in regional partial duration series index-flood modeling

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1 WATER RESOURCES RESEARCH, VOL. 33, NO. 4, PAGES , APRIL 1997 Generalized least squares and empirical Bayes estimation in regional partial duration series index-flood modeling Henrik Madsen and Dan Rosbjerg Department of Hydrodynamics and Water Resources, Technical University of Denmark, Lyngby, Denmark Abstract. A regional estimation procedure that combines the index-flood concept with an empirical Bayes method for inferring regional information is introduced. The model is based on the partial duration series approach with generalized Pareto (GP) distributed exceedances. The prior information of the model parameters is inferred from regional data using generalized least squares (GLS) regression. Two different Bayesian T-year event estimators are introduced: a linear estimator that requires only some moments of the prior distributions to be specified and a parametric estimator that is based on specified families of prior distributions. The regional method is applied to flood records from 48 New Zealand catchments. In the case of a strongly heterogeneous intersite correlation structure, the GLS procedure provides a more efficient estimate of the regional GP shape parameter as compared to the usually applied weighted regional average. If intersite dependence is ignored, the uncertainty of the regional estimator may be seriously underestimated and erroneous conclusions with respect to regional homogeneity may be drawn. The GLS procedure is shown to provide a general framework for a reliable evaluation of parameter uncertainty as well as for an objective appraisal of regional homogeneity. A comparison of the two different Bayesian T-year event estimators reveals that generally the simple linear estimator is adequate. Introduction is closely related to the AMS/GEV index-flood model since application of the GP distribution for modeling exceedance Recent research on frequency analysis of extreme hydrologic magnitudes together with the assumption of a Poisson distribevents has proven that the combination of hydrologic informa- uted number of threshold exceedances implies the annual maxtion from different sites in a region provides a viable means for ima to be GEV distributed with the same shape parameter as improving quantile estimates at a specific site. The use of in the GP distribution [e.g., Madsen et al., this issue (a)]. Madregional information reduces the sampling errors, and, in ad- sen et al. [this issue (b)] compared the AMS/GEV and PDS/GP dition, it facilitates the choice of an appropriate statistical regional estimation procedures and found that the PDS/GP distribution. Furthermore, regionalization forms the basis for model is more robust with respect to violation of the basic making inferences at ungauged sites. homogeneity assumption of the index-flood method. A widely used regional estimation method for annual max- Another way of inferring regional information is the empirimum series (AMS) is the index-flood method. In this method ical Bayes method. In the Bayesian approach, beliefs or knowlthe data at different sites in the region are divided by the at-site edge about the parameters of a specified extreme value distriscale parameter (index-flood parameter), and the normalized bution are expressed in terms of a probability distribution, the data are jointly used to estimate the parameters of the regional prior distribution. Using Bayes' theorem, the prior information distribution. The at-site quantile estimator is then obtained by is combined with site-specific information to obtain an updated multiplying the regional normalized quantile estimator by an distribution, the posterior distribution. While Bayesian inferestimate of the site-specific index-flood parameter. For estima- ence generally allows the prior distribution to be determined tion of the regional parameters, Wallis [1980] and Greis and by subjective means, empirical Bayes theory is more objective Wood [1981] proposed a procedure based on regional averages in the sense of inferring the prior parameters from empirical of normalized probability weighted moments (PWM). The data [Kuczera, 1982a]. In a regional context the prior param- PWM index-flood algorithm with a generalized extreme value eters for a specific location are estimated from regional data in (GEV) distribution for annual floods has been advocated in terms of either observations (data-based model) or physical charseveral regional studies [e.g., Hosking et al., 1985; Wallis and acteristics (regression model) at the other sites in the region. Wood, 1985; Lettenmaier et al., 1987; Hosking and Wallis, 1988; Most applications of the empirical Bayes theory in regional Potter and Lettenmaier, 1990; Stedinger and Lu, 1995]. Reanalyses of annual floods have used regional regression models cently, Madsen and Rosbjerg [this issue] introduced an indexthat relate prior parameters to catchment characteristics. For flood method based on partial duration series (PDS) with instance, Cunnane and Nash [1971] presented an empirical generalized Pareto (GP) distributed exceedances. This model Bayesian T-year event estimator based on the Gumbel distribution where the mean and the coefficient of variation were 1Also at Danish Hydraulic Institute, H0rsholm, Denmark. Copyright 1997 by the American Geophysical Union. Paper number 96WR /97/96WR expressed in terms of catchment area, average annual rainfall, and catchment slope. Kuczera [1982a] compared a data based model and a regional regression model for inferring the log variance in the lognormal distribution and concluded, in that

2 772 MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING particular application example, that the regression model was Regional Index-Flood Estimator superior. Rosbjerg and Madsen [1995] applied Kuczera's model The PDS regional index-flood model with GP distributed and found that a data-based prior distribution implied an efexceedances was introduced by Madsen and Rosbjerg [this isficient T-year event estimator as compared to other regional sue]. A brief outline of this model is given in the following. estimators, especially when regional homogeneity may be Denote by X the stochastic variable of the exceedance magniquestioned. tude. It is assumed that X follows the GP distribution with In PDS modeling, Rousselle and Hindie [1976] introduced a probability density function Bayesian estimator for the classical PDS model with exponentially distributed exceedances. This model was also considered by Rasmussen and Rosbjerg [1991], who expressed the prior f(x)=/x(l+k) 1-K /z(1 + (1) knowledge of the scale parameter in the exponential distribuwhere/z is the mean value and c is the shape parameter. The tion in terms of catchment characteristics, whereas a noninforrange ofxis 0 -< x < c for c-< 0, and 0 -< x -< mative prior was adopted for the Poisson parameter. Recently, for c > 0. For c = 0 the exponential distribution is obtained. Madsen et al. [1994, 1995] presented a Bayesian PDS model The annual number of threshold exceedances is assumed to be based on the GP distribution. They applied the model in a Poisson distributed with parameter ;t that equals the expected regional analysis of extreme precipitations and used datanumber of exceedances per year. based prior distributions for both the scale and the shape Consider now a region of M sites with PDS records xo, parameter in the GP distribution as well as for the Poisson where i = 1, 2,..., M;j - 1, 2,..., Ni; andn is the parameter. number of observations in t years of recording at site i. The Although the above cited Bayesian models incorporate the regional T-year event estimator is given by index-flood concept of scaling, an important distinction between index-flood and empirical Bayes estimators must be emphasized. The basic hypothesis of the index-flood method is = g, (2) that data at different sites in the region are identical, except for scale. This homogeneity assumption implies that the coefficient where is the at-site sample mean, i = Ni/t is the at-site estimate of the Poisson parameter, and kr is a regional estiof variation (Cv) and higher order dimensionless moments (or mate of the shape parameter. Madsen and Rosbjerg [this issue] distribution parameters related to these moments) are condetermined kr on the basis of regional average L-moment stant in the region. The empirical Bayes method does not estimates. In the present analysis, allowing a more convenient assume strict homogeneity with respecto the regional paramparameterization of the Bayesian T-year event estimator, a eters. In fact, specifying a prior distribution for these paramslightly modified estimator is adopted based on a regional eters acknowledgesome degree of heterogeneity. The indexweighted average of k, that is, flood method may thus be treated as a special case of a general empirical Bayes model, that is, when the prior distribution is a delta function concentrating all its probability mass in a single k = w k w (3) point. i=1 i=1 In practice, the homogeneity assumption of the index-flood where w is the weight, usually taken to be equal to the sample method is almost certain not to hold. For instance, there seems size, and to be some evidence that C v of annual floods varies with catchment area, and this variation can be modeled by multi- 1 scaling theory [Smith, 1992; Gupta et al., 1994]. In this context, = 2 T2i (4) the Bayesian method may be seen as an alternative approach to meet the same objective of describing the regional variabil- In (4), r 2 is the L coefficient of variation (L-C,, which is ity of C [Ribeiro-Corr a et al., 1995]. In general, however, it estimated on the basis of unbiased PWM estimators [Landwehr may be difficulto verify any structured variation of a regional et al., 1979; Hosking and Wallis, 1995]. The T-year event estiparameter from empirical data because of sampling errors. In mator of the basic variable is finally obtained as /ri = œr + q0i where q oi is the threshold level. such cases an adequate estimator of the regional parameter is The problem of threshold selection for the regional PDS given by the regional average of the at-site parameter estimamodel was discussed by Madsen and Rosbjerg [this issue], who tors, and the prior distribution then reflects the uncertainty of recommended the use of a standardized regional procedure the regional average. Thus the Bayesian method offers a genrather than using site-specific and more subjective methods. In eral framework for an assessment of the uncertainty of a rea regional flood study in New Zealand, Madsen et al. [this issue gional estimator. (b)] determined the threshold level as a certain quantile of the The objective of the present study is to introduce and evaldaily flow duration curve. The resulting PDS records were uate the performance of an empirical Bayes estimation proce- shown to better capture the underlying structure of flood series dure for the PDS regional index-flood model based on GP with respect to describing differences in flood characteristics distributed exceedances. The prior distributions of the PDS between regions as compared to the AMS records. The parameters are inferred from regional data using generalized method adopted by Madsen et al. [this issue (b)] is used to least squares regression in order to take sampling errors and define the PDS in the application example presented below. intersite dependence into account. Combination of prior and site-specific information for estimating T-year events is illustrated by an application example based on PDS flood records from 48 catchments in New Zealand. The index-flood method presumes that c is constant in the region. Madsen and Rosbjerg [this issue] analyzed the effect of heterogeneity by comparing the performance of the regional T-year event estimator with that of a site-specific estimator.

3 MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING 773 The analysis revealed that the regional estimator is more efficient than the at-site estimator even in strongly heterogeneous regions. This conclusion, however, is based on a regional average performance index, and sites with characteristicsignificantly different from the average site will perform worse than this index indicates. In fact, regional information may be counterproductive at these sites. The empirical Bayes theory provides an estimator that makes it possible to take regional heterogeneity into account and hence produce more efficient T-year event estimators. Another aspect to be considered concerns the estimation at ungauged sites. In this case also the threshold level and the site Estimation of Prior Parameters where i = Oi '-{" ei i = 1, 2,..., M (5) cov{ e, e E{ei} = 0 j} = cr 2 i=j i - j 0-,i0-,jP,ij (6) 2 is the at-site sampling error variance of i, and P ii In (6), 0- i is the sampling error correlation coefficient due to concurrent observations stations i and j (intersite correlation). It is assumed that Oi can be determined from physical characteristics by the following linear relationship: p Oi = [30'-[- E [3kAik '-{" ai i = 1, 2,..., M (7) k=l where A ik (k = 1, 2,..., p) are the considered physical characteristics and 15is the error term owing to lack of fit of the regression model (model error). The properties of 8 i read E{ai} = 0 0-a i =j cov {8i, 8 } = 0 i4:j where 0-2 a is the model error variance. The elements of the specific PDS parameters / and X have to be inferred from covariance matrix A of the total errors r i = ei + 8 are given regional data. If the threshold level is determined as a certain by quantile of the daily flow duration curve, the method proposed 2 by Fennessey and Vogel [1990] can be used to estimate the Aij = COV {Tli, Tlj} "- 0-ei '- 0'28 i -j threshold level from physical data. They approximated the (9) empirical flow duration curve by a lognormal distribution and A i = coy {r, r j} = 0' i0' ip ii i j estimated the two lognormal parameters from catchment area Note that the assumption of homoscedasticity in OLS regresand a basin relief parameter. The/ and X parameters can also sion requires both 0' 2 and 0'2 a to be independent of i. In the be estimated from physical data. For instance, since the mean GLS procedure, only 0'2 a is assumed to be independent of i. of the exceedances in flood analysis is a typical scale parame- In matrix notation the system of equations (5) and (7) can be ter, it is to be expected that catchment area explains a large written as O = X/3 + r where part of the regional variability of g. In a Bayesian context the relationships between/ and X and physical characteristics in the region are inferred as prior information. At gauged sites this information is then combined with the site-specific inforo = = mation. r = (r r 2' r M) r (1 1 Am All''' A The GLS estimator of/3 is determined by solving Prior information of the PDS parameters,, and X are inferred from regional data. Generalized least squares (GLS) regression provides an efficient method for estimating the moments of the prior distribution. When the residuals of a regres- = ( sion model are heteroscedastic and cross-correlated, the GLS where 0'2a, using the method of moments estimation, is obmethod provides more accurate estimates of the regression tained from the solution of model parameters than the ordinary least squares (OLS) procedure [Stedinger and Tasker, 1985], and, in addition, it pro- (t9 - X/3)rA- (O - X/3) = m - p - 1 (12) duces a reasonable and nearly unbiased estimator of the model The solution of (11) and (12) requires an iterative scheme. In error variance [Stedinger and Tasker, 1986]. In this section, the some cases one may find that no positive value of 0'2 a can satisfy GLS regression model is described. Particular emphasis is (11) and (12). In these instances the sampling errors more than given to the important special case of a regional mean model, account for the difference between t9 and X/3, and 0'2 a is then corresponding to the above mentioned data-based model. taken to be zero. Having estimated the model parameters ( ) GLS Regression Model and the model error variance (&2a), the prior mean and variance of 0i can be taken as Denote by Oi a PDS parameter at station i. The estimator b i is subjected to a random sampling error s i, that is, a0, = x,3 a0, = + (8) ( 0) where x]' is the ith row in X, and ( ) = [XrA-1X]-1 is the covariance matrix of the estimated model parameters. The prior variance in (13) is the mean square error of prediction which includes both the sampling uncertainty of the estimated regression model parameters and the model error variance. When the model error is large compared to the sampling errors, the GLS solution is close to that obtained by OLS regression. If p i = 0, the GLS procedure corresponds to a weighted least squares (WLS) approach [Tasker, 1980]. For modest intersite correlations, the WLS and GLS procedures are essentially similar [Stedinger and Tasker, 1985]. In hydrologic applications a log linear model is often applied, and in

4 774 MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING this case regression is based on In ( i) versus In (Aik). The prior mean and variance of In (0i) is obtained from (13), and assuming that In (Oi) is lognormally distributed, these mo- N. = int, i=1 Ni } 14) where var { fl,4 N,4 } is the at-site sampling error variance using In (19) the first term of &20 can be interpreted as the variability the regional average of i and the average number of obser- due to regional heterogeneity, whereas the second term repvations. Formulae for calculation of var { fi. ln. } for k and resents the sampling variability of the mean valuestimator/3o are given by Madsen and Rosbjerg [this issue]. For the mean corrected for intersite correlation. In practice, if no outlying value a log linear regression model is employed. The sampling values of at-site sampling error variances or intersite correlaerror variance of In (fii) reads tion coefficients are present, the homogeneous error structure can be approximately achieved by letting ^ 2 equal the average I C of 2 and letting e equal the average of )ei. Stedinger and 00 2 i = var {In( /)} '/_1,/2 var { i} Ni (15) Tasker [1986] determined the model error variance estimator in (18) as a bias correction for the OLS estimator &2 = s 2. and hence in this case an appropriate estimator of frei 2 is given Recently, Mikkelsen et al. [1996] extended the regional mean by model with approximately homogenousampling errors by including geostatistical correlation, that is, coy { 8, 8 } = pa 0.2 _ 2 for i j in (8), where the correlation coefficients Pai are ei -- Ni estimated by variogram analysis corrected for sampling errors (16) and intersite d pendence. i M Estimation of the correlation coefficients in the covariance matrix is performed as follows. Since X is the mean annual number of exceedances, the intersite correlation coefficient between estimated X values can be calculated as the correlation coefficient between the number of exceedances in concurrent years. For In (/z) and k the intersite correlation coefficients can be expressed in terms of the correlation coefficient between concurrent exceedances Pii as, respectively, cor {ln (fii), i=1 Regional Mean Model The special case of a regional mean model arises when only the intercept /30 is included in the regression equation (see 2 ments are easily converted into moments of O i. (7)). In the case of homogeneousampling errors, that is, 00ei Application of the GLS procedure requires an estimate of e 2 and P e ij =P e, where i = 1, 2, *'*, M and j = 1, the sampling error covariance matrix. This estimator should be 2,..., M, an explicit solution of (11) and (12) exists [Madsen independent, or nearly so, of b i [Stedinger and Tasker, 1985]. et al., 1994]: Since O' 2i for i and X i is expressed in terms of the population values (see work by Madsen and Rosbjerg [this issue] for de- 1 M t ails), application of the variance formulae by inserting i and 0-'- i (17) i=1 Xi seriously violates this independence criterion. Following the approach by Tasker [1980!, a reasonable estimator of 00ei 2 that is nearly independent of 0i is given by max {0; s 2- (1 - (18) 2 N. var {. N. } 00 ei -- Ni S 2 1 M M_ 1 (fii-] 0)2 i=1 Estimates of the prior mean and variance are obtained from (13) as M+i 1 M 1)] In (fii)} cor {jcli, t j} = Pij and cor { i, j} = pi2j which corresponds to the weighted index-flood estimator given [Stedinger, 1983; Madsen and Rosbjerg, this issue]. Estimation by (3) with weights equal to wi = [00 2i ]-1. Assuming of Pi is complicated by the fact that observations occur at that 00e 2 = c/ni (see(14) and (16)), wi in the case of regional irregularly times. An estimation procedure that deals with this homogeneity (002 = 0) is given by wi = Ni, which is the weight problem is given by Mikkelsen et al. [1996]. Tasker and Stedinger usually adopted in index-flood modeling. In the case of re- [1989] argued that the use of samplestimates of Pi due to the gional heterogeneity, if the model error variance is expressed large sampling uncertainties often encountered in practice may as 002 = c/nr ' one obtains W i = NiNR/(N i + NR). This result in a matrix A that cannot be inverted. To overcome this weight was proposed by Stedinger et al. [1993] in order to limit problem, they proposed a method based on smoothing the the weight assigned to sites with very long records since such estimates of Pi by relating them to the distance between gaug- sites by using w i = Ni may have undue influence on the ing stations. This method is also adopted here. regional estimator. (19) In general, rio is not equal to the simple regional average because the GLS algorithm weights the estimated parameters according to the covariance matrk of the errors A. For instance, the WLS case (p ; = 0) the estimator reads M 0i[ 00 i q- 0028] -1 f0-.-1 (20) M 2 [00ei-I" 0028]-1 i=1

5 MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING 775 Important to note at this point is that the regional K estima- b = s b a + (1 - s)b ^s tor in the index-flood approach (see(3)) is a special case of a (22) general regional model based on GLS regression. In the case var {b AS} s-- of regional heterogeneity and intersite dependence, the re- var { AS} + var { REa} gional GLS estimator may be virtually identical to that obwith variance tained from (3); however, the uncertainty of the estimator, by disregarding heterogeneity and intersite dependence, may be var { b e } = s var { b ea} (23) significantly underestimated (see work by Rosbjerg and Madsen [1996] for a discussion of this aspect). Furthermore, if the The weighting factor s (often denoted the shrinkage factor) regional variability can be modeled from physical characteris- expresses the relative weight assigned to, respectively, regional tics, the GLS regression model provides a more et cient esti- and at-site information, depending on the uncertainties of the mate of K than the regional weighted average value. Finally, it two information sources. The estimator in (22) is an exact should be noted that the regional mean model estimator of o-2 Bayesian estimator only for normal prior and sample distribucan be interpreted as a heterogeneity measure; that is, if?r2 > 0, tions. However, under quadratic loss, (22) is the best linear the hypothesis of regional homogeneity may be questioned. estimator irrespective of the type of prior and sample distributions [Kuczera, 1983]. The posterior T-year event estimator œ.b and the associated variance var { œ B) are obtained from Regional T-Year Event Estimators (2) and (21) by inserting the pooled parameter estimators and variances given by (22) and (23). Madsen and Rosbjerg [this The estimated prior mean and variance of the PDS paramissue] provided expressions of var {k As) and var { AS), and eters derived in the previous section form the basis for incluvar {b As) = o'2/n where 0 '2 is the variance of the exceedances. sion of regional information in T-year event estimation at both The posterior variance var { E ) in (23) is smaller than gauged and ungauged sites. At gauged sites the Bayesian proeither var { s) or var { REO), implying that pooling at site cedure provides an updating scheme in which prior and sample and regional information always lead to more precise estimainformation of 'the PDS parameters are combined using Bayes' tors. In practice, however, var { AS) and var { REO) have to be theorem. In this section, two different T-year event estimators inferred from observed data, and if one source of information are described: a linear Bayes estimator that requires that only has a large uncertainty relative to the other, pooling may be some moments of the prior distributions are specified and a counterproductive. For instance, Kuczera [1982b] observed parametric Bayes estimator in which the families of prior disthat in the case of small at-site records, estimation methods tributions have to be defined. The estimators refer to a specific that use only regional information are superior to methods that site; however, for the sake of simplicity, index i is omitted in pool regional and at-site information. Kuczera [1983] studied the following. Theoretically, application of Bayes' theorem dethe effect of sampling uncertainty on the pooled estimator in mands prior and sample information to be independent, imdetail and provided the following indicator for pooling inforplying that the Prior information at the site being considered mation: should be obtained from the (M - 1) other sites in the region. In practice, however, the prior information is usually inferred [ b AS breg] 2 once using all M sites in the region and is then combined with a= var.. { bas} + ' r {'breo}--i (24) the sample information at the specific site. If A < 0, pooling is preferable, whereas for A > 0 pooling may Linear Bayes Estimator be counterproductive, especially if A is large and the estimates To make inferences at ungauged sites, the prior T-year event estimator and the associated uncertainty is determined on the basis of the estimated prior moments of the PDS parameters (denoted b REø and var {breo in the following). The T-year event estimator œt REa is obtained from (2) by inserting fr REa, k REa, and REa. An approximate expression of the variance of (OXT 2 (OXT 2 var (œt --, - j var ( +, 0h J var ( +, - / var (21) An estimate of the variance OfœT REG is then obtained from (21) by inserting the estimated prior moments. An empirical linear Bayes estimator of the PDS parameters that combines prior and site specific information (quantified by AS and var { AS}) reads of var {b As} and var {firea} are very uncertain. In this case, and if s is close to 1, only regional information should be used, and if s is close to 0, the at-site estimator is preferable. In the former case, A can be interpreted as a discordancy measure to identify those sites that diverge significantly from the group as a whole. œr can be obtained from a Taylor series expansion of (2). If a regional standardized procedure is used to determine the Parametric Bayes Estimator threshold level, it is reasonable to assume that the flood trig- In the parametric Bayesian approach, the PDS parameters gering process is independent of the distributional character- Ix,, and A are treated as stochastic variables, each with a istics of the peak magnitudes; that is, the regional distribution specified family of prior distribution. In the following the noof A can be assumed independent of the regional distributions tation (m, k, ) is used for possible realizations of (/x,, A). of/x and. Furthermore, the index-flood concept of scaling The parameters/x, c, and A are described by, respectively, an prescribes the regional properties of/x and to be indepen- inverse gamma, a beta, and a gamma distribution [Madsen et dent. Thus a first-order Taylor series expansion of (2) yields al., 1994]: 1 (26) fk(k) = p{ )p( j k+ -k T f, ( e ) = r- ( e T)v-1 exp (- t r) (27)

6 776 MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING Since the GP distribution has an infinite variance for K = -1/2 and since it reduces to the triangular distribution for K = 1/2, the variation of is restricted to the interval -1/2 < < 1/2. The prior parameters (/3, %,, v, r) are estimated on the basis of the prior mean and variance obtained from GLS regression. The prior distribution of the T-year event x r is deduced by a change of variables: LCx) = j where 2 the transformation f (m) f (k) f (f) Im--g(x) d ' dk m = #(x) (28) =,] (29) is obtained from (2). The integral in (28) has to be solved numerically. A point estimator œr REc and the associated uncertainty var {œr REc} can be determined as the mean and the variance in the prior x r distribution. Posterior distributions of the PDS parameters are obtained by combining the prior distribution and the sample likelihood function using Bayes' theorem. Since a conjugate prior distribution is adopted for the 3 parameter, the posterior distribution of 3, f[(e), is also a gamma distribution with updated parameters ld t -- ld -3- N and T t --' T -3- t. The posterior probability density function of (tz, ) using Bayes' theorem reads where f (m) f (k)l, (m, k) f, (m, k): (30) d -1/2 N 1 ( i=1 Xt) (1/k)-I l, (m ' k)=l--[ m(l+k) 1-k m(1 + k) (31) given by Mikkelsen et al. [1996] was applied. Their approach requires the definition of concurrent exceedances. For any pair of stations, stations A and B, with observations x.4i, where i -- 1, 2,..., N.4; and xbi, where j = 1, 2,..., NB, the sample of concurrent exceedances was defined as (XAi' XBj)' tei ---2-; tei q-- f-) tej ---2-; tej :/= 0 these distributions are given by Hosking [1990] and Stedinger et al. [1993]. Hosking [1991] and Vogel and lgqlson [1996] provided polynomial approximations which are sufficiently accuis the sample likelihood function of (tz, ). For k > 0 the GP rate in most applications. The L-moment ratio diagrams show distribution has an upper bound, and lg,,(m, k) is given by two virtually distinct groups of points corresponding to each (31) only if Vxi: xi -< m(1 + k)/k. Otherwise, lg,,(m, k) is region. Region A stations have generally higher L-C v, L skewequal to zero. Finally, by substituting fx( ) with f[(f) and ness, and L kurtosis than region B stations. The record-lengthfg(m)f,(k) withf,,(m, k) in (28), the posterior distribution,, EB weighted average point of region A is close to the GP line and of x r is obtained. The posterior T-year event estimator x r indicates a distribution with a very small (negative) shape paand the associated variance var {œ½b} are determined as the rameter, whereas the weighted average of region B is very close mean and the variance in the posterior xr distribution. to the EXP point (K = 0). Application of Hosking and Wallis' [1993] homogeneity test based on L-Cv reveals that all 48 Application The regional estimation procedure was applied to flood stations form a very heterogeneous region, while the division with respect to AAR yields two acceptably homogeneous regions (see Table 1). For both regions, Hosking and Wallis' records from 48 New Zealand South Island catchmerits with [1993] goodness-of-fit test indicates that the GP distribution is recording periods ranging from 21 to 42 years. The PDS records were defined by using the 2% quarttile of the daily flow duration curve as the threshold level. To ensure independence, only those exceedances were retained that fulfilled the United States Water Resources Council [1982] recommendations of (1) a separation distance between two successive peak flows of at least 5 + In (AREA) days, where AREA is the catchment area in square miles, and (2) an interevent discharge below 75% of the lowest of the two peaks. adequate. Note that to discriminate between various twoparameter distributional alternatives, which is common in PDS analysis, the L-C JL skewness diagram is sufficient, and, as can be seen in Figure 1, it is easier to interpret than the L skewness/l kurtosis diagram. Also note that Hosking and Wallis' [1993] goodness-of-fitest, which is based on the difference between the regional average L kurtosis and the L kurtosis of the fitted distribution, is designed for three-parameter distributions. A goodness-of-fit measure for a two-parameter distri- To estimate the intersite correlation structure, the method bution should rather be based on the difference between the A i- I, 2,...,NA, j= I, 2,...,NB B (32) where t e is the date of occurrence of the peak flow and At is a lag time that is introduced to account for, respectively, moving weather patterns and different rainfall-runoff relationships of the catchments. In this study At = 5 days was used. From a physical point of view, intersite correlations, if any, are expected to be positive, and hence negative estimates of the correlation coefficient were set to zero. Initially, the 48 catchments were divided into two regions according to their annual average rainfall (AAR) [see Madsen et al., this issue (b)]. Low-AAR stations (AAR < 1300 mm) form a region of 18 sites (hereafter denoted region A) where all but one are situated east of the main divide (a mountain range running southwest to northeast along the island). Of the 30 high-aar stations (AAR _> 1300 mm) in region B, 18 are situated west and 12 east of the main divide. Especially in the southwestern part of the island, very high amounts of AAR are observed with AAR = 7400 mm being the maximum. In Figure 1 estimated L-moment ratios, respectively, L-Cv versus L skewness and L skewness versus L kurtosis, are shown together with the theoretical relationships for a number of parent distributions that have been proposed for modeling exceedances in PDS. These include the GP, the lognormal (LN), the gamma (GAM), the Weibull (WEI), and the exponential (EXP) distributions (note that the GP, GAM, and WEI distributions all include the EXP distribution as a special case). The theoretical relations between the L-moment ratios for

7 . MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING /,,oo (a) ' ) / o' REGAobs.,,, REGA era 0.60 /',,,' / REG Bobs. /, REGB era ",- --GP o.o '. J 5' / '... WEI fi' $ /// EXP o L-$kewness 0.40 [] ""' a REG A obs. [] [] ooo øø REG A average a r-i" *. "* o REG Bobs., 0.30 ".""."' REG B average []O* * []. *..* * --GP... LN (b)... GAM... WEI EXP L-Skewness Figure 1. L-moment ratio estimates of region A and region B data compared to the theoretical relationships for the generalized Pareto (GP), lognormal (LN), gamma (GAM), Weibull (WEI), and exponential (EXP) distributions. (a) L-C, versus L skewness. (b) L skewness versus L kurtosis. regional average L skewness and the L skewness of the fitted number of exceedances in concurrent years), the correlationdistribution. distance relationship has no apparent trend. In this case a In Table 1 summary statistics for the two regions are given homogeneous correlation structure was found to be appropritogether with those obtained when all 48 stations are consid- ate; that is, p - where is the regional average correlation ered jointly. These results also supporthe division into two coefficient (see Table 1). regions on the basis of AAR. The two regions are seen to differ Regional regression analyses were performed for In (/ ), K, not only with respec to the shape parameter but also with and X in the two separate regions. Available catchment charrespecto the Poisson rate, the average rate being largest in acteristics included geologic, physiographic, and meteorologiregion B. Moreover, the average intersite correlation is larger cal characteristics, soil properties, and land-use parameters in the two regions than when all stations are considered jointly. (see Table 2). Three different regression models were em- The correlation structure within each region was analyzed in ployed: (1) WLS regression, (2) approximate GLS regression more detail by relating the estimated correlation coefficients to using a regional average correlation coefficient (denoted GLS1 the distance between stations. First, the correlation between in the following), and (3) GLS regression where the intersite concurrent exceedances is considered. In region A (see Figure correlation coefficient is related to the distance between sta- 2a) there is a strong correlation-distance relationship, the cor- tions (GLS2). The regression procedures were applied includrelation coefficient being a decreasing function of the distance. ing all combinations of the explanatory variables. The final In region B (see Figure 2b), the relationship is more scattered, choice of regression model was made by using the average which to some extent can be explained by the bad correlation mean square error of prediction (see(13)) as a performance between east and west coast stations. To be used in the GLS index; that is, the model that produced the lowest value of regression analyses, in both regions a function of the form p = ;6-20i was generally chosen. However, because of the principle exp (- ado), where d is the distance, was fitted by eye to the of parameter parsimony, if only a minor improvement was observed data. With respect to the intersite correlation be- obtained by including an additional catchment characteristic, tween estimated h values (i.e., the correlation between the that characteristic was not included. Since focus in this study is

8 778 MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING Table 1. Summary Statistics Region All Stations A B (a) Region A o Number of stations Average number of observations Mean (Iz) Average correlation coefficient Shape Parameter (K) Mean Standard deviation Average correlation coefficient Poisson Parameter ()t) 0.0 Mean Standard deviation Average correlation coefficient Heterogeneity measure H* *Based on the regional variability of L-Cv. H < 1, acceptably homogeneous; I -< H < 2, possibly heterogenous; H > 2, definitely heterogeneous [Hosking and Wallis, 1993]. o 0.8 :.%' :. '1 *., *...,....* _. =._._... _-. -,'.. 1 '.-. 0! Distance between (b) Region B stations on medium and large quantile estimation (T > 10 years), the variability of the h parameter has only a small impact on the uncertainty of the T-year event estimator, and hence no attempt was made to describe the regional variability of h from catchment characteristics, that is, a regional mean model was chosen in this case. In region A, one catchment (71122 Maryburn) was found to differ significantly from the group as a whole and was excluded in the regression analyses of In (/x). The catchment, which was also identified as an outlier in a previous study of mean annual floods by McKerchar [1991], has a remarkably low flood response compared to other catchments, which is mainly due to its very high storage capacity (high HG value). Moreover, it is virtually uncorrelated to all other catchments in region A. The results from the regression analyses are shown in Table 3. With respect to regression of In (/x), AREA is the most important variable in both regions followed by AAR, whereas slope (S) is included as the third variable in region A only. The regression coefficients in the two regions differ significantly, a result that supports the division of the 48 stations into the two regions. It is seen that WLS and GLS1 regressions yield virtually identical regression coefficients, and the only difference between the two approaches is the estimates of the model error variance and the prior variances. When the correlation structure is taken into account (GLS2), slightly different regression coefficients are obtained. Interestingly, intersite correlation has the most pronounced impact on the regression results in region A, although region B has the largest average correlation coefficient. Also shown in Table 3 is the average shrinkage factor of the linear Bayes estimator of/x (see(22)), which can be interpreted as a measure of the relative gain of including regional information. The gain can also be expressed in terms of an equivalent sample size, that is, the sample size that would provide a sample variance of/x equal to the prediction error variance of the regression equation. The equivalent sample size N E expressed in terms of the shrinkage factor s and the at-site sample size N is obtained from (22) as Nœ = Ns/(1 - s). Inserting the average number of observations from Table 1 and the average shrinkage factor of the GLS2 regression from Table 3, one obtains Nœ 14 in both regions, corresponding O Distance between stations Figure 2. Estimated correlation coefficients of concurrent exceedances as a function of the distance between stations. (a) Region A. (b) Region B. to 4-5 years of record. For comparison, Hebson and Cunnane [1987] found that the mean annual flood estimated by regional regression is less precise than that estimated from only! year of record. With respect to regression of K, a regional mean model was found appropriate in all cases. The results from the WLS regression indicate that the two regions are homogeneous (model error variance equal to zero), which is consistent with Table 2. Catchment Characteristics Characteristic Notation Catchment area, km 2 AREA Average annual rainfall, mm AAR Soil drainage index* D Depth-weighted macroporosity, % DWP Average elevation, m ELEV Proportion of bare land, % BL Hydrogeology index ' HG Minimum porosity, % MP Average catchment slope, deg S Vegetation index: VEG *Ranging from 1 for very poor drainage to 7 for excessive drainage. 'Ranging from 1 for low to 8 for high bedrock infiltration capacity and transmissibility. $Ranging from 1 for low to 2 for high vegetation. Areas with no vegetation are not included in the calculation of VEG.

9 MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING 779 Table 3. Results of the Regression Analyses Parameter Type* Regression Equation Model Average Average Error Prior Shrinkage Variance Variance Factor Region A /x WLS In (/x) = In (AREA) x x In (AAR) In (S) GLS1 In (/x) = In (AREA) x x In (AAR) In (S) GLS2 In (/x) In (AREA) x x In (AAR) In (S) ( WLS ( = X GLS1 ( = X GLS2 K = X X WLS X GLS1 X GLS2 X = Region B /x WLS In (/x) = In (AREA) x x In (AAR) GLS1 In (/x) = In (AREA) x x In (AAR) GLS2 In (/x) In (AREA) x x In (AAR) ( WLS ( = X GLS1 ( = X X GLS2 ( = X X X WLS X = GLS1 X = GLS2 X = *WLS, weighted least squares regression; GLS1, approximate generalized least squares regression using a regional average correlation coefficient; GLS2, generalized least squares regression based on the correlation-distance relationship. Hosking and Wallis' [1993] homogeneity test. In region A the equivalent recording period of 260 years in region A and 40 presence of intersite correlation is seen to have a significant years in region B. impact on the results. On the basis of the average correlation coefficient, one would not expect correlation to be a problem, and this is also reflected in the GLS1 regression results. However, when the correlation structure is taken into account, a different conclusion applies. A significantly smaller regional With respect to regression of X, the GLS procedure yields a slightly larger regional estimate of X than the WLS procedure. The regional information corresponds to an equivalent recording period of about 8 years in both regions. On the basis of the GLS2 regression results, the prior 100- estimate of is obtained in this case. This is due to the fact that year event was estimated using the linear and the parametric the three stations having the smallest k values in the region (k for all three) are highly correlated, and hence the weights assigned to these sites in the GLS2 procedure are much smaller as compared to the weights in the WLS and GLS1 procedures. Thus, to deal with such correlated data, GLS regression should be applied. In region B the effect of intersite correlation is different. In this case, all three regresempirical Bayes procedure, respectively. In region A the two procedures provide essentially identical results, the parametric approach producing slightly larger prior 100-year event estimates, œ oo, and standard deviations, S { œ oo ), at all sites. Also in region B the parametric approach produces larger œ oo and S {œ oo) at all sites; however, more pronounced differences between the two procedures are observed in this case. sion methods provide virtually the same regional estimate of, The relative difference between the two estimates is about but the estimates of the model error variance and the prior variance diverge significantly. When intersite dependence is taken into account, the estimate of the model error variance indicates that region B is heterogeneous as opposed to Hosking and Wallis' [1993] homogeneity test. The test assumes independence between sites, and the lack of power of the test in the case of intersite dependence may lead to the erroneous conclusion with respecto regional homogeneity. When the region is heterogenous, the weight assigned to the regional estimator in the empirical Bayes estimation procedure is much smaller 2.5% for œ oo and 5% for S { œ oo ). In Figure 3 the posterior estimates of œ oo and S { œ oo ) are compared. In general, the parametric procedure yields the largest estimates. The relative difference between the two estimates ofx x oo has a maximum of 18% and an average of 5%. For S{œ oo) the maximum and average differences are, respectively, 30% and 11%. The main reason for the two estimators being different is that the prior distributions of the PDS parameters have positive skewness which is not accounted for in the linear Bayes method, i.e. in this respect the parametric Bayes procedure provides a more than would be expected in the case of regional homogeneity. correct result. From a computational point of view, however, Thus GLS regression provides a more reliable assessment of the uncertainty of the regional estimator. The regional information of based on GLS2 regression corresponds to an the linear Bayes method has a great advantage. The method is easily implemented, for instance in a spreadsheet, whereas the parametric model requires implementation of numerical inte-

10 , MADSEN AND ROSBJERG: REGIONAL INDEX-FLOOD MODELING looo o E loo lo (a) T-year event estimator Region A Region B / _J,,,,,,,,!,,,,,,,,,,,,,,,,,i 1 O Linear estimator (b) Standard deviation looo u 100 a Region A O Unear estimator Figure 3. Comparison of linear and parametric Bayes estimation procedures. (a) Posterior T-year event estimator. (b) Standard deviation of posterior T-year event estimator. gration techniques and a powerful CPU. Since the relative difference between the two estimators of x loo generally is less than the uncertainty of the estimator (S{ œ1oo} equals 15-22% of œ loo), the linear Bayes method is adequate in most practical applications. However, if one is interested in the probability density function ofxr, and not only the mean and the variance, the parametric Bayes procedure should be applied. Conclusions A regional model for estimation of extreme hydrologic events based on the PDS approach with GP-distributed exceedances has been introduced. The model combines the in- dex-flood procedure with an empirical Bayes method for inferring regional information. To describe the regional variability of the PDS parameters, a regional GLS regression model that allows the parameters to be determined from catchment characteristics has been applied. It has been shown that takes the correlation structure into account. With respect to regression of the mean value, catchment area was found to explain a large part of the regional variability. The information content of the regional data was seen to be significant in this case, corresponding to an equivalent record length of 4-5 years. For the shape parameter a regional mean model was found to be appropriate. Application of the different regression procedures clearly illustrated the importance of taking intersite dependence into account. In the case of a highly heterogeneous correlation structure, as was observed in region A, the WLS and GLS estimators differ significantly, and in such cases erroneous results are obtained if the record-lengthweighted average procedure is applied for estimating the regional parameter. As was observed in region B, application of the WLS procedure may also lead to erroneous conclusions with respect to regional homogeneity, implying a serious underestimation of the uncertainty of the regional estimator. Thus the GLS procedure provides a general framework for a reliable assessment of parameter uncertainty as well as for an objective appraisal of regional homogeneity. Two different empirical Bayes estimators have been compared: a linear Bayes estimator in which only the mean and the variance of the prior distributions have to be specified and a parametric Bayes estimator that requires the families of prior distributions to be defined. 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