18. Total probability methods for problems in flood frequency estimation

S. Rocky Durrans Department of Civil and Environmental Engineering The University of Alabama, Tuscaloosa, Alabama U.S.A.

Abstract

The theorem of total probability, when applied in concert with deterministic methods of flood routing, yields an integrated deterministic-stochastic tool which may be employed to salve some difficult problems in flood frequency estimation. Most notably, the integrated modeling approach cari be employed for flood frequency estimation at regulated sites, and it cari also be employed to study the suitabilities of implied structures in schemes that have been proposed for regional flood frequency analysis. Because the method involves a deterministic comportent, it cari also be used in a predictive, and even predscriptive, fashion. It is the put-pose of this paper to present the integrated modeling approacb, and to illustrate its application. N&s and opportunities for additional research are also identifkd.

RQumC

Le th&xbme des probabilit& totales, lorsqu’on l’applique de concert’des mathodes d6terministes de transfert de crues, donne un outil intkgr6 d6terministe et stochastique qui peut étre employ6 pour rbsoudre quelques problémes difficiles pour l’estimation des fn?quences de crues. Plus particulibrement, l’approche de modt5lisation intCg& peut être employ& pour l’estimation des f%quences de cmea sur des sites r@ul& et elle peut Ure aussi appliqut?e pour &udier I’ad6quation des structures impliqu6es dans les schbmas propos& pour l’analyse rbgionale des frQuences de crues. Puisque la m&hode cornprt& une composante dkterministe, elle peut aussi are utiliske sous un mode prtiictif et m&me prescriptif. L’objectif de ce papier est de prbsenter l’approche de modélisation intCgr& et d’en ilhrstrer l’application. Les besoins et les pistes de recherches additionelles sont aussi indentif%s.

18.1 Introduction

18.1.1 General

It is a great pleasure and honor for me to have had the opportunity to visit Paris for the purpose of attending and presenting my work at the Tntemational Conference on Statistical and Bayesian Methods in Hydrology, which conference was held in honor of Professor Jacques Bemier. 1 fïrst met Professor Bernier at the conference which was held at the University of

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11 I I I l I I I l

1900 1910 1920 1930 1940 1950 1960 1970 1980 YEAR

Figure 18.1 fiktoricd trend in aruruul V.S. losses (1983 dollars) due tojbod damage Source: Nationul Weatlrer Service

Waterloo in Ontario, Canada, in the summer of 1993, and 1 have corne to know several of bis colleagues and fîiends (most notably the group at INRS-Eau in Québec City) rather well since that time. Unfortunately, however, 1 have not had the opportunity to collaborate Jirectly with Prof. Bemier himself. 1 am certainly aware of his signifïcant contributions to statistical hydrology, and especially to Bayesian methods, and 1 am greatly impressed by bath their quality and by their depth and breadth. It is my hope that the work presented in the following pages W ill be interpreted as being logically connected to Prof. Bemier’s efforts (through the well-known and fundamental connection between the theorem of total probability and Bayes’ theorem ). It is the intent of this presentation to establish and lay down the framework for an integrated determ inistic-stochastic approach to flood frequency analysis, with the hope that the inclusion of a determ instic comportent in a problem which has historically been treated usually in only a statistical way W ill increase the credibility of flood quantile estimates deriving therefrom .

The problem of estimation of the magnitudes and corresponding probabilities of floods is one of considerable importance. Despite efforts to control the effects of floods by means of bath structural and nonstructural measures, statistics on their effects in the U.S. demonstrate that they are exacting a continually increasing economic and flnancial drain on society. Figure 18.1 shows the historical trend in annual U.S. losses due to flood damage, and indicates that over the time period from 1900 to 1980 annual damages have increased by over an order of magnitude. Hoyt and Langbein (1955) have suggested that the lion’s share of the increase is due to increased property values, as well as the continued development of flood-prone lands. Improved flood loss reporting, as well as possible climatic changes, may have some effect as well. The cost of flooding in terms of loss of life is also of significant concem. When compared to the population-adjusted death rates caused by three other natural hazards (lightning, tomadoes, and tropical cyclones), that due to flooding has shown that little real progress bas been made. Figure 18.2 demonstrates that death rates in the U.S. due to the three compared lnuards bave either dropped dramatically or remained nearly constant over time whereas that due to flooding appears to be slightly increasing. Other effects of flooding relate

300

2.8 r

2.6 R \

0

2.4

i

3, Lighlnmg

i= 2.2 a \

01 l I I I I l 1 1941 1946 1951 1956 1961 1966 1971 1976

to 10 10 10 10 10 10 to 1945 1950 1955 1960 1965 1970 1975 1980

Figure 18.2 Population-a&sted deatll rates in the V.S. from four stom irazardr Source: National Chatic Data Center

to riparian ecosystems and geological processes. The nutrients in sediments which are naturally deposited by floods are essential for biological production and habitat regeneration in the riparian zone. The selective degradation and aggradation of river reaches has far- reaching effects in terms of changes to landforms and river meander pattems.

As already noted, the objective of this paper is to present an integrated deterministic- stochastic approach which has been devised as a consistent framework for approaching various problems in flood frequency analysis. A key component of the approach consists of an application of the theorem of total probability, which is a comerstone of Bayesian theory The motivation behind the development of this framework is that of providing a consistent and physically meaningful basis for flood frequency estimation.

18.2 Total probability applications

The theorem of total probability is an elementary result of an application of the classical axioms of probability (Stuart and Ord, 1987) to a set of mutually eurlusive and collectively exhaustive events. Despite the intrinsic merit of the theorem, however, some would argue that little has corne of it. This is evidently due to the difficulty, in some applications, of evaluation of both the mixture coefficients (the probabilities of the collectively exhaustive and mutually exclusive events) and the conditional probability distributions in an objective and meaningful way.

Within the field of flood frequency analysis, there have been several types of applications of the total probability idea. By far the most common of these is that in which flood events are viewed as arising from differing causal mechanisms. That is, flood events are viewed as being caused by rainfall events, or by snowmelt, or by other similar effects. Mixture models

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are built as a weighted combination of probability distributions, each of which is descriptive of flood events arising from a single causative mechanism. Examples of this application are widespread, examples of which are given by Hazen (1930), Singh and Sinclair (1972), Waylen and Woo (1982), Jarrett and Costa (1982), Hirschboeck (1985,1986), and Diehl and Potter (1986).

Another widespread application of mixture models to flood frequency analysis arises when one must consider ephemeral streams, where there is a finite probability that an annual streamflow maximum Will be equal to zero. In such cases the mixture mode1 Will consist of a combination of both a discrete probability distribution (to represent the single spike of probability mass at zero), and at least one continuous distribution to represent the probability density for peak discharges greater than zero. Examples of this application are given by Jennings and Benson (1969) and Haan (1977).

A third area in which total probability ideas have been applied in flood frequency analysis is that in which they are embedded in applications of Bayesian theory. This type of application is not nearly as widespread as the others that have been mentioned above, primarily because of the general inability to objectively specify a prior distribution. An area in which there has been some work done, however, is that of developing bias correctors for estimators of the coefficient of skewness. Lall and Beard (1982) and Durrans (1994) are two examples of this.

In this paper it is intended to demonstrate how the theorem of total probability cari be coupled with deterministic simulation tools to develop consistent and physically meaningful solutions to two classes of problems in flood analysis. The flrst problem lypc considcred is that of development of flood frequency curves for regulated sites, such as cioLvIlstrcarn of dams. The second application concems the regionalization of flood frequency information.

18.1.3 Outline of paper

Methods of estimating flood frequencies have a long history which dates to at least the early part of the 20th Century. Section 18.2 of this paper provides a very brief summary of the various types of methods that have been developed, and also contains a detailed discussion of some of the fundamental statistical properties of regulated flood peaks. The formal development of an integrated deterministic-stochastic approach to flood frequency analysis is presented in Section 18.3, as is an application of the method for the development of a regulated flood frequency curve. Section 18.4 presents remarks on the way in which the integrated approach may be applied to regionalize flood frequency information. In particular, it is shown how it may be employed to validate (or invalidate) the very rigid and rather ad hoc assumptions that are intrinsic to current regionalization schemes, most notably the index flood method. Conclusions and additional research needs are presented in the closing Section 18.5 of this paper.

18.2 Flood Frequency Analysis

18.2.1 Overview

Flood frequency analysis involves the estimation of exceedance probabilities corresponding to flood peaks of various magnitudes, or vice-versa. Data used to support the estimation process usually consist of the maximum instantaneous discharge rates from each water year of record (an annual series), which is sometimes approximated by the annual maximum average daily discharge. Other data types of interest may consist of flood volumes, maximum stages, or of

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a11 flood discharge peaks which are greater than some threshold. The peaks-over-threshold (POT), or partial duration series, approach is based on the recognition that the second- or even third-largest peaks in some years may be greater than the largest peaks in other years. The brief reviews presented in the following Sections 18.2.2 and 18.2.3, as well as the techniques that are presented later in Sections 18.3 and 18.4, relate to the annual series approach, though the modeling approach could be applied to distributions developed from partial duration series.

18.2.2 Statistical methods

Early approaches to flood frequency analysis were all statistical in nature. That is to say, they involved the fitting of a probability distribution to an observed series of flood peak data. Statistical methods of flood frequency analysis cari be classified into at-site estimation techniques and into regionahzation techniques. They cari also be classified as to whether they are parametric or nonparametric. The discussions in the bulk of this paper are focused on the problem of at-site estimation; a discussion of issues associated with regionalization is delayed until Section 18.4.

The parametric approach to statistical flood frequency estimation is the classical one and is undoubtedly the most widely applied. In this approach, one must Select a probability distribution for modeling of the data, and one must also choose a procedure for estimation of the pamrneters of the distribution. Integration of the fïtted density may then be accomplished to estimate the various quantiles of interest.

The need to Select both a distribution and a parameter estimation method in parametric methods of flood frequency analysis leads to a certain amount of subjectivity in the resulting quantile estimates. In the tails in particular, where little if any data are available, the choice of one probability mode1 over another cari have a significant impact on the resulting quantile estimates. Most models perform quite compambly to one another in their mid-ranges, and this tends to make it very diffïcult to discriminate one from another. The concept of robustness (Kuczera, 1982) is a way in which some of these selection difficulties may be overcome, but a demonstration of robustness cari often involve a time-consuming and costly simulation study.

An application of a pasametric method of flood frequency analysis involves making some assumptions pertaining to the statistical properties of the data being described. In particular, the data should be random, independent, homogeneous, and stationary. A number of tests have been presented in the literature for judging the quality of data in terms of these attributes. A description of a number of these tests are provided by Kite (1977); Loucks, Stedinger and Haith (1981); and Bob& and Ashkar (1991).

Even before the widespread use of parametric methods of flood frequency analysis, there was a good deal of use of’nonparametric methods. The early nonparamehic approaches involved primarily the use of plotting position formulas and probability paper, followed by the sketching of a frequency curve to smooth the trend of the data. The subjectivity of the sketching, as well as the diffrculty of extrapolating a sketched curve, are what ultimately led to the demise of this method, and it was replaced by the more objective methods involving estimation of the parameters of a parametric distribution.

As noted in the previous subsection, however, the parametric approach to flood frequency estimation is not entirely objective either. A measure of subjectivity is introduced by the need to choose the distribution and estimation method. Partly because of the difficulties and uncertainties that are inherent to these choices, but also because of the growing belief that no one parametric distribution is adequate to represent all cases (or even the full range of flood values at a single site), there has been a surge of interest in the past decade on nonparametric

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‘methods of density estimation (Adamowski, 1985). These methods of density estimation are typically based on a superposition, or convolution, of kemel functions, and cari provide very good fits to observed data samples, though they do experience problems when one must extrapolate beyond the range of the data sample.

The nonparametric approach to estimation, like the parametric one, also requires that some choices be made. First one must choose a kemel type that is desired to be used, and one must then decide on how best to estimate the kemel bandwidth. These problems are directly analogous to the choices that must be made in the parametric approach, but Silverman (1986) bas indicated that there is really very little to choose between the various kemels, at least on the basis of the integrated mean square error. Adamowski and Feluch (1990) have considered the use of a skewed kemel (the Gumbel kemel) in an attempt to reduce the bias of quantile estimates in extrapolation, but found that little was to be gained by this. Moon and Lall (1994) have adopted a different approach, and have employed SO-called kemel quantile estimators.

Estimation of the kemel bandwidth-in nonparametric density estimation has usually been accomplished by minimizing the integrated mean square error (IMSE) ofthe density estimator over the full range of the ‘distribution. It is argued here that one should instead focus on minimization of either the mean square et-roc (MSE) or the bias of estimators for particular quantiles. This, of course, is motivated by the observation that the interest in flood analysis is the prediction of quantiles, not density functions, and the fact that minimization of the IMSE does not imply that MSEs and/or biases of quantile estimators are also minimized.

A signifïcant aspect of nonparametric methods of density estimation when compared to parametric methods is that the observations do not necessarily need to be homogencous. Because of the flexibility that is inherent to kemel-based estimators, they cari exhibit the unusual, and sometimes multimodal, density shapes that arise when mixtures of populations are’ present. W ith respect to the qualities of randomness, independence, and stationarity, however, nonparametric methods are subject to the same limitations as are parametric methods.

An additional advantage of nonparametric estimators arises when one must consider multivariate modeling. In flood frequency analysis, this would occur if one were interested in both flood peaks and volumes simultaneously. Parametric modeling using multivariate densities is tractable only in the few cases where multivariate distributions are known, or when the variables are statistically independent of one another. The multivariate normal distribution bas been widely used, but it cari be very difficult to put multivariate flood data into this form, even through the use of normalizing transformations, and this has become a major stumbling block in attempts to mode1 more than one random variable at a time. Multivariate kemel- based density estimators, like their univariate counterparts, are very flexible and cari describe the joint behavior of variables in a nonrestrictive way. Some applications are described by Lall and Bosworth (1994) and Silverman (1986). Silverman also indicates that kemel-based multivariate densities cari be estimated with much less data than cari multivariate histograms or other characterizations of the joint behavior; this is particularly attractive in hydrologie applications where there is often a paucity of data available.

18.2.3 Runoff modeling methods

The runoff modeling approach to flood frequency estimation has developed primarily as a consequence of the continued development of computers and hydrologie simulation codes. TO some degree, however, estimates of flood quantiles were available through the use of runoff models long before these modem accomplishments. A case in point is that of the use of the

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rational method for peak runoff estimation. A fundamental assumption in that case is that the rainfall and nmoff rates have the same frequency of occurrence. It is known that this is not generally truc, but the mtional method continues to be one of the most widely applied methods in day-to-day engineering practice.

Analytical solutions for the derivation of flood frequency distributions from rainfall distributions have also been applied. Eagleson (1972) was the pioneer in this area. A number of other investigators have followed this path, but Moughamian, McLaughlin and Bras (1987) have concluded that these methods do not perform very well. They suggest that fundamental improvements are needed before any confidence cari be assigned to these methods.

Rainfall-runoff simulation models may be classifïed as being either event-based or continuous. For the purpose of simulating flood frequency relationships, however, models of the continuous type are the most widely applied. This is due to the diffculty in practice of specifying appropriate antecedent conditions for event-based models. Inputs to continuous simulation models may consist of historical records if they are available and of sufficient length, but they are probably more frequently obtained as the output of stochastic simulation models. Peaks in the continuous streamflow hydrograph which are generated by the runoff simulation mode1 are subjected to statistical analyses as described in Section 18.2.2. Examples of this approach are provided by Bras et al. (1985); and Franz, Kraeger and Linsley (1986).

An attractive aspect of the runoff modeling approach to flood frequency estimation, like other approaches such as those afforded by the geomorphic instantaneous unit hydrograph (Rodrfguez-Rurbe and Valdes, 1979), is that it represents an attempt to understand and mimic the physical processes that are important in the transformation of rainfall to runoff. It has bcen suggested by the National Research Council (NRC, 1988, p. 56) that runoff models might be useful for regionalization of flood frequency behavior, the thought being that differences in flood frequency curves from one site to another are due only to differences in the catchments, and not in the meteorology. That is, if meteorological variables could be regionalized, then runoff models could be used to account for the runoff response differences due to the catchment properties.

The goal of obtaining flood frequency estimates which are physically based is certainly a laudable one, but it appears as though the runoff modeling approach is simply unable to achieve the desired performance. The complexity of the runoff generation processes, combined with the spatial and temporal heterogeneities and variabilities in the forcing and catchment system variables, conspire to yield a runoff response behavior which is beyond the abilities of models to reproduce. Indeed, when flood frequency curves developed using rainfall-runoff models are compared with those based on actual historical data, the inadequacies of models become quite apparent. Figures 18.3 and 18.4 present results obtained by Thomas (1982) and Muzik (1994), and demonstrate that distributions generated from the outptit of rainfall-runoff models display a variante that is smaller than that exhibited by historical data. Thomas referred to this as a “10s~ of variante” problem; it is analogous to a s’imilar problem in time series synthesis and forecasting where highs and lows are consistently under- and over-predicted.

Given the problems with runoff modeling and derived distribution methods that have been highlighted above, and the objective of the present work to develop a physically meaningful approach to some problems in flood frequency analysis, one is left to question what is being offered here that would surmount the diffïculties discussed. The answer lies in the use of the derived distribution concept, but to derive flood frequency curves from other flood frequency curves rather than from precipitation frequency curves. The physical linkage between the streamflow discharges at different points along a river or stream is much better understood,

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Figure 18.3 Observed and simulated (Qhetic)fiood frequency curves Source: Thomas (1982)

1.003 1.05 1.25 2 5 10 50 100 500 RECURRENCE INTERVAL (years)

Figure 18.4 Comparison of observed and syntheticfloodfrequency curves for tlle Link Red Deer River Source: hfuzik (1994)

and is subject to much less intrinsic variability due to antecedent conditions and the like, than is the linkage between precipitation and the resulting storm runoff. In other words, the transformation of discharge from one site to another along a stream is much more determ inistic

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than is the transformation from rainfall to runoff, at least in terms of our current abilities to describe these processes. Early work by Laurenson (1973,1974) along the same lines as that presented here demonstrates the promise of this ides.

18.2.4 Effects of regulation

Whereas one of the applications of the total probability methods that are presented in this paper is directed to the determination of flood frequency curves at locations downstream of regulating structures, it is appropriate before proceeding to review some of the statistical characteristics of regulated flood peak sequences. Of particular interest are the qualities of randomness, independence, homogeneity, and stationarity. The observations that are made with respect to these qualities are employed in Section 18.3 to develop a modeling framework which is consistent with them.

The first quality of concem is that of randomness. In a hydrologie context, it is generally accepted that randomness means essentially that the fluctuations of the variable of interest arise from natural causes. It is therefore generally considered by hydrologists that flood flows which have been appreciably altered by the operation of a regulating structure are not random. It is argued here, however, that this is not truc. Because flood events occur randomly in time (even though they tend to occur in particular seasons), and because of the randomness associated with the stage (and other conditions) of a regulating reservoir when flood events occur (due to the randomness of antecedent conditions), the regulated flood events downstream of the regulating structure must also be random. This is truc because a function of a random variable is also a random variable, and it must be truc even if the reservoir were operated in exactly the same way every time a flood event occurred (which is not very likely).

The property of independence, in the context of the at-site approach to flood frequency analysis, relates to whether the annual flood event in year t has any predictive ability with respect to flood events in years t+ 1, t+2, and SO on. That is, it refers to the lack of serial cor-relation. In regional analyses, the effects of spatial cor-relation must be considered as well. It is generally true in flood frequency analysis, especially when annual as opposed to partial duration series are being modeled, that sequential flood events are independent of one another in time. Exceptions to this may occur in cases where this is a significant amount of storage present upstream of the location of interest. Lye (persona1 communication, 1993) has considered such problems for Canadian rivers. It is assumed in the sequel that annual flood events cari be considered to be independent of one another; additional work is needed to generalize the results that are presented.

Because of the effect of initial reservoir conditions when flood events occur, as well as the effects of operating the reservoir in different ways, regulated flood events cannot be considered to be homogeneous. That is, regulated flood peaks derive from different population distributions, which may be indexed by the initial and boundary conditions pertinent to the reservoir when flood events occur. A graphical depiction of this is provided by figure 18.5, which shows conditional regulated flood frequency distributions downstream of a reservoir. That figure was generated for the same hypothetical reservoir discussed in Section 18.3.3 using a Monte Carlo procedure. The dotted curve represents the unregulated flood frequency distribution upstream of the regulating facility, and the solid curves show some of the conditional distributions that result. The first of the two numbers shown for each conditional distribution represents the (dimensionless) initial stage of the reservoir (0 = empty, 1 = full), and the second represents the (dimensionless) outlet gate opening amount (0 = closed, 1 = fully open). It bas been assumed that the gate opening amount is held constant throughout the

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~._ .“. -. . .

95 90 80

Exceedance probability, percent

Figure 18.5 Unregulated (dotted curve) and conditional regulated (solid curves) flood freqrretlcy distributions bared on sinrulation of a hypothetical reset-voir

duration of the flood event; this would be true for an unattended reservoir. In other words, the curve with the label (0.9;O) represents the regulated flood frequency distribution that would arise if, every time a flood event were to occur, the reservoir had an initial dimensionless stage of 0.9 and a zero outlet gate opening amount. Of course, real reservoirs, because of the effects of antecedent conditions and operating policies, have initial and boundary conditions (gate openings) that vary from one time to another. For any possible combination of initial and boundary conditions, there is a regulated flood frequency distribution that is conditional on that combination.

It is clearly evident in figure 18.5 that the population distribution from which a regulated flood event derives is very much dependent on the conditions of the reservoir when the flood event occurs. This observation is the basis for the use of the total probability theorem in the integrated determ inistic-stochastic approach presented in Section 18.3.2. A point which may also be noted from figure 18.5, however, is that it tends to yield rather nonsensical results on the left-hand-side of the diagram ; i.e. when the exceedance probability is large. In particular, it indicates flood magnitudes of zero over considerable portions of some of the conditional distributions, pa.rticularly those in which the initial reservoir stage is considerably below the crest of the emergency spillway. This behavior is apparent because the Monte Carlo simulation was accomplished in an event-based rather than continuous manner. The results make sense from a conservation of mass viewpoint, but they do not make sense from a flooding viewpoint. This is true because even low-flow releases made during the year would be greater than zero.

Regardless of the behavior of the left-hand-side of figure 18.5, the right-hand-side does make sense and it is in that region that one is primarily interested anyway. The problems in the’left-hand-side are therefore not believed to be of any significant concem, and this is reinforced by the fact that the nonsensical results arise only when the initial reservoir stage is

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very low. Since the likelihood of this occurring in real reservoirs is usually very small, except perhaps in extremely arid regions, the limitations are not believed to be of too much concem. The net consequence of ail of this is that there Will be an implicit assumption in the developments to follow that annual floods upstream of the regulating reservoir cause annual floods downstream of the regulating reservoir. This is certainly true for the most extreme events, and such is evident in figure 18.5. A POT, or partial duration series, approach to the problem may be able to be applied to lift this assumption, and future work should address this possibility.

The final statistical characteristic of interest is that of stationarity. In the present context, flood sequences Will be taken to be stationary if the reservoir operating policy is stable. A stable operating policy, according to Loucks et al. (1981), is one in which the operating rules are consistent from one year to the next, even though there are within-year variations due to the annual streamflow cycle. Nonstationary regulated flood peak sequences arise when a stable operating policy is not in effet; i.e., when there have been changes made in the way in which the reservoir is operated.

In summary, regulated flood sequences may be considered to be random but they are not homogeneous. Whether they are independent or stationary dcpends on the circumstances of individual cases. For the purpose of this presentation, however, it W ill be assumed that they are both independent and stationary. The issue of independence is an area in which additional work is needed. Where flood sequences are nonstationary due to operational changes, the total flood sequence should be subdivided into subsequences which are intemally stntinnary. This cari be accomplished on the basis of recorded changes in the operating policy.

18.3 Frequency estimation for regulated sites

18.3.1 Overview

Section 18.2.4 provided an exposé of the fundamental statistical characteristics of regulated flood peak sequences. It is the purpose of the next Section 18.3.2 to present a generalized flood frequency modeling framework that is consistent with those characteristics, and which preserves the physical linkage that must exist between flood frequency relationships at different locations along a stream. Section 18.3.3 then provides a detailed example of an application of the developed method, and Section 18.3.4 discusses SO~C of its inherent attributes.

The same integrated deterministic-stochastic modeling framework that is presented in Section 18.3.2 for treatment of regulated flood frequency problems cari also be applied for problems in regionalization. Discourse on this latter application area is contained in Section 18.4.

18.3.2 Integrated modeling framework

There are a number of previous investigators who have presented methods for estimation of regulated flood fîequency relationships. What is believed to be a fairly comprehensive list is Langbein (1958); Laurenson (1973,1974); Sanders et al. (1990), and Bradley and Potter (1992). Al1 of these approaches have involved the theorem of total probability, though in different sorts of ways. Other methods which may be used for regulated frequency estimation derive from the theory of storage (Moran, 1959), as well as from the application of various types of mathematical programming techniques (Loucks et al., 1981). These latter methods yield the probability distributions of release volumes instead of peaks,‘however, and they are

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- ~- ---

therefore not as useful as methods that cari yield the distribution of peaks directly. The method presented by Bradley and Potter (1992) is also fundamentally based on modeling of flood volumes, and obtains peaks on the basis of an observed relationship between the two variables.

As the title of this paper suggests, the theorem of total probability is also used here to permit the modeling of regulated flood frequency behavior. The approach used here is rather unique in comparison with the previous approaches, however, and it tends to emphasize the physical properties of the regulating reservoir that are important determinants of the regulated flood frequency behavior. An introduction to these physical effects, and the way in which they induce heterogeneity into regulated flood sequences, was presented in Section 18..2.4.

In addition to the theorem of total probability (the stochastic component), the integrated modeling approach presented here also involves a deterministic component. It is because of the presence of this deterministic component, of course, that the modeling approach enjoys some physical meaning, and it is also because of this component that the physical linkages between flood frequency relationships are able to be preserved. In application, the deterministic component amounts to no more than a hydrologie (or hydraulic) routing algorithm.

The framework and example presented in this and the subsequent section are intended to establish the regulated flood frequency relationship immediately downstream of a regulating reservoir, based on knowledge of the unregulated flood frequency relationship upstream of the reservoir. If the flood frequency relationship is needed some distance downstrcam of the regulating reservoir, then the techniques presented in this section must be combincd with those presented in Section 18.4. As already noted, there are also several assumptions that are intrinsic to the framework that is presented. Recapping, these assumptions are:

(1) regulated annual floods downstream of a dam are caused by the unregulated annual floods occurring upstream of the dam;

(2) regulated floods are independent events; and (3) the reservoir operating policy is stable.

Because of the need to route flood hydrographs through the regulating reservoir, which involves volume as well as peak discharge considerations, it is necessary to treat flood analysis in this work in a multivariate way. This need also arises because of the several different but interrelated variables that must be considered in order to quantify the initial and boundary conditions pertinent to the reservoir itself. Because of the need to work with multivariate distributions, and because of the complications and inadequacies that arise with multivariate normal modeling, the use of nonparametric methods is believed to be called for.

In the following, let x = [xi x, .-lT denote a random vector of unregulated flood characteristics. Also let y = bl yZ -.]r denote a corresponding random vector of regulated flood characteristics. The individual elements ‘ii and yi of these vectors represent the instantaneous peak flow, the flood volume, and possibly other but more difficult to quantify hydrograph characteristics such as hydrograph shape (multi-peakedness, etc.). Defme FAX) = Pr(X, <x,, X*<x*, -*) as the joint distribution function of the unregulated flood characteristics, and define F,,(y) analogously as the desired unconditional joint distribution function of regulated flood characteristics. In actuality, F’Jy) is dependent on the operating policy that is in effect for the reservoir, but as long as the operating policy is stable that distribution may be viewed as an unconditional one.

The random vectors x and y pertain to the flood variables of interest. TO account for the reservoir, one also needs to introduce a random vector A = [A1 A2 -.]’ and corresponding

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density yA(l) of initial and boundary conditions relevant to the reservoir. The individual elements li of this vector represent the initial reservoir stage (at the beginning of a flood event), outlet gate opening amounts, and possibly other variables as well such as the rate of change of outlet gate openings during the passage of a flood event.

It is necessary in applications to quantify the distributions of the random vectors x and A, and to judge whether they are correlated with one another. That is, one is required to develop estimators for F.&) and fA(L) as well as the ‘correlation matrix between x and 1. If the correlations are judged to be suffrciently large that a hypothesis of independence cannot be supported, then one should develop an estimator for the joint distribution of x and A, which Will be denoted as F,(x,l).

The deterministic component of the integrated procedure involves routing of flood hydrographs through the regulating reservoir to develop a distribution function F,,,,,QlA) of regulated flood characteristics conditioned on a particular combination A of reservoir conditions. This deterministic component of the procedure cari be summarized in a general form as

&&l~) = WXWI (18.la)

for the case where x and A are independent, or as

FY,*cylv = GVL(~, 91 (18.lb)

for the case where x and A are correlated. In these expressions, G, and G, are functions which map the unregulated flood frequency relationship into a conditional regulated one. Actual performance of this mapping must be accomplished using a Monte Carlo method. It is clear from these expressions that the conditional regulated flood frequency distribution is a derived distribution, but that it has been derived from another flood distribution rather than from a rainfall distribution as is done by Eagleson (1972) and others.

The theorem of total probability permits determination of the unconditional distribution F,(y) of regulated flood characteristics as

c w9 =

I FY,AYl wL(oQ (18.2)

where the integration is performed over the complete space of feasible reservoir conditions. A discrete analogue of this application of total probability may be written in the form

fXY) = aIF,@) -t- azF2Cy) + ... + a.F,,(y) (18.3)

where {ai) is a set of weighting factors that sum to unity, and where ~ioI> may be regarded as a component distribution. In other words, a; is the probability of the reservoir conditions being in the i-th of a total of n discrete states, and Fio) is the conditional regulated flood distribution corresponding to that reset-voir state.

The dimensionalities of the vectors x, y and I is an issue that is certainly of some concem in applications. Clearly, the smaller are these dimensionalities, the easier Will it be to determine the regulated flood frequency relationship. However, the unjustifkd use of dimensionalities that are too small W ill obscure some of the important physical determinants of the regulated flood frequency behavior and Will lead to a result which may not accord with reality. It is suggested that the minimum dimension of the vectors x and y be equal to 2, with

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the elements representing the instantaneous peak and the flood volume. With respect to the reservoir conditions, the required dimensionality of the vector I W ill depend on the particulars of each application. In the case of a reservoir with an uncontrolled outlet, only the initial reservoir stage would need to be considered. In the case of a reservoir with a controllable outlet, but in which the outlet gate settings are not adjusted during the passage of a flood (an unattended reservoir), two dimensions.would be necessary (see the discussion in Section 18.2.4). More complex reservoirs with multiple outlets and in which gate settings may be modifîed during the passage of a flood Will require correspondingly greater dimensionalities in the vector 1. A goal in practice should be to make the vector dimensionalities as small as possible for computational reasons without adversely affecting the net result. This cari be accomplished in an iterative way by successively adding to the dimensionalities of the vectors and checking to see whether the derived unconditional flood distribution appreciably changes,

18.3.3 Example application

An example application of the integrated modeling framework to develop a regulated flood frequency curve downstream of a hypothetical reservoir is illustrated in this section. For simplicity of presentation, the reservoir is assumed to have a controllable outlet gate, but the gate settings are not modified during the passage of flood events; that is, the reservoir is an unattended one. It is also assumed in this example that the vectors x and A are independent. It is not the intent of this section to solve an actual real-world r:oblem, but ratfli:r to demonstrate how the procedure may in fact be implemented, and to illustrate the various types of information that are required. The overall procedure is presented in a number of subsections, each of which de& with a specific aspect of the problem.

(i) Marginal distribution of unregulatedjlood peaks

It may be observed that the integrated deterministic-stochastic framework permitting the estimation of regulated flood frequency curves that was described in Section 18.3.2 is nonparametric in nature. That is, there are no assumptions made with respect to the forms of either the unregulated or regulated flood frequency distributions, nor are there any assumptions made as to the form of the distribution of reservoir conditions. For the purpose of this illustrative example, however, it is assumed that the marginal distribution of unregulated flood peaks is the Gumbel, or extreme value Type 1 (EVl) distribution. This assumption is an expedient only, as it is a simple matter to draw random samples from that distribution using methods of simulation. The EV1 distribution is also widely regarded as being reasonably flood-like.

Denoting unregulated annuel flood peaks by the random variable X,, and expressing the EV1 distribution function in inverse form, i.e. as a quantile function, one cari generate synthetic unregulated flood peaks for simulation purposes as

Xl = rn - Q In(-ln u) (18.4)

where u is a uniformly distributed random variable on the interval (0,l) and a and m, respectively, are scale and location parameters of the EV1 distribution.

In the present example, E(X,) = 300 m3S’ and the coefficient of variation of X, is 0.3. The parameters a and M in equation (18.4) are therefore equal to 70.2 and 260, respectively. These assumptions make the probability of generation of negative values of x, extremely small.

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Negative values, if and when generated in the simulations, were discarded and replaced by a subsequently generated positive value.

In mal-world applications, estimation of the unregulated flood frequency distribution must be accomplished using streamflow data observed upstream of the reservoir. If a gaging site is some distance upstream of the reservoir, then the procedures discussed in Section 18.4 cari be employed. In other cases, it may be possible to use data for the reservoir itself, such as stages and releases, to derive the reservoir inflow hydrograph and hence the unregulated flood frequency distribution as well.

(ii) Conditional distriblttion of unregulatedjlood volumes

Denote the random variable representative of unregulated flood volumes by X,, and condition the distribution of flood volumes on the magnitude of flood peaks. Rogers (1980,1982), Rogers and Zia (1982), Mimikou (1983), and Singh and Aminian (1986) have concluded that a relationship between flood peaks and volumes cari be expressed by

h21, (Qpm = b + r log,, V (18.5)

where Qp = x,/A is the peak discharge rate per unit area, V = x,/A is the runoff volume per unit area, and A is the area of the drainage basin. Singh and Aminian (1986) considered x, and x, as the peak and volume of the direct runoff hydrograph. Base flow needs to be ndded separately, and has been assumed to be a constant 20 rn’s-’ in this illustrative example.

Equation (18.5) was originally established by means of a linear regression of log(Q,,#) on log V. Bradley and Potter (1992) have also used simulation and the nonparametric LOWESS smoother (Cleveland, 1979) to develop a relationship between flood peak and volume. The intent of these previous studies has been to predict flood peaks from flood volumes. In the present example it is intended to do the opposite; that is, it is intended to predict flood volumes from flood peaks. Because of the analytic form of equation (18.5), as well as the desire to keep the example relatively simple, that expression Will be employed here.

Shictly speaking, a relationship developed by regressing a variable y on another variable x should not be inverted to develop a predictor for x as a function of y. This is SO because there is not in general a “reverse causality”, and also because the parametcrs in the functional relationship Will in general be different for the inverse relationship than for the forward one. Equation (18.5), however, does not imply a causal relationship (flood peaks are not caused by flood volumes); it is simply the consequence of an empirical observation. The linearity of the logarithmic plot of the data would have been present regardless of which of the variables had been taken as the predictor. It is for this reason that it is assumed here that equation (18.5) cari be ïnverted and rearranged, and that the resulting expression given as follows cari be interpreted as the expected value of log,, Vgiven log,, Qp:

al%lo v) = c(lcg Y = (log,, Q, - W(r + 2)

It is also assumed for the present example that the conditional distribution of log,, V given log,, Qp is normal with a standard deviation of ab, v = 0.1, that the drainage basin has an area ofA = 1300 km’, and that the values of the parameters in equation (18.6) are b -= -1.75 and r = -1. These values, based on the work of Singh and Aminian (1986), are reasonable, even though their original relationship has been inverted.

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Under the foregoing assumptions, an unregulated flood volume x1 may be generated for simulation pur-poses as

x2 = A antilog(p, V + tar, J (18.7)

where z is a standard normal variate with zero mean and unit variante. Note that because of the log transformation, it is not possible to generate negative flood volumes using this relationship.

The unregulated flood peaks x1 and flood volumes x2 determined based on the procedures discussed in this and the previous subsection are used in this example to quantify reservoir inflow hydrographs. For pur-poses of illustration, some rather analytical expressions have been used for these variables, but this should not be construed to imply that the assumptions made to achieve those expressions are necessary. In actual applications, it may be preferable to mode1 the joint distribution of xi and x2 using nonparametric multivariate kemel methods. Silverman (1986) and Lall and Bosworth (1994) provide examples of this technique.

(iii) Reservoir inflow hydrographs

Based on the values of x, and x2 generated as described in the previous subsections, one must construct a synthetic direct runoff hydrograph. This hydrograph, when combined with the base flow, may then be routed through the reservoir to obtain the outflow hydrograph properties. Naturally, the outflow hydrograph properties Will be conditional, based on the initial and boundary conditions pertaining to the reservoir.

As an expedient, the U.S. Soi1 Conservation Service (SCS) dimension& triangular l:\Pdrograph (SCS, 1969) is used in this example as a standard shape to represent the direct runoff component of the reservoir inflow hydrograph. The triangular hydrograph is characterized by linear rising and receding limbs, with a hydrograph base time Tb equal to 2.67 times the time to peak TP. Since the peak of the direct runoff hydrograph is equal to x,, and since the volume of the direct runoff hydrograph must be equal to x2, the direct runoff hydrograph base time is Tb = 2x2/x, and its time to peak is TP = 3x2/4x,.

Use of the SCS triangular hydrograph in this way implies that flood hydrographs Will always have only a single peak. Should it be desired to permit the possibility of multiple peaks, a greater dimensionality would need to be considered for the random vector x.

(iv) Reset-voir initial and boundary conditions

The reservoir considered in this example is of a very simple nature, but is adequate to serve the demonstration purposes of this presentation. The reservoir is considered to have vertical sides, a single outlet gate whose opening is controllable, and an emergency overflow spillway which is modeled as a weir. For ease of presentation, the outlet gate opening amount is assumed to be fixed throughout the passage of a flood event through the reservoir. As noted earlier, this is not a limitation of the method described in this paper as additional reservoir variables could be included to account for the rates and/or times of change of gate opening amounts.

For the case of this simple reservoir, the reservoir variables comprising the vector 1 are the initial reservoir depth (an initial condition), denoted by 1,, and the outlet gate opening area (a boundary condition), denoted by AZ. Modeling of the distributions of these variables is accomplished by defining a dimensionless initial reservoir depth D. and a dimensionless gate

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opening area A. defined by

D. = &lD, (18.8)

and

A. = 12/A, (18.9)

The terms D, and A, in these expressions denote, respectively, the full reservoir depth (to the crest of the emergency spillway), and the full gate opening area. Reservoir depth is measured with respect to the outlet gate opening, whose hydraulic behavior is modeled as an orifice.

Whereas both of the variables D. and A. are defined only on the inter-val [O,l], they are modeled arbitrarily in this example using the beta distribution. For illustrative pur-poses, the marginal density of the dimensionless depth is taken to be

fD(D.) = 30.2 (18.10)

It is clear from this definition of the marginal density that a full reservoir is the most probable initial condition of the reservoir when a flood event occurs.

The distribution of the dimensionless gate opening amount is assumed in this example to depend only on the dimensionless depth. Its conditional density function is assumed to have the form

&,(A. ID.) = A.=-‘( 1-A.)‘-?(a + p)/lr(a)r(p)]

where

a = 1 + 9D.

(18.11)

(18.12)

P = 10 - 9D. (18.13)

This specitïcation of the conditional distribution of outlet gate opening amounts states that when the reservoir is empty, a zero gate opening amount is the most probable situation. When the reset-voir is full; a full gate opening amount is the most probable situation, and when the reset-voir is half full, the most probable gate opening amount is also one-half of the full amount. Figure 18.6 is an illustration of a histogram that is representative of the joint distribution of D. and A. as defined by equations (18.10) through (18.13). The heights of the columns; i.e. the (ai) values for use in equation (18.3), were determined by numerical integration. Modeling of the joint distribution of D. and A. in this way is again only an expedient that has been employed for this illustrative example. In applications it would likely be preferable to mode1 the joint distribution using a nonparametric kemel method.

It is clear that the distributions of the random variables D. and A., and hence of the variables )Li and A,, W ill depend on the operating policy in effect for the reservoir. Changes in the operating policy, if and when they occur, W ill result in changes in these distributions and hence in changes in the downstream regulated flood frequency relationship. Where actual data relevant to reset-voir conditions are not available to permit the estimation of the joint distribution of reservoir conditions, or in cases where one might be interested in predicting the effects that would occur as a consequence of operational changes, one cari resort to methods

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- -.

0 41

i

Figure 18.6 Joinf dett.siry&rtction of A. and D. for example probkm

of simulation to derive the necessary data. Note, however, that the inabilities of simulators to accurately depict the whole range of streamflow responses is not an issue in this case. This is SO because reservoir conditions at the beginning of flood events are c3ntrolled by antecedent conditions, and these in tum tend to be dominated by relatively average streamflow conditions. Hydrologie simulators are quite good at being able to reproduce system behaviors in such situations.

The discharge from an orifice with an opening area 1, and a discharge coefficient C, when the head on the orifice, i.e. the reservoir depth, is equal to h is

QO = C,A,J(2gh) (18.14)

The discharge from a rectangular weir of length L with a weir coefficient C, and a head h, is

Q, = C,&h;12 (18.15)

For the purposes of this example, C,, = 0.6 is used in the orifice equation (18.14) for a representation of the reservoir’s principal spillway (a conduit type of spillway), and the weir equation (18.15) with C, = 3 and L = 50 m is used for the overflow spillway. Weir flow is assumed to occur only if the reservoir is surcharged during a flood event such that the depth h becomes greater than the full depth DP In such cases the head on the weir is taken to be h,

= h’- D,. Other variables pertinent to the reservoir used in the simulations are presented in table 18.1.

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Tahle 18.1 Reset-voir propoerties for example problem

Property Symbol

Full reset-voir depth Dl Full gate opening area 4 Reset-voir surface area A, Overflow weir length L Orifice discharge coefficient c, Weir coefficient cw

Value employed

60 m 5 m’ 1.05 x 10’ m’ 50 m 0.6 3.0

(Y) Simulation procedure

The simulation procedure that should be employed to compute the regulated flood frequency curve for a given reservoir operating policy depends on whether the random vectors x and I are independent or correlated. Since it has been assumed throughout this example application that they are independent, that procedure Will be given first. The procedure for the case where they are correlated Will then be given.

A step-by-step procedure which may be followed for the case where the independence of x and A is true is as follows:

(1) Develop an estimator of the distribution FAX) of the upstream unregulated floods. Also develop an estimator of the density fA(A) of reservoir initial and boundary conditions. These estimators may be developed using either parametric or nonparametric techniques.

(2) Randomly sample values of x, and x2 from the distribution of unregulated flood characteristics. Construct the direct runoff component of a synthetic reservoir inflow hydrograph using these two values, and add base flow to obtain the total synthetic inflow hydrograph.

(3) Randomly sample values of D. and A. from the distribution of dimensionless reservoir conditions, and compute values of A1 and A2 using equations (18.8) and (18.9).

(4) Route the inflow hydrograph through the resexvoir using the continuity equation

dhldt = [I(t) - Q(h)]/A, (18.16)

where h is the reservoir depth at time t, I(t) is the synthetic inflow hydrograph developed in step (2), Q(h) is the depth-dependent reservoir outflow rate, and A, is the reservoir surface area. Integration of equation (18.16) was accomplished for this example using a predictor-corrector, or Heun, method (Chapra and Canale, 1988) with a time step of At = TJlO.

(5) Repeat steps (2) through (4) many times (say N times) to obtain N outflow hydrograph peaks. Rank and assign plotting positions to these values and use them to empirically define the regulated flood frequency distribution Fyo1). The value of Nshould be chosen sufficiently large that the empirical distribution is not sensitive to small variations in N; it is suggested that N should be at least several thousand.

When performing steps (2) through (4) in the above procedure, one could also obtain N regulated flood hydrograph volumes as well. One would then have the necessary information

317

to empirically quantify the joint distribution of both regulated flood peaks and volumes. It may be noted that this procedure applies the theorem of total probability in a rather

implicit sort of way. An alternative and more explicit application of the theorem may be accomplished through discretization of the joint density of reservoir conditions in a manner similar to that shown in figure 18.6, and use of equation (18.3). This alternative procedure was used to generate figure 18.5 as it yields the conditional distributions, which may be of interest in some applications, as well as the final unconditional distribution.

The simulation procedure that should be used when the random vectors x and A are correlated is essentially the same as !!tat given above for the independent case. The primary difference is that one would first develop an estimator for the joint distribution F,(x,À) of both flood characteristics and reservoir conditions. The values of xi, xz, Ii, and AZ would then all be sampled from that distribution. The remaining steps of the procedure would be the same as for the independent case.

18.3.4 Discussion

The result of the application of the step-by-step procedure discussed in the previous subsection is shown in figure 18.7. The dotted curve shown there is the marginal distribution of unregulated flood peaks upstream of the reservoir, and the solid curve is the marginal distribution of regulated flood peaks immediately downstream of the reset-voir. For reasons discussed in Section 18.2.4, it is not clear that the regulated flood distribution shown is sensical in the left-hand portion of the figure. However, the right-hand portion of tlrc ti~urc, which is the region of prime interest in applications, does make sense. Indeed, it may be observed that the two frequency curves Will converge to one another as the flood magnitude increases, i.e. as the exceedance probability decreases. This must be SO because of the diminishing effect of a reservoir in flood peak attenuation as the flood magnitude increases. The fact that this consistency is attained is made possible only because of the integrated nature of the approach. In effect, the integrated approach is able to preserve the physical linkage that must exist between the two flood frequency relationships.

An additional point worthy of note is that the simulation procedures described above are very well suited to implementation in parallel processing environments. This is clearly desirable because of the computational intensiveness of the required Monte Carlo simulations.

18.4 Regionalization of Frequency Information

18.4.1 Overview

Regionalization techniques in the field of flood frequency analysis are motivated by the recognition that quantile estimates based only on at-site data, because of the shortness of streamflow records and the need to extrapolate to long recurrence inter-vals, have large degrees of variability, and hence uncertainty, because of sampling variations. The use of historical data cari be employed to. ameliorate these problems to some degree, but the practice of regionalizing flood fmquency behavior is likely the more common approach. Where possible, the use of both historical data and regionalization should be employed.

TO a certain degree, the use in hydrology of the term regionalization has corne to refer to two different but related techniques. This is rather unfortunate, and it has likely led to some confusion among practitioners. In the first type of regionalization, one is interested in predicting flood quantiles at ungaged sites. While this cari be accomplished using rainfall and

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3 20 10 5 2 1

Exceedanco probability, percent

Figure 18.7 Regulated (solid) and unregulated (dotted) jloodji-equency distributions

runoff modeling methods (see Section 18.2.3), the term regionalization usually refcrs to the use of multivariate regression models (Benson, 1962). The U.S. Geological Survey has devoted a considerable amount of effort to develop such models for use throughout the United States. The second type of regionalization, which is the more prominent one in the recent flood frequency literature that has appeared in the archiva1 joumals, involves the use of information at gaging sites remote from the one of primary interest to improve the statistical properties of quantile estimators. The focus here is on improving estimates at gaged sites, though it is recognized that this should ultimately enable improved estimates at ungaged sites to be obtained as well. There are a number of methods that have been proposed for accomplishment of this second type of regionalization. The most prominent among them are the index flood method (DaQmple, 1960) and regionalization of distributional parameters (Houghton, 1978a,b) and statistics (namely skewness) (Hardison, 1974; Tasker, 1978), though this latter method may be counterproductive (Landwehr, Matalas and Wallis, 1978).

It is the objective of this section of this paper to show how the integrated modeling framework developed in Section 18.3.2 may be employed for regionalization. The issues motivating this additional application area are discussed in the following Section 18.4.2, and Section 18.4.3 provides an overview of the extension of the approach to the problem of regionalization. It is noted here at the outset that the regionalization method suggested here cari be employed for both types of regionalization problems mentioned above. That is to say, it cari be employed for the estimation of flood frequency relationships at ungaged sites, and it cari also be employed to improve the estimates at gaged sites. Section 18.4.4 remarks on the statistical estimation gains which may be realized in the latter type of regionalization.

18.4.2 Motivation

There are two primary issues that are motivating the extension of the integrated modeling framework to permit it to be employed in a regionalization context as well. The first issue

319

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motivating this discussion stems from the recognition made in Section 18.3.4 that the integrated modeling approach cari preserve the physical linkage that must exist between flood frequency relationships at different spatial locations (in that case, at locations upstream and downstream of a regulating reservoir). This leads one immediately to ponder whether the same approach might be useful for regionalization of flood frequency information. It is maintained by this author that the answer to this must be in the affirmative, and that the integrated modeling framework which has been devised is essentially a “comprehensive statistical model” as called for by the National Research Council (NRC, 1988).

The second motivating issue stems from some perceived shortcomings in the currently applied regionalization procedures, most notably in the index flood method. This method is purely statistical and makes use of some very rigid and ud hoc assumptions which tend to be very difficult to rationalize and validate based on physical and hydrologie reasoning. In particular, the index flood method presumes that the flood frequency distributions at all sites in a homogeneous region are identical except for scale. In other words, it is assumed that all sites in the region have the same coefficients of variation and skewness. Other statistical methods of regionalization involve similar assumptions as to the spatial stationarity of one or more statistical characteristics. Lettenmaier, Wallis and Wood (1987) and Hosking and Wallis (1988) have shown that the index flood method is reasonably robust to departures from truly homogeneous regions, but this is still not very comforting in view of the lack of any physical or hydrologie reasoning to support it. In fact, it is argued shortly that physical reasoning implies that the index flood method is not suitable for flood frequency regionalization, despite the fact that it sometimes seems to work reasonably well.

An additionaI issue confounding statistical methods of regionalization, and one of the most difficult to overcome in practice, is that of the need to identify homogeneous regions of gaging sites. A number of methods have been presented in the literature for accomplishing this task, but they again tend to be purely statistical in character. Most frequently, the pooling of sites into homogeneous regions is based on whether significant differences cari be discemed between like statistics computed for different sites. Unfortunately, the statistics of interest in this respect are usually the moment or L-moment ratios of relatively high order, and these tend to have sufficiently large sampling variantes that any tests for discrimination which might be devised are necessarily not very powerful. In effect, subtle differences in statistics from one ,site to another cari be very difficult to detect. Such methods of pooling sites into regions are based entirely on statistical considerations, and take no account of the physics of flood events. The only ways in which the most common assumptions used in regionalization cari be justified are based on statistical arguments, and these must be considered to be weak because of the lack of power of discriminating tests.

TO illustrate the type of problem that cari arise, consider a gaging site for which the random variable representing annual flood peaks is denoted as X. Consider also an additional site downstrearn and along the same stream, and denote the same random variable there as Y. Because these two sites are along the same stream, and therefore are nearly identical in terms of their flood frequency behavior, most would agree that these two sites should be ‘pooled together into thesame homogeneous region. Inde& it is difficult to imagine a case where two sites would be considered more homogeneous. Now, if the random variable Y at the downstream site is a simple linear function of the random variable X at the upstream site, i.e. if Y = cX, where c is a constant, then it is easy to show that the coefficients of variation and skewness for the two sites are identical. In this case the common index flood assumption would be justifiable, at least on statistical grounds. If, on the other hand, however, the physical linkage that must exist between the two sites indicates that the relationship is more

320

likely nonlinear, such as Y = axb, b # 0,1, then the index flood assumption would be invalid; Given that the hydrologie and hydraulic behaviors of real rivers and streams are generally nonlinear, this observation casts some serious doubt on the suitability of the index flood method.

18.4.3 Extension of integrated modeling

As already noted in the previous section, the recognition of the ability of the integrated deterministic-stochastic modeling framework to preserve the physical linkage between different sites leads one to ponder its potential application for regionalization as well. In the present section is considered an approach which may be employed for development of a flood frequency relationship for an ungaged site. This is the first of the two types of regionalization that were discussed in Section 18.4.1.

TO accomplish this estimation at an ungaged site, it W ill be necessary (at least initially) to consider sites only on streams on which there is also a gaged site. Denote the gaged location as site X, and denote the flood frequency distribution which may be estimated from the records for that site,as FAX). Denote the ungaged location as site Y, and denote the desired flood frequency distribution at that site as Fr@). Denote the joint density of initial and boundary conditions relevant to the stream reach between the two sites asfA(

It is clear that this notation is virtually identical to that which was employed for development of regulated flood frequency curves in Section 18.3. The elements of the random vectors x and y will again refer to instantaneous flood peaks, flood volumes, and possibly other flood hydrograph characteristics. In the present regionalization case, however, the elements of the vector A of initial and boundary conditions will have somewhat different meanings. One of the elements in this vector Will be the initial stage or discharge in the river reach at the beginning of flood events, and the remaining elements Will correspond to both boundary conditions and forcing relevant to the stream mach behveen the two sites. Boundary conditions may exist within the reach or may exist somewhere outside of the reach, but they should be chosen such that they do in fact have an effect on the hydraulic behavior of the reach. An example of a boundary condition outside of the reach would be one in which the stream discharges into a large Me, and in which the lake causes a backwater effect within the stream reach of interest. Forcing that would be relevant to the reach would consist of lateral inflows and/or outflows to and from the reach. These could be accounted for using a runoff model.

The procedure for deriving the desired flood frequency distribution E’Jy) in this regionalization, or information transfer, application would be essentially the same as that used to derive a regulated flood frequency distribution in Section 18.3. The only real difference is that channel routing would be used instead of reservoir routing. One could also choose between hydrologie and hydraulic routing schemes (this is true as well for the reservoir case, but there one would almost always choose a simple hydrologie router). If the ungaged site were upstream of the gaged site, then inverse flood routing would need to be accomplished.

An extension of this information transfer idea for ungaged sites could also be extended to ungaged sites at other locations within a drainage network. That is, it is not absolutely necessary that the ungaged site be on the same link in the overall network as is the gaged site. This type of an application would, however, require the consideration of the complicating factors at confluences of streams. As shown by Dyhouse (1985), however, this is yet another area in which the theorem of total probability finds application. In effect, the theorem of total probabihty, when used in conjunction with other, deterministic hydrologie and hydraulic tools, cari be employecl to facilitate the prediction of the flood frequency behavior almost anywhere

321

in a stream network based on knowledge of the behavior at one or more other locations in the network. This modeling framework is therefore extremely powerful, but its potential is currently limited by the loss of variante problem associated with the runoff modeling tools that would be necessary to account for lateral inflows and outflows.

18.4.4 Variante reduction through optimal interpolation

The problems that arise in the regionabzation, or information transfer, problem as a consequence of the need to use runoff models cari be overcome by considering gaged sites only. That is, rather than employing one gaged site and one ungaged site in the modeling effort, one cari employ two gaged sites.

W ithout loss of generality, consider two sites X and Y on the same stream link, and assume that site X is upstream of site Y. Because both sites are gaged and hence have streamflow records that have been collected for some period of time, one cari estimate their respective flood frequency distributions F&) and FJy) using standard methods of statistical analysis. One cari also, because of the records available, quantify the joint densityf,(A) of streamflow conditions and incremental flows between the two sites. This would require some use of a routing mode1 to account for peak attenuation within the reach, but it would obviate the requirement of a runoff simulation model.

Now, given the flood frequency distribution F&) and the joint densityf,(l), one could again employ the integrated modeling framework to develop a flood frequency distribution at site Y. Since this derived distribution Will be different from the distribution determined from the records at site Y, it will be denoted here as FAz). The net result of this exercise is that one will have two estimators for the flood frequency distribution at site Y. That is, one Will have redundant estimators for various flood quantiles. Based on the ideas of optimal interpolation (Gelb, 1989), one could then combine the redundant estimators for any desired quantile in a linear fashion SO as to develop a quantile estimator with a smaller variante than that possessed by either of the two original estimators. The improved flood frequency distribution at site Y might then be employed with a reverse application of modeling to improve the distribution at site X. This might then be used again to improve the estimator at site Y, and SO on in an iterative way.

The net result of this application is that one cari accomplish the most fundamental objective of regionalization, namely that of improving the statistical properties of quantile estimators by permitting information at sites remote fi-om the one of immediate interest to have some bearing on the estimation process. In contrast to purely statistical methods of regionalization, however, the integrated modeling approach accomplishes the task in a meaningful way.

18.5 Summary

It bas been argued in this paper that an integrated deterministic-stochastic modeling framework may be employed to consistently and effectively approach some of the more difficult and elusive problems in the field of flood frequency analysis. In particular, it cari be employed to develop flood frequency curves at regulated sites downstream of dams and reservoirs, and it cari also be used for the transfer of information from one spatial location to another. It combines the best features of both statistical and deterministic modeling tools, and moulds them into a new tool whose power is arguably greater than that of the sum of its component parts. In effect, it establishes a framework for a “comprehensive statistical model” (NRC,

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1988) which car-t be employed to resolve the differences between the statistical and runoff modeling approaches to flood frequency analysis.

It is important to recognize that the developmental aspects of the integrated modeling tool are by no means complete. Several assumptions have been made in the discussions in this paper, and more work is necessary to generalize the method even further. Of particular relevance in this respect are the issue of independence of regulated armual floods, as well as the treatment of partial duration series. Additional work related to regionalization (information transfer) should also be given a high priority.

Flood frequency modeling with the integrated tool involves the use of multivariate probability distributions. These distributions are considerably more difficult to work with than are univariate models, and are therefore more exacting in terms of the educational background requirements on the part of mode1 users. Multivariate modeling is also more demanding in terms of data requirements (the amount of data needed), and this is certainly a cause for some concem, particularly in an application area such as flood frequency analysis where there never seems to be enough data. Planet Earth Will continue to tum, however, and data Will continue to be collected. At the same time, more and more rivers and streams Will become regulated, and the need to be able to estimate regulated flood frequency relationships Will become more acute. But what are the most important types and quantities of data that Will be needed to accomplish this estimation? The integrated modeling approach presented here is a tool which cari be applied in a systematic way to answer this question. Use of this modeling approach cari therefore be employed as a guide to point the way in future data collection and archival efforts.

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Bibliography

Adamowski, K. (1985) Nonparametric kemel estimation of flood frequencies. Water Resour. Res., 21, 18851890.

Adamowski, K., and W. Feluch. (1990) Nonparametric flood-frequency analysis with historical information. ASCE Jour. Hydr. Engr., 116, 10351047.

Benson, M. A. (1962) Evolution of methods for evaluaring the occurrence of jloods. U.S. Geological Survey Water Supply Paper 1580-A, Washington, D.C.

Bob& B., and F. Ashkar. (1991) The gamma family ami derived distributions in hydrology. Water Resources Publications, Littleton, Colorado.

Bradley, A.A., and K.W. Potter. (1992) Flood frequency analysis of simulated flows. Water Resour. Res., 28, 23752385.

Bras, R.L., D.R. Gaboury, D.S. Grossman, and G.J. Vicens. (1985) Spatially varying rainfall and floodrisk analysis. ASCE Jour. Hydr. Engr., 111, 754-773.

Chapra, S.C., and R.P. Canale. (1988) Numerical methods for engineers. 2nd ed. McGraw- Hill, New York.

Cleveland, W.S. (1979) Robust locally-weighted regression and smoothing scatterplots. Jour. Amer. Stat. ASSOC., 74, 829-836.

Dalrymple, T. (1960) Floodfiequency analyses. U.S. Geological Survey Water Supply Paper 1543-A, Washington, D.C.

Diehl, T., and K.W. Potter. (1986) Mixed flood distributions in Wisconsin, paper presented at the International Symposium on Flood Frequency ami Risk Analysis, Louisiana State University, Baton Rouge, Louisiana.

Durrans, S.R. (1994) Bayesian approach to skewness bias correction for Pearson Type 3 populations. Jour. Hydrol., 161, 155-168.

Dyhouse, G.R. (1985) Stage-frequency analysis at a major river junction. ASCE Jour. Hydr. Engr., 111, 565-583.

Eagleson, P.S. (1972) Dynamics of flood frequency. Water Resour. Res., 8, 878-898. Franz, D.D., B.A. Kraeger, and R.K. Linsley. (1986) A system for generating Iong

streamflow records for study of floods of long retum period, paper presented at the International Symposium on Flood Frequency ami Risk Analysis, Louisiana State University, Baton Rouge, Louisiana.

Gelb, A. (ed.) (1989) Applied optimal estimation. The MIT Press, Cambridge, Mass. Haan, CT. (1977) Statistical methodr in hydrology. Iowa State University Press, Ames. Hardison, CH. (1974) Generalized skew coefficients of annual floods in the United States and

their application. Water Resour. Res., 10, 745. Hazq A. (1930) Floodflows: A study offiequencies and magnitudes. John Wiley and Sons,

New York.

324

--~ -..

Hirschboeck, K.K. (1985) Hydroclimatology offlow events in the Gila River basin, central and southem Arizona, Ph.D. Dissertation, University of Arizona, Tucson, Arizona.

Hirschboeck, K.K. (1986) Hydroclimatically-defined mixed distributions in partial duration flood series, paper presented at the International Symposium on Flood and Risk Analysis, Louisiana State University, Baton Rouge, Louisiana.

Hosking, J.R.M., and J.R. Wallis. (1988) The effect of intersite dependence on regional flood frequency analysis. Water Resour. Res., 24, 588-600.

Houghton, J.C. (1978a) Birth of a parent: The wakeby distribution for modeling flood flows. Water Resour. Res., 14, 1105-l 110.

Houghton, J.C. (1978b) The incomplete means estimation procedure applied to flood frequency analysis. Water Resour. Res., 14, 111 l-l 115.

Hoyt, W.G., and W.B. Langbein. (1955) Floods. Princeton University Press, New Jersey. Jarrett, R.D., and J.E. Costa. (1982) Multi-disciplinary approach to the flood hydrology of

foothill streams in Colorado, in International Symposium on Hydrometeorology, AI. Johnson and D.A. Clark (eds.), 565-569, Amer. Water Resour. Assoc., Bethesda, Maryland.

Jennings, M.E., and M.A. Benson. (1969) Frequency curves for annual series with some zero events or incomplete data. Water Resour. Res., 5, 276-280.

Kite, G.W. (1977) Frequency and risk analyses in hydrology. Water Resources Publications, Littleton, Colorado.

Kuczera, G. (1982) Robust flood frequency models. Water Resour. Res., 18, 315-324. Lall, U., and L.R. Beard. (1982) Estimation of Pearson Type 3 moments. Water Resour. Res.,

18, 1563-1569. Lall, U., and K. Bosworth. (1994) Mutivariate kemel estimation of functions of space and

time, in T%e Series Analysis in Hydrology and Environmental Engineering, K. W. Hipel, A.I. McLeod, U.S. Panu, and V.P. Singh (eds.), 301-315, Kluwer Academic Publishers, Dordrecht, The Netherlands.

Landwehr, J.M., N.C. Matalas, and J.R. Wallis. (1978) Some comparisons of flood statistics in real and log space. Water Resour. Res., 14, 902.

Langbein, W.B. (1958) Queuing theory and water storage. Jour. Hydr. Div., ASCE, 84, 181 l- 1 to 1811-24.

Laurenson, E.M. (1973) Effect of dams on flood frequency. Proc., International Symposium on River Mechanics, 9-12 January, Bangkok, Thailand, International Association for Hydraulic Research.

Laurenson, E.M. (1974) Modeling of stochastic-deterministic hydrologie systems. Water Resour. Res., 10, 955-961.

Lettenmaier, D.P., J.R. Wallis, and E.F. Wood. (1987) Effect of regional heterogeneity on flood frequency estimation. Water Resorrr. Res., 23, 3 13-323.

Loucks, D.P., J.R. Stedinger, and D.A. Haith. (1981) Water Resource Systems Planning and Analysis. Prentice Hall, Englewood Cliffs, New Jersey.

Mimikou, M. (1983) A study of drainage basin linearity and nonlinearity. Jour. Hydrol., 64, 113-134.

Moon, Y-.-I., and U. Lall. (1994) Kernel quantile function estimator for flood frequency analysis. Water Resour. Res., 30, 3095-3 103.

Moran, P.A.P. (1959) The theory of storage. Methuen, London. Moughamian, M.S., D.B. McIaughlin, and R.L. Bras. (1987) Estimation of flood frequency:

an evalualion of two derived distribution procedures. Water Resour Res., 23, 1309-1319.

325

Muzik, 1. (1994) Understanding flood probabilities, in atreme Values: Floods and Droughts, K.W. Hipel (ed.), 199-207, Kluwer Academic Publishers, Dordrecht, The Netherlands.

NEK (1988) Estimatingprobabilities of extremefloods: methods and recommended research. National Research Council, Washington, D.C.

Rodriguez-Iturbe, I., and J.B. Valdés. (1979) The geomorphologic structure of hydrologie response. Water Resour. Res. , 15, 1409- 1420.

Rogers, W.F. (1980) A practical mode1 for liiear and nonlinear runoff. Jour. Hydrol., 46, 5 l- 78.

Rogers, W.F. (1982) Some characteristics and implications of drainage basin linearity and nonlinearity . Jour. Hydrol., 55, 247-265.

Rogers, W.F., and H.A. Zia. (1982) Linear and nonlinear runoff from large drainage basins. Jour. Hydrol., 55, 267-278.

Sanders, CL., Jr., H.E. Kubik, J.T. Hoke, Jr., and W.H. Kirby. (1990) Floodfrequency of the Savannah River at Augusta, Georgia. U.S. Geological Survey Water Resour. Invest. Rpt. 90-4024.

SCS (1969) National Engineering Handbook, Section 4, Hydrology. U.S. Soi1 Conservation Service, Washington, D.C.

Silverman, B.W. (1986) Density estimation for stafistics and data anai’ysis. Chapman and Hall, New York.

Singh, K.P., and R.A. Sinclair. (1972) Two-distribution method for flood-frequency analysis. ASCE Jour. Hydr. Engr., 98, 29-45.

Singh, V.P., and H. Amin&. (1986) An empirical relation between volume and peak of direct runoff. Water Resour. Bull., 22, 725-730.

Stuart, A., and J.K. Ord. (1987) Kendall’s advanced theory of statistics. Oxford University Press, New York. ,

Tasker, G.D. (1978) Flood frequency analysis with a generalized skew coefficient. Water Resour. Res., 14, 373.

Thomas, W.O., Jr. (1982) An evaluation of flood frequency estimates based on rainfall/runoff modeling. Water Resour. Bull., 18, 22 l-230.

Waylen, P., and M.-K. Woo. (1982) Prediction of annual floods generated by mixed processes. Water Resour. Res., 18, 1283-1286.

326