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An estimation of the probable maximum precipitation for river basins in the Czech Republic
Atmospheric Research 77 (2005) 407 – 421 www.elsevier.com/locate/atmos
An estimation of the probable maximum precipitation for river basins in the Czech Republic Daniela RezacovaT, Petr Pesice, Zbynek Sokol Institute of Atmospheric Physics, Acad. Sci. Czech Republic, Bocni Str. II/1401, 141 31 Prague, Czech Republic Received 31 March 2004; received in revised form 17 September 2004; accepted 15 October 2004
Abstract The hydro-meteorological evaluation of a flood event in July 1997 (the Odra flood in Central Europe) demonstrated that new procedures to estimate design floods for the reservoir outflow structures in the Czech Republic (CR) were needed. Therefore, the techniques of the estimation of Probable Maximum Precipitation (PMP) were developed in a national research project (1998–2000), and the activity focused on the improvement of the area related PMP estimates was going on within a present national project. In the frame of the evaluation of the next extreme precipitation event in August 2002 (the Labe flood in the CR and Germany), we compared the catchments related precipitation with the PMP estimates. In this paper, an outline of the PMP estimation techniques is given and the use of data from the Czech gauges is described, the aim being the statistical derivation of the point and area PMP estimates for precipitation duration of 1 to 5 days. The use of radar data in assessing the maximized area reduction factor is discussed and the relationship resulting from the radar measurements over the CR territory is presented. An evaluation of the radar-based area rainfall enabled us to transform the point PMP to the area PMP estimate designed for the river basins in CR. In the last part of the paper, the results obtained by comparing the rainfalls in 1997 and 2002 flood events with the PMP estimates are presented. The comparison showed that the maximum area rainfalls over small Czech catchments (the 3rd order river basins) did not exceed 63% of the corresponding PMP values. D 2005 Elsevier B.V. All rights reserved. Keywords: Precipitation; Probable maximum precipitation; Flood; Area reduction factor; Radar-estimated rainfall
T Corresponding author. Fax: +420 2 72763745. E-mail address: [email protected] (D. Rezacova). 0169-8095/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.atmosres.2004.10.011
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1. Introduction Two extended floods occurred in Central Europe during the last 10 years, and both events hit the territory of the Czech Republic (CR). The Odra flood caused flooding of eastern parts of the CR (Moravia region, the Odra and Morava river basins) in July 1997, and a western part of the CR (Bohemia region, Labe river basin) was widely stricken by the flood in August 2002 (the Labe flood in the CR and Germany). The description of the floods in 1997 and 2002 can be found in the paper by Rezacova et al. (2005). After each of both flood episodes, the Ministry of Environment of the CR stimulated the research community to make greater efforts in a comprehensive evaluation of many factors and circumstances related to the flooding. Naturally, the evaluation included analyses of meteorological and hydrological conditions before, during and after the floods. The hydro-meteorological evaluation of the Odra flood indicated that new procedures and concepts were needed to estimate design floods for the reservoir outflow structures in the CR. In the beginning, design floods were estimated on the basis of 150-year rainfall values in the CR. However, observed rainfalls for duration times of 2–5 days exceeded those estimates in 1997. Therefore, methods of the estimation of design precipitation and/ or design flood were investigated in a national research project in 1998–2000. The project was supported by the Czech Ministry of Environment and focused on techniques capable to estimate design floods including the concepts not yet employed. As a result, the first set of probable maximum precipitation (PMP) estimates was proposed for a variety of precipitation durations and catchments of various sizes over the CR (Rezacova et al., 2001). The work is going on within a present national project, focused on an improvement of the area related PMP estimates. Hydrologists use the PMP values in assessing the probable maximum flood (PMF) related to a catchment of a dam. The PMF is one of conceptual flood events employed in designing hydrological structures. It can be used to design the spillway, which minimizes the risk of dam overtopping, or applied to checking the flood resistance of the hydrological structures built in the past time. The PMF concept was examined by a series of calculations. Nevertheless, up to now no decision has been arrived at on its operational application in the CR. The manual of the World Meteorological Organization (WMO, 1986) represents methodical basis of the techniques and principles of PMP estimation. The manual defines PMP as bthe greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a particular time of year. Long-term climatic trends are not considered when preparing PMP estimates.Q Consequently, the PMP value represents a physically or statistically sound upper limit of the precipitation amount closely related to a given area or river basin, and a given precipitation duration. In principle, the expected global warming may cause an increase in extreme rainfall, especially where a small increase in air temperature leads to a relatively large increase in the precipitable water content (Clark and Rakhecha, 2002). However, potential effects of global change are not considered in this paper. In accord with the WMO manual, there is no generally recommended technique of PMP estimation. We have to take into account present knowledge of precipitation
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processes as well as extreme precipitation events that have occurred over the area of interest. From this point of view, the extent and quality of available precipitation data are extremely important, too. Any PMP value is considered to be an estimate, which relies on the employed technique and on the input data used. This is the reason why several independent methods should be employed to estimate the PMP. In our studies we found two techniques of PMP estimation suitable for the CR territory, and we applied them with respect to the precipitation duration. The PMP was estimated by storm model approach (Collier and Hardaker, 1997) for precipitation duration shorter than 1 day, and the WMO statistical technique (WMO, 1986; Hershfield, 1961, 1965) was modified in order to determine the PMP for the precipitation duration of 1 day and longer. In theory, the exceedence probability of the PMP is zero in accord with the PMP definition. It means that the direct verification of the PMP estimates is impossible in principle, and the estimating procedure chosen has to be thoroughly revised only when a real precipitation is larger than the PMP estimate. That approach was accepted in the hydro-meteorological evaluation of the extreme precipitation event in August 2002 (the Labe flood). We compared the point and area PMP estimates, which had been determined in the year 2000, with the rainfalls measured during the Labe flood in August 2002. The same comparison of PMP and the measured rainfalls from Odra flood was implemented. It enabled us to express the rainfalls measured during both these extreme events relative to corresponding PMP values. In this paper we summarize the results obtained at estimating the PMP for the territory of the CR and we are focusing on the PMP values for the precipitation duration of 1–5 days. Firstly, we are briefly describing the techniques applied to get point PMP and, secondly, a statistical method, used to determine point PMP for longer precipitation duration in the Czech territory, is introduced (Section 2). The method of determining a maximized area reduction factor by using radar data is outlined in Section 3. In Section 4 we present the results obtained at the comparison of the PMP estimates with the extreme precipitation amounts from flood events in August 2002 and July 1997. Finally, the conclusions and outlook for future work are summarized in Section 5.
2. Methods of PMP estimation When estimating the PMP, we primarily determine a point PMP value for a given station or a grid point. The area estimate can be obtained from the grid point values covering the area of interest or by using an area reduction factor, which expresses a maximized relationship between point and area precipitation. The approaches to the PMP estimation can be divided into three basic categories: (a) Statistical analysis of extreme precipitation follows from the precipitation measurement and strongly depends on the extent and quality of ground precipitation data. (b) Transposition of real extreme precipitation events from the place of occurrence to the area of interest is the technique recommended by the WMO manual. Maximization of meteorological parameters follows the transposition. Main
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disadvantages of this method consist in determining the transposition limits and expressing the effect of orography. The last factor can be of relevance over the CR particularly for prolonged precipitation duration. (c) Storm model approach bases the PMP estimation on the physical parameterization of precipitation process and on the maximization of its components. The statistical techniques (a) are based entirely on measured precipitation values, whereas the (b) and (c) put emphasis on the meteorological analysis of the conditions responsible for the development of extreme precipitation. The development and testing of the techniques of the PMP estimation still continue. Recently, an approach applying multifractal analysis techniques to the calculation of PMP estimates from observations in the Eastern United States was suggested by Douglas and Barros (2003). In their article, a comprehensive discussion can be found, which reviews techniques for PMP estimation and discusses the applicability of the PMP and PMF concepts. After considering our data resources, we decided to adopt two methods of PMP estimation to the Czech territory. In order to estimate the PMP values for the time duration up to 12 h, we applied the convective storm model (Collier and Hardaker, 1997) as heavy rainfalls with short time durations occurred in connection with summer convection in the CR. The storm model approach maximizes the physical factors controlling the convective precipitation development. The model determines the point PMP using the concept of precipitation efficiency, E(t), i.e. the PMP estimate is given by the product of E(t) and the maximized precipitable water in a vertical column, MWP. The MWP value is determined by the pseudoadiabatic ascent from the maximized surface dew point temperature, which is calculated by the maximization of the radiative and orographic components. The development of deep stratiform clouds of a large area extent can cause heavy precipitation with the duration of 24 h and more. Such an intensive widespread rain developed during the extreme precipitation events in July 1997 and August 2002. In order to estimate PMP values for the rainfall durations of 1–5 days, we found suitable to employ a statistical method because an extensive data set of daily precipitation was available from the gauge measurements in the CR. The statistical technique, described in the next Section 2.1, was also applied to durations shorter than 1 day. For that application, we could use only a limited data set from 40 ombrographs. Unfortunately, the ombrographic measurements were highly unevenly distributed over the territory of the CR. The PMP estimates for 1-, 3-, 6-, and 12-h duration time resulting from both, storm approach and statistical technique, were compared at several test river basins. Thus we obtained two independent PMP values for each duration and river basin. The storm model values were accepted as resulting estimates of the PMP for precipitation duration shorter than 24 h as they systematically exceeded the statistical estimates. Uncertainties in the storm model approach are mainly due to the maximization of orographic components, i.e. the maximization of the wind velocity in the storm inflow region. At present the storm model results are considered preliminary and worth refining in the future. The statistical technique, providing point PMP values for the duration 1 day and longer, has been considered as final and is described in the next subsection.
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2.1. Statistical approach to PMP estimation Statistical methods are useful for the PMP estimation if adequate precipitation data is available. Once a statistical model is constructed, its application is simple and fast. The statistical method aims at the determination of the point PMP for a given gauge position or grid point. In order to determine an area PMP estimate, a correction that transforms the point estimate to an area rainfall (WMO, 1986) is needed. The comparatively large extent of daily precipitation measurements, available in the CR, allowed us to estimate PMP statistically for the rainfall durations from 1 day up to 5 days. The statistical technique, described in the manual (WMO, 1986), consists in the PMP estimation based on the method by Hershfield (1961, 1965). The procedure uses the site related series of annual maximum rainfalls and expresses maximum point rainfall X n at the site with the measurement covering n years by: Xn ¼ Mcn þ Kn Scn ;
ð1Þ
where M cn and S cn are the mean value and standard deviation of the series, after correcting the corresponding sample values, M n and S n , for the limited extent of series, rainfall outliers, and the fixed time of rainfall onset. K n in Eq. (1) is the scale factor. The corrections are given in the WMO manual by nomograms which aim at expressing the value X n = X, where X is the absolute maximum in the whole data set. In order to determine the scale factor, Hershfield (1961) recommended a site independent value K = K n = 15 in his first work. This was the largest K value obtained from his data comprising 2645 stations. Later, he showed that K varied according to the rainfall duration and mean value (Hershfield, 1965). The WMO manual contains nomograms, which can be used to determine the K value for various rainfall durations and M cn values. In the first stage of our work, we applied the statistical procedure from the WMO manual to estimate daily PMP over the Czech territory. The input data contained maximum annual daily rainfalls from 834 gauges (Rezacova et al., 1999). After using the nomograms from the WMO manual, we obtained the daily PMP values that appeared to be excessively high. At several stations, the daily PMP exceeded 700 mm and occasionally even 800 mm. The explanation could be found in the fact that the correction factors from the WMO manual were derived from the data collected mostly in the United States. This is why we modified the method and developed the statistical model by using the data from the CR. Measurements from 849 gauges, which were distributed nearly evenly over the Czech territory (Fig. 1) and reported daily rainfalls, were used to determine the annual maximum rainfalls for the precipitation duration of 1–5 days. Only gauges with the length of gauge record exceeding 10 years were included in the data set. The set consisted of 178, 156, 523, and 2 stations with length records up to 19, 29, 39, and 49 years, respectively. Basic characteristics of annual maximum daily rainfalls and the corresponding frequency of station are shown in Fig. 2. 2.2. Modification of the WMO statistical technique for the Czech territory When modifying the statistical method, we accepted the WMO two-step procedure, which firstly corrected the sample mean, M n , and sample standard deviation, S n , and
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Fig. 1. The territory of the CR with the orography illustrated by the shading in the map (see the scale with altitude given in meters). The positions of Czech capital Prague, and of both meteorological radars operating in the CR are shown. The crosses mark the positions of 849 gauges reporting daily rainfalls.
then estimated the scale factor K. In correcting the sample parameters, we also accepted that the series were characterized by four parameters only. Those were the M n , S n , and the mean, M nx , and standard deviation, S nx , obtained after excluding one maximum
Fig. 2. Number of stations with given characteristics of annual maximum daily rainfalls. The notation: mean annual maximum (Mean), absolute annual maximum (Max), and station related PMP estimate (PMP). Two nearby mountain stations with PMP 473 mm and 512 mm were not included in the PMP histogram.
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value from the series of n annual maximum rainfalls. Under this assumption, we tested whether there was not any significant geographical or topographical difference among the series. In order to divide the stations in groups, we applied cluster analysis with various cluster variables like station geographical location, station altitude, and station related M n , S n , and X n values. However, these attempts failed to provide meaningful results as to the regionalization. Therefore, we supposed that there might not be any significant difference among the station related series, and the data from all gauges were treated as a single data set. Consequently, the values M n and S n were corrected in order to be in agreement with the mean, M, and standard deviation, S, of the whole data set. From the whole set of annual maximum rainfalls, we randomly selected 1000 subsets each containing n elements, and from each subset we determined the values M n , S n , M nx , S nx . We used linear regression models M ¼ ax Mn Mnx þ bx Mn þ cx Mn Snx þ dx ;
ð2Þ
S ¼ as Sn Snx þ bs Sn þ cs Sn Mnx þ ds ;
ð3Þ
and determined regression coefficients a x , b x , c x , d x , a s, b s, c s and d s. In this way the regression coefficients were calculated for n = 10, 15, 20,. . ., 50 elements. The stability and accuracy of Eqs. (2) and (3) were verified on an independent selection of another 1000 subsets using the RMSE and bias. Eqs. (2) and (3) were used to determine corrected values, M cn and S cn , for each station. Eqs. (2) and (3) were applied to generalize the relationships from WMO manual where M cn was considered to be a function of M n and M nx only and similarly S cn depended only on S n and S nx . In order to determine parameter K, we firstly used the method similar to the WMO manual. A regression model expressing the dependence of K on M cn and S cn was put together. However, we obtained too low PMP estimates, in particular in the mountainous areas, where the PMP values did not exceed the historical maximum rainfalls. Finally, we accepted a simple assumption that the K value was constant over the Czech territory. We determined K value so as the site-related PMP estimate just exceeded the absolute historical maximum reported in the CR. For instance, value K = 11.54 ensured that the daily PMP at the station Nova Louka (50849VN, 15809VE, 780 m a.s.l.) exceeded the historical maximum daily rainfall of 345.1 mm, reported there in 1897. It is worth mentioning that the assumption of constant K over the whole Czech territory has to be considered a first approximation. Efforts to find a suitable division of the Czech territory and include the stations located in cross-border regions are issues to be dealt with in future work. Point PMP estimates were determined for all stations in the data set and for the rainfall duration 1, 2, . . . , 5 days. As an example, the area distribution of daily PMP values is shown in Fig. 3. 2.3. Conversion of point PMP values into area PMP estimate Area reduction factors (ARF) convert the point precipitation into the area average value for a given duration, probability of exceedance, and area size (e.g. Sivaplan and Blo¨schl,
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Fig. 3. Area distribution of daily point PMP as obtained by a simple interpolation of gauge-related PMP values. The scale of grayness gives PMP values in millimeters. The borders of the 3rd order catchments are highlighted.
1998; Kottegoda and Rosso, 2001). In order to convert the point PMP to its area value, we estimated maximized ARF values, ARFx. In the first stage of the work, we analyzed the area distribution of rainfalls during several extreme precipitation events to determine ARFx as a function of the area size and precipitation duration. For each event and a given area size, we looked for the maximum area precipitation value by changing the shape of the area. Such an approach was found strongly dependent on the event and there was only a limited number of rainfall fields available for each duration. For that reason, we decided to derive ARFx values by using radar data accessible in the CR.
3. Derivation of area reduction factor using radar data The utilization of radar data in hydrology is a relatively new area of research or application (e.g. Tilford et al., 2002; Durrans et al., 2002). However, the radar data use in hydrological applications is expected to be increasing as the accumulation of radar data records grows and the understanding of the quantitative interpretation of radar-based rainfall values is improving. Durrans et al. (2002) used radar-based rainfalls to derive the depth–area relationships, and discussed error sources potentially involved in the determination of radar-rainfalls. We used radar-based rainfalls to estimate the maximum area reduction factor, ARFx, as a function of area size and rainfall duration. For the determination of ARFx values, we used a data set consisting of maximum column radar reflectivity Zmax measured by the weather radar Skalky (Gematronik METEOR 360AC, Doppler C band radar with the wavelength of 5.3 cm). The radar Skalky has been operated by the Czech weather service (Czech Hydrometeorological Institute-CHMI) since 1996 and we use the data collected during warm seasons (April to
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September) 1996–2001. It is important that the comparatively limited data set contained data from the large-scale flood in July 1997. Except for the historical daily maximum rainfall in 1897, the maximum rainfalls for 2–5 days were reached just in 1997. Apart from the 1997 flood, the radar data set included also the local flood event in July 1998 with more then 200 mm daily rainfall values over a north-eastern region of the CR. Second meteorological radar Brdy (EEC DWSR-2501 C, Doppler C band radar with the wavelength of 5.3 cm) has been operating in CR since 2000 (Fig. 1). The Brdy measurements were not included in this work. Maximum reflectivity values, Zmax, were available from 10-min scans at 256 256 pixels of the size 2 km 2 km. We considered only the pixels in the distance between 20 km and 200 km from the radar position, to reduce the bias of radar measurement. Further, similarly to Durrans et al. (2002), we assumed that the biases of radar-rainfalls tended to be canceled due to the non-dimensional ARFx. Hourly radar-rainfalls were determined from maximum reflectivity values, Zmax, by the procedure routinely applied in the CHMI and described, e.g. in (Kracmar et al., 1998). The Zmax values were converted into rain rate, R, using standard Marshall–Palmer relationship Zmax = 200R 1.6, and integrated in time to obtain hourly rainfall values. The hourly accumulation time started at t0 = 00, 01, . . . , 23 UTC. For each of about 4000 sets of hourly rainfalls, we determined the maximum hourly rainfall Px[xx,yx], where xx and yx are the pixel coordinates. Two maximum values were considered for each set of hourly rainfalls. In the next step, we limited the ARFx calculation to the highest 500 values of Px. The radar derived ARF was expressed by ARFð Px; PnÞ ¼ PA½ Að Px; PnÞ=Px;
ð4Þ
where PA[A(Px,Pn)] is the mean hourly rainfall over the A(Px,Pn) area. That area contains the [xx,yx] pixel and all the pixels with hourly precipitation larger or equal to the value of Pn under the constraint that all pixels inside A(Px,Pn) were connected at least by one corner. The ARF values are calculated for several Pn values. The same procedure was applied to the radar-rainfalls with the duration d = 1, 2, . . . , 6, 12, 24, 48,. . ., 120 h. The corresponding pixel rainfalls were determined by summing the Table 1 Characteristics of maximum point radar-based rainfalls, Px [mm], for various rainfall durations, d [h] d
Mean
St_dev
Max
Min
1 3 6 12 24 48 72 96 120
79.8 113.3 138.2 169.9 212.9 283.4 337.3 374.0 392.9
23.4 31.7 33.4 31.0 51.6 103.9 134.4 149.7 146.4
202.2 254.6 276.7 285.8 421.7 623.5 773.5 869.3 911.0
54.8 77.8 99.9 128.6 168.0 195.3 233.5 263.3 285.0
In total the highest 500 Px values were considered for each d to determine the mean Px value (mean), standard deviation (st _ dev), maximum (max) and minimum (min) in mm.
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Table 2 Characteristics of the areas with the mean area radar-rainfall PA b Pn(i) and PA z Pn(i + 1), where Pn [mm] is a defined minimum rainfall (see the text for more detailed explanation) i Pn _ 1 N Amean Amax Pn _ 24 N Amean Amax 1 2 3 4 5 6 7 8 9 10 11 12
0.2 1 2 5 10 30 50 75 100 120 130 140
500 500 500 500 500 500 500 225 87 36 20 11
22,629 8559 4961 1700 575 101 44 13 9 8 8 7
150,768 85,000 47,912 21,016 12,616 10,948 10,824 120 44 32 28 24
5 10 30 50 75 100 125 150 175 200 225 250
500 500 500 500 500 500 500 500 398 228 139 94
61,880 39,173 6848 2076 430 123 45 18 6 5 4 4
186,528 162,840 92,444 40,052 8068 1856 812 320 32 12 8 4
The left part of the table shows the values for 1-h rainfall (Pn_1), and the right part relates to 24-h rainfalls (Pn_24). The N gives the number of areas, which were identified by the procedure described in the text. The Amean and Amax give the mean, and maximum area extent, respectively. The Amean and Amax are in square kilometers.
hourly rainfall values. Like the hourly values, area A, and rainfall PA were determined for each duration and for each start time t0. The event with the duration d was defined by the decrease of hourly rainfalls at (t0 1) and at (t0 + d + 1). We requested the hourly values PA{t0 1} and PA{t0 + d + 1} to be smaller than 0.1% of hourly rainfalls PA{t0} and PA{t0 + d}, respectively. Basic characteristics of maximum rainfalls Px as a function of rainfall duration d [h] are shown in Table 1, and the areas defined by the selection procedure are characterized in Table 2. The maximum value, ARFx, depending on area size A was expressed as an upper envelope of the ARF(A) values by the relationship ARFx ¼ exp aX 2 þ bX 4 ; for X Nd; ð5aÞ ARFx ¼ 1; for 0bX Vd;
ð5bÞ
where the independent variable X is given by X ¼ log10 ð AÞ d;
ð5cÞ
and parameters a, b, and d were determined empirically as a function of precipitation duration. The value d was set for 0.9, and the resulting values of a, and b are given in Table 3. The resulting relations (5a), (5b) and (5c) are depicted in Fig. 4. Table 3 Coefficients a, and b, entering the ARFx formulae (5a), (5b) and (5c) for various rainfall durations (d) d [h]
a
b
d [h]
a
b
1 2 3
0.1501 0.1616 0.1445
0.0115 0.0020 0.0006
6 12 z 24
0.0852 0.0603 0.0501
0.0013 0.0010 0.0
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Fig. 4. The dependence of ARFx values on the area extent. The rainfall duration is denoted.
In order to determine an area-related PMP for a catchment, we considered a dense grid of 1 km 1 km covering a catchment. The grid point PMP values were calculated as a weighted average of PMP values for the gauges inside the catchment and in its surroundings. The distance between the grid point and the gauge served as a weight. Then the relationships (5a), (5b) and (5c) were applied to the maximum PMP value over all grid points inside the catchment.
4. Comparison of PMP and heavy flood rainfalls in 1997 and 2002 The comparison between the PMP estimates and the extreme precipitation amounts falling during two extreme flood events (July 1997 and August 2002) was implemented. The extreme rainfalls in August 2002 and July 1997 occurred in two time periods of several days duration. In 2002, the first precipitation episode took place on August 6 and 7, the second episode followed on August 11–13. Both precipitation episodes hit the area of the Labe river basin, which covers several urban areas and towns in the CR, including Prague. In Germany, a part of Dresden was flooded, for example. In 1997, the precipitation fell out in the Moravia region where the heavy rain affected two main catchments (the Morava and Odra river basins). The episodes were separated by a longer period nearly without rainfall. The first episode occurred on July 4–7, and was not followed by a second precipitation period until July 17–21. In 1997 the first rainfall event was markedly dominant while in 2002, the precipitation amount was higher during the second period. The paper by Rezacova et al. (2005) presents a more detailed description of both flood situations.
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The area PMP estimates were determined for the catchments of various area extents and locations by the statistical method described in Sections 2 and 3. We used the station-related point PMP estimates that were derived from the historical data up to the year 2000, and the conversion to the area estimates was done by radar-based ARFx. Main attention was devoted to the PMP estimates for the river basins of the 3rd order, which covered the territory of the CR and reflected the area distribution of the flood rainfalls. Altogether 63 Bohemian catchments belong to the Labe river basin, the Moravian river basins Odra and Morava consist of 8 and 21 basins, respectively. The daily PMP values are shown in Fig. 5 for various catchment areas. The highest area-related PMP estimates (over 250 mm) were found in catchments where mountain regions dominated. However, not all mountainous river basins show such high PMP values. On
Fig. 5. Daily area PMP estimates for the river basins of the 3rd order (see Fig. 3) as a function of the basin area. The PMP values belonging to the Labe river basin (Bohemia) and to the Odra or Morava river basins (Moravia) are distinguished by full and blank circles, respectively.
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the contrary, there are several river basins hit by the flood precipitation in 1997, which show high PMP but do not belong to the mountainous catchments. Obviously, the altitude does not play the decisive role and orographic influences are highly complex phenomena (precipitation enhancement). The CHMI provided us with the daily area precipitation over the 3rd order catchments for both flood events. The rainfalls for longer rainy periods were calculated from daily values. For each catchment, we selected the maximum area rainfall for a given rainfall duration (1–5 days) and compared its value with the corresponding PMP estimate. The results are summarized in Fig. 6. In August 2002, the highest area precipitation amounts relative to the PMP values were reached at in one of the river basins for the durations 2 and 3 days (64%) and in another basin the value reached 62% of the corresponding PMP for 2 days. In 1997, the highest area precipitation relative to PMP occurred in the upper
Fig. 6. Maximum area rainfalls relative to area PMP estimates for the Czech river basins of the 3rd order. The left (right) column corresponds to flood in 2002 (1997). The rows correspond to the rainfall duration of 1 to 5 days.
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Morava river basins for the duration of 3, 4, and 5 days (62% maximum) and the maximum of 63% was attained in other two basins (the Opava and Becva river basins). From Fig. 6, it is obvious that the area precipitation amounts during both events arrived at similar values with respect to the PMP estimates. The maximum values were observed for the duration of 2 and 3 days in 2002 while the maximum precipitation amounts in 1997 shifted towards longer rainfall durations. The statistical models used to estimate the point PMP values were developed in the year 2000 with historical rainfall data up to 2000. Therefore, the revision of the models with data including also the 2002 rainfalls should be performed in spite of the fact that there is not any area PMP exceeded in 2002. The evaluation of point PMP values also showed that the point estimates were not exceeded in the Czech territory. The only site, for which we obtained more than 100% of the daily point PMP value, was the German station Zinnwald, which was located close to the Germany–CR border. This finding indicates the need to extend our study area by adding crossborder regions.
5. Conclusion In this paper we described the technique used to estimate the point and area PMP values over the Czech territory. We concentrated on the description of the statistical technique, which was designed to the precipitation duration of 1, 2, . . . , 5 days. The conversion of the point PMP to the area PMP estimate was based on evaluating radarbased rainfalls. We determined the maximized area reduction factor, ARFx, and the formula, designed to calculate the ARFx according to the area extent, is presented in this paper. The PMP estimates were calculated at the gauge positions and for the system of small river basins covering the territory of the CR (the 3rd order basins). The area PMP estimates were compared with heavy rainfalls that had occurred during the flood events in August 2002 and July 1997. The comparison showed that the maximum area precipitation did not exceed 63% of the PMP estimates. Present state of PMP estimates should be considered a prerequisite stage. The results show that the next work should concentrate on the following topics: (i) We need to revise the PMP estimates by extending the precipitation data up to the year 2002 and the data from cross-border regions. (ii) The stability of the ARFx values should be verified with gauge adjusted radar-based rainfalls, and the data reported by both Czech radars should be considered. (iii) The gauge-related PMP values are determined for each station independently on the remaining stations. We should consider the correlation between PMP values from adjacent gauges and look for local orography effects. (iv) A study focused on the distribution of radar-based rainfalls according to the duration within the day is under way. Preliminary results show that it enhances efforts to regionalize the Czech territory, and takes into account the effect of orography. It should improve a more detailed determination of point and area PMP estimates.
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Acknowledgements The work was supported by the grant VaV510/3/97 (the Ministry of Environment, CR) and the grant QD1368 (the Ministry of Agriculture, CR). The Czech Hydrometeorological Institute is thanked for the provision of the radar and precipitation data for this research. Two referees commented the text and we highly appreciate their comments and recommendations.
References Clark, C., Rakhecha, P.R., 2002. Areal PMP distribution of one-day to three-day duration over India. Meteorol. Appl. 9, 399 – 406. Collier, C.G., Hardaker, P.J., 1997. Estimating probable maximum precipitation using a storm model approach. J. Hydrol. 183, 277 – 306. Douglas, E.M., Barros, A.P., 2003. Probable maximum precipitation estimation using multifractals: application in the eastern United States. J. Hydrometeorol. 4, 1012 – 1024. Durrans, S.R., Julian, L.T., Yekta, M., 2002. Estimation of depth–area relationships using radar-rainfall data. J. Hydrol. Eng. 7, 356 – 367. Hershfield, D.M., 1961. Estimating the probable maximum precipitation. J. Hydraul. Div. vol. 87. Amer. Soc. Civ. Eng., pp. 99 – 106. Hershfield, D.M., 1965. Method for estimating probable maximum precipitation. J.-Am. Water Works Assoc. 57, 965 – 972. Kottegoda, D.M.C., Rosso, R., 2001. The derivation of areal reduction factor of storm rainfall from its scaling properties. Water Resour. Res. 37, 3247 – 3252. Kracmar, J., Joss, J., Novak, P., Havranek, P., Salek, M., 1998. First steps towards quantitative usage of data from Czech weather radar network. COST75 Advanced Weather Radar Systems. COST75 Final Seminar, Locarno, Switzerland, 23–27 March 1998, pp. 91 – 101. Rezacova, D., Sokol, Z., Kveton, V., 1999. Estimation of probable maximum precipitation over catchments in the Czech Republic. Extended Abstract in Proc. 4th European Conf. on Applications of Meteorol. (ECAM99), Norkopping, Sweden, 13–17 September, pp. 228 – 229. Rezacova, D., Sokol, Z., Pesice, P., 2001. Deterministicke´ a statisticke´ odhady pravdeˇpodobne´ maxima´lnı´ sra´zˇky pro u´zemı´ Cˇeske´ republiky (in Czech). Proc. of the Seminar to the Results of the Grant Project VaV 510/3/97. ISBN: 80-02-01440-59, pp. 24 – 35. Rezacova, D., Kaspar, M., Muller, M., Sokol, Z., Kakos, V., Hanslian, D., Pesice, P., 2005. A comparison of the flood precipitation episode in August 2002 with historic extreme precipitation events on the Czech territory. Atmos. Res 77, 354–366 (this issue). Sivaplan, M., Blo¨schl, G., 1998. Transformation of point rainfall to areal rainfall: intensity–duration–frequency curves. J. Hydrol. 204, 150 – 167. Tilford, K.A., Fox, N.I., Collier, C.G., 2002. Application of weather radar data for urban hydrology. Meteorol. Appl. 9, 95 – 104. WMO, 1986. Manual for estimation of probable maximum precipitation. Operational hydrology, Rep. 1, WMO-No. 332. 269 pp.
An estimation of the probable maximum precipitation for river basins in the Czech Republic Daniela RezacovaT, Petr Pesice, Zbynek Sokol Institute of Atmospheric Physics, Acad. Sci. Czech Republic, Bocni Str. II/1401, 141 31 Prague, Czech Republic Received 31 March 2004; received in revised form 17 September 2004; accepted 15 October 2004
Abstract The hydro-meteorological evaluation of a flood event in July 1997 (the Odra flood in Central Europe) demonstrated that new procedures to estimate design floods for the reservoir outflow structures in the Czech Republic (CR) were needed. Therefore, the techniques of the estimation of Probable Maximum Precipitation (PMP) were developed in a national research project (1998–2000), and the activity focused on the improvement of the area related PMP estimates was going on within a present national project. In the frame of the evaluation of the next extreme precipitation event in August 2002 (the Labe flood in the CR and Germany), we compared the catchments related precipitation with the PMP estimates. In this paper, an outline of the PMP estimation techniques is given and the use of data from the Czech gauges is described, the aim being the statistical derivation of the point and area PMP estimates for precipitation duration of 1 to 5 days. The use of radar data in assessing the maximized area reduction factor is discussed and the relationship resulting from the radar measurements over the CR territory is presented. An evaluation of the radar-based area rainfall enabled us to transform the point PMP to the area PMP estimate designed for the river basins in CR. In the last part of the paper, the results obtained by comparing the rainfalls in 1997 and 2002 flood events with the PMP estimates are presented. The comparison showed that the maximum area rainfalls over small Czech catchments (the 3rd order river basins) did not exceed 63% of the corresponding PMP values. D 2005 Elsevier B.V. All rights reserved. Keywords: Precipitation; Probable maximum precipitation; Flood; Area reduction factor; Radar-estimated rainfall
T Corresponding author. Fax: +420 2 72763745. E-mail address: [email protected] (D. Rezacova). 0169-8095/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.atmosres.2004.10.011
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1. Introduction Two extended floods occurred in Central Europe during the last 10 years, and both events hit the territory of the Czech Republic (CR). The Odra flood caused flooding of eastern parts of the CR (Moravia region, the Odra and Morava river basins) in July 1997, and a western part of the CR (Bohemia region, Labe river basin) was widely stricken by the flood in August 2002 (the Labe flood in the CR and Germany). The description of the floods in 1997 and 2002 can be found in the paper by Rezacova et al. (2005). After each of both flood episodes, the Ministry of Environment of the CR stimulated the research community to make greater efforts in a comprehensive evaluation of many factors and circumstances related to the flooding. Naturally, the evaluation included analyses of meteorological and hydrological conditions before, during and after the floods. The hydro-meteorological evaluation of the Odra flood indicated that new procedures and concepts were needed to estimate design floods for the reservoir outflow structures in the CR. In the beginning, design floods were estimated on the basis of 150-year rainfall values in the CR. However, observed rainfalls for duration times of 2–5 days exceeded those estimates in 1997. Therefore, methods of the estimation of design precipitation and/ or design flood were investigated in a national research project in 1998–2000. The project was supported by the Czech Ministry of Environment and focused on techniques capable to estimate design floods including the concepts not yet employed. As a result, the first set of probable maximum precipitation (PMP) estimates was proposed for a variety of precipitation durations and catchments of various sizes over the CR (Rezacova et al., 2001). The work is going on within a present national project, focused on an improvement of the area related PMP estimates. Hydrologists use the PMP values in assessing the probable maximum flood (PMF) related to a catchment of a dam. The PMF is one of conceptual flood events employed in designing hydrological structures. It can be used to design the spillway, which minimizes the risk of dam overtopping, or applied to checking the flood resistance of the hydrological structures built in the past time. The PMF concept was examined by a series of calculations. Nevertheless, up to now no decision has been arrived at on its operational application in the CR. The manual of the World Meteorological Organization (WMO, 1986) represents methodical basis of the techniques and principles of PMP estimation. The manual defines PMP as bthe greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a particular time of year. Long-term climatic trends are not considered when preparing PMP estimates.Q Consequently, the PMP value represents a physically or statistically sound upper limit of the precipitation amount closely related to a given area or river basin, and a given precipitation duration. In principle, the expected global warming may cause an increase in extreme rainfall, especially where a small increase in air temperature leads to a relatively large increase in the precipitable water content (Clark and Rakhecha, 2002). However, potential effects of global change are not considered in this paper. In accord with the WMO manual, there is no generally recommended technique of PMP estimation. We have to take into account present knowledge of precipitation
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processes as well as extreme precipitation events that have occurred over the area of interest. From this point of view, the extent and quality of available precipitation data are extremely important, too. Any PMP value is considered to be an estimate, which relies on the employed technique and on the input data used. This is the reason why several independent methods should be employed to estimate the PMP. In our studies we found two techniques of PMP estimation suitable for the CR territory, and we applied them with respect to the precipitation duration. The PMP was estimated by storm model approach (Collier and Hardaker, 1997) for precipitation duration shorter than 1 day, and the WMO statistical technique (WMO, 1986; Hershfield, 1961, 1965) was modified in order to determine the PMP for the precipitation duration of 1 day and longer. In theory, the exceedence probability of the PMP is zero in accord with the PMP definition. It means that the direct verification of the PMP estimates is impossible in principle, and the estimating procedure chosen has to be thoroughly revised only when a real precipitation is larger than the PMP estimate. That approach was accepted in the hydro-meteorological evaluation of the extreme precipitation event in August 2002 (the Labe flood). We compared the point and area PMP estimates, which had been determined in the year 2000, with the rainfalls measured during the Labe flood in August 2002. The same comparison of PMP and the measured rainfalls from Odra flood was implemented. It enabled us to express the rainfalls measured during both these extreme events relative to corresponding PMP values. In this paper we summarize the results obtained at estimating the PMP for the territory of the CR and we are focusing on the PMP values for the precipitation duration of 1–5 days. Firstly, we are briefly describing the techniques applied to get point PMP and, secondly, a statistical method, used to determine point PMP for longer precipitation duration in the Czech territory, is introduced (Section 2). The method of determining a maximized area reduction factor by using radar data is outlined in Section 3. In Section 4 we present the results obtained at the comparison of the PMP estimates with the extreme precipitation amounts from flood events in August 2002 and July 1997. Finally, the conclusions and outlook for future work are summarized in Section 5.
2. Methods of PMP estimation When estimating the PMP, we primarily determine a point PMP value for a given station or a grid point. The area estimate can be obtained from the grid point values covering the area of interest or by using an area reduction factor, which expresses a maximized relationship between point and area precipitation. The approaches to the PMP estimation can be divided into three basic categories: (a) Statistical analysis of extreme precipitation follows from the precipitation measurement and strongly depends on the extent and quality of ground precipitation data. (b) Transposition of real extreme precipitation events from the place of occurrence to the area of interest is the technique recommended by the WMO manual. Maximization of meteorological parameters follows the transposition. Main
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disadvantages of this method consist in determining the transposition limits and expressing the effect of orography. The last factor can be of relevance over the CR particularly for prolonged precipitation duration. (c) Storm model approach bases the PMP estimation on the physical parameterization of precipitation process and on the maximization of its components. The statistical techniques (a) are based entirely on measured precipitation values, whereas the (b) and (c) put emphasis on the meteorological analysis of the conditions responsible for the development of extreme precipitation. The development and testing of the techniques of the PMP estimation still continue. Recently, an approach applying multifractal analysis techniques to the calculation of PMP estimates from observations in the Eastern United States was suggested by Douglas and Barros (2003). In their article, a comprehensive discussion can be found, which reviews techniques for PMP estimation and discusses the applicability of the PMP and PMF concepts. After considering our data resources, we decided to adopt two methods of PMP estimation to the Czech territory. In order to estimate the PMP values for the time duration up to 12 h, we applied the convective storm model (Collier and Hardaker, 1997) as heavy rainfalls with short time durations occurred in connection with summer convection in the CR. The storm model approach maximizes the physical factors controlling the convective precipitation development. The model determines the point PMP using the concept of precipitation efficiency, E(t), i.e. the PMP estimate is given by the product of E(t) and the maximized precipitable water in a vertical column, MWP. The MWP value is determined by the pseudoadiabatic ascent from the maximized surface dew point temperature, which is calculated by the maximization of the radiative and orographic components. The development of deep stratiform clouds of a large area extent can cause heavy precipitation with the duration of 24 h and more. Such an intensive widespread rain developed during the extreme precipitation events in July 1997 and August 2002. In order to estimate PMP values for the rainfall durations of 1–5 days, we found suitable to employ a statistical method because an extensive data set of daily precipitation was available from the gauge measurements in the CR. The statistical technique, described in the next Section 2.1, was also applied to durations shorter than 1 day. For that application, we could use only a limited data set from 40 ombrographs. Unfortunately, the ombrographic measurements were highly unevenly distributed over the territory of the CR. The PMP estimates for 1-, 3-, 6-, and 12-h duration time resulting from both, storm approach and statistical technique, were compared at several test river basins. Thus we obtained two independent PMP values for each duration and river basin. The storm model values were accepted as resulting estimates of the PMP for precipitation duration shorter than 24 h as they systematically exceeded the statistical estimates. Uncertainties in the storm model approach are mainly due to the maximization of orographic components, i.e. the maximization of the wind velocity in the storm inflow region. At present the storm model results are considered preliminary and worth refining in the future. The statistical technique, providing point PMP values for the duration 1 day and longer, has been considered as final and is described in the next subsection.
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2.1. Statistical approach to PMP estimation Statistical methods are useful for the PMP estimation if adequate precipitation data is available. Once a statistical model is constructed, its application is simple and fast. The statistical method aims at the determination of the point PMP for a given gauge position or grid point. In order to determine an area PMP estimate, a correction that transforms the point estimate to an area rainfall (WMO, 1986) is needed. The comparatively large extent of daily precipitation measurements, available in the CR, allowed us to estimate PMP statistically for the rainfall durations from 1 day up to 5 days. The statistical technique, described in the manual (WMO, 1986), consists in the PMP estimation based on the method by Hershfield (1961, 1965). The procedure uses the site related series of annual maximum rainfalls and expresses maximum point rainfall X n at the site with the measurement covering n years by: Xn ¼ Mcn þ Kn Scn ;
ð1Þ
where M cn and S cn are the mean value and standard deviation of the series, after correcting the corresponding sample values, M n and S n , for the limited extent of series, rainfall outliers, and the fixed time of rainfall onset. K n in Eq. (1) is the scale factor. The corrections are given in the WMO manual by nomograms which aim at expressing the value X n = X, where X is the absolute maximum in the whole data set. In order to determine the scale factor, Hershfield (1961) recommended a site independent value K = K n = 15 in his first work. This was the largest K value obtained from his data comprising 2645 stations. Later, he showed that K varied according to the rainfall duration and mean value (Hershfield, 1965). The WMO manual contains nomograms, which can be used to determine the K value for various rainfall durations and M cn values. In the first stage of our work, we applied the statistical procedure from the WMO manual to estimate daily PMP over the Czech territory. The input data contained maximum annual daily rainfalls from 834 gauges (Rezacova et al., 1999). After using the nomograms from the WMO manual, we obtained the daily PMP values that appeared to be excessively high. At several stations, the daily PMP exceeded 700 mm and occasionally even 800 mm. The explanation could be found in the fact that the correction factors from the WMO manual were derived from the data collected mostly in the United States. This is why we modified the method and developed the statistical model by using the data from the CR. Measurements from 849 gauges, which were distributed nearly evenly over the Czech territory (Fig. 1) and reported daily rainfalls, were used to determine the annual maximum rainfalls for the precipitation duration of 1–5 days. Only gauges with the length of gauge record exceeding 10 years were included in the data set. The set consisted of 178, 156, 523, and 2 stations with length records up to 19, 29, 39, and 49 years, respectively. Basic characteristics of annual maximum daily rainfalls and the corresponding frequency of station are shown in Fig. 2. 2.2. Modification of the WMO statistical technique for the Czech territory When modifying the statistical method, we accepted the WMO two-step procedure, which firstly corrected the sample mean, M n , and sample standard deviation, S n , and
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Fig. 1. The territory of the CR with the orography illustrated by the shading in the map (see the scale with altitude given in meters). The positions of Czech capital Prague, and of both meteorological radars operating in the CR are shown. The crosses mark the positions of 849 gauges reporting daily rainfalls.
then estimated the scale factor K. In correcting the sample parameters, we also accepted that the series were characterized by four parameters only. Those were the M n , S n , and the mean, M nx , and standard deviation, S nx , obtained after excluding one maximum
Fig. 2. Number of stations with given characteristics of annual maximum daily rainfalls. The notation: mean annual maximum (Mean), absolute annual maximum (Max), and station related PMP estimate (PMP). Two nearby mountain stations with PMP 473 mm and 512 mm were not included in the PMP histogram.
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value from the series of n annual maximum rainfalls. Under this assumption, we tested whether there was not any significant geographical or topographical difference among the series. In order to divide the stations in groups, we applied cluster analysis with various cluster variables like station geographical location, station altitude, and station related M n , S n , and X n values. However, these attempts failed to provide meaningful results as to the regionalization. Therefore, we supposed that there might not be any significant difference among the station related series, and the data from all gauges were treated as a single data set. Consequently, the values M n and S n were corrected in order to be in agreement with the mean, M, and standard deviation, S, of the whole data set. From the whole set of annual maximum rainfalls, we randomly selected 1000 subsets each containing n elements, and from each subset we determined the values M n , S n , M nx , S nx . We used linear regression models M ¼ ax Mn Mnx þ bx Mn þ cx Mn Snx þ dx ;
ð2Þ
S ¼ as Sn Snx þ bs Sn þ cs Sn Mnx þ ds ;
ð3Þ
and determined regression coefficients a x , b x , c x , d x , a s, b s, c s and d s. In this way the regression coefficients were calculated for n = 10, 15, 20,. . ., 50 elements. The stability and accuracy of Eqs. (2) and (3) were verified on an independent selection of another 1000 subsets using the RMSE and bias. Eqs. (2) and (3) were used to determine corrected values, M cn and S cn , for each station. Eqs. (2) and (3) were applied to generalize the relationships from WMO manual where M cn was considered to be a function of M n and M nx only and similarly S cn depended only on S n and S nx . In order to determine parameter K, we firstly used the method similar to the WMO manual. A regression model expressing the dependence of K on M cn and S cn was put together. However, we obtained too low PMP estimates, in particular in the mountainous areas, where the PMP values did not exceed the historical maximum rainfalls. Finally, we accepted a simple assumption that the K value was constant over the Czech territory. We determined K value so as the site-related PMP estimate just exceeded the absolute historical maximum reported in the CR. For instance, value K = 11.54 ensured that the daily PMP at the station Nova Louka (50849VN, 15809VE, 780 m a.s.l.) exceeded the historical maximum daily rainfall of 345.1 mm, reported there in 1897. It is worth mentioning that the assumption of constant K over the whole Czech territory has to be considered a first approximation. Efforts to find a suitable division of the Czech territory and include the stations located in cross-border regions are issues to be dealt with in future work. Point PMP estimates were determined for all stations in the data set and for the rainfall duration 1, 2, . . . , 5 days. As an example, the area distribution of daily PMP values is shown in Fig. 3. 2.3. Conversion of point PMP values into area PMP estimate Area reduction factors (ARF) convert the point precipitation into the area average value for a given duration, probability of exceedance, and area size (e.g. Sivaplan and Blo¨schl,
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Fig. 3. Area distribution of daily point PMP as obtained by a simple interpolation of gauge-related PMP values. The scale of grayness gives PMP values in millimeters. The borders of the 3rd order catchments are highlighted.
1998; Kottegoda and Rosso, 2001). In order to convert the point PMP to its area value, we estimated maximized ARF values, ARFx. In the first stage of the work, we analyzed the area distribution of rainfalls during several extreme precipitation events to determine ARFx as a function of the area size and precipitation duration. For each event and a given area size, we looked for the maximum area precipitation value by changing the shape of the area. Such an approach was found strongly dependent on the event and there was only a limited number of rainfall fields available for each duration. For that reason, we decided to derive ARFx values by using radar data accessible in the CR.
3. Derivation of area reduction factor using radar data The utilization of radar data in hydrology is a relatively new area of research or application (e.g. Tilford et al., 2002; Durrans et al., 2002). However, the radar data use in hydrological applications is expected to be increasing as the accumulation of radar data records grows and the understanding of the quantitative interpretation of radar-based rainfall values is improving. Durrans et al. (2002) used radar-based rainfalls to derive the depth–area relationships, and discussed error sources potentially involved in the determination of radar-rainfalls. We used radar-based rainfalls to estimate the maximum area reduction factor, ARFx, as a function of area size and rainfall duration. For the determination of ARFx values, we used a data set consisting of maximum column radar reflectivity Zmax measured by the weather radar Skalky (Gematronik METEOR 360AC, Doppler C band radar with the wavelength of 5.3 cm). The radar Skalky has been operated by the Czech weather service (Czech Hydrometeorological Institute-CHMI) since 1996 and we use the data collected during warm seasons (April to
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September) 1996–2001. It is important that the comparatively limited data set contained data from the large-scale flood in July 1997. Except for the historical daily maximum rainfall in 1897, the maximum rainfalls for 2–5 days were reached just in 1997. Apart from the 1997 flood, the radar data set included also the local flood event in July 1998 with more then 200 mm daily rainfall values over a north-eastern region of the CR. Second meteorological radar Brdy (EEC DWSR-2501 C, Doppler C band radar with the wavelength of 5.3 cm) has been operating in CR since 2000 (Fig. 1). The Brdy measurements were not included in this work. Maximum reflectivity values, Zmax, were available from 10-min scans at 256 256 pixels of the size 2 km 2 km. We considered only the pixels in the distance between 20 km and 200 km from the radar position, to reduce the bias of radar measurement. Further, similarly to Durrans et al. (2002), we assumed that the biases of radar-rainfalls tended to be canceled due to the non-dimensional ARFx. Hourly radar-rainfalls were determined from maximum reflectivity values, Zmax, by the procedure routinely applied in the CHMI and described, e.g. in (Kracmar et al., 1998). The Zmax values were converted into rain rate, R, using standard Marshall–Palmer relationship Zmax = 200R 1.6, and integrated in time to obtain hourly rainfall values. The hourly accumulation time started at t0 = 00, 01, . . . , 23 UTC. For each of about 4000 sets of hourly rainfalls, we determined the maximum hourly rainfall Px[xx,yx], where xx and yx are the pixel coordinates. Two maximum values were considered for each set of hourly rainfalls. In the next step, we limited the ARFx calculation to the highest 500 values of Px. The radar derived ARF was expressed by ARFð Px; PnÞ ¼ PA½ Að Px; PnÞ=Px;
ð4Þ
where PA[A(Px,Pn)] is the mean hourly rainfall over the A(Px,Pn) area. That area contains the [xx,yx] pixel and all the pixels with hourly precipitation larger or equal to the value of Pn under the constraint that all pixels inside A(Px,Pn) were connected at least by one corner. The ARF values are calculated for several Pn values. The same procedure was applied to the radar-rainfalls with the duration d = 1, 2, . . . , 6, 12, 24, 48,. . ., 120 h. The corresponding pixel rainfalls were determined by summing the Table 1 Characteristics of maximum point radar-based rainfalls, Px [mm], for various rainfall durations, d [h] d
Mean
St_dev
Max
Min
1 3 6 12 24 48 72 96 120
79.8 113.3 138.2 169.9 212.9 283.4 337.3 374.0 392.9
23.4 31.7 33.4 31.0 51.6 103.9 134.4 149.7 146.4
202.2 254.6 276.7 285.8 421.7 623.5 773.5 869.3 911.0
54.8 77.8 99.9 128.6 168.0 195.3 233.5 263.3 285.0
In total the highest 500 Px values were considered for each d to determine the mean Px value (mean), standard deviation (st _ dev), maximum (max) and minimum (min) in mm.
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Table 2 Characteristics of the areas with the mean area radar-rainfall PA b Pn(i) and PA z Pn(i + 1), where Pn [mm] is a defined minimum rainfall (see the text for more detailed explanation) i Pn _ 1 N Amean Amax Pn _ 24 N Amean Amax 1 2 3 4 5 6 7 8 9 10 11 12
0.2 1 2 5 10 30 50 75 100 120 130 140
500 500 500 500 500 500 500 225 87 36 20 11
22,629 8559 4961 1700 575 101 44 13 9 8 8 7
150,768 85,000 47,912 21,016 12,616 10,948 10,824 120 44 32 28 24
5 10 30 50 75 100 125 150 175 200 225 250
500 500 500 500 500 500 500 500 398 228 139 94
61,880 39,173 6848 2076 430 123 45 18 6 5 4 4
186,528 162,840 92,444 40,052 8068 1856 812 320 32 12 8 4
The left part of the table shows the values for 1-h rainfall (Pn_1), and the right part relates to 24-h rainfalls (Pn_24). The N gives the number of areas, which were identified by the procedure described in the text. The Amean and Amax give the mean, and maximum area extent, respectively. The Amean and Amax are in square kilometers.
hourly rainfall values. Like the hourly values, area A, and rainfall PA were determined for each duration and for each start time t0. The event with the duration d was defined by the decrease of hourly rainfalls at (t0 1) and at (t0 + d + 1). We requested the hourly values PA{t0 1} and PA{t0 + d + 1} to be smaller than 0.1% of hourly rainfalls PA{t0} and PA{t0 + d}, respectively. Basic characteristics of maximum rainfalls Px as a function of rainfall duration d [h] are shown in Table 1, and the areas defined by the selection procedure are characterized in Table 2. The maximum value, ARFx, depending on area size A was expressed as an upper envelope of the ARF(A) values by the relationship ARFx ¼ exp aX 2 þ bX 4 ; for X Nd; ð5aÞ ARFx ¼ 1; for 0bX Vd;
ð5bÞ
where the independent variable X is given by X ¼ log10 ð AÞ d;
ð5cÞ
and parameters a, b, and d were determined empirically as a function of precipitation duration. The value d was set for 0.9, and the resulting values of a, and b are given in Table 3. The resulting relations (5a), (5b) and (5c) are depicted in Fig. 4. Table 3 Coefficients a, and b, entering the ARFx formulae (5a), (5b) and (5c) for various rainfall durations (d) d [h]
a
b
d [h]
a
b
1 2 3
0.1501 0.1616 0.1445
0.0115 0.0020 0.0006
6 12 z 24
0.0852 0.0603 0.0501
0.0013 0.0010 0.0
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Fig. 4. The dependence of ARFx values on the area extent. The rainfall duration is denoted.
In order to determine an area-related PMP for a catchment, we considered a dense grid of 1 km 1 km covering a catchment. The grid point PMP values were calculated as a weighted average of PMP values for the gauges inside the catchment and in its surroundings. The distance between the grid point and the gauge served as a weight. Then the relationships (5a), (5b) and (5c) were applied to the maximum PMP value over all grid points inside the catchment.
4. Comparison of PMP and heavy flood rainfalls in 1997 and 2002 The comparison between the PMP estimates and the extreme precipitation amounts falling during two extreme flood events (July 1997 and August 2002) was implemented. The extreme rainfalls in August 2002 and July 1997 occurred in two time periods of several days duration. In 2002, the first precipitation episode took place on August 6 and 7, the second episode followed on August 11–13. Both precipitation episodes hit the area of the Labe river basin, which covers several urban areas and towns in the CR, including Prague. In Germany, a part of Dresden was flooded, for example. In 1997, the precipitation fell out in the Moravia region where the heavy rain affected two main catchments (the Morava and Odra river basins). The episodes were separated by a longer period nearly without rainfall. The first episode occurred on July 4–7, and was not followed by a second precipitation period until July 17–21. In 1997 the first rainfall event was markedly dominant while in 2002, the precipitation amount was higher during the second period. The paper by Rezacova et al. (2005) presents a more detailed description of both flood situations.
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The area PMP estimates were determined for the catchments of various area extents and locations by the statistical method described in Sections 2 and 3. We used the station-related point PMP estimates that were derived from the historical data up to the year 2000, and the conversion to the area estimates was done by radar-based ARFx. Main attention was devoted to the PMP estimates for the river basins of the 3rd order, which covered the territory of the CR and reflected the area distribution of the flood rainfalls. Altogether 63 Bohemian catchments belong to the Labe river basin, the Moravian river basins Odra and Morava consist of 8 and 21 basins, respectively. The daily PMP values are shown in Fig. 5 for various catchment areas. The highest area-related PMP estimates (over 250 mm) were found in catchments where mountain regions dominated. However, not all mountainous river basins show such high PMP values. On
Fig. 5. Daily area PMP estimates for the river basins of the 3rd order (see Fig. 3) as a function of the basin area. The PMP values belonging to the Labe river basin (Bohemia) and to the Odra or Morava river basins (Moravia) are distinguished by full and blank circles, respectively.
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the contrary, there are several river basins hit by the flood precipitation in 1997, which show high PMP but do not belong to the mountainous catchments. Obviously, the altitude does not play the decisive role and orographic influences are highly complex phenomena (precipitation enhancement). The CHMI provided us with the daily area precipitation over the 3rd order catchments for both flood events. The rainfalls for longer rainy periods were calculated from daily values. For each catchment, we selected the maximum area rainfall for a given rainfall duration (1–5 days) and compared its value with the corresponding PMP estimate. The results are summarized in Fig. 6. In August 2002, the highest area precipitation amounts relative to the PMP values were reached at in one of the river basins for the durations 2 and 3 days (64%) and in another basin the value reached 62% of the corresponding PMP for 2 days. In 1997, the highest area precipitation relative to PMP occurred in the upper
Fig. 6. Maximum area rainfalls relative to area PMP estimates for the Czech river basins of the 3rd order. The left (right) column corresponds to flood in 2002 (1997). The rows correspond to the rainfall duration of 1 to 5 days.
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Morava river basins for the duration of 3, 4, and 5 days (62% maximum) and the maximum of 63% was attained in other two basins (the Opava and Becva river basins). From Fig. 6, it is obvious that the area precipitation amounts during both events arrived at similar values with respect to the PMP estimates. The maximum values were observed for the duration of 2 and 3 days in 2002 while the maximum precipitation amounts in 1997 shifted towards longer rainfall durations. The statistical models used to estimate the point PMP values were developed in the year 2000 with historical rainfall data up to 2000. Therefore, the revision of the models with data including also the 2002 rainfalls should be performed in spite of the fact that there is not any area PMP exceeded in 2002. The evaluation of point PMP values also showed that the point estimates were not exceeded in the Czech territory. The only site, for which we obtained more than 100% of the daily point PMP value, was the German station Zinnwald, which was located close to the Germany–CR border. This finding indicates the need to extend our study area by adding crossborder regions.
5. Conclusion In this paper we described the technique used to estimate the point and area PMP values over the Czech territory. We concentrated on the description of the statistical technique, which was designed to the precipitation duration of 1, 2, . . . , 5 days. The conversion of the point PMP to the area PMP estimate was based on evaluating radarbased rainfalls. We determined the maximized area reduction factor, ARFx, and the formula, designed to calculate the ARFx according to the area extent, is presented in this paper. The PMP estimates were calculated at the gauge positions and for the system of small river basins covering the territory of the CR (the 3rd order basins). The area PMP estimates were compared with heavy rainfalls that had occurred during the flood events in August 2002 and July 1997. The comparison showed that the maximum area precipitation did not exceed 63% of the PMP estimates. Present state of PMP estimates should be considered a prerequisite stage. The results show that the next work should concentrate on the following topics: (i) We need to revise the PMP estimates by extending the precipitation data up to the year 2002 and the data from cross-border regions. (ii) The stability of the ARFx values should be verified with gauge adjusted radar-based rainfalls, and the data reported by both Czech radars should be considered. (iii) The gauge-related PMP values are determined for each station independently on the remaining stations. We should consider the correlation between PMP values from adjacent gauges and look for local orography effects. (iv) A study focused on the distribution of radar-based rainfalls according to the duration within the day is under way. Preliminary results show that it enhances efforts to regionalize the Czech territory, and takes into account the effect of orography. It should improve a more detailed determination of point and area PMP estimates.
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Acknowledgements The work was supported by the grant VaV510/3/97 (the Ministry of Environment, CR) and the grant QD1368 (the Ministry of Agriculture, CR). The Czech Hydrometeorological Institute is thanked for the provision of the radar and precipitation data for this research. Two referees commented the text and we highly appreciate their comments and recommendations.
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