Flood frequency prediction for data limited catchments in the Czech Republic using a stochastic rainfall model and TOPMODEL

Journal of

Hydrology Journal of Hydrology 195 (1997) 256-278

Flood frequency prediction for data limited catchments in the Czech Republic using a stochastic rainfall model and TOPMODEL Sarka Blazkovaa**, Keith Bevenb ‘TheT.G. Masaryk Water Research Institute, Prague, Poa%abskd30, 160 62 Praha 6, Czech Republic bbncaster University, Centrefor Research on Environmental Systems and Statistics, IEBS, Bailrig& Luncaster LA1 4YQ, UK Received 3 March 1995; revised 15 July 1995; accepted 15 July 1995

A continuing problem in hydrology is the estimation of peak discharges for design purposes on catchments with only limited available data. A promising and elegant approach to this problem is the derived flood frequency curve pioneered by Bagleson (1972. Water Resow. Res., 8(4): 878-898). A number of studies using this approach have been published over the last 20 years but only a few have compared the predicted curves with observations. One exception used a simple stochastic rainfall model to drive a version of TOPMODEL (Beven, 1987, Earth Surj Processes Landforms, 12: 69-82). The present study describes a new version of the stochastic rainfall simulator previously used with TOPMODBL and its application on three small catchments (l-87,4.75 and 25.8 1 km*) in the Jizera Mountains in the Czech Republic. The rainfall model differentiates between high and low intensity events. The resulting rainfall statistics were checked by comparisons with measured data. The flood frequency curves predicted by the combined model were constrained by the regional estimates or a measured series for short return periods and used to predict longer return period flood magnitudes. Only one TOPMODEL parameter has to be adjusted-an effective average transmissivity. For the two smaller catchments also a rainfall parameter has to be adjusted depending on the size of the catchment. It is shown that the random sequence of rainstorms can have a significant effect on the predicted 100 year return period event, even for 1000 year simulations and without allowing for uncertainty in the rainfall model parameters.

* comsponding

author.

0022-1694H7/$17.00

Q 19!&‘- Elsevier Science B.V. All rights reserved

PfI SOO22- 1694(96)03238-6

S. Blazkova, K. Beven/Journai of Hydrology 195 (1997) 256-278

257

1. Introduction

A continuing problem in hydrology is estimation of peak discharges for design purposes on ungauged catchments. One approach to this problem is to regionalize the flood frequency characteristics of a number of gauged basins; another is to estimate the flood frequency distribution on the basis of more readily available long rainfall records or intensity duration frequency (idf) rainfall statistics, a third is to use a specified design storm. The last two methods require the estimation of an effective runoff coefficient or percentage runoff for each storm, a particularly difficult problem for ungauged catchments. One set of methods for estimating percentage runoff, closely related to the development of unit hydrograph technique, is also based on regionalized regression relationships (e.g. NERC, 1975). Attempts to develop a more physically based approach started with the derived distribution technique of Eagleson (1972), which assumed a constant runoff coefficient for all storms. Extensions of this work, using the Geomorphological Unit Hydrograph, have been pnbfished by Hebson and Wood (1982), Cordova and Rodriguez-Iturbe (1983), and Diaz-Granados et al. (1984), while Eagleson (1972), Shen et al. (1990) and Cadavid et al. (1991) have used a kinematic wave routing algorithm. All of these extensions were based on the infiltration excess concept to predict effective rainfalls and neglected any heterogeneity in rainfalls or antecedent moisture conditions, which were assumed to be the same for all storms. Moughamian et al. (1987) and Raines and Valdes (1994) have published comparisons of some of these methods and suggest that all have deficiencies and in particular that the results are sensitive to both the infiltration parameters (which may be difficult to estimate) and the nature of the rainfall distributions used. In fact, field studies of catchment response have shown that the simplified approach to effective rainfall used in these studies is often inadequate (see for examples the reviews by Dunne (1987) and Beven (1983), and papers in Anderson and Burt (1990)). The conversion of rainfall into streamflow is a complex nonlinear combination of processes involving partial surface and subsurface contributing areas, which depends strongly on the antecedent moisture status of the catchment, the pattern of rainfall intensities and, in some cases, the spatial distribution of rainfalls. More complex models of effective rainfall have been used in flood frequency predictions by Beven (1986a, b, 1987) and Sivapalan et al. (1990) based on variants of TOPMODEL that allow for variations in effective rainfalls as a result of the effects of topography, soil heterogeneity, antecedent storage and, in the case of Sivapalan et al. (1990), spatial variability in rainfalls. These studies allowed for the prediction of runoff generation by infiltration excess, saturation excess and subsurface flow process mechanisms. With the exception of the study of Beven (1987), this latter set of TOPMODEL based studies have been concerned with sensitivity studies rather than applications to actual catchments. Beven (1987) showed that his model could reproduce the flood frequency curve of the Wye catchment in mid-Wales, using rainfall model parameters derived from observed rainfalls and with some calibration of one parameter of the rainfall-runoff model. Contributing areas predicted by the model approached 100% only for the most extreme events. In the study reported here the stochastic rainstorm version of TOPMODEL is developed

258

$. Blazkova, K. Beven/Journal of Hydrolosy 195 (1997) 256-278

further to allow for more complex rainstorm inputs in an application to three catchments in the Jizera Mountains in the Czech Republic. Previous work has suggested that the variable contributing area runoff production that underlies the TOPMODEL predictions is appropriate in this region. Regionalized flood frequency and flow duration curves are available for the area based on relatively short records from gauged catchments. These are used to constrain the predictions of the model for short return periods. Longer rainfall records are available for constraining the stochastic rainstorm generator. The aim is to use the combined rainfall and runoff generation models, constrained by the data available, to improve the estimates of flood magnitudes at longer return periods taking account of the effects of antecedent conditions and topo~aphic controls on the dynamics of contributing areas. Such data limited situations are common. In trying to improve flood frequency estimates when data are limited it is necessary to make simplifying assumptions with the aim of keeping the number of parameters to be identified to a minimum, in the hope that the resultant model will be relatively robust. In this study, a particular form of model has been used to take account of the nonlinearities in the runoff generation mechanisms that result from the interaction of rainfall volumes and intensities, antecedent conditions and topographic controls on contributing areas. However, only a small number of parameters must be estimated for each application.

2, Rainfall models Cox and Isham (1994) recognized at least three broad types of mathematical models of rainfall: (1) empirical statistical.models; (2) models of dynamic meteorology; (3) intermediate stochastic models with a modest number of parameters which are intended to relate to underlying physical phenomena such as rain cells. They subdivided storm systems into convective and nonconvective (see also Robinson (1994)) and described approximately temporal and spatial scales of the physical processes involved. For Western Europe, the convective type is the short-lived and extremely intense rainfall of a typical summer thunderstorm. For nonconvective (cyclonic) systems the precipitation field is a region of perhaps 104 km 2 or more, lasting 12 h or so, and consisting of a series of lowpressure fronts. These frontal rainfall areas or rainbands are regions of high intensity rainfall lasting for between about 90 min and 4 h, and have a spatial size of the order of 103-104 km 2. Within the rainbands are the rain cells, which are the smallest precipitation elements observable by radar, and are represented by small intense ~ O r echoes of the order of 10-50 km 2. These cells last on average for about 40 rain and tend to cluster within the rainbands. Frontal precipitation may contain a mixture of the convective and nonconvective type (Cox and Isham, 1994). Peters-Lidard and Wood (1994) distinguished a small mesoscale area (a cluster of rain cells) and a large mesoscale area (a rainband or a synoptic scale front). For flood frequency estimation, which requires the generation of large numbers of storms and storm profiles of intensity, the third type of model is the most appropriate. There are a number of variants on these models ranging from the simple minimal parameterization of exponential distributions of storm arrivals, durations and volumes to complex models that attempt to represent explicitly the clustering of cells by Neyman-Scott or Barlett-Lewis processes (Cox and Isham, 1994). The spatial scaling of rainfalls may also

$. Biafra, K. Beven/Journalof Hydrology 195 (1997) 256-278

259

be an important consideration in modelling for flood estimation (see, e.g. Gupta and Dawdy, 1995; Lovejoy and Schertzer, 1995) although Obled et al. (1994) have suggested that, at least in some circumstances, runoff simulations are sensitive to spatial pattern of storm rainfall more in respect of the determination of the volume of input and average intensity, than the pattern itself. All the more complex models suffer from requiring a large number of parameters to be calibrated against the observed rainfall data, which were limited in the present study (see below). However, initial analyses showed that the simple rainfall model based on exponential distributions, previously used with TOPMODEL by Beven (1986a, b, 1987) and other derived distribution methods of flood frequency estimation (Eagleson, 1972; Hebson and Wood, 1982; Diaz-Granados et al., 1984), could not adequately reproduce the observed rainstorm behavionr, in particular the high intensity events important to the flood hydrology of the Jizera Mountains. Thus a new variant on this model was developed, as described below.

3. Study catchments in the Jizera Mountains The calculations described in this study have been carried out on three catchments, two of them being experimental catchments of the Czech Hydrometeorological Institute (CHMI, 1995): the Uhlirsk~ catchment (UHL in Fig. 1) on the Cern~ Nisa (1.87 km 2) and Jezdeck~ (JZD in Fig. 1) on the Cern~ Desn~ (4.75 km2). Both these catchments have a series of observations too short (only about 10 years, which were moreover a dry period) to derive flood frequency curves from measured data. Regionalized estimates of frequency curves for return periods of between 2 and 20 years and flow duration curves derived by CHMI were employed to constrain the model predictions. Experimental studies in parts of the smaller catchment (Uhlirsk~;) have shown that the pattern and dynamics of the saturated areas are broadly in agreement with the pattern of the topographic index used by TOPMODEL. A third catchment is the Josefuv Dul catchment (JD in Fig. 1) on the Kamenice River (25.81 kin2). It has an observed series of 68 years and historical flood information. In the Jizera Mountains, snow cover lasts usually from December to April, when it melts away slowly, but it can also entirely disappear any time during winter. Important floods occur in the summer half-year. Exceptionally, a large rainfall event can occur during winter. At the Josefuv DUl site during the 68 year period of observations two such events occurred, with peaks o f 66.3 m 3 s -t and 49.8 m 3 s -t caused by rainfall totals of 76.5 mm in 24 h and 81.1 mm in 48 h, with some residual snowmelt. Other winter floods were in the range from 7 to 35 m 3 s -t, and the annual floods of return period 1, 2 and 5 years evaluated from measured data (CHMI) are 19.5 m 3 s -1, 31.6 m 3 s -t and 46.5 m 3 s -t, respectively (Novick~ et al., 1993). Within the range from a 2 to 100 year return period both the empirical and the GEV (generalized extreme value; fitted using Hosldng et al. (1985)) dislributions for annual and summer floods are almost the same, whereas for the winter floods they are vastly different (lower by about .50%). It is therefore possible to model flood frequencies for return periods greater than 2 years using only the months from May to October. Earlier work with TOPMODEL coupled to the original rainfall simulator in the Jizera

260

S. Biazkova, K. Bcven/Journal of Hydrology 195 (1997) 256-278

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Mountain's has shown that the modelled frequency curves are less steep than the frequency curves determined statistically in the Jizera Mountains region. Fig. 2 and Fig. 3 show the CHMI estimates of frequency curves for the UhlirskA and JezdeckA catchments. Results of the original rainfall simulator, having only one type of storms, and those of simulations with the new version, which predicts both high intensity and low intensity events, are also shown. The original simulator does not model high intensity events often enough to produce a flood frequency with the steep curvature that is appropriate for this region. When the TOPMODEL parameter To (effective transmissivity) is adjusted the whole curve moves up or down but does not significantly change in curvature. These results gave the impetus to the revision of the rainfall simulator used in this study.

4. The rainfall model

4.1. Description of the rainfall model The new version of the rainfall model was intended to take into account two types of rainfall, i.e. frontal rains and convective storms, within a simple parameterization, an extension of the approach used in the previous TOPMODEL based frequency predictions

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Fig. 2. Flood frequency curves on the Uhlirsk~ (UHL) catchment. Continuous lines, simulated with the new version of the rainfall model (LI and HI storms) for three values of the threshold parameter of Hi storms THRIm (ram h-'): lower line has the THRI m estimated from data; middle and upper lines me runs of 3000 years with values of THRIm constrained by the maximum hourly rainfall intensity (mm h-'); dotted lines, simulated with the old version (only one type of storms) for two values of the effective transmissivity parameter TO(m 2 h-'). *, Regional estimates of CHMI used for adjustment; circle, regional estimates of CHMI not used for adjustment; evl, the extreme value Type 1 or Cmmbel distribution reduced variate. (Beven, 1986a, b, 1987), which was itself b a s e d on the Eagleson (1972) stochastic rainfall model. Both types o f event m a y also contain relatively short-lived cells o f higher intensity. However, it is impossible without additional information to distinguish between those generation mechanisms in a historical rainfall record. Therefore for the purpose o f this study two types of events will be defined simply as low intensity (LI) and high intensity (HI) events. Each type o f e v e n t has its o w n set o f parameters: The duration o f an L I rainstorm with a threshold o f I h is generated as NDLx-- 1 - T B A R u l n ( 1 - U) (h)

(1)

where T B A R u is the mean storm duration o f L I events, and U is a random number taken from a uniform [t3,1] distribution. The duration o f the high intensity rainfall N D m uses random sampling from a discrete cumulative distribution to preclude the possibility o f events longer than 6 h. Probabilities for the occurrence o f durations in the range 1 - 6 h were determined empirically from the data available.

262

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K. Beven/Journal of Hydrology 195 (1997) 256-278

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Fig. 3. Flood fTequency curves on the Jezdeckai (JZD) catchment. Continuous lines, simulated with the new version of the rainfall model (LI and HI storms) for three values of the threshold parameter of Hi storms THRIm (ram hll): lower line has the THRIm estimated from data; middle and upper fines are runs of 3000 years with values of THRIm constrainedby the maximum hourly rainfall intensity (nun h-I); dotted lines, simulated with the old version (only one type of storms) for two values of the effective transmissivity parameter To (m2 h-t). *, Regional estimates of CHMI used for adjustment; circle, regional estimates of CHMI not used for adjustment; evl, the extreme value Type 1 or Gumbel distribution reduced variate. The mean rainfall intensities for the two types o f events are generated independently from duration b y RINT u = 0.001 - R B A R u l n ( 1 - U) (m h - l)

(2a)

RINTm = T H R I m - RBARmln(1 - U) (m h - i)

(2b)

where R B A R is the mean storm intensity, and T H R I m is a threshold value which is a parameter computed from observed data. F o r L I events the threshold is equal to 1 ram. The interstorm periods for both types o f events are generated from N A T u = 1 - A T B A R u l n ( 1 - U) (h)

(3a)

N A T m = 1 - A T B A R m l n ( 1 - U) (h)

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where A T B A R is the mean storm interarrival time. A minimum interstorm period o f 1 h was used, which means that no zero rainfall periods within a rainfall event are allowed. The random storm profile requires the specification o f a

S. Biaekom, K. Beven/Joumal of Hydrology 195 (1997) 256-278

263

Table 1 Sununa~ of the rainfall modelparameters Low intensity (LO e~nt3 TBARu RBARu ATBARu RSPH High intensity (HI) events RBARm THRIm ATBARm

RSPm

Mean storm duration of LI events (no zero rainfall within event allowed) Mean storm intensity of LI events Mean storm interarrival lime of LI events Mean dimensionless stormprofile Standard deviations of the storm profile randomcomponentfor each ordinate First-order autocorrelmioncoefficientfor the randomc ~ t Empirical cumulative distribution of HI durations Mean storm intensity of HI events Threshold of mean storm intensity of HI events Mean storm interarrival time of HI events (set to once per s ~ half of the year in this study) Mean dimensionless storm profile Standard deviations of the stormprofile randomcomponentfor each ordinate First-order autocorrelationcoefficientfor the randomcomponent

mean dimensionless storm profile calculated from the observed data scaled by duration and rainfall volume, standard deviations of the storm profile random component for each ordinate calculated from the observed dimensionless profiles, and a first-order autocorrelation coefficient for the random component (RSPu, RSPm) also calculated from the observed storm profiles. Both profiles and parameters are different for each type of event and have been calculated directly from the available observed storm data. A summary of the parameters required is given in Table 1. A large number of very short events are modelled this way, some of which can be considered as convective. The HI events are inserted into the LI events following the randomly selected period between HI storms. Those can be considered as either fluctuations of intensity of frontal rains or convective cells of higher intensity associated with frontal rains. Events of large intensity are generated more frequently with this new version. Fig. 4 shows a comparison of the maximum hourly intensities simulated in multiple 50 year series for different model formulations. Twenty series were modelled. For a model without HI storms (top left figure) the maximum intensity is most often lower than 50 mm h q and the maximum intensity in the 1000years is about 7 0 m m h q. The higher intensities generated by the new model are illustrated in the bottom left picture. 4.2. Computing parameters f o r the rainfall model In this application a continuous version of TOPMODEL is used to model complete summer half-years with an hourly time step. Ideally, the rainfall parameters should be then computed from a continuous hourly rainfall record. Because such data were not available for this area, a set of 80 selected rainfall periods with hourly time step data prepared by Krejcov6 (1992) for the Smed~ Basin (site Bfl~ Potok (BP) in Fig. 1) from the period 1957-1986 was used instead. These data had been originally chosen for analysis on the

264

S. Blazkova, K. Beven/Journal of Hydrology 195 (1997)256-278

Table 2 Rainfall intensifies with duration I h at the stations Sou.~ (SOUS) and Ostrava (Trupl, 1958)

Retarll period (years)

Illtensity (mm h -I) SOUS

Intensity (ramh -I) Ostrava

1 2 5 10 20 50 100

17.71 21.82 27.14 31.25 35.42

15.91 20.20 26.32 31.32 36.72 43.56 48.96

basis of being wet periods containing a flood peak. The rainfall is an areal average over 26.13 km 2 intended for rainfall-runoff modelling. The corresponding runoff comprises a range of flood peaks with various estimated return periods from much less than 1 year to 50 years. The lengths of the events varied from 1 to 10 days including some intervals with no rainfall. If a minimum interstorm period of 1 h is used there are 526 events available within this sample.

4.2.1. Selection of high intensity storms High intensity events were defined as sections of data for which the maximum intensity reached or exceeded a certain limit. Differentiating between high and low intensity events also has to be to some extent arbitrary, especially with limited data sets. In this study, the limit between low and high intensity events was based on an analysis of short time intensities evaluated for the purposes of urban hydrology by Tmpl (1958). For the rainfall station Sou~ (SOUS in Fig. 1) the intensities with the duration 1 h are given in Table 2. Choosing a return period of 1 year (ATBARm = 4416 h for the summer half-year) yields a corresponding intensity of 17.71 mm h -I. The beginning and end of the HI sections were determined in such a way that the intensity does not drop under 6 mm h -l. In this way, 18 HI sections were identified (including newer data of CHMI added later). From these, the other HI parameters required (RBARtn, THRItn and profile parameters) were computed, with the threshold for average intensity of HI events THRIm being set at the lowest average intensity of a HI storm observed in the available sample (11.7 mm h-l). For the computation of the autocorrelation coefficients of the storm profile random component there were 33 pairs available. The autocorrelation coefficient is negative (--0.305). 4.2.2. Parameters of low intensity storms After leaving out the HI sections, the LI event parameters CI'BARu, RBARu and parameters of the dimensionless storm profile) were computed from the characteristics of the remaining periods. It should be recalled that these rainfall periods are a censored sample, being only those associated with flood peak discharges. Thus a large number of small events are missing from the computation. It can therefore be expected that there is some error in the estimation, particularly of TBARu, and the interarrival time cannot be estimated at all from these data.

S. Blazkova, K. Beven/Journal of Hydrology 195 (1997)256-278

265

Table 3 Comparison of modelled rainfall characteristics with characteristics obtained from observed data Volume (mm)

RBARu a (mill h -I)

TBARu" 0l)

A'I"BARu" 01)

Av. max. (ram h -t )

Low intensity (I.2) events Modelled Observed

8.09 Volume (ram)

0.93 0.92 RBARm m (ram h -t )

8.56 8.65 Duration (h)

40.6 40.9 ATBARma (h)

2A9 2.65 Av. max. ( m m h -t )

High intensity (HI) events Modelled Observed

56.5 50.3

16.2 16.1

3.47 3.33

4254 4416

25.8 24.0

Volume

Intensity

Duration

Interan'ival

Av. max.

(ram)

(nun h-b

(h)

time (11)

(nun h -I)

40.6 40.9

2.78 3.00

Total events Modelled Observed

8.85 -

0.98 0.97

8.61 8.75

• Parameters.

The interstorm period ATBARLI was computed using the average rainfall total of the summer half of the year (May to October) compotedas the areal average on the catchmerits by CHMI using isolines constructed from long-term data. For the catchments Uhlirsk~ and Josefuv Dul it is 771 mm, for the Jezdeck~i catchment 870 mm. Given the RBARLI and TBARu parameter values, the ATBARu value is then equal to 40.86 h and 34.54 h, respectively. For the computation of the autocorrelation coefficients of the storm profile random component there were 3977 pairs available. The high value of autocorrelation (0.389) can probably be explained by the fact that the rainfall is an areal average and likely to be smoother than point rainfall.

4.3. Checking the assumptions and performance of the rainfall model It must be stressed that this simplified rainfall model was developed in the context of severely constrained data availability. The aim was to implement a model that would respect the apparent characteristics of the summer rainstorms in the region, with a minimum number of parameter values to be constrained by the available data. In confirming the validity of the model, plots of the average and maximum intensities, durations and volumes of observed LI events suggest that the assumption of exponential distributions is reasonable. In Table 3 a comparison between rainfall characteristics in a modelled 1000 years series and observations is given for the LI and HI events as well as for the total storms. Parameters (i.e. average intensities, durations and interarrivai times) are reproduced correctly. The modelled average rainfall for the period from May to October agreed with the long-term average.

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$. Blazkova,K. Beven/Yournalof Hydrology 195 (1997)256-278

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4.3.1. Maximum hourly intensities The station SOUS mentioned above (Table 2) did not have a series of observations long enough to estimate maximum short-terra intensities of longer return periods. However, tables presented by Trupl (1958) show that the short duration maximum intensities (1 h or so) do not change dramatically from place to place in the Czech Republic. For a longer series at the station Ostrava the highest hourly intensities are also given in Table 2 (Trupl, 1958). The hourly intensity with return period 100 years is about 49 m m h -~. The simulated maximum intensities in a 1000 year period (20 series each of 50 years) are in shown Fig. 4 (bottom left). The intensities of catastrophic events can be much higher. On the Jflovslc) Brook during an event in 1987 the rainfall total falling in about 90 min on 6 km 2 was 190 nnn, and on 18 km 2 was 136 nun in 2 h. The r e m m period of this storm was estimated as I 0 0 0 10000 years (Ka~pdrek et al., 1989). 4.3.2. Comparison with observed daily rainfall Data on short time intensities of sufficient length to validate predictions of higher return period rainfalls will rarely, if ever, be available. It is somewhat easier to validate the predictions of daily rainfall frequencies, for which data series tend to be longer. However,

267

S. Blazkom, K. Beven/lownal of Hydrology 195 (1997) 256-278

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Fig. 5. Comparisonof ten modelled9 year exceedencecurves of daily rainfall (continuous lines) with the curve plotted from 9 year series of observedvalues (circles). for the Uhlirsk~i catchment only a 9 year series of dally areal precipitation was available from the CHMI. Observed daily rainfall exceedences are plotted in Fig. 5. It should be noted that these are data from a dry period 1982-1990, when the average summer rainfall was only 656 mm. Ten 9 year series were simulated with the new model with R B A R u adjusted in such a way that the average summer rainfall was 656 mm. The simulations agree reasonably well with the observed distribution (Fig. 5). In addition, within the region a much longer series of annual maxima of daily rainfall (May to October) is available (station Nov~i Louka NL in Fig. 1 within the Josefuv Dul catchment). The original record was extended by regression with the station Korenov (KO in Fig. 1) to a length of 83 years. The common period of observation was 41 years, with a correlation coefficient between the two series of 0.818 (S. Kol(trov~i, unpublished data, 1995). The exceedence curve from these observed data is shown in Fig. 6 together with ten modelled curves of the same length. In this simulation no parameters were adjusted in any way. The agreement of six of the series is very good whereas four series overestimate for longer return periods. One of them models an event comparable with the observed historical extreme (345.1 nun in 1897). This has been plotted separately as a horizontal line in the range between return period 100 and 1000 years as it is twice as large as the next highest value and its probabifity of occurrence is uncertain. This extreme value was

268

S. Biafra, K. Beven/Journal of Hydrology 195 (1997) 256-278

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measured at station Nov~ Louka. At the other station, Korenov, a value of 300 mm was observed the same day. The maximum value mentioned above and other observed 1, 2 and 3 day maxima in the region are given in Table 4. In various 1000 year simulations with different seeds the largest totals of 1, 2 and 3 days usually do not exceed 300 ram. One exceptional modelled sequence (different seeds from that in Fig. 5) gave 352 ram, 378 mm and 381 mm for 1, 2 and 3 day totals, respectively. These comparisons show that the new version of the rainfall model reproduces the available observed rainfall characteristics in the region reasonably well. The effect of possible error in the rainfall model parameters on the flood frequency curve itself will be discussed below.

5. TOPMODEL The full background to TOPMODEL has been described by Beven et al. (1995). It is a simple semi-distributed model based on an assumption that the effective hydraulic gradient at a point can be approximated by the local slope angle. This leads to the effects of

S. Bia~om, K. Boen/Journal of Hydrology 195 (1997) 256-278

269

Table 4 Maximum observed 1, 2 and 3 day rainfall totals for the whole period of observation up to 1980 (Bubenfckov~iet el., 1985) Station

Altitude (m a.s.l.)

Bedfichov Novt Louim (NL)* Bedfichov dam site(BE)' Bedfichov Kristi~mov(KRI)• Josnfuv Dul (JDU)* Dean~ Son~ (SOUS)' Bfl~ Potok Jizerka (J1Z)' Stork (SMRK)" Bfl~ Potok u Smd~u~-y(ST)" Bfl~ Potok (BIL)a Bfl~ Potok cottage Smbiava (SM)'

Number of years of observation

Totals (nun) 1 day

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780

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topography and soil information being represented in the form o f the distribution o f the soil-topographic index, a/T0tan ~, where a is the area drained per unit contour length at a given point, fl is the slope angle, and To is the lateral transmissivity when the soil is just saturated (m 2 h-l). Assuming an exponential decline o f transmissivity with depth o f the water table, the lateral flow at any point (q~) is given b y qi = rotan fie - sdm

(4)

Following the steady-state relationships on which T O P M O D E L is based, current local soil moisture deficit can then be calculated from

where S B A R is the m e a n storage deficit for the whole catchment, 3' is the areal mean value o f the logarithm o f the soil-topographic index, St is local storage deficit (m), and m is a scale parameter (m). Different forms o f transmissivity function lead to different soil-topographic indices (see A m b r o i s e et al., 1996). F o r the exponential case the T O P M O D E L theory leads to a

270

S. Blazkova, K. Beven/Journal of Hydrology 195 (1997)256-278

subsurface storage deficit-discharge function of the form

Q=Qoexp(-SBAR/m)

(6)

where Q0 = exp(-3'). Evapotranspiration losses are controlled by potential evapotranspiration and storage in the root zone with the additional parameter SRMAX (effective available water capacity of the root zone). The potential evapotranspiration estimation uses the same seasonal sine curve as Beven (1986a, b, 1987) with a single mean daily potential evapotranspiration parameter. In this application the results were not sensitive to the value of SRMAX, within a realistic range of values, and therefore it was kept constant at 0.04 m. The initial storage deficit at the beginning of May is sampled from a cumulative distribution derived by inverting Eq. (6) to obtain a value of SBAR given discharge, where the discharges are taken just after the end of snowmelt for each of the 9 years of data. Antecedent conditions before individual rainstorm events are produced by the continuous hourly simulation in each summer period. The model includes a channel muting algorithm based on the network width function and a velocity-discharge relationship that yields a constant wave velocity (Beven, 1979). Following Beven (1979), a wave velocity of 1 m s -l (3600 m h -t) is assumed. The lengths of the longest stream in the Uhlirsk~i and Jezdeck~ catchments are about 2000 m and 3500 m, respectively, so that runoff reaches the outlet of the catchment in one time step. In the Josefuv Dul catchment a channel length based width function was derived from the map of the network, with a maximum delay of 2 h.

5.1. Constraining TOPMODEL parameter ranges One of the great advantages of TOPMODEL is the modest number of parameters. In some applications (e.g. Romanowicz et al., 1994) only two parameters are used: the scale parameter m and the average transmissivity To. Those parameters can, in principle, be determined from data. The value of m can be estimated from recession curves. Lateral transmissivities can be measured in the field or estimated from conductivity data under certain assumptions. In practice, one obtains rather wide ranges of possible values (see discussions in Beven et al. (1995)). In the data limited case being investigated here, it is necessary to make use of whatever data are available for the estimation of parameter values. In the Czech Republic regionalized estimates are available on any ungauged catchments for the annual water balance, the flow duration curve and the flood frequency curve, which may be reasonably reliable for short return periods. These data have been used to constrain the model parameters. It is intended in future work to look at the additional information from hydrographs and measured saturated areas.

5.1.1. Using the recession and flow duration curves The regime of low flows is important information because it will control the initial conditions for individual high flow events. From flow duration curve of the summer halfyears a quantile was selected on which the parameter m was adjusted. The estimated values

$. B l a ~ a , K. Be~nlJournal of Hydrology 195 (1997) 256-278

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are 0.016 m, 0.016 m and 0.035 m for Uhlirsl~, ]ezdeckA and Iosefuv Dul, respectively. The parameter m controls depletion. In other applications of TOPMODEL it has been calibrated on individual events or estimated by the analysis of depletion curves. Such a (preliminary) analysis of 1982 hourly recessions curves for the Uhlirsk~ catchment gave a value of the parameter m of 0.0135 m, not very far from the value from the flow duration Curve.

5.1.2. Transmissivities TOPMODEL makes separate calculations for each discrete increment of the soiltopographic index distribution. For the catchments under study there are 1 0 - 1 6 classes of topographic index (derived from an elevation map using the procedures of Quinn et al. (1995)) and five classes of transmissivity, which is assumed to be distributed log normally. Assuming independence of transmissivity and topographic index, the calculations therefore use, e.g. 16 × 5 = 80 cells. It is recognized that this is a very coarse discretization of the transmissivity distribution, but this has been necessitated by computational constraints. There is no information on lateral transmissivity available on the catchments.

272

$. Blazkova, K. Beven/Journal of Hydrology 195 (1997) 256-278

Transmissivifies were adjusted by changing the mean of the log normal distribution in such a way that the modelled curve reproduced correctly the regional estimate or empirical flood frequency curve for each catchment in the area between 2 and 20 year flood (corresponding to the extreme value Type 1 or Gumbel distribution reduced variate; evl = 0.37 and 2.97, respectively). After some preliminary runs, the standard deviation of the discretized log normal distribution was set at a value of three for all the catchments. The adjusted mean T Ovalues are 5.96 m 2 h -I, 2.19 m 2 h -1 and 13.0 m 2 h -1 for Uhlirsk~, JezdeckA and Josefuv Dul, respectively. The results of this fitting using the measured peaks at the Josefuv Dul are shown in Fig. 7. In Fig. 2 and Fig. 3 for Uhlirskd and Jezdeckd the result of this step is the lower continuous line going through the CHMI estimates for 2 and 20 year return periods. The return periods of 5 and 10 years axe slightly overestimated, especially for the Uhlirskd catchment. In Fig. 2 and Fig. 3 the sensitivity of the flood frequency curve to the transmissivity parameter To can be seen for the case of the old rainfall model with one type of rainfall (dotted lines). The lower dotted lines have the same To as the continuous lines whereas the transmissivity of the upper dotted line is substantially lower.

6. Constraining flood frequencies using rainfall parameters The HI rainfall parameters were computed from areally averaged rainfalls on a catchment of 26.13 km 2 (Site BP in Fig. 1), which has a similar character, in terms of the maximum rainfall, to the catchments under study (isolines of average annual maxima of daily rainfall totals of CHMI in Fig. 1). For the Josefuv Dul catchment (25.81 km 2) the parameters computed from the data were used without any adjustment. Rainfall maxima on smaller catchments would be closer to the (larger) point rainfall values because not so much spatial averaging takes place as the storm moves over the catchment. For the two smaller catchments the threshold parameter for the HI storms, THRItn, was adjusted. The value estimated from the data is 11.7 mm h -~, i.e. the lowest average intensity which occurred in the data set. As the number of HI storms was only 18 it can be assumed that the error in this value might be large. The average rainfall characteristics discussed above (averages of the modelled mean and maximum intensities) do not depend on the parameter THRIm. However, the absolute maxima are substantially affected. If the THRIHI value in Eq. (2) is close to RBARm, all the RINTtn values will be close to each other and the absolute maximum intensities will not be too high. On the other hand, if THRIHI is low there will be some low and also some high RINT[] values to keep the balance. This is demonstrated in Fig. 4 (fight side as compared with the left bottom picture with the THRItn estimated from data). Several runs with various lower THRItn were computed and the maximum modelled hourly rainfall intensities were compared with the extreme case observed in the Czech Republic (referred to in Section 4.3.1), i.e. 190 mm per 90 min over 6 km 2. Fig. 2 and Fig. 3 show two of the computed runs of length 3000 years for each catchment with the parameter THRIrn and the maximum hourly intensity. The choice of the curve depends on the judgement of the realistic reduction of the 90 rain rainfall extreme to 60 rain, the consideration of the (slightly)

S. Blazkova, K. Beven/Joun,ud of Hydrology 195 (1997) 256-278

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smaller catchment areas (4.75 and 1.87 km 2) and the probability given to the observed extreme event. The regional CI-IMI estimates of 50 and 100 return period floods (evl = 3.9 and 4.6), which were not used for any adjustments, are shown for comparison (circles). In the case of the two smaller catchments, which do not have an observed series of a reasonable length, the extreme observed rainfall was used not only for checking the performance of the model (as in the case of the Josefuv Dul catchment) but for constraining the range of the possible predictions. In many cases, there will be fewer rainfall data available than in this study. The sensitivity of the model to errors in parameter estimation is therefore important. The parameters of LI storms were changed to values shown in Fig. 8 while keeping the average seasonal rainfall total the same (771 ram). The change of storm duration from 8.7 h to 5 or 14 h (continuous lines) as well as reducing the intensity from 0.92 to 0.46 m m h -t or increasing to 1.8 m m h -I (dotted lines) produced only a small change in the frequency curve. In spite of producing the 'right' frequency curve, some of these cases would have to be rejected using some of the constraints available. The curve with T B A R u = 14 h (upper continuous line) produced unrealistically high maximum 3 - 6 days rainfall totals (for 3 days 601 ram, which can be compared with measured data in Table 4; for 6 days

274

S. Blazkova, K. Beven/Journal of Hydrolosy 195 (1997) 256-278

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Fig. 9. Three modelled flood frequencycurves (out of ten) of the length of I000 years differing only by the sequence of HI storms (the highest, lowest and a central curve); evl, the extreme value Type 1 or Gumbel distribution reducedvariate. 832 ram). The second largest volume in a 1000 year simulation was 515 mm for 6 days. This set of parameters also produces low flows that are too small because of the long average interstorm period A T B A R u = 66 h. The cases with shorter A T B A R u (23 h, lower continuous line; 32 h, lower dotted line), on the other hand, produce low flows that are too large. The wrong proportion of low flows would lead to the adjustment of either the TOPMODEL parameter m or the duration of storms depending on which information would seem more reliable. The only curve in Fig. 8 which is without problems with constraints is the upper dotted line. The reason is that this set of parameters gives on average a similar volume of events in a similar average time spacing as the 'true' set.

7. Sensitivity of predicted flood frequencies to random realization In 1000 summer half-years about 9 0 0 0 0 LI storms are modelled. Using different random sequences has a very small effect. In contrast, the seeds used for the realizations of HI storms (which occur once per year on average) affect the flood frequency considerably at longer return periods. Fig. 9 shows the frequency curve with plotted modelled points from

S. Bkukova, K. 3evedJmmal

of Hydtology

195 (1997) 256-278

275

1000 year simulations. The series were selected from ten cases with different HI seeds: the lowest, the highest and a central simulation. The several largest floods of each series affect the fitted distribution in the same way as outliers in an observed series would do. It is important to note that for the estimation of floods of longer return periods than 100 years a longer simulation than 1000 years should be used. In Fig. 7, a 28 000 year simulation (modelled points) on the Josefuv Dul catchment is plotted together with the observed 68 year series (circles). On an Intel DX4-100 processor the 28000 summer half-years took four nights of computation. The largest recorded historical good of 1897 is plotted as a horizontal line in the range between 100 and 1000 year return period. The region of scattered points, which for a 1000 year simulation starts at about the 100 year Wurn period (evl = 4.6), starts here at about 1000 years (evl = 6.91). From Fig. 7 it can be seen that a robust estimation of a flood of 10000 years (evl = 9.21) consistent with any given model parameterixation would require still longer simulation. Additional uncertainty will be associated with different sets of model parameters that might be consistent with the data used to constrain the model (see, e.g. Beven, 1993; Freer et al., 1996).

three

8.

-on

and conclusions

This paper has explored the use of continuous simulation for the estimation of flood frequency curves given limited data. Increasing computing power has made continuous simulation a viable alternative to the assumption of a particular distributional form for extreme events, particularly when only short discharge records ate available. In this way, the estimation of flood frequencies can make use of the generally longer historical rainfall records to condition the parameters of a rainfall model as well as representing the topographic, soil and antecedent condition controls on runoff generation more explicitly. However, any such procedure remains dependent on the accuracy of the assumed distributional forms for the stochastic rainfall model and the accuracy of the runoff generation concepts employed. Given the limited data available in the Czech Republic, in both cases we have attempted to use models requiring a minimum number of parameters to be consistent with those observed data available. In the Jixera Mountain region the available data suggested that the stochastic rainstorm generator required at least two components to represent low intensity and high intensity storms. The low intensity storm parameters do not have a great effect on the modelled flood frequency curves and should also be easily estimated for the region, given mapped estimates of long-term average summer period rainfall totals. The high intensity parameters were derived from a much smaller sample and will require further investigation, including the way in which the parameters may vary with catchment size. The runoff generation model parameters were also constrained by the limited available data. The parameter m which, in controlling the shape of the recession curve, is important in setting up the initial conditions before each event, was determmed from the regionalixed flow duration curve. Effective lateral transmissivities were constrained by the regional estimates of the flood frequency curve, which is thought to be reasonably accurate for short

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return period events. The combined stochastic rainfall-runoff generation model is then used to predict the growth curve for longer return periods. This study has been primarily an exercise in constraining a very simple flexible model representation of rainfall and runoff processes with limited data. One advantage of the technique is that it allows the sampling variance for low-frequency events to be explored directly over long simulation periods (up to 28 000 years in this study). It should be noted that such long runs are intended only to improve the precision of frequency estimates consistent with current conditions. Considerable variability between 1000 year simulations has been demonstrated. It is recognized that there will also be additional uncertainty associated with the parameter values of the combined model that will be the subject of additional investigation. It is also hoped to extend the study to consider the value of different types of data in constraining the flood frequency estimates and to incorporate the annual snowmelt period that may be important on larger catchments in this region.

Acknowledgements The research has been supported by CEC Contract ERB-CIPA-CT-92-0321 issued under the programme for Co-operation in Science and Technology with Central and Eastern European Countries, and by the Ministry of Environment of the Czech Republic in projects 043 of The T.G. Masaryk Water Research Institute and GA 1533/94. S.B. is grateful to Jim Freer for carrying out the digital terrain analysis, to Rob Lamb for the preliminary recession analysis, to Paul Quinn and Renata Romanowicz for help with TOPMODEL during extended visits in Lancaster, to Katrin Krejcova and Svetlana Kol(trova of The T.G. Masaryk Water Research Institute, and to Alena Kulasov~l and Libuse Bubenfckova of the Czech Hydrometeorological Institute for providing data. The critical comments of three anonymous referees led to a great improvement in the final version of this paper.

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