Natural uncertainty stems from the fact that nutrient loading is triggered by random natural events, such as storms. The use of the stochastic model fully accounts for natural uncertainty.
4.3.2 Parameter Uncertainty
Parameter or sample uncertainty originates from the generally limited statistical information available for estimating the parameters 8_L1 and
eLz
of the probability den- sity functions of _L, and L z . This uncertainty can be accounted for by considering these statistical parameters as random variables with respective pdfs p ( a l ) and p ( Q 2 ) . Natu- ral and sample uncertainty can then be considered jointly by using Bayesian distributions (Benjamin and Cornell 1970)for
h1
and a similar distribution forh2.
Concerning parameter uncertainty in the estimation of the pdfs for concentrations c1 and:,, two cases can be distinguished:
-
(i) No observed data on concentrations are available. This is the situation we face in the present example. As shown earlier, models are available to estimate the mean values of c and:,, and the variances of; and:, can also be obtained. In a Bayesian framework,-
these statistics permit the construction of two-parameter prior distributions of statisti- cal parameters of and_cz
; Bayesian distributions can then be estimated by simulation (Szidarovszky et al. 1976).(ii) Some observed data are available on
c l
and:, . These data-based estimates can be generated by Iikelhood functions l(xlf), where x is the data set. Prior and data-based information can then be combined by means of Bayes' rule to obtain posterior estimates.4.3.3 Model Uncertainty
Model uncertainty is present in several parts of the methodology; some of the more important sources are as follows:
- procedures for the estimation of runoff, sediment yield, and concentration;*
- selection of the type of pdf for phosphorus loading;
- consideration of a finite - and often small - number of control alternatives;
- number and type of criteria used to characterize the objectives of watershed activity and eutrophication control;
- failure to take into account the dynamic nature of decision making or transient effects following the implementation of control measures;**
- choice of MODM technique.
Model uncertainty can cause considerable distortion of the results. No general "cure" is available, but three approaches for minimizing the difficulties may be mentioned. First, sensitivity analysis can be used to ascertain the effects of important sources of model un- certainty. Second, the methodology can be developed in the direction of more sophisticated
*It is known that the connection between rainfall, runoff, sediment, and phosphorus loading is very complex and difficult to model. We have used here the universal soil loss equation modified for ap- plication to short-term events as in Smith et al. (1977), but the main principles of our approach would still be applicable if a more sophisticated sediment yield model were used.
**For example, contour farming techniques yield long-term benefits to the farmers in terms of reduc- tion or elimination o f topsoil losses. Accounting for the costs o f contouring without including the benefits o f topsoil conservation would be valid only for short- or at most medium-term planning.
models, such as the one for runoff estimation. Thud, observed data can be used to calibrate various submodels such as those for runoff, sediment, and phosphorus concentrations.
Snowmelt runoff may release substantial quantities of dissolved phosphorus from crop residues left on the surface of the soil after the fall harvest (Timmons et al. 1970, Meta Systems Inc. 1978). since this dissolved phosphorus may not be in equilibrium with frozen surface soils, the relationships between c2 and cl generally used may not hold. The tillage method used has a critical impact on phosphorus loading in snowmelt; methods not involving plowing tend to leave crop residues on the surface and thus create a greater potential for snowmelt phosphorus loading. A relatively simple relationship is available to estimate average annual snowmelt phosphorus loading (Meta Systems Inc. 1978); also sto- chastic models of snowmelt (Cary et al. 1977) can be used within the event-based frame- work of phosphorus loading. These two approaches can be coupled, leading to a simulation of loading per snowmelt event; the inclusion of snowmelt phosphorus loading in the control model is then straightforward.
4.5 Lake Modeling
It is assumed in our example that phosphorus loading, as described in eqn. (3), ade- quately characterizes the trophic state of the lake. Coefficients k l and k 2 (in eqn. 3) can be estimated with the same precision as other parameters of a limnological model, that is, with at least 10% error. The sensitivity of the objective functions f1 and fi to these coef- ficients should be investigated further. In reality, there are a great number of other physi- cal and chemical variables, such as nitrate concentrations, lake sediment, light, and tem- perature, affecting primary algal production. Detailed simulation models are available to describe the eutrophication process in a water body (for example lmboden and Gachter 1975, Herodek and Csaki 1980). These are, however, difficult to use for forecasting and/
or control, since many model parameters are unknown, while nonstationarity of the lake makes calibration impossible; furthermore, a number of the input variables or parameters may be stochastic. To cope with these difficulties, several semiempirical lake models have been developed. One of the simplest of these (Vollenweider 1968) forms the basis of our example. We believe, however, that further progress is possible in the direction of stochastic control models based on more sophisticated lake representations, such as those proposed by Schlinder et al. (1978), Walker (1979), and Reckhow and Chapra (1979).
4.6 Major Steps in the Analysis
We provide here a step-by-step summary of the methodology in order to facilitate future real-life applications:
a. Evaluate present and expected future socioeconomic activities in the area of the watershed.
Evaluate the main environmental problems affecting the water body.
Select appropriate indicators, criteria, or measures of effectiveness for socio- economic activities (benefits, costs, population, energy requirements, etc.) and environmental problems (pollution loading, trophic index, primary algal pro- duction, etc.).
Select possible control actions considering the pollution sources, the transport process, and the water body itself. Both structural and nonstructural measures should be taken into account.
Derive relationships between socioeconomic activities, possible control measures, and environmental problems o n the basis of the criteria selected in step (c).
Define the elements (input, state, state transition, output, output function) of the system model (see Section 3.1).
Select a suitable MODM technique using the "elimination" procedure recom- mended in the Appendix.
Find the solution t o the problem.
Evaluate the results by considering uncertainties (see Section 4.3), using sensi- tivity analysis and analysis of observed data.
Note that several feedbacks or iterations may be necessary between various steps, for example between MODM selection (g) and criteria selection (c), or between steps (i) and (d).
5 CONCLUSIONS
The t r a d e a f f between the environmental protection of a water body and the eco- nomic interests such as agriculture in the area of the associated watershed can be usefully considered within a multiobjective decision-making framework. In the example studied, two objectives, each composed of several criteria, have been considered, namely the re- duction of dissolved and sediment phosphorus loading and the costs and economic losses connected with proposed control measures. For situations involving multiple objectives, MODM techniques are briefly reviewed in the Appendix.
Stochastic modeling is necessary for eutrophication control planning, especially if both point and non-point sources contribute t o nutrient loading. The stochastic model for phosphorus loading estimation described here can be incorporated into a decision model in order t o identify the most effective control alternative. The model can also b e used for other nutrients and for heavy metal pollutants.
Both structural (tertiary treatment, runoff regulation, sediment trapping) and non- structural measures (fertilizer control, land-use change) have been considered in designing a finite number of realistic control alternatives. T o account for the stochastic character o f phosphorus loading, we have used the relative frequency u, with which alternative j results in the greatest reduction in loading as our measure of control effectiveness. Finally, under case study conditions, the combined control measure consisting of storage, sediment silt- ing, and tertiary treatment seems to represent the best compromise.
ACKNOWLEDGMENTS
The research leading t o this report was mostly carried out during short-term visits of the authors t o the International Institute for Applied Systems Analysis, where the en- couragement and useful advice of Oleg Vasiliev, Janusz Kindler, and Laszlo Somlyody was highly appreciated. Partial support from the US National Science Foundation under grant INT 78-1 2 184, from the Hungarian IIASA Committee, and from the team of SZTAKI sci- entists led by A. Prekopa, is gratefully acknowledged. The simulation algorithm was de- veloped and run with the help of Jim Kempf, whose biosystem knowledge has been a precious asset throughout the research.
APPENDIX: GUIDELINES FOR THE SELECTION O F A MULTIOBJECTIVE DECISION-MAKING (MODM) TECHNIQUE
Arrays of Systems versus Criteria
The core, basis, or keystone of most MODM schemes is a display of alternative sys- tems versus the chosen criteria. There is no unified vocabulary t o describe or categorize this display, which may be called an impact matrix (Nijkamp and Rietveld 1976, O'Brien et al. 1977), a cost-effectiveness (CE) array (Kazanowski 1968), or a payoff table. An im- pact matrix corresponds t o the matrix of efficient (marginal) points or nondominated extreme points (Benayoun et al. 1971) or Pareto set. Following similar lines, the alterna- tives can be called alternative systems, actions, decisions, variants, or choices. One fairly common method of display is shown in Table A l , in which the alternatives are represented b y columns I, 11,111, etc. The rows represent the criteria, also known as measures of effec- tiveness (Kazanowski 1968, 1972), figures of merit or performance indices (Wymore 1976), ratings, losses, payoffs, and even objectives (Keeney et al. 1976, Keeney and Wood 1977).
This last designation is somewhat confusing within the framework of the so-called cost- effectiveness approach, in which "objectives" are distinguished from "criteria".
Table A1 or any similar representation cannot be set u p when the set of alternatives is continuous over the decision space (e.g., amount of water treated or pumped from a well); in this case, the table is replaced b y a generator of alternatives. The row entries or TABLE A1 Example of a cost-effectiveness array.
Criterion or
Alternative system measure o f
effectiveness 1 11 111
. . .
Cost 88 82 9 1
. . .
Benefit 95 9 2 103
...
Probability of
water shortage 0.05 0.10 0.02
...
Dissolved oxygen
level 8 6 4
. . .
Hectares irrigable 125 150 120
. . .
criteria define the decision space to be used for MODM between these alternatives. Note MODM methods run into difficulties because of the combinatorial nature of the task. Any dichotomous classification with n categories - discrete or continuous state or decision variable, deterministic or stochastic, single or multiple period, prior articulation of prefer- ences or interactive, etc. - will yield 2" categories. Furthermore, hardly any method will fit exactly in one category; for example, most methods can be made interactive.
Our advice here is to ask questions about the desiderata and capabilities of the deci- sion maker on the one hand, and about the nature of the problem (including availability of resources and data) on the other, and then to match the answers to the two setsof ques- tions. By the end of this procedure the number of remaining applicable MODM approaches is generally very small.
It is useful to begin with two questions concerning the decision maker:
What are the decision makerS desiderata?
The decision maker may wish to be presented with a single "best" system; in this case a method that collapses all the objectives into a single index, such as multiattribute utility theory (MAUT) (Keeney and Raiffa 1976), one that always leads to complete order- ing, such as ELECTRE I1 (Duckstein and Gershon 1981), or cooperative game theory with a given starting or "status quo" point (Szidarovszky et al. 1978) are possible contenders.
Alternatively, he may require a decision-making aid that reduces the choice set; ELECTRE I (Benayoun et al. 1966) can be used for discrete systems and compromise programming (CP) (Zeleny 1973) for continuous ones.
Another possible response is that the decision maker wants to participate actively in the decision-making process, in which case interactive methods such as SEMOPS (Monarchi et al. 1973) or TRADE (Goicoechea et al. 1976) can be used. Finally, he may wish to introduce the effects of uncertainty. Statistical or economic uncertainty can be incorpo- rated in one or more criteria (Duckstein 1978), and for continuous problems it is possible to introduce uncertainty into multiobjective linear programs (Sengupta et al. 1973, Goicoechea et al. 1979).
What are the decision maker's capabilities?
As pointed out by Lettermaier and Burges (1978), real-life decision makers may not always have a strong theoretical background, and they are unlikely to have much time available for interaction with analysts, computers, or models. It is thus important not to choose overly complicated models that require the decision maker to possess sophisticated
analytical knowledge and to spend much of his time in calibration. This latter point is discussed in some detail in Major (1977), Krzysztofowicz el al. (1977), Cohon and Marks (1977), and Cohon (1978).
Turning now to the problem itself, two broad questions suggest themselves:
Are there qualitative criteria involved that should preferably not be quantified?
If the answer is yes, this eliminates all multiobjective programming approaches, such as surrogate worth tradeaff (SWT) (Hairnes et al. 1975), STEM (Benayoun et al. 1971), SEMOPS (Monarchi et al. 1973), TRADE (Goicoechea 1977), MAUT, PROTRADE, co- operative game theory, goal and compromise programming, the iterative algorithm of Neuman and Krzysztofowicz (1977) or the approach developed by Wierzbicki (1980). On the other hand, it would indicate methods such as metagame theory (Hipel et al. 1974, 1976, Ragade et al. 1976a, 1976b), ELECTRE I, 11, and 111 (Roy 1977), and concordance analysis (Nijkamp and Rietveld 1976).
If all criteria can be quantified, are the alternative systems defined in a discrete or continuous mode?
If a finite set of distinct alternatives is used, then again most multiobjective pro- gramming methods are ruled out. On the other hand, the use of goal and compromise programming, cooperative game theory, MAUT, or ELECTRE may well be indicated. For example, ELECTRE I has been used as a decision-making aid for problems with quantita- tive and qualitative criteria and discrete alternatives by David and Duckstein (1976) and Gershon et al. (1 980).
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