Water level [m+ NAP]

Transcription

1 Quantiles of Generalised Gamma Distributions from a Bayesian Point of View Jan M. van Noortwijk September 7, 998 Abstract In this paper, a Bayesian approach is proposed to estimate ood quantiles while taking the statistical uncertainties into account. Predictive exceedence probabilities of annual maximum discharges are obtained using the threeparameter generalised gamma distribution. The parameters of this distribution are assumed to be random quantities rather than deterministic quantities and to have a prior joint probability distribution. On the basis of observations, this prior joint distribution can be updated to the posterior joint distribution by using Bayes' theorem. An advantage is that the generalised gamma distribution ts in well with the stage-discharge rating equation being an approximate power law between water level and discharge. Furthermore, since the generalised gamma distribution has three parameters, it is exible in tting data. Many wellknown probability distributions, which are commonly used to estimate quantiles of hydrological random quantities, are special cases of the generalised gamma distribution. As an axample, a Bayesian analysis of annual maximum discharges of the Rhine at Lobith is performed to determine ood quantiles including their uncertainty intervals. Keywords Bayesian analysis; Discharge; Flood quantiles; Generalised gamma distribution; Rating equation. HKV Consultants, P.O. Box 220, 8203 AC Lelystad, The Netherlands.

2 Introduction The Dutch river dikes have to withstand water levels and discharges with an average return period which is less than or equal to,250 years, where a downstream water level can be determined on the basis of the upstream discharge by using a river ow model (see WL & EAC/RAND [6] and Walker et al. [5]). Therefore, a design discharge is dened as the annual maximum river discharge for which the probability of exceedence is /,250 per year. A usual problem in obtaining design discharges is that there is a limited amount of observations available (for example, with respect to the Dutch river Rhine there are only 94 annual maximum discharges available). The design discharge is usually estimated by extrapolating a probability distribution which is tted to the observed discharges. The two-parameter gamma distribution (Pearson type III distribution) is among the distributions commonly used. When more shape exibility is needed to t the data, Ashkar & Ouarda [] suggest to use the three-parameter generalised gamma disribution. In order to take the uncertainties into account when there is a small amount of data, they developed approximate condence intervals for quantiles of the gegeneralised gamma distribution. These condence intervals have been obtained by applying techniques from classical statistics. In this paper, an alternative way to take the statistical uncertainties into account is proposed: regarding the statistical parameters as being random quantities rather than deterministic quantities. On the basis of the observed annual maximum discharges, the prior joint distribution of these random quantities can be updated to the posterior joint distribution by using Bayes' theorem. In the framework of tting a generalised gamma distribution to discharge data, we further point at the relation between this distribution and the stage-discharge rating equation (an approximate power law in which the discharge is expressed in terms of the water level). The observations that have been analysed are the annual maximum discharges of the Dutch river Rhine at Lobith (near the Dutch-German border). This paper is set out as follows. In Section 2, an approximate rating equation is derived for the river Rhine at Lobith. The mathematics needed to apply a Bayesian analysis to the observed annual maximum discharges can be found in Section 3. Results for the river Rhine are presented in Section 4 and conclusions are gathered in Section 5. 2 Rating equation Although a ood wave is a gradually varied unsteady non-uniform ow, the uniform- ow condition (on the average) is frequently assumed in the computation of ow in rivers. The results obtained from this assumption are understood to be approximate and general, but they oer a relatively simple and satisfactory solution to many practical problems. 2

3 Under the condition of uniform open-channel ow, the average discharge can be determined on the basis of Manning's equation: Q = V A = R2=3 A p S n = A5=3p S P 2=3 n ; wherein Q = discharge [m 3 /s], V = mean velocity [m/s], A = cross-sectional water area [m 2 ], P = wetted perimeter [m], R = A=P = hydraulic radius or hydraulic mean depth [m], S = channel slope [-], and n = Manning roughness coecient [s/m =3 ] (see e.g. Chow [3, Ch. 5-6] or Chow, Maidment & Mays [4, Ch. 2]). Due to extreme discharges, a river overows its main channel and inundates the ood plain. Since the total width of the ood plain of the Rhine at Lobith is about,50 m (including the main channel), the inundated ood plain can approximately be regarded as a wide rectangular cross-section with a width w [m] much larger than its water depth d [m]. In this situation, the hydraulic radius can be approximated by and, accordingly, R = A P = wd w + 2d d Q wd5=3p S : () n 2 x Measured & Extrapolated Fitted: q-q0=a(h-h0)^b Discharge [m^3/s] Water level [m+ NAP] Figure : Actual (measured and extrapolated) and tted rating curve of the river Rhine at Lobith. 3

4 The stage-discharge rating equation () suggests that the rating curve can be approximated by the following power law between water level and discharge (see Shaw [, Ch. 6] and Chow, Maidment & Mays [4, Ch. 9]): q? q 0 a [h? h 0 ] b ; q q 0 ; h h 0 ; a; b > 0; (2) wherein q = discharge [m 3 /s], q 0 = threshold value discharge [m 3 /s], h = water level [m +nap], h 0 = threshold value water level [m +nap], and d = h? h 0 ; the level nap represents the so-called normal Amsterdam level. The thresholds q 0 and h 0 can best be chosen as such that they coincide with a change in slope of the rating curve; for example, a change related to the normal bankful level above which the rating curve diers from the within-banks curve owing to the dierent hydraulics of ood plain ow. By applying rating equation (2), regularly observed or continuously recorded water levels can be converted to corresponding discharge estimates. Hydrological statistical analyses are mostly performed on the basis of discharge data. As a result of Eq. (2), we t a probability distribution to those discharges which are larger than the bankful discharge q 0 = 4; 397 m 3 /s (or, in other words, water levels which are higher than the bankful level h 0 = 2:6 m +nap). With the aid of a least-squares method, a and b in Eq. (2) have been tted to measurements and extrapolations of the Dutch Ministry of Transport, Public Works, and Water Management [0]. As it can be seen in Figure, the estimated values a = 952 and b = :48 result in a very good approximation of the actual rating curve of the Rhine at Lobith. 3 Bayesian analysis of discharges In this section, we show that the generalised gamma distribution ts in well with the stage-discharge rating equation in terms of the power law between water level and discharge. As a matter of fact, if the discharge is assumed to have a generalised gamma distribution and if the power law holds, then the water level has a generalised gamma distribution as well (and the other way round). Furthermore, we regard the statistical parameters of the generalised gamma distribution to be unknown having a joint probability distribution. Since the rating equation (2) holds for the thresholds q 0 and h 0, the following three questions must be answered:. What is the probability that the annual maximum discharge Q exceeds the threshold q 0 m 3 /s? (or, in other words, what is the probability that the annual maximum water level H exceeds the threshold h 0 m +nap?) 2. What is the conditional probability distribution of the annual maximum discharge Q given that Q q 0 m 3 /s? (or, in other words, what is the condi- 4

5 tional probability distribution of the annual maximum water level H given that H h 0 m +nap?) 3. What are the statistical uncertainties involved? These three questions can be answered using Bayesian statistics (see e.g. Bernardo & Smith [2]). The probability that Q q 0, including the uncertainties involved, has been determined in the same way as in Van Noortwijk, Kok & Cooke [4]. Let us denote \the probability that the annual maximum discharge exceeds q 0 " by the random quantity. Suppose we observe z annual maximum discharges of which n exceed q 0, then the likelihood function is given by l(n; z? nj ) = n (? ) z?n I (0;) (); where I A (x) = if x 2 A and I A (x) = 0 if x 62 A. Furthermore, we assume the prior uncertainty such large that the posterior conditional probability density function of - when the observations are given - can be written as p(j n; z? n) / l(n; z? nj )? (? )? : The function [(? )]?, 0 < <, can be interpreted as a non-informative, improper, prior probability density function of (see Bernardo & Smith [2, Ch. 5]). The Discharge [m^3/s] Year Figure 2: Observed annual maximum discharges of the river Rhine at Lobith during

6 posterior probability density function of - when n out of z observations exceed q 0 - is then given by a beta distribution with parameters n and z? n: p (j n; z? n) =?(z)?(n)?(z? n) n? (? ) z?n? I (0;) () = Be (j n; z? n) for z > n > 0. Accordingly, the expected probability that the discharge Q is larger than or equal to q 0 is simply PrfQ q 0 g = R 0 Be (j n; z? n) d = n=z: (3) The conditional probability distribution of the annual maximum discharge at Lobith, given that Q q 0, is chosen as such that the type of distribution remains the same when the transformation Eq. (2) is applied from discharge to water level. If the probability density functions of the discharge Q and the water level H are given by p(q? q 0 ) and ~p(h? h 0 ), respectively, then ~p (h? h 0 ) = ab [h? h 0 ] b? p a [h? h 0 ] b Table : Cross-sectional parameters, as well as the prior and posterior estimates of,,, and. parameter description value dimension w width main channel and ood plain,47 m a parameter rating equation b parameter rating equation q 0 threshold value discharge 4,397 m 3 /s h 0 threshold value water level 2.6 m +nap z number of observations 94 - n number of observations exceeding q L lower bound for 0 - U upper bound for 4 - k number of subdivisions for 00 - L lower bound for 0 - U upper bound for 6 - m number of subdivisions for 00 - E(j n; z? n) posterior expectation of E(j q? q 0 ) posterior expectation of.3 - E(j q? q 0 ) posterior expectation of.79 - E(j q? q 0 ) posterior expectation of 9:24 0?4-6

7 for h h 0 or, in terms of the discharge, p (q? q 0 ) = a? b [q? q b 0 ] b? ~p a? b [q? q 0 ] b for q q 0. A probability density function of Q for which p() and ~p() belong to the same type of distribution is the generalised gamma distribution (see Stacy [3], Lienhard & Meyer [7], and Johnson, Kotz & Balakrishnan [5, Ch. 7]): l (q? q 0 j ; ; ) = (4) =?( ) [q? q 0]? exp n? [q? q 0 ] o I (0;)(q? q 0 ) with unknown parameters > 0, > 0, and > 0. Indeed, the probability density function of h is also a generalised gamma distribution: l h? h 0 b; ; a = = (a ) b?( ) [h? h 0 ] b? exp n? a [h? h 0 ] bo I (0;)(h? h 0 ): The generalised gamma distribution has been succesfully tted to annual maximum discharges by Ashkar & Ouarda []. Since it has three parameters, it includes many well-known probability distributions like the exponental distribution ( = and = ), the gamma distribution ( = ), the chi-square distribution ( =, = t=2, and = =2), the Weibull distribution ( = ), the l -isotropic distribution ( = ), the Rayleigh distribution ( = 2 and = ), and the Maxwell distribution ( = 3 and = 2=3). In addition, the lognormal distribution is a limiting special case when tends to zero, from above. An interesting physical characterisation of gegeneralised gamma distributions on the basis of statistical mechanics can be found in Lienhard & Meyer [7]. As a special case Lienhard [6] derived a gegeneralised gamma distribution (with = 2) to describe rainfull run-o from a watershed. Subsequently, we determine the moments and the exceedence probability of the generalised gamma distribution. For this purpose, the incomplete gamma function is dened as?(x; y) = R t=y tx? e?t dt; x > 0; y 0: By means of the gamma function,?(x) =?(x; 0), the nth moment of Q? q 0 can be written as E ([Q? q 0 ] n j Q q 0 ; ; ; ) =? n? n +.? for n 0. Similarly, the conditional exceedence probability follows: Pr fq > qj Q q 0 ; ; ; g =?? ; [q? q 0].? 7 (5)

8 for q q 0. Note that the gegeneralised gamma distribution takes its name from the fact that Y = [Q? q 0 ] has a gamma distribution with shape parameter and scale parameter. A random quantity X has a gamma distribution with shape parameter > 0 and scale parameter > 0 if its probability density function is given by: Ga(xj ; ) = [ =?()] x? expf?xg I (0;)(x): Recall that n annual maximum discharges exceeding q 0 have been observed. Since these discharges are conditionally independent when the values of the parameters, en are given, the likelihood function of the observations q j, j = ; : : : ; n, can be written as l (q? q 0 j ; ; ) = Q n j= l (q j? q 0 j ; ; ) ; where q = (q ; : : : ; q n ). In order to quantify the uncertainty in the parameters,, and, we assume that they have a prior probability distribution and that they are a priori mutually independent. On the basis of the observations, we obtain the conditional probability density function of, when the values of and are given, as well as the joint probability density function of and. Analogous to determining the posterior probability density function of, we assume the posterior conditional probability density function of - when the values of,, and the observations q are given - to have the form p (j ; ; q? q 0 ) / l (q? q 0 j ; ; )? : The function? can be interpreted as a non-informative, improper, prior probability density function of. The main advantage of this approach is that the posterior distribution of, given (; ; q? q 0 ), results in a gamma distribution (see Bernardo & Smith [2, Ch. 5]): p (j ; ; q? q 0 ) = Pn j= [q j? q 0 ] n = Ga? n n? exp n? P n j= [q j? q 0 ] o n ; P n j= [q j? q 0 ] : (6) In a similar manner, the likelihood function of the observations, given (; ), can be expressed in explicit form: l (q? q 0 j ; ) = (7) Z = l (q? q 0 j ; ; )? d =? n Q n j= h i n [q j? q 0 ]?? Pn j= [q j? q 0 ] : n =0 8

9 Since the posterior joint probability density function of and cannot be expressed in explicit form, we have to resort to approximations. For this purpose, it is customary to dene discrete distributions in the same way as they were applied to quantify the uncertainty in the shape parameter of a Weibull distribution in Soland [2] and Mazzuchi & Soyer [8, 9]. Using Eq. (7) and Bayes' theorem, it follows that where p ( h ; i j q? q 0 ) = l (q? q 0 j h ; i ) p ( h ) p ( i ) P k h= P m i= l (q? q 0j h ; i ) p ( h ) p ( i ) ; (8) h = L + [(2h? )=2] [( U? L )=k]; h = ; : : : ; k; i = L + [(2i? )=2] [( U? L )=m]; i = ; : : : ; m; with L and U being the lower and upper bound for, respectively, and L and U being the lower and upper bound for, respectively. As non-informative prior probability functions for and, we can best use two uniform distributions: p( h ) = Pr f = h g = =k; h = ; : : : ; k; p( i ) = Pr f = i g = =m; i = ; : : : ; m: We note that alternative forms for the prior distributions of,,, and are also possible and that they can be informative, rather than non-informative, as well. Eventually, we are able to determine the predictive expected exceedence probability of the annual maximum discharge by integrating Eq. (5) over the random quantities,, and : where Pr fq > qj Q > q 0 g = (9) = P k P m R h= i= =0 PrfQ > qj Q > q 0 ; h ; i ; g p ( h ; i ; j q? q 0 ) d; p ( h ; i ; j q? q 0 ) = p (j h ; i ; q? q 0 ) p ( h ; i j q? q 0 ) (0) (the product of Eqs. (6) and (8), respectively) for h = ; : : : ; k and i = ; : : : ; m. Note that the prior independence between the random quantities,, and converts into posterior dependence given the observations. By using Eqs. (3) and (9), it follows that Pr fq > qg = Pr fq > qj Q q 0 g Pr fq q 0 g () for q q 0. The Bayesian approach presented in this paper extends the approach of Van Noortwijk, Kok & Cooke [4]. 9

10 Probability lambda Figure 3: Prior probability function of : p( h ), h = ; : : : ; k Probability lambda Figure 4: Posterior probability function of : p( h j q? q 0 ), h = ; : : : ; k. 0

11 Probability mu Figure 5: Prior probability function of : p( i ), i = ; : : : ; m Probability mu Figure 6: Posterior probability function of : p( i j q? q 0 ), i = ; : : : ; m.

12 Probability Discharge [m^3/s] x 0 4 Figure 7: Empirical and predictive cumulative probability distribution of the annual maximum discharge of the river Rhine at Lobith, including their 90 per cent uncertainty interval. 0 0 Probability / Discharge [m^3/s] x 0 4 Figure 8: Empirical and predictive probability of exceedence of the annual maximum discharge of the river Rhine at Lobith, including their 90 per cent uncertainty interval. 2

13 4 Results: Design discharge of river Rhine The Bayesian updating approach has been applied to the annual maximum discharges of the river Rhine at Lobith during the period (see Figure 2). The four largest discharges, sorted in descending order, are 2,280 m 3 /s (926),,93 m 3 /s (995),,365 m 3 /s (920), and,00 m 3 /s (993). The prior and posterior probability functions of and are shown in Figures 3-6. The parameters of the prior probability distributions of and, as well as the posterior expectations of,,, and can be found in Table. Figure 7 presents the empirical cumulative probability distribution, the predictive cumulative probability distribution, and the 90 per cent uncertainty intervals. These 90 per cent uncertainty intervals (the 5th and 95th percentile) are determined using Monte Carlo simulation on the basis of 0,000 samples. The empirical and predictive probability of exceedence, according to Eq. (), are displayed in Figure 8 including their 90 per cent uncertainty intervals. Note that the empirical probabilities of exceedence are calculated with the aid of Chegodayev's formula: Pr fq > q z?u+:zg = [u?0:3]=[z +0:4], where q z?u+:z is the uth largest observation among the sample of z observations ordered by descending magnitude (see e.g. Chow, Maidment & Mays [4, Ch. 2]). The Bayes estimate of the design discharge (i.e. the discharge with an exceedence probability of /,250) equals 5,550 m 3 /s with a 90 per cent uncertainty interval of (2,600; 8,00). To take account of possible future changes in the German basin of the Rhine, a correction term of 500 m 3 /s should be added resulting in a discharge of 6,050 m 3 /s (see WL & EAC/RAND [6]). 5 Conclusions In this paper, the discharge of the Rhine at Lobith with an average return period of,250 years has been determined on the basis of a Bayesian analysis. A three-parameter generalised gamma distribution has been used to obtain predictive exceedence probabilities of annual maximum discharges. In order to take account of the statistical uncertainties involved, the parameters of the generalised gamma distribution are assumed to be unknown and to have a prior joint probability distribution. On the basis of the observed annual maximum discharges at Lobith, this prior joint distribution has been updated to the posterior joint distribution by using Bayes' theorem. An advantage is that the generalised gamma distribution ts in well with the stage-discharge rating equation being an approximate power law between water level and discharge. Furthermore, since the generalised gamma distribution has three parameters, it is exible in tting data. Many well-known probability distributions, which are commonly used to estimate quantiles of hydrological random quantities, are special cases of the generalised gamma distribution. While taking account of possible future changes in the German Rhine basin, the Bayesian estimation procedure results in a predictive design discharge of 6,050 m 3 /s with a 90 per cent uncertainty interval of (3,00; 8,600). 3

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