fig. 4). Firstly, the simulation is carried out to compute the discharge without considering any uncertainty in the precipitation. The simulated discharge together with the corresponding observed discharge is presented in Fig. 5. The figure also shows the basin average precipitation on the negative y-axis. It is to be noticed that the simulation was carried out taking the accumulated precipitation in each subbasin, and not with the basin average precipitation. Secondly, the methodology is applied to estimate the uncertainty in the forecasted discharges due to the

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S. Maskey et al. / Advances in Water Resources 27 (2004) 889–898 Time (hours)

Discharge (m3/s)

Precipitation (mm)

0

24

48

72

96

120

144

168

0.0 10.0 20.0 30.0 2000 Simulated

1500

Observed

1000 500 0 0

24

48

72

96

120

144

168

Time (hours)

Fig. 5. Observed and simulated discharges without considering uncertainty in the forecasted precipitation for the period 4 to 11 July 1997. The hour 0 refers to 4 July 1997 at 06:00 h.

uncertainty in the forecasted precipitation. The results are characterised as ‘‘with disaggregation’’ and ‘‘without disaggregation’’. Whereas ‘‘with disaggregation’’ considers uncertainty due to the unknown temporal distribution of the precipitation by using disaggregated signals as inputs, ‘‘without disaggregation’’ uses a constant signal, equal to the mean value, over the entire subperiods. Since each subbasin precipitation is represented using a fuzzy membership function for uncertainty, the former considers both temporal and spatial uncertainty and the latter considers only the spatial uncertainty. In Fig. 5, the simulated discharge is the result of the precipitation for a = 1 without disaggregation.

without disaggregation. This suggests that the uncertainty due to the unknown temporal distribution can be a predominant factor over the uncertainty in the magnitude of the precipitation. The results shown in Figs. 6 and 7 correspond to a = 0 and 1 only. For a complete membership function (MF) of the output to be constructed, the computation needs to be carried out at various a-cuts. Three more a-cuts at a = 0.25, 0.5, and 0.75 were selected. The MFs obtained for the forecasts at 81, 84 and 87 h with and without disaggregation are shown in Fig. 8(a–c). These figures further illustrate that the uncertainty due to the unknown temporal distribution is predominant over the uncertainty due to the quantity of precipitation. The support of disaggregated discharges is also greatly increased. For a = 1, there is no upper and lower bounds of ða¼1Þ ða¼1Þ ða¼1Þ the input precipitation, i.e. P i;LB ¼ P i;UB ¼ P i . This means that for a = 1 without disaggregation there is only one set of input precipitation resulting in only one output. Consequently, there are no upper and lower bounds of the output. In fact, this particular case (i.e. a = 1, without disaggregation) considers neither temporal nor spatial uncertainty. Whereas, in the case with disaggreða¼1Þ gation, the one set of input precipitation, i.e. P i , will

4.1. Results with three subperiods Fig. 6 shows the uncertainty bounds in the forecasted discharges, i.e. QUBQLB, with disaggregation for a = 0 and 1 as a function of the forecast time. The comparison of the uncertainty bounds (QUBQLB) with and without disaggregation is presented in Fig. 7. For a = 1 without disaggregation, there is only one possible value Pi,mc for the precipitation. Therefore, there is no uncertainty in this case. It is clearly seen from Fig. 7 that in all forecasts the output uncertainty is much larger with than

QUB - QLB (m 3/s)

150

α

100

α=0

=1

50

0 42

54

66

78

90

102

Time (hours)

Fig. 6. Uncertainty range in forecasted discharges (QUBQLB) with disaggregation.

S. Maskey et al. / Advances in Water Resources 27 (2004) 889–898

895

QUB - Q LB (m /s)

150 Without disaggregation With disaggregation

3

100

50

0 42

54

66

(a)

78

90

78

90

102

Time (hours)

3

QUB - Q LB (m /s)

150 With disaggregation 100

50

0 42

54

66

102

Time (hours)

(b)

Fig. 7. Uncertainty range in forecasted discharges (QUBQLB): (a) for a-cut level 0, and (b) for a-cut level 1. There are no upper and lower bounds for a-cut level 1 in the case of without disaggregation.

Membership

1.00

0.50

0.00 600

(a)

700

800

Discharge (m 3/s)

Membership

1.00

0.50

0.00 800

(b)

4.2. Results with three and six subperiods 900

1000

Discharge (m 3/s)

Membership

1.00

0.50

0.00 1150

(c)

of temporal uncertainty. These conclusions are clearly seen in Fig. 8. Also of significance are the shapes of the MFs with and without disaggregation. The shapes of the MFs without disaggregation are fairly triangular showing that if the temporal uncertainty is neglected the triangular MFs of the inputs (precipitation) result in a triangular MF of the output (discharge). Alternatively, with disaggregation the shapes are rather uniform suggesting that the additional uncertainty due to the uncertainty in the temporal distributions is almost independent from the a-cut level.

1250

1350

Discharge (m3/s)

Fig. 8. Membership functions of forecasted discharges without disaggregation (grey line) and with disaggregation (dark line): (a) forecast at 81 h, (b) forecast at 84 h, and (c) forecast at 87 h.

have a number of temporally disaggregated precipitation sets resulting in a number of output discharges which means that there exist upper and lower bounds for the outputs even for a = 1 due to the consideration

A key issue in the application of this methodology is the selection of an appropriate number of subperiods. Increasing the number of subperiods leads to increase the number of parameters for the search space and consequently the computational requirement. On the other hand, intuitively, increasing the number of subperiods may widen the uncertainty bounds in the output. To verify this, the results with three subperiods are compared with the results with six subperiods. With six subperiods and nine subbasins, the number of parameters increases from 18 to 45. In order to reduce the number of parameters, the temporal pattern is assumed to be identical for all the subbasins. This assumption significantly reduces the number of parameters from 45 to 5 in the case of six subperiods and 18 to 2 in the case of three subperiods. It is however important to check the effect of this assumption. Therefore, the three subperiods case is also repeated applying this assumption. The comparison of

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S. Maskey et al. / Advances in Water Resources 27 (2004) 889–898 150

QUB - Q LB (m /s)

Same temporal pattern for all subbasins

3

100

Varying temporal pattern over subbasins

50

0 42

54

(a)

66

78

90

102

78

90

102

Time (hours)

3

QUB - Q LB (m /s)

150 Same temporal pattern for all subbasins Varying temporal pattern over subbasins

100

50

0 42

54

(b)

66 Time (hours)

Fig. 9. Uncertainty range in forecasted discharges (QUBQLB) with disaggregation into three subperiods for: (a) a-cut level 0, and (b) a-cut level 1.

200 6 Subperiods

3

QUB - Q LB (m /s)

3 Subperiods 150 100 50 0 42

54

66

78

90

102

78

90

102

Time (hours)

(a) 200

6 Subperiods

3

Q UB - Q LB (m /s)

3 Subperiods 150 100 50 0 42

(b)

54

66 Time (hours)

Fig. 10. Uncertainty range in forecasted discharges (QUBQLB) with disaggregation for: (a) a-cut level 0, and (b) a-cut level 1.

this result with the previous result with varying temporal patterns over subbasins is presented in Fig. 9. The differences are minor except for forecasts at 84 and 87 h. In this particular example, assuming identical temporal patterns over all the subbasins resulted in larger uncertainty bounds than with the varying patterns. Fig. 10 shows the comparison of uncertainty bounds of cases with three and six subperiods for a = 0 and 1. It can be observed that the increase in the uncertainty bound is not very significant compared to the increase in the number of subperiods. This means that the number of subperiods can be taken reasonably small without underestimating the uncertainty in the output unacceptably.

5. Conclusion This paper has presented a methodology for the treatment of precipitation uncertainty in rainfall-runoff-routing models in the framework of fuzzy set theory assisted by genetic algorithms. The methodology uses the random disaggregation of precipitation into subperiods to take into account uncertainty due to the unknown temporal distribution of the precipitation. The principle of disaggregation presented in this paper for uncertainty assessment can also be used in the framework of the Monte Carlo technique if the forecast of precipitation is probabilistic. This methodology is independent of the structure of the forecasting model. In other words,

S. Maskey et al. / Advances in Water Resources 27 (2004) 889–898

897

it can be used with any rainfall-runoff-routing type of deterministic model. The results show the great potential of the fuzzy extension principle combined with a genetic algorithm for the propagation of uncertainty. The results also show that the output uncertainty due to the uncertainty in the temporal and spatial distributions can be significantly dominant over the uncertainty due to the uncertain magnitude of precipitation. This suggests that using space- and time-averaged precipitation over the catchment may lead to erroneous forecasts. The estimated uncertainty in the output may seem small compared to the magnitude of the flood. This is due to relatively short forecast period (3 h). Obviously increasing the forecast period significantly increases the uncertainty in the forecasted precipitation and thereby increases the output uncertainty. Moreover, it is to be noted that the estimated uncertainty is only due to the uncertainty in the precipitation. It does not include the parameter and model uncertainty. This paper has also attempted to answer the question concerning an appropriate number of subperiods. While the estimated uncertainty without disaggregation and disaggregated with three subperiods differed significantly (Fig. 9a), the results with three and six subperiods showed little difference (Fig. 10). Therefore, in this particular example three subperiods seem enough. In general, the determination of the number of subperiods should be governed from the considerations that: (i) the uncertainty should not be underestimated beyond a reasonable limit, (ii) the computational requirements should not be too large, (iii) the subperiods should be large enough for the disaggregated precipitations to remain realistic, and (iv) the length of the subperiod should be of the same order of magnitude as the correlation time of the precipitation signal.

A.1. Fuzzy set and fuzzy number

Acknowledgments

A.3. Alpha-cut of a fuzzy set

This research was initiated by the EC project Operational Solutions for the Management of Inundation Risks in the Information Society (OSIRIS), contract IST-1999-11598. The authors highly appreciate and extend their gratitude to the IMWM, Poland for providing the data and the model. The authors are also grateful to Prof. Mike J. Hall, UNESCO-IHE Institute for Water Education, The Netherlands for providing valuable remarks on the manuscript.

An a-cut (alpha-cut) of a fuzzy set A, denoted as Aa is the set of elements x of a universe of discourse X for which the membership function of A is greater than or equal to a. That is,

Let X be a universe set of x values (elements). Then A is called a fuzzy (sub)set of X, if A is a set of ordered pairs: A ¼ fðx; lA ðxÞÞ; x 2 X ; lA ðxÞ 2 ½0; 1g

ðA:1Þ

where lA(x) is the grade of membership (or degree of belief) of x in A. The function lA(x) is called the membership function of A. A fuzzy set is called a fuzzy number if it is normal (i.e. the maximum of the memberships is 1) as well as convex. A.2. Fuzzy extension principle Consider a function of several uncertain variables: y ¼ f ðx1 ; . . . ; xn Þ

ðA:2Þ

Let fuzzy sets A1, . . ., An be defined on the universes X1, . . ., Xn such that x1 2 X1, . . ., xn 2 Xn. The mapping of these input sets can be defined as a fuzzy set B: B ¼ f ðA1 ; . . . ; An Þ

ðA:3Þ

where the membership function of the image B is given by 8 maxfmin½lA1 ðx1 Þ; . . . ; lAn ðxn Þ; > > > < y ¼ f ðx ; . . . ; x Þg 1 n ðA:4Þ lB ðyÞ ¼ > 0 if there is no x1 2 X 1 ; . . . ; xn 2 X n > > : such that f ðx1 ; . . . ; xn Þ ¼ y Eq. (A.4) is the mathematical expression for the extension principle of fuzzy sets. The above equation is defined for a discrete-valued function f. If the function f is continuous-valued then the max operator is replaced by the sup (supremum) operator (the supremum is the least upper bound).

Aa ¼ fx 2 X ; lA ðxÞ P a; a 2 ½0; 1g

ðA:5Þ

The a-cut provides a convenient way of performing arithmetic operations on fuzzy sets and fuzzy numbers µ X (x)

Appendix A. Definitions on fuzzy sets and operations The mathematical definition of the extension principle and some other relevant definitions on fuzzy sets and operations are briefly presented here. More detailed coverage of the topic can be found in [1,8,13,25,33].

1 α

a x1

b x2

α-cut -cut level X

Fig. 11. Illustration of an a-cut of a fuzzy set.

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including in applying extension principle. Let us consider a fuzzy number (a membership function) as shown in Fig. 11. And let a a-cut level intersects at two points a and b on the membership function. The values of the variable x corresponding to points a and b are x1 and x2 (x1, x2 2 X), respectively. Then the set Aa contains all possible values of the variable X including and between x1 and x2. The x1 and x2 are commonly referred as lower and upper bounds of the a-cut.

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