Separation of year and site effects by generalized linear models in regionalization of annual floods

Transcription

1 WATER RESOURCES RESEARCH, VOL. 37, NO. 4, PAGES , APRIL 2001 Separation of year and site effects by generalized linear models in regionalization of annual floods Robin T. Clarke Instituto de Pesquisas Hidrfiulicas, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil Abstract. This paper explores the utility of generalized linear models (GLMs) and generalized linear mixed models (GLMMs) for regionalization and record augmentation of flood data. Because both models allow the separation of site and year effects, each gives an estimate of a site's mean annual flood free from effects of the particular years in which floods were recorded. In addition, a GLM gives a measure of the extent to which a short flood record can be regarded as representative of a longer period. The way in which GLMs and GLMMs are formulated effectively unites two problems that have largely been regarded as separate: namely, regional regression and data augmentation. GLMM model structure implicitly includes correlation between flood records at neighboring sites, and both models contain facilities for testing whether time trends exist in flood data, for testing whether such trends are regionally uniform when they are found to exist, and for testing which climatological and/or physiographic variables are useful for information transfer. By appropriate selection of probability distribution and link function, biases are avoided that arise where log transformed data are back transformed to the scale on which flood flows are recorded. The paper illustrates GLMs and GLMMs by fitting them to flood records of variable length from 12 sites in the Ibicui drainage basin in southern Brazil. While the models are extremely flexible and are easily fitted for gamma-distributed data (and to data following certain other probability distributions), the assumptions on which they are based are arguably more complex than in more familiar methods, requiring careful checking procedures. 1. Introduction postulated for flood characteristics other than Q. There are various problems with this approach, some of which are men- A central problem in hydrology, particularly in developing tioned in section 2. countries, is the transfer ("regionalization") of information The case where flow records exist but are short has historiabout hydrological regimes, in particular, of flow characteris- cally been often treated as a separate problem, generally called tics, from sites with long flow records to sites where records are "record extension" or, where it is necessary only to improve short or nonexistent. The broad range of activities embraced by estimates of mean and variance at the short-record site, hydrological regionalization includes the transfer of informa- "record augmentation" [Stedinger et al., 1993]. On the basis of tion about flood characteristics to a point on a drainage net- normal theory (or at least assuming that the logarithm of work where knowledge of the frequency and magnitude of annual flood (ln Q) is normally distributed [National Environflood flows is required for planning purposes. Regionalization mental Research Council (NERC), 1975]), Matalas and Jacobs of hydrologic data has a long history, extending back for >40 [1964] derived the augmented estimates of mean and variance years [Matalas and Benson, 1961; Benson, 1962; Matalas and of observations at the short-record site, using a longer record Gilroy, 1968; Thomas and Benson, 1970; Hardison, 1971]. correlated with it. Their results were extended to the multivar- For sites where flow records are absent, a widely used techiate case [Moran, 1974], again assuming normal theory. Cornique [e.g., Mosley and McKerchar, 1993; Stedinger et al., 1993] responding results for nonnormal cases do not appear to exist, for the transfer of information about flood characteristics from and back transformation from the log to linear scales intro- P sites with continuous records of discharge, not necessarily of duces bias (see section 2). As an alternative to the record the same length, to a site without records is the following. Let extension and augmentation procedure, Bayes' theorem has be the annual flood at stationj (j = 1...P),A be been used to combine an estimate of Q : obtained from redrainage area, and S be channel slope, where (A, S,... is gional regression with the estimate of Q : obtained from a a set of physiographic and climatic variables thought to detershort record available at site K [NERC, 1975]. mine the magnitudes of the ; let the suffix K denote the site This paper presents an alternative approach to those dewhere no flow record exists. Then (1) the "regional regression" scribed above, based on the use of generalized linear models In Oi = a + /3 In A i + D2 In S i e i is calculated (GLMs) [McCullagh and Nelder, 1989], in which site and year from the P sets { Oi, Ai, Si,... }; (2) this regression is used to effects are separated. One advantage of this approach is that estimate In O : using lna :, In S :...; and (3) exp [ln O :] is since each year effect is calculated, it is possible to assess the taken as an estimate of Q :. Analogous regressions may be degree to which a short record is "representative" of a longer Copyright 2001 by the American Geophysical Union. Paper number 2000WR /01/2000WR period of record. However, there are also other advantages as well as some disadvantages, as set out below. The paper shows that the flexibility of GLM structure includes both regional 979

2 980 CLARKE: GLM IN REGIONALIZATION OF ANNUAL FLOODS Table 1. Details of 12 Flow-Recording Stations in the River Ibicui Drainage Area (Southern Brazil) Code River Station Area, km 2 of variable precision since, in general, the P stations will have records of differing lengths. The difficulty can be partially resolved [Tasker, 1980] by a weighted multiple regression using number of years of record as a weighting function. However, since headwater catchments often tend to attract human set- 4. No allowance is made for the fact that since the means Oi are derived from periods of (usually) different length, each mean will be influenced by the particular climatic conditions pertaining during the period of observation. A short record may include by chance a relatively high proportion of wet or Toropi Vila Clara 2, Toropi Ponte Toropi 3,323 dry years, so being "unrepresentative" of long-run climatic Santa Maria Rosfirio do Sul 12,210 conditions. This should be distinguished from the problem of Jaguari Jaguari 2, Jaguarzinho Ernesto Alves 921 spatial correlation mentioned in Point 3, although the two are Jaguari Passo do Loreto 4,574 related. Short records can include sequences of years that are Ibicui Jacaqufi 27,260 wetter or drier than average, whether or not spatial correlation Arroio Miracatfi Ponte de Miracatfi 380 exists between floods recorded at different sites. The possible Ibicui Manoel Viana 28, Itfi Passo da Cachoeira 2,451 unrepresentative nature of short records is one justification for Ibirapuitfi Alegrete 5,776 the device, used later in the paper, to separate year effects Ibicui Passo Mariano Pinto 35,935 from site effects and to estimate them separately. Although GLMs and their extended form generalized linear mixed models (GLMMs) may be relatively unfamiliar to engiregression and data augmentation as particular cases. Furthermore, biases due to back transformation from log to linear neers concerned with practical details of flood data analysis, these models can be very easily fitted by present-day statistical scales are avoidable, a wide range of probability distributions packages when the probability distribution of floods belongs to for annual flood data can be used (although there are distinct advantages to be gained by using a distribution from the exponential family as defined below), and standard GLM hypothesis-testing procedures make it easy to test whether trends exist the exponential family (defined below); computing time rarely takes more than a few seconds. The following sections explore and illustrate the utility of these models for the analysis and regionalization of flood data. in annual flood sequences. In concluding this section, we note that although the title and content of this paper refer to annual flood data, the general approach is equally valid for mean annual flow, annual 3. Data GLM and GLMM characteristics are illustrated using data rainfall, and other variables of hydrological interest, although there will be additional points to consider if, say, records confrom the River Ibicui, a tributary in southern Brazil of the River Uruguai, which joins the la Plata drainage system. Antain a marked serial correlation structure. nual maximum mean daily discharges ("annual floods") were available from the 12 gauging stations shown in Table 1. The duration of records extends from 9 to 41 years. The 12 drainage 2. Problems With Regional Regression basins range in area from 380 to 35,935 km 2, through almost 2 and Data Augmentation orders of magnitude. It could be argued that records from Among problems associated with regional regression, at small and large basins should be analyzed separately. The least in the way it is frequently used in some developing coun- reasons for not doing so are (1) that the work reported here tries, are the following. formed part of a planning study for development of the Ibicui 1. No allowance is made for the facthathe ( i are usually basin as a whole, so that there was an administrative requirement to analyze records from all available sites jointly, and (2) that with such a limited number of sites a breakdown into groups classified by area would result in some classes with very few sites. tlement later than downstream areas, a weighted regression may give insufficient weight to records from upstream sites 4. Preliminary Analysis where hydrological records are shorter. Classical statistical models commonly require constant vari- 2. Here exp [ln O :] is an underestimate of the true mean ance either in the scale of measurement or in a scale to which annual flood at the site K that is of interest. It can be shown data can be transformed [McCullagh and Nelder, 1989]. The log that if E[( ] = /xi, the mean annual flood at site j, and the transform used in the regional regression model, given above, coefficient of variation C is small enough for terms of order not only secures linearity in the model parameters (the coefhigher than C 2 to be neglected, thene[ln ( i] = In/x i - C2/2 ficients of In A i, In S i,... ) but also goe some way toward' approximately, where C 2 is E[(Q - g/)2]/g. This bias will achieving homogeneous variance, as the multiple regression be introduced wherever In Q (or In Qo, the log of the annual model requires, although the log transform of data introduces flood in year i at site j) enters the analysis. bias when results are back transformed. In their account of 3. No allowance is made for the correlation between an- GLMs, of which classical models such as regional regression nual floods at different stations, arising where in a particular year a long period of heavy rainfall over an extensive area results in large floods at many flow-recording stations within it (or, conversely, where an extended drought results in annual maximum discharges which are low at many sites). This problem has been widely studied [Matalas and Benson, 1961; Kuczera, 1983; Stedinger and Tasker, 1985, 1986a, 1986b; Hosking and Wallis, 1988]. are particular cases, McCullagh and Nelder [1989, chapter 8] discuss models for data in which the variances of data groups show a quadratic relation when plotted against their means: that is, models for data with constant coefficient of variation (CV). The form of the relation between mean and variance is (together with considerations of statistical independence of data and asymmetry of their distribution) one of the determinands leading to the concept of quasi-likelihood [McCullagh

3 CLARKE: GLM IN REGIONALIZATION OF ANNUAL FLOODS 981 Table 2. Mean Annual Floods with Standard Errors, Variances of Mean Annual Floods, and Coefficients of Variation for 12 Sites in River Ibicui Drainage Basin Station Mean (_ SE), Variance, Coefficient of Years a m 3 s -1 m 6 s -2 Variation, % Vila Clara _ (x 103) 38.4 Ponte Toropi _ Rosfirio do Sul _ Jaguari _ Ernesto Alves _ Passo do Loreto _ Jacaqufi _ Ponte de Miracatfi _ Manoel Viana _ Passo da Cachoeira _ Alegrete _ Passo Mariano Pinto _ anumbers of complete years (not necessarily consecutive) of flow record from which annual maximum mean daily discharges were derived. and Nelder, 1989, chapter 9], used where there is no theory available on the random mechanism by which the data are generated. Explicit formulation of a likelihood function is questionable in such circumstances. The starting point in the analysis of the Ibicui data was therefore a plot of the relation between site means and variances. For the sites listed in Table 1, the means, variances, and CV of annual flood are shown in Table 2. However, these statistics are of variable precision, some being based upon many more years of record than others; therefore it was necessary to take an account of the very different numbers of years from which means and variances were calculated. Figure 1 shows the station variances together with a fitted quadratic curve, the fit taking account of the different number of years of record at each station by use of weighted regression. Ignoring a nonsignificant constant and linear term, the fitted curve is variance = C(mean)2 with C = _ with R 2 = 97.8% (residual d.f. = 10). Clearly, the quadraticurve 3.5 x 106 Variances (o) of annual floods, and fitted quadratic curve _ I I I O Coefficients of variation within 12 gauge sites 55 5O.-e 45 > o , I I, i, I Mean annual flood, cumecs Figure 1. (top) Quadratic relation between variance of annual flood (vertical axis) and mean annual flood (horizontal axis) for 12 gauge sites in the River Ibicui drainage basin, southern Brazil. (bottom) Plot of coefficients of variation (CV) against mean annual flood for the same basins.

4 982 CLARKE: GLM IN REGIONALIZATION OF ANNUAL FLOODS Table 3. Analysis of Records From Each Station Individually: Estimates of the Shape Parameter, from 12 Sites and Deviance Measures of Goodness of Gamma Distribution Fit Site, SE (,) Deviance a Vila Clara Ponte Toropi Rosfirio do Sul Jagbari Ernesto Alves Passo do Loreto Jacaqufi Ponte de Miracatfi Manoel Viana Passo da Cachoeira Alegrete Passo Mariano Pinto Pooled b aall with 3 d.f. bpooled estimate of,, obtained by weighting individual values inversely as their variances. shows a strong relation between variance and mean; however, Figure 1 (bottom) show that when the CV is plotted against mean annual flood, the quadratic trend is no longer evident. CV values are consistent with the hypothesis of zero correla- tion with mean annual flood. The conclusion is that a model assuming constant coefficient of variation is appropriate for the data. 5. A GLM With Gamma-Distributed Annual Floods A quadratic relation between site variance and mean suggests a gamma distribution for annual flood data [McCullagh and Nelder, 1989]. Denoting the annual flood by random variable Q with observed values q, this distribution is G(/x, v) given by G(Ix,,) = [11r(,)](,q/ix) v exp (-uq/lx)d(lnq) ordinary least squares arises as maximum likelihood from the q->0,>0 t >0. (1) In classicalinear modeling the residual sum of squares In this form the mean and variance of the gamma distribution (RSS) gives a measure of goodness of model fit; in GLMs the are/x and/x2/u, respectively; the CV is 1/v /2. With the annual corresponding measure of goodness of fit is the deviance D, floods at the 12 sites arrayed in a table with 12 columns (sites) which reduces to RSS if a normal distribution is substituted for and 41 rows (years) containing many missing values, the GLM G(ixo, ) in point 2 above. Table 3 shows the deviances D for the annual flood Qii in year i at site j is defined as follows: obtained from likelihood considerations and defined for the (1) E[Qo] = i&ij; (2) Qij is distributed as G(I. ij, 1 ); (3) gamma distribution [McCullagh and Nelder, 1989] by r ii = xi5/3, where xi5 is a vector of explanatory variables (such asa i, Ss, or some measure of precipitation Po intensity year D = -2 ] {ln(qo/12o) - (Q,j -/20)//2o}. (2) i at station j) and/3 is a vector of coefficients (discussion of the i j form of r o, which is termed the linear predictor, is given The deviance is interpreted as a measure of goodness of fit below); (4) r is = #(/xo), where #( ) is a known, monotonic between the observedata Qo and the fitted values generated differentiable "link" function relating the expected value by the model, with large deviance indicating poor fit. More E[ Qo] to the linear predictor is. McCullagh and Nelder [1989] precisely, for the data Qo from station j, deviance is calculated point out that the log link function #(/xo) = In (/xo) achieves as the difference between two log likelihoods relevant to diflinearity without the need to abandon the original scale of measurement by transforming the data. They show that with the combination of log link function and quadratic varianceferent models of the data: (1) the log likelihood when (in this case) a gamma distribution with/x o different for each item of data is fitted, so that as many parameters are fitted as there are mean relationship, fitting the GLM is equivalento assuming data items, giving no deviations (this gives the maximum that Qo has the gamma distribution with constant shape parameter, independent of the mean, in the same sense that achievable log likelihood) and (2) the log likelihood obtained when (in this case) a single gamma distribution with mean/x is normal distribution. Since the model assumes that the shape parameter, is constant between sites, it was necessary to test the hypothesis that, did not vary significantly from site to site. A twoparameter gamma distribution of the form (1) was therefore fitted to the data from each site separately, and the 12 estimates of the shape parameter, were as shown in Table 3. The pooled estimate of,, averaged over all 12 sites with weights equal to the inverses of their variances, was _ Since u- - CV 2, the square of the coefficient of variation of the gamma distribution, we have CV 2 = 1/4.927 = with approximate (large sample) standard error _+0.017, agreeing reasonably well with the value C = _ reported above, found when variance is regressed on the squared mean. It cannot be ruled out that the fact that, can be taken as constant for the Ibicui data may be just a fortunate coincidence, despite the fact that the 12 sites have drainage basins ranging over almost 2 orders of magnitude (from 380 to nearly 36,000 km2). However, even where the hypothesis of constant, must be rejected, procedures similar to that of Aitkin [1987] are available to model such heterogeneity in terms of one or two additional parameters: for example, by writing 5 = 'o(1 + OA ) if, is thought to depend on areaa and 'o and 0 are constant. The informal argument above can be supplemented by a formalikelihood ratio test. When the gamma parameters/x and, are estimated each site separately, the log likelihood function at site j is {/,j[ln( 'Qo/12 ' - Qo/12 ')] - In [F(.)]}, i where the estimates/,i are as shown in Table 3 and/2 are the raw means. Summed over j, the station suffix, the total log likelihood is Under the null hypothesis that all are equal to,, say, the log likelihood is reduced to , where the pooled, = , not too different from the pooled value of shown in Table 3. The reduction in log likelihood, with 11 d.f., is distributed approximately as X 2 with 11 d.f. and is not statistically significant. This confirms that there is no evidence of differences between the 12 shape pa- rameters.

5 CLARKE: GLM IN REGIONALIZATION OF ANNUAL FLOODS 983 fitted to all the data from station j. The log likelihood for case 2 will be smaller than the log likelihood for case 1, and the adapted to test whether the trend varied significantly between sub-basins by introducing dummy variables [e.g., Clarke, 1994] deviance measures the difference. Associated with each of the into the vector of explanatory variables deviances in Table 3 is a number of degrees of freedom, three 2. The second case is a i constant and s of the form at each station; since the deviance is distributed approximately /3ofo(As, Ss,...) + /33C (As, Ss,... ) +..., with fo( ), as X 2, the deviances in Table 3 can be regarded as approximate fl( ),..., known functions of physical or climatological ex- X 2 variates with mean 3 (= d.f.) and variance 6 (= 2 d.f.). planatory variables, such as area A s, channel slope Ss, etc. This Significant values of X 2 would indicate that the gamma distri- model is broadly analogous to the "regional regression" model bution gave a poor fit since the deviance measure of discrepancy is then large and statistically significant; only 1 of the 12 deviances in Table 3 is statistically significant at P < 5 %; this mentioned earlier but avoids the bias introduced by back transformation from logs to the original scale of measurement. The simplest cases are fo = 1, fl = Aj, f2 = Sj,..., and fo = 1, is not far from what would be expected where 12 independent f = In As, f2 = In Ss,... If/30,/31,..., are estimated, then significance tests are made at the 5% level, showing that the gamma gave a good fit at all stations. As a final justification of the two-parameter gamma distribution, attempts were made to fit a three-parameter gamma at each site. When this distribution was fitted by maximum likelihood, the iterative calculation failed to converge at any of the 12 sites. the model can be used to estimate E[QK] at a site K without records. It is also possible to includ explanatory variables on the right-hand side of (3) having a different value in each cell of the years times sites table. Where the annual flood is a consequence of snowmelt, for example, the variable Zis might be a measure of the water equivalent of snowpack shortly before it melted; the site term s then would be of the form tdo(4,... ) + tffl(4,... ) GLMs in Which the Linear Predictor q Separates Site and Year Effects We now consider appropriate forms for the linear predictor rt / = x /9. The expected value E[Q /] = /x / of the annual flood in year i at site j depends both upon the year of measurement and the site at which it was observed. To make this explicit, the linear predictor of the GLM can be written in the form *l,j = I & + ai + sj, (3) 3. The third case is a i a random variable and s constant. This corresponds to the case in which the effects a i of the years are regarded as a random sample from a population of year effects, the purpose of the analysis being to estimate mean annual flood at each gauge site, averaged over the population of years of which the years of record are a sample. The year effect a i is taken to have zero mean and variance tr a, 2 and attention is focused on the estimation of/x + si, the mean value for gauge j. Fitting the GLM yields estimates of the constants/x and s and the variance ga' 2 Since annual floods Q is and Q ik in the same year i but at different sites j and k contain the same random component a i, the GLM builds in the correlation where /x is a regional mean annual flood, a i is a component specific to the ith year of record, and s i is a component specific between them; with the log link function used in this paper, cov to the jth gauging site. This is of the general form r/is = x0r./3, [Q s, Q,] = exp (2/x + s + S )ga. 2 Inclusion of both in which a i and s are elements of the vector/3 of parameters, constant and random effects in addition to the constant/x in with the elements of x0 r. equal to 0 or 1:0 for data Q is not in the linear predictor/i s makes the GLM into a GLMM. Thus year i or from site j and 1 otherwise. In (3), the mean/x is a the site terms s would also be regarded as random if it were constant, but a i and s may be regarded either as constants or required to explore the spatial structure of annual flood magas random variables, depending on the purpose of the analysis. nitudes within a region, using a correlation function p(ss, s,) Various cases can be distinguished as follows. in terms of a distance measure between sites j and k. 1. The first case is a i constant and s i constant. This is broadly analogous to the "record augmentation" procedures mentioned above. The fixed quantities/x, a, and si are esti- 7. Separation of Site and Year Effects in the Ibicui Data mated by fitting the GLM; then by using the year effects a i the degree to which a short period of record is representative can To recapitulate, analysis of the Ibicui data shows that (1) be assessed. Summed over all years (41 years for the Ibicui annual floods at the 12 sites can be taken as gamma-distributed data), the total of the a will be zero; if, therefore, the sum of with shape parameter v, (2) a GLM with log link function a i for a much shorter record is negative (positive), annual avoids log transformation of annual flood data and the consefloods in those years will be lower (higher) than the full period quent bias when data are back transformed to the scale of of record requires for that site. Assuming a log link function, measurement, and (3) linear predictor of the GLM can sepathe corrected mean for the site j will be estimated as exp ( + rate year and site effects, with site effects described, if and s)' A variation of the model in (3) is obtained by putting a i = when appropriate, in terms of basin characteristics. For the [30fo(ti) q- /31f1(ti) q-... [3kfk(ti)withfo(ti), fl(ti),..., as Ibicui data the gamma-distributed Q s for year i at site j has known functions of time. The linear predictor of the GLM is E[Q s] = / s, and we now consider the three cases (1) r/i = then still of the same form, and the model can be used to / + a + ss, with a and s; constant, as in (3);(2) r/ ; = / + explore whether time trends exist in annual floods. In its sim- a i + fla;, a simplified form of case (2) in section 6; and (3) plest form, fo(ti) = 1, fl(t ) = t i to explore a linear trend. If r/ ; = / + a i + ss, with a random and s fixed. Cases 1 and a significantrend of any kind were found, it would obviously 2 are GLMs; case 3 is a GLMM. be inappropriate to try to estimate flood frequencies using 1. For case 1, linear predictor of form r/ s = / + a i + s;: methods that assume stationarity in annual flood sequences. If year effects a and site effects; are constant. For the effects of a general, basin-wide time trend in annual floods were de- year (a ) and station (ss) both constant, Table 4 shows the tected, yet another variant of the basic GLM model could be fitted site means exp (/ + ss). The raw means are also

6 984 CLARKE: GLM IN REGIONALIZATION OF ANNUAL FLOODS Table 4. Raw Means of Annual Maximum Floods and Estimates of Site Means exp (/ + si) Given by a Log Link Function Q ii with Gamma Distribution, Shape Parameter v Constant Over Sites Station Raw Mean, exp Record m 3 s - ( + ) Difference Length Vila Clara Ponte Toropi Rosfirio do Sul J agu ari Ernesto Alves Passo do Loreto Jacaqufi Ponte de Miracatfi Manoel Viana Passo da Cachoeira Alegrete Passo Mariano Pinto /3A : year effects a i and site effects for basin j taken as proportional to drainage basin area Ai. Standard GLM fitting procedures give the estimate of/3 as/3 = _+ 0.0 s 251, and an approximate t test gives t = on 239 d.f., showing the expected high significance. The residual deviance, given in (2) above, is D 2 = on 239 d.f., showingood evidence of model fit. The year effects a i (not presented in section 6 for reasons of space) are shown in Figure 2. As where constants are estimated in an analysis of variance table, these effects are constrained to sum to zero over the 41 years of record; for any record shorter than the full 41 years, the difference between zero and the sum of the year effects a i for the period of record measures how far the site's flood record is representative of the period as a whole. As explained above, the year effects can also be used to detect time trends in annual flood records, which may occur in regions where land use or climate regime has changed. Although Figure 2 shows no visual evidence of time trends, it can be seen that the standard errors, shown as error bars, are larger in the early years of record, as a consequence reproduced for comparison. It can be seen that while some of of the fewer data then available. Thus for any drainage areaa c the larger differences between the raw and fitted means occur the annual flood in any year L, say, would be estimated by where records are short, this is not always the case. It is of / L c = exp (/ + z. + / A c). If the fitted values / c are interest to calculate the regressions of both sets of means on obtained for each year and each site, including entries for drainage basin area A ; the regression of the raw means gives which data are missing, the averages for the 12 stations are r 2 = 93.3% on 10 d.f. with a residual standard deviation of shown in Table 5. Clearly, the differences between the two sets _+275 m 3 s-i; using the estimates of/ + s, r 2 = 94.5 %, with of means are now much greater than in the previous model, but a residual standardeviation of _+266 m 3 s-], the gain in terms if the two sets of means are each regressed on drainage area, of explained variance being very slight in this instance. the raw means (as before) gave r 2 = 93.3% with residual 2. For case 2, linear predictor of form r i = / + a i q- standardeviation +_275 m 3 s-l, while the values of exp (/ + 1-s I 12 Ibicui stations:annual effects across stations and standard errors Year Figure 2. Plot of year effects, free from effects of the 12 stations within the Ibicui drainage basin. Standard errors of year effects are also shown; those in earlier years are larger as a consequence of fewer data.

7 CLARKE: GLM IN REGIONALIZATION OF ANNUAL FLOODS 985 Table 5. Comparison of Raw Means With Means of Fitted exp (al + /3AK) Raw Mean, exp m 3 s -1 (a + /3Ag) Vila Clara Ponte Toropi Rosfirio do Sul Jaguaff Ernesto Alves Passo do Loreto Jacaqufi Ponte de Miracatfi Manoel Viana Passo da Cachoeira Alegrete Passo Mariano Pinto recorded. As applied to the Ibicui data, the basis of the method is the two-parameter gamma distribution G( ij, '), with shape parameter, constant from site to site (although the method could be adapted to deal with nonconstant,). The gamma mean/ ij varies from site to site and from year to year and is related to a linear predictor of the form rhj = xi fl, where fl is a vector of parameters and xij is a vector of explanatory variables. By suitable choices for the variables in xi, the site and year effects present in each annual flood can be disentangled, and/or the relation between site effects and basin characteristics can be established for regionalization purposes. Use of a log link function in the GLM to relate ij to r i j has the desirable consequence that log transformation of flood data for regionalization purposes is avoided. While the usual regional regression expresses the expected value E [In Q] to basin characteristics, the GLM relates In E[Q] to them. The constant, used in the analysis of the Ibicui data has an analogy in the commonly used index flood method of flood frequency analysis, in which it is assumed that the distributions of floods at different sites in a region are the same except for a scale or index flood parameter which reflects the size, rainfall, and runoff characteristics of each drainage basin. This index flood is usually taken as the mean annual flood at each site, and the data Qij/Oj are then pooled over sites. In the gamma distribution given in (1) above, the random variable Q representing the annual flood is also scaled by dividing by the mean/. The paper also suggests that the separation of site and year effects is desirable because (1) it allows mean annual flood at a site with short record to be estimated free from the effects of the particular sequence of years in which floods were recorded; (2) it provides a direct measure of how far the sequence of years in the record is representative of the longest period of record at any site, when all records of annual flood can be assumed stationary and free from trend; (3) by modeling the year effects si in terms of known functions of time the existence of time trends in flood records can be explored and their gl + / AK) gave r 2 = 96.3 % with residual standard deviation significance tested (the estimation of long-term flood frequen- _+242 m 3 s -1. The reduction in standardeviation might be cies at sites where trends exist being a fruitless exercise); and considered useful rather than dramatic. (d) where time trends in flood data are shown to exist, the 3. For case 3, linear predictor of form r i j = / + a i q- $j: extent to which they are spatially homogeneous can be exyear effects a i are random and gauging station effectsj are plored by introducing dummy variables [e.g., Clarke, 1994]. fixed. The GLMM model has 13 parameters: 12 site effectsj Fitting GLMs and GLMMs to annual flood data requires an subject to the constraint Zaj = 0, which reduces the 12 to 11, iterative calculation to maximize a log likelihood function. the variance o- a 2 of the year effects, and the common shape Provided that the probability distribution of annual floods beparameter,. Their estimates are calculated by iteratively weighted nonlinear least squares [McCullagh and Nelder, 1989], and convergence was achieved after five iterations. The estimate of %2 was _ The 12 site means exp ( + j) are shown in Table 6, and the estimate of the shape parameter, was If the estimated site means are regressed upon drainage area, the percentage variance aclongs to the exponential family (for each member of which a set of sufficient statistics exist for the distribution parameters 0) with general form [Cox and Hinckley, 1974] f(q; o) - exp {a(o)b(q) + c(o) + d(y)} and to which the gamma, normal, binomial, Poisson, and inverse normal distributions belong, the calculation is relatively counted for is r 2 = 94.3 % with 10 d.f., the residual standard trouble-free and is achieved by iterative weighted least squares deviation being _+250 m 3 s -1. This represents a small improve- [McCullagh and Nelder, 1989]. There is no reason why other ment on the r 2 = 93.3%, with residual standard deviation well-known distributions, such as the generalized extreme _+275 m 3 s -1, obtained when the raw means are regressed on drainage area. value (GEV) distribution, should not be used instead of,), but this has not been explored in the present paper. It is conjectured that problems of failure to converge and conver- 8. Discussion gence to nonunique optima are likely to be encountered where distributions not from the exponential family are used. This paper suggests that the GLMs and GLMM discussed Since the data analyzed in the paper are sequences of annual above constitute an alternative approach to regionalization floods, the usual assumption has been made that no serial and information transfer between sites where annual floods are correlation exists between elements of the time series that constitute the columns of the years times site matrix of annual flood records. In fact, the GLMs used in this paper require the Table 6. Comparison of Raw Means With Fitted Station Effects: Year Effects Random Raw Mean, Estimates of exp m 3 S -1 (.L + Sj) Vila Clara Ponte Toropi Rosfirio do Sul Jaguari Ernesto Alves Passo do Loreto Jacaqufi Ponte de Miracatfi Manoel Viana Passo da Cachoeira Alegrete Passo Mariano Pinto

8 986 CLARKE: GLM IN REGIONALIZATION OF ANNUAL FLOODS stronger assumption of statistical independence between an- Anderson, R. J., N. F. Ribeiro, and H. F. Diaz, An analysis of flooding nual floods in successive years, but this assumption is almost in the Parana/Paraguay River Basin, The World Bank Latin Am. Tech. Dep., Washington, D.C., universal in flood frequency studies. This paper notes, how- Benson, M. A., Evolution of methods for evaluating the occurrence of ever, that GLMs may also be appropriate for information floods, U.S. Geol. Surv. Water Supply Pap., 1580-A, 30 pp., transfer and regionalization of other hydrological variables, for Clarke, R. T., Statistical Modelling in Hydrology, John Wiley, New which the assumption of statistical independence may be less York, Cox, D. R., and D. V. Hinckley, Theoretical Statistics, Chapman and tenable; it may be expected that mean annual flows, for exam- Hall, New York, ple, show some degree of serial correlation, particularly in Fahrmeir, L., and G. Tutz, Multivariate Statistical Modelling Based on basins with large annual carryover storage, such as those of the Generalized Linear Models, Springer Ser. in Stat., Springer-Verlag, Amazon and la Plata. Extension of GLM and GLMM to the New York, analysis of correlated data and to the analysis of correlated Hardison, C. H., Prediction error of regression estimates of streamflow characteristics at ungauged sites, U.S. Geol. Surv. Prof. Pap., 750-C, data is, at present, an active field of research. C228-C236, Hosking, J. R. M., and J. R. Wallis, The effect of intersite dependence 9. Conclusions on regional flood frequency analysis, Water Resour. Res., 24, , This paper explores the use of generalized linear models Kuczera, G., Effect of sampling uncertainty and spatial correlation on (GLMs) and generalized linear mixed models (GLMMs) as an an empirical Bayes procedure for combining site and regional information, J. Hydrol., 65(4), , alternative to existing procedures for information transfer of Laraque, A., J. C. Olivry, D. Orange, and B. Marieu, Variation in space flood characteristics by regional regression and record aug- and time of rainfall and hydrological regimes in central Africa from mentation. The following conclusions are made. (1) GLMs and the beginning of the century (in Portuguese), in XII Symposium of GLMMs allow the effects of site and year effects (both of the Brazilian Water Resources Association, Anais vol. 3, pp , Assoc. Bras. de Recursos Humanos, S o Paulo, Brazil, which enter the observed annual flood Qi at a site j in year i) Lettenmaier, D. P., J. R. Wallis, and E. F. Wood, Effect of regional to be separated, with the desirable consequence that the mean heterogeneity on flood frequency estimation, Water Resour. Res., 23, annual flood at a site with short record can be adjusted for the , particular sequence of years in which floods were recorded. (2) Matalas, N. C., and M. A. Benson, Effects of interstation correlation By the same token, calculation of the year effects gives a on regression analysis, J. Geophys. Res., 66(10), , Matalas, N. C., and E. J. Gilroy, Some comments on regionalization in quantitative measure of how far a short flood record can be hydrologic studies, Water Resour. Res., 4, , considered representative of flood characteristics determined Matalas, N. C., and B. Jacobs, A correlation procedure for augmenting over a longer record. (3) GLMs and GLMMs combine into a hydrological data, U.S. Geol. Surv. Prof. Pap., 434-E, El-E7, single formulation two problems that have usually been treated McCullagh, P., and J. A. Nelder, Generalized Linear Models, 2nd ed., Chapman and Hall, New York, separately in flood frequency analysis: namely, regional regres- Moran, M. A., On estimators obtained from a sample augmented by sion and record augmentation. (4) GLMMs implicitly allow for multiple regression, Water Resour. Res., 10, 81-85, the inclusion of seasonal correlation between recorded floods: Mosley, M.P., and A. I. McKerchar, Streamflow, in Handbook of that is, the tendency for annual floods to be large (small) at Hydrology, edited by D. R. Maidment, Chap. 8, pp , McGraw-Hill, New York, neighboring sites in wet (dry) years. (5) The statistical structure Natural Environmental Research Council (NERC), NERC flood studof both GLMs and GLMMs allows formal tests to be made for ies report, vol. 1, chap. 3 and 4, London, detecting time trends in annual flood records and for assessing Stedinger, J. R., and G. D. Tasker, Regional hydrologic analysis, 1, which climatological and physiographic variables are useful for Ordinary, weighted, and generalized least squares compared, Water regional information transfer. (6) Appropriate choice of GLM Resour. Res., 21, , Stedinger, J. R., and G. D. Tasker, Correction to "Regional Hydrolink function and probability distribution avoids biases intrologic Analysis, 1, Ordinary, Weighted, and Generalized Least duced by back transformation from the log scale, used in re- Squares Compared," Water Resour. Res., 22, 844, 1986a. gional regression, to the scale of flood measurement (i.e., m 3 Stedinger, J. R., and G. D. Tasker, Regional hydrologic analysis, 2, s- ). (7) For the data used in this paper to explore GLMs and Model-error estimators, estimation of sigma and log-pearson type 3 distributions, Water Resour. Res., 22, , 1986b. GLMMs, two-parameter gamma distributions with constant Stedinger, J. R., R. M. Vogel, and E. Foufoula-Georgiou, Frequency shape parameter, were appropriate. This may not generally be analysis of extreme events, in Handbook of Hydrology, edited by the case. While the models can be adapted to nonhomoge- D. R. Maidment, Chap. 18, pp , McGraw-Hill, New neous, if the nonhomogeneity can be parsimoniously de- York, scribed by one or two parameters, GLMs and GLMMs would Tasker, G. D., Hydrologic regression with weighted least squares, Water Resour. Res., 16, , have little to offer where nonhomogeneity cannot be so mod- Thomas, D. M., and M. A. Benson, Generalization of streamflow eled. characteristics from drainage-basin characteristics, U.S. Geol. Surv. Water Supply Pap., 1975, 55 pp., Acknowledgment. The author thanks the referees for some constructive suggestions. References Airkin, M., Modelling variance heterogeneity in normal regression using GLIM, Appl. Stat., 36, , R. T. Clarke, Instituto de Pesquisas Hidrfiulicas, UFRGS, Caixa Postale 15029, Avenida Bonto Goncalves 9500, Porto Alegre, RS CEP , Brazil. (clarke@if. ufrgs.br) (Received August 21, 2000; revised November 13, 2000; accepted November 13, 2000.)