WATER RESOURCES RESEARCH, VOL. 37, NO. 6, PAGES , JUNE 2001

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1 WATER RESOURCES RESEARCH, VOL. 37, NO. 6, PAGES , JUNE 2001 A Bayesian joint probability approach for flood record augmentation Q. J. Wang Agriculture Victoria, Tatura, Victoria, Australia Abstract. At-site flood frequency prediction is often subjecto large uncertainties because of the limited length of record available at the gauging station of interest. In many cases, however, one can find a nearby gauging station that has a considerably longer record. In this paper, we introduce a two-site joint probability approach for the transfer of flood information between two stations. We assume that the annual maximum floods at the two stations follow a bivariate generalized extreme value distribution. We use a Bayesian procedure to estimate the parameters of the distribution by jointly using all the flood data available at the two stations. Efficient transfer of information is achieved by simultaneously carrying out (1) the inference of correlation of floods at the two stations, (2) the transfer of information between the two stations, and (3) the inference of the frequency distributions for the two stations. 1. Introduction Annual maximum floods are, in general, much more variable than annual or monthly stream flow volumes and therefore Statistics derived from hydrological records are often subject to large uncertainties because of the limited lengths of records available in contrast to the large natural variability of hydrological events in time. However, hydrological variables are, in general, highly correlated in space. As a result, one can often obtain information about the hydrological characteristics at a location from data collected nearby. For example, if there are periods during which data were collected at gauging station A but not at gauging station B, the extra period of records at A can in some cases be used to improve our knowledge of hyrequire much longer records to obtain reliable statistics. More importantly, one is often interested in the estimation of large return period floods for the purpose of hydrological design and assessment. When the record length is short, as is often the case, the uncertainty in the estimation can be considerable. This in turn can have large economic, social, and environmental consequences. Therefore the development of an efficient procedure for record augmentation of annual maximum floods is highly desirable. In this paper, we introduce a procedure for augmenting drological characteristics at B. flood peak records. We adopt a bivariate extreme value distri- Vogel and Stedinger [1985] used the term "record augmen- bution for annual maximum floods at two gauging stations. We tation" to describe procedures for transferring information use a Bayesian framework to formulate and solve the inference between gauging stations to better estimate summary statistics, problem. We demonstrate by a case study the benefit of jointly using all flood data for two stations. such as the mean and standard deviation. They used the term "record extension" to describe procedures for infilling missing data or extending data series. For record augmentation, Fiering [1963] and Matalas and Jacobs [1964] provided maximum like- 2. Joint Probability Model lihood estimators of mean and standard deviation. Vogel and We denote annual maximum floods at stations A and B as Stedinger [1985] derived estimators of mean and standard de- random variables X and Y, respectively. X and Y must occur in viation that have minimum variances. the same water year but need not correspond to the same All available augmentation procedures are based on an as- storm event. We denote the joint distribution function of X sumption that variables, with or without any transformation, and Y by Fx, v(x, y) = Pr (X < x, Y < y), the marginal follow a joint bivariate normal distribution. As a result, these distribution for X by Fx(x) = Pr (X < x), and the marginal augmentation procedures have been applied to quantities, distribution for Y by Fv(y ) = Pr (Y < y). The marginal distributions can be estimated separately for each of the stasuch as annual or monthly stream flow volumes, for which the tions by using "at-site" data only. We aim to improve the at-site bivariate normality is often achievable by an application of a estimation through transferring information between the two suitable transformation, such as the logarithmic transformastations by using a "two-site" joint probability approach. tion or the more general Box-Cox transformation [Box and One of the commonly used (marginal) distributions for de- Cox, 1964]. These procedures have also been adopted for augscribing annual maximum floods at a station is the generalized mentation of low-flow and flood data [Vogel and Kroll, 1991], extreme value (GEV) distribution (see, for example, Martins but further studies are needed to determine the extent to which and Stedinger [2000] for a summary of available estimation the assumption of bivariate normality following a transforma- methods). For X the distribution function is given by tion can be applied. exp {-[1 - Kx(X- x)/ax] } Kx - O This paper is not subjecto U.S. copyright. Published in 2001 by the American Geophysical Union. Fx(x) = exp [-e-(x- x)/ x] 1/ x = 0, (1) Paper number 2000WR with a density function

2 1708 WANG: BAYESIAN JOINT PROBABILITY APPROACH fx(x) = [Fx(x)(ax)-ie-(X-,X)/ x Kx = 0. (2) Similar distributions and density functions can be given for Y. For convenience later, X is transformed to U by -(Kx) - In [1- x(x- x)/ax] 4:0 u = (x- = 0, (3) so that U follows a standard extreme value type I (EVI) distribution with Fv(u) = exp (-e-u), (4) fu(u) = exp (-e-u)e -u. (5) Similarly, Y can be transformed to V, which also follows a standard EVI distribution. Having chosen the marginal distributions, we now need a dependency (copula) function to describe the nature of the association between U and V (and thus between X and Y). One of the bivariate extreme value distributions is the Gumbel type B [Gumbel and Mustaft, 1967; Johnson and Kotz, 1972] given by Fu, v(u, v) = exp [_½-mu q- e-my)l/m], (6) fu, v(u, v) = Fu, v(u, v)e-m(u+v)(e-mu + e-mv) -2+l/m [m - I + (e-ran q- e-my)l/mi. (7) U and V are independent of each other when m = I and completely dependent of each other when m =. In general, P,i/= 1 - m -2, (8) where Pu, v- is the correlation coefficient between U and V [Gumbel and Mustaft, 1967]. The joint density function of X and Y is given by Bayes' theorem states that given a prior distribution p(0) of the model parameters and the observed data {x, y}, the posterior distribution of 0 is given by p(01x, y) r p(o)p(x, y10), where p(x, y[0) is the likelihood function, which is the probability of the occurrence of {x, y} given 0. Under the assumption that the annual maximum floods are independent from year to year, the likelihood function is given by K I J p(x, yl0)= 1-[ fx, v(xk, yklo) 1-[f(xilo) 1-[fCy1o). k=l i=1 j=l ( ) ( 2) The right-hand side of (11) can, in theory, be normalized to give an exact posterior distribution p( 0Ix, y). This can then be used to find, by using the marginal distribution functions, the posterior distributions p[x(f)lx, y] and p[x(f)lx, y] of flood magnitudes for a specified nonexceedance probability F or the posterior distributions p[fx(x)lx, y)] and p[fv(ylx, y)] of nonexceedance probabilities for specified flood magnitudes x and y, respectively Prior Distribution In the Bayesian formulation of (11), it is necessary to specify a prior distribution p(0). This gives an opportunity to incorporate into the analysis any knowledge about the parameters prior to the use of the data at the specific stations. The prior knowledge may come from regionalization analysis (see review article by Groupe de Recherche en Hydrologic Statistique (GREHYS)[1996]) or simply from experience about the range, variation, and even the correlation of parameters [e.g., Martins and Stedinger, 2000]. When prior knowledge is not available, a noninformative prior distribution may be used. In this study, we used a noninformative prior distribution, du dv fx, r(x, y) = fv,v(u, v) dx dy = fs, v(u, v) [(ax- gx(x- x)]- [(av - gr(y- v)]-'. (9) 3. Inference Method 3.1. Bayesian Formulation There are a total of seven parameters in the proposed joint probability model. We denote them by 0 = ( x, ax, Kx, v, av, gv, m). ( o) These parameters need to be estimated from the data available at both stations. We use the following notations to distinguish three groups of data: (1)x,, y,, k = 1, 2,..., K, for annual maximum floods in years that have records available at both station A and station B; (b)xi, i = 1, 2,..., I, for annual maximum floods in years that have records available at station A but not station B; and (c) y, j = 1, 2,..., J, for annual maximum floods in years that have records available at station B but not station A. The data in each group need not be consecutive. In addition, we use {x, y} to denote all the three groups of data put together. We use a Bayesian framework to infer the model parameters. With the recent advances in computational methods for Bayesian inference the Bayesian framework is well suited for solving complex statistical problems [Gelman et al., 1995]. p ( 0 ) constant, (13) within the parameter zone that is mathematically feasible {ax > O, av > O, x > -1,. > -1, x(x - iix)/ax < I for all observed x, <v(y- s%-)/ - < I for all observed y, and -1 _< p,.. _< 1 }, with Pu, - being further limited to nonnegative values only. Care needs to be taken in the specification of the noninformative prior distribution to ensure that it does not lead to improper posterior distributions. In the case study described in section 5 the prior (13) did not seem to lead to an improper posterior distribution p(0lx, y) as different runs of random sampling from p(0lx, y) appeared to converge. However, a thorough investigation of alternative and, perhaps, more reasonable prior distributionshould be pursued in future studies Numerical Implementation: Markov Chain Monte Carlo Simulations We employ a technique called Markov chain Monte Carlo simulations to draw a large random sample of 0 from the posterior distribution p( 0Ix, y). This sample can be converted, by using the marginal distribution functions, to a sample of x(f) and y(f) or Fx(x ) and Fv(y ) to numerically represent their corresponding posterior distributions p[x(f)lx, y] and p[x(f)lx, y] or p[fx(x)lx, y] and p[fy(y)lx, y]. The technique of Markov chain Monte Carlo simulations allows one to draw representative random samples from any distributions as long as their densities can be computed [Gelman et al., 1995].

3 WANG: BAYESIAN JOINT PROBABILITY APPROACH E! B Figure 1. Annual maximum flood data available at and In hydrology and water resources literature, Qian and Richardson [1997], Kuczera and Parent [1998], and Campbell et al. [1999] used this technique. Specifically, we use the Metropolis-Hastings algorithm which consists of the following steps: 1. Choose a starting point 0 ø for which p(0øl x, y) > Select a proposal distribution q(0*10). 3. Fort = 1, 2,..., Sample a candidate point 0* from Calculate the following quantity: Set Year p(o*lx, y)/q(o*l o - ) r = p(0 _Xlx, y)/q(o _Xlo,). Ot = { 0* 0 t- with otherwise. probability min (r, 1) The algorithm creates a sequence of a large number of points stations for 14 years, available only at for 13 years, and (0, 02,... ) which converge to p(01 x, y). These points form a available only at for 19 years. The annual maximum sample to numerically represent p(01 x, y). The proposal disfloods are highly correlated as shown by the display of the tribution q(0*l 0t- ) may have any form, although the samconcurrent data in Figure 2. Thus we would expect that the pling convergence rate may vary. As its notation indicates, the joint use of data from the two stations would improve inferproposal distribution may depend on the last point 0 t'. We ence of the flood frequency distributions at the two stations. use a multivariate normal distribution with mean equal to 0 t- We applied the Bayesian procedure to infer the flood freand a fixed covariance matrix. One may run the algorithm with an initially chosen covariance matrix to generate a sample of points. These points are then used to construct a revised covariance matrix for final simulations to ensure a reasonable 120 rate of convergence. Gelman et al. [1995] and Tanner [1996] provide useful discussions constructing efficient simulation -,'" 100 algorithms and on assessing convergence At-site Estimation The formulation and implementation presented above can also be used for at-site estimation. For X, (10)-(12) are reduced respectively to 0 = ( x, ax, x), (14) p (0Ix) p(0)p(xl 0), K I p(xl0) = I-If (x lo)i-[f (x, lo). (16) k=l i=1 4. Checking the Bivariate Distribution It is important that the assumed probability model for annual maximum floods at two stations be reasonable. The fol- lowing is a simple method for checking whether the association (dependency) structure between annual maximum floods at two stations can be considered to follow the Gumbel type B bivariate extreme value distribution. 1. Transform the concurrently observed data. Use (3) to transformx to u and the equivalent of (3) to transform y to v, k = 1, 2,..., K. In the transformation the mean estimates of parameters obtained by the two-site approach may be used. 2. Compute the difference t, = u, - vk, k = 1, 2,..., K. Rank t, in ascending order. Use a plotting position formula to assign a nonexceedance probability F to each t. 3. Plot t versus -ln (F- - 1). If the points follow a straight line well, it is considered that the use of the Gumbel type B bivariate extreme value distribution of (6) is reasonable. The slope of the plot should be approximately (l/m). This method was originally proposed by Gumbel and Mustaft [1967] based on the property that if U and V follow the Gumbel type B bivariate extreme value distribution of (6) and (7), the difference T = U - V should follow a logistic distribution 5. Example 5.1. Data Fr(t) = (1 + e-mr) -. (17) We used as a case study data from two gauging stations in Victoria, Australia: Ovens River at Bright (403205) and Buckland River at Buckland (403206). The catchments of the two stations are adjacento each other with areas of 495 and 303 km 2, respectively. The two rivers meet at -- 7 km downstream of Bright and 15 km downstream of Buckland. Figure 1 shows the time series of annual maximum instan- taneous floods at the two stations. Data are available at both o E C3 20,, Discharge at (m3s q) Figure 2. Concurrent annual maximum floods at and

4 1710 WANG: BAYESIAN JOINT PROBABILITY APPROACH Table 1. Precision Ratios of the Two-Site Approach to the At-site Approach Distribution Parameters Flood Ouantiles of Return Periods, years Site s c a K quency distributions at two stations. For the purpose of com- For station , there were 33 years of data. The ratio of parison, we used both the two-site and the at-site approaches. the precision, defined earlier as the inverse of variance, of the 5.2. Precision and Approximate Equivalent Record Lengths The accuracy of an inferred quantity may be measured by its precision, defined as the inverse of its variance. Precision estimates can be affected by both record lengths and estimated population parameter values. As the latter can both increase or two-site approach to the precision of the at-site approach was 1.37 for the 2-year flood. Thus one would need an equivalent record length of 1.37 x 33 = 45.2 years for the at-site approach to achieve the same precision as the two-site approach. For station , there were 27 years of data. The precision ratio was 1.64 for the 2-year flood. Thus one would need an equivdecrease the estimated precision, we shall, in our presentation alent record length of 1.64 x years for the at-site of results, attribute any changes in precision to changes in equivalent record lengths only. The precision ratios of the two-site approach to the at-site approach are given in Table 1 for the parameters of the distributions and a number of quantiles. The large precision ratios approach to achieve the same precision as the two-site approach. These two equivalent record lengths (45.2 and 44.3 years) are very close to the total observation period (46 years) covered by at least one of the two stations. We therefore conclude that the two-site approach was highly efficient. in Table 1 show the considerable benefit from using the twosite approach, especially for larger quantiles. A total of Posterior Distributions years of nonconcurrent data at were used to augment One of the advantages of the Bayesian approach via Markov the 27 years of data observed at , while a total of 13 chain Monte Carlo simulations is that an inference of the years of nonconcurrent data at were used to augment uncertainty of a quantity or quantities is attainable. Figure 3 the 33 years of data observed at This explains the larger gain in inference precision at We explored the use of equivalent record lengths [Stedinger (a) 50 and Cohn, 1986] as a measure of information gain. We derived equivalent record lengths that would be needed for the at-site 40- approach to achieve similar precision as the two-site approach. It should be emphasized that the derivation presented below is based on a number of rough assumptions and should be 30- treated as approximate only. For a sample from a normal distribution the variance of the 20- sample mean is known to be inversely proportional to the sample size. As this result is reasonably extendable to samples 10- from other distributions, we assumed that it held for the annual maximum floods. In addition, because the 2-year flood 0 I I derived from an annual maximum flood series should be close to the mean of the floods, we also assumed that the variance of the 2-year flood estimator was inversely proportional to the sample size. (b) S0 40-,-'- 3 - At-site estimation... Two-site estimation :E_ , -' Figure 3. Probability density curves of the inferred generalized extreme value (GEV) shape parameter for Figure 4. (a) A total of 100,000 sampling points of the posterior joint distribution of the GEV location and scale parameters (at-site approach). (b) A total of 100,000 sampling points of the posterior joint distribution of the GEV location and scale parameters (two-site approach).

5 ._ WANG: BAYESIAN JOINT PROBABILITY APPROACH At-site estimation 0.6 r = 0.973, ;', Two-site estimation ' , _. i i i i Flow (m3s " ) Figure 5. Probability density curves of the inferred 100-year flood for z In(I/F-I) Figure 7. A logistic plot for checking the Oumbel type B bivariate extreme value distribution. gives the density curves of the inferred GEV shape parameter < for The more peaky curve obtained by using the two-site approach implied less inference uncertainty. There was also a shift in the mean < in this case. Figure 4a shows 100,000 sampling points of the posterior joint distribution, obtained by the at-site approach, of the GEV location and scale parameters, and a, for This is in contrast to Figure 4b, in which a significant reduction in scatter was achieved by using the two-site approach. Figure 5 gives the density curves of the inferred 100-year flood. Again, the twosite approach led to much more precise inference. Another advantage of our approach was that the uncertainty in the sample estimate of the correlation coefficient was addressed. Figure 6 shows the density curve of the inferred Pt,v, the correlation coefficient between the transformed variables U and V, of floods at the two stations. The uncertainty in the Pt,v inference reflected the fact that there were only 14 years of concurrent observations. Given the posterior distributions p[fx(x)[x, y] and p[fy(y)lx, y], the expectations of Fx(x ) and Fy(y) are the composite predicted probabilities of floods not exceeding x and y, respectively. Readers are referred to Stedinger [1983, 1997], and Al-Futaisi and Stedinger [1999] for discussions on how to incorporate uncertainties into flood risk analyses for design and management purposes o o Checking the Bivariate Distribution To check whether the use of the Gumbel type B bivariate extreme value distribution was reasonable, we produced the logistic plot shown in Figure 7. The points roughly followed a line with some deviation in the middle. As there were only 13 points of concurrent data, the deviation from a straight line is not unreasonable. Thus the use of the Gumbel type B bivariate extreme value seems acceptable. 6. Conclusions We introduced a joint probability method for the transfer of flood information between two gauging stations. We assumed that the annual maximum floods at the two stations followed a bivariate extreme value distribution. We used a Bayesian framework to find the posterior distribution of the parameters of the bivariate extreme value distribution by using jointly all the data available at the two stations. We then used a sampling technique based on Markov chain Monte Carlo simulations, namely, the Metropolis-Hastings algorithm, to obtain the posterior distributions of flood quantiles and exceedance proba- bilities for both stations. Efficient transfer of information was achieved by simultaneously carrying out (1) the inference of correlation of floods at the two stations, (2) the transfer of information between the two stations, and (3) the inference of the frequency distributions for the two stations. With proper software support the method could be used as a routine engineering tool for flood frequency analysis. By more efficiently using the data that have been collected at large public and private expense we could produce more reliable flood estimates and thus bring considerable economic benefit to engineering designs. i r i i i i Pu, v Figure 6. Probability density curve of the correlation coefficient between the two transformed variables U and V of floods at stations and , respectively. Acknowledgments. A large component of the work presented in this paper was completed while I was employed by the Department of Civil and Environmental Engineering, University of Melbourne, Australia. I would like to thank my former colleagues in that department for their support, in particular, Andrew Western and Rob Argent for their useful comments on an early draft of the paper. I am most grateful to reviewers Bryson Bates, Eduardo Matins, Peter Rasmussen and, in particular, to Associate Editor Jery Stedinger for their valuable comments.

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