Historical information in a generalized maximum likelihood

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1 WATER RESOURCES RESEARCH, VOL. 37, NO. 10, PAGES , OCTOBER 2001 Historical information in a generalized maximum likelihood framework with partial duration and annual maximum series Eduardo $. Martins Funda ao Ccarcnse de Meteorologia e Recursos Hidricos, Fortaleza, Ceara, Brazil School of Civil and Environmental Engineering, Cornell University, Ithaca, New York, USA Jery R. Stedinger School of Civil and Environmental Engineering, Cornell University, Ithaca, New York, USA Abstract. This paper considers use of historical information with partial duration series (PDS) and annual maximum series (AMS) flood risk models. A generalized Pareto distribution for exceedances over a threshold combined with the Poisson arrival model yields a three-parameter generalized extreme value (GEV) distribution for the AMS. When fitting three-parameter GEV models using generalized maximum likelihood estimators, the average gains from use of historical information are about the same with both AMS and PDS frameworks, though the exact values depend upon the shape parameter K. The effect of the arrival rate X is modest. In general, average gains are higher when K = 0.0 as opposed to when < -< When fitting two-parameter models (exponential-poisson and Gumbel), the average gains are less than those observed with the corresponding three-parameter models with - 0. Fitting a two-parameter AMS lognormal distribution to lognormal data yielded higher average gains with use of historical information than were obtained with the two-parameter AMS/Gumbel distribution. 1. Introduction to restrict this parameter to a statistically/physically reasonable range of values, as well as assigning reasonable prior Systematic flood records often can be augmented with his- probabilities to the values of within that range. The comtorical flood data, describing the magnitudes of floods that bined use of this prior and the at-site information has resulted occurred before (or after) the systematic record period. The in more accurate T-year event estimators with small and modmagnitude of large historical floods can be determined because erate sample sizes over the range of the K values of interest these extraordinary events result in physical evidence and his- (-0.30 <- <- 0.0) with both AMS/GEV and PDS/GP models torical accounts, which are still available many years later in than is obtained with MLEs without a prior or with L moment the form of human records. One could also obtain paleoflood and moment estimators [Martins and Stedinger, 2000, this isdata by analyzing physical evidence of the occurrence of large sue]. floods (e.g., sediment deposit and botanical records) [Stedinger This paper extends the GML estimators for systematic data and Baker, 1987; Baker, 2000]. Jarrett and Tomlinson [2000] in both AMS and PDS frameworks for use with historical present a review of paleoflood studies in the western United information in both frameworks. Previous investigations of the States, particularly Colorado. They discuss field errors in iden- value of historical information [Leese, 1973; Stedinger and tifying paleoflood information and illustrate the value of pa- Cohn, 1986, 1987; Cohn, 1984, 1986; Franc s et al., 1994] used leoflood data. Because only large values are recorded, histor- only the AMS framework. The use and value of historical ical flood and paleoflood data constitute censored data. information in both PDS and AMS frameworks employing The difficulty of estimating the shape parameters K for both GML estimators with the GEV distribution is explored here in the generalized extreme value (GEV) [Hosking et al., 1985; Lu a series of Monte Carlo experiments. They illustrate the perand Stedinger, 1992; Martins and Stedinger, 2000] and the gen- formance of GML estimators for the three-parameter AMS/ eralized Pareto (GP) distributions [Rosbjerg et al., 1992; Du- GEV and three-parameter PDS/GP flood models with both puis, 1996; Madsen et al., 1997; Dupuis and Tsao, 1998; Martins systematic and historical information. The performance of the and Stedinger, this issue] is well known. Thus there should be three-parameter models relative to the corresponding twosignificant value to using additional regional and historical parameter models is also evaluated. The gains with the GEV information concerning K to improve quantile estimators. Ear- models are compared with those obtained with two- and threelier studies have used generalized maximum likelihood (GML) parameter lognormal distributions. estimators for systematic data in annual maximum series (AMS) [Martins and Stedinger, 2000] and in partial duration 2. series (PDS) frameworks [Martins and Stedinger, this issue]. Literature Review The GML estimators employ a Bayesian prior distribution for Several studies showed that historical flood information is of value when fitting AMS data [Benson, 1950; Leese, 1973; Con- Copyright 2001 by the American Geophysical Union. Paper number 2000WR /01/2000WR die and Lee, 1982; Condie, 1986; Stedinger and Cohn, 1986, 1987; Jin and Stedinger, 1989; Franc s et al., 1994; Stevens, 1994; Cohn et al., 1997]. The greatest gains were observed when 2559

2 2560 MARTINS AND STEDINGER: HISTORICAL INFORMATION IN A GML FRAMEWORK estimating distributions with a third shape parameter [Cohn and Stedinger, 1987], as is also done here. The magnitude of historical events can be difficult to determine, so Stedinger and Cohn [1986] also considered cases wherein all that was known is whether an event exceeded a perception threshold. In many cases, maximum likelihood methods efficiently employ historical/paleoflood information, thereby augmenting existing systematic records. Hald [1949] and Cohen [1950, 1976] ams [1984] use graphical methods to display flood data and to estimate flood frequency relationships. NERC [1975, pp ] illustrates the use of historical information in a PDS (exponential-poisson) framework. Stedinger et al. [1986, 1988] develop maximum likelihood models for PDS with historical information; however, no theoretical or experimental studies were performed. Bernlet et al. [1987] discuss the use of historical flood information with nonparametric introduced censored sample maximum likelihood estimators flood frequency estimation procedures within a PDS frame- (MLEs) for normal and lognormal distributions. Leese [1973], work. They compare nonparametric plotting positions and Condie and Lee [1982], and Condie and Pilon [1983] adapted their confidence intervals based on a normal approximation those estimators to combine systematic and historical informa- with a fitted Weibull model for the Garonne River and Rh6ne tion in an/x VIS framework, with a Gumbel, a three-parameter River records. While the PDS approach is not as popular as the lognormal, and a log Pearson type III distribution, respectively. annual maximum series approach, it has many advocates, who The National Environmental Research Council (NERC) [1975, pp ] applied the equations derived by Leese [1973] for continue to address problems with its implementation [Lang et al., 1999]. three gauge stations, estimating the parameters and their large sample standard errors. The GEV distribution has often been employed. Jin [1988] 3. PDS and AMS Analyses With Systematic and Historical Information and Jin and Stedinger [1989] employ historical/paleoflood information in a regional flood frequency analysis with a GEV distribution by using a combination of L moments and MLEs. This hybrid estimator attempted to take advantage of the good performance of the L moments GEV estimator in small samples, as well as introducing historical information through the likelihood function. Hosking and Wallis [1986a, 1986b] and Phien and Emma [1989] also report studies of the use of historical and paleoflood information with GEV distributions. Alternative methods have been proposed by the U.S. Water If the magnitudes of historical flood data can be precisely determined, then this information can be incorporated into the likelihood function as censore data (CD). Often the magnitudes of historical flood data cannot be precisely estimated, and the only information available is the number of years in which the largest flood exceeded a perception threshold xr [Stedinger and Cobh, 1986, 1987]. This case is called binomial censore data (BCD). In practice, several thresholds may exist which correspond to Resources Council (USWRC) [1982], Wang [1990], Russell different periods in the historical record [see Stedinger et al., [1982], Ostenaa et al. [1996, 1997], and Cohn et al. [1997]. USWRC [1982] proposed a procedure that weights moments of 1986, 1988; England, 1998]. The maximum likelihood approach described here can include cases with multiple thresholds in the historical and systematic data to obtain adjusted moments addition to lower and upper bounds and uncertainty ranges. for the Pearson type III distribution. Tasker and Thomas [1978] and Pilon et al. [1987](with a type II censoredata) explore its performance. Sho et al. [2000] recently compared adjusted 3.1. Partial Duration Series (PDS) Analysis The GP-Poisson model employs a Poisson model for flood moments and MLEs for the Gumbel distribution when histor- arrivals with independent exceedances that have a generalized ical information is observed with error. Kuczera [1996, 1999] provide a sophisticated treatment of rating curve error with Pareto (GP) distribution, whose cumulative distribution function (CDF) is systematic and historical data in a likelihood framework. Wang [1990] extends the concept of probability weighted F(xlx ->x0) = 1 - [1 - (K/a)(x -x0)] TM, (1) moments to censored samples drawn from the GEV distribution. In a Bayesian framework, Russell [1982] also considers the combined use of systematic records and additional information, such as historical information, for estimating design where a is a scale parameter and Xo is a low bound parameter and when for the shape parameter ( > 0 the CDF has an upper boundxmax = Xo + a/ ( and for ( < 0 the CDF is unbounded. The exponential distribution is obtained as ( --> 0. floods. The independent prior distribution for the mean and If m is the number of floods {Xl, x2,..., Xm} that exstandard deviation assigned probabilities to five alternatives ceeded a base level Xo in S years of systematic record, the log ranging from low to high values. The posterior probabilities are computed by weighting each element of the likelihood function with its corresponding initial probability. Ostenaa et al. [1996, 1997] considers the use of paleoflood likelihood function for the GP-Poisson model is given by [Hosking and Wallis, 1987]: LeDs(Olx) - m In (x) - xs - m In (a) information in a rigorous Bayesian framework with the log- Pearson type III distribution. Noninformative priors were em- + ployed, and the predictive distribution, used to describe flood risk, ( )m In (Yi), K i=1 was obtained by numerical integration over the three-dimensional (2) parameter space for each exceedance probability of interest. Cohn et al. [1997] introduce the expected moment algorithm, which is an iterative procedure that computes moment estiwhere 0 = (X, a, ) and Yi ( c/ot)(x i -- Xo). In addition to systematic information, historical information may be available. Here two cases are considered (as given by mates for the log-pearson type III distribution with both sys- Stedinger and Cohn [1986, 1987]). First, for a historical period tematic and censored data [see also England, 1998; Cohn et al., of H years it is known that m * floods exceed a threshold xr, 2001]. Hirsch and Stedinger [1987] consider plotting positions but the exceedance magnitudes are not well known (BCD for historical information in an AMS framework [see also Bardsley, 1989]. Gerard and Karpuk [1979] and Wang and Adcase). Second, the number and magnitudes of floods that exceed xr are well known (CD case). Here the threshold xr for

3 MARTINS AND STEDINGER: HISTORICAL INFORMATION IN A GML FRAMEWORK 2561 the historical period corresponds to the T-year flood. However, in practice, only the magnitude ofx r would be known and not its exceedance probability BCD case: Number of floods that exceed xr known. If m * floods exceeded the perception threshold x r over a H-year historical period, then the log likelihood for the GP- Poisson model becomes Lucr to PDS \ x) = LpDs(Ox ) + m* In (X*)- X'H, (3) wherey. = 1 - ( c/a*)(x.- ). The conditional probability density function f(x. O n, x -> xr) is given by the expression analogous to (5), wherein O n replaces 0. where X* = X(yr) / andyr = 1 - (g/a)(xr - Xo) CD case: Number and magnitudes of floods that exceed xr known. If the magnitudes of the historical floods can be precisely determined, then this information adds three 4. Generalized Maximum Likelihood Estimators additional terms to the log likelihood in (3) to yield Martins and Stedinger [2000] examined the behavior of MLEs for the GEV distribution in small samples and demonm v LpDs(Olx) ci - r cdr a._, PDS k 0 x) -- In (Yr) - m* In (a) strated that absurd values of the GEV shape parameter and distribution quantiles can be generated. This paper uses the GML estimators developed by Martins and Stedinger [2000, this issue] to restrict g to a statistically/physically reasonable range In (y) (4) using a prior distribution for g. Their geophysical prior is a.= beta distribution, rr(g) = (0.5 + g)p- (0.5 - g)q- /B(p, q), between [-0.5, +0.5], with p = 6 and q = 9, wherein wherey.: 1 - (g/a)(x.- Xo) and {x.} are the magnitudes B(p, q) = F(p)F(q)/F(p + q). It has expectation E[g] = of the historical floods with conditional probability density and Vat [g] = (0.122) 2. This prior restricts the MLE g function f(x.lo, x -> mr) given by f(x O) f(x O, x -> xr): 1 - F(xr 0)' estimator of both GEV and GP shape parameters to a plausible range, which is thought to be consistent with worldwide experience for such geophysical phenomena as rainfall depths (5) and flood flows. Moreover, the prior assigns reasonable prior This conditional distribution is employed in the reported probabilities to the values of g within that range. For g outside the range -0.4 to +0.2, the resulting GEV distributions do not Monte Carlo simulations Annual Maximum Series (AMS) Analysis The GP distribution combined with the Poisson model for the number of exceedances yields a GEV distribution for the AMS [Stedinger et al., 1993]. The log likelihood function for a set { x i} of independent and identically distributed observations from a GEV distribution with sample size S is m^s(0lx) = -s In (a*) In (y i) -- (Yi) lb( i=1 K where O n = (, a*, g) andyi = 1 - (g/a*)(xi- ) [Hosking et al., 1985] BCD case: Number of floods that exceed xr known. If K floods exceeded a perception threshold x r during a H-year historical period, one can modify (6) to incorporate this information, obtaining L CI ro AMSk x) = LAMS(0 I x) + K In {1 - exp [-[yr] /K]} ' (6) - (H - K)(y r) /K, (7) whereyr = 1 - (g/a*)(xr - ). The second and third terms correspond to the H years of historical record during which K floods are above xr and (H - K) are below xr, respectively CD case: Number and magnitudes of floods that exceed xr known. If, in addition to the number of ex- ceedances, the magnitudes { x.} of the exceedance events are also observed, the log likelihood function becomes L 0 '4 x) -- LAMS(0 A X) -- KIn (a)* +.= --1 ( ln(y)-(y) / - (H- K)(yr) TM, (8) have density functions consistent with flood flows and rainfall [Martins and Stedinger, 2000, p. 739]. Other estimators implicitly have similar constraints. L moments restrict g to the range g > -1, and the method of moments estimator employs the sample standard deviation so that g > Use of the sample skew introduces the constraint that g > Once the prior is defined, the generalized likelihood (GL) is computed as GL(01x) - L(01x)w( ). Thus In [GL(01x)] equals the log likelihood equation for the case of interest (PDS or AMS with either systematic data, BCD, or CD) plus the additional term In [w(g)]. The generalized maximum likelihood estimators (GMLEs) for AMS and PDS data can be identified by maximizing the generalized log likelihood function, which corresponds to the mode of the Bayesian posterior distribution of the parameters [Berger, 1985, p. 133]. The Newton-Raphson method was used to compute the GMLEs. The asymptotic properties of MLEs hold for g < 0.5 [Smith, 1985]. According to Lehmann and Casella [1996], in general, GMLEs will have the desired asymptotic properties if and only if both the likelihood (f) and the prior (w) satisfy several regularity conditions. Under these regularity conditions on f and w, as the sample size increases the likelihood function dominates the prior distribution [Robert, 1994, p. 138], and, asymptotically, both MLEs and GMLEs have the same properties. Therefore our GMLE will inherit the desirable asymptotic optimal properties of the MLEs for -0.5 < g < 0.5, the bounds of our prior. 5. Monte Carlo Experiments This section reports a series of Monte Carlo experiments that explore the performance of MLEs and GMLEs for two-

4 2562 MARTINS AND STEDINGER: HISTORICAL INFORMATION IN A GML FRAMEWORK and three-parameter flood models using either PDS or AMS systematic and historical information Experimental Design 300 From the relationship between the GEV parameters and those of the PDS model [Stedinger et al., 1993, equation 200 (18.6.6)] the parameters of the PDS model (Xo, a, K) can be 100 determined once the arrival rate ;t is specified. Values of K in the range (-0.3, 0.0) are considered. The arrival rate ;t was 0 varied from 0.8 to 5, which is a reasonable range [Taesombut and Yevjevich, 1978]. A systematic record of length S = 30 years was employed. The AMS and PDS experiments were run independently. The 30 years of systematic record were combined with historical records of length H = 0, 50, 100, 150, , and 500 years. The threshold level XT for the historical flood information for both AMS and PDS was either the 10- or year annual flood level, corresponding to exceedance rates in a PDS series of 1/9.5 and 1/99.5 floods per year, respectively. 150 This follows from 100 x* = -ln (1 - l/ta), (9) where Ta is the return period of a level in an AMS analysis [see Stedinger et al., 1993, equation (18.6.3b)] Simulation Procedure The Monte Carlo simulation experiment for the PDS analysis proceeded as follows: 1. For v = XS, generate a Poisson variate M with mean v corresponding to the number of floods larger than Xo in S years. M flood peaks were then generated from F(xlo, x -> Xo) in (1). 2. For v* = X'H, generate a Poisson variate M*. M* historical flood peaks were then generated from the conditional distribution in (5). 3. Use the generated data to calculate the GML estimators &, k, and and then calculate the pth quantile œp for the AMS as a function of those parameters using [Stedinger et al., 1993, equations (18.6.6) and ( )] p = x0 + {1 - [-ln (p)/x] ). (10) For the AMS analysis the experiment proceeded as follows: 1. Generate S flood events from the GEV distribution, F(x10). 2. Generate a binomially distributed variate K with parameterspe (probability of exceedance either 0.10 or 0.01) and n = H, which corresponds to the number of floods greater than the perception threshold xr during the historical period. K historical floods were then generated from the conditional distribu- tion f(x.l 3. Use these data to calculate the GML estimators }, &*, and k and then calculate the p th quantile œp using [Stedinger et al., 1993, equation ( )] &* p = } + - {1 - [-ln (p)] }. (11) Both analyses were repeated 2000 times to compute the bias, variance, and mean square error (MSE) of 100-year flood quantile estimators (p ). Similar simulation procedures were employed for the exponential-poisson and the Gumbel models corresponding to PDS and AMS with g = 0. 5oo T = 10yrs K = Historical Record Length T = 100 yrs K = (a) His torical Re cord Length BCD O AMS- BCD [] 5.0- BCD 5.O- CD BCD O AMS- BCD [] 5.0- BCD 5.O - CD Figure 1. Effective record length (ERL) for g = as a function of H and ;t when estimating the 100-year annual flood with S = 30 years and perception thresholds xr at (a) T = 10 years and (b) T = 100 years Effective Record Length (ERL) and Average Gain (AG) for AMS and PDS The effective record length (ERL) for both AMS and PDS analyses is computed as the number of AMS years that would result in the same MSE as a given combination of systematic and historical information. Cohn [1984] and Cohn and Stedinget [1987] used this definition in their analysis of the value of historical information in an AMS framework. Those authors assumed that the MSEs are inversely proportional to the ERLs. For the three-parameter AMS/GEV and PDS/GP- Poisson models this assumption is not satisfactory, as can be seen in Figures lb and 2b. Therefore, in order to determine the ERL it was necessary to determine the relationship between the MSE for the AMS analysis and the record length. Because the relationship between the MSE and the record length was not a simple one, the model a MSE = N(l+,/N+c/g2+a/N3 ) = a exp {-(1 + b/n + c/n 2 + d/n 3) log (N)) (12) was fitted to a set of {ni, MSEi } pairs, wherein rt i varied from 20 to 500 years. The ERL is then determined by solving for the N in MSEA S( pis = N, H = 0) that results in the same MSE as a given combination of systematic and historical data, which is described by MSEAMS(. p S, H) and MSEPDS(. p S, H), for AMS and PDS analyses, respectively. Table A1 (Appendix) reports the parameters for (12), developed as part of this study.

5 MARTINS AND STEDINGER: HISTORICAL INFORMATION IN A GML FRAMEWORK OO O T = 10yrs... [ -f BCD Historical Record Length (a) His torical Re co rd Length (b)! -h- O AMS- BCD 5.0- CD -f- -h BCD O AMS- BCD [] 5.0- BCD 5.O- CD Figure 2. Effective record length for K = 0.0 as a function of H and, when estimating the 100-year annual flood with S = 30 years and perception thresholds xr at (a) T = 10 years and (b) T = 100 years. The average gain associated with historical information is defined as the average increment in the effective AMS record length per year of historical record, AG(H) = [ERL(H) - ERL(0)]/H, (13) where ERL( ) refers to either AMS or PDS and ERL(0) is the effective record length without historical information. (ERL(0) = S for the AMS framework.) The ERL(0) in a PDS framework also includes the record length gains or losses that result from using a PDS with the given, instead of an AMS over the S-year systematic record period. For the case H = 0 the AG was defined as 6. Results AG(0) = ERL(0)/S. (14) Monte Carlo results are reported for binomial censore data GMLE (GMLE/BCD) and censored data GMLE (GMLE/ CD) for both the AMS and PDS data sets. Both threeparameter and two-parameter (K = 0) models (AMS and PDS) are considered. Figures 1-3 and Tables i and 2 give the effective record lengths and average gains in terms of the effective AMS record length AMS Versus PDS With Systematic Data Only In Table 1, for systematic data only, the AG(0) values computed using (14) are close to 1. Thus, for systematic data only, the AGs for the PDS analysis are not sensitive to X for the three-parameter GP-Poisson model. A year of PDS data with X in the range considered was generally worth from 0.9 to 1.1 of a systematic year in terms of the estimated decrease in the MSE of estimators for the three-parameter PDS model. This suggests that gains from use of PDS instead of AMS analyses are generally not substantial (as shown by Martins and Stedinger [this issue]). However, in Table 2 for the two-parameter exponential-poisson model the AG(0) values in a PDS framework are sensitive to X. For the two-parameter model the AG for systematic data alone showed greater variability and exceeded unity for X -> 1.65, as suggested by the asymptotic estimates of the variances [Cunnane, 1973]. Martins and Stedinger [this issue] draw similar conclusions using an efficient index, which is a ratio of the MSEs Fitting Three-Parameter AMS and PDS Models Figures 1 and 2 illustrate the ERL of estimators using both AMS and PDS data sets as a function of H when estimating the 100-year annual flood with a perception threshold for historical information at the 10- or the 100-year annual flood levels. The systematic record length (S) is 30 years. The parent GEV distributions had < = -0.3 and 0.0 and thus are unbounded above. For Figures i and 2 the Poisson arrival rates were 0.8 and 5.0 floods per year for the PDS models. AG(H) values are reported in Table 1. One can see the following. 1. In all cases, historical information resulted in substantial increases in the ERL. 2. There is a relatively modest increase the ERL eds and the AG(H) in Table 1 with an increase in the Poisson arrival rate h when T = 100 years for both CD and BCD. 3. For CD historical information and BCD with T = 100 years the ERL increases rapidly with H; however, with BCD and T = 10 years the ERL shows a decreasing slope with increasing H, as was expected. (See Stedinger and Cohn's [1986] Figures 3 and 5 and their discussion of this limiting behavior.) 4. For both BCD and CD historical information the ERL AMs is generally between the ERL eds values for h and h = 5. The spread between these three values is smallest for the higher censoring threshold T = 100 years. 5. The BCD provided almost as much information as the CD when the censoring threshold was T years, which is near the quantile of interest. 6. For both AMS and PDS data, AG(H) values in Table 1 with historical information are much higher when g (a thin-tailed distribution) as opposed to negative values of g (a thicker-tailed distribution). Table 1 supplements Figures i and 2 by reporting the AG(H) values when estimating the 100-year flood with H = 100 and S = 30 for a larger set of h and values. One hundred years of historical information will be equivalent to 100AG(H) years of systematic data. For < K -< 0 with three-parameter models the largest gains from use of CD historical information for both T = 10 and T = 100 were obtained when <- 0. For binomial censored data the largest gains were observed when g = 0 only for the higher perception threshold (T = 100). The asymptotic results for the three parameters of the GP- Poisson model are given by Martins [2001]. For AMS data, Jin [1988] presents the asymptotic results for the three-parameter GEV distribution, and Cohn [1986] presents the results for the three-parameter lognormal distribution, whereas Cohn et al. [2001] consider the log Pearson type III distribution. These authors show that the agreement between Monte Carlo and

6 2564 MARTINS AND STEDINGER: HISTORICAL INFORMATION IN A GML FRAMEWORK Table 1. Average Gains From Use of Historical Information With the Three-Parameter GML Estimators When Estimating the 100-Year Annual Flood With S - 30 Years and H = 100 years a GMLE/BCD GMLE/CD Systematic Only, AG(0) = AMS b PDS c PDS c PDS c PDS c = AMS b PDS c PDS c PDS c PDS c = AMS b PDS c PDS c PDS c PDS c = 0.00 AMS b PDS c PDS c PDS c PDS c adefinitions are as follows: AG, average gains; GML, generalized maximum likelihood; GMLE, generalized maximum likelihood estimator; BCD, binomial censore data; CD, censore data; AMS, annual maximum series; GEV, generalized extreme value; PDS, partial duration series; and GP, generalized Pareto. bgev is used. cgp/poisson is used. asymptotic results are mixed in realistic length samples. Asymptotic variances and asymptotic average gains for these three-parameter distribution should be used with caution Fitting Two-Parameter AMS and PDS Models The PDS/exponential-Poisson and AMS/Gumbel models were also tested. Figure 3 displays the ERL of 100-year flood estimators as a function of H and X when fitting the correct two-parameter model for a perception threshold at the 10- and the 100-year annual flood levels. For X -> 1.65 the ERL eds is greater than 30 years when S = 30 and H = 0. These findings agree with those of Taesombut and Yevjevich [1978]. Figure 3 includes results for the two-parameter AMS model. Table 2 reports the AGs. One can see the following. 1. In all cases, historical information resulted in substantial increases in the ERL, though less than were obtained with the corresponding three-parameter models which had K There is a substantial increase in the ERL eds and AG(H) in Table 1 with an increase in the Poisson arrival rate X when T = 10 years. This is not the case for T = 100 years. 3. For CD historical information and BCD with T = 100 years the ERL increases almost linearly with H; however, for BCD with T = 10 years the ERL shows a decreasing slope with increasing H, as was expected [see Stedinger and Cohn, 1986, Figures 3 and 5]. The effect is most apparent for smaller X. 4. Again, for both BCD and CD historical information the ERL ^Ms is between the ERL ei s values for X = 0.8 and X = 5. Table 2. Average Gains From Use of Historical Information With the Two-Parameter ML Estimators When Estimating the 100-Year Annual Flood With S = 30 Years and H = 100 Years For K = 0.00 GMLE/BCD GMLE/CD Systematic Only, AG(0) loo AMS PDS b PDS b PDS b PDS b agumbel values are used. bexponential/poisson is used.

7 MARTINS AND STEDINGER: HISTORICAL INFORMATION IN A GML FRAMEWORK T_--_I_0 zrs - 36ø / Exp/Poson Historical Record Length 160 ] T=100yrs 120 E_xp/Poisson 40 0 (a) Historical Record Length (b) A -' O A -&- O 0.8- BCD 0.8- CD AMS - BCD 5.O- BCD 5.O- CD BCD AMS - BCD 5.0- BCD 5.0- CD Figure 3. Effective record length for two-parameter models (K - 0.0) as a function of H and X when estimating the 100- year annual flood with S = 30 years and perception threshold XT at (a) T = 10 years and (b) T = 100 years. The spread between these three values is smallest for the higher censoring threshold T = 100 years. 5. Again, the BCD provide almost as much information as the CD when the censoring threshold is T = 100 years, which is near the quantile of interest. 7. Lognormal Experiments A Monte Carlo experiment was conducted to evaluate average gains associated with historical information when floods are lognormally distributed and to compare such results with those presented above for the GEV distribution. Two thousand replicates with S = 30 and H = 100 were generated from a three-parameter lognormal distribution (/x = 0, cr = 0.833, and r = 0) as given by Cohn and Stedinger [1987]. This distribution has a real-space coefficient of variation (CV) equal to 1.0. The maximum likelihood method was used when esti- mating one and two parameters, and a hybrid MLE employing a quantile lower bound (QLB) (MLE/QLB) was used when estimating three parameters. (See Stedinger [1980] and Cohn and Stedinger [1987] for details.) Cohn and Stedinger [1987], who conducted a similar experiment, computed the ERL as MSE^MS(œplS, H = 0) ERLAMS(H) = USEAMS( IS, H) S (15) and computed the AGs by (13). This assumes that MSEs are inversely proportional to the effective record lengths. This assumption is reexamined here. Table 3 reports AGs computed assuming that MSEs are inversely proportional to the ERL (AG-linear (15)) and using the nonlinear model (AG-nonlinear) in (12) calibrated to lognormal results. Although the difference between the AGlinear and AG-nonlinear increases as the number of parameters to be estimated increases, the largest difference between corresponding values is only on the order of 10% (threeparameter case). The assumption of linearity (b = c = d = 0 in (12)), which was not satisfactory when floods were assumed to be GEV distributed, is reasonable when floods have a Gumbel distribution or a lognormal distribution. The average gains for the two-parameter lognormal distribution are greater than the values obtained for the AMS/ Gumbel model. This is perhaps because fitting the twoparameter lognormal distribution with MLEs is equivalent to fitting a normal distribution, which has a thinner tail than the Gumbel distribution; in Table 1 with CD GEV data, larger AGs were associated with < values closer to 0 (thinner tails). For the three-parameter lognormal distribution (CV = 1) the AGs were generally in the range of the values obtained for the AMS/GEV models. 8. Conclusions This paper demonstrates that historical data can be employed with PDS data to substantially improve estimators of Table 3. Average Gains From Use of Historical Information With the One-, Two-, and Three-Parameter Lognormal Distribution When Estimating the 100-Year Annual Flood With S = 30 Years and H = 100 Years a T = 10 Years T = 100 Years Number of Parameters Lognormal Lognormal Linear Nonlinear GEV/Gumbel Linear Nonlinear GEV/Gumbel BCD (0.23; 0.67) b (0.22; 0.45) b CD '" '" (0.74; 0.91) b (0.22; 0.61) b aag/linear is the average gain computed assuming that the relationship between MSE and sample size is linear, whereas AG/nonlinear uses equation (12). bvalues are from Table 1 depending upon K.

8 2566 MARTINS AND STEDINGER: HISTORICAL INFORMATION IN A GML FRAMEWORK parameters and quantiles. The value of historical information is greater when three rather than two parameters must be Table A2. Parameters of the Variance Equation for Models Used in This Study a estimated in a PDS framework. This pattern was seen before with lognormal distributions in an AMS framework [Cohn and Model a b c d Stedinger, 1987]. When the shape parameter is to be estimated, 2P lognormal all the information about the frequency of large floods is ob- 3P lognormal tained from the data itself, and a year of historical information Gumbel ( = 0.00) GEV (K = -0.30) (CD) is of almost as much value as a systematic year of record. GEV ( = -0.20) When precise historical information is not available, substan- GEV ( ) tial improvements were still obtained from use of binomial GEV ( = 0.00) censored data (BCD). When the perception threshold is near the quantile of interest, the BCD were worth almost as much aequation for the variance is of the same form as equation (12). as the CD when estimating three parameters. When fitting three-parameter PDS models (PDS/GPa = 1. The functions defined by Tables A1 and A2 are denoted Poisson), the effective record length (ERL) and average gains as MSE(Nln) and Var respectively. For any a and n (AGs) increase very little with the arrival rate X. In contrast, the MSE and variance can be computed as the ERL and the AGs are sensitive to,x when fitting twoparameter models. In general, the average gains with historical MSE(Nla, ) = a 2 MSE(NI ) information with either type of data (AMS and PDS) are much higher when K = 0.0 (a thin-tailed GEV distribution) as op- Var (NI-, ) -.2 Var (NI ). posed to when < K -< -0.1 (thicker-tailed GEV distri- The location parameter does not affect the MSE or variance butions). The AGs obtained with historical information for the of the estimators. three-parameter PDS and AMS GEV models are generally relatively close, though the exact values depend upon the shape parameter. For both BCD and CD cases with either two- or Acknowledgments. We thank the Brazilian Government agency three-parameter models the ERL ^ 4s is generally between the CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecno- ERL eds with X = 0.8 and X - 5 for both BCD and CD cases. logico) and the Ceara State Government (Brazil), who provided support for much of this research, and M. Jin for pioneering the trail. We For the two-parameter PDS and AMS models (exponential- also thank two anonymous referees and the Editor, who provided Poisson and Gumbel, respectively) the AGs obtained with his- reviews and useful comments. torical information are very close when the perception threshold is near the quantile of interest. Otherwise, the gains with the AMS/Gumbel model are in the range of the gains with the References PDS/exponential-Poisson model. Other experiments found the Baker, V. R., Paleoflood hydrology and the estimation of extreme floods, in Inland Flood Hazards: Human, Riparian, and Aquatic Comaverage gains for the two-parameter lognormal model were munities, edited by E. E. Wohl, pp , Cambridge Univ. Press, greater than the values obtained from the AMS/Gumbel New York, model. Overall, the results demonstrate that historical information Bardsley, W. E., Using historical information in nonparametric flood estimation, J. Hydrol., 108, , can be of great value and that the choice of an AMS or PDS Benson, M. A., Use of historical information data in flood frequency analysis, Eos Trans. AGU, 31(3), , framework does not make much difference. Similarly, for the Berger, J. 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