Regional estimation of rainfall intensity-duration-frequency curves using generalized least squares regression of partial duration series statistics

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1 WATER RESOURCES RESEARCH, VOL. 38, NO. 11, 1239, doi: /2001wr001125, 2002 Regional estimation of rainfall intensity-duration-frequency curves using generalized least squares regression of partial duration series statistics Henrik Madsen DHI Water and Environment, Hørsholm, Denmark Peter Steen Mikkelsen, Dan Rosbjerg, and Poul Harremoës Environment and Resources DTU, Technical University of Denmark, Lyngby, Denmark Received 5 December 2001; revised 3 April 2002; accepted 11 April 2002; published 15 November [1] A general framework for regional analysis and modeling of extreme rainfall characteristics is presented. The model is based on the partial duration series (PDS) method that includes in the analysis all events above a threshold level. In the PDS model the average annual number of exceedances, the mean value of the exceedance magnitudes, and the coefficient of L variation (LCV) are considered as regional variables. A generalized least squares (GLS) regression model that explicitly accounts for intersite correlation and sampling uncertainties is applied for evaluating the regional heterogenity of the PDS parameters. For the parameters that show a significant regional variability the GLS model is subsequently adopted for describing the variability from physiographic and climatic characteristics. For determination of a proper regional parent distribution L moment analysis is applied for discriminating between the exponential distribution and various two-parameter distributions in the PDS model. The resulting model can be used for estimation of rainfall intensity-duration-frequency curves at an arbitrary location in a region. For illustration, the regional model is applied to rainfall data from a rain gauge network in Denmark. INDEX TERMS: 1854 Hydrology: Precipitation (3354); 1869 Hydrology: Stochastic processes; 1833 Hydrology: Hydroclimatology; 1821 Hydrology: Floods; KEYWORDS: rainfall extremes, regional estimation, partial duration series, generalized least squares regression, L moments Citation: Madsen, H., P. S. Mikkelsen, D. Rosbjerg, and P. Harremoës, Regional estimation of rainfall intensity-duration-frequency curves using generalized least squares regression of partial duration series statistics, Water Resour. Res., 38(11), 1239, doi: /2001wr001125, Introduction [2] The rainfall intensity-duration-frequency (IDF) relationship is widely applied in hydraulic and hydrological engineering for design of structures that control storm runoff and flooding. For a site where rainfall measurements are available frequency analysis can be performed for development of the IDF relationship. For design purposes national maps of IDF relationships have been constructed, e.g., in the United States [Hershfield, 1961], UK and Ireland [Natural Environment Research Council (NERC ), 1975], and Denmark [Danish Water Pollution Control Committee (DWPCC ), 1974]. These maps are based on pooling information from regional stations and simple interpolation between the sites. [3] The construction of rainfall IDF maps in the traditional manner has two major deficiencies. First, if the length of the at-site rainfall record is small as compared to the design return period, the estimated IDF relationship is very uncertain. Secondly, since the spatial variability of extreme rainfall characteristics may be large even within Copyright 2002 by the American Geophysical Union /02/2001WR small areas [e.g., Harremoës and Mikkelsen, 1995] simple pooling may provide unreliable IDF estimates. Presently, the guidelines for estimation of IDF curves are being revised in a number of countries, e.g., Canada [Adamowski et al., 1996], United States [Fennessey, 1998; Schaefer, 1998], and Denmark [Mikkelsen et al., 1998]. More and longer rainfall time series are now available and recent developments in regional frequency analysis procedures provide the basis for more comprehensive statistical analyses of these data. [4] In regional rainfall frequency analysis, rainfall measurements from several sites in a region are combined for estimation of regional IDF curves. By utilizing regional data the estimation uncertainty can be reduced significantly. Furthermore, by relating the extreme rainfall characteristics to relevant climatic and physiographic variables, IDF curves can be estimated at an arbitrary site in the region. The regional modeling includes three basic elements: (1) delineation of regions, (2) estimation of regional parameters, and (3) determination of a regional distribution. [5] Regions used in precipitation analyses are often defined as geographically coherent areas with similar climatic and physical characteristics. Such regions, however,

2 21-2 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES may not be homogenous with respect to the extreme rainfall statistics (e.g., mean, coefficient of variation and higher order moments). For instance, considerable geographical variability was observed in extreme rainfall characteristics in Denmark even within small areas having a relatively flat topography [Madsen et al., 1994; Harremoës and Mikkelsen, 1995; Mikkelsen et al., 1996]. Schaefer [1990] found that the initial division of Washington State into 5 regions according to geography and topography provided statistically heterogeneous regions. Moreover, he found that the extreme rainfall characteristics varied systematically with the mean annual precipitation (MAP). In the analysis of rainfall extremes in Denmark [Mikkelsen and Harremoës, 1993] and Canada [Adamowski et al., 1996] it was also found that certain statistical characteristics varied systematically with MAP. [6] For estimation of regional parameters and determination of a regional parent distribution procedures based on L moments [Hosking, 1990] have shown to be efficient. As compared to product moment estimates, L moment estimates are unbiased and relatively insensitive to outliers; properties that are extremely important in regional studies. Regional parameter estimation procedures based on L moments, or equivalently probability weighted moments [Greenwood et al., 1979], were introduced by Wallis [1980] and Greis and Wood [1981]. Hosking and Wallis [1993] developed a heterogeneity measure for evaluating the significance of the regional variability of second and higher order L moments. Hosking and Wallis [1993] also developed a goodness of fit measure based on L moments for determining the regional distribution. The L moment based procedures have been widely applied in regional precipitation analyses [e.g., Schaefer, 1990; Cong et al., 1993; Naghavi and Yu, 1995; Adamowski et al., 1996; Fennessey, 1998; Schaefer, 1998]. [7] Most rainfall frequency studies have been based on analyzing records of annual maxima. As an alternative to the annual maximum series (AMS) approach, the partial duration series (PDS) method that includes all events above a threshold level has been advocated. Recent studies by Madsen et al. [1997a, 1997b] revealed that the widely applied PDS model with generalized Pareto distributed exceedances, in general, is more efficient than the corresponding AMS model based on the generalized extreme value distribution for both at-site and regional quantile estimation. The PDS method has been applied in regional rainfall frequency analyses by Van Montfort and Witter [1986], Fitzgerald [1989], Madsen et al. [1994], and Mikkelsen et al. [1995, 1996]. [8] In this paper a general framework for regional analysis and modeling of extreme rainfall characteristics is presented. The model is based on an extension of the regional PDS index-flood method developed by Madsen and Rosbjerg [1997a, 1997b]. The regional rainfall frequency model includes a generalized least squares regression methodology for analyzing regional homogeneity and describing any regional variability of the PDS parameters from climatic and physiographic characteristics. For determination of a regional parent distribution L moment analysis is applied. The resulting model provides an estimate of the IDF curve and the associated uncertainty at an arbitrary site in the region. For illustration, the developed model is applied to a regional data set of Danish rainfall extremes. 2. Modeling Framework [9] The rainfall intensity for a given duration is described by a stochastic variable Z with observations {z i, i =1,2,..., m} where m is the total number of rain events in the historical time series. The extreme value model is based on the PDS method where the population includes all events above a threshold level. Basically, two different methods are available for choosing the threshold level. In type I sampling, a threshold level z 0 is explicitly defined and the events {z i > z 0, i =1,2,..., n} are included in the PDS. In type II sampling, the n largest events are included in the PDS {z (1) z (2)... z (n) }. [10] In the present study type I sampling is applied, implying that the average annual number of threshold exceedances l becomes a regional variable. In the case of type II sampling using a constant l in the region to define the PDS, the threshold level becomes a regional variable. An example of this approach is given by Mikkelsen et al. [1995, 1996]. The regional modeling procedure described in the following can be applied also in the case of type II sampling, simply by considering z 0 as a regional variable instead of l. Furthermore, if neither z 0 nor l are considered constant in the region, the regional modeling procedure is easily extended by including an additional parameter to be regionalized. [11] The exceedances in the PDS are described by a stochastic variable X = Z z 0,Z>z 0 with observations {x i, i =1,2,..., n} where n is the PDS sample size. It is generally assumed that the occurrence of exceedances can be described by a Poisson process with constant or one-year periodic intensity, implying that the number of exceedances is Poisson-distributed with intensity l. In the basic PDS model the one-parameter exponential distribution was applied for modeling the exceedance magnitudes [Shane and Lynn, 1964; Todorovic and Zelenhasic, 1970]. Several alternative two-parameter distributions have been proposed, including the generalized Pareto [e.g., Hosking and Wallis, 1987; Rosbjerg et al., 1992a; Madsen and Rosbjerg, 1997a, 1997b; Madsen et al., 1997a, 1997b], gamma [Zelenhasic, 1970], Weibull [Ekanayake and Cruise, 1993], and lognormal [Rosbjerg et al., 1991] distributions. [12] In a PDS context the T-year event is usually defined as the (1-1/(lT ))-quantile in the distribution of the exceedances [e.g., Rosbjerg, 1985]. Denote by F(x; A) the cumulative distribution for the exceedance magnitudes, the T-year event can be written z T ¼ z 0 þ F lt ; A ð1þ where A are the parameters of the exceedance distribution. An estimate of the T-year event is obtained from (1) by inserting estimates of the PDS parameters. An estimate of the Poisson parameter l is given as the average number of observed exceedances per year, i.e., ˆl ¼ n=t where t is the record length. For estimation of the parameters of F(x) the method of L moments [Hosking, 1990] is adopted. For a two-parameter exceedance distribution the T-year event estimate is then given by

3 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES 21-3 ^z T ¼ z 0 þ F ^lt ; ^m; ^t 2 ¼ z 0 þ g ^l; ^m; ^t 2 where ^m is the sample mean (equal to the first L moment estimate), and ^t 2 is an estimate of the coefficient of L variation (LCV). For estimation of L moments unbiased probability weighted moment estimators are used [Landwehr et al., 1979; Hosking and Wallis, 1995]. [13] The regional analysis of extreme rainfalls is based on the parameterization defined above. That is, the Poisson parameter, the mean exceedance, and the LCV are considered as regional variables. The aim of the regional modeling is to provide estimates of the three parameters and their associated uncertainties at an arbitrary site in the region. The regional T-year event estimate can be written ẑ R T ðþ¼z s 0 þ g ˆl R ðþ; s ˆm R ðþ; s ˆt R 2 ðþ s where s refers to the location of the site, and R denotes a regional estimate. The uncertainty of the regional T-year event estimate is approximately Varfẑ R T ðþg¼ 2 Varfˆl R ðþg þ 2 Varfˆm R ðþgþ 2 Varfˆt 2 2 ðþg s ˆlR where the partial derivatives are evaluated at (l, m, t,) = ðþ; s ˆm R ðþ; s ˆt R 2 ðþ s. In (4) it has been utilized that the mutual dependence between the regional parameter estimates can be neglected [Madsen and Rosbjerg, 1997a]. [14] For estimation of the regional IDF relationships using (3) (4) the modeling framework presented herein includes the following elements: (1) evaluation of regional homogeneity for each of the three PDS parameters, (2) for the parameters that show a significant regional variability, evaluation of the potential of describing the variability from relevant physiographic and climatic characteristics, and (3) determination of a regional distribution for the exceedances. In the following sections the different modeling elements are described. 3. Regional Modeling of PDS Parameters [15] To investigate the regional variability of the PDS parameters, a regression analysis is carried out. This analysis has a twofold purpose. First, the regression model is applied for assessment of the regional heterogeneity of the PDS parameters. Secondly, for parameters that show a significant regional variability, the regression model is subsequently adopted to evaluate the potential of describing the variability from physiographic and climatic characteristics. The regression model is based on a generalized least squares (GLS) estimation method that explicitly accounts for sampling uncertainty and intersite dependence. The GLS model is briefly described below. A more detailed description of the model is given by Stedinger and Tasker [1985, 1986] and Madsen and Rosbjerg [1997b] GLS Regression Model [16] Denote by ˆq i an estimate of a PDS parameter at station no. i, i =1,2,..., M, where M is the number of sites in the region. The following model is considered ð2þ ð3þ ð4þ ˆq i ¼ b 0 þ Xp k¼l b k A ik þ e i þ d i where A ik are the considered physiographic and climatic characteristics, b k are the regression parameters, e i is a random sampling error, and d i is the residual model error. The sampling error and the residual model error are assumed to have zero mean and covariance structures 8 < s 2 ei ; Covfe i ; e j g¼ : s ei s ej r eij ; i ¼ j i 6¼ j ð5þ 8 < s 2 d ; i ¼ j ; Covfd i ; d j g¼ : 0; i 6¼ j ð6þ where s 2 ei is the sampling error variance, r eij is the correlation coefficient due to concurrent observations at stations i and j (intersite correlation coefficient), and s 2 d is the residual model error variance. It should be noted that the intersite correlation defined above is due to the sampling of concurrent extreme rainfall events at the different locations in the region. Thus it does not consider any physical or causal dependence in extreme rainfall patterns in the region. For instance, if two records do not overlap in time, the intersite correlation due to sampling extreme events from the two records is equal to zero. [17] If the sampling error covariance matrix is known, the GLS estimates of the regression parameters and the residual model error variance are obtained from the following set of equations ½A T 1 AŠB ¼ A T 1 ; ð ABÞ T 1 ð ABÞ ¼ M p 1 ð7þ where ¼ ˆq 1 ; ˆq 2 ;...; ˆq T M T B ¼ 0 b 0 ; b 1 ;...; b p 1 1 A 11 A 1p A ¼ : : : B A 1 A M1 A Mp and is the error covariance matrix of the total errors 8 < s 2 ei þ s2 d ; ij ¼ Covfe i þ d i ; e j þ d j g¼ : s ei s ej r eij ; i ¼ j i 6¼ j The solution of (7) requires an iterative scheme. In some cases, no positive value of s d 2 can satisfy (7). In these instances, the sampling errors more than account for the difference between and AB, and s d 2 is set equal to zero [Madsen and Rosbjerg, 1997b]. The GLS (regional) estimate of the PDS parameter and the associated variance at an arbitrary site are given by ˆq R ðþ¼a s ðþ s T ˆB; VarfˆqR ðþg¼a s ðþ s X T A s B ðþþŝ2 d ð8þ ð9þ ð10þ where A(s) T = (1, A 1 (s), A 2 (s),..., A p (s)), and B = (A T 1 A) 1 is the covariance matrix of the estimated

4 21-4 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES regression parameters. The variance of the regional estimate accounts for sampling errors, corrected for intersite dependence, and residual model errors. [18] When only the intercept b 0 is included in the regression equation (referred to as the regional mean model), the GLS regression model provides an estimate of the regional average PDS parameter and the associated uncertainty (note that, in general, the regional estimate is different from the arithmetic regional average because the GLS algorithm weights the PDS parameter estimates according to the covariance matrix of the errors). In this case, the GLS estimate of the residual model error variance can be interpreted as a measure of regional heterogeneity; that is, if ^s 2 d ¼ 0, the region can be considered homogeneous. Madsen and Rosbjerg [1997b] showed that the regional average GLS estimator is a general extension of the record-length-weighted average commonly applied in the index-flood procedure [e.g., Stedinger et al., 1993; Stedinger and Lu, 1995; Madsen and Rosbjerg, 1997a]. The record-length-weighted average estimator considers neither intersite correlation nor regional heterogeneity. [19] If ^s 2 d > 0, the region is heterogeneous, and the GLS regression model in (5) can subsequently be applied to evaluate the potential of describing this variability. Alternatively, the region can be divided into distinct subregions according to similarities in station characteristics. To evaluate the homogeneity of the defined subregions and for estimation of the subregional average parameters the regional mean model is adopted. It should be noted that the subregional approach provides discontinuities in extreme value characteristics across subregional boundaries Estimation of the Error Covariance Matrix [20] The solution of the GLS regression equations, cf. (7), requires an estimate of the sampling error covariance matrix. In general, approximate expressions of the sampling error variances of the PDS parameters can be formulated in terms of the population parameters (see, e.g., Madsen and Rosbjerg [1997a] for expressions for the PDS model with generalized Pareto distributed exceedances). Estimates of the sampling error variances can then be obtained by substituting the population parameters by the sample estimates. However, for solving the GLS regression equations, the error covariance estimator should be independent, or nearly so, of the PDS parameter estimator ^q i [Stedinger and Tasker, 1985]. Following the approach by Tasker [1980], estimates of the sampling error variances for the three PDS parameters that fulfill this independence criterion are presented below. [21] For the Poisson parameter, the sampling error variance is given by s 2 ei = l i /t i. An estimate of the sampling error variance that is virtually independent of the parameter estimate ˆl i can be obtained by substituting the population parameter l i by the regional average of the parameter estimates, i.e. ^s 2 ei ¼ c t i ; c ¼ 1 M X M i¼1 ˆl i ð11þ For the mean exceedance, the sampling error variance is s 2 ei = s 2 i /n i where s 2 i is the population variance. A reasonable estimate of s 2 ei is then given by ^s 2 ei ¼ c n i ; c ¼ 1 M X M ^s 2 i i¼1 ð12þ For estimation of the sampling error variance of the LCV estimator a Monte Carlo simulation procedure is applied, following the lines of calculation of different regional statistics by Hosking and Wallis [1993]. In this case, a large number of regional samples are generated from a kappa distribution based on the regional average L moment statistics (regional average LCV, L skewness and L kurtosis) and regional average number of threshold exceedances n. From these simulations the variance of the LCV estimates V 2 is calculated. The estimate of s 2 ei for the LCV estimate can then be determined by ^s 2 ei ¼ nv 2 n i ð13þ The flexible four-parameter kappa distribution is used in the simulations to avoid choosing a particular distribution (such as the exponential distribution and various two-parameter distributions) as a regional parent at this stage of the analysis. [22] For estimation of the intersite correlation between parameter estimates two types of correlation are considered (1) correlation between the annual number of exceedances, and (2) correlation between concurrent exceedance magnitudes. The correlation coefficient between the estimated Poisson parameters, r lij, is equal to the correlation coefficient between the annual number of threshold exceedances [Mikkelsen et al., 1996]. The correlation coefficient between the sample mean values, r mij, is equal to the correlation coefficient between concurrent exceedances r ij. The correlation between higher order sample moments depends on the order of the moment [Stedinger, 1983]. For the LCV estimates the intersite correlation coefficient is given by r tij = r 2 ij [Madsen and Rosbjerg, 1997a]. Thus the effect of intersite dependence is less severe when estimating higher order moments. [23] Calculation of the correlation between the annual numbers of exceedances is based on the concurrent observation years at the two stations. For estimation of the correlation between the exceedances a series of concurrent exceedances at the two stations is defined based on the time of onset and termination of the extreme events. From this series a conditional correlation coefficient can be calculated (conditional upon extreme events occurring at the two stations at the same time). To account for extreme events that do not overlap temporally, an unconditional correlation coefficient is calculated (see Mikkelsen et al. [1996] for details). It should be noted that due to moving rain cells and frontal systems two extreme events may be concurrent in meteorological terms without overlapping in time. However, to take this kind of correlation into account detailed meteorological information of each extreme rain event in the region is needed, and this information was not available in the present project. In the work of Mikkelsen et al. [1996] the movement of rain cells was conceptually addressed and it was shown that this did not have a significant impact on the estimation of the intersite correlation. [24] In general, the estimated intersite correlation coefficients have relatively large sampling uncertainties. A

5 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES 21-5 Figure 1. Empirical probability distributions of the 10-min rain intensity for the 41 stations. direct use of the sample estimates may result in an error covariance matrix that can not be inverted, and hence provides an ill-posed solution of (7) [Tasker and Stedinger, 1989]. To overcome this problem the correlation coefficients are smoothed by relating the sample estimates to the distance between stations. In this case an exponential correlation function is used r ij ¼ j h d ij wd ij þ1 i ¼ exp d ij wd ij þ 1 where d ij is the distance between stations. 1nj ð14þ 4. L Moment Analysis [25] For determination of a regional parent distribution L moment analysis is applied. The goodness of fit measure proposed by Hosking and Wallis [1993] was formulated for discriminating between various three-parameter distributions. For specific use in PDS modeling where the candidate distributions are the one-parameter exponential distribution and different two-parameter distributions, the test statistic has been reformulated. First, consider a two-parameter distribution. Since the L skewness of a two-parameter distribution is determined by the LCV estimate, the distance between the regional estimate of the L skewness and the theoretical L skewness for the considered distribution can be used as a measure of the goodness of fit. To test the significance of this distance, it is related to the sampling uncertainty of the regional L skewness estimate. Thus the goodness of fit measure can be formulated G 3 ¼ t 3 t DIST 3 s 3 ð15þ where t 3 is the regional (record-length-weighted) average L skewness, t DIST 3 is the theoretical L skewness for the considered distribution, and s 3 is the standard deviation of the regional L skewness estimate. The goodness of fit measure for the one-parameter exponential distribution can be formulated in a similar way. In this case the goodness of fit is based on the distance between the regional LCV estimate and the theoretical LCV, i.e. G 2 ¼ t 2 t EXP 2 s 2 ð16þ where t 2 is the regional (record-length-weighted) average LCV, t EXP 2 is the theoretical LCV for the exponential distribution (equal to 1/2), and s 2 is the standard deviation of the regional LCV estimate. [26] The standard deviations of the regional LCV and L skewness estimates are determined by simulating a large number of homogenous regions from a kappa distribution by using the regional average L moment statistics and number of observations corresponding to the observed series. A significance test for the goodness of fit measure can be formulated, assuming that the L moment estimates are independent, homogenous and normally distributed. In this case, the G statistic in (15) (16) is approximately normally distributed. The goodness of fit measure of Hosking and Wallis [1993] based on the L kurtosis includes a bias correction. However, since LCV and L skewness have negligible biases, bias corrections have not been included in (15) (16). [27] Hosking and Wallis [1993] also proposed a test statistic based on L moment estimates for assessing regional homogeneity. It should be noted, however, that the L moment statistic does not account for intersite dependence. The lack of power of the test in the case of significant intersite dependence may lead to erroneous conclusions with respect to regional homogeneity [Madsen and Rosbjerg, 1997b]. The GLS model used here explicitly accounts for intersite correlation, and, in addition, it provides an estimate of the uncertainty of the regional mean. 5. Application Example 5.1. Rainfall Data [28] In 1979 a new system of high-resolution automatic rain gauges was introduced in Denmark. The measuring network covers a total area of 43,000 km 2, and the distances between gauging stations range between 1 and 300 km. All

6 21-6 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES Figure 2. Spatial structure of the intersite correlation due to concurrent exceedances: (top) 10-min intensity and (bottom) 24-hour intensity. stations are equipped with tipping bucket gauges with a bucket size of 0.2 mm. The raw data that consist of the number of tips per min are transformed into one-min intensities for individual rain events. The preliminary separation of rain events is defined as periods exceeding one hour without precipitation. At present, 90 stations have been connected to the measurement system, and the longest records consist of more than 18 years of data. In the regional study, 41 stations with more than 10 years of data have been included, corresponding to a total of about 650 station years. Details of the measurement system and the quality control of the data are given by Jørgensen et al. [1998]. [29] From the original data of one-min intensities, maximum intensities averaged over different durations were extracted. Denote by i(t) the one-min rain intensity at time t. The mean intensity at time t, y(t), with duration t is defined as yðþ¼ t R tþt=2 t t=2 iðþdx x t ð17þ A rain event corresponding to the considered duration t is defined as an uninterrupted sequence of positive y(t). Denote by t 0i and t 1i the onset and the termination of the defined rain event, the maximum mean intensity of the event is then given by z i ¼ MaxfyðÞ; t t 0i t t 1i g ð18þ With the above definition two rain events are independent if the time between two consecutive tips of the rain gauge is larger than t. However, due to the preliminary separation of rain events, for t < 60 min independent rain events are separated by at least one hour without precipitation. Maximum mean intensities (for simplicity denoted intensities in the following) were abstracted for 8 different durations ranging between 10 min and 48 hours. [30] For a preliminary visual assessment of the extreme rainfall characteristics in the region, the empirical probability distributions from the 41 stations have been plotted. As an example, the 10-min intensities are shown in Figure 1

7 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES 21-7 Table 1. Explanatory Variables Used in the Regression Analysis Characteristic Mean annual precipitation Altitude Geographical location Distance to the sea or a large lake Shelter index Units mm m eastings, northings km degrees where the exceedance probabilities, or equivalently the return periods, of the observations are calculated using the median plotting position formula [e.g., Rosbjerg et al., 1992b]. At a first glance, the extreme intensities show a pronounced regional variability. The first stage in the regional modeling is to analyze whether this variability is significant (i.e. if it reflects true regional differences) or can be explained by sampling uncertainties. If the variability is significant, in the second stage the potential for describing the variability from physiographic and climatic characteristics is analyzed. [31] The PDS for the 41 stations were defined by using the same threshold level at all stations (see Figure 1). The threshold level was chosen on the basis of a preliminary sensitivity analysis of regional average extreme value characteristics as a function of the threshold level, implying a regional average number of exceedances per year in the range for the analyzed rainfall variables Intersite Correlation Analysis [32] In Figure 2 the intersite correlation structure due to concurrent exceedances is shown for two of the analyzed rainfall variables, the 10-min and the 24-hour intensity. In Figure 2 the sample estimates, a moving average curve based on a moving window including 10 data points, and the fitted exponential correlation function are shown. The two parameters j and w of the correlation function were determined by a visual adjustment of the function to the moving average curve. In general, the intersite correlation is a decreasing function of the distance. Furthermore, the correlation structure depends strongly on the considered duration, the correlation being larger for larger durations. This structure reflects the fact that extreme intensities for large durations are mainly caused by moving frontal rain systems with a large spatial coverage, whereas extreme intensities for small durations are caused by convective rain cells with a limited spatial extent. [33] The intersite correlation between the annual number of threshold exceedances has no apparent spatial structure, and both large positive and negative correlations are observed. In this case a constant function equal to the regional average correlation coefficient was fitted to the data GLS Regression Analysis [34] The characteristics used as explanatory variables in the regression analysis are shown in Table 1. The mean annual precipitation (MAP) is determined by interpolation of MAP for the standard normal period from 300 locations in Denmark [Frich et al., 1997] based on daily measurements. The shelter index is calculated as the average of the angle between the gauge orifice and the horizon for 8 directions. This parameter is included to analyze the bias of the measurements due to different shelter conditions for the 41 gauges. Any significant correlation between a PDS parameter and the shelter index has to be treated as an additional source of uncertainty (measurement error). [35] In the regression analysis the Cook s D statistic has been calculated [Tasker and Stedinger, 1989]. For the GLS regional mean model large values of Cook s D indicate stations that diverge significantly from the group as a whole, and hence the statistic can be used to identify possible outliers. Stations being identified as possible outliers should be carefully examined for gross errors in their data. If the data seems acceptable, the possibility of moving the station to another region should be considered. [36] Results of the regression analysis for the Poisson parameter are shown in Table 2. For all analyzed rainfall variables the Poisson parameter has a pronounced regional variability. A large part of this variability can be explained by the mean annual precipitation. In this case the Poisson parameter is an increasing function of MAP, i.e. the larger MAP the more events above the threshold are observed (see Figure 3). The correlation between the Poisson parameter and MAP is more pronounced for intensities with large durations. Inclusion of other characteristics in the regression equation did not improve the regional model. [37] The results for the mean exceedance are shown in Table 3. For small durations (t 1 hour) the region can be considered homogeneous (note that for the 10-min rain intensity one station is identified as an outlier and is excluded from the analysis). For larger durations a significant metropolitan effect was observed. The average extreme intensities for durations between 1 and 12 hours are significantly larger in the Western part of the Copenhagen area than in the rest of the country. For durations between 12 and 48 hours also the eastern part of the Copenhagen area has significantly larger average extreme intensities than in the rest of the country. A regional model is defined that divides the country into three subregions: (1) the western Copenhagen area, (2) the eastern Copenhagen area, and (3) the rest of the country. For durations between 1 and 12 hours regions (2) and (3) are pooled into one region, whereas for durations larger than 12 hours regions (1) and (2) are pooled. The mean exceedance in subregion (3) can be considered homogeneous. In the Copenhagen area, however, the mean exceedance has a significant variability, but none of the considered climatic and physiographic characteristics provide any significant explanations of this varia- Table 2. Results of the GLS Regression Analysis for the Poisson Parameter Duration Mean, years 1 Regional Mean Model Residual Variance, (years 1 ) 2 Regression Model Based on MAP, Residual Variance, (years 1 ) 2 10 min min min h h h h h

8 21-8 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES Figure 3. GLS estimate of the Poisson parameter and corresponding 95% confidence interval compared with observed values: (top) 10-min intensity and (bottom) 24-hour intensity. Table 3. Results of the GLS Regression Analysis for the Mean Exceedance a Regional Mean Model Subregional Mean Models Subregion 1 Subregion 2 Subregion 3 Duration Mean, mm/s Residual Variance, (mm/s) 2 Mean, mm/s Residual Variance, (mm/s) 2 Mean, mm/s Residual Variance, (mm/s) 2 Mean, mm/s Residual Variance, (mm/s) 2 10 min b min min h h h h h a Subregions: 1, western Copenhagen area; 2, eastern Copenhagen area; 3, the rest of the country. b One outlier station excluded from the analysis (large Cook s D statistic).

9 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES 21-9 Table 4. Results of the GLS Regression Analysis for the LCV Duration Mean Residual Variance 10 min a min a min a h h h h h a One outlier station excluded from the analysis (large Cook s D statistic). bility. Thus in all the subregions the mean exceedance is modeled by a regional mean model. [38] For some of the rainfall variables the mean exceedance and the shelter index were found to be slightly correlated, indicating a tendency that gauges with better shelter conditions measure larger intensities. As mentioned above, the variability due to different shelter conditions has to be considered as an additional error that is included in the resultant uncertainty measure in the regional model (i.e. this portion of the regional variability cannot be modeled from climatic and physiographic characteristics). Alternatively, the measurements should be corrected to eliminate the effect, see e.g., examples of precipitation correction formulae of Allerup et al. [1997]. [39] Results of the regression analysis for the LCV are shown in Table 4. For the 10, 30 and 60-min intensity one station diverges significantly from the group as a whole (large Cook s D statistic). If this station is excluded in the analysis, LCV can be considered homogeneous for all analyzed rainfall variables, except for the 48-hour intensity. None of the considered climatic and physiographic characteristics were able to explain the regional variability of the 48-hour intensity. Thus for all rainfall variables a regional mean model was adopted for modeling LCV L Moment Analysis [40] For the analyzed rainfall variables L moment diagrams were constructed. In the L moment diagram LCV and L skewness estimates are compared to the theoretical Table 5. Goodness of Fit Measures for the Gamma (GAM), Weibull (WEI), Lognormal (LN), Generalized Pareto (GP), and Exponential (EXP) Distributions Duration GAM WEI LN GP EXP 10 min a a min 1.1 a 0.5 a min 1.2 a 0.2 a a 4.7 3h a a 3.2 6h a a h a a h a a h a a 4.9 a Distribution cannot be rejected at a 5% level of significance (jgj < 1.96). relationships for a number of candidate distributions, including the generalized Pareto (GP), lognormal (LN), gamma (GAM), Weibull (WEI), and exponential (EXP) distributions. As an example the L moment diagram for the 10-min intensity is shown in Figure 4. The theoretical L moment relationships for the considered distributions are given by Hosking and Wallis [1997]. Note that the GP, GAM, and WEI distributions contain the exponential distribution as a special case. [41] The goodness of fit statistics for the 5 candidate distributions are compared to the quantiles of a standard normal distribution in Table 5. For all rainfall variables the LN and EXP distributions are rejected at a 5% level of significance. The GAM distribution is rejected for 6 of the variables, whereas the GP and the WEI distribution are generally accepted. Analysis of the L moment diagrams reveals that the cloud of points in the diagram is better described by the theoretical GP line than the WEI line, and hence the GP distribution should be preferred (see Figure 4 as an example). Thus it is concluded that the GP distribution can be adopted as a regional parent for all rainfall variables Regional Estimation Model [42] Based on the above findings a regional model can be formulated for estimation of T-year intensities and associated uncertainties at an arbitrary site in the region. For a GP- Figure 4. L moment ratio estimates for the 10-min intensity compared to the theoretical relationships for a number of candidate distributions.

10 21-10 MADSEN ET AL.: REGIONAL ESTIMATION OF RAINFALL IDF CURVES Figure 5. IDF curve and corresponding 68% confidence interval for a location in the Copenhagen area (region 1) with a mean annual rainfall of 600 mm. Return period T is given in years. distributed parent, the regional T-year event estimate, cf. (3), reads 2 3!ˆk R ẑ R T ðþ¼z s 0 þ ˆm R ðþ s 1 ðþ s þˆkr ðþ s 4 1 ˆk R 1 5; ðþ s ^l R ðþt s ð19þ ˆk R ðþ¼ s 1 ˆt R 2 ðþ s 2 The uncertainty of this estimate can be found from (4). [43] The estimation procedure can be summarized as follows. 1. Based on an estimate of MAP at the site in question, the mean annual number of exceedances and the associated uncertainty are estimated from the GLS regression equation. 2. For intensities with small durations, t 1 hour, the mean value of the exceedances and the associated uncertainty are obtained from the regional mean model covering the whole region. For larger durations, the regional mean model for the relevant subregion is adopted. 3. The regional estimate of LCV and associated uncertainty is obtained from the regional mean model covering the whole region. 4. The regional PDS parameter estimates and estimates of the uncertainties are finally inserted in (19) and (4). An example of a regional IDF curve is shown in Figure Conclusions [44] A general framework for regional analysis and modeling of extreme rainfall characteristics has been developed. The model is based on a PDS parameterization, which includes the average annual number of exceedances, the mean value of the exceedance magnitudes, and the LCV to be assessed from regional data. A GLS regression model that explicitly accounts for intersite correlation and sampling uncertainties has been introduced for evaluating the regional heterogenity of the PDS parameters. For the parameters that show a significant regional variability, the GLS model is subsequently applied for describing the variability from physiographic and climatic characteristics. The resulting GLS models can then be applied for estimation of the PDS parameters and associated variances at an arbitrary location in the considered region. [45] For determination of a proper regional parent distribution L moment analysis is applied. In this respect, regional goodness of fit measures have been formulated that specifically apply for discriminating between the exponential distribution and various two-parameter distributions in the PDS model. [46] The regional model was applied to rainfall data from a high-resolution rain gauge network in Denmark. The GLS regression analysis revealed that a large part of the regional variability of the mean annual number of exceedances in the PDS can be explained by MAP; that is, for larger MAP the more extreme events are observed. The mean value of the exceedance magnitudes can be assumed constant in the region for small durations (less than one hour). For larger durations a significant metropolitan effect was observed, the mean intensities in the Copenhagen area being significantly larger than in the rest of the country. For LCV a regional mean model was adopted. The L moment analysis revealed that the GP distribution is an appropriate regional distribution. The analysis led to IDF-curve estimates and associated uncertainties at arbitrary locations in Denmark. [47] Acknowledgments. The research was supported by the Den Kommunale Momsfond and the Water Pollution Control Committee of the Danish Society of Engineers. The rainfall data was provided by the Water Pollution Control Committee and the Danish Meteorological Institute. References Adamowski, K., Y. Alila, and P. J. Pilon, Regional rainfall distribution for Canada, Atmos. Res., 42, 75 88, Allerup, P., H. Madsen, and F. Vejen, A comprehensive model for correcting point precipitation, Nord. Hydrol., 28, 1 20, Cong, S., Y. Li, J. L. Vogel, and J. C. Schaake, Identification of the underlying distribution form of precipitation by using regional data, Water Resour. Res., 29(4), , Danish Water Pollution Control Committee (DWPCC), Estimation of IDF curves (in Danish), Publ. 16, Danish Soc. of Eng., Teknisk Forlag, Denmark, Ekanayake, S. T., and J. F. Cruise, Comparisons of Weibull- and exponen-

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