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OCR for page 99
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
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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
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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)
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Representative terms from entire chapter:
climatic change
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
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, S—Marginal 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)
Robustness—A`/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 fence——A 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.
0
REFERENCES
1. A. Maass et al. (1962). Design of Water Resource Systems,
Harvard U. Press, Cambridge, Mass.
2. M. B Fiering (1967). Streamflow Synthesis, Harvard U.
Press, Cambridge, Mass.
3. M. B Fiering and B. Jackson (1971). Synthetic Stream.flo'vs,
Water Resources Monograph 1, American Geophysical
Union.
4. M. G. Kendall and A. Stuart (1961). The Advanced Theory of
Statistics, Vol. 2, Hafner Publishing Co., New York.
5. C. S. Holling (19731. Resilience and stability of ecological
systems, Ann. Re?;. Ecol. Systematics 4, 1.
6. M. B Fiering arid C. S. Holling (1974~. Management and
standards for perturbed ecosystems, Agro-ecosystems 1, 301.
7. R. A. Bryson (1974). A perspective on climatic change,
Science 184, 753.
8. W. S. Broecker (1975). Are we on the brink of a pronounced
global warming? Science 189, 460.
9. E. Schulman (1956). Dendroclimatic Changes in Semiarid
America, U. of Arizona Press, Tucson, Ariz.
10. C. S. Russell et al. (1970). Drought and Water Supply, The
Johns Hopkins Press, Baltimore, Md.
11. H. Chernoff and L. Moses (1959). Elementary Decision
Theory, John Wiley and Sons, New York.
12. J. R. Slack et al. (1975). On the value of information to flood
frequency analysis, Water Resources Res. 11, 629.
13. J. R. Wallis et al. Effect of sequence length on the choice of
NICHOLAS C. MATALAS a nd MYRON B FIERING
assumed distribution of floods, Water Resources Res., in
press.
14. J. R. Wallis et al. (1974). Just a moment, Water Resources
Res. 10, 211.
15. N. C. Matalas et al. (1975). Regional skew in search of a
parent, Water Resources Res. 11, 815.
16. M. B Fiering (1965). An optimization scheme for gaging,
Water Resources Res. 1, 463.
17. N. C. Matalas (1968). Optimum gaging station location, Pro-
ceedings of the IBM Scientific Computing Symposium on
Water and air Resource Management, IBM, White Plains,
N.Y.
18. T. Maddock III (1974). An optimum reduction of gages to
meet data program constraints, Hydrol. Sci. Bull. 19, 337.
19. M. E. Moss and M. R. Karlinger (1974). Surface water net-
work design by regression analysis simulation, Water Re-
sources Res. 10, 427.
20. N. C. Matalas and H. H. Barnes, Jr. (1976). A Bicentennial
Reflection: Stream Gaging in the United States, Interna-
tional Symposium on Organization and Operation of Hydro-
logical Services, Ottawa, Ont., Canada, July 1~16.
21. R. Dorfman et al. (1972). Models for Managing Regional
Water Quality, Harvard U. Press, Cambridge, Mass.
22. J. Pratt (1964). Risk aversion in the small and in the large,
Econometrica 32.
23. J. Schaake and M. B Fiering (1967). Simulation of a national
flood insurance fund, Water Resources Res. 3, 913.
