Flood frequency estimation by continuous simulation for a catchment treated as ungauged (with uncertainty)

Transcription

1 WATER RESOURCES RESEARCH, VOL. 38, NO. 8, 1139, /2001WR000500, 2002 Flood frequency estimation by continuous simulation for a catchment treated as ungauged (with uncertainty) Sarka Blazkova T. G. Masaryk Water Research Institute, Prague, Czech Republic Keith Beven Institute of Environmental and Natural Sciences, Lancaster University, Lancaster, UK Received 9 March 2001; revised 28 February 2002; accepted 19 March 2002; published 13 August [1] A general methodology for flood frequency estimation based on continuous simulation is here applied to a gauged site in the Czech Republic treated as if it was ungauged. In this implementation, stochastic temperature and precipitation models are used to drive TOPMODEL to simulate stream discharges. The coupled model parameters are varied randomly across specified ranges using Monte Carlo simulation. The results from a sample of 48,600 simulations each of length 100 years using an hourly time step are conditioned on low return period regionalized flood frequency, snow water equivalent, and flow duration curve information. Performance measures for each predicted variable are combined using fuzzy inference and simulations considered as nonbehavioral are rejected. 10,000-year simulations are made with the remaining 2281 behavioral simulations to produce prediction limits for flood magnitudes and other response variables at different return periods. The results are checked against a historical series of annual maximum discharges available at the site for a period before it was destroyed by the construction of a dam. The results compare well and appear to give more realistic prediction bounds than statistical extrapolations based on the Wakeby distribution, particularly at longer return periods. INDEX TERMS: 1821 Hydrology: Floods; 1854 Hydrology: Precipitation (3354); 1860 Hydrology: Runoff and streamflow; 1863 Hydrology: Snow and ice (1827); KEYWORDS: flood frequency, ungauged watershed, TOPMODEL, GLUE, snowmelt, fuzzy inference 1. Introduction [2] The estimation of flood frequency statistics for ungauged catchments is a continuing problem of great practical interest. The dominant methodology used in the past has been the regionalization of statistics from gauged catchments by a regression analysis with catchment characteristics [e.g., Natural Environment Research Council (NERC), 1975; Pilgrim, 1987; Institute of Hydrology, 1999]. The question of how far catchment responses can be represented by regression relationships has been discussed by Beven [2000]. Recent work has had the aim of making estimates with a more robust statistical basis [Hosking and Wallis, 1997] or with a firmer physical basis, either by seeking alternative forms of regionalization, such as the network related scaling relationships of Gupta and Dawdy [1995]; derived distribution techniques using analytical or storm by storm simulation [e.g., Eagleson, 1972; Hebson and Wood, 1982; Cordova and Rodriguez-Iturbe, 1983; Sivapalan et al., 1990; Cadavid et al., 1991; Raines and Valdez, 1994], or by continuous hydrograph simulation driven by historical rainfalls [Calver et al., 1999; Lamb, 1999] or a stochastic rainfall model [e.g., Beven, 1986, 1987; Blazkova and Beven, 1995, 1997; Cameron et al., 1999, 2000a, 2001b]. [3] One argument for using the simulation approaches is that rainfall records are more widely available and tend to Copyright 2002 by the American Geophysical Union /02/2001WR have longer periods of records than stream data. However, as pointed out by Beven [1986], Raines and Valdes [1994], Moughamian et al. [1987] and Cameron et al. [2000b, 2001a] the results will be sensitive to both the form of distributions used in the stochastic rainfall and the process representations and parameter values of the model used to transform the rainfalls into stream discharges and particularly discharge peaks. Both may be subject to considerable uncertainty in the representation of extremes, and in the case of an ungauged catchment the data available for constraining that uncertainty will be limited. [4] This paper explores the use of continuous simulation as a complement to regionalized regressions using catchment characteristics as a way of estimating flood frequencies for an ungauged catchment in the Czech Republic. The hydrological model TOPMODEL is driven by stochastic inputs representing both rainfall and snow so that full yearround simulations can be run thereby incorporating the effects of the seasonal variation in antecedent conditions on runoff production in a consistent way. As far as we know, this is the first attempt at flood frequency estimation by continuous simulation to include the possibility of snowmelt and rain-on-snow flood events. The analysis is carried out within a Monte Carlo simulation uncertainty estimation framework to explore the value of different types of data in constraining the resulting estimates of flood frequency. A similar analysis for a number of gauged catchments has been recently completed by Cameron et al. [1999, 2000a]. This paper is different in addressing what

2 14-2 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT Figure 1. The Joseful Dul catchment (outlet JD), the weather stations used for computing parameters of the temperature and precipitation model (BE, NL, UH, SOUS), and the snow observation sites (solid triangles, within forest; open triangles, on the clearing). might be achieved in the case of an ungauged catchment and in evaluating the Monte Carlo simulations using fuzzy performance measures which seem more appropriate given the imprecision of the information available for model evaluation in this catchment when it is treated as ungauged. 2. Study Site [5] The Joseful Dul catchment (25.81 km 2 ) is a catchment situated in the Jizera Mountains of the Czech Republic (Figure 1). The only discharge records available for this catchment are 68 annual maximum peaks ( ) prior to the observations being discontinued as a result of the construction of a dam at the site. These have not been used in the study except in a final evaluation of the methodology so that the catchment has essentially been treated as ungauged. Standard regionalized information produced by the Czech Hydrometeorological Institute (CHMI) for ungauged catchments is available for flood frequencies up to 100 year return periods and for the flow duration curves represented as 13 quantile values (discharges exceeded on intervals ranging from 30 to 364 days per annum). CHMI were asked to exclude the data from this site in making the regionalized estimates of flood frequency and flow duration for this study. The regional estimate of average annual rainfall is 1534 mm. There is a practical need for flood frequency estimates of inflows to the reservoir at this site as a contribution to the reevaluation of dam safety currently being carried out in the Czech Republic. [6] For computing the parameters of the rainfall model for the summer months (May to October), a point rainfall series from the station at Sous was available (Figure 1) for the period This was augmented by information on high-intensity rainstorms (see below). Some information is also available on frequencies of annual maxima of snow water equivalent. Six series of 10 to 13 years in length from various altitudes with various vegetation covers have been used to calculate an areally weighted catchment average snow water equivalent for comparison with the catchment average predictions of the model. 3. Hydrological Model [7] TOPMODEL is a simple semi-distributed model of catchment hydrology that predicts storm runoff from a combination of variable saturated surface contributing area and subsurface runoff [Beven and Kirkby, 1979; Beven et al., 1995; Beven, 1997a, 1997b, 2001]. The predictions of the dynamics of the contributing areas as the catchment wets and dries are based on the pattern of a soil-topographic index that includes the effects of a stochastic variation in downslope transmissivity in the same way as Blazkova and Beven [1997]. Model calculations are based on 75 increments of the soil-topographic index distribution. An infiltration excess overland flow component, used in the flood frequency studies of Beven [1986, 1987], has not been included in the current study to reduce the number of parameters to be specified. On the vegetated soils of the study catchment it is expected that infiltration excess runoff will not be commonly

3 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT 14-3 Table 1. List of Parameters Used in the Model a Fixed Parameters GLUE Parameters TOPMODEL a/tan B distribution m depletion parameter 0.01 to m std b of t0 1 as log of m 2 /h t0 average transmissivity 2 to 10 as log of m 2 /h routing velocity 1m/s srmax max. root zone stor to 0.20 m network width function Snow model fraction of free water 0.08 memory of free water 168 h dfmin min mm/h/ C dfmax max. degree hour factor 0.15 to 0.26 mm/h/ C Temperature model Evapotranspiration Low-intensity storms High-intensity storms degree hour factor average temperature 4.4 C Fourier/ model A1 c B1 c. daily mean daily std hourly mean autoregressive model a1 d a2 d a3 d std b. daily hourly Thornthwaite coeff. for temper. sine function of daylight hours jt = e, at = e seasonal parameters aver. season. vol. season average duration average intensity. Nov. April mm Nov. April 2.19 to h 0.44 to 0.83 mm/h. May mm May 3.76 to h 0.82 to 0.92 mm/h. June Aug mm June Aug to h 0.99 to 1.10 mm/h. Sept. Oct mm Sept. Oct to h 0.73 to 0.81 mm/h storm profile. cumulative ordinates. std for each ordinate. autocorrelation coeff monthly probabilities of occurrence distrib. of aver. intensities (exponential with threshold) cumulative distrib. of duration storm profile. cumulative ordinates. std for each ordinate. autocorrelation coeff Table 2 threshold on average intensity 11.7 mm/h average intensity 15.0 to 17.2 mm/h a Fixed parameters are those held constant in the Monte Carlo simulations; GLUE parameters were selected from uniform distributions within the specified ranges. b Here std is standard deviation. c A1 and B1 are first Fourier coefficients for cosine and sine terms, respectively. d Here an is autoregressive coefficient. e Here jt and at are coefficients for Thornthwaite formula. generated except under the most extreme events when soil saturation will normally already be high. [8] TOPMODEL, driven by a stochastic rainfall model, was used in estimating the summer flood frequency characteristics of three catchments in the Czech Republic with various lengths of record by continuous simulation [Blazkova and Beven, 1997]. It has also been used in studies of 5 catchments in the U.K. [Cameron et al., 1999, 2000a]. Here, that work is extended to the problem of estimating the flood frequency characteristics for an ungauged catchment, taking account of the uncertainties inherent in parameter estimation. Ways in which this uncertainty might be constrained for an ungauged site are considered. In addition a snow component has been added in this study to allow whole year simulations. A summary of the parameters required by the model and the ranges considered in this study are given in Table Stochastic Components [9] As in the model used by Blazkova and Beven [1997] the stochastic rainfall model allows for distributions of two types of storms with high and low intensity. In the present study, the year has been divided into four characteristic seasons of different lengths (see Table 1). Every season has its own average seasonal total, average duration, and average intensity of low-intensity events. Storm durations and intensities are assumed to have independent exponential distributions (r = for the storm data at the Sous site). An average length of interstorm periods in each season is computed so as to ensure that the seasonal average volume is preserved. Again an exponential distribution of arrival times is assumed. The low-intensity event model is completed by a storm profile component that is based on a mean dimensionless storm profile, scaled by rainfall volume and duration, calculated from the observed storms and a first order autocorrelated stochastic component based on the variance of the observed profiles at different dimensionless durations. [10] In the conditions of the Czech Republic storms with high intensity occur within longer events of lower intensity as convective cells connected to frontal rainfall or as fluctuations of the intensity of frontal rains. For the purposes of this rainfall simulator a high-intensity event is

4 14-4 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT Table 2. Number of Observed Low-Intensity Events, Average Rainfall Intensity and Average Duration for the Period March to November for Point Rainfalls at Sous ( ) March April May June July August September October November Events Average intensity, mm/h Average duration, hours defined as such a period of rainfall (in hourly time step) for which the maximum hourly intensity is larger than the hourly intensity with the return period of one year and the hourly intensity does not drop under a specified threshold (for Czech applications 6 mm/h seems suitable). The highintensity storms have a probability of occurrence which varies with the season and is near to zero in the winter months. On average, one high-intensity storm is modeled in a year. Based on the observed data, an exponential distribution is used for the high-intensity storm rainfalls above the threshold. Durations are sampled from the empirical cumulative distribution function. The scaled storm profile for these storms contains an autocorrelated stochastic component. [11] A fuller description and tests of the rainfall model are reported by Blazkova and Beven [1997]. The model is simpler than some others that are available but there appears to be little advantage in using more complex (and parameter rich) model structures, even in the estimation of extreme events [see Cameron et al., 2000b, 2001a] Precipitation Event Parameters [12] For the point rainfall series at Sous, the low-intensity events have average intensity mm/h and average duration 3.98 h. Averages of the observations for individual months are given in Table 2. These records have been used as the basis for estimating the parameter ranges for the lowintensity events in each season, taking account of the regional estimates for areal reduction factors so that the model will estimate catchment average rainfalls (Table 1). [13] For the high-intensity events, Trupl [1958] put together extensive tables of maximum intensities of rainfall in the Czech Republic which show that the maximum hourly intensities of return period 1 and 20 years of three long series in Prague are very similar to other areas of the Czech Republic. These data have been used to estimate the probabilities of occurrence in individual months computed in the observed summer months as given in Table 3. It can be expected that a high-intensity storm can occur with a very low probability also in the winter half year. Some (low) probabilities were assumed and the summer probabilities recomputed in order for all the probabilities to sum up to one. Some limited information on high-intensity events was available for the region of this study. This included the 18 high-intensity storms used by Blazkova and Beven [1997] and 9 additional storms from the Sous station, both adjusted for catchment area. The average intensity of the 27 storms is mm/h and the high-intensity threshold is not changed with the additional data. The average intensity is considered as a GLUE parameter with the range 15.0 to 17.2 mm/h. The average duration is 2.85 h. [14] Figure 2a shows a comparison of observed and modeled daily totals for a thousand different realizations of the same length as the observed series (32 years) of simulated hourly rainfalls. In each simulation, full years were modeled. In Figure 2b the hourly totals are compared for the summer period for which observed data are available. Both figures show reasonable functioning of the precipitation model, the observed values are well within the range of the simulated series of the same length Snow Accumulation and Melt Component [15] The stochastic snow precipitation component is based directly on the low-intensity rainfall model, with a temperature threshold parameter to decide whether the precipitation is treated as either rain or snow. In order to keep the number of model parameters to a minimum, a simple degree-day snow routine was employed, taking account of melt associated with rainfall inputs, with a degree-day factor that is allowed to vary seasonally [e.g., World Meteorological Organization (WMO), 1986]. Meltwater is allowed at the base of the pack once the free water storage of the pack has been exceeded (here fixed at 8%). A memory component for refreezing is fixed at 7 days. When the pack starts refreezing again, free water can remain in the snow for some time. A new period of melt can readily start in the same time step when the temperature is again higher than the critical base temperature TB. When the memory of free water is exceeded (during an extended period of temperature below TB), the free water percentage becomes again zero and the next melt period is delayed. [16] TOPMODEL has been used before in conjunction with such a snow accumulation and melt component by Ambroise et al. [1996]. Here, to reduce the number of parameters that would need to be considered in model calibration, the only snowmelt parameter varied in conditioning the model was the maximum value of a seasonal Table 3. Observed Probability of a High-Intensity Rainfall Event for Prague and Probabilities Assumed in the Rainfall Model for the Joseful Dul Catchment November December January February March April May June July August September October Observed Assumed

5 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT 14-5 Figure 2. (a) Comparison of observed (pluses and dashed line) and modeled (shaded lines) daily precipitation totals. (b) Comparison of observed (pluses and dashed line) and modeled (shaded lines) hourly summer precipitation totals. sine curve variation in the degree-hour factor dfmax (Table 1). It is shown below that the resulting snow component, albeit simple, can satisfactorily reproduce the long-term distribution of maximum annual snow water equivalent Temperature and Evapotranspiration Components [17] The addition of the snow model requires the provision of a temperature input to the model. The generation of a temperature series has been based on an analysis of 17 years of daily temperature data for the station at Bedrichov- Nova Louka (average temperature 4.4 C). Hourly temperature data were taken from a short series at the station Uhlirska. The temperature model includes the following components: the average temperature; the first sine and cosine components of a Fourier analysis for the deviations of individual daily temperatures away from the average; a similar 1st order Fourier model for the standard deviations of the daily deviations away from the average; a third order autoregressive model for daily residual temperatures; a 1st order Fourier model for the hourly temperature variation during the day; and a second order autoregressive model for the hourly temperature residuals. [18] No account is taken of any correlation between temperature variation and the occurrence of a rainstorm at this stage. The inclusion of a temperature component has also allowed a temperature dependent potential evapotranspiration function to be included to replace the simple annual average sine curve used by Beven [1986] and Blazkova and Beven [1997]. However, again this has been kept simple, using the Thornthwaite formula in which potential evapotranspiration is dependent on a function of temperature and the seasonal pattern of hours of daylight [Thornthwaite, 1948]. Actual evapotranspiration is calculated within the TOPMODEL component as a function of potential evapotranspiration and soil moisture status up to a limit of an effective maximum available water capacity parameter, srmax. [19] A full list of the parameters for each of the different components, showing those which were varied in the calibration process, is given in Table Model Calibration and Uncertainty Estimation Within a GLUE Framework 5.1. GLUE Methodology [20] The generalized likelihood uncertainty estimation methodology of Beven and Binley [1992] is a flexible Monte Carlo simulation based approach to model conditioning and uncertainty estimation. It is based on rejecting the idea that there is a unique optimum parameter set in a model calibration in favor of multiple parameter sets and even multiple model structures that may be acceptable in simulating the system under study. In this situation it is only possible to evaluate the relative likelihood of a given model and parameter set in reproducing the data available to test the models. In particular, those models that are deemed unacceptable or nonbehavioral may be rejected by being given a likelihood of zero. Uncertainty in the predictions may then be estimated by calculating a likelihood weighted cumulative distribution of a predicted variable based on the simulated values from all the retained simulations (those with a likelihood value greater than zero). Thus, for any model predicted variable, Z, P ^Z t < z X i¼n ¼ LM ð i Þj^Z t;i < z i¼1 where P(^Z t < z) are prediction quantiles, ^Z t;i is the value of variable Z at time t simulated by model M( i ) with parameter set i and likelihood L[M( i )]. Accuracy in estimating such prediction quantiles will then depend on having an adequate sample of models to represent the behavioral part of the model space. Within this framework the values of the parameters are always treated as a set with their associated likelihood value so that any interactions ð1þ

6 14-6 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT between parameter values in fitting the available observations are included implicitly in the conditioning process. [21] Traditional likelihood functions, based on residual error models, may be used within this framework (as in the related study of gauged catchments by Cameron et al., [1999, 2000a]) but there is also the flexibility to use more ad hoc likelihood measures or fuzzy belief or possibility measures. The only requirements, in fact, are that a measure should increase with increasing goodness of fit (or belief), and that nonbehavioral models should be rejected (given a likelihood value of zero). The GLUE methodology has been applied in rainfall-runoff modeling by Beven and Binley [1992], Beven [1993], Romanowicz et al. [1994], Freer et al. [1996], Fisher and Beven [1996], Seibert [1997], Piñol et al. [1997], Lamb et al. [1998], Dunn [1999], and Beven and Freer [2001] and in flood frequency estimation by Cameron et al. [1999, 2000a, 2001b]. The study of Franks et al. [1998] has previously demonstrated the utility of a fuzzy measure in assessing the predictions of imprecise estimates of saturated contributing area in an application of TOPMODEL within the GLUE framework. [22] The GLUE methodology focuses attention on the value of different types of data in evaluating different possible models and constraining the predictive uncertainty. In the current application we can easily make Monte Carlo simulations with different choices of parameter values across the ranges shown in Table 1. The ranges have been chosen to be reasonable in respect of what is known about the rainfall and hydrological characteristics of the catchment. The result is a very wide range of flood frequency predictions. A priori it may be very difficult to specify which of the resulting combinations of randomly chosen parameters are not feasible predictors of the catchment frequency characteristics. Even for a model that is widely used, little or nothing is known about the interactions amongst the parameters within the (nonlinear) model structure that leads to either behavioral or nonbehavioral simulations in a particular application. In addition, the Joseful Dul catchment is being treated here as ungauged, so that even a nominal check on the performance of the rainfallrunoff modeling is not possible. [23] So, what sort of data could be used to constrain the resulting range of Monte Carlo predictions in this type of situation? We have available the CHMI regional flood frequency estimates that might be expected to be more reliable for low return period peaks than for higher return periods. Here only the estimates of up to the 10-year return period peak have been used in conditioning the Joseful Dul simulations. We also have the CHMI regionalized estimate of flow duration characteristics of the catchment and estimates of the distribution of maximum annual snow water equivalents for some stations in the region. All of these data sources may be expected to be in error with respect to the actual values for the ungauged Joseful Dul catchment. They are used here only as indices as to what might be acceptable simulations and the rejection of some of the possible models as nonbehavioral Conditioning Process Within GLUE [24] Each model realization was initially run for a period of 100 years using hourly time steps. Simulations of 100 years will give an adequate estimate of the 10 year flood which is the longest return period of the regional frequency estimate used in evaluating each model parameter set. A single realization takes approximately 1 minute on a 450 MHz Pentium PC using the Gnu Fortran compiler running under Linux. Using 20 PCs in parallel we have made 48,600 runs of the model using different randomly chosen sets of parameter values within the GLUE framework. The full range of 1000 of those different realizations, conditioned only on the prior estimates of the parameter ranges, is shown in Figure 3. It is clear that the Monte Carlo experiments give an extreme range of predictions at this point before any conditioning in respect of the relative likelihoods calculated for individual realizations. [25] It is worth noting at this point that there is also a realization effect associated with each of these runs due to the limited length of each simulation [Beven, 1987]. Figure 4 shows the results of 20 (100 year) runs of the model with a single set of parameter values. The differences between the runs arise only from using different random seeds to initialize the rainfall model. In Figure 4a, the resulting flood frequency estimates are compared with the Wakeby distribution estimates as fitted to the observed annual maximum series. Two of the simulations lie outside the prediction limits of the Wakeby procedure. The differences are less for both flow duration curves (Figure 4b) and frequencies of maximum snow water equivalent (Figure 4c). [26] As always in any flood frequency analysis, this type of realization effect also affects the observed flood peaks on which the regionalized estimates are based. The variability of observed floods is such that the observed sequence cannot be assumed to be an adequate sample of the asymptotic distribution of floods at a site under current climate conditions. Thus in conditioning the predicted flood series on information derived from the observed data, it must be remembered that we are evaluating the likelihood of a particular parameter set and rainstorm realization in reproducing the behavior indicated by a particular and limited sample of observations. In this study, for the ungauged Joseful Dul catchment, the observations are associated with additional uncertainty since they are based on best estimates of distributions derived from data collected elsewhere. [27] From the outputs saved for each Monte Carlo realization, three goodness-of-fit criteria or performance measures were computed as follows: 1/sum of absolute errors between a Wakeby distribution fitted to modeled annual peaks for each realization and the regional CHMI estimate. The errors are computed on the first four quantiles (1-, 2-, 5-, and 10-year return periods) for which we expect the regionalized estimates to be most robust. 1/sum of absolute errors between the regional flow duration curve and modeled curve on all the 13 quantiles available. 1/sum of absolute errors between Wakeby distributions fitted to maximum annual snow water equivalents observed and modeled on five quantiles (up to the 20-year return period estimate). [28] The conditioning process within the GLUE framework will give greater weight to predictions from parameter sets that perform well in terms of these different performance

7 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT 14-7 Figure 3. Flood frequency curves of a sample of 1000 realizations of 100 years from the total of 48,600 generated (shaded lines); Wakeby estimate with uncertainty bounds (dashed lines); regionalized estimate of CHMI up to 10 years return period (asterisk); ev1, extreme value reduced variate (Gumbel); 4.6, return period 100 years; 6.91, 1000 years; 9.21, 10,000 years. measures. The different performance measures may be transformed or combined in different ways. Here we have chosen to explore a method of fuzzy combination, treating each of the three measures as fuzzy variables in a way consistent with the vague knowledge available for conditioning at this ungauged site. This allows a possibilistic, rather than probabilistic, interpretation of the resulting combined measure that can still be used as a likelihood value or Figure 4. Twenty simulations of 100 years with the same set of parameters but different Monte Carlo seeds (shaded lines). (a) Flood frequency curve, Wakeby estimate with uncertainty bounds (dashed lines); regionalized estimate of CHMI up to 10 years return period (asterisks); (b) flow duration curve, regional estimate (asterisks); (c) frequency curve of maximum snow water equivalents, observed (asterisks); ev1, extreme value reduced variate (Gumbel); 4.6, return period 100 years.

8 14-8 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT Figure 5. Fuzzy inference system for the combination of different likelihood measures; 3 inputs: flood frequency (3 membership functions linguistically called hwrongi, hreasonablei, and hgoodi, from left to right), flow duration (3 membership functions hwrongi, hreasonablei, hgoodi), snow water equivalent (2 membership functions hwrongi, hgoodi); 1 output: fuzzy relative likelihood (4 membership functions hvery lowi, hlowi, hacceptablei, hhighi); 18 rules (rows); centroid defuzzification. Vertical lines show the evaluation for one particular set of inputs (0.392 is 1/sum of absolute errors on flood frequency curve, etc.; 350 is the resultant likelihood after defuzzification; degree of membership is indicated by shading). prediction weight in the GLUE methodology (equation (1) above). We therefore call the combined possibility measure a fuzzy relative likelihood in what follows and do not expect the resulting possibilistic range of predictions to be an estimate of the true probability of an outcome. However, we consider the latter impossible to estimate given the potential for model structure error and the nature of the data available for calibration for this ungauged site. [29] There are many possible ways in which the different model realizations could be evaluated in terms of a fuzzy relative likelihood. In this study we have made use of the fuzzy toolbox available for MATLAB. The three goodnessof-fit criteria mentioned above are described by 2 or 3 membership functions each (linguistically called hwrongi, hreasonablei, or hgoodi). In Figure 5, for example, the class ofhwrongi simulations is described by the triangular membership function on the left in the first column,hreasonablei by the approximately centrally placed triangle,hgoodi by the trapezoid on the right. [30] The output (relative likelihood) can be hvery lowi, hlowi, hacceptablei or hhighi. The first membership function hvery lowi at the left side is very narrow (can hardly be seen). It ensures the rejection of the cases when any of the criteria is hwrongi. [31] Combining the different measures uses all the 18 rules (rows) as shown in Figure 5. If any of the criteria is hwrongi the combined fuzzy likelihood will be hvery_lowi and that model realization can be rejected as nonbehavioral (rules 1, 2, 3, 5, 7, 9, 10, 11, 12, 13, 15, 16, 17, 18). If all three are hgoodi, the likelihood is hhighi (rule 4). The flood frequency and flow duration criteria are considered to be more important in evaluating the models and the resultant relative likelihood has been made more sensitive to them. If either of these criteria is hhighi and the other hreasonablei the resultant likelihood is hacceptablei (rules 6 and 8). If both are hreasonablei the resultant likelihood is hlowi (rule 14). In order to derive a final fuzzy relative likelihood measure for each model evaluated in this way a centroid defuzzification method was used. Note that on the original performance measures there is a smooth transition between the best and poorer models. The fuzzy rules provide a means of getting a clearer discrimination between what might be considered behavioral and nonbehavioral parameter sets, although it was found that the application of the

9 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT 14-9 Figure 6. Dotty plots of all the behavioral 100-year simulations (2281 out of a total of 48,600 realizations) for each of the randomly varied parameters showing some sensitivity to conditioning based on the fuzzy rules. Likelihoods of Figure 5 have been rescaled to sum up to one. rules in this case still left a large number of simulations with low (but not zero) relative weight. 6. Results 6.1. Simulations of 100 Year Periods [32] Figure 6 shows plots of the resulting fuzzy relative likelihood values for all 2281 parameter sets for which the relative likelihood was nonzero (out of the total 48,600 simulations run), for the 7 most sensitive parameters. The average duration and intensity of the November to April lowintensity storms proved always to be insensitive. These dotty plots represent projections of the samples of the fuzzy relative likelihood surface in the high dimensional parameter space onto each parameter axis. The better simulations have higher relative likelihood values. Each dot represents a single run of the model. It can be seen that both behavioral and nonbehavioral simulations can occur almost anywhere within the chosen ranges of the effective parameter values. They show little or no sensitivity to variations in the maximum snowmelt factor dfmax. Only the T0 parameter shows any strong conditioning of high likelihood simulations to only a limited part of the original range, and even then some occasional relatively good simulations occur elsewhere. [33] These 2281 simulations of 100-year periods are used for the computation of 5 and 95% prediction limits using the fuzzy measures as the likelihood weighting in equation 1 (Figure 7). The fuzzy likelihood weighting narrows the initial range shown in Figure 3. Remember that all the simulations that contribute to these prediction limits are considered behavioral in terms of the fuzzy measures as defined above, although the best simulations contribute greater weight in the application of equation (1). The higher upper bounds are primarily reflecting the simulated magnitude-frequency characteristics of the rainfall that cannot be taken account of in the simple statistical extrapolation of the single (short) realizations of observed floods if the site were gauged. The prediction limits are not uncertainty bounds in the sense of giving an estimate of the probability of an observed discharge having a true probability of exceedence within the range indicated. They are rather conditional prediction limits: conditional on the choice of rainfall and hydrological model used; the ranges of parameter values considered; the actual realizations simulated and the evaluation measures used. The prediction limits have value in the sense that all the elements must be made explicit and can be discussed or justified. Within the context of estimating flood frequency for an ungauged catchments we would

10 14-10 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT Figure 7. Flood peak prediction limits (5 and 95%) for 100 year behavioral simulations using fuzzy likelihood weights (solid lines); Wakeby estimate with uncertainty bounds on the basis of observed flood peak statistics (dashed lines); regionalized estimate of CHMI up to 10 years return period (asterisks); ev1, extreme value reduced variate (Gumbel); 4.6, return period 100 years; 6.91, 1000 years; 9.21, 10,000 years. suggest that an estimate of the true probability of exceedence is not possible and that the choices we have made are reasonable Using Long-Period Realizations to Estimate Asymptotic Frequencies [34] The use of a 100-year simulation period, as presented above, was chosen as an approximate estimate of the effective record length of the regionalized estimates derived from shorter observed time series. It represents a compromise between sampling the parameter space reasonably well and having excessively long run times, even on the parallel computer system used here. There is still, however, a significant realization effect on the extremes of 100 year simulations that will affect the estimates of floods with return periods greater than about 10 years. Ideally longer simulation periods would be used to minimize this realization effect in calculating the uncertainties in, for example, the 1 in 50 year event. [35] For the 2281 behavioral parameter sets from the 100- year simulations, 10,000-year simulations were then generated to approximate the asymptotic distributions for the parameter sets. They should be robust in integrating over the realization effect for return periods of up to the order of 1000 years. The prediction limits are further constrained for these 10,000-year simulations (Figure 8a). The fuzzy relative likelihood weights associated with the parameter set are retained from the 100 year simulations in each case for the application of equation (1), since the real realization effect of the observed data and regionalized curves will be implicitly reflected in these weights. The flow duration curves and frequencies of snow water equivalents can be seen in Figures 8b and 8c. 7. Evaluation Against the Historical Data [36] Up to this point the Joseful Dul catchment has been treated as ungauged. However, as noted above, there is a long annual maximum peak record available prior to inundation of the site. Table 4 compares the observed numbers of annual maxima occurring in each seasonal period with the likelihood weighted mean and ranges from the behavioral predictions. With the exception of the September- October period, the single realization that is the observed series falls within the predicted ranges. In fact, it might be expected that the simulations, predicting hourly average runoff values, might underestimate the recorded instantaneous annual maxima from short intense events in summer and autumn [see, e.g., Beven, 1987]. Thus comparing these annual maxima has its limitations, but they are the only observed data that are available. [37] Figure 8a compares the prediction limits arising from the current study, with those for a Wakeby distribution fitted by L moments to the 68 annual maximum peaks by the methods of Hosking [1997]. It is seen that the current study estimates a quite similar upper prediction limit as the Wakeby statistical procedure. The lower prediction limit is much higher than that of the statistical estimate and appears more reasonable in this case in that it rises in a way consistent with the physical process. The statistical estimate, of course, is based on an assumption that the fitted distribution holds for return periods much longer than the length of the observed record. The current model estimate includes

11 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT Figure 8. (a) Flood peak prediction limits (5 and 95%) as in Figure 7 but using 10,000 year behavioral simulations (full lines) compared to same Wakeby estimates of limits and the mean estimate (dashed lines), observed peaks (circles), regionalized estimate of CHMI up to 10 years return period (asterisks); (b) flow duration prediction limits (5 and 95%) based on fuzzy likelihood weights (full lines) compared to regionalized estimates (asterisks); (c) Peak annual snow water equivalent prediction limits (5 and 95%) (solid lines) compared to weighted average of observed data (asterisks and dashed line); ev1, extreme value reduced variate (Gumbel); 4.6, return period 100 years; 6.91, 1000 years; 9.21, 10,000 years. the effects of the nonlinear interactions between rainfalls and flood peaks, in so far as they are represented by the chosen model structures and parameter sets identified as behavioral within the GLUE methodology. 8. Interpreting the Predictions in Terms of Runoff Processes [38] One of the referees for this paper suggested that our brute force approach to model evaluation would not lead to any physical meaningful conclusions about the nature of the runoff processes in this catchment. This is not strictly correct. Despite the significant uncertainties in modeling the response of a catchment treated as ungauged, we can derive likelihood weighted estimates of seasonal flood predictions and other response variables. As an example, Figure 9 shows the predicted cumulative distributions for the 10-year and 100-year return period discharges in different seasons of the year. These are likelihood weighted distributions over all the 10,000 year behavioral simulations. Also shown in Figure 9 are the predicted distributions of 10 and 100 year return period average antecedent snow water equivalent in the catchment prior to the maximum seasonal flood each year. This gives some indication of the role of snow in flood producing events in this catchment. Clearly many other response variables could be saved and displayed in this way. 9. Conclusions [39] This paper has shown how continuous rainfall-runoff modeling can be used for the estimation of flood frequency characteristics in a practical application to an ungauged catchment. A demonstration that the methodology can provide good simulations of both hydrographs and frequency characteristics at gauged sites has been previously

12 14-12 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT Table 4. Comparison of Observed and Simulated Numbers of Annual Maxima Occurring in Different Seasons (Simulated Values are Based on 10 Thousand Years Simulations Normalized to the 68 Years of the Observed Data) Season Observed Number of Annual Peaks Normalized Weighted Average Number Normalized Lower Limit Normalized Upper Limit November April May June August September October shown by Cameron et al. [1999, 2000a]. The estimates have been made within the GLUE framework, using minimal regional information to condition a fuzzy measure associated with the model predictions for each randomly chosen parameter set. This measure is then used as a relative likelihood weight in the application of equation (1). The result is a range of frequency curves that can be interpreted in terms of a distribution of predictions for any chosen return period which might then be used in a risk assessment for dam safety evaluation or other purpose. The range of predictions derived in this way using 10,000-year simulations has resulted in roughly similar upper prediction limits of flood magnitudes at longer return periods as those of a direct estimate based on the historical annual maximum Figure 9. Predicted cumulative distributions for (a) 10 years and (b) 100 years return period discharges in different seasons; predicted distributions of (c) 10 and (d) 100 year return period average antecedent snow water equivalent in the catchment prior to the maximum seasonal flood each year; November to April (dashed line), May (dotted line), June to August (solid line), October to November (dashed-dotted line).

13 BLAZKOVA AND BEVEN: FLOOD FREQUENCY FOR AN UNGAUGED CATCHMENT series for the site. The lower prediction limit appears to be more physically reasonable than that of the statistical estimate. [40] The advantages of the current approach are that more information about the rainfall statistics and the characteristics of the catchment is included in the estimation procedure in a way that is consistent with an overall water balance and flow duration simulation for the catchment. The approach can also reflect the fact that the observations available for conditioning are but one (generally short) realization from the range of possible events. This single realization may not be a good estimator of the asymptotic distribution of events as is always assumed in statistical extrapolations. In the methodology proposed here the realization effect can be explicitly taken into account, subject to the limitations of computer time in obtaining adequate samples. Longer runs (here 10,000-year simulations) will be required to achieve stability in the prediction limits for long return period events having conditioned on an estimated effective record length of the observations. The computer time required will become less of an issue in future in comparison with the availability of data with which to constrain the prediction bounds. [41] Clearly, the predictions are dependent on the models of rainfall and runoff used. However, the results suggest that application of a similar approach to other ungauged catchments for which TOPMODEL is a suitable rainfall-runoff model might give reasonable flood frequency estimates. There is no theoretical reason why different rainstorm, snowmelt and rainfall-runoff models should not be included within the estimation methodology presented, only the practical reasons of available computer time. It is also possible within this framework, once a set of behavioral simulations has been identified, to evaluate the predicted distributions of different response variables for the catchment (as in Figure 9) and to estimate the effects of future climatic scenarios on frequency distributions [Cameron et al., 2001b]. [42] Acknowledgments. The authors are grateful to Jim Freer for setting up part of the computations on the Lancaster parallel PC system; to Jon Hosking for providing software for calculating the regionalized flood frequency estimates and statistical uncertainty bounds; to André Musy for support while the second author was visiting EPFL, Lausanne; and to Josef Sobota for providing the Prague rainfall series. Hydrological and meteorological data and regional estimates were provided by the Czech Hydrometeorological Institute (CHMI). The work was carried out within the grants of the Ministry of Environment of the Czech Republic VaV 510/3/97 and VaV 510/1/99 and the grant OK373 of Ministry of Education of the Czech Republic, which made it possible to participate in the EUROTAS project of the European Union. We are grateful also to the two referees who helped to clarify the presentation. References Ambroise, B., J. Freer, and K. J. Beven, Application of a generalized TOPMODEL to the small Ringelbach catchment, Vosges, France, Water Resour. Res., 32, , Beven, K. J., Runoff production and flood frequency in catchments of order n: An alternative approach, in Scale Problems in Hydrology, edited by V. K. Gupta, I. Rodriguez-Iturbe, and E. F. Wood, pp , D. Reidel, Norwell, Mass., Beven, K. J., Towards the use of catchment geomorphology in flood frequency predictions, Earth Surf. Process. Landforms, 12, 69 82, Beven, K. J., Prophecy, reality and uncertainty in distributed hydrological modelling, Adv. 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