3. modelling methods for evaluating alternatives

3. Modelling Methods for Evaluating Alternatives

1. Introduction 59

1.1. Model Components 60

2. Plan Formulation and Selection 61

2.1. Plan Formulation 61

2.2. Plan Selection 63

3. Modelling Methods: Simulation or Optimization 64

3.1. A Simple Planning Example 65

3.2. Simulation Modelling Approach 66

3.3. Optimization Modelling Approach 66

3.4. Simulation Versus Optimization 67

3.5. Types of Models 69

3.5.1. Types of Simulation Models 69

3.5.2. Types of Optimization Models 70

4. Model Development 71

5. Managing Modelling Projects 72

5.1. Create a Model Journal 72

5.2. Initiate the Modelling Project 72

5.3. Selecting the Model 73

5.4. Analysing the Model 74

5.5. Using the Model 74

5.6. Interpreting Model Results 75

5.7. Reporting Model Results 75

6. Issues of Scale 75

6.1. Process Scale 75

6.2. Information Scale 76

6.3. Model Scale 76

6.4. Sampling Scale 76

6.5. Selecting the Right Scales 76

7. Conclusions 77

8. References 77

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1. Introduction

Water resources system planners must identify andevaluate alternative water resources system designs ormanagement plans on the basis of their economic, eco-logical, environmental, and social or political impacts.One important criterion for plan identification and evalu-ation is the economic benefit or cost a plan would entailwere it to be implemented. Other criteria can include theextent to which any plan meets environmental, ecologicaland social targets. Once planning or management per-formance measures (objectives) and various general alter-natives for achieving desired levels of these performancemeasures have been identified, models can be developedand used to help identify specific alternative plans thatbest meet those objectives.

Some system performance objectives may be inconflict, and in such cases models can help identify theefficient tradeoffs among these conflicting measures ofsystem performance. These tradeoffs indicate what com-binations of performance measure values can be obtainedfrom various system design and operating policy variablevalues. If the objectives are the right ones (that is, they are

59

Modelling Methods for Evaluating Alternatives

Water resources systems are characterized by multiple interdependentcomponents that together produce multiple economic, environmental, ecologicaland social impacts. Planners and managers working to improve the performanceof these complex systems must identify and evaluate alternative designs andoperating policies, comparing their predicted performance with the desired goalsor objectives. These alternatives are defined by the values of numerous design,target and operating policy variables. Constrained optimization together withsimulation modelling is the primary way we have of estimating the values of thedecision variables that will best achieve specified performance objectives. Thischapter introduces these optimization and simulation methods and describes whatis involved in developing and applying them in engineering projects.

3

what the stakeholders really care about), such quantita-tive tradeoff information should be of value during thedebate over what decisions to make.

Regional water resources development plans designedto achieve various objectives typically involve investmentsin land and infrastructure. Achieving the desired eco-nomic, environmental, ecological and social objectivevalues over time and space may require investments instorage facilities, including surface or groundwaterreservoirs and storage tanks, pipes, canals, wells, pumps,treatment plants, levees and hydroelectric generatingfacilities, or in fact the removal of some of them.

Many capital investments can result in irreversibleeconomic and ecological impacts. Once the forest of avalley is cleared and replaced by a lake behind a dam, itis almost impossible to restore the site to its originalcondition. In parts of the world where river basin orcoastal restoration activities require the removal of engi-neering structures, water resources engineers are learningjust how difficult and expensive that effort can be.

The use of planning models is not going to eliminatethe possibility of making mistakes. These models can,however, better inform. They can provide estimates of the

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different impacts associated with, say, a natural unregu-lated river system and a regulated river system. Theformer can support a healthier ecosystem that provides ahost of flood protection and water quality enhancementservices. The latter can provide more reliable and cheaperwater supplies for off-stream users and increasedhydropower and some protection from at least smallfloods for those living on flood-prone lands. In short,models can help stakeholders assess the future conse-quences, the benefits and costs, and a multitude of otherimpacts associated with alternative plans or managementpolicies.

This chapter introduces some mathematical optimiza-tion and simulation modelling approaches commonlyused to study and analyse water resources systems. Themodelling approaches are illustrated by their applicationto some relatively simple water resources planning andmanagement problems. The purpose here is to introduceand compare some commonly used methods of (orapproaches to) modelling. This is not a text on the state-of-the-art of optimization or simulation modelling. Insubsequent chapters of this book, more details will begiven about optimization models and simulation meth-ods. More realistic and more complex problems usuallyrequire much bigger and more complex models thanthose developed in this book, but these bigger and morecomplex models are often based on the principles andtechniques introduced here.

Regardless of the problem complexity or size, the mod-elling approaches are the same. Thus, the emphasis here ison the art of model development: just how one goes aboutconstructing a model that will provide information neededto solve a particular problem, and various ways modelsmight be solved. It is unlikely anyone will ever use any ofthe specific models developed in this or other chapters,simply because they will not be solving the specific exam-ples used to illustrate the different approaches to modeldevelopment and solution. However, it is quite likely that water resources managers and planners will use themodelling approaches and solution methods presented in this book to analyse similar types of problems. Theparticular problems used here, or any others that couldhave been used, can be the core of more complex modelsaddressing more complex problems in practice.

Water resources planning and management today isdominated by the use of predictive optimization and

60 Water Resources Systems Planning and Management

simulation models. While computer software is becomingincreasingly available for solving various types of opti-mization and simulation models, no software currentlyexists that will build those models themselves. What andwhat not to include and assume in models requiresjudgement, experience and knowledge of the particularproblem being addressed, the system being modelled andthe decision-making environment. Understanding thecontents of, and performing the exercises for, this chapterwill be a first step towards gaining some judgement andexperience in model development.

1.1. Model Components

Mathematical models contain algebraic equations. Theseequations include variables that are assumed to be knownand others that are unknown and to be determined.Known variables are usually called parameters, andunknown variables are called decision variables. Modelsare developed for the primary purpose of identifying thebest values of the latter. These decision variables caninclude design and operating policy variables of variouswater resources system components.

Design variables can include the active and floodstorage capacities of reservoirs, the power generatingcapacity of hydropower plants, the pumping capacity ofpumping stations, the efficiencies of wastewater treatmentplants, the dimensions or flow capacities of canals andpipes, the heights of levees, the hectares of an irrigationarea, the targets for water supply allocations and so on.Operating variables can include releases of water fromreservoirs or the allocations of water to various users overspace and time. Unknown decision variables can alsoinclude measures of system performance, such as neteconomic benefits, concentrations of pollutants, ecologicalhabitat suitability values or deviations from particularecological, economic or hydrological targets.

Models describe, in mathematical terms, the systembeing analysed and the conditions that the system has to satisfy. These conditions are often called constraints.Consider, for example, a reservoir serving various watersupply users downstream. The conditions included in amodel of this reservoir would include the assumption thatwater will flow in the direction of lower heads (that is,downstream unless it is pumped upstream), and thevolume of water stored in a reservoir cannot exceed the

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Modelling Methods for Evaluating Alternatives 61

reservoir’s storage capacity. Both the storage volume overtime and the reservoir capacity might be unknown. If the capacity is known or assumed, then it is among theknown model parameters.

Model parameter values, while assumed to be known,can often be uncertain. The relationships betweenvarious decision variables and assumed known modelparameters (i.e., the model itself) may be uncertain. Inthese cases the models can be solved for a variety ofassumed conditions and parameter values. This providesan estimate of just how important uncertain parametervalues or uncertain model structures are with respect tothe output of the model. This is called sensitivity analysis.Sensitivity analyses will be discussed in Chapter 9 inmuch more detail.

Solving a model means finding values of its unknowndecision variables. The values of these decision variablescan define a plan or policy. They can also define the costsand benefits or other measures of system performance

associated with that particular management plan orpolicy.

While the components of optimization and simulationmodels include system performance indicators, modelparameters and constraints, the process of model devel-opment and use includes people. The drawing shown inFigure 3.1 illustrates some interested stakeholders busystudying their river basin, in this case perhaps with theuse of a physical model. Whether a mathematical modelor physical model is being used, one important consider-ation is that if the modelling exercise is to be of any value,it must provide the information desired and in a form thatthe interested stakeholders can understand.

2. Plan Formulation and Selection

Plan formulation can be thought of as assigning particularvalues to each of the relevant decision variables. Planselection is the process of evaluating alternative plans andselecting the one that best satisfies a particular objectiveor set of objectives. The processes of plan formulation andselection involve modelling and communication amongall interested stakeholders, as the picture in Figure 3.1suggests.

The planning and management issues being discussedby the stakeholders in the basin pictured in Figure 3.1could well include surface and groundwater water alloca-tions, reservoir operation, water quality management andinfrastructure capacity expansion.

2.1. Plan Formulation

Model building for defining alternative plans or policiesinvolves a number of steps. The first is to clearly specifythe issue or problem or decision(s) to be made. What arethe fundamental objectives and possible alternatives?Such alternatives might require defining allocations ofwater to various water users, the level of wastewater treat-ment, the capacities and operating rules of multipurposereservoirs and hydropower plants, and the extent andreliability of floodplain protection from levees. Each ofthese decisions may affect system performance criteria orobjectives. Often these objectives include economic meas-ures of performance, such as costs and benefits. They mayalso include environmental and social measures not

Figure 3.1. These stakeholders have an interest in how theirwatershed or river basin is managed. Here they are using aphysical model to help them visualize and address planningand management issues. Mathematical models often replacephysical models, especially for planning and managementstudies. (Reprinted with permission from Engineering News-Record, copyright The McGraw-Hill Companies, Inc.,September 20, 1993. All rights reserved.).

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expressed in monetary units. (More detail on performancecriteria is contained in Chapter 10.)

To illustrate this plan formulation process, considerthe problem of designing a tank to hold a specific amountof water. The criterion to be used to compare differentfeasible designs is cost. The goal in this example is to findthe least-cost shape and dimensions of a tank that willhold a specified volume, say V, of water.

The model of this problem must somehow relate the unknown design variable values to the cost of thetank. Assume, for example, a rectangular tank shape. Thedesign variables are the length, L, width, W, and height,H, of the tank. These are the unknown decision variables.The objective is to find the combination of L, W, and Hvalues that minimizes the total cost of providing a tankcapacity of at least V units of water. This volume V will beone of the model parameters. Its value is assumed knowneven though in fact it may be unknown and dependent inpart on the cost.

The cost of the tank will be the sum of the costs ofthe base, the sides and the top. These costs will dependon the area of the base, sides and top. The costs per unit area may vary depending on the values of L, W andH; however, even if those cost values depend on the values of those decision variables, given any specificvalues for L, W and H, one can define an average cost-per-unit area. Here we will assume these averagecosts per unit area are known. They can be adjusted ifthey turn out to be incorrect for the derived values of L,W and H.

These average unit costs of the base, sides and top willprobably differ. They can be denoted as Cbase, Cside andCtop respectively. These unit costs together with the tank’svolume, V, are the parameters of the model. If L, W, andH are measured in metres, then the areas will beexpressed in units of square metres and the volume willbe expressed in units of cubic metres. The average unitcosts will be expressed in monetary units per squaremetre.

The final step of model building is to specify all therelations among the objective (cost), function and deci-sion variables and parameters, including all the condi-tions that must be satisfied. It is often wise to first statethese relationships in words. The result is a word model.Once that is written, mathematical notation can bedefined and used to construct a mathematical model.


The word model for this tank design problem is tominimize total cost where:

• Total cost equals the sum of the costs of the base, thesides and the top.

• Cost of the sides is the cost-per-unit area of the sidestimes the total side area.

• Cost of the base is the cost-per-unit area of the basetimes the total base area.

• Cost of the top is the cost-per-unit area of the toptimes the total top area.

• The volume of the tank must at least equal some spec-ified volume capacity.

• The volume of the tank is the product of the length,width and height of the tank.

Using the notation already defined, and combining someof the above conditions, a mathematical model can bewritten as:

Minimize Cost (3.1)

Subject to:

Cost � (Cbase � Ctop)(LW) � 2(Cside) (LH � WH) (3.2)

LWH � V (3.3)

Equation 3.3 permits the tank’s volume to be larger thanthat required. While this is allowed, it will cost more if thetank’s capacity is larger than V, and hence the least-costsolution of this model will surely show that LWH willequal V. In practice, however, there may be practical,legal and/or safety reasons why the decisions with respectto L, W and H may result in a capacity that exceeds therequired volume, V.

This model can be solved a number of ways, whichwill be discussed later in this and the next chapters. Theleast-cost solution is

W� L � [2Cside V/(Cbase � Ctop)]1/3 (3.4)

and H � V/[2Cside V/(Cbase � Ctop)]2/3 (3.5)

or H � V1/3[(Cbase � Ctop)/2Cside]2/3 (3.6)

The modelling exercise should not end here. If there isany doubt about the value of any of the parameters, a sen-sitivity analyses should be performed on those uncertainparameters or assumptions. In general these assumptionscould include the values of the cost parameters (e.g., thecosts-per-unit area) as well as the relationships expressedin the model (that is, the model itself). How much does

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the total cost change with respect to a change in any of thecost parameters or with the required volume V? Howmuch does any decision-variable change with respect to changes in those parameter values? What is the percentchange in a decision-variable value given a unit percentchange in some parameter value (what economists callelasticity)?

If indeed the decision-variable values do not changesignificantly with respect to a change in the value of anuncertain parameter value, there is no need to devotemore effort to reducing that uncertainty. Any time andmoney available for further study should be directedtoward those parameters or assumptions that substan-tially influence the model’s decision-variable values.

This capability of models to help identify what data areimportant and what data are not can guide monitoringand data collection efforts. This is a beneficial attribute ofmodelling often overlooked.

Continuing with the tank example, after determining,or estimating, all the values of the model parameters andthen solving the model to obtain the cost-effective valuesof L, W and H, we now have a design. It is just one of anumber of designs that could be proposed. Anotherdesign might be for a cylindrical tank having a radius andheight as decision-variables. For the same volume V andunit area costs, we would find that the total cost is less,simply because the areas of the base, side and top are less.We could go one step further and consider the possibilityof a truncated cone, having different bottom and topradii. In this case both radii and the height would be thedecision-variables. But whatever the final outcome of ourmodelling efforts, there might be other considerations orcriteria that are not expressed or included in the modelthat might be important to those responsible for plan(tank design) selection.

2.2. Plan Selection

Assume P alternative plans (e.g., tank designs) have been defined, each designated by the index p. For eachplan, there exist np decision variables xj

p indexed with the letter j. Together these variables and their values,expressed by the vector Xp, define the specifics of the pth plan. The index j distinguishes one decision-variablefrom another, and the index p distinguishes one planfrom another. The task at hand, in this case, may be to

find the particular plan p, defined by the known values of each decision-variable in the vector Xp, that maximizesthe present value of net benefits, B(Xp), derived from the plan.

Assume for now that an overall performance objectivecan be expressed mathematically as:

maximize B(Xp) (3.7)

The values of each decision-variable in the vector Xp thatmeet this objective must be feasible; in other words, theymust meet all the physical, legal, social and institutionalconstraints.

Xp feasible for all plans p. (3.8)

There are various approaches to finding the ‘best’ plan orbest set of decision-variable values. By trial and error, onecould identify alternative plans p, evaluate the net bene-fits derived from each plan, and select the particular planwhose net benefits are a maximum. This process couldinclude a systematic simulation of a range of possiblesolutions in a search for the best. When there is a largenumber of feasible alternatives – that is, many decision-variables and many possible values for each of them – itmay no longer be practical to identify and simulate allfeasible combinations of decision-variable values, or evena small percentage of them. It would simply take too long.In this case it is often convenient to use an optimizationprocedure.

Equations 3.7 and 3.8 represent a discrete optimiza-tion problem. There are a finite set of discrete alternatives.The set could be large, but it is finite. The tank problemexample is a continuous optimization problem having, at least mathematically, an infinite number of feasiblesolutions. In this case optimization involves findingfeasible values of each decision-variable xj in the set ofdecision variables X that maximize (or minimize) someperformance measure, B(X). Again, feasible values arethose that satisfy all the model constraints. A continuousconstrained optimization problem can be written as:

maximize B(X) (3.9)

X feasible (3.10)

While maximization of Equation 3.7 requires a comparisonof B(Xp) for every discrete plan p, the maximization ofEquation 3.9, subject to the feasibility conditions requiredin Equation 3.10, by complete enumeration is impossible.If there exists a feasible solution – in other words, at least

wrm_ch03.qxd 8/31/2005 11:50 AM 3

one that satisfies all the constraints – mathematically thereare likely to be an infinite number of possible feasiblesolutions or plans represented by various values of thedecision-variables in the vector X.

Finding by trial and error the values of the vector Xthat maximizes the objective Equation 3.9 and at the sametime meet all the constraints is often difficult. Some typeof optimization procedure, or algorithm, is useful in suchcases. Mathematical optimization methods are designedto make this search for the best solution (or better solu-tions) more efficient. Optimization methods are used toidentify those values of the decision-variables that satisfyspecified objectives and constraints without requiringcomplete enumeration.

While optimization models might help identify thedecision-variable values that will produce the best plandirectly, they are based on all the assumptions incorpo-rated in the model. Often these assumptions are limiting.In these cases the solutions resulting from optimizationmodels should be analysed in more detail, perhapsthrough simulation methods, to improve the values of thedecision-variables and to provide more accurate estimatesof the impacts associated with those decision-variablevalues. In these situations, optimization models are usedfor screening out the clearly inferior solutions, not forfinding the very best one. Just how screening is performedusing optimization models will be discussed in the nextchapter.

The values that the decision-variables may assume arerarely unrestricted. Usually various functional relation-ships among these variables must be satisfied. This is whatis expressed in constraint Equations 3.8 and 3.10. Forexample, the tank had to contain a given amount of water.In a water-allocation problem, any water allocated to andcompletely consumed by one user cannot simultaneouslyor subsequently be allocated to another user. Storagereservoirs cannot store more water than their maximumcapacity. Technological restrictions may limit the capaci-ties and sizes of pipes, generators and pumps to thosecommercially available. Water quality concentrationsshould not exceed those specified by water qualitystandards or regulations. There may be limited fundsavailable to spend on water resources developmentprojects. These are a few examples of physical, legal andfinancial conditions or constraints that may restrict theranges of variable values in the solution of a model.


Equations or inequalities can generally express anyphysical, economic, legal or social restrictions on thevalues of the decision-variables. Constraints can alsosimply define relationships among decision-variables. For example, Equation 3.2 above defines a new decision-variable called Cost as a function of other decision-variables and model parameters.

In general, constraints describe in mathematical termsthe system being analysed. They define the system com-ponents and their inter-relationships, and the permissibleranges of values of the decision-variables, either directlyor indirectly.

Typically, there exist many more decision-variablesthan constraints, and hence, if any feasible solution exists,there may be many such solutions that satisfy all the con-straints. The existence of many feasible alternative plans isa characteristic of most water resources systems planningproblems. Indeed it is a characteristic of most engineeringdesign and operation problems. The particular feasiblesolution or plan that satisfies the objective function – thatis, that maximizes or minimizes it – is called optimal. It isthe optimal solution of the mathematical model, but itmay not necessarily be considered optimal by any decision-maker. What is optimal with respect to some model maynot be optimal with respect to those involved in a plan-ning or decision-making process. To repeat what waswritten in Chapter 2, models are used to provide infor-mation (useful information, one hopes), to the decision-making process. Model solutions are not replacements forindividuals involved in the decision-making process.

3. Modelling Methods: Simulationor Optimization

The modelling approach discussed in the previous sectionfocused on the use of optimization methods to identifythe preferred design of a tank. Similar methods can beused to identify preferred design-variable values and oper-ating policies for multiple reservoir systems, for example.Once these preferred designs and operating policies havebeen identified, unless there is reason to believe that aparticular alternative is really the best and needs no fur-ther analysis, each of these preferred alternatives can befurther evaluated with the aid of more detailed and robustsimulation models. Simulation models address ‘what if ’

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questions: What will likely happen over time and at oneor more specific places if a particular design and/oroperating policy is implemented?

Simulation models are not limited by many of theassumptions incorporated into optimization models. Forexample, the inputs to simulation models can include amuch longer time series of hydrological, economic andenvironmental data such as rainfall or streamflows,water supply demands, pollutant loadings and so onthan would likely be included in an optimization model.The resulting outputs can better identify the variationsof multiple system performance indicator values: that is, the multiple hydrological, ecological, economicand environmental impacts that might be observed overtime, given any particular system design and operatingpolicy.

Simulating multiple sets of values defining the designsand operating policies of a water resources system cantake a long time. Consider, for example, only 30 decision-variables whose best values are to be determined. Even ifonly two values are assumed for each of the 30 variables,the number of combinations that could be simulatedamounts to 230 or in excess of 109. Simulating and com-paring even 1% of these billion at a minute per simulationamounts to over twenty years, continuously and withoutsleeping. Most simulation models of water resourcessystems contain many more variables and are much morecomplex than this simple 30-binary-variable example. Inreality there could be an infinite combination of feasiblevalues for each of the decision-variables.

Simulation works when there are only a relatively fewalternatives to be evaluated, not when there are a largenumber of them. The trial and error process of simulationcan be time consuming. An important role of optimiza-tion methods is to reduce the number of alternatives for simulation analyses. However, if only one method of analysis is to be used to evaluate a complex waterresources system, simulation together with human judge-ment concerning which alternatives to simulate is often,and rightly so, the method of choice.

Simulation can be based on either discrete events ordiscrete time periods. Most simulation models of waterresources systems are designed to simulate a sequence ofdiscrete time periods. In each discrete time period, thesimulation model converts all the initial conditions andinputs to outputs. The duration of each period depends in

part on the particular system being simulated and thequestions being addressed.

3.1. A Simple Planning Example

Consider the case of a potential reservoir releasing waterto downstream users. A reservoir and its operating policycan increase the benefits each user receives over time byproviding increased flows during periods of otherwiselow flows relative to the user demands. Of interest iswhether or not the increased benefits the water usersobtain from an increased flow and more reliabledownstream flow conditions will offset the costs of thereservoir. This water resources system is illustrated inFigure 3.2.

Before this system can be simulated, one has to definethe active storage capacity of the reservoir and how muchwater is to be released depending on the storage volumeand time period; in other words, one has to define thereservoir operating policy. In addition, one must alsodefine the allocation policy: how much water to allocateto each user and to the river downstream of the usersgiven any particular reservoir release.

For this simple illustration assume these operating andallocation policies are as shown in Figure 3.3. Also forsimplicity assume they apply to each discrete time period.

The reservoir operating policy, shown as a red line inFigure 3.3, attempts to meet a release target. If insufficientwater is available, all the water will be released in the timeperiod. If the inflow exceeds the target flow and the reser-voir is at capacity, a spill will occur. This operating policyis sometimes called the ‘standard’ operating policy. It isnot usually followed in practice. Most operators, asindeed specified by most reservoir operating policies, willreduce releases in times of drought in an attempt to save

river QtX 1t

user 1

user 2

user 3

X2t X3t

reservoir

E020108

w

Figure 3.2. Reservoir-water allocation system to be simulated.

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some water in the reservoir for future releases in case of an extended period of low inflows. This is called ahedging policy. Any reservoir release policy, including ahedging policy, can be defined within the blue portion ofthe release policy plot shown in Figure 3.3. The dash–dotline in Figure 3.3 is one such hedging function.

Once defined, any reservoir operating policy can besimulated.

3.2. Simulation Modelling Approach

The simulation process for the three-user system shownin Figure 3.2 proceeds from one time period to the next.The reservoir inflow, obtained from a database, is addedto the existing storage volume, and a release is determinedfrom the release policy. Once the release is known, thefinal storage volume is computed and this becomes the


initial volume for the next simulation time period. Thereservoir release is then allocated to the three downstreamusers and to the river downstream of those users asdefined by the allocation policy. The resulting benefitscan be calculated and stored in an output database.Additional data pertaining to storage volumes, releasesand the allocations themselves can also be stored in theoutput database, as desired. The process continues for theduration of the simulation run. Then the output data canbe summarized for later comparison with other simula-tion results based on other reservoir capacities and oper-ation policies and other allocation policies. Figure 3.4illustrates this simulation process.

It would not be too difficult to write a computer programto carry out this simulation. In fact, it can be done on aspreadsheet. However easy that might be for anyone familiarwith computer programming or spreadsheets, one cannotexpect it to be easy for many practicing water resourcesplanners and managers who are not doing this type of workon a regular basis. Yet they might wish to perform a simula-tion of their particular system, and to do it in a way thatfacilitates changes in many of its assumptions.

Computer programs capable of simulating a wide vari-ety of water resources systems are becoming increasinglyavailable. Simulation programs together with their inter-faces that facilitate the input and editing of data and thedisplay of output data are typically called decision supportsystems. Their input data define the components of thewater resources system and their configuration. Inputsinclude hydrological data and design and operating policydata. These generic simulation programs are now becom-ing capable of simulating surface and groundwater waterflows, storage volumes and qualities under a variety ofsystem infrastructure designs and operating policies.

3.3. Optimization Modelling Approach

The simple reservoir-release and water-allocation plan-ning example of Section 3.1 can also be described as anoptimization model. The objective remains that of maxi-mizing the total benefits that the three users obtain fromthe water that is allocated to them. Denoting each user’sbenefit as Bit (i � 1, 2, 3) for each of T time periods t, thisobjective, expressed symbolically is to:

maximize total benefits � (3.11){ }B B Bt t tt

T

1 2 3+ +∑

initial storage and inflow

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E020

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allo

catio

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6

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user

river

useruser

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Figure 3.3. Policy defining the reservoir release to be madeas a function of the current storage volume and current inflowand the reservoir release allocation policy for the river flowdownstream of the reservoir. The blue zone in the reservoirrelease policy indicates the zone of feasible releases. It isphysically impossible to make releases represented by pointsoutside that blue zone.

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The benefit, Bit, for each user i in each time period tdepends on the amount of water, Xit, allocated to it.These benefit functions, Bit � Bit(Xit), need to be known and expressed in a form suitable for solutionusing the particular optimization solution algorithmselected. The unknown variables include the allocations,Xit, and associated reservoir releases Rt for all periods t � 1, 2, 3, . . . , T. Assuming there is no significant incre-mental runoff between the upstream reservoir and thesites where water is diverted from the river, the amountsallocated to all users, the sum of all Xit in each period t,cannot exceed the amount of water released from thereservoir, Rt, in the period. This is one of the optimiza-tion model constraints:

Rt � X1t � X2t � X3t (3.12)

The remaining necessary constraints apply to the reser-voir. A mass balance of water storage is needed, alongwith constraints limiting initial storage volumes, St, to thecapacity, K, of the reservoir. Assuming a known time-series record of reservoir inflows, It, in each of the timeperiods being considered, the mass-balance or continuityequations for storage changes in each period t can bewritten:

St � It – Rt � St�1 for t � 1, 2, . . . , T;

If t � T, then T�1 � 1. (3.13)

The capacity constraints simply limit the unknown initialstorage volume, St, to be no greater than the reservoircapacity, K.

St � K for t � 1, 2, . . . , T. (3.14)

A one-year analysis period with T � 12 time periods of onemonth each in combination with three allocation vari-ables, Xit, a storage variable St, and a release Rt, variable ineach period t, includes a total of sixty unknown decision-variables. The job of the optimization solution procedureis to find the values of these sixty variables that will satisfythe objective, Equation 3.11, that is to say, maximize totalbenefits, and at the same time satisfy all of the thirty-sixconstraint equations and inequalities as well.

In this example the reservoir inflows, It, its storage capac-ity, K, and each user’s benefit functions, Bit, are assumedknown. In some cases such information is not known. Norwere other purposes, such as hydropower, flood control,water quality or recreation considered in this example, tomention only a few possible extensions. Such conditionsand extensions will be considered in later chapters.

3.4. Simulation Versus Optimization

Unlike simulation models, the solutions of optimizationmodels are based on objective functions of unknowndecision-variables that are to be maximized or minimized.

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(t)

S(t)

R(t)

i(t)X

Q(t)

Bi(t)

- inflow

- initial storage volume

- reservoir release

- allocation to user i

- allocation to stream

- benefit for user i

- time periodt

Figure 3.4. Flow diagram of thereservoir-allocation systemsimulation process. Thesimulation terminates aftersome predefined number ofsimulation time steps.

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The constraints of an optimization model containdecision-variables that are unknown and parameterswhose values are assumed known. Constraints areexpressed as equations and inequalities. The tank model(Equations 3.1, 3.2 and 3.3) is an example of an opti-mization model. So is the reservoir water-allocationmodel, Equations 3.11 to 3.14. The solution of an opti-mization model, if one exists, contains the values of all of the unknown decision-variables. It is mathematicallyoptimal in that the values of the decision-variables satisfyall the constraints and maximize or minimize an objectivefunction. This ‘optimal’ solution is of course based on theassumed values of the model parameters, the chosenobjective function and the structure of the model itself.

The procedure (or algorithm) most appropriate forsolving any particular optimization model depends in part on the particular mathematical structure of the model. There is no single universal solution procedurethat will efficiently solve all optimization models. Hence,model builders tend to model water resources systems by using mathematical expressions that are of a formcompatible with one or more known solution procedures.Approximations of reality, made to permit model solutionby a chosen optimization solution procedure (algorithm),may justify a more detailed simulation to check andimprove on any solution obtained from that optimization.Simulation models are not restricted to any particular formof mathematics, and can define many relations includingthose not easily incorporated into optimization models.

One of many challenges in the use of optimizationmodelling is our inability to quantify and express


mathematically all the planning objectives, the technical,economic, and political constraints and uncertainties, andother important considerations that will influence thedecision-making process. Hence at best a mathematical-model of a complex water resources system is only anapproximate description of the real system. The optimalsolution of any model is optimal only with respect to theparticular model, not necessarily with respect to the realsystem. It is important to realize this limited meaning of theword ‘optimal,’ a term commonly used by water resourcesand other systems analysts, planners and engineers.

Figure 3.5 illustrates the broad differences betweensimulation and optimization. Optimization models needexplicit expressions of objectives. Simulation models do not.Simulation simply addresses ‘what-if ’ sceanarios – what mayhappen if a particular scenario is assumed or if a particulardecision is made. Users must specify the values of design and operating decision-variables before a simulation can beperformed. Once these values of all decision-variables aredefined, simulation can help us estimate more precisely theimpacts that may result from those decisions. The difficultywith using simulation alone for analysing multiple alterna-tives occurs when there are many alternative, and potentiallyattractive, feasible solutions or plans and not enough time orresources to simulate them all. Even when combined withefficient techniques for selecting the values of each decision-variable, an enormous computational effort may still lead toa solution that is still far from the best possible.

For water resources planning and management, it is often advantageous to use both optimization andsimulation modelling. While optimization will tell us

water resources system

water resources system

simulation

optimization

system design and operating policy

system design and operating policy

system inputs

system inputs

system outputs

system outputs

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o

Figure 3.5. Distinguishingbetween simulation andoptimization modelling.Simulation addresses ‘what if ’questions; optimization canaddress ‘what should be’questions. Both types of modelsare often needed in waterresources panning andmanagement studies.

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what we should do – what the best decision is – thatsolution is often based on many limiting assumptions.Because of this, we need to use optimization not as a wayto find the best solution, but to define a relatively smallnumber of good alternatives that can later be tested,evaluated and improved by means of simulation. Thisprocess of using optimization to reduce the large numberof plans and policies to a few that can then be simulatedand better evaluated is often called preliminary screening.

3.5. Types of Models

3.5.1. Types of Simulation Models

Simulation models can be statistical or process oriented,or a mixture of both. Pure statistical models are basedsolely on data (field measurements). Pure process-oriented models are based on knowledge of the funda-mental processes that are taking place. The examplesimulation model just discussed is a process-orientedmodel. It incorporated and simulated the physicalprocesses taking place in the system. Many simulationmodels combine features of both of these extremes.

The range of various simulation modelling approachesapplied to water resources systems is illustrated in Figure 3.6.

Regressions, such as that resulting from a least-squaresanalysis, and artificial neural networks are examples ofpurely statistical models. As discussed in Chapter 6, arelationship is derived between input data (cause) andoutput data (effect), based on measured and observeddata. The relationship between the input and the outputvariable values is derived by calibrating a black-box orstatistical model with a predefined structure unrelated tothe actual natural processes taking place. Once calibrated,the model can be used to estimate the output variablevalues as long as the input variable values are within therange of those used to calibrate the model.

Hybrid models incorporate some process relationshipsinto regression models or neural networks. These rela-tionships supplement the knowledge contained in thecalibrated parameter values based on measured data.

Most simulation models containing process relationshipsinclude parameters whose values need to be estimated. Thisis called model calibration. Calibration requires measuredfield data. These data can be used for initial calibration andverification, and in the case of ongoing simulations, forcontinual calibration and uncertainty reduction. The latter is sometimes referred to as data assimilation.

Other simulation model classifications are possible.Simulation models can be classified based on what the

regressionandneuralnetworks

hybridmodels

processmodels& dataassimilation

processbasedsimulationmodels

fully process orientedfully data oriented

processknowledge

measureddata

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s

Figure 3.6. Range of simulationmodels types based on theextent to which measured fielddata and descriptions of systemprocesses are included in themodel.

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model simulates: on the domain of application. Todayone can obtain, or develop, computer programs written tosimulate a wide variety of water resources system compo-nents or events. Some of these include:

• water quantity and/or quality of rivers, bays, estuariesor coastal zones

• reservoir operation for quantity and/or quality• saturated and/or unsaturated zone groundwater quan-

tity and/or quality• precipitation runoff, including erosion and chemicals• water system demands, supply distribution and treat-

ment• high-water forecasting and control• hazardous material spills• morphological changes• wastewater collection systems• wastewater purification facilities• irrigation operations• hydropower production• ecological habitats of wetlands, lakes, reservoirs and

flood plains• economic benefits and costs.

Simulation models of water resources systems generallyhave both spatial and temporal dimensions. These dimen-sions may be influenced by the numerical methods used,if any, in the simulation, but otherwise they are usuallyset, within the limits desired by the user. Spatial resolu-tions can range from 0 to 3 dimensions. Models are some-times referred to as quasi 2- or 3-dimensional models.These are 1 or 2-dimensional models set up in a way thatapproximates what takes place in 2- or 3-dimensionalspace, respectively. A quasi-3D system of a reservoircould consist of a series of coupled 2D horizontal layers,for example.

Simulation models can be used to study what mightoccur during a given time period, say a year, sometime inthe future, or what might occur from now to that giventime in the future. Models that simulate some particulartime in the future, where future conditions such asdemands and infrastructure design and operation arefixed, are called stationary or static models. Models thatsimulate developments over time are called dynamicmodels. Static models are those in which the externalenvironment of the system being simulated is not chang-ing. Water demands, soil conditions, economic benefit


and cost functions, populations and other factors do notchange from one year to the next. Static models provide asnapshot or a picture at a point in time. Multiple years ofinput data may be simulated, but from the output statis-tical summaries can be made to identify what the valuesof all the impact variables could be, together with theirprobabilities, at that future time period.

Dynamic simulation models are those in which theexternal environment is also changing over time.Reservoir storage capacities could be decreasing due tosediment load deposition, costs could be increasing dueto inflation, wastewater effluent discharges could bechanging due to changes in populations and/or waste-water treatment capacities, and so on.

Simulation models can also vary in the way they aresolved. Some use purely analytical methods while othersrequire numerical ones. Many use both methods, asappropriate.

Finally, models can also be distinguished according tothe questions being asked and the level of informationdesired. The type of information desired can range fromdata of interest to policy-makers and planners (requiringrelatively simple models and broader in scope) to that ofinterest to researchers desiring a better understanding of the complex natural, economic and social processestaking place (requiring much more detailed models andnarrower in scope). Water management and operationalmodels (for real-time operations of structures, for exam-ple) and event-based calamity models are somewherebetween these two extremes with respect to model detail.The scope and level of detail of any modelling study willalso depend on the time, money and data available forthat study (see Chapter 2).

3.5.2. Types of Optimization Models

There are many ways to classify various types of con-strained optimization models. Optimization models canbe deterministic or probabilistic, or a mixture of both.They can be static or dynamic with respect to time. Manywater resources planning and management models arestatic, but include multiple time periods to obtain astatistical snapshot of various impacts in some planningperiod. Optimization models can be linear or non-linear.They can consist of continuous variables or discrete orinteger variables, or a combination of both. But whatever

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type they are, they have in common the fact that they aredescribing situations where there exist multiple solutionsthat satisfy all the constraints, and hence, there is thedesire to find the best solution, or at least a set of verygood solutions.

Regardless of the type of optimization model, they allinclude an objective function. The objective function ofan optimization model (such as Equation 3.11 in theexample problem above) is used to evaluate multiplepossible solutions. Often multiple objective functionsmay be identified (as will be discussed in Chapter 10). Butat least one objective must be identified in all optimizationmodels. Identifying the best objective function is often achallenging task.

Optimization models can be based on the particulartype of application, such as reservoir sizing and/or opera-tion, water quality management, or irrigation develop-ment or operation. Optimization models can also beclassified into different types depending on the algorithmto be used to solve the model. Constrained optimizationalgorithms are numerous. Some guarantee to find the best model solution and others can only guarantee locallyoptimum solutions. Some include algebraic ‘mathematicalprogramming’ methods and others include deterministicor random trial-and-error search techniques. Mathematicalprogramming techniques include Lagrange multipliers,linear programming, non-linear programming, dynamicprogramming, quadratic programming, fractional pro-gramming and geometric programming, to mention afew. The applicability of each of these as well as otherconstrained optimization procedures is highly dependenton the mathematical structure of the model. The follow-ing Chapter 4 illustrates the application of some of the most commonly used constrained optimization tech-niques in water resources planning and management.These include classical constrained optimization usingcalculus-based Lagrange multipliers, discrete dynamicprogramming, and linear and non-linear programming.

Hybrid models usually include multiple solutionmethods. Many generic multi-period simulation modelsare driven by optimization methods within each timeperiod. (The CALSIM II model used by the State ofCalifornia and the US Bureau of Reclamation to allocatewater in central California is one such model.)

Each of a variety of optimization modelling types and solution approaches will be discussed and illustrated

in more detail in subsequent chapters. In some cases, wecan use available computer programs to solve optimiza-tion models. In other cases, we may have to write our own software. To make effective use of optimization, andeven simulation, models one has to learn some modelsolution methods, since those methods often dictate thetype of model most appropriate for analysing a particularplanning or management problem or issue.

To date, no single model type or solution procedurehas been judged best for all the different types of issuesand problems encountered in water resources planningand management. Each method has its advantages andlimitations. One will experience these advantages andlimitations as one practices the art of model developmentand application.

4. Model Development

Prior to the selection or development of a quantitativesimulation model, it is often useful to develop a concep-tual one. Conceptual models are non-quantitative repre-sentations of a system. The overall system structure isdefined but not all its elements and functional relation-ships.

Figure 3.7 illustrates a conceptual model, withoutindicating what each box represents, defining relation-ships between what land and water managers can do andthe eventual ecological impacts of those actions.

Once a conceptual model has been quantified(expressed in mathematical terms), it becomes a mathe-matical model. The model’s equations typically includestate and auxiliary variables, parameters and other modelcomponents.

The values of the model’s parameters need to be deter-mined. Model calibration involves finding the best valuesfor these parameters. It is based on comparisons of themodel results with field measurements. Optimizationmethods can sometimes be used to identify the values ofall model parameters. This is called model calibration oridentification. (Illustrations of the use of optimization forestimating model parameter values are contained inChapters 4 and 6).

Sensitivity analysis (Chapter 9) may serve to identifythe impacts of uncertain parameter values and showwhich parameter values substantially influence the

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model’s results. Following calibration, the remaininguncertainties in the model predictions may be quantifiedin an uncertainty analysis as discussed in Chapter 9.

In addition to being calibrated, simulation modelsshould also be validated or verified. In the validation orverification process the model results are compared withan independent set of measured observations that werenot used in calibration. This comparison is made to verifywhether or not the model describes the system behavioursufficiently correctly.

5. Managing Modelling Projects

There are some steps that, if followed in modellingprojects, can help reduce potential problems and lead tomore effective outcomes. These steps are illustrated inFigure 3.8 (Scholten et al., 2000).

Some of the steps illustrated in Figure 3.8 may not be relevant in particular modelling projects. If so, theseparts of the process can be skipped. Each of thesemodelling project steps is discussed in the next severalsections.


5.1. Creating a Model Journal

One common problem of model studies once they areunderway occurs when one wishes to go back over aseries of simulation results to see what was changed, whya particular simulation was made or what was learned. Itis also commonly difficult if not impossible for thirdparties to continue from the point at which any previousproject terminated. These problems are caused by a lackof information on how the study was carried out. Whatwas the pattern of thought that took place? Which actionsand activities were carried out? Who carried out whatwork and why? What choices were made? How reliableare the end results? These questions should be answerableif a model journal is kept. Just like computer-programmingdocumentation, this study documentation is oftenneglected under the pressure of time and perhaps becauseit is not as interesting as running the models themselves.

5.2. Initiating the Modelling Project

Project initiation involves defining the problem to bemodelled and the objectives that are to be accomplished.

land and water management practices and policies

hydrological consequences

ecosystem habitat & function impacts

impacts on specific species

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t

Figure 3.7. An example of aconceptual model without itsdetail, showing the cause andeffect links betweenmanagement decisions andspecific system impacts.

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There can be major differences in perceptions betweenthose who need information and those who are going toprovide it. The problem ‘as stated’ is often not the prob-lem ‘as understood’ by either the client or the modeller. Inaddition, problem perceptions and modelling objectivescan change over the duration of a modelling project.

The appropriate spatial and time scales also need to beidentified. The essential natural system processes must beidentified and described. One should ask and answer the question of whether or not a particular modellingapproach, or even modelling in general, is the best way toobtain the needed information. What are the alternativesto modelling or a particular modelling approach?

The objective of any modelling project should beclearly understood with respect to the domain and theproblem area, the reason for using a particular model, the questions to be answered by the model, and thescenarios to be modelled. Throughout the project theseobjective components should be checked to see if anyhave changed and if they are being met.

The use of a model nearly always takes place within abroader context. The model itself can also be part of a

larger whole, such as a network of models in which manyare using the outputs of other models. These conditionsmay impose constraints on the modelling project.

Proposed modelling activities may have to be justifiedand agreements made where applicable. Any client at anytime may wish for some justification of the modellingproject activities. Agreement should be reached on howthis justification will take place. Are intermediate reportsrequired, have conditions been defined that will indicatean official completion of the modelling project, is verifi-cation by third parties required, and so on? It is particu-larly important to record beforehand the events or timeswhen the client must approve the simulation results.Finally, it is also sensible to reach agreements with respectto quality requirements and how they are determined or defined, as well as the format, scope and contents ofmodelling project outputs (data files) and reports.

5.3. Selecting the Model

The selection of an existing model to be used in any proj-ect depends in part on the processes that will be modelled(perhaps as defined by the conceptual model), the dataavailable and the data required by the model. The avail-able data should include system observations for compar-ison of the model results. They should also includeestimates of the degree of uncertainty associated witheach of the model parameters. At a minimum this mightonly be estimates of the ranges of all uncertain parametervalues. At best it could include statistical distributions ofthem. In this step of the process it is sufficient to knowwhat data are available, their quality and completeness,and what to do about missing or outlier data.

Determining the boundaries of the model is an essen-tial consideration in model selection. This defines what isto be included in a model and what is not. Any modelselected will contain a number of assumptions. Theseassumptions should be identified and justified, and latertested.

Project-based matters such as the computers to beused, the available time and expertise, the modeller’s per-sonal preferences, and the client’s wishes or requirementsmay also influence model choice. An important practicalcriterion is whether there is an accessible manual foroperating the model program and a help desk available toaddress any possible problems.

report model results

step 7

interpret model results

implement the model

analyse the model

select the model

initiate modeling project

establish project journal

step 6

step 5

step 4

step 3

step 2

step 1

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u

Figure 3.8. The modelling project process is an iterativeprocedure involving specific steps or tasks.

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The decision to use a model, and which model to use,is an important part of water resources plan formulation.Even though there are no clear rules on how to select the right model to use, a few simple guidelines can bestated:

• Define the problem and determine what information isneeded and what questions need to be answered.

• Use the simplest method that will yield adequate accu-racy and provide the answer to your questions.

• Select a model that fits the problem rather than tryingto fit the problem to a model.

• Question whether increased accuracy is worth theincreased effort and increased cost of data collection.(With the advances in computer technology, compu-tational cost is rarely an issue except perhaps for somegroundwater management problems.)

• Do not forget the assumptions underlying the modelused and do not read more significance into the simu-lation results than is actually there.

5.4. Analysing the Model

Once a modelling approach or a particular model hasbeen selected, its strengths and limitations should bestudied in more detail. The first step is to set up a plan fortesting and evaluating the model. These tests can includemass (and energy) balance checks and parameter sensitivityanalyses (see Chapter 9). The model can be run underextreme input data conditions to see if the results are asexpected.

Once a model is tested satisfactorily, it can be cali-brated. Calibration focuses on the comparison betweenmodel results and field observations. An important prin-ciple is: the smaller the deviation between the calculatedmodel results and the field observations, the better themodel. This is indeed the case to a certain extent, as the deviations in a perfect model are only due to measure-ment errors. In practice, however, a good fit is by nomeans a guarantee of a good model.

The deviations between the model results and the fieldobservations can be due to a number of factors. Theseinclude possible software errors, inappropriate modellingassumptions such as the (conscious) simplification ofcomplex structures, neglect of certain processes, errors inthe mathematical description or in the numerical methodapplied, inappropriate parameter values, errors in input


data and boundary conditions, and measurement errorsin the field observations.

To determine whether or not a calibrated model is‘good’, it should be validated or verified. Calibratedmodels should be able to reproduce field observations notused in calibration. Validation can be carried out forcalibrated models as long as an independent data set hasbeen kept aside for this purpose. If all available data areused in the calibration process in order to arrive at the best possible results, validation will not be possible.The decision to leave out validation is often a justifiableone especially when data are limited.

Philosophically, it is impossible to know if a model ofa complex system is ‘correct’. There is no way to prove it.Experimenting with a model, by carrying out multiplevalidation tests, can increase one’s confidence in thatmodel. After a sufficient number of successful tests, onemight be willing to state that the model is ‘good enough’,based on the modelling project requirements. The modelcan then be regarded as having been validated, at least forthe ranges of input data and field observations used in thevalidation.

If model predictions are to be made for situations orconditions for which the model has been validated, onemay have a degree of confidence in the reliability of thosepredictions. Yet one cannot be certain. Much less confi-dence can be placed on model predictions for conditionsoutside the range for which the model was validated.

While a model should not be used for extrapolationsas commonly applied in predictions and in scenarioanalyses, this is often exactly the reason for the modellingproject. What is likely to happen given events we have notyet experienced? A model’s answer to this questionshould also include the uncertainties attached to thesepredictions. Beck (1987) summarized this dilemma in thefollowing statement: ‘using scientifically based models,you will often predict an incorrect future with great accu-racy, and when using complex, non-identifiable models,you may be capable of predicting the correct future withgreat uncertainty’.

5.5. Using the Model

Once the model has been judged ‘good enough’, it may beused to obtain the information desired. One shoulddevelop a plan on how the model is to be used, identifying

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the input to be used, the time period(s) to be simulated,and the quality of the results to be expected. Again, closecommunication between the client and the modeller isessential, both in setting up this plan and throughout itsimplementation, to avoid any unnecessary misunderstand-ings about what information is wanted and the assump-tions on which that information is to be based.

Before the end of this model-use step, one shoulddetermine whether all the necessary model runs havebeen performed and whether they have been performedwell. Questions to ask include:

• Did the model fulfill its purpose?• Are the results valid?• Are the quality requirements met?• Was the discretization of space and time chosen well?• Was the choice of the model restrictions correct?• Was the correct model and/or model program chosen?• Was the numerical approach appropriate?• Was the implementation performed correctly?• Are the sensitive parameters (and other factors) clearly

identified?• Was an uncertainty analysis performed?

If any of the answers to these questions is no, then thesituation should be corrected. If it cannot be corrected,then there should be a good reason for this.

5.6. Interpreting Model Results

Interpreting the information resulting from simulationmodels is a crucial step in a modelling project, especiallyin situations in which the client may only be interested inthose results and not the way they were obtained. Themodel results can be compared to those of other similarstudies. Any unanticipated results should be discussedand explained. The results should be judged with respectto the modelling project objectives.

The results of any water resources modelling projecttypically include large files of time-series data. Only themost dedicated of clients will want to read those files, sothe data must be presented in a more concise form.Statistical summaries should explicitly include anyrestrictions and uncertainties in the results. They shouldidentify any gaps in the domain knowledge, thus generat-ing new research questions or identifying the need formore field observations and measurements.

5.7. Reporting Model Results

Although the results of a model should not be the sole basisfor policy decisions, modellers have a responsibility totranslate their model results into policy recommendations.Policy-makers, managers, and indeed the participatingstakeholders often want simple, clear and unambiguousanswers to complex questions. The executive summary of areport will typically omit much of the scientifically justifieddiscussion in its main body regarding, say, the uncertain-ties associated with some of the data. This executivesummary is often the only part read by those responsiblefor making decisions. Therefore, the conclusions of themodel study must not only be scientifically correct andcomplete, but also concisely formulated, free of jargon, and fully understandable by managers and policy-makers.The report should provide a clear indication of the validity, usability and any restrictions of the model results.The use of visual aids, such as graphs and GIS, can be veryhelpful.

The final report should also include sufficient detail toallow others to reproduce the model study (including itsresults) and/or to proceed from the point where this studyended.

6. Issues of Scale

Scaling aspects play an important role in many modellingprojects. Four different types of scales can be distin-guished: the process scale, the information scale, themodel scale and the sampling scale. Each of these isdiscussed below.

6.1. Process Scale

Most hydrological processes vary over space and time.The scale on which the process variations manifest them-selves is referred to as the process scale. Spatially, processscale can vary from the movement of small granules ofsediment, for example, to the flooding of large river basinsor coastal zones. All kinds of intermediary scale processescan be found, such as drainage into ditches of runoff fromparcels of land, transport of sediment in brooks and flowmovements in aquifers.

Various temporal scales can also be distinguished,varying from the intensity of rain in less than a minute to

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the change in landscape in geological time. Many processdescriptions require a spectrum of scales. Such is the case,for example, in the simulation of interdependent surfaceand groundwater quantity and quality processes takingplace in a watershed.

6.2. Information Scale

Information scale is the spatial and temporal scale of theinformation required. Generally, a strategic water resourcesmanager (for example the local, regional or nationalgovernment) needs information on a scale relative to theirresponsibilities and authorities. This level of information islikely to differ from the level desired by operational watermanagers dealing with day-to-day issues.

Information at scales smaller than what is needed isseen as being ‘noise’. Information at scales larger thanwhat is needed is not relevant or helpful. For localorganizations (e.g., water boards) concerned with runoff,for example, there is no need to collect information onindividual raindrops. The important spatial variances are usually within a range varying from hundreds ofmetres to hundreds of kilometres. Larger-scale variations(differences between precipitation in the Netherlands and Russia or between North and South America, forexample) are rarely if ever relevant. The informationscale depends on the task set for the water planner ormanager.

6.3. Model Scale

Model scales refer to their spatial and temporal discretiza-tion. The model scales determine the required data inter-polation and aggregation.

If the temporal and spatial scales of the problem havenot been defined clearly enough, this can affect the laterphases of the modelling process negatively. If the modelscale chosen is too large, this may result in too general aschematization and relevant details might not be derivedfrom the results. If the chosen model scale is too small,irrelevant small-scale variations can lead to non-optimalcalibrations for the large-scale variations.

In large, spatially distributed models in particular, it isvital that the scale and the number of independentparameters (degrees of freedom) are chosen on the basisof the available data. If too many parameters are included


in a model, there is a risk of it appearing to work well butbeing unsuitable for interpolation or prediction. This canactually only be determined if adequate measuring dataare available having a measurement frequency at leastequal to the chosen modelling time step. This must betaken into account when selecting or constructing themodel, as there is otherwise the risk that the modelcannot be calibrated adequately.

6.4. Sampling Scale

Sampling scale is the scale at which samples are taken.The sampling spatial scale can vary from ‘point observa-tions’ (for example, a temperature measurement at acertain location at a certain time) to area observations(using, for example, remote sensing images). The densityof the measuring network and the sampling or measuringand recording frequency determine the sampling spatialand temporal scales.

6.5. Selecting the Right Scales

Modellers must choose the model scale in such a mannerthat the model provides information at the requiredinformation scale, taking into account the process scalespresent, in combination with the spatial and temporalsampling scales. It is possible that situations will occurthat are impossible to model just because of these scaleissues.

The relationship between the types of scales is repre-sented in Figure 3.9. The relative level of detail is given onthe horizontal axis, from considerable detail on the left tomuch less detail on the right.

To show that the various types of scales may not bemutually compatible, consider the following threeexamples.

samplingscale

small scale

processscale

informationscale

large scale

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v

Figure 3.9. Relationships among various scale types.

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Example 1: The information scale is different from the processscale

Water resources planning studies are typically carried outat a river basin or watershed scale. From a hydrological(process) point of view, this makes sense because it enablesa comprehensive analysis (including upstream and down-stream impacts), makes it easier to develop a water balancefor the study area, and reduces the amount of informationneeded at the borders of the study area. However, mostdecision-makers are not interested in results at river basinor watershed scale; they want to know what these resultsmean for their province, municipality or city. This conflictof scales can be solved by a well-considered selection of the(sub)watersheds that will be considered in the study and apost-processor that translates the results at these processscales into the required administrative scales.

Example 2: The information scale is smaller than the processscale

Imagine that a water resources manager wants to evaluatealternative anti-dehydration measures on the groundwaterlevel over a period of five years. Thus the requiredinformation scale is a five-year period. However, thegroundwater level is characterized by a very slow response.The relevant temporal groundwater-level process scale isaround fifteen to twenty years. Whatever managementalternative is implemented, it will take fifteen to twentyyears to determine its impact. Thus, regardless of the choiceof measuring frequency (sampling scale) and the modelscale, it is impossible within the five-year period of interestto arrive at information on the groundwater-level changes asa consequence of anti-dehydration measures.

Example 3: The sampling scale is larger than the process scaleand information scale

Assume it is necessary to estimate the change in concen-trations of certain substances in the soil and groundwaterin an urban area. The information spatial scale is one totwo decimetres. This corresponds to the spatial variationof cohesion processes that take place in the soil andgroundwater aquifer. However, logistic and budgetaryconsiderations make it impossible to increase the spatialsampling density to less than a measurement site everyfew hundred metres. As the spatial sampling scale ismuch larger than the spatial process scale, useful inter-polations cannot be made.

7. Conclusions

This chapter has reviewed some basic types of models andpresented guidelines for consideration when undertaking a modelling project. Generic models for water resourcessystem analyses are increasingly becoming available, savingmany organizations that need model results from having todevelop their own individual models. While many readers ofthis book may get involved in writing their own models,most of those involved in water resources planning and man-agement will be using existing models and analysing andpresenting their results. The information provided in thischapter is intended to help model users plan and manageeffective modelling projects as well as improve the repro-ducibility, transferability and usefulness of model results.

8. References

BECK, M.B. 1987. Water quality modeling: a review ofthe analysis of uncertainty. Water Resources Research, No. 23, pp. 1393–1442.

SCHOLTEN, H.; WAVEREN, R.H. VAN; GROOT, S.;GEER, F.C. VAN; WÖSTEN, J.H.M.; KOEZE, R.D. andNOORT, J.J. 2000. Good modelling practice in water manage-ment, proceedings hydroinformatics 2000. Cedar Rapids, Iowa,International Association for Hydraulic Research.

Additional References (Further Reading)

ABBOTT, M.B. 1999. Introducing hydroinformatics.Journal of Hydroinformatics, Vol. 1, No. 1, pp. 3–20.

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3. modelling methods for evaluating alternatives

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