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Journal of Hydrology 307 (2005) 175–195 www.elsevier.com/locate/jhydrol

Frequency analysis of nonidentically distributed hydrologic ?ood data

V.P. Singha,*, S.X. Wangb, L. Zhanga

a

Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803-6405, USA b Bureau of Hydrology, Yangtze Valley Planning Of?ce, Wuhan, Hubei 430010, China Received 9 July 2003; revised 30 August 2004; accepted 1 October 2004

Abstract The curve-?tting method of ?ood frequency analysis, based on a plotting position formula, does not accommodate nonidentically distributed ?ood sequences. On the other hand, the methods of combining component-frequency distributions may overestimate the exceedance probability of a given ?ood quantile. This study develops a method for estimating the exceedance probability of a speci?ed ?ood magnitude for nonidentically distributed ?ood sequences. The method is applied to several data sets, and its results are found to be in good agreement with observed values. q 2004 Elsevier B.V. All rights reserved.

Keywords: Exceedance probability; Flood frequency; Flood quantile; Log-Pearson type 3 distribution; Probability distribution

1. Introduction The value of a design ?ood estimated from annual maximum ?ood ?ow series with conventional methods of ?ood frequency analysis is based on the assumption that the ?ow series is statistically independent and identically distributed (iid) from year to year. However, the iid assumption is not valid in many cases (Todorovic and Rousselle, 1971). After carefully analyzing the logarithmic probability plots of observed ?ood sequences at many stream gaging stations from different parts of the world, Singh (1968) observed that these plots exihibited reverse

* Corresponding author. Fax: C1 225 388 8652. E-mail address: cesing@lsu.edu (V.P. Singh). 0022-1694/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2004.10.029

curvatures, and he attributed the curvatures to the heterogeneity of the ?ood populations. According to Singh (1987), mixed populations and hence the mixed distributions occur because of a host of factors, such as different types of ?ood-producing storms, rainfall and snowmelt ?oods, dominance of within-thechannel or ?oodplain ?ow, antecedent basin soil moisture, and vegetal cover conditions. Waylen and Woo (1982, 1983) reported that in the Cascade Mountains of the Paci?c northwestern US and southwestern British Columbia, Canada, high ?oods occurred due to heavy winter rainfall or snowmelt in the spring. They further observed that the annual ?ood series was produced by more than one hydrometeorologic process and hence suggested that the individual frequency distribution of snowmelt ?ood and that of the rain-generated ?ood should be

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compounded to provide a better ?t of the frequency distribution to the annual ?ood series. Kirkham (2002) described the peak ?ow causative mechanisms and separated peak ?ows by seasonal mechanisms. Causative mechanisms included rain, rain-on-snow, rain on frozen ground in winter, clear sky or rain-onsnow in spring, and thundershowers in summer. Frequencies were determined for ?oods that occurred in winter, prior to snowmelt, in spring during snowmelt, and in summer after the snowmelt period. Stratifying peak ?ows based on the season of occurrence provided a more accurate estimate of peak ?ow frequency distributions than would be obtained if the annual peak ?ow data were employed. Diehl and Potter (1987) found that the ?oods in Wisconsin were of two types that were hydrologically distinct and the annual ?ood series was a mixture of two or more populations. Hirschboech (1987) analyzed more than 2000 ?ood events from the Gila River basin in central and southern Arizona spanning the period 1950–1980, and linked them to climatic characteristics by analyzing daily synoptic weather maps. The analyses demonstrated that ?oods in the Gila River basin resulted from a variety of atmospheric processes that varied seasonally and from year to year, as well as spatially. When the ?oods were grouped based on their meteorological causes such that each group of ?oods was homogeneous, then the ?ood frequency distributions exhibited means and variances that differed from those of the overall frequency distribution of the entire ?ood series. Alila and Mtiraoui (2002) investigated long-term hydroclimatic records of the Gila River basin in southeast and central Arizona using the assumption of the annual maximum discharge record being drawn from an independent and identically distributed population as well as using the mixed population method. The least squares method was applied for the estimation of parameters of the mixed population method. In the estimation procedure, the ?rst and second order moments of the annual discharge series were kept constant. Their results showed that the mixed population method ?tted the annual discharge series better than did the conventional direct ?tting method. The annual maximum ?oods in southern Louisiana are caused by frontal rainfall in December to May, whereas those caused by the Gulf tropical depressions

(GTD, tropical storms) usually occur in May through October. From the ?ood data of the Amite River basin obtained from the USGS, it was found that during a period 1962 to 2000, 24 annual maximum ?ood events (at Darlington) and 31 ?ood events (at Denham Springs) were generated by frontal rainfall during December through May. Still there were some severe ?ood events (3 events at Darlington and 8 events at Denham Springs) that were generated by the GTD during May through October. The annual maximum river ?oods in southern China are caused by frontal (monsoon) rainfall in May to July, and those in the southeastern coastal area of China in June to July, whereas those caused by the western Paci?c tropical storms (typhoons) may occur in July to September. In the northern regions of northeast China, the annual maximum ?oods may be produced by snowmelt in spring and by rainfall in summer. In the southwestern part, the upstream region of the Yangze Valley, many stream?ow hydrographs within a year exhibit seasonality. The annual maximum ?ood may occur in summer from June to early August when the monsoon fronts advance from south to north or in the fall from late August to early October when the fronts withdraw from north to south. Although both summer and fall ?oods result from frontal rains, their hydrologic characteristics are distinctly different, for the intensity of the rainproducing systems vary with season. The above discussion shows that ?oods are generated by different mechanisms and as a result there exist nonidentically distributed annual maximum ?ood series in different regions of the world (Beran et al., 1986; Rossi et al., 1984). Thus, for the planning, design and management of water resources projects, ?oods within a year may not be adequately represented by their maximum and the design ?ood or rainfall value corresponding to a speci?ed frequency should, therefore, vary with season. Furthermore, the annual maximum ?ood or rainfall series may not be identically distributed and the design ?ood value, based on the annual maximum ?ood (or rainfall) series and current empirical estimation methods, then become questionable. The objective of this paper is to develop a method for estimating a design ?ood from nonidentically distributed series and provide an estimation procedure applicable for practical use.

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177

2. Review of existing methods The methods of statistically obtaining a design ?ood value from nonidentically distributed ?ood series, proposed in the hydrologic literature, can, in general, be classi?ed into two groups: estimation based on the annual maximum series (unclassi?ed data) directly, or that based on the seasonal maximum (classi?ed) data. Singh (1968, 1987) and Singh and Sinclair (1972) developed a mixed distribution model p(x) considering the observed annual maximum ?ood series, X, (or their logarithms in a non-normal case) drawn from two normal populations with probability distributions p1(x) and p2(x) having means m1 and m2, variances s2 and s2 , 1 2 and relative weights a and 1Ka, respectively: p?x? Z ap1 ?x? C ?1 K a?p2 ?x? (1)

and the EM algorithm did not guarantee convergence to the global maximum. Rossi et al. (1984) applied the two-component extreme value distribution to analyze 39 annual ?ood series of Italian basins. This distribution assumed individual ?oods to arise from a mixture of two exponential components. Its parameters were estimated by the maximum likelihood (ML) method. Fiorentino et al. (1987) estimated the parameters by using the principle of maximum entropy (POME). The POME method was found simpler than the ML method, and was applicable in both the site-speci?c and regional cases. In contrast, Waylen and Woo (1982) suggested that the overall annual maximum ?ood distribution FT may be represented by FT ?x? Z F1 ?x?F2 ?x? (2)

Singh (1987) estimated parameters (m1, m2, s1, s2, and a) from the observed annual maximum ?ood series with a nonlinear programming algorithm by minimizP ing the objective function jDzj subject to ?ve constraints, where Dz is the difference between the standard normal deviate corresponding to the observed probability pm equal to (mK0.38)/(nC0.24) and that obtained from the ?tted mixed-distribution equation corresponding to p, m is the rank of the ?ood series, and n is the sample size. By analyzing six annual ?ood series from rivers in Japan, the USSR, Poland, Czechoslovakia, Italy, and USA, Singh (1987) showed the superiority of the mixed distribution method in yielding reliable ?ood estimates. Johnson and Kotz (1970) recommended the method of moments to estimate the ?ve parameters of the Singh model where ?ve moments are needed. However, this procedure is not viable for practical use due to the susceptibility of higher-order moments to data errors. Leytham (1984) derived the maximum likelihood estimates with the use of the expectationmaximization algorithm (EM algorithm). He investigated small sample properties of the parameter estimates by Monte Carlo simulation and concluded that the parameter estimates from unclassi?ed data were found to be inaccurate and greatly inferior to the estimates from the corresponding classi?ed data, whereas the properties of quantiles estimated from classi?ed and unclassi?ed samples were found to be in reasonable agreement. He, further, pointed out that there might be singularities on the likelihood surface

where ?oods, produced by two independent processes, are characterized by distributions F1 and F2. Representing F1 and F2 by the Gumbel distribution in Eq. (2), the procedure to obtain FT consisted of three steps: (1) identi?cation of two independent subpopulations, the snowmelt- and rainfall-generated; (2) estimation of parameters for each subsample; and (3) application of Eq. (2) to obtain the probabilities with which ?oods of various magnitudes would occur. USWRC (1981), CHNMWREP (1983), and USSRNHI (1984) stressed the need to classify the annual maximum ?oods caused by different generating mechanisms, based on hydrometeorologic and statistical considerations. CHNMWREP (1983); USSRNHI (1984) recommended combining the seasonal ?ood frequency distribution p1(x) and p2(x) (if only 2 classes) into p(x), the ?ood frequency distribution of the annual maximum, by the probability law: p?x? Z p1 ?x? C p2 ?x? K p1 ?x?p2 ?x? (3) assuming the seasonal maximum variables X1 and X2 are mutually independent and identically distributed. The estimation method of Singh (1987) is actually a curve-?tting procedure. The results of such a method, as discussed by Wang and Singh (1992), depend, to a large degree, on the plotting position formula adopted. The formula, PmZ(mK0.38)/(nC 0.24) due to Blom (1958), used by Singh (1987) when

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3/8z0.38, gives a good approximation to the probability of the expectation of the mth order statistic from the normal population (Cunnane, 1978). However, similar to many other types of plotting position formulae, this formula is derived from the order statistics of independent identically distributed samples. Thus, the nonidenticality of distributions of the annual maximum ?ood series may invalidate this formula. Consequently, the ?tted distribution and its parameters estimated as above become questionable. A plotting position formula applicable to a nonidentically distributed sample is a pre-requisite for such a ?t, which however, does not appear to have been developed in the hydrologic literature yet. When using classi?ed data (seasonal maxima), the ?ood frequency distribution of the annual maximum ?ood obtained from Eq. (2) is the distribution of max (X1, X2), or the joint probability p{X1%x,X2%x}. If the equality F(x)Z1Kp(x) is used, Eq. (2) immediately turns into Eq. (3), i.e. Eqs. (2) and (3) are essentially the same; whereas Eq. (3) represents the probability of the sum of two independent events, {X1Ox} and {X2Ox}, i.e. p{(X1Ox)g(X2Ox)}, meaning that the probability that at least one of these two events occurs. However, of concern in ?ood frequency analysis is the exceedance probability of the annual maximum, the largest of the seasonal maxima within a year, regardless of whether others exceed x or not, i.e. the probability of either {X1Ox} if {X1OX2} or {X2Ox} if {X2OX1}. Furthermore, to be the annual maximum, X1 and X2 are obviously exclusive of each other because the annual maximum is unique and only one of them can be the maximum. Thus, the frequency distribution given by Eqs. (2) and (3) may be overestimated. Moreover, if p1(x)Z p2(x)Zp0(x), i.e. the case of identical distributions, it should be expected that p(x)Zp1(x)Zp2(x)Zp0(x), while Eq. (3) leads to p?x?Z 2p0 ?x?K p2 ?x? sp0 ?x?. 0 This clearly shows the inadequacy of existing curve ?tting methods and hence the need for their improvement.

which, in general, can be discerned through hydrometeorologic analyses of ?ood-generating processes. The annual maximum ?ood variable X can, then, be de?ned as X Z maxfQ?t?; t 2Tg Z maxfmax?Q?ti ?; ti 2DTi ?g Z maxfXi g

i i2s

(4)

where Q(t) stands for the stream?ow hydrograph, and Xi, iZ1,.,s, is the maximum in the ith time interval (regarded as the ith seasonal maximum, from now onwards). 3.1. Basic assumptions For development of the methodology, the following assumptions are made (1) The seasonal maximum ?ood series, Xi, of any ith season in a year is identically distributed as pi(x) (exceedance probability) since it results from the same ?ood-generating process, while the maxima of different seasons are nonidentically distributed, i.e. pi ?x? spj ?x?; i sj (5)

(2) The seasonal maximum ?ood series are statistically independent of each other, i.e. p?Xi R x; Xj R x? Z pi ?x?pj ?x?; i sj pfXi R x; Xj R x; Xk R xg Z pi ?x?pj ?x?pk ?x?; i sj sk ? p?X1 R x; .; Xs R x? Z p1 ?x?.ps ?x? (6) (3) The annual maximum ?ood X may occur in different seasons with different probabilities. Taking a simple example, let X be the annual maximum ?ood series, X1 the winter maximum ?ood series, and X2 the summer maximum ?ood series. Then, X(i), the maximum ?ood in ith year would either happen in the winter maximum ?ood series X1 or the summer ?ood series X2. Since X1 and X2 usually do not belong to the same population, the probability of X(i) being drawn from X1 may be different from that if it were drawn from X2. Thus, If an

3. Development of methodology Let the ?ood period T within a year be partitioned into s time intervals (or seasons) DT1,.,DTs, without overlap according to ?ood-generating mechanisms,

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179

element is de?ned as: fAi g Zfannual maximum flood that occurs in the i K seasong; i Z 1; .; s then 0! pfAi g! 1; i Z 1; .; s;

s X iZ1

the probability multiplication theorem: p?x? Z

s X iZ1

?7?

pfAi gpfXR xjAi g Z

s X iZ1

p?Ai ?p?xjAi ? (13)

pfAi g Z 1

(8)

(4) If Ai is further referred to as a subset of the annual maximum ?ood series that occurs in the ith season, then events {Ai}, iZ1,.,s, are mutually exclusive since only one of s seasonal maximum ?oods can be the annual maximum, i.e. Ai h Aj Z F; i sj (9)

Eq. (13) is the total probability law and expresses the frequency distribution of the annual maximum ?ood as the sum of the frequency distributions of those annual maximum ?oods that are conditioned on the maxima occurring in the ith season with the probability weight p(Ai) that an annual maximum occurs in the ith season, iZ1,.,s. When p(xjAi)Zp(xjAj), isj, i, jZ1,.,s, i.e. the annual maxima occurring in different seasons are identically distributed, the conditional frequency distributions p(xjAi), iZ1,.,s, are free of Ai, then Eq. (13) leads to p?x? Z p0 ?x?

s X iZ1

where F is referred to as the impossible event, or an empty set and

s X iZ1

p?Ai ? Z p0 ?x?

(14)

Ai Z U

(10)

indicating that A1,., As partition the sample space U of the annual maximum ?ood (X). 3.2. Derivation of the frequency distribution Based on the above assumptions, the frequency distribution of the annual maximum ?ood X can be derived as follows Considering that the occurrence of an event BZ{XRx} must be associated with one of the events {Ai}, iZ1,.,s, fXR xg Z

s X iZ1

where p0(x) is a ?xed frequency distribution indicating that the overall annual maxima are identically distributed. If p(xjAi), iZ1,.,s, are continuous and differentiable, then the probability density function of X exists and takes the form f ?x? Z where fi ?xjAi ? Z dp?xjAi ? dx (15)

s X iZ1

p?Ai ?fi ?xjAi ?

fBh Ai g

(11)

3.3. Parameter estimation The moments and parameters of the annual maximum ?ood frequency distribution given by Eq. (13) can be derived as follows. The rth moment about the origin can be de?ned as ? ? P mr Z xr dp?x? Z xr p?Ai ?dp?xjAi ? P Z s p?Ai ?mir (16) iZ1 in which mir is the rth original moment of the conditional frequency distribution p(xjAi).

From the assumption (4), the joint events {BhAi}, iZ1,.,s, should also be mutually exclusive. From Eq. (11), and summing the probabilities, the following is obtained: p?XR x? Z

s X iZ1

p?Bh Ai ?

(12)

Finally, one obtains the frequency distribution for the annual maximum ?ood ?ow by using

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When rZ1, one obtains the mathematical expectation of the annual maximum ?ood X: E?X? Z m1 Z

s X iZ1

p?Ai ?E?Xi ?

(17)

x(p0) when p(Ai) and p(xjAi), iZ1,.,s, are known. However, because of the nonlinearity in the x–p relationship of the conditional frequency distribution, an iterative procedure (for instance, the Newton– Raphson method) may be necessary: xkC1 Z xk K where g 0 ?x? Z

s X iZ1

where E(Xi), iZ1,.,s, is the mathematical expectation of Xi or p(xjAi). Using the relationship between the original and central moments, Eq. (16) yields: variance Z var?X? Z m2 Z m2 K E ?X? Z

s X iZ1 2

g?xk ? ; k Z 0; 1; 2; . g 0 ?xk ?

(21)

p?Ai ?

p?Ai ??var?Xi ? C E2 ?Xi ?? K E2 ?X? (18) or

s X dp?xjAi ? ZK p?Ai ?fi ?xjAi ? dx iZ1

(22) ?xk K xkK1 ?g?xk ? ; k Z 0; 1; . g?xk ? K g?xkK1 ?

where var(Xi) is the variance of Xi or p(xjAi), iZ 1,.,s, and the third central moment: m3 ?X? Z m3 K 3E?x?m2 C 2E3 ?X? Z

s X iZ1

xkC1 Z xk K

(23)

3.5. Estimation procedure To determine the exceedance probability from the annual maximum ?ood involves estimating the probability p(Ai) and the conditional frequency distributions p(xjAi), iZ1,.,s (19) 3.5.1. Estimation of p(Ai), iZ1,.,s According to its de?nition and Eq. (11), p?Ai ? Z pfXi Z max?X1 ; X2 ; .; Xs ?g s Z p h?Xi O Xj ? ; i Z 1; .; s

jsi

p?Ai ??mi3 C 3var?Xi ?E?Xi ?

C E3 ?Xi ?? K 3E?X??var?X? C E2 ?X?? C 2E3 ?X?

where mi3, iZ1,.,s, is the third central moment of Xi or p(xjAi). From Eqs. (17)–(19), the coef?cient of variation and the coef?cient of skewness, Cv and Cs, respectively, can then be obtained in the usual manner. 3.4. Estimation of quantiles In general, the exceedance probability of the annual maximum ?ood, and hence the design ?ood value can be graphically obtained when p(Ai) and p(xjAi), iZ1,.,s, have been estimated; whereas for a more accurate design ?ood value, a numerical algorithm might be necessary Given the design exceedance probability p 0, Eq. (13) can be rewritten as g?x? Z p0 K

s X iZ1

(24)

p?Ai ?p?xjAi ? Z 0

(20)

Eq. (24) indicates that p(Ai) can be estimated by comparison between the seasonal maximum ?ood series. After having drawn s seasonal maximum ?ood series of size n, the probability p(Ai) can be estimated as follows: (a) The simplest case: sZ2: Following the Bernoulli concept in Mises (1964), suppose (x11,.,x1n) and (x21,.,x2n) are independent, identically distributed (iid) samples of X1 and X2, respectively. Let ( ) 1; xik O xjr ?1? Zkr ?i? Z r; k 0; otherwise Z 1; .; n; i sj; i; j Z 1; 2 (25)

with design event value x(p0) as an unknown parameter. In general, Eq. (20) can be solved for

V.P. Singh et al. / Journal of Hydrology 307 (2005) 175–195

?1? Obviously, Zkr ?i? is a random variable and follows the Bernoulli distribution with ?1? pfZkr ?i? Z 1g Z pfXi O Xj g ?1? pfZkr ?i? Z 0g Z 1 K p?Xi O Xj ?

181

(26)

P ?2? The summation s Zkr ?i? is the total number of kZ1 the elements of the Xi ?ood series being simultaneously larger than those of the other two ?ood series in the total n2 times of comparison of the elements of the Xi ?ood series with the pairs of the other two. Thus, the statistic

?2? T?i? Z n n 1 X X ?2? Z ?i?; i Z 1; 2; 3 n2 kZ1 rZ1 kr

Hence,

?1? E?Zkr ?

Z pfXi O Xj g

(27)

(33)

Since every element of the Xi ?ood series would be compared with all P elements of the Xj ?ood series, the ?1? it is expected that n Zkr ?i? is the total number of kZ1 the elements of the Xi series being larger than those of P ?1? the Xj series. n Zkr ?i? is the total number of the rZ1 elements of the Xj series being larger than those of the Xi series. We construct the statistic

n n 1 X X ?1? T ?1? ?i? Z 2 Z ?i?; i Z 1; 2 n kZ1 rZ1 kr

is obtained with E2 ?T ?2? ?i?? Z

n n 1 XX ?2? E?Zkr ?i?? n2 kZ1 rZ1

Z pfXi O Xj1 ; Xi O Xj2 g

(34)

(28)

with the expectation: E?T ?1? ?i?? Z

n n 1 XX ?1? E?Zkr ?i?? Z pfXi O Xj g n2 kZ1 rZ1

(29)

Therefore, Eq. (28) provides an unbiased estimation of probability p(Ai). (b) Case: sZ3: Let 3 seasonal maximum series of size n be from a 3xn dimensional matrix Y. Each ?ood series becomes a row of the matrix Y. Let ( 1; xik O yir ?2? Zkr ?i? Z i Z 1; 2; 3 (30) 0; otherwise where yir is the rth column of the 2xn dimensional submatrix Yi of Y, which is obtained by deleting the ith row, the Xi ?ood series from Y, and xik>yir means that xik is larger than both components of yir, xj1r, and xj2r, i.e. xik>xj1r and xik>xj2r, j1sj2, respectively. ?2? Similar to that in Eq. (25), the statistic Zkr is a discrete random variable, also following the Bernoulli distribution with

?2? pfZkr ?i? Z 1g Z pfXi O Xj1 ; Xi O Xj2 g ?2? pfZkr ?i?

Therefore, T(2)(i) in Eq. (33) is an unbiased estimator or p(Ai) in case sZ3. (c) Case: any positive integer s: An unbiased estimator of the probability p(Ai), iZ1,.,s, can be similarly constructed by considering Y as an s!n dimensional matrix and yir in Eq. (30) as the (sK1)th column (sK1 dimensional vector) of (sK1)!n dimensional submatrix Yi of Y obtained by deleting the ith row from Y. Such probability estimators p*(Ai), iZ1,.,s, should satisfy the set of Eq. (8), where the ?rst would certainly be met because of the assumption (2), but the second needs further justi?cation which can be inductively made as follows. (i) The simplest case, sZ2: Since every element of the X1 ?ood series is compared with all the elements of the X2 ?ood series, there are in total n2 times of P ?1? comparisons. Denote n1 Z n Zkr ?1?, the number kZ1 of the elements of X1 ?ood series being larger than those of the X2 series, and thus the remainder of the ?1? total n2 times of comparison in counting Zkr ?1? and ?1? 2 Zkr ?2?, n Kn1 would be the number of the X2 ?ood elements being larger than those of the X1 ?ood P ?1? elements, i.e. n Zkr ?2?, thus it follows that kZ1 p? ?A1 ? Z T ?1? ?1? Z which results in p? ?A1 ? C p? ?A2 ? Z 1 (36) n1 ; n2 p? ?A2 ?

2

ZT ?1? ?2?Z

n2 Kn1 n2

(35)

Z 0g Z 1 K pfXi O Xj1 ; Xi O Xj2 g

(31)

Hence,

?2? E?Zki ?i?? Z pfXi O Xj1 ; Xi O Xj2 g

(32)

(ii) Any positive integer s, sR3: In a manner P ?sK1? similar to the above, denote n1 Z n Zkr ?1?, i.e. kZ1 there are n1 items of X1OX2, X1OX3, ., X1OXs, in

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total, n2 times of comparisons. Therefore, there should be n2Kn1 times or at least one of the elements of the X2, X3, ., Xs ?ood series being larger than those of the X1 ?ood series. Of these n2Kn1 times, suppose P ?sK1? there are n2 times, i.e. n2 Z n Zkr ?2? of the kZ1 elements of the X2 ?ood series being larger than those of the X3,., Xs ?ood series, and hence being larger than the X1 series; thus analogously, one can reach that n2Kn1Kn2K.KnsK1 should be the elements of the Xs series being larger than those of the X1, X2,.XsK1 series and the following equations can be deduced: p? ?A1 ? Z ? nsK1 n2 2 n K n1 K . K nsK1 p? ?As ? Z n2 p? ?AsK1 ? Z with

s X iZ1

n1 n2 (37)

p? ?Ai ? Z 1

(38)

(1) Analyze the ?ood-generating processes within the whole ?ood period and see if they are homogeneous. If not, partition the total ?ood period into s ?ood seasons, if necessary. With each season, the ?ood-generating factors should be the same in order for the seasonal ?ood sequence to be identically distributed. (2) Draw s seasonal maximum ?ood series from observed data within DTi, iZ1,2,.,s, and pick the largest ?ood value every year to form the annual maximum ?ood series in which those values from the ith season are gathered to form Ai, iZ1,.,s. (3) Estimate the probability p(Ai) using Eqs. (25) and (28), or Eqs. (30) and (32),. by comparison between the seasonal maximum ?ood series derived in step (2). (4) Fit continuous distributions to the Ai ?ood series derived in step (2) by using a regular frequency analysis procedure to obtain the conditional frequency distribution p(xjAi), iZ1,.,s. (5) Combine the results of steps (3) and (4) to obtain the ?ood frequency distribution of the annual maximum. (6) Compute the parameters by Eqs. (17)–(19). 3.5.4. Direct curve ?tting method The direct curve ?tting method involved the following steps: (1) Arranging the values of X in increasing order; (2) computing the cumulative probability (called observed) using a plotting-position formula; and (3) ?tting a theoretical probability distribution to the probability values.

Thus, the probability estimation of p(Ai), iZ1,.,s, can be obtained. 3.5.2. Estimation of p(xjAi), iZ1,.,s As discussed in the preceding sections, the conditional frequency distribution p(xjAi), iZ1,.,s, describes the probabilistic nature of that part of the annual maximum ?ood that occurs in the ith season, i.e. Ai. As an event set, Ai is the cross of the annual and seasonal maximum ?ood ?ow sample spaces. Its elements belong to both spaces. Thus, the conditional ?ood frequency distribution p(xjAi) should be estimated from those observed values of the Xi ?ood series that are picked as the annual maximum ?oods. After picking those from the Xi series with size less than n, the ?tted distribution p(xjAi) can be obtained by any standard estimation procedure. 3.5.3. Step by step estimation Based on the above discussion, a step by step working estimation procedure can be summarized as follows:

4. Application 4.1. Data To validate the proposed method for estimating the ?ood frequency distribution of nonidentically distributed annual maximum ?ood sequences, four sets of data were employed: (1) data from the Reynolds Creek watershed in southwestern Idaho, (2) data from the Amite River basin in Louisiana, (3) data from 2 stations of Gila River basin at southeastern and central Arizona, and (4) data from four stations including one rainfall

V.P. Singh et al. / Journal of Hydrology 307 (2005) 175–195 Table 1 Relevant information on hydrologic data Station AMF* Winter East Outlet Tollgate 34 34 30 34 33 29 Generating process Reyholds Creek (US) Spring 34 32 30 Summer 31 16 23

183

Number of dominant generating records due to AMF Winter 1 9 9 Frontal 24 31 Spring 32 25 21 Summer 1 0 0 GTD 3 8

Amite River Basin (US) Frontal GTD* Darlington Denham Springs 27 39 27 39 27 39

Gila River Basin (US) Winter and Spring Clifton Tucson 70 72 70 72 Summer and Fall 70 72 Winter and Spring 21 7 Frontal 21 21 16 26 Summer and Fall 49 65 Typhoon 54 8 12 24

Yangzi River (China) Frontal Typhoon Wenzhou Diqing Zhushan Lianpanshan 75(AMR) 29(AMF) 28(AMF) 50(AMF) 75 29 28 50 75 29 28 50

AMF refers to the annual maximum ?ood (peak discharge); GTD refers to the Gulf tropical depression.

and three stream?ow gages in China Relevant information on data is shown in Table 1. Idaho data. Kirkham (2002) analyzed ?ood frequencies of peak ?ows from the Reynolds Creek watershed in southwestern Idaho and the data used here are from her study. It is a dry upland range basin with rocky outcrops scattered throughout the watershed and is maintained by the US Department of Agriculture–Agriculture Research Service. The data were available for three stream?ow gages: Reynolds Mountain East (drainage areaZ0.40 sq. km), Reynolds Creek at Tollgate (drainage areaZ55 sq. km), and Reynolds Creek outlet (239 sq km). Hydrometeorologic analysis of precipitation data shows that most of the precipitation is from winter storms moving over the watershed from west to southwest. Snow constitutes about 20% of the annual precipitation in the lower basin, whereas in the higher elevations about 76% of the precipitation is snow. Reynolds Creek receives the greatest amount of precipitation during the months of November, December, and January. The precipitation during

these months is primarily in the form of snow in the high elevation areas and stored in the southwest section of the basin. The months of July, August and September receive the least amount of precipitation. Based on the precipitation data analysis, the causal factors of annual peak discharge can be classi?ed into 3 mechanisms: (1) pre-snowmelt, (2) snowmelt, and (3) post-snowmelt. At Reynolds Mountain East, annual peak ?ows primarily occurred during spring (April through June), whereas at the other two gages they occurred during both spring and winter (December through June). Louisiana data. The Amite River basin, located in southeastern Louisiana and southeastern Mississippi, was selected for this study. The basin encompasses an area of approximately 5180 sq. km and includes portions of East Baton Rouge, Ascension, Livingston, East Feliciana, St Helena, Iberville, St. James and St. John the Baptist parishes within Louisiana and the Amite County within Mississippi. The Amite River originates in the southern part of Lincoln County, Mississippi; and ?ows in the south and

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southeastern direction, a distance of 273.6 km to the eastern side of Lake Maurepas in southeastern Louisiana. In the Amite River basin, ?ooding occurs mostly during December to May (Amite River Basin Commission, 1995). The rainfall producing mechanisms mainly are frontal and GTD (Gulf Tropical Depression, which occurs mostly in late spring, summer and fall). The annual maximum ?ood ?ow sequences were available for two discharge stations located at Darlington and Denham Springs. The data for the Darlington gage is from the water year 1962 to the water year 1988, and the data for the Denham Springs gage is from the water year 1962 to the water year 2000. The data both at Darlington and Denham Springs show that ?ooding during December to May is caused by frontal rains. Arizona data. The Gila River basin is located in southeastern and central Arizona, USA, which is an arid to semi-arid area. The annual maximum ?ood ?ow sequences at Clifton on the Gila River (from year 1914–2002) and at Tucson on the Santa Cruz River (from year 1911–2002) was employed. The data at both of these stations were classi?ed into two subgroups: winter and spring, and summer and fall. Chinese data. Of the four stations, two are in Zhejiang, an east coast province of China. According to hydrometeorologic analyses by the Provincial Hydrologic Service, the annual maximum ?oods due to frontal weather systems occur in early summer (before 20 July, on the average) or due to typhoon storms (the northwest Paci?c hurricanes) in late summer (from July 20 to September). In coastal areas, where the Wenzhou raingage is located, the most and the strongest rainfall storms are caused by typhoons; in its mountainous western area, such as the watershed above the Diqing stream?ow gaging station, the largest ?ows are produced by frontal weather systems. The other two stations are the Zhushan station on the Du River and Lianpanshan station on the Han River located in the Hubei Province in central China. Although all ?oods are produced by frontal rainfall, large ?oods may occur either in summer (July to early August, when the monsoon fronts advance towards north China) or in fall (from late August to early October when the monsoon fronts withdraw from the north). Usually, the summer ?ood has a higher peak discharge and shorter duration; in

contrast, the fall ?ood is larger in volume and longer in duration. Based on the available physical evidence, the annual maximum ?ood (or rainfall) series for these four stations were classi?ed into two types, typhoon and frontal rains, summer and fall ?oods. Relevant information on the four data sets is shown in Table 1. 4.2. Computation of frequencies 4.2.1. Frequencies by direct curve ?tting The ?ood series was analyzed for all 4 data sets using the direct curve ?tting method mentioned earlier The observed probabilities were computed from the Gringorten plotting-position formula (Gringorten, 1963): P?i? Z i K 0:44 m C 0:12 (39)

where i is the ith smallest observation in the data set arranged, and m is the sample size. A theoretical distribution was ?tted to the values obtained from Eq. (39). The probability distributions obtained by the direct curve ?tting method were compared with those obtained from the proposed method. 4.2.2. Frequencies by the proposed method The proposed method was applied to all four data sets Idaho data. Direct curve ?tting was done for annual, winter, spring and summer peak ?ow series at 3 stations which are Reynolds East, Outlet and Tollgate. The log-Pearson Type III probability distribution was selected for different seasons at these stations as shown in Table 2. For the direct curve ?tting of the annual, winter, spring and summer maximum series at these stations, parameters were estimated based on the method of moments; also the events corresponding to the design return periods were obtained using the selected probability distribution (shown in Table 3). According to the proposed method described above, the conditional frequency distribution p*(xjAi) was obtained ?rst. Here, p*(A1), p*(A2) and p*(A3) were obtained for these stations by using Eq. (37). Based on Eqs. (17)–(19), the ?rst 3 moments were obtained for the annual series, according to the proposed method, at these stations. The results of ?tting are shown in Table 3

V.P. Singh et al. / Journal of Hydrology 307 (2005) 175–195 Table 2 Moments and LP III parameters for the data at Reynolds Creek watershed Station Variable Record length 34 34 31 Outlet Discharge 33 32 16 Tollgate Discharge 29 30 23 Rainfall mechanisms Moments X (m3/s) 0.023 K4.714 0.128 K2.232 0.013 K5.143 13.85 2.182 7.42 1.882 5.16 0.213 3.28 0.824 4.61 1.413 0.43* K1.539

185

S2 (m3/s)2 0.003 1.434 0.004 0.363 0.0007 1.385 518.62 2.816 30.61 0.905 67.02 2.474 9.55 1.283 7.45 0.545 0.64 1.186

Cs 4.65 0.949 0.66 K1.055 3.88 0.715 2.78 K0.996 1.17 K2.39 2.35 0.212 1.46 K0.889 0.39 K1.322 2.77 1.334

Cv (%) 240.24 K25.414 46.83 K26.986 194.55 K22.882 164.43 76.916 74.54 50.551 158.64 737.598 94.21 137.435 59.23 52.284 186.26 K66.502

Kurtosis 24.29 1.06 3.75 1.086 18.61 0.669 10.86 2.435 4.41 7.813 8.06 K1.456 4.75 0.428 2.62 1.562 8.99 1.667

East

Discharge

Winter rain Spring rain Summer storm* Winter rain Spring rain* Summer storm* Winter rain* Spring rain* Summer storm*

Original Log Original Log Original Log Original Log Original Log Original Log Original Log Original Log Original Log

Parameters of ?tted Log-Pearson III (LP III) distribution Station East Variables Discharge Annual Winter Spring Summer Annual Winter Spring Summer Annual Winter Spring Summer a (m3/s) 0.353 0.631 0.228 0.646 1.427 1.145 0.355 0.298 0.235 0.627 0.27 0.298 b 2.713 3.609 6.958 3.321 0.516 2.224 8.048 13.359 11.133 3.976 7.537 13.36 r (m3/s) K3.142 K6.989 K3.821 K7.288 K1.621 K0.996 K1.212 K5.621 K1.162 K1.856 K0.685 K5.621

Outlet

Discharge

Tollgate

Discharge

Some data are missing for the generating mechanism. a, b, and r represent the parameters of log-Pearson type III distribution.

and Figs. 1 and 2. It is shown that the design events from the direct curve-?tting method are somewhat different from those for the proposed method. The design events corresponding to certain return periods show that the smaller the peak discharge, the more similar these two methods are. Both methods ?tted the observed data reasonably well, although there are some uncertainties in the data set itself. Louisiana data. The direct curve ?tting method was applied to annual, frontal and Gulf Tropical Depression series for the data at Darlington and Denham Springs. The log-Pearson type III

distribution was found to be the best-?tted distribution for all the data series (Fig. 3). The frequency distributions selected for annual and different generating mechanisms are shown in Table 4. For the proposed method, the conditional frequency distribution p*(xjAi) was obtained ?rst. Here, p*(A1), p*(A2) were obtained for all the data series. Using Eqs. (17)–(19), the ?rst three moments were obtained for the mixed annual series. The results of calculations are shown in Table 5 and Fig. 4. It is seen that for both stations the design events from the direct curve-?tting method are different from those for

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Table 3 Comparison of design events computed by the proposed method and the curve ?tting method Main characteristics Reynolds Creek (US) East Curve ?tting (LPIII) Mechanisms Proportion S p(A1)(%) p(A2)(%) p(A3)(%) Mean (m3/s) Variance (m3/s)2 skew x(1%) x(2%) x(5%) Proposed method 3 4.7 92.2 3.1 0.12 0.004 0.47 0.296 0.278 0.245 Outlet Curve ?tting (LPIII) Proposed method 3 49.7 47.6 2.7 10.56 284.99 3.91 113.29 89.35 56.34 Tollgate Curve ?tting (LPIII) Proposed method 3 31.3 67.6 1.1 4.15 8.56 0.69 11.568 10.654 10.231

Parameters Design events (m3/s)

0.13* 0.005 0.82 0.306 0.285 0.251

16.53 513.28 2.61 149.62 104.91 62.78

5.33 8.70 0.34 10.716 10.492 9.942

p(1) (A1): winter; p(A2): spring; p(A3): summer. (2) the moments calculated by taking logarithm which causes the mean appeared to be negative.

the proposed method. The difference at Darlington is larger than that at Denham Springs. For the Darlington station, the direct curve-?tting procedure ?tted the observed data better than did the proposed method. The reason may be because for this station the daily discharge from the water year 1989 to the

water year 2000 was missing. For the whole procedure, the lengths of annual, frontal and GTD data series are 27, 27, 13, respectively, indicating that the sample sizes are small for statistical analysis. For Denham Springs Station, both methods ?tted the data reasonably well.

Fig. 1. Peak discharge frequency curves for different seasons at Reynolds Creek ((a) East, (b) Outlet, (c) Tollgate).

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Fig. 2. Peak discharge frequency curves for annual maximum series at Reynolds Creek ((a) East, (b) Outlet, (c) Tollgate).

Arizona data. The direct curve ?tting method was applied to annual, winter and spring, and summer and fall series data at Clifton and Tucson. The log-Pearson type III distribution was found to be the best-?tted

distribution for all the data series (Fig. 5). The frequency distributions selected for annual and two combined seasons are shown in Table 6. For the proposed method, the conditional frequency

Fig. 3. Peak discharge frequency curves for frontal and GTD generating mechanism of Amite River ((a) Darlington, (b) Denham Springs).

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Table 4 Moments and LP III parameters for the data of the Amite River basin Station Variable Record length 27 27 Denham Springs Discharge 39 39 Rainfall mechanisms Moments X (m3/s) 726.88 6.22 306.37 5.42 1181.4 6.83 417.66 5.67

S2 (m3/s)2 348560 0.85 96378 0.62 593390 0.61 133210 0.82

Cs 0.84 K0.19 2.42 0.54 0.89 K0.85 1.36 K0.11

Cv (%) 81.22 14.86 101.33 14.51 65.20 11.39 87.38 16.02

Kurtosis 2.64 K1.20 8.13 1.65 3.39 0.054 4.10 K0.904

Darlington

Discharge

Frontal GTD* Frontal GTD

Original Log Original Log Original Log Original Log

Parameters of ?tted Log-Pearson III (LP III) distribution Station Darlington Variables Discharge Annual Frontal GTD Annual Frontal GTD* a (m3/s) 1.054 2.413 0.732 0.336 0.456 4.141 b 0.643 0.146 1.152 4.934 2.922 0.048 r (m3/s) 5.681 5.867 4.571 5.24 5.498 5.469

Denham Springs

Discharge

GTD: gulf tropical depression.

distribution p*(xjAi) was obtained ?rst. Here, p*(A1), p*(A2) were obtained for all the data series by Eq. (35). Using Eqs. (17)–(19), the ?rst three moments were obtained for the mixed annual series. The results of calculations are shown in Table 7 and Fig. 6. It is seen that for both stations the design events from the direct curve-?tting method are somewhat different from those for the proposed method. The design events corresponding to certain return periods show that the smaller the peak discharge, the more similar

these two methods are. For high return periods, say, 50 and 100 years, the design events obtained by the proposed method were higher than for the direct method. Since there are uncertainties in the data sets themselves, both methods ?tted the observed data reasonably well. Chinese data. The Pearson type III distribution, as recommended by CHNMWREP (1983) and the least squares ?tting procedure were used to ?t the A1 and A2 series to obtain the conditional frequency distribution

Table 5 Comparison of design events computed by the proposed method with the direct curve ?tting method Main characteristics Amite River (US) Darlington Curve ?tting (LPIII) Mechanisms Proportion s p(A1)(%) p(A2)(%) Mean (m3/s) Variance (m3/s)2 Skew x(1%) x(2%) x(5%) Proposed method 2 88.89 11.11 680.16 338000 0.96 2312.8 1926.6 1464.8 Denham Springs Curve ?tting (LPIII) Proposed method 2 89.5 10.5 1191.2 599870 0.94 3922.0 3341.5 2626.3

Parameters Design events (m3/s)

783.55 329570 0.74 2866.23 2494.37 1985.31

1229.60 558520 0.85 3669.77 3193.65 2562.82

p(A1): Frontal; p(A2): GTD.

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Fig. 4. Peak discharge frequency curves for annual maximum series of Amite River ((a) Darlington, (b) Denham Springs).

p*(xjAi), iZ1, 2. These were then combined with p*(A1) and p*(A2), according to Eq. (3), to obtain the frequency distribution p*(x) of the annual maximum ?ood ?ow (or rainfall). For comparison, estimates

were also made by ?tting the Pearson type 3 distribution to the annual maximum series with the use of the least squares curve-?tting procedure to the plot obtained by the Weibull formula and by Eq. (3)

Fig. 5. Peak discharge frequency curves for winter and spring; and summer and fall data of Gila River basin. ((a) Clifton, (b) Tucson).

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Table 6 Moments and LP III parameters for the data of the Gila River basin Station Variable Record length 70 70 Tucson Discharge 39 39 Rainfall mechanisms Moments X (m3/s) 72.91 3.38 193.17 4.59 53.05 0.37 189.41 3.43

S2 (m3/s)2 19040 3.21 30292 1.17 26377 6.18 34077 1.16

Cs 3.56 0.25 2.28 K0.46 4.35 0.003 4.78 K0.37

Cv (%) 189.26 53.05 90.12 23.57 306.17 667.26 97.46 31.39

Kurtosis 16.85 2.18 8.92 2.84 23.65 2.39 33.01 3.48

Clifton

Discharge

Winter and Spring Summer and Fall Winter and Spring Summer and Fall

Original Log Original Log Original Log Original Log

Parameters of ?tted Log-Pearson III (LP III) distribution Station Clifton Variables Discharge Annual Winter and spring Summer and fall Annual Winter and spring Summer and fall a (m3/s) 1.138 3.103 1.046 1.4169 432.74 1.2904 b 0.969 0.333 1.071 0.5396 0.00003 0.696 r (m3/s) 2.759 2.342 3.474 2.708 0.358 2.531

Tucson

Discharge

where the component frequency distributions p? ?x? 1 and p? ?x? were estimated from the seasonal maximum 2 series (frontal and typhoon rains (?oods); summer and fall ?oods). The results of calculation are given in Tables 8 and 9. Among three sets of results, Eq. (3) yielded the largest probability value for a given ?ood magnitude in most cases, with the exception of the Zhushan station when the return period was larger than 100 years which might be the result of statistical uncertainty of the series. Design ?ood values from

the direct curve-?tting method with the use of usual plotting positions were underestimated for three of the four stations (Diqing, Zhushan, Lianjpanshan stations), whereas for the other station, they were almost the same as those of the proposed method. In the middle and lower parts of the frequency distributions, the three methods were comparable. However, more extensive data analysis needs to be undertaken to demonstrate the superiority of the proposed method on a wider scale. Nevertheless, by analyzing the data of four stations, the soundness

Table 7 Comparison of design events computed by the proposed method with the curve ?tting method Main characteristics Gila River Basin (USA) Clifton Curve ?tting (LPIII) Mechanisms Proportion s p(A1)(%) p(A2)(%) Mean (m3/s) Variance (m3/s)2 Skew x(1%) x(2%) x(5%) Proposed method 2 35.71 64.29 150.22 29594 2.34 644.1139 474.6392 300.2386 Tucson Curve ?tting (LPIII) Proposed method 2 13.31 86.69 171.26 35198 4.34 362.8245 273.1948 178.4998

Parameters Design events (m3/s)

127.49 19857 2.89 429.17 350.63 253.5

202.35 25639 3.92 273.72 221.67 159.06

p(A1): winter and spring; p(A2): summer and fall.

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Fig. 6. Peak discharge frequency curves for annual maximum series of Gila River basin ((a) Clifton, (b) Tucson).

and applicability of the proposed method, Eq. (13), and the estimation procedure are shown. 4.3. Comparison of the proposed method with the curve ?tting method In order to evaluate the performance of the proposed method in comparison with the curve ?tting method, the Kruskal–Wallis test and the Akaike Information Criterion (AIC) were applied.

The Kruskal–Wallis test was applied to determine whether the ?ood series came from same population. It is a nonparametric version of the classical one-way ANOVA, which determines the p-value for the null hypothesis that all sample populations are drawn from same continuous probability distribution, and all samples are mutually independent. The Kruskal– Wallis test statistic can be expressed as follows:. Let ni (iZ1, 2,., k) represent the sample sizes for each of the k groups (i.e. samples) in the data. Next,

Table 8 Parameters of the distributions ?tted to the seasonal maximum series for Chinese data Station Variable Length of record (year) 75 75 29 29 28 28 50 50 Cause Parameters X 110 175 2670 1700 3530 3130 13200 12700

Cv 0.35 0.578 0.4 0.85 0.598 0.573 0.555 0.708

Cs 1.35 1.34 1.29 1.744 1.92 1.15 1.67 1.5

Wenzhou Diqing Zhushan

3-day rain (mm) Peak discharge (m3/s) Peak discharge (m3/s) Peak discharge (m3/s)

Lianpanshan

Frontal rain Typhoon rain Frontal rain Typhoon rain Frontal rain (summer) Frontal rain (fall) Frontal rain (summer) Frontal rain (fall)

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Table 9 Comparison of design events computed by the proposed method with those by two other methods for the Chinese data Main characteristics Mechanisms s Yangzi River (China) Wenzhou Curve ?tting (PIII) Proportion p(A1)(%) p(A2)(%) Mean (m3/s) Variance (m3/s)2 Skew x(1%) x(2%) x(5%) S Proposed method 2 29.6 70.4 188.7 8135.8 1.561 670 495 365 Proposed method 2 55.1 44.9 4182 4495700 1.701 16200 11400 8200 Eq. (3) Diqing Curve ?tting (PIII) Proposed method 2 75.9 24.1 2902 0.436 1.311 9100 6900 5300 Proposed method 2 51.9 48.1 16353 66346000 1.268 56000 42500 32000 Eq. (3)

Parameters Design events (m3/s) Mechanisms

191 8300.5 1.54 673 497 365 Zhushan Curve ?tting (PIII)

685 505 375 Eq.(3)

2880 1401100 1.1 8410 6530 5120 Lianpanshan Curve ?tting (PIII)

9800 7150 5400 Eq. (3)

Proportion Parameters

Design events (m3/s)

p(A1)(%) p(A2)(%) Mean Variance Skew x(1%) x(2%) x(5%)

4120 3894700 1.48 14300 10600 7960

15900 11300 8300

16400 68320000 1.04 54300 41600 32000

62000 46500 34000

the combined sample is ranked. Then, RiZthe sum of the ranks for group i is computed. Then the Kruskal– Wallis test statistic is computed as: HZ

k X R2 12 i K 3?n C 1? n?n C 1? iZ1 ni

lack-of-?t of the model and the unreliability of the model due to the number of model parameters, and can be expressed as: AICZK2 log?maximised likelihood for model?

(40)

C2?no: of fitted parameters?

(41)

where H denotes the Kruskal–Wallis test statistic, and n denotes the combined sample size. The Kruskal–Wallis test statistic approximates a c2 distribution with kK1 degrees of freedom if the null hypothesis of equal populations is true. If the p-value obtained from the Kruskal–Wallis test is greater than the critical value, i.e. aZ0.05 (which is mostly applied in statistical testing), then the null hypothesis, i.e. the sample populations drawn from same continuous probability distribution, will be accepted. Otherwise if the p-value is less than the critical value the null hypothesis will be rejected. The Akaike Information Criterion (AIC), developed by Akaike (1974), was employed to identify which method was better. AIC includes two parts:

Thus, the best model is the one which has the minimum AIC value. Idaho data. The Kruskal–Wallis test (Eq. (40)) showed that the Idaho ?ood data from winter, spring and summer did not come from the same population (i.e. Table 10). This test also showed that there was no signi?cant difference between the curve ?tting method and the proposed method, even though there were minor differences in the values of certain return periods obtained from the curve ?tting method and the proposed method (i.e. Table 11). Since it was dif?cult to discern which method performed better for computing the design ?ood, the AIC criterion (Eq. (41)) was applied. The AIC values of the direct curve ?tting method and the proposed method for

V.P. Singh et al. / Journal of Hydrology 307 (2005) 175–195 Table 10 Kruskal–Wallis statistical test of seasonal series Station Reynolds Creek East p-value !0.001 Outlet !0.001 Tollgate !0.001 Amite River Darlington !0.001 Denham Springs !0.001 Gila River Clifton !0.001 Tucson !0.001

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The critical criterion is aZ0.05.

the Idaho data are shown in Table 12, which indicate that the proposed method had lower AIC values than did the curve ?tting method. Thus the proposed method is more reliable based on the AIC criterion. Louisiana data. The Kruskal–Wallis test and AIC criterion were applied to data at Darlington and Denham Springs. The Kruskal–Wallis test shows that the frontal series and Tropical Depression series for both sites are not from the same population, as shown in Table 10. The Kruskal–Wallis test showed that ?ood series obtained by the direct curve ?tting method and that by the proposed method were from the same population, as shown in Table 11. The AIC values were computed for ?ood series at both Darlington

and Denham Springs, as shown in Table 12. For Denham Springs, the proposed method had lower AIC values, but for Darlington the curve ?tting method had lower AIC values. The reason for the higher value that proposed method had for Darlington may be because of missing data. The curve ?tting method and the proposed method yielded signi?cantly different design event values, showing that different ?ood generating mechanisms exercised a signi?cant in?uence on the design event values. Arizona data. The Kruskal–Wallis test and AIC criterion were applied to the data at Clifton and Tucson. The Kruskal–Wallis test shows that the seasonal series for both sites are not from the same

Table 11 Kruskal–Wallis statistical test of the curve ?tting method and the proposed method Reynolds Creek East p-value 0.98 Difference (%) x(1%) X(2%) X(5%) 3.27% 2.46% 2.39% Outlet p-value 0.97 Difference (%) X(1%) X(2%) X(5%) 24.28% 14.83% 16.26% Tollgate p-value 0.98 Difference (%) X(1%) X(2%) X(5%) 7.95% 1.54% 2.91%

Amite River Darlington p-value 0.33 Difference (%) X(1%) x(2%) x(5%) 19.30 22.70 26.21 Denham Springs p-value (%) 0.98 Difference (%) X(1%) x(2%) x(5%) 0.04% 6.44% 14.92%

Gila River Clifton p-value 0.31 Difference (%) x(1%) x(2%) x(5%) 50% 35.36% 18.40% Tucson p-value 0.44 Difference (%) x(1%) x(2%) x(5%) 32.55% 23.24% 12.22%

(1) The critical value for the statistical test of whether the estimation from the proposed method and the curved ?tting method can be obtained from the same population is aZ0.05; (2) Difference is the difference in percentage between the curved ?tting method and the proposed method.

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Table 12 AIC values of the curve ?tting method and the proposed method Reynolds Creek East LP III K97.952 Amite River Darlington LP III 745.1487 Gila River Basin Clifton LP III 841.565 Proposed 703.301 Tucson LP III 756.613 Proposed 666.704 Proposed 806.7037 Denham Springs LP III 1350.1 Proposed 1228.1 Proposed K102.617 Outlet LP III 414.3481 Proposed 218.029 Tollgate LP III 179.342 Proposed 152.936

population, as shown in Table 10. Also the same test showed that the ?ood series obtained by the direct curve ?tting method and that by the proposed method were from the same population, as shown in Table 11. The AIC values were computed for ?ood series at both Clifton and Tucson as shown in Table 12, which indicate that the proposed method had lower AIC values than did the directive curve ?tting method. Thus the proposed method is more reliable. The design event values produced by the curve ?tting method and the proposed method were signi?cantly different, implying that ?ood generating mechanisms had a signi?cant in?uence on ?ood frequency analysis and the consequent design event values.

(2) To properly estimate the frequency distribution of the annual maximum ?ood sequence when it is nonidentically distributed, more data are needed, including not only the annual maximum ?ood series which should be divided into several subsets with identical distributions but also the seasonal maximum series in order to estimate the probability p(Ai), iZ1,., s.. (3) The proposed method re?ects the nature of the ?ood phenomenon, and can be used by practitioners. (4) The design event values obtained by the proposed method and the direct curve ?tting method may be signi?cantly different, i.e. for the Gila river basin, for a 100-year ?ood event this difference may be 50% at Clifton, and 32.55% at Tucson. This difference might be caused by two reasons: (i) During the relatively dry climatic periods the peak discharge might more likely fall into the lower and middle quantiles during the curve ?tting method; and (ii) for the composite ?ood series, the peak values available are instantaneous peak values. However, when the data are partitioned into different seasons, enough instantaneous peak values are not available for each season or each mechanism. This de?ciency is then made up by employing daily peak values. In each case the ?tting is compared with the observed composite instantaneous peak values.

5. Summary and concluding remarks A method for determining the exceedance probability of nonidentically distributed annual maximum ?ood sequences has been developed, and applied to four real-world samples. The following conclusions can be drawn from this study: (1) For an accurate estimation of design ?oods, the ?ood-generating processes should be carefully analyzed if they are homogeneous or not within a year. When these processes vary with season, the effect of their heterogeneity should be taken into account.

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Thus, the data de?ciency might also contribute to the large difference between the design events obtained by two methods. Regardless of the reason, these differences show that different ?ood generating mechanisms may exercise a signi?cant in?uence on the design ?ood value. (5) Table 11 shows that the difference for the Amite River basin in Louisiana and Gila River basin in Arizona are of opposite nature. This may be caused by the differences in their respective climates to which the two river basins belong: the Amite River basin belongs to a sub-humid tropical climate whereas the Gila River basin belongs to a dry arid climate. The ?ood generating mechanisms are different in the two cases. (6) The existing approaches to deal with nonidentically distributed annual maximum series are not entirely valid. For example, Eq. (3) may overestimate the exceedance probability of the annual ?ood maximum; whereas the direct curve-?tting procedure based on the plotting position does not accommodate nonidentically distributed samples.

Acknowledgements The authors are grateful to Ms Tracie Kirkham for providing the data of Idaho watersheds. References

Akaike, H., 1974. New look at the statistical model identi?cation. IEEE Transactions on Automatic Control AC-19 (6), 716–722. Alila, Y., Mtiraoui, A., 2002. Implication of heterogeneous ?oodfrequency distributions on traditional stream-discharge prediction techniques. Hydrological Processes 16, 1065–1084. Beran, M., Hosking, J.R.M., Arnell, N., 1986. Comment on "Twocomponent extreme value distribution for ?ood frequency analysis, by F. Rossi, M. Fiorentino, P. Versace. Water Resources Research 22 (2), 63–266. Blom, G., 1958. Statistical estimates and transformed beta variables. Wiley, New York, N.Y. Cunnane, C., 1978. Unbiased plotting position—a review. Journal of Hydrology 37, 205–222. Diehl, T., Potter, K.W., 1987. Mixed ?ood distribution in Wisconsin, in: Singh, V.P. (Ed.), Hydrologic Frequency Modelling, pp. 213–226.

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