RELATIVE ACCURACY OF LOG PEARSON III PROCEDURES

Transcription

1 RELATIVE ACCURACY OF LOG PEARSON III PROCEDURES By James R. Wallis 1 and Eric F. Wood 2 Downloaded from ascelibrary.org by University of California, Irvine on 09/22/16. Copyright ASCE. For personal use only; all rights reserved. ABSTRACT: The U.S. Water Resources Council (WRC) has suggested that the log-pearson III distribution, fitted by the method of moments, should be used in flood frequency analysis. A Monte Carlo simulation assessment of the WRC procedures shows that the flood quantile estimates obtainable by these procedures are poorer than those obtainable by using an index flood type approach with either a generalized extreme value distribution or a Wakeby distribution fitted by probability weighted moments. It is suggested that the justification for using the WRC Bulletin 17B guidelines is in need of reevaluation. INTRODUCTION The foreword to the U.S. Water Resources Council's well-known guidelines for flood frequency analysis, Bulletin 17B (5), contains the following important statement: "An accurate estimate of the flood damage potential is a key element to an effective, nationwide flood damage abatement program." Bulletin 17B recommended the use of log-pearson III for the estimation flood risk, and the USGS prepared a computer code to facilitate the estimation procedure. This paper reports upon an investigation into accuracy of the flood estimates that are obtainable with this approach and compares the results with three other alternative estimation procedures. Annual flood risk is, usually estimated using the annual extreme peak series. Let 2 be the maximum value from a set of values x 1,..., Xt, in which x lr,.., x k = independent flood peaks within the year. Fisher and Tippett (2) derived the asymptotic distribution for 2 as k > ; there are three possible distributions: the Fisher-Tippett Extreme values types I, II, and III. The Fisher-Tippett type I is best known as the Gumbel distribution (6), which has been widely applied to flood data. The UK Flood Studies Report (13) uses the generalized extreme value (GEV) distribution which combines into a single form the three possible extreme value distributions (9). While the extreme value distribution has a certain theoretical appeal for fitting flood data, it is not obvious that the number of independent flood producing events within a year is sufficiently large for the distribution of annual flood maxima to attain that of an extreme value distribution. In fact, the observed distribution of annual floods may well lie somewhere in between the unknown distribution of all runoff peaks and an asymptotic extreme value distribution. It should be recognized that under these conditions longer records will better define 'Hydro., Inst, of Hydr., Wallingford, England; on leave from IBM Research, Yorktown Heights, N.Y. 2 Hydro., Inst, of Hydr., Wallingford, England; on leave from Princeton Univ., Princeton, N.J. Note. Discussion open until December 1, To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on October 8, This paper is part of the Journal of Hydraulic Engineering, Vol. Ill, No. 7, July, ASCE, ISSN /85/ /$ Paper No

2 this intermediate distribution but longer records do not imply that an asymptotic extreme value distribution has been attained or that a GEV distribution is an appropriate choice for flood frequency analysis. For flood peak probability distributions, the convergence properties to an extreme value distribution has not been well-researched. There remains for the engineer or hydrologist the problem of fitting a flood frequency model to the flood peak data. In practice, a desirable flood frequency model and fitting procedure would estimate the T-year peak in a robust manner. By robust we mean that the estimated design flood should be reasonably accurate over a range of possible distributions with flood-like properties. In particular, for the engineer we need an estimation procedure for which small changes in the data, like the addition or removal of one datum point, does not lead to a large change in the estimated design floods. Further, as hydrologic records tend to be short, it is essential that the procedure chosen should perform as well as possible (i.e., small variance) with small samples, and if the procedure yields biased estimates, that the magnitude and direction of the bias attributable to the methodology be known or estimable. In Bulletin 17B, eight different probability models were considered and compared; the objective being to find a methodology that best estimated future flood risks. Bulletin 17B used split sample historical records from 300 stations with an average record length of 47 yr. Bulletin 17B recommended the use of the log-pearson III distribution even though the results were reported as not definitive. That the Bulletin 17B test did not provide a definite answer is scarcely surprising. The true distribution of U.S. floods is unknown. However, it would appear that the observed flood sequences have skews greater than 1.14 (the Gumbel skew), are highly kurtotic (larger than for a comparable GEV distribution), and have coefficients of variation in the range (10). It can be shown by computer simulation using data generated from distributions that have the preceding flood-like distributional properties that goodness-of-fit tests with short records cannot provide unequivocal answers about the nature of the underlying distribution or a necessarily reliable guide to the problem of estimating the extreme quantiles with maximum accuracy, which emphasizes the need for robust estimation techniques. Almost from its inception, the Water Resources Council guidelines recommending the log-pearson III distribution were criticized and fully analyzed in the literature (1,10,12,14). While they comprise a theoretical explanation of why the log-pearson III approach to flood frequency analysis should not be expected to give accurate estimates, they do not provide an objective empirical test of actual performance. In this paper, we present results from testing the ability of the WRC log-pearson III methodology, and competitive procedures, to accurately estimate extreme quantiles relating to the 20 to 1,000 yr flood events using carefully controlled computer experiments. ANALYSIS Comparisons of flood estimation procedures using real world data is often interesting, but the interpretation of differences, if observed, may 1044

3 be tricky. With only a single data set (realization) per site, no measure of the relative efficiencies of the different estimation procedures is possible. Further, since the true flood probabilities are also unknown there is no a priori way of knowing which of differing estimation procedures is best. For the foreging reasons, we tested the performance of competitive flood frequency estimation procedures using computer simulation. In general, the computer simulations consisted of the following steps: 1. From a known distribution (or probability law) a sequence of psuedorandom numbers was generated (equivalent to a single hypothetical flood record for a site). 2. Each hypothetical flood record was fit using each estimation procedure. 3. Using the estimated parameters of the fitted distributions, the estimated design floods (upper estimated quantile) were compared to the known true values. 4. Steps 1-3 were repeated many times and the statistical properties of the alternative estimation procedures assessed. We will refer to the steps 1-4 as a Monte Carlo simulation. Since the design variable of importance is the discharge associated with a particular return period, T, the accuracy and consistency in estimating the T-year flood forms the basis for comparing alternative methodologies. Clearly, the identification of the true form of the underlying flood distribution is an issue that cannot be resolved here. We conducted our experiments over a fairly wide range of possible underlying distributions, and found that the general pattern of the results was the same. To prevent the reader being inundated in tables and graphs of results, we will present the results for only one distribution, log-pearson III. Thus, we will assume, as does the WRC, that floods are distributed as log- Pearson III, and we accept that this choice influences the results to be presented here in favor of the WRC methodology. There are an infinite number of hypothetical regions that could be tested in a computer simulation study such as this. We investigated a fairly wide range of possible choices although only the results for three of these hypothetical regions are reported here. Results for the nonreported regions did not depart markedly from the repeated results, and our conclusions appear to be quite general. The first hypothetical region used consisted of 20 stations, N s = 20, with records ranging from yr. Table 1 gives the at-site flood statistics. They are representative of the statistics reported in Ref. 10 derived from 1,351 stations using the 14 USGS hydrological regions of the continental U.S. The site characteristics were generated from Normal independent distributions with the statistics as given in Table 2. On the basis of extensive simulations of log-pearson III data, which are summarized in Fig. 1, the expected value of the estimated sample skew (in real space) can be seen to vary from Similar downward biases in the moment estimate of skew for small finite samples have been reported for other distributions (15). 1045

4 TABLE 1 True Statistics for 20 Station Hypothetical Region Downloaded from ascelibrary.org by University of California, Irvine on 09/22/16. Copyright ASCE. For personal use only; all rights reserved. Site (1) Record length (2) Mean (3) Standard deviation (4) Skew (5) Coefficient variation (6) TABLE 2 Statistics Used to Generate At-Site Statistics for Hypothetical Region 1 At-site flood probability characteristics 0) Mean annual flood Coefficient of variation Coefficient of skew Mean (2) Standard deviation (3) Statistical Indices. As stated earlier, the comparisons among flood frequency models focused upon their relative accuracy in estimating the T-year flood discharges, where the return periods considered were T = 20, 50, 100, 500, and 1,000 yr. Since the gaging sites in our hypothetical _l if) < a: > < SAMPLE LENGTH POPULATION SKEW - J.O 0 IO FIG. 1; Average Skew versus Sample Lengths for True Skews of 3.0 and 2.0 for log-pearson III Data 1046

5 region represent a variety of basins, the estimated discharges, Q T, have been scaled by the true at-site T-year discharge, Q T, using the function QT~ QT/QT- The four statistics calculated for each T-year discharge at each site are: the average bias, the average root mean square error (rmse) and two confidence limits, ucl and lcl. These limits represent the 90% confidence limits (5% on each tail) under the assumption that the quantile estimation errors are normally distributed. The four statistics calculated for each site are given below. The index i represents the site, T the T-year flood discharge, and k the Monte Carlo repetition. For ease of presentation, the i index on Q and Q has been dropped. t ^-i U Yk Q T average bias: b, r = 2, ~ (1) N rk =l QT N r /A ^ \ 2 1/2 average rmse: _Ly Qn ~ Q N,.M Q it [(Qr-Qr) S T ] average ucl: u lt = (3) QT [(Q T -Q T )-1.645S r ] average lcl: L, T = (4) QT where for each site i 1 Nr QT = T7^QT k (5) Sr = (j7 t QT) - Or (6) in which N r = number of Monte Carlo repetitions. The regional average statistics were estimated from the N s sites in a similar manner. These statistics were j N s average regional bias: b T = ZJ^IT (7) 1 N s \ 1/2 average regional rmse: r T = I 2J rf T I (8) N s i=1 1 ^ average regional ucl: U T = 2J ^«T (9) average regional lcl: 1 Ni L T = ZJ L it (10) N s ;=i in which N s = the number of sites. For a flood frequency procedure to be performing accurately, it should have zero bias and low rmse, resulting in confidence bounds positioned tightly about zero on the scaled discharge ordinate. (2) 1047

6 Alternative Flood Frequency Procedures. The statistical performance of six alternative flood frequency models/procedures are considered. For alternatives 1-3, the procedures are flood data analyzed using the Water Resources Council guidelines (5), differing from each other in the manner of defining the log-skew value. For alternative 1, skew was based on only the at-site data; for 2, the skew was calculated by weighting the regional average log-skew and at-site log-skew by the weighting equation given in Bulletin 17B (Eq. 5, p. 12); and for 3 only the regional (average) log-skew is used. The USGS computer program J407 (USGS Watstore Volume 4) was used to fit the flood data according to Bulletin 17B guidelines (W. O. Thomas, personal communication). Procedures 4 and 5 fit the annual extreme flood data to the generalized extreme value (GEV) distribution (9) by probability weighted moments (PWM) (8). The GEV distribution was included due to its theoretical appeal, as mentioned in the introduction, and its use in Great Britain for flood risk estimation (13). The GEV distribution has the form F (x) = exp exp i exp k(x - u) a (*-») a, k = 0 in which x bounded by u + a/k from above if k > 0 and from below if k < 0. Here, u and a are location and scale parameters, respectively, while the shape parameter k determines which extreme-value distribution is represented. Fisher-Tippett types I, II, and III correspond to k = 0, k < 0, and k > 0, respectively. When k = 0 the GEV distribution reduces to the Gumbel distribution. It can be shown (6) that the distribution of extreme maxima from any distribution asymptotically approaches one of the three extreme value distributions. The PWM method has been shown to provide more accurate X-year flood discharge estimates than maximum likelihood techniques using sample sizes commonly found in hydrology (8). Procedure 4 estimates the at-site T-year flood discharges using only data from the individual site, i.e., there is no pooling of data from other sites. Pooling has the potential to transfer information from other sites. To what extent depends upon how the data is pooled, the correlation among sites, and the statistical homogeneity of the sites. Detailed analyses of these factors is beyond the scope of this paper. Procedure 5 carries out a regional analysis by pooling the data, estimating a regional flood frequency curve using PWM estimators, and applying the regional curve to each site. Since the data represent a variety of basin sizes, the first step in the regionalization is to scale the data by dividing the atsite data by the at-site sample mean so that the regional flood curve will have a mean of 1. Flood discharge estimates for each site are obtained by scaling the regional estimates by the at-site mean. The approach taken herein is essentially an index flood method. Details of the approach are given in Refs. 4 and 16 with the slight modification that herein each site is weighted by its record length. Subsequently, we shall refer to this (11) 1048

7 methodology as the GEV/PWM algorithm. Recently the Wakeby distribution, fitted by PWM, has been proposed for estimating flood quantile values (16). Therefore, procedure 6 fits the Wakeby distribution using regional PWM in the same manner as described above for procedure 5. The Wakeby distribution has the form q = m + fl'[l - (1 - Ff] - c[l - (1 - F)-"] (12) in which F = F(q) = Prob [Q < q]. The properties of the Wakeby distribution can be found in Ref. 11 and references therein. Two points are significant with respect to this procedure: 1. The Wakeby is an extremely flexible distribution in that particular combination of the parameters m, a, b, c, and d result in a distribution similar or identical to those most commonly used in flood frequency estimation. 2. Since the Wakeby has five parameters, estimation based upon a single site data is poor and so only regional estimation similar to procedure 5 is of interest to flood frequency analysis. The procedure will be referred to as WAK/PWM in the sections that follow. RESULTS Accuracy in Quantile Estimation. The first set of Monte Carlo runs investigated the performance of the six procedures when the hypothetical region consisted of the mix of basins having the statistical characteristics given in Table 1. Flood data were generated from a log-pearson III distribution and the sample statistics given earlier were averaged over 1,000 repetitions. Once again, to save the reader from being inundated with tables and figures of results, only a selected few are presented in detail, and through analysis, extended to the conditions not presented in detail. Figs. 2(a-/) present the average regional bias, Eq. 7, average 5% upper confidence limit, Eq. 9, and the average 5% lower confidence limit, Eq. 10, for the six flood estimation procedures over the quantile range 0.95 (T = 10 yr flood) to (T = 1,000 yr flood). A comparison between the two at-site procedures 1 and 4 show that they have significantly wider confidence bounds than the regional procedures, as a group. Within the regionalization procedures, the Wakeby performed best, having virtually zero bias and very tight confidence bounds. Next was the GEV/PWM procedure with confidence bounds comparable to the Wakeby but with higher bias. The two regional WRC/ log-pearson III procedures performed significantly worse with confidence bounds between 150% and 240% wider than the Wakeby confidence bounds. Also notice how the Wakeby confidence bounds are essentially constant across the quantiles. Most of the uncertainty in the flood frequency curve for the WAK/PWM arises from the error in estimating the at-site mean which is used to standardize the data prior to regionalization; in noted contrast to the WRC procedures, which show a near exponential increase in the width of the confidence bounds with increasing return period. 1049

8 =5* (a) /,*' ^ / / lb) _._ 5%UCL BIAS 57.LCL, i 1 ' ' ' Id) 5%UCL BIAS 5%LCL RETURN - M PERIOD,T _-.---, RETURN PERIOD, T FIG. 2. Bias and Upper and Lower 5% Confidence Bounds Based upon Sample Functions Generated as Log-Pearson III According to Specification of Table 1: (a) Procedure 1, At-Site Skew; (b) Procedure 2, At-Site Regional Weighted Skew; (c) Procedure 3, Regional Skew; (d) Procedure 4, At-Site GEV/PWM; (e) Procedure 5, Regional GEV/PWM; (f) Procedure 6, Regional WAK/PWM For specific sites within the region, the procedures may provide more accurate quantile estimates than those summarized in Figs. 2{a-f), and a study of these differences is instructive. For the log-pearson III results, it appears that sites with negative log-skew have significantly poorer performance than sites with a positive log-skew statistic (see Table 3). Furthermore, there is a linkage between the magnitude of the log-skew, the percentage of the simulations that detected low outliers and poor performance. The relationship between low outlier detection and performance is given in Table 3 for the 500-yr flood (0.998 quantile). In particular, notice the poor WRC performance on sites 8-10, 13, and 18, which had a high percentage of data sets with one or more detected low outliers. To test whether the observed poor WRC algorithm behavior was caused by the heterogeneity of hypothetical region 1, two alternative regions were analyzed (identified below as regions 2 and 3). These two regions each had 20 sites. The at-site statistics for region 2 were all equal to those of site 4 of Table 1, while the at-site statistics in region 3 were equal to site 10 of Table 1. Regions 2 and 3 can be thought of as mimicking the proverbial "peasin-a-pod" sets of experimental catchment hydrology, with each basin id- (r) 1050

9 TABLE 3. Performance in Estimating 500-Year Flood for Three Regional Estimation Techniques in 20 Site Heterogeneous Region Downloaded from ascelibrary.org by University of California, Irvine on 09/22/16. Copyright ASCE. For personal use only; all rights reserved. Site (1) Average "Equation 1. 'Equation 2. Log-skew (2) Repetitions detecting outliers (%) (3) << ) , REGION 3 u REGIONS bi REGiON2rr REGION 2 bi Log-Pearson III (Regional Skew) Bias a (4) (W rmse b (5) O 100 IOOO RETURN PERIOD, T GEV (Regional) Bias (6) L5 [M rmse (7) Wakeby (Regional) Bias (8) rmse 0) FIG. 3. Bias and Associated Root Mean Square Error, for Homogeneous 20 Site Regions; Region 2 Is Equivalent to Site 4 of Table 1; and Region 3 Is Equivalent to Site 10 of Table 1: (a) Log-Pearson III Weighted Regional Skew with Outlier Algorithm; (h) Regional GEV/PWM; (c) Regional WAK/PWM 1051

10 having equal flood potential as the observed samples as being random manifestations of a stationary uniform climate. Below, we will refer to regions 2 and 3 as "homogeneous" regions. Figs. 3(o-c) present the results for the three regional procedures: regional WRC with regional log-skew weighting and outlier detection, regional GEV/PWM and WAK/PWM for both of the preceding homogeneous regions. The following observations can be made: 1. The regional WRC procedure gave the poorest quantile estimates in terms of normalized root mean square error, rmse, (Eq. 2). The Wakeby procedure WAK/PWM performed significantly better in both the bias and the rmse performance criteria as can be seen from Figs. 3(a-c). The GEV/PWM procedure performed poorer at high quantiles (T > 500 yr) for homogeneous region 3 than for site 10 of the heterogeneous region with regard to the bias. In fact, the increase in bias for the high quantile homogeneous region, when compared io the heterogeneous region, was sufficient to negatively influence the rmse performance criteria. 2. Except for the very high quantile values estimated by the GEV/PWM procedure, all regional procedures performed better on homogeneous regions than they did on heterogeneous regions. The percentage improvement in normalized rmse is shown in Table 4. Notice that the WRC and WAK/PWM procedures have larger improvements in region 3 (a negative log-skew region based on site 10 of Table 1) than for region 2 (a positive log-skew region). A heterogeneous region of negative log-skews could, therefore, be expected to result in relatively poorer quantile estimates than would be observed for a homogeneous region of negative log-skews, an important thought to be kept in mind when using the WRC guideline with U.S. flood data for which varying negative skews form the rule not the exception. Effect of Low Outlier Algorithm. The WRC procedure has both low and high outlier detection algorithms. In the case of the low outlier al- TABLE 4. Percentage Improvement in Normalized rmse Due to Regional Homogeneity Region (1) 2 3 Return period (2) , ,000 WRC (3) GEV/PWM (4) a -14.7" WAK/PWM (5) "Negative values indicate a decrease in performance. Note: Region 2 quantiles compared to site 4 (Table 1); Region 3 quantiles compared to site 10 (Table 1). 1052

11 Downloaded from ascelibrary.org by University of California, Irvine on 09/22/16. Copyright ASCE. For personal use only; all rights reserved. gorithm, once detected the computer program eliminates the datum point and recalculates the site statistics. For high outliers, the data point is retained in the analysis if no historical information is used. (Note that in the analyses presented here no historical information was used.) In the last section, it was observed that sites which had the propensity for low outliers had quantiles that were poorly estimated by the WRC algorithm. To further test this observation, the low outlier algorithm was by-passed and the Monte Carlo experiments repeated. The results show that the bias in the quantile estimates is slightly (about 15%) lower without the outlier algorithm for those sites with high C v and negative logskew but shows no change for the lower C v, positive log-skew sites. There was virtually no change from the confidence limits of the quantile estimates shown in Fig. 2 as a result of either the inclusion or exclusion of the outlier algorithm. Therefore, the poor performance of the WRC procedure on sites with C v high enough to result in negative log-skew is not related to the low outlier algorithm specifically but is related to the basic estimation procedure in total. Effect of Record Length. The effect of varying record length on quantile estimate was investigated for two sites. One site, identified as LENGTH OF RECORD (years) FIG. 4. Bias, Root Mean Square Error, and Upper and Lower 5% Confidence Bounds for Site Having Positive Log-Skew = 0.239, within Region of 40 Sites Having Total Region Size of 1,500 Station-Years of Data: (a) Procedure 2, WRC Log- Pearson II, with Regional Weighted Skew; (b) Procedure 6, Regional WAK/PWM I0D- I0C Q[ i.o (a) UCLioO _ 60 >-CLl LENGTH OF RECORD ' (b) *^ years) FIG. 5. Bias, Root Mean Square Error, and Upper and Lower 5% Confidence Bounds for Site Having Negative Log-Skew = , within Region of 40 Sites Having Total Region of 1,500 Station-Years of Data: (a) Procedure 2, WRC Log- Pearson III, with Regional Weighted Skew; (b) Procedure 6, Regional WAK/PWM 1053

12 site A, had statistics: mean = 150, C v = 0.687, skew = 3.0, and log-skew = 0.239; the second site, site B, had statistics: mean = 158, C v = 0.964, skew = 3.0, and log-skew = Record length was varied from yr in increments of 15. The sites were part of a 40 site heterogeneous region with a total record of 1,500 station yr. Results are given in Figs. 4(a-b) and 5(a-b) for the T = 100 yr flood discharge. Notice that the WAK/PWM procedure has significantly lower rmse than the regional WRC procedure. In fact, the performance of the WAK/PWM procedure at a record length of 15 yr is not equaled with the WRC procedure until a record length of 60 yr. The decrease in scaled rmse for all methods appears to be about n~ 1/2, in which n = record length, but the WAK/ PWM started with much lower rmse than the WRC procedures, and this superiority appears to be maintained over a wide set of experimental conditions. The results for a regional GEV/PWM (not shown) were similar to those for the WAK/PWM algorithm. SUMMARY AND CONCLUSIONS The performance of the Water Resources Council, WRC, flood frequency procedures, and computer algorithm was tested and compared to alternative flood frequency procedures under the assumption that flood frequency follows a log-pearson III probability distribution. It was found that among the WRC procedures, the one that performed best used a weighted log-skew coefficient based upon the average regional log-skew and the at-site log-skew (Eq. 5 of Bulletin 17B, Ref. 5). Using only the average regional log-skew was essentially as good. The alternative regional procedures GEV/PWM and the WAK/PWM performed significantly better. Specifically, for the heterogeneous log-pearson III region of Table 1, the average 90% confidence band for the 100 yr event were as follows. 1. WRC at-site skew procedure: Q I00 < Qwo < Q wo. 2. WRC weighted skew procedure: Q m < Q 100 < Q wo. 3. WRC regional average procedure: Q wo < Q W o < Q GEV/PWM at-site procedure: Q 100 < Q 100 < 1.50 Q GEV/PWM regional procedure: Q 1Q0 < Q 100 < Q WAK/PWM regional procedure: Q 100 < Q 100 < Q 100. For homogeneous regions, the GEV/PWM and WAK/PWM again outperformed the WRC procedures. It was found that the WRC procedures provided especially poor estimates for basins with negative skews in the log transformed data. Since a significant portion of flood data exhibit negative skews in log-space, the use of the WRC procedures on such data is an additional cause of concern. In the work reported herein, the computer experiments were based upon log-pearson III data. Additional computer work using data from a variety of other probability models (Weibull, GEV, and Wakeby) gave the same results with one modification, i.e., the WRC performed rela- 1054

13 tively poorer than for the reported log-pearson III data. Therefore, it must be concluded that the WRC flood frequency procedures, as outlined in Bulletin 17B, should not be used as a basis for engineering design because significantly more accurate estimates can be obtained by other currently available statistical procedures. The Water Resources Council recognized that the U.S. national guidelines would need to be periodically re-evaluated and updated in light of future research findings. Given the results presented herein and related research from other studies (7,14,16,17,19), we suggest that the current national guidelines now need to be reassessed. In the interim, it is suggested that design engineer could avail themselves of the permissible, and legal, option of substituting a more accurate and consistent methodology. In fact, given the weight of current evidence failure to do so might well be considered by a court as construing professional negligence. ACKNOWLEDGMENTS This work was supported in part by the Institute of Hydrology (UK). The writers are greatly indebted to Drs. J. S. G. McCulloch of IH and P. E. O'Connell of the University of Newcastle-upon-Tyne for sustaining this research effort. The writers are also grateful to Dr. D. Lettenmaier for his timely comments during the research and on earlier drafts. APPENDIX. REFERENCES 1. Bobee, B., "The log-pearson Type III Distribution and Its Application in Hydrology," Water Resource Research, Vol. 11, No. 5, 1975, pp Fisher, R. A., and Tippett, L. H. C, "Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample," Proceedings of the Cambridge Phil. Society, Vol. 24, 1928, 24 pp. 3. Galanbos, J., "The Asymptotic Theory of Extreme Order Statistics," lohn Wiley and Sons, New York, N.Y., Greis, Noel P., and Wood, E. F., "Regional Flood Frequency Estimation and Network Design," Water Resource Research, Vol. 17, No. 4, 1981, pp "Guidelines for Determining Flood Flow Frequency," Bulletin 17B of the Hydrology Subcommittee, Interagency Advisory Committee on Water Data, U.S. Department of the Interior, Geological Survey. 6. Gumbel, E. ],, Theory of Extremes, Columbia University Press, New York, N.Y., Harter, H. L., "A New Table of Percentage Points of the Pearson Type III Distribution," Technometrics, Vol. 11, No. 1, Feb., 1969, pp Hosking, I. R. M., Wallis, J. R., and Wood, E. F., "Estimation of the Generalized Extreme-Value Distribution by the Method of Probability Weighted Moments," MRC Technical Report 2674, Mathematics Research Center, University of Wisconsin, Apr., 1984, p lenkinson, A. F., "The Frequency Distribution of the Annual Maximum (or Minimum) of Meteorological Elements," Quarterly Journal of Royal Meteorological Society, Vol. 81, 1955, pp Landwehr, I. M., Matalas, N. C, and Wallis, J. R., "Some Comparisons of Flood Statistics in Real and Log Space," Water Resource Research, Vol. 14, No. 5, 1978, pp Landwehr, I. M., Matalas, N. C, and Wallis, I. R., "Quantile Estimation 1055

14 with More or Less Floodlike Distributions," Water Resource Research, Vol. 16, No. 3, 1980, pp Matalas, N. C, Slack, J. R., and Wallis, J. R., "Regional Skew in Search of a Parent," Water Resource Research, Vol. 11, No. 6, 1975, pp Natural Environmental Research Council (NERC) Floods Studies Report, Vol. I-V, Natural Environment Research Council (UK), London, England, Reich, B., "Lysenkoism in U.S. Flood Determinations," Transactions of the American Geophysical Union, Vol. 58, No. 11, Nov., 1977, 1135 pp. 15. Wallis, J. R., Matalas, N. C, and Slack, J. R., "Just a Moment!," Water Resource Research, Vol. 10, No. 2, Apr., Wallis, J. R., "Risk and Uncertainties in the Evaluation of Flood Events for the Design of Hydraulic Structures," Piene e Siccita, Guggino, Rossi, and Todim, eds., Fondazine Politechnica Del Mediterraneo Gatania, Italy,