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Water-Resource Systems Planning 6 INTRODUCTION NICHOLAS C. MATALAS U.S. Geological Survey MYRON B FIERING Harvard University Given the stochastic nature of streamflow, the planning, design, and operation of water-resource systems are necessarily subject to uncertainty. This is particularly so when dealing with design criteria that incorporate extreme. The future inflows to which the system is meant to respond are unknown and not predictable with a rea- sonable degree of reliability. Nonetheless, stochastic models of streamflow may be constructed and used to assess the risks associated with alternative system de- signs.i Basically, this entails the construction of stochastic flow models not for predictive purposes but to generate many sequences of synthetic flows such that each se- quence may be regarded as being an equally likely reali- zation of future sets of flows. With the set of synthetic flow sequences, a better assessment may be made (usu- ally by simulation) of the expected performance of a system design than with the historical flows alone.2 3 A variety of synthetic flow-generating models has been developed, and although they differ in their con- struct, they are all based on the assumption that 99 streamflow is a stationary process so that the values of the models' parameters are time invariant. The assumption of stationarity may be questioned, particularly in a region where land use has changed and the water resources have undergone development. Apart from economic activities, the assumption of stationarity may be questioned in terms of climatic change. Recent climatic literature has pointed out that the past several decades have been a period of rather mild and stable climate but that the future may be less so. If this is indeed true, then more severe floods and droughts may be expected in the relatively near future. Whether or not the climate is changing is subject to debate concerning the time scale over which change is defined; if it is changing, the nature of the change and how and when it would impact on hydrology are uncer- tain. If climatic changes are reflected in a decrease in pre- cipitation or prolongation of drought periods, then water- supply systems in years ahead may be stressed. The uncertainties as to the nature and magnitude of climatic changes and of consequent impacts on water resources compound the problems of uncertainty associated with
100 the planning, design, and operation of water-resource systems. Although it may not be possible at present quan- titatively to define climatic change and its hydrologic impacts even in probabilistic terms, the uncertainty can be dealt with explicitly in the development and manage- ment of water-resource systems. The manner in which this might be done is discussed below. We introduce the concepts of robustness and resili- ence; these terms, which originated in statistics4 and ecology,5~ have not heretofore been used in a water- resource context. Several definitions are proposed, but no definitive version is reached. The proposals are not in- consistent or contradictory- they emphasize different as- pects of the concepts. CLIMATIC CHANGE Climatic change may be realized in a number of different wayside If climate is regarded as a stochastic process, then change would be manifest in the parameters of the proba- bility distributions and in the specification of the appro- priate density function of such variables as temperature and precipitation. In addition, change may be reflected in the measures of climatic persistence or the extent to which climatic events in one time period are related to those in another. The difficulty in measuring climatic change is due largely to the fact that a definition of change, at least an operational definition, is yet to be widely accepted. Change implies a trend, real or appar- ent. Classical statistical literature has pointed out the pitfalls in detecting trends in short historical records. Even if climate is a stationary process from the long-run point of view, a climatic anomaly is, from an operational perspective, a change if the anomaly persists over the economic planning horizon. While the long-run nature of climate is of major scientific interest, immediate interest in water-resources planning lies in the short run, say 50 to 100 years. It is difficult to discern trends from historical records of temperature and precipitation, particularly as the records are short as measured against geologic time and the qual- ity of the records may have been affected by changes in the location of stations and by natural or man-induced changes in ambient conditions. Statistical analysis of trends in climate are based not so much on historical records as on a variety of long-term surrogate measures of climate such as tree rings, mud varves, and evidences of glacial advances and retreats, as well as historical ac- counts of past climatic events. The appeal of tree-ring records is their continuity and length spanning several centuries. The records are relatively easy to obtain on a wide geographical basis. The tendency for tree-ring widths to decrease in abso- lute value and to become less variable with time is as- cribed to the mechanics of growth and is essentially removed by transforming a nonstationary sequence of ring widths into a stationary sequence of ring indices.9 NICHOLAS C. MATALAS a nd MYRON B FIERING The oscillator.v character of sequences of tree-ring indices is ascribed to temporal changes in precipitation excesses and deficiencies. Much has been done in reconstructing estimates of past climates from tree-ring indices, but less effort has been given to the reconstruction of streamflows. In the absence of causal tree growth-streamflow models, regression of streamflows on tree-ring indices is a basis of flow reconstruction. Regression, however, may disturb the statistical properties of the reconstructed flows relative to the his- torical flows. Tree-ring indices are more normally dis- tributed and more highly autocorrelated than streamflow. These properties of tree-ring indices are passed to (and embedded in) the reconstructed flows, and, in addition, regression renders the reconstructed flows less variable than the historic ones unless random components are introduced specifically to preserve higher moments. The differences in the statistical properties between recon- structed and historical flows may not be attributed en- tirely to regression; the manner in which tree-ring widths are transformed into indices may also contribute. Whether the differences in statistical properties are suff~- cient to offset the utility of the reconstructed flows in planning and management of water-resource systems re- mains to be determined. Apart from a few efforts to reconstruct flow sequences on the basis of tree-ring and other geochronologic rec ords, hydrologic modeling has been concerned mainly with short-range forecasting of runoff events rather than predicting long-run hydrologic impacts of climatic changes. Thus, at present, there is little in the hydrologic literature to guide water-resource planners and managers as to the effects of climatic changes. From the climatic literature, it is possible to construct a number of climatic scenarios that might be regarded as realizable in the near future, say, over the economic time horizon for project design. Among the scenarios might be one of increased variability of precipitation, one of declin- ing temperatures, and others of a more complex nature. Although alternative scenarios can be constructed, it is unlikely that climatologists would be willing to assign probabilities to them. Their reluctance to do so is not without reason. The causal arguments favoring any one scenario are not strong. Moreover, models for making reliable long-range predictions of climate do not exist, and the prospects do not appear to be good for making reliable short-range, say weekly or monthly, meteorolo.gi- cal forecasts. Meteorological variability generally is ac- commodated by flexible operating rules for reservoir sys- tems; this is not a primary concern of this chapter. But the various statistical studies supporting climatic trends can- not be discounted in planning water-resource systems without first evaluating their economic and operational consequences. A climatic scenario is not easily mapped into a water- resource scenario, except perhaps in a gross descriptive way. Changes in temperature and precipitation regimes cause changes in cloud cover and radiation transfer that
Water-Resource Systems Planning can initiate shifts in patterns of human settlement and development. Changes in streamflow patterns cause further geomorphological changes and are themselves affected by the extent to which regional flora and fauna become adapted to new climatic regimes. Among hydrologic phenomena, it is the extremes the floods and minimal flows that exert the greatest stress and exact the greatest penalties on the area's economy. Perhaps a small decrease in temperature could result in heavier snowpacks and delay in melting, thus altering the timing of snowmelt floods and perhaps increasing the magnitude of the floods if melting is delayed until heavy rains occur. Changes in the timing and magnitude of floods would impact the operation of reservoirs for flood control, hydropower, or water supply. If climatic change is in the form of longer periods of rainfall deficiency relative to long-term regional averages, then droughts measured in terms of low flows may be intensified in terms of both flow deficiencies and their duration. Severe droughts (as measured by intensity or duration) may in some cases overtax existing water-supply systems. Large systems typically have substantial redundancy and robustness that enable them technologically and in- stitutionally to adapt to large stresses. Recent studies of the northeast drought,~° 1961-1965, show the remarkable extent of short-run adaptation by the community to phenomena that could, in fact, be manifestations of an undetected climatic shift. If hydrologic consequences of questionable origin persist, institutional measures (insur- ance, subsidies, zoning, for example) are available as an alternative to precipitous and irreversible structural mea- sures (reservoirs, pipelines, well fields, for example). To assess fully the consequences of climatic shifts, the tradeoffs among these measures must be evaluated and articulated. The exact way in which complex changes in climate would impact on water resources has not been addressed; and until research along these lines is undertaken, there is no way analytically to map a climatic scenario into its water-resource consequence. And even if this could be done, the probability of realization within the economic time horizon of the water-resource scenario would be conditioned on that of the climatic one, the latter proba- bility still to be defined. WATER-RESOURCE SYSTEM DESIGN A long streamflow record, the basis for design of most water-resource systems, constitutes fragile information in that it represents a combination of deterministic and stochastic elements whose fluctuations cannot readily be associated with climatic shifts. However strong might be the evidence that climate is changing or that its popula- tion parameters are different than heretofore, the noise in the "black boxes" that map climate into flow are so large that it may be extremely difficult to detect climatic shifts by examining hydrologic data alone, and it might there- 101 D, 1 D2] D3 I iDIn lI On I I I I 1 TO ~ ~ Max. FIGURE 6.1 Optimal system design conditioned on ranges of A. fore be still more difficult to modify existing systems or specify new designs on the basis of climatic change. The difficulties noted above are elaborated here. It is assumed initially that design of the water-resource system at hand can be optimized on the basis of a single parame- ter, namely, the population mean of annual streamflow at some gauging location. It is further assumed that the design decision (in however many dimensions or vari- ables, such as the types and sizes of projects and their appurtenant structures, their sequencing, and their opera- tion) is divided into a number of discrete design choices designated Do, D2, . . ., Dn, so that for any value of the population mean ,u there is a unique design Di that op- timizes the system objective function. This is shown in Figure 6.1, for which the line Respace is divided into segments through which design Di is optimal. Of course, the value of ,u is never available. Nonetheless, our con- struct is based on the reduction in performance attributed to less than perfect information, so that it is appropriate to assume here that Di is conditioned upon ,u. Some designs are more robust than others in that they are applicable over a wider range of,u-values, while some are optimal for narrow ranges of the population mean. This is the most elementary definition; maximal robust- ness would be associated with some Di optimal for all values of,u. Bayes's theorem provides another approachii to the robustness of a particular design Di: Pr (Di is chosen) Pr (Di is optimal ~ Di is chosen) = Pr (Di is optimal Pr (Di is chosen ~ Di is optimal). Both products are the joint probability that Di is chosen and is optimal. The robustness could be given by either conditional probability, but both have shortcomings. Pr (Di is optimal ~ Di is chosen) could be very close to unity if Di is in fact optimal whenever chosen, however in- frequently; Pr (Di is chosen ~ Di is optimal) could also be close to unity if Di is chosen whenever it is optimal, however infrequently. The former conditional probability is a measure of robustness of the system, the latter a measure of robustness of the design process. Economic issues introduced later clarify the distinction. Consider the loss of information attached to estimating the mean A, for the case in which design depends only on estimates of that parameter. Suppose only two designs are available, Do and D2, and that the associated ranges of ,u are as shown in Figure 6.2. No other values of ,u can be obtained. Because of the loss of information inherent in the sampling (information) program, x (the sample mean)
102 | Range of x~D2 at l Range of x~D, at Level p I level p 1- -1 X1 X3 X2 X4 ~: ~2 ~3 FIGURE 6.2 Impact of information loss on choice of system design. is not a perfect estimator of ,u. The figure shows that a wide range of x-values could, at probability level p, be attached to populations characterized by ranges ~1 ~ ~ ~ ~2 and ~2 ~ ~ ~ ~3, respectively.* The di- vergence associated with each "funnel" is measured by the efficacy of the sampling program; the funnels need not be identical or symmetrical. Clearly Do should be chosen for x ~ x ~ x2 and D2 for X3 ~ X ~ X4; the range x2<x<x3 provides some difficulties, and we discuss below how the concept of regret can help to choose between Do and D2. This choice, which depends on eco- nomics, is importantly different from the mathematical choice between ranges of,u. Another possibility is that we identify a new design alternative D3, to be chosen whenever the sample mean x falls in the double-hatched interior funnel. D3 would never be optimal if ,u were known; hence the funnel converges to a point on the ,u-axis. In the discussion that follows, it should be noted that such designs, which are promoted to optimality only because of the uncertainty in estimating ,u, can be among the finite set of design choices subject to analysis. Another approach to defining robustness at probability level of design Do is given by the ratio ~2 - /3 - Xt), and that of D2 by ~4 - X3)I(X4 - X2), while the p-level robustness of the system is [(x-2 - A) + ~4 - X3~/(X4 - I. The geometric interpretations are evident from Figure 6.2 The above concept can be generalized to encompass the mean, ,u, and the standard deviation, A, of the annual flows. Figures 3(a) and 3(b) show how the several designs Di carve the decision space in the (,u,~-plane. The plane *The set (X~,X3) iS the union of all the probability intervals for each ,u in the interval [,U,i,~2]. Therefore for any ~ subject to , then with probability p or greater, x ~ x ~ x2. NICHOLAS C. MATALAS a nd MYRON B FIERING might be divided into a number of discrete zones, each associated with a design Di [Figure 3(a)], or there might be smooth contourlike loci [Figure 3(b)] that define com- binations along which design Di should be chosen. The important questions are the extent to which information about the parameters' impacts on economic and institu- tional issues and the consequent likelihood that a new design choice would be required to meet system per- formance criteria under those new parameter values. In traditional water-resource design methodologies there is uncertainty as to the mean flow, more uncertainty about the standard deviation, and even more uncertainty about the higher moments of the flow probability density functions. But if examination of Figures 3(a) and 3(b) indicates that the same design Di (or some slight modifica- tion thereof would serve over an area of the (,u,~-plane attached to some climatic scenario, precise specification of those parameter values becomes unimportant in the design of the water-resource system. In other words, if design Di is optimal within a particular sector of the (,u,cr)-plane, and if there is confidence that current and modified parameter values lie within that sector even or ~ ~ D3 1 Dl D2 D6 1 D7 D4 1 D5 ~~\~ ( b ) FIGURE 6.3 (a) Optimal system design zones conditioned on ranges of,u and A. (b) Optimal system design loci conditioned on ranges of ," and A.
Water-Resource Systems Planning though they cannot be specified exactly, then nothing would be gained by collecting more information or even by identifying whether changes in the population mo- ments are due to climatic shifts, oscillations, cycles, or other forms of nonstationarity. Generalization to the third and higher moments is con- ceptually trivial but numerically subtle. If the decision space is augmented by a third dimension, say, the coeffi- cient of skewness, By, and the autocorrelation of order k, Pk. then the space may be carved into disjoint segments, each of which may be warped and to each of which is attached a design choice Di. Experience in design of water- resource systems suggests that robustness and orientation of design contours or segments are strongly related to system objectives. For example, if the system priority is to serve agricultural and water-supply purposes, the design is likely to depend primarily on the lower moments or measures of central tendency of the flow probability dis- tribution, and the design will tend to be stable or robust along the axes of the higher moments. Put another way, if one is designing on the basis of the mean flow alone, even modest storage facilities will generally remove enough variability from the tails of the flow probability distribu- tion to render the optimal design relatively insensitive to skewness, y. It thus becomes less important to identify closely the population skewness, to mount the necessary gauging program that would define it more precisely, or to be concerned over whether By has been affected by climatic changes. On the other hand, if the system is designed against extreme, more extensive information about By and the higher moments might be indicated because the design choice would then be expected to be more sensitive to the value of By. This would be manifest in Figure 3(b) by more closely spaced contours. i3 It becomes more difficult when we move from flow parameters and ask instead about the effect of climatic changes on water-resource system designs. These changes are manifest as (filtered) changes in precipitation and temperature patterns, which must be filtered or mapped into apparent or potential changes in flow regime, which, in turn, dictate potential changes in system de- sign. Thus potential shifts in the climatic parameters must be mapped through two filters before their effects on design choices can be evaluated. And, unfortunately, the filters are very noisy. Because of the complex delaying phenomena that are part of the hydrologic cycle and that are expressed in runoff and storage relations, we can only imprecisely map climatic shifts into changes in flow pat- terns (and even less well can we impute or detect climatic changes from flow changes). Even if there were to be a verifiable change in precipi- tation, say, a small increase in the mean annual value, it is not clear that the increment would be reflected in flow measurements over the short run coincident with the economic planning horizon. Typically, there would be a change in vegetative cover so that only some of the incremental precipitation would appear as incremental 103 runoff, the rest being diverted to modified interception and evapotranspiration. Changes in temperature, whether due to changes in precipitation or to independent causes, might occur; these might produce further shifts in the vegetative cover or in land-use patterns (which might result from changes in cropping patterns induced by small changes in the thermal regime). In any case, how- ever induced, changes in cropping patterIls and land use imply new runoff coefficients for the region, so that with limited hydrometeorologic data the incremental precipi- tation cannot reliably be mapped directly into incremen- tal flow. The same unreliability governs for decreases in mean precipitation. It is interesting to consider the rate at which regions adapt to new climatologic characteristics. The evolution of new vegetative patterns, the development of residen- tial or commercial properties, and other long-term ad- justments such as geomorphologic changes do not occur instantaneously. Adaptation to new precipitation patterns can be presumed to occur at about the same rate manifest by the precipitation, so it might be quite difficult to detect significant changes in runoff moments due to changes in precipitation and temperature. Traditional descriptive hydrology is that branch of the subject that converts fundamental processes (precipita- tion and temperature) into flows and their moments, where estimates of the moments may be subject to large sampling errors.~4 i5 Early efforts to study transfer of hydrologic information made little reference to economic criteria but were based mainly on maximization of hydro- logic information. -20 Stochastic hydrology is that branch of Me discipline that converts statistical parameters of flows into designs or into an array of technologically feasible design choices, which, upon economic analysis, lead ultimately to a final choice. Both conversions add statistical noise to the signals generated by previous analyses. Part of the design problem is to identify the types of climatic shift that might be anticipated and to determine if they are sufficiently precipitous with respect to flow characteristics to dictate a change in system de- sign. It is not necessary for this purpose to know or to try to determine whether there is a true climatic shift. This may be an interesting scientific question, important in its own right, but it is virtually meaningless for the design of water-resource systems. It is also unimportant to know if the population moments of the flow distribution are mod- ified, because, again, while this might be an important hydrologic matter, it is important for water-resource de- sign if, and only if, the changes, when coupled with economic criteria, lead to a new design. Let Di be the design that optimally meets the system objective given the ith combination of values of flow parameters. There are n different designs available, cor- responding to the subspaces into which the flow parame- ter space, say (lo, (r, lye, is divided and, possibly, the zones of overlap in the sample space. These are ignored for Me moment. The problem of continuous decisions is not treated here. One outcome of the Paretian analysis de-
104 scribed at Me end of this copter is a snow set of design options that form the basis of fort her negotiation. Thus there is strong precedent for using a discrete number of design choices. Di is Me optimal design corresponding ~ a particular point in Me (,u, or, Respace. D,* is Me design actually chosen when the designer perceives Me sample estimates of the population parameters to be those of Me ith combination. This may not require 1 hat Me sample estates themselves lie in the ranges spared lay be, Audi. For example, as shown in Figure 6~2, the range of x that leads to Do is not congruent to Mat for A, and similarly for D2. If the designer could always identity correctly the population parameters, Me design problem would be Vivian and the correct decision wood always be made. But Me mapping from sample parameters to design contains opportunities for oveHap and error. Thus Me use of mathematical surfaces to separate Me several decision options shoed be modiBed to accommodate zones of ambiguity, a Complication dealt wig later. It is assumed that the designer always makes the correct decision based on ~e available climatic evidence; that is, design rules lead unambiguously from estimates A, s, go to design Di. Suppose there are only two designs or decisions available: to build (Do) or not to build (D2) Me system. The parameter space for flows is divided into two segments: Bat segment for which the structure should be built (S~' climatic shift) and that for which it should not be built (S2, no climatic shift). Me designer cannot ol3serve S directly but makes measurements, trend analyses, projec- tions, and other climatic studies from which Me evidence indicates (but not with certainty) Mat state So or state S2 governs. The two sets of climatic evidence are E, for sate S ~ and E2 for state S2, and while the evidence Carl lead to a wrong decision, it can never lead to "no decision." If evidence Ei is obtained, Men state Si is assumed to govern and decision D2* is made. The decision Di* is optimal if the evidence E2 points to the correct state Si, and non- optimal otherwise. Let evidence Ei be available so design D'* is made. If the evidence is correct and state S ~ obtains' Ten idle decision is correct and Me optimal design Is Do. Conversely, if the evidence leads to an incorrect assess- ment of the system state, Me choice is Did (to build), whereas *he optimal design should be D2 (not to builds. Analysis of the uncertainty -can be compressed into a few compact statements concen~ing Me conditional prom abilities that relate Me availability of evidence Ei and Me occurrence of states Si. The tighter Me relationship bet tween climatic and flow variables, the more likely ~t is Mat Me correct flow description and design are extracted from climatic evidence. A mathematical formalism handles some of the statisU- Cal issues. Again resorting to Bayes's theorem, the joint probability Mat evidence E' and state Si occur jointly is given by Pr(Ei' Si) = Pr(Ei | S.] Pr(S) = Pr(Si | E j Pr(E). It is interesting to consider `'the probability oLmaking die NICHOLAS C. MATALAS and MYRON B FIEBING nit decision" and to tie this to Mother candidate donation of robustness. Pr(Ei ~ Si) is the conditional probability Mat the evidence points to sate Si given that Si governs, the sum ~ Pr(Ei ~ Si) is the probability of a i correct Outcome because Di is selected for Si. Pr(Si ~ E'3 Is Me probability that state Si obtains given Me evidence Ei; Me sum ~ Pr(Si ~ Ei), is the probability of a correct i decision Di. Robustness cannot lie deduced from these probabilities alone, or in summation, because these Agues do not include the notion of a nonoptimal design performing '~reasonably well" under different conditions 5r Me evidence is an unbiased estimator of the states so that the marginal densities of S and E are identical, the conditional probabilities are equal. The notion Mat the probability densities of Ei end Si are equal does not imply Mat Ei is a good (in some sense) indicator of Si. It merely states that Ei is observed as often as Si, but it may happen that Ei is observed when Sj, some over state, obtains. In other words, Me joint or simultane- ous occurrence of Ei and Si may be rare even though Probe = Pr(Si3. The essence of the conditional occurrence is contained in what are known as matrices of conditional probability, which may be written as Pr(Ei ~ SO or Pr(Si ~ Eli, depending on the conditioned variable. These canon be deduced Dom each other unless more informa- hen, in Me form of the marginal or unconditioned prom abilities Prom and Pr(S0, or the joint density Pr(E2, Si), is available. The probabilities are arrayed in a square matrix whose elements are Me conditional probabilities that Dates S. and S2 will be realized given that evidence E ~ is available. The row sums are unity because one or the other state must occur. If the marginal probabilities of S2 and Ei are unequal, it is important to determine whether the design objective is to maximize the probability of a good outcome. A technique for dealing with this issue is the decision-theoretic concept lcnown as regret (see Matrix l). Suppose Tree discrete designs are available, Mat each corresponds to a set of population moments, and that net benefits can be arrayed in a matrix whose elements repre- sent net benefits (however calculated and discounted) Magic 1 E, SMarginal Probability Matrix State E. v'- dence ~1 s2 1 E1 Prosy ~ E] ) Pr(S2 I E,,) 1 E2 Pr(S1 I E2) Pr(S2 I E2) 1
Water-Resource Systems Planning 105 associated with selection of design Di* when design Dj situation that can occur if a particular decision is made; a would have been optimal (see Matrix 2~. There is no conservative design technique would be to minimize design reserved for the case in which it is impossible to over all the row maxima by selecting that row for which discriminate among population moments (as in Figure the maximal regret is smallest. 6.2, where D3 was optional in the range of overlap but not Other objectives can be used, including minimizing the elsewhere). Elements along the main diagonal are maxi- expected regret. The probabilities required to estimate ' ~ ~ - '' the expected regret are derived from Bayesian analysis, perhaps incorporating subjective probabilities, where- upon it is straightforward to identify an optimal decision. It is also reasonable to use regret analysis to identify the expected gain associated with improving our ability to make correct estimates of the system state (climate). In other words, it might happen that an investment in infor- mation transfer or data collection would increase the diagonal elements of the E, S marginal probability matrix, implying thereby a higher probability that the climate state Si will be realized when climatic evidence Ei is available. The result of the increase would be smaller probabilities associated with off-diagor~al elements in the regret matnx, whose result in turn would be smaller expected losses associated with the decisions Die (see Matrix 4), and the regret matrix would be as shown in Matrix 5. mat tor their columns because the optimal design (by definition) returns net benefits that are larger than those that would accrue to any other decision. It does not follow that elements along the main diagonal are the largest elements in their rows. Matrix 2 Net Benefit Matrix Optimal Design Actual - - Design D, D2 D3 D' 11 8 2 D2* 6 9 4 D3* 10 5 12 If each element in a column is subtracted from the maximal value in that column, the difference is a measure of the opportunity loss or regret associated with having made decision Di* when D3 would have been optimal. (See Matrix 3.) Mere are several criteria for extracting a decision from the regret matrix. One particularly conservative objective is to minimize the maximal regret. The maximal value in any row of the regret matrix identifies the worst benefit Matrix 3 Regret Matrix Optimal Design Actual Design Dt D2 D3 D * 1 0 1 10 D * 2 5 0 8 D3* 1 4 0 Matrix 4 Net Benefit Matrix Optimal Design Actual Design D t D2 D3 Q 1 11 8 2 D2* 6 9 D3 10 5 12 D4* 6 6 6 Clearly D3* minimizes expected regret for a uniforms prior distribution attached to Me states. Equally clear is the fact that under no realized state So does ~4* maximize net benefits. Its value lies In its robustness in -the in- sensitiv~ty of its performance to the true optimal selec- don. D4* would never be chosen if Si were known; it can be interpreted as a "hedge', in real problems, for which E i is known but no unique S i is unarnbiguousIy indicated. Another candidate that naturally presents itself as a mea- sure of robustness of design D: is aim, where cri is the standard (Ieviation of the elements in row ~ of the regret matrix. Robustness by itself does not imply a good design. The mean and standard deviation of the rows of the regret matrix in this example are (3 7, 4.5), (4.3, 3.3), (1.7,
106 Matrix 5 Regret Matrix Optimal Design Actua I Design D ~ D2 D3 D1* 0 1 10 D2* 5 0 8 D3* 1 4 0 D4* 5 3 6 1.9), and (4.7, 1.2~. In the final section we deal with how to choose among such combinations of expectation and standard deviation. THE 3 R'S: ROBUSTNESS, REGRET, AND RESILIENCE The above sections introduce the concepts of robustness, regret, and resilience. These concepts are important, even in the absence of climatic concerns, and therefore merit further elaboration. Robustness refers to the insensitivity of system design to errors, random or otherwise, in the estimates of those parameters affecting design choice. For example, suppose the design of a water-resource system is dependent only on the mean, ,u, and the standard deviation, cr. of the annual inflows to the system. The optimal design as- sociated with ,u and or is denoted Di. The design Di is said to be robust at probability level p if sample estimates of ,u and ~ lead to the choice of Di with probability p. There is no meaning attached to robustness without an associated probability level. A geometric representation of robust- ness is as follows. Let (,u,~ - Di denote the set of all pairs of values of ,u and is for which the optimal system design is Di. This set is shown as a footprintAi on the (,u,~-population plane in Figure 6.4. The range of sample values x and s associated with a given (population) point in the set, perhaps derived from a particular climatic scenario, are unknown, but they may be estimated from available hydrometeorologic data. The ranges of the estimates, x and s, about ,u and cr. are shown on the (x,x)-sample plane in Figure 6.4. All sample values ~,s) bounded by probability p yield the set Di' of designs, presumed to be optimal for the sample estimates (x,s). The level of information is functionally related to the sample size, the assumed population model, the probabil- ity contour or p-level, the estimating techniques, and the NICHOLAS C. MATALAS a nd MYRON B FIERING Probability p envelope of all (x,s) ~ Dj' derived from all (,u,a) ~ Dj / Probability p envelope of all (x,s) ~ Dj' /~ derived from (,u,cr) ~ Dj f`_~'j Ti : t 7Sample plane / ( Level of information) RobustnessA`/A'' a function of (,u,a) and p p~, _ information in the sample ~!< 4~ Population plane FIGURE 6.4 Robustness of system design given imperfect in- forrnation. values of ,~ and (r.129~3920 For all points (~,~) ~ Di, the (probability) p-envelope of all (f,s) > Di' is delineated on the (x,x)-sample plane and projected downward onto the (,u,~-population plane. This projection is not merely a vertical transfer of the envelope. The design mechanism or algorithm, the operating policy, ~e number of poten- tial design decisions, and other factors dictate the nature of the reflection back to the (,u,~-plane. The robustness of design Di can be measured by the ratio Ai to Ai', where Ai denotes the area containing Di on the (,u,~)- population plane and Ai' denotes the area on the (,u,cr)- population plane contained with the projected p-enve- lope. In general, the area Ai is contained within Ai'; if not, the desired ratio is of that portion of A contained within A' to Ai'. Robustness is at a maximum, A2/Ai' = 1, if, as illustrated in Figure 6.5, the sample estimates are perfect ~ = ,u and s = cr), in which case Di' ~ Di. It would be instructive to attach a probabilistic interpretation to the robustness ratio A~/Ai', but none is readily available. If all the points within the p-level envelope in the ~,s) or (,u, ~) planes were equally likely, the ratio would approxi- mate the conditional probability Pr(Di is chosen ~ Di is optimal), thus showing the equivalence of the analytical and graphical interpretations of robustness. Further the robustness could be evaluated by simula- tion as follows:
Water-Resource Systems Planning s/ ~ ~D', / W~ 1x r l 1 / Sample plane (Level of Information) D'j~Dj l I Robustness-Aj/A'j=1 1 1 ! l Population plane FIGURE 6.5 Robustness of system design given perfect infor- mation- 1. Identify all feasible designs Di, ~ = 1, 2 . . ., n. 2. Pick a (,u,cr,y, . . .) point and the associated optimal design D2 3. Generate a long trace of flows, which are Den grouped into many replications wig each characterized by sample estimates (x, s, g, Ji. 4. Each replication yields a design. If the design is the optimal design Di, score a success; otherwise, score a failure. 5. Calculate directly the conditional probabilities asso- ciated with design error, noting that ~ Pr(Di is opti- mal ~ Di is chosen) is a measure of robustness of the system design while ~ Pr(Di is chosen ~ Di is optimal) i is a measure of robustness for the design process. If the available information yields a design belonging to We set Di' when in fact the optimal design is Dj, then Were is a resulting opportunity loss, referred to as regret, which is measured by the intersection of We surface (,u,~) ~ Dj and the surface (x,s) ~ Di', as shown in Fig- ures 6.6 and 6.7. With perfect information, Di'~ Di, in which case (,u,cr) ~Dj and (x,s) Audi' are disjoint, so there is no regret. The system Di, if it has the built-in buffering and redundancy typical of large water-resource systems, can be operated technically and institutionally to simulate another system Dj such that the resulting economic losses would be small and bounded by some fraction, say ~ percent. This capability is referred to as resilience at level a. Geometrically resilience is depicted in Figure 6.8 by the extension of the (,u,~) ~ Di surface. The ratio of the area contained within the extended surface to the area i07 Probability p enevelope of all (x,s) IDES :~derived from all (,u,a) ~ Dj S / '' / Sample plane / ~~~ - - ~~ 1 / (Level of Information) ,, _ x 1 1 Design regret (error) I l at probability | level pa | ~ ;7 Population plane FIGURE 6.6 Regret in system design. _ S / ,D5,~ / / D3 // D4 / Sample Plane x Population Plane ~ D2 C/ross-hatched areas / D3 ~ D4 / imply Regret ~ D ~ / ~ Projection of D5/ FIGURE 6.7 Regret in system design.
108 Projection of probability p envelope of all ~ (x s) ~ Di derived from all (I,) ~ D; / /k ;~7opulation plane Regiontin which Dj can be operated (technically and institutionally) to simulate Dj => small economic loss System resi l fenceA j' /A; at level ICY FIGURE 6.8 System resilience. (level 0~) contained within the projected p-envelope is the measure of resilience.* If the argument for, say, Di is extended to cover all Do, perhaps weighted by the a pr~ori probabilities for each Dz' then the sum of weighted resilience measures is a mea- sure of system resilience. Extensions of this argument lead to conclusions that many small reservoirs may for a significant range of cost functions somehow be "better" than a single large reservoir and that the economic cost of losing the economies of scale is equivalent to an insur- ance premium that purchases resilience. For example, if husband and wife elect to travel in two separate airplanes, they lose economies of scale leg, family plan fares, taxicabs, etc. but gain resilience in that they drive essen- tially to zero the probability that their children will be orphaned! Another way to draw the distinctions between robust- ness of system design and robustness of system outcome is to note the role of sensitivity analysis in design of water-treatment facilities.T In one study, four plants were to be built over a number of years to meet growing demands. The least-cost solutions were identified, but 11 other solutions (any of which might have been reached by an experienced designer) lay within 3.3 percent of the minimal cost. Indeed, it would be reasonable to posit that the range of costs lies well within the noise that might be anticipated from the design algorithm. The outcome is thus insensitive to the decision; we say it is robust with respect to the solution. However, if the economic parameters of the decision model are changed, Harrington shows how one solution remains optimal even though the value of the outcome changes significantly; this exemplifies sensitivity of the outcome with respect to model parameters and insensitiv- ity (robustness) of the decision with respect to these same parameters. * The use of p,c~ to define resilience is consistent with specifica- tion of confidence and tolerance limits in statistics. The concepts are identical. ~ J. J. Harr~ngton, personal communication. NICHOLAS C. MATALAS a nd MYRON B FIERING CONFLICTS OF INTEREST IN WATER- RESOURCES PLANNING Conflicts in system design are the rule rather than the exception. Typically, two or more parties dispute the algorithm for calculation of benefits, and the issue is resolved only after long and costly litigation. Courts of law are called upon to render judgments in areas for which they could hardly be less well equipped. Another class of system-design objectives incorporates a priori assignment of probabilities to the several states So, the net effect of which is weighting the various regret elements and tipping the decision. This is a political and social reality, not necessarily immoral or unethical; it reflects the fact that the priority assessments of the deci- sion maker inevitably align themselves more closely with those of one participant (in the decision-making process) than another. For most resource development programs, particularly those characterized by multiple-purpose use, it is virtually impossible to imagine that all the partici- pants, all the vested economic and social interest groups, all the impacted public and private agencies will have the same perception of objectives, benefits, and costs for the system. Conflicting interests are the rule rather than the exception, and because the decisions must somehow be made, by someone or some agency, it is better to anticipate these interests than to be surprised by their occurrence and thus to be unprepared to deal with them in a systematic, disciplined, and rational way.2i Climatic shifts may not produce conflicts of interest, but they accentuate existing conflicts. Note that specifica- tion of the parameter p is important in design, and two agencies or conflicting parties may have different levels of risk aversion and hence may propose different "optimal designs." Somehow the conflict between agency con- straints and objectives must be resolved. In some of the western states, conflicts between agricultural and energy interests are anticipated over the use of available water supplies. The conflicts might be aggravated by climatic shifts resulting in a decrease in water supplies. On the other hand, a farmer might be willing to relinquish some of his water rights if he were convinced that climatic shifts would lead to increased water availability or in- creased precipitation whereby dry-land farming would be profitable. There are system objective functions that, when applied to the regret matrix, can be used to identify nondominated design alternatives for future negotiation and tradeoff. This is known as Paretian analysis. It em- phasizes that many solutions are admissible in that they represent output combinations from which it is impossi- ble to improve the position of one participant without worsening the position of at least one other. In other words, the participants agree that impoverishment of each other is not their objective (as it is in classical two-person, zero-sum games) but that each would be happy to have everyone else do well as long as this does not occur at his own expense. If one participant perceives that his inter-
Water-Resource Systems Planning ests are jeopardized by continued improvement in the position of another, he can threaten to terminate the negotiations or otherwise derail the decision-making pro- cess. He could agree to a less desirable position if side payments were made. Our social and institutional struc- ture is particularly weak in arranging for such side pay- ments because they smack of bribery and extortion, but moral rectitude is not an inherent human trait so much as it is a social tradition, and we could improve our perfor- mance in this regard. The implication is that we can derive from the analysis a set of potential solutions that are not dominated; each represents a point on the Paretian frontier. Consider Figure 6.9, which shows the net benefits perceived by each of two participants in a decision- making process. In the general case, with more than two participants, the dimensionality of the decision space would exceed two. The coordinate axes represent the values assigned by the several participants (in this case, only 2) to each potential decision. All those designs that lie to the south and west of another design are dominated because both participants could improve their positions by moving to the north and east. As long as the move does not require that participant X move westward, or that participant Y move southward, the design is dominated. All nondominated designs are connected by an envelope along which tradeoffs or negotiations can take place. This envelope is called the Paretian frontier and contains all points that are Pareto-admissible. The solution will lie along that frontier; generally speaking, it will lie closer to the design preferred by that participant with the greatest political clout, the largest army, the most sheep or goats, the most money, or the most of whatever currency is recognized by the parties to negotiate. The final negotiated position may not accord with the objective function of any of the participants or of the administrative body responsible for implementing the decision. In this sense, Paretian analysis is different from traditional benefit/cost analysis, which imposes an objective func- NBy . . Points on arc are undominated and define negotiation frontier. . . FIGURE 6.9 Pareto negotiation frontier. 109 tion (by force of arms, legislation, or whatever) and pro- ceeds to optimize the entire system, for all prospective users, on the basis of that imposed objective. Another kind of conflict involves the tradeoff between return and risk; all of us face and resolve such issues daily.22 23 In terms of a reservoir design problem, it is appropriate to ask how expected (or some other measure of) net benefits should be combined with variation (say, the standard deviation) of regret to form a negotiation set. This is the matter of comparing the four outcomes calcu- lated from the regret matrix developed earlier. Variation in regret (or net benefits) is undesirable so that the two axes of the Paretian analysis would be mean (abscissa) and deviation (ordinate) of benefits, and it would be advantageous to move to the southeast (maximize the mean, minimize the deviation) to define the Paretian negotiation frontier. The concept of dominance still gov- erns; only the direction changes. The means and standard deviations of net benefits for Do through D4 are (6.0, 3.65), (6.0, 2.16), (8.5, 3.51), and (1.0, 01. On this evidence, Do is dominated by D2, but no choice can be made among the three remaining candi- dates without explicit consideration of the tradeoffs, negotiations, and side payments. On the basis of expected regret, D3 dominates all other solutions. If specific prior densities were assigned to the Di (or Si), other results would obtain. The point here is that the decision-making process, or what we have also called the design process, is depen- dent to an important degree on political, institutional, economic, military, social, and other nontechnologic fac- tors. To attempt to assess the effect of climate change, natural or man-made, without recourse to these non- technologic issues would be irresponsible. CONC L U S I O N S All points within the arc are dominated and it is technologically infeasible to be outside the arc. NBx This chapter does not resolve any issues by providing statistical tests and algorithms for adapting standard de- sign rules to the case in which climatic shift is a potential perturbation on system design. Rather, it lays out a for- malism for economic, institutional, and social adaptation to a variety of political perceptions on the value of hy- drometeorologic information in reducing risk of error in anticipation of climatic shifts. At the present state of the meteorologic and clima- tologic arts, it is not likely that more definitive conclu- sions can be reached. But it is comforting to recog- nize that most large systems contain so much buffering and redundancy that resilient design can be operationally achieved without recourse to sophisticated or elaborate projections about the climate. When not designing for extreme, the problem is more subtle, but the concept of regret could be applied through the system and testing o different climatic scenarios and of their effect on hydro- . . . logic regimes.
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