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Hydrological Sciences Journal ISSN: 0262-6667 (Print) 2150-3435 (Online) Journal homepage: http://www.tandfonline.com/loi/thsj20 he use of resampling for estimating confidence intervals for single site and pooled frequency analysis / Utilisation d'un rééchantillonnage pour l'estimation des intervalles de confiance lors d'analyses fréquentielles mono et multi-site DONALD H. BURN o cite this article: DONALD H. BURN (2003) he use of resampling for estimating confidence intervals for single site and pooled frequency analysis / Utilisation d'un rééchantillonnage pour l'estimation des intervalles de confiance lors d'analyses fréquentielles mono et multi-site, Hydrological Sciences Journal, 48:1, 25-38, DOI: 10.1623/hysj.48.1.25.43485 o link to this article: https://doi.org/10.1623/hysj.48.1.25.43485 Published online: 19 Jan 2010. Submit your article to this journal Article views: 432 View related articles Citing articles: 23 View citing articles Full erms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalinformation?journalcode=thsj20

Hydrological Sciences Journal des Sciences Hydrologiques, 48(1) February 2003 25 he use of resampling for estimating confidence intervals for single site and pooled frequency analysis DONALD H. BURN Department of Civil Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada dhburn@sunburn.uwaterloo.ca Abstract A balanced resampling approach is presented for estimating confidence intervals for extreme flow quantiles determined from data at a single site. he approach is also adapted to provide resampled estimates for confidence intervals for extreme flow quantiles obtained from pooled frequency analysis. he balanced resampling approach does not require assumptions, in contrast to conventional approaches that are typically based on an asymptotic formula and require a distributional assumption. he approach is demonstrated to provide useful information in the context of both single site and pooled frequency analysis. he application of the approach also demonstrates the benefits of employing a pooled frequency analysis approach for estimating extreme flow quantiles. Key words confidence limits; uncertainty; resampling; bootstrap; flood frequency analysis, extreme flows; L-moments Utilisation d un rééchantillonnage pour l estimation des intervalles de confiance lors d analyses fréquentielles mono et multi-site Résumé Nous présentons une approche de rééchantillonnage équilibré pour évaluer les intervalles de confiance lors de l estimation des quantiles de crues extrêmes à partir des données d un site unique. L approche est également adaptée en cas d estimation des quantiles à partir d une analyse fréquentielle multi-site. L approche de rééchantillonnage équilibré ne nécessite aucune hypothèse, contrairement aux approches conventionnelles qui sont généralement basées sur des ajustements asymptotiques et qui nécessitent de choisir une fonction de distribution. Il apparaît que cette approche fournit une information utile dans les deux cas d analyse fréquentielle mono et multi-site. Son application montre également l intérêt de s appuyer sur une analyse fréquentielle multi-site pour estimer les quantiles de crues extrêmes. Mots clefs limites de confiance; incertitude; rééchantillonnage; bootstrap; analyse fréquentielle de crues; écoulements extrêmes; L-moments INRODUCION Many water resources applications require an estimate of a design flow, in the form of the magnitude of an extreme flow quantile, as well as an indication of the uncertainty associated with the estimate. he uncertainty measure is often expressed in the form of a confidence interval for the quantile estimate. Confidence intervals have traditionally been calculated using a formula that depends on the distribution function that has been fitted to the extreme flow data as well as the fitting method used to estimate the extreme flow quantiles (see, for example, Kite, 1977; Stedinger et al., 1993; Lu & Stedinger, 1992). here are several shortcomings to this approach. First, the formulae that are generally used are asymptotic formulae implying that they may not be accurate for the short data records that are often available when extreme flow quantiles must be estimated. It should be noted that exact confidence intervals are available for some Open for discussion until 1 August 2003

26 Donald H. Burn probability distributions combined with specific parameter estimation techniques. Second, the approaches generally involve making a distribution assumption for the estimates of the extreme flow quantiles. ypically, the extreme flow estimates are assumed to be normally distributed. he distribution assumption is then used to estimate the confidence intervals from a variance estimate obtained from an asymptotic formula. he assumed distribution may or may not be appropriate for a given data set. An alternative approach is the calculation of confidence intervals through a resampling, or bootstrap, approach. his approach avoids having to make a distributional assumption and does not rely on the available sample size being large enough to ensure that the asymptotic behaviour of the approach applies. Furthermore, as a nonparametric estimation approach, resampling is easy to use and involves the same calculation approach regardless of the extreme flow cumulative distribution function to be fitted to the extreme flow data or the fitting method. It is the intent of this paper to explore the use of resampling approaches for calculating confidence intervals for extreme flow quantiles for both single site and pooled analysis. he latter application appears to represent a unique contribution of this research. he next section of the paper provides an overview of the resampling methods that are used. his is followed by a comparison of the resampling approach to the use of an asymptotic formula for estimating confidence intervals for the Generalized Extreme Value (GEV) distribution. he application of the resampling approach to a number of single site and pooled frequency analysis cases and an exploration of the characteristics of the confidence intervals obtained follows. he final section of the paper summarizes the important conclusions from the research. RESAMPLING APPROACH Single site analysis Confidence intervals can be calculated for the estimate of an extreme flow quantile at a single site using a balanced resampling approach (Reed, 1999). Resampling approaches involve creating new samples from the original sample by a bootstrapping process which involves randomly selecting data points, with replacement, from the original sample and then estimating the extreme flow quantile from each of the resampled data sets. An empirical distribution for the extreme flow quantile can be obtained from the resulting collection of estimates. In balanced resampling, first introduced by Davison et al. (1986), each data point appears the same number of times in the union of the resampled data sets. his is accomplished by creating B copies of the original sample and then concatenating the samples to obtain a sample of length B n, where B is the number of resamples desired and n is the original data set length. he elements of the concatenated sample are randomly permutated and the permutated sample is then divided into B samples of length n. A total of B estimates for the quantile of interest are then obtained and are used to estimate the desired confidence intervals. he procedure to find the 100(1 2α)% confidence intervals, following Faulkner & Jones (1999), is: 1. Generate B resampled data sets using balanced resampling, as described above. 2. For each sample, estimate the extreme flow quantile of interest, Qˆ i, where Qˆ i is

he use of resampling for estimating confidence intervals 27 the estimate of the -year flow quantile from the ith sample. Quantile estimates are obtained by the method of L-moments calculated using unbiased estimates of the probability weighted moments. 3. Calculate bootstrapped residuals, E i, which are the deviations of each estimated quantile from the quantile estimate for the original sample. he calculation of E i is through E i = Qˆ i Qˆ sam, where Qˆ sam is the sample estimate for the -year flow quantile. 4. Rank the deviations from smallest to largest and find E (m) and E (p) where m = α(b + 1) and p = (1 α)(b + 1). For B = 999, this implies choosing the 25th and 975th values to construct 95% confidence intervals. 5. Construct the confidence interval for the unknown quantile as: ( Qˆ E (p), sam Qˆ sam E (m) ) he confidence interval defined in step 5 uses what is known as a test-inversion approach (for details, see Faulkner & Jones, 1999; Carpenter, 1999). Pooled analysis In many situations, information from more than one streamflow location is pooled to improve the estimation of extreme flow quantiles at a site of interest. his approach can be applied to sites which are ungauged, or to sites which have a gauging record that is not sufficiently lengthy to afford an accurate estimate of the quantile of interest. he approach involves forming a collection of gauging stations, called a pooling group, where the members of the pooling group are in some way considered to be similar in terms of hydrological response (Burn, 1997; Cunderlik & Burn, 2002). In pooled frequency analysis, an estimate of the uncertainty associated with an extreme flow quantile is also typically desired. Although attempts have been made to derive regional estimates for the confidence intervals, these approaches generally suffer from the same limitations as the asymptotic approach for single site analysis plus additional complications arising from the pooling of information from the collection of gauging stations. An example of an asymptotic approach is the work of De Michele & Rosso (2001) who applied an asymptotic formula for the GEV distribution, based on the work of Lu & Stedinger (1992), for estimating confidence intervals in a pooled frequency analysis context. However, this technique involves determining the sample size as the total number of station-years of record in the pooling group, an assumption that ignores the impact of spatial correlation in the data within the pooling group (see De Michele & Rosso, 2001, p. 458). Since a pooling group will generally exhibit spatial correlation, a procedure that preserves the existing spatial correlation structure is required. A resampling approach can again be used to estimate the confidence intervals for the case of pooled frequency analysis. o preserve the spatial correlation structure of the data in the pooling group, a vector bootstrap approach (GREHYS, 1996) can be used. In vector bootstrapping, resampling is done on years such that selecting a year implies that all sites with a data value for that year have the corresponding data value included in the bootstrap sample. his approach ensures that the spatial correlation structure in the original data set is preserved in the resampled data sets. he approach

28 Donald H. Burn can be implemented by applying a balanced resampling to the years in a manner analogous to the approach used for single site analysis for flow values. he use of balanced resampling implies that all years from the collection of years with data will appear the same number of times in the union of the resampled data sets. Once the pooled data set has been assembled, quantiles can be estimated in accordance with the pooling method employed and confidence intervals calculated following the approach outlined above for single site analysis. Faulkner & Jones (1999) presented a similar approach for rainfall frequency analysis. APPLICAION OF HE RESAMPLING APPROACH Comparison with an asymptotic formula Lu & Stedinger (1992) present a formula for calculating the asymptotic variance of a quantile estimate for the GEV distribution when the parameters are estimated using the method of L-moments. he cumulative distribution function for the GEV distribution is: 1 κ Q ξ F ( Q) = exp 1 κ for κ 0 (1) α where ξ is a location parameter, α is a scale parameter, and κ is a shape parameter. he variance can be estimated from (Lu & Stedinger, 1992): var { [ κ ]} 2 [ ˆ 2 exp a0 ] ( ) + a1( ) exp( κ) + a2 ( ) κ + a3( ) α = (2) n Q 3 where Q is the estimate for the -year flow event; a 0 (), a 1 (), a 2 (), and a 3 () are coefficients that depend on the return period, ; and n is the number of observations in the sample. he values for coefficients for different return periods are tabulated in Lu & Stedinger (1992). With the asymptotic variance calculated through equation (2), the confidence intervals can be determined by assuming that the extreme flow estimates follow the Gaussian distribution. Stedinger et al. (1993) report that equation (2) provides estimates of the variance that are relatively accurate for record lengths of 20 70 years and for κ values less than or equal to 0.2. o compare the resampling and formula approaches for calculating confidence intervals, 18 data sets of annual maximum daily flow were considered. he data sets were assembled to reflect varying hydrological conditions and also varying record lengths. he data were drawn from three sources. he first source is data from the UK contained in the Flood Estimation Handbook (FEH, 1999), from which eight rivers were selected. he second source is data from rivers on the Canadian Prairies and is from data examined in Burn (1997). Seven rivers were selected from this data set. he third set of rivers is located in north-central Italy and is based on data investigated in Castellarin et al. (2001). hree Italian rivers were analysed. able 1 summarizes the important characteristics of the rivers examined. In able 1, UKi refers to the ith station from the UK data set, CAi refers to the ith station from the Canadian data set and Ii refers to the ith station from the Italian data set. For

he use of resampling for estimating confidence intervals 29 able 1 Stations used in the comparison of single site estimation approaches. Site name Record length GEV parameters: Average ratio of formula over resampled interval width Agreement* α κ UK7 38 42.76 0.122 0.98 G UK4 52 38.02 0.256 1.02 G CA3 52 20.59 0.244 1.05 G CA7 52 16.28 0.290 1.17 R I2 25 4.782 0.088 1.18 R I1 74 218.0 0.182 1.19 R UK1 33 71.05 0.091 1.21 R UK8 71 79.50 0.016 1.26 R CA1 29 5.259 0.099 1.37 R CA5 65 6.972 0.307 1.40 R CA6 33 6.637 0.321 1.49 R UK6 28 15.96 0.090 1.50 R UK3 59 42.86 0.122 1.56 R UK5 35 6.605 0.027 1.60 R CA2 43 136.0 0.165 1.61 R I3 42 118.5 0.167 1.67 R UK2 26 7.144 0.203 1.75 R CA4 31 5.409 0.227 1.87 R * G indicates that the agreement between the two estimates is good, R indicates that the resampling approach resulted in narrower confidence intervals. each of the rivers, the GEV distribution was visually confirmed to provide a reasonable fit to the observed extreme flow data. Also reported in able 1 are the values for α and κ for the GEV distribution with the parameters fitted using the method of L-moments (Hosking & Wallis, 1997). hese values are required for estimating the confidence intervals using equation (2). he final information contained in able 1 is an assessment of the agreement between the two estimates for the distribution of the estimates of extreme flow. he ratio of the confidence interval width for the formula approach divided by the width from resampling was calculated and averaged over four return periods. Calculations were performed for the 10-, 20-, 50-, and 100-year extreme flow quantiles. he agreement was classified as being either a good agreement, implying that both approaches provided similar distributions, or the resampling approach was considered to have a narrower distribution. Note that there was only one case where the formula approach yielded a slightly narrower distribution than the resampling approach, perhaps implying that the formula approach may overestimate the uncertainty in the extreme flow quantile estimates. hree of the data sets yielded a good agreement between the approaches while 15 data sets resulted in a narrower distribution using the resampling approach. here appears to be no pattern in terms of the record length or the GEV parameters for the sites that yielded a good agreement in comparison to those that resulted in a narrower distribution for the resampling approach. Figures 1 and 2 provide a comparison of the probability density functions for the 10- and 100-year extreme flow quantiles for two sites. For the resampling calculations, the number of resamples was set to 999. he resampled probability density functions plotted in Figs 1 and 2 were obtained by fitting a nonparametric distribution to the 999

30 Donald H. Burn Frequency 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 Resampled distribution Normal distribution Resampled distribution Normal distribution 0.000 200 300 400 500 600 700 800 Fig. 1 Probability density functions for the 10- and 100-year flow quantiles for site UK4. Frequency 0.10 0.08 0.06 0.04 0.02 Resampled distribution Normal distribution Resampled distribution Normal distribution 0.00 20 40 60 80 Fig. 2 Probability density functions for the 10- and 100-year flow quantiles for site CA4. estimates of the extreme flow quantile. he nonparametric fitting was obtained using an approach described by Adamowski (1985). he intent with the nonparametric fitting was to produce a smoothed representation of the probability density function for comparison purposes. Figure 1 presents the results for UK4, a site that exhibits a good agreement between the estimated probability density function for the two approaches. he agreement between the distributions that is observed in Fig. 1 is typical of the results for the other sites that were classified as having a good agreement. Figure 2 presents the results for CA4, a site that results in a narrower probability density function for the resampling approach. Apparent from Fig. 2 is that the resampling results lead to a dramatically narrower probability density function for the 100-year quantile, while the two approaches are in closer agreement for the shorter return period. his behaviour was found to be characteristic of the results for the other sites for which the resampling results led to a narrower distribution. Clearly the two approaches can lead to very different estimates for the confidence intervals, especially for the longer return periods.

he use of resampling for estimating confidence intervals 31 Single site analysis In this section, the characteristics of the resampling method for the estimation of confidence intervals at a single site are explored further. In the previous section of this paper, the method of L-moments was used to estimate extreme flow quantiles in order to allow a fair comparison between the resampling approach and the formula approach. In the remainder of this paper, the L-median approach (FEH, 1999) to quantile estimation is adopted. his choice is made here to be consistent with the approach taken in the pooled analysis that will follow. he basis for estimating extreme flow quantiles with the L-median approach is: Q = QMED x (3) where QMED is the estimate of the median flood for the site of interest and x is the estimated dimensionless growth curve for the site. Equation (3) represents an implementation of the so-called index flood approach. raditionally, the flow magnitude that is used for the index flood has been the mean of the annual flood series. However, FEH (1999) has adopted the median flood, QMED, as the index flood since estimates of QMED are more robust than are estimates of the mean annual flood. In the FEH approach, the growth curve is defined so that the 2-year growth factor is equal to 1, implying that the median of the growth curve distribution is 1. his is in contrast to a more traditional approach wherein it is the mean of the growth curve that is equal to 1. he parameters of the growth curve can be estimated by matching the sample L-moment ratios to the L-moment ratios for the selected distribution function and matching the sample median to the theoretical median (i.e. the 2-year event). In this work, the distribution function is selected based on a pooled analysis following procedures outlined in Hosking & Wallis (1997). Following this approach, the data series from the UK are seen to be well characterized by the Generalized Logistic (GLO) distribution, the Canadian data sets are observed to follow the GEV distribution, and the Italian data sets follow either the GLO distribution (two sites) or the GEV distribution (one site). he quantile function for the GLO distribution can be defined as: Q { ( ) } k α = ξ + 1 1 k 0 (4) k where ξ is the location parameter, α the scale parameter, and k the shape parameter. he median value is the value corresponding to a return period of 2 years (i.e. probability of exceedence of F = 0.5). Substituting = 2 in equation (4) results in (Robson & Reed, 1999): QMED = ξ (5) he GLO growth curve is then defined by substituting x = Q/QMED in equation (4) resulting in: x { 1 ( ) } k β = 1 + 1 k 0 (6) k

32 Donald H. Burn where β = α/ξ. he parameters k and β can be estimated from the sample L-moment ratios as (Robson & Reed, 1999): k = t 3 (7) t2k sin( πk) β = (8) kπ( k + t2 ) t2 sin( πk) where t 2 and t 3 are the sample L-moment ratios, which can be calculated as (Hosking & Wallis, 1997): l l t = 2 3 2 = t3 (9) l1 l2 where the l i are sample L-moments that can be calculated from the available data set (Hosking & Wallis, 1997). he L-median estimates for the parameters of the GEV distribution can be derived in a similar manner (Robson & Reed, 1999). L-median estimates for the parameters were used with the resampling approach for each of the data series presented in able 1. Selected results are presented in the section of the paper dealing with pooled frequency analysis, where the results for single site and pooled analysis are compared. In addition to calculating the confidence intervals for each site, an investigation of the impact of the record length on the confidence intervals, and the corresponding extreme flow quantiles, was performed. wo sites with long gauging records were examined. he sites selected were CA7 and I1 with record lengths of 52 and 74 years, respectively. For each site, the effect of varying the record length was evaluated by selecting three record lengths shorter than the entire record length and conducting an analysis on randomly selected data sets each with the same, shorter, record length. he process can be summarized as follows: 1. Randomly select a sample of size n s, where n s < n, the full record length. Random selection is without replacement so that a single extreme flow event can occur no more than once in an individual resampled data set. 2. For each sample, determine extreme flow quantiles, and the corresponding 95% confidence intervals, for a selection of return periods. he confidence intervals are estimated using the resampling approach with B = 999 resamples. 3. Repeat steps 1 and 2 a total of N times (in this work N = 500). 4. Determine, from the N values, the median of the extreme flow quantile estimates and the median for the upper and lower 95% confidence intervals. he median value is selected here because it is a robust measure of central tendency. Figure 3(a) and (b) displays the results for sites CA7 and I1, respectively. he three sets of lines show the extreme flow quantiles (the middle set of lines) and the upper and lower 95% confidence intervals. Figure 3(a) shows results for the 10-, 20-, 50-, 100-, and 200-year return periods for site CA7 that have been fitted to the GEV distribution. he available record length for the site is 52 years and subsets of lengths 30, 40, and 50 years have been selected. he results reveal that both the quantile estimates and the confidence intervals are quite sensitive to the available record length. he perils of estimating even the 50-year return period flow event from a data set with a record length of only 30 years are readily apparent. It is further observed that the uncertainty associated with the 100-year event estimated with a 30-year record length is roughly equivalent to the uncertainty associated with the 200-year event estimated

he use of resampling for estimating confidence intervals 33 (a) 400 300 200 Data Set Size of 30 Data Set Size of 40 Data Set Size of 50 100 (b) 0 4000 3000 2000 1000 10 20 50 100 200 Data Set Size of 30 Data Set Size of 50 Data Set Size of 70 0 10 20 50 100 200 Return Period Fig. 3 Sensitivity of at-site quantiles and confidence intervals to the record length for (a) site CA7 with an available record length of 52 years; and (b) site I1 with an available record length of 74 years. with a 50-year record. Finally, it is noted that the upper confidence interval exhibits greater variation with changes in the record length than is the case for the lower confidence interval. Figure 3(b) displays corresponding results for site I1 that have been fitted to the GLO distribution. he available record length for the site is 74 years and subsets of lengths 30, 50, and 70 years have been selected. his site displays minimal variability in the quantile estimates, but does exhibit record length sensitivity in the confidence intervals. he 70-year record length is seen to offer substantively narrower confidence intervals, even in comparison to the results from a record length of 50 years. Note that, even with a record length of 70 years, the uncertainty associated with the 100-year event, and especially the 200-year event, is quite large. Reducing these uncertainties is the intent of pooled frequency analysis, which is discussed below. Pooled frequency analysis A fundamental task in pooled frequency analysis is identifying the pooling group of gauged locations from which extreme flow information will be combined to derive a

34 Donald H. Burn quantile estimate for the site of interest. Many approaches to this task have been proposed including the definition of a focused pooling group (see, for example, Burn, 1990; FEH, 1999; Burn & Goel, 2000; Castellarin et al., 2001; Cunderlik & Burn, 2002). Most approaches involve determining a similarity measure that defines a distance, in an appropriate space, between the target site and every potential member of the pooling group. he pooling group is then defined to consist of all gauging stations that are sufficiently close to the target site. he FEH (1999) advocates setting a target number of station-years of data to include in a pooling group where the target number is a function of the return period of interest. he suggested guideline is to include 5 station-years of data, where is the return period of interest. he proximity of each station to the target site is then defined in a three-dimensional space with attributes consisting of the catchment area, catchment rainfall, and a measure of the soil type characteristics. he information from stations in the pooling group is combined using a weighted average approach as defined by (Robson & Reed, 1999): M i wi tr pooled i= 1 tr = r = 2, 3 (10) M w i= 1 i pooled where t r is a pooled L-moment ratio, M is the number of stations in the pooling group, and w i is a weighting factor defined by: w i = s i n i (11) where n i is the number of years of record for site i, and s i is a similarity ranking factor defined as: s M j= i i = M j= 1 n n j j where the denominator is the total number of station-years of record in the pooling group and the numerator gives the number of station-years in the pooling group provided by sites that are no more similar to the target site than is site i, where the sites have been ordered by their similarity to the target site. he pooling procedure noted-above is recommended in the FEH (1999) and was applied herein to the data sets in able 1 from the UK. For the remaining data sets, a similar approach was applied with the sole difference being in the attributes used to define the similarity between sites. For the Canadian data, similarity was defined in terms of seasonality measures described in Burn (1997), while for the Italian data similarity was defined in terms of different seasonality measures as described in Castellarin et al. (2001). Seasonality measures are defined in terms of the timing of flood and/or rainfall events and have been demonstrated to result in effective pooling groups. Selected results from applying the resampling approach to confidence interval estimates based on pooled frequency analysis are presented in Fig. 4(a) (c). (12)

he use of resampling for estimating confidence intervals 35 (a) 150 100 50 Observed Data At-site Fit Pooled Fit At-site 95% C.I. Pooled 95% C.I. 0 Return Period 2 5 10 20 50 100 200 (b) 300 250 200 150 100-1 0 1 2 Observed Data At-site Fit Pooled Fit At-site 95% C.I. Pooled 95% C.I. 3 4 5 50 0 Return Period 2 5 10 20 50 100 200-1 0 1 2 3 4 5 (c) 600 400 Observed Data At-site Fit Pooled Fit At-site 95% C.I. Pooled 95% C.I. 200 Return Period 2 5 10 20 50 100 200 0-4 -2 0 2 4 Reduced Variate Fig. 4 Comparison of at-site and pooled quantiles and confidence intervals for (a) site UK7, (b) site CA6, and (c) site CA7. Figure 4(a) presents the results for site UK7 from able 1. his site has a record length of 38 years, representing a reasonable amount of at-site information. he pooling group that is formed for this site was designed for estimating the 100-year extreme flow quantile and thus contains 500 station-years of data. he pooling group comprises

36 Donald H. Burn 18 stations, including the target site. he pooling group can be considered acceptably homogeneous in accordance with the H 2 criterion of Hosking & Wallis (1997) with H 2 = 0.17. Note that a value of H 2 < 1 signifies a pooling group that is acceptably homogeneous. Figure 4(a) reveals a good agreement between the at-site and pooled fit of the GLO distribution to the observed data. Both the at-site and pooled 95% confidence intervals are plotted in Fig. 4(a) with the pooled confidence intervals reflecting a considerable reduction in uncertainty associated with the extreme flow quantiles, especially for the longer return periods. Note that by definition, both estimates of the 2-year event are equal to the estimate of the at-site median flood, and therefore the estimated 95% confidence intervals, for the 2-year event, are also equal. Figure 4(b) presents the results for site CA6. his site has a record length of 33 years. he pooling group that is formed for this site was also designed for estimating the 100-year extreme flow quantile and contains 503 station-years of data. he pooling group comprises 12 stations, including the target site. he pooling group can be considered acceptably homogeneous with H 2 = 0.53. Figure 4(b) reveals a divergence between the at-site and pooled fit of the GEV distribution to the observed data with the pooled fit resulting in lower estimates for extreme flow quantiles for return periods longer than two years. he at-site fit is more heavily influenced by the largest observed flood event than is the pooled fit. he at-site and pooled 95% confidence intervals plotted in Fig. 4(b) again reveal that the pooled confidence intervals are considerably narrower with the differences again especially noticeable for the longer return period events. Figure 4(c) presents the results for site CA7. his site has a record length of 52 years. he pooling group that is formed for this site contains 12 stations, including the target site, and 507 station-years of data. he pooling group can be considered acceptably homogeneous with a value of H 2 = 1.09. Figure 4(c) reveals a slight divergence between the at-site and pooled fit of the GEV distribution to the observed data. he at-site and pooled 95% confidence intervals plotted in Fig. 4(c) again reveal that the pooled confidence intervals are considerably narrower with the differences especially noticeable for the longer return period events. his is true even though the available at-site record is 52 years, a record length that would be considered lengthy in comparison to what is available at many gauging stations. hese results emphasize the importance of adopting a pooled frequency approach especially for the estimation of quantiles with a return period that is close to or longer than the at-site record length. he final investigation with the pooled frequency analysis explores the sensitivity of the estimates for extreme flow quantiles and the 95% confidence intervals. As noted above, the pooling groups in this work were defined to have approximately 500 station-years of data. his section explores how the pooled fit and the confidence intervals change if the target number of station-years of data is increased or decreased by approximately 10%. An example of the results obtained is presented in Fig. 5 for site UK7. he original pooling group contained 500 station-years of data and the reduced and expanded pooling groups contained 441 and 559 station-years of data, respectively. Figure 5 displays the upper end of the extreme flow quantile vs return period plot for the original pooling group and a reduced and an expanded pooling group. he plot reveals only minor sensitivity to the pooling group size for the quantile plot and modest sensitivity for the 95% confidence intervals. he confidence intervals are narrower for the larger pooling group, particularly for the longer return period

he use of resampling for estimating confidence intervals 37 700 600 500 400 Reduced Pooling Group Fit Pooling Group Fit Expanded Pooling Group Fit Reduced Pooling Group C.I. Pooling Group C.I. Expanded Pooling Group C.I. 300 200 10 20 50 100 200 Return Period Fig. 5 Sensitivity of pooled quantiles and confidence intervals to the pooling group size for site UK7. events. his confirms the logic imbedded in the 5 guideline of the FEH (1999) that recommends increasing the pooling group size as the return period of interest increases. he results in Fig. 5, and the results for other stations that are not shown here, indicate that the precise size of the pooling group is not critical but that the general concept of an increased pooling group size for longer return periods is sound. his conclusion is apparent from the behaviour of the confidence intervals for the different sized pooling groups. CONCLUSIONS AND RECOMMENDAIONS A balanced resampling approach for estimating confidence intervals for extreme flow quantiles has been presented. he results from applying the approach to individual extreme flow data sets reveal that the resampling approach has advantages in comparison to the formula approach. he resampling approach does not require assumptions, is easy to implement, and, for the example data sets examined, resulted in either narrower confidence intervals or similar results to those from the application of an asymptotic formula. It appears as though the confidence intervals estimated using an asymptotic formula may tend to overestimate the uncertainty associated with the estimates of extreme flow quantiles. he resampling approach was applied to both single site records and in the context of a pooled frequency analysis. he results indicate considerable reductions in uncertainty associated with pooled frequency analysis, especially for longer return period events. he confidence intervals for single site analysis were used to explore the effect of the available record length on the uncertainty associated with extreme flow quantiles. he results revealed that short record lengths lead to considerable uncertainty in the estimates for quantiles with a long return period. Finally, the confidence interval estimates for the pooled analysis were used to demonstrate that pooled frequency analysis is preferred to single site analysis and that it is advantageous to increase the pooling group size when estimating longer return period events.

38 Donald H. Burn Acknowledgements he research described in this paper has been partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). his contribution is gratefully acknowledged. Helpful comments and suggestions from two reviewers have resulted in an improved manuscript. REFERENCES Adamowski, K. (1985) Nonparametric kernel estimates of flood frequencies. Water Resour. Res. 21(11), 1585 1590. Burn, D. H. (1990) Evaluation of regional flood frequency analysis with a region of influence approach. Water Resour. Res. 26(10), 2257 2265. Burn, D. H. (1997) Catchment similarity for regional flood frequency analysis using seasonality measures. J. Hydrol. 202, 212 230. Burn, D. H. & Goel, N. K. (2000) he formation of groups for regional flood frequency analysis. Hydrol. Sci. J. 45(1), 97 112. Carpenter, J. (1999) est inversion bootstrap confidence intervals. J. Roy. Statist. Soc. B61, 159 172. Castellarin, A., Burn, D. H. & Brath, A. (2001) Assessing the effectiveness of hydrological similarity measures for flood frequency analysis. J. Hydrol. 241, 270 285. Cunderlik, J. M. & Burn, D. H. (2002) he use of flood regime information in regional flood frequency analysis. Hydrol. Sci. J. 47(1), 77 92. Davison, A. C., Hinkley, D. V. & Schechtman, E. (1986) Efficient bootstrap simulation. Biometrika 73, 555 566. De Michele, C. & Rosso, R. (2001) Uncertainty assessment of regionalized flood frequency estimates. J. Hydrol. Engng ASCE 6(6), 453 459. Faulkner, D. S. & Jones, D. A. (1999) he FORGEX method of rainfall growth estimation, III: Examples and confidence intervals. Hydrol. Earth System Sci. 3(2), 205 212. FEH (1999) Flood Estimation Handbook. Institute of Hydrology, Wallingford, UK. GREHYS (1996) Inter-comparison of regional flood frequency procedures for Canadian rivers. J. Hydrol. 186, 85 103. Hosking, J. R. M. & Wallis, J. R. (1997) Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge University Press, Cambridge, UK. Kite, G. W. (1977) Frequency and Risk Analyses in Hydrology. Water Resources Publications, Littleton, Colorado, USA. Lu, L.-H. & Stedinger, J. R. (1992) Variance of two- and three-parameter GEV/PWM quantile estimators: formulae, confidence intervals, and a comparison. J. Hydrol. 138, 247 267. Reed, D. W. (1999) Flood Estimation Handbook, vol. 1: Overview. Institute of Hydrology, Wallingford, UK. Robson, A. J. & Reed, D. W. (1999) Flood Estimation Handbook, vol. 3: Statistical Procedures for Flood Frequency Estimation. Institute of Hydrology, Wallingford, UK. Stedinger, J. R., Vogel, R. M. & Foufoula-Georgiou, E. (1993) Frequency analysis of extreme events. In: Handbook of Hydrology (ed. by D. R. Maidment), Chapter 18. McGraw-Hill, New York, USA. Received 1 March 2002; accepted 30 September 2002