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Models Quantify the Total Maximum Daily Load ProcessJoseph V. DePinto1; Paul L. Freedman, P.E.2; David M. Dilks3; and Wendy M. Larson4

Abstract: Mathematical models have been used for many years to assist in the management of water quality. The total maxiload ~TMDL ! process is no exception; models represent the means by which the assimilative capacity of a water body can beand a waste load allocation can be determined such that the assimilative capacity is not exceeded. Unfortunately, in many TMDof models has not always adhered to the best modeling practices that have been developed over the past half-century. This pawhat are felt to be the most important principles of good modeling practice relative to all of the steps in developing and applyinfor computing a TMDL. These steps include: Problem definition and setting management objectives; data synthesis for use inmodel selection; model calibration and, if possible confirmation; model application; iterative modeling; and model postaudmathematical modeling of aquatic systems is not an exact science, it is essential that these steps be fully transparent tostakeholders through comprehensive documentation of the entire process, including specification of all inputs and assumoverriding consideration is that data richness and quality govern the level of model complexity that can be applied to a given symodel should never be more complex than the data allow. Also, in applying a model, one should always attempt to quuncertainty in predictions. In general, quantifying uncertainty is easier with simple models, which is another reason to begin withframework.

DOI: 10.1061/~ASCE!0733-9372~2004!130:6~703!

CE Database subject headings: Water pollution; Water quality; Mathematical models; Wasteload allocation.

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Introduction

Mathematical models of watersheds and receiving water bare often an integral part of the total maximum daily l~TMDL ! process. All water bodies that have been placed onstate 303~d! priority water body list require the development oTMDL. The TMDL process requires the determination ofpoint source~PS! waste load and nonpoint source~NPS! loadallocations for a water body that is necessary to meet spewater quality objectives. This determination must be madethat there is a margin of safety between the allocated PS andloads and the assessment of the maximum total daily loadpollutant that the water body can receive and still meet the wquality criteria for its designated uses~i.e., its assimilative capaity!. One of the best tools available for determining the quantive relationship between pollutant sources and water qualityteria is a water quality model.

Models can serve multiple purposes in the TMDL procThey can be used to calculate watershed loads for existing

1Senior Scientist, Limno-Tech, Incorporated, 501 Avis Dr., Ann ArMI 48108.

2President, Limno-Tech, Incorporated, 501 Avis Dr., Ann Arbor,48108.

3Vice President, Limno-Tech, Incorporated, 501 Avis Dr., Ann ArMI 48108.

4Senior Project Scientist, Limno-Tech, Incorporated, 501 AvisAnn Arbor, MI 48108.

Note. Associate Editor: Ray Whittemore. Discussion open untilvember 1, 2004. Separate discussions must be submitted for indipapers. To extend the closing date by one month, a written requesbe filed with the ASCE Managing Editor. The manuscript for this pwas submitted for review and possible publication on January 7, 2approved on May 14, 2003. This paper is part of theJournal of Envi-ronmental Engineering, Vol. 130, No. 6, June 1, 2004. ©ASCE, ISS
0733-9372/2004/6-703–713/$18.00.
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ditions, relate loads to water quality response, and evaluateffectiveness of proposed control alternatives in reducingand improving water quality to meet standards. But, many inwater quality field believe that TMDL modeling is not beproperly implemented, and that improvements are needed. Ineral, the abuse of models in the TMDL process often occurcause of limited data, budget, and schedule inherent to the cTMDL process, coupled with poor training and inadequate erience on the part of model users. Recognizing these probseveral reviews and evaluations of the TMDL process ovepast several years have identified the need for improvementmodeling applications in the TMDL process~NRC/NAS 2001Water Environment Federation 2001; Limno-Tech, Inc. e2002; U.S. EPA 2002!.

This paper presents a set of guiding principles for using mels to support TMDLs. These principles have been developanswer the most basic questions a TMDL developer mayregarding the use of models, including:

• ‘‘ How do I pick a model for my TMDL, and how complex dit need to be?’’

• ‘‘ How much and what kind of data do I need to supportmodel?’’

• ‘‘ How do I apply the model and calculate a TMDL?’’This paper is not meant to be a comprehensive reference fuse of models in general, or even in the TMDL process. Theis far too broad to cover in a single section, and other hirespected books and technical guidance manuals already pthis type of information~e.g., Schnoor et al. 1987; ThomannMueller 1987; Donigan and Huber 1991; Chapra 1997; U.S.1997; Fitzpatrick et al. 2001!. Our intent is rather to provide geeral guidance that a practitioner can employ when specificonducting TMDLs. This guidance is more philosophical or cceptual. It consists of principles rather than specific technica
rection.
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Use of Models in the Total Maximum Daily LoadProcess

Models are an integral part of the TMDL process, in that tprovide a quantitative link between pollutant sources and reing water quality. One should always remember that a modemathematical abstraction of real-world processes. Its functito relate inputs to outputs: Stressors and loads to water qresponse. The simulation of natural processes in a model ceither empirical or mechanistic or a mixture of both.

The National Research Council~NRC/NAS 2001! has statethe importance of models in the TMDL development proces

Fig. 1. Application of a model to calculate a total maximum daload ~TMDL !

Fig. 2. Conceptual modeling framework

704 / JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / JUNE 200

‘‘Because they represent our scientific understanding ofstressors relate to appropriate designated uses, models playtral role in the TMDL program. Models are the means of mapredictions—not only about the TMDL required to achieve wquality standards, but also about the effectiveness of diffactions to limit pollutant sources and modify other stressoreach attainment of a designated use.’’

The use of modeling to link loading and water quality inTMDL process is simply characterized in Fig. 1, which represhow a model might define the relationship between the totallutant load and water quality response. Where the relationexactly meets the water quality objective~standard! is defined athe assimilative capacity. The model defines the maximum pant load that results in attainment of the desired water quobjective~i.e., the TMDL!. The load that is allocated to PS aNPS is generally less than the TMDL, reflecting a marginsafety. It should be noted that the functionality in Fig. 1 is oillustrative; the shape of the curve will vary depending onpollutant and the nature of the system response. Additionallyrigorous analysis, one would want an error band around the mrelationship, to establish model uncertainty and a margisafety.

The above example provides a simple description of how mels are used to calculate a TMDL. Fig. 2 provides an illustraof the conceptual linkages and data flow through the tyTMDL modeling process. Separate models can be used tolate both watershed source loads and the load-response re

e total maximum daily load~TMDL ! process
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ship in the receiving water. The watershed model can be folated over a wide range of complexity~model complexity will bediscussed later!, but requires basic hydrometeorological, hydgraphic, and land use/cover inputs as well as coefficientsallow parameterization of the process relationships used imodel. Output from the watershed model becomes input ofloads to the receiving water body model. The type and complof the receiving water body model can also vary widely, buwill have its own set of basic input~including PS loads! andparameterization requirements. The final output, however,be the suite of water quality parameters that are listed asparameters in the TMDL.

Basic Principles for Model Selection

This section addresses the question, ‘‘How do I pick the rmodel to support the development of a TMDL?’’ To help ansthis question, the TMDL developer should refer to several gmodel selection compendiums available~U.S. EPA 1997; LimnoTech, Inc. 1999; Shoemaker et al. 1999; Fitzpatrick et al. 20!.The basic principles for model selection are:• There is no one best model for all TMDLs; model selec

should be driven by an explicit consideration of:~1! management objectives;~2! site-specific characteristics; and~3! re-source constraints.

• Management questions drive desired model complexitydata constrain it.

• Select the simplest model that adequately addresses theagement objectives.

Model Selection Requisites

A wide range of models exists to support TMDL developmU.S. EPA’s Compendium~U.S. EPA 1997! lists dozens of modeapplicable to the TMDL process. No single model is applicfor all situations. The success~or failure! of TMDL developmencan depend upon the selection of the appropriate model.

To facilitate proper selection, three separate but interrefactors must be given explicit consideration:1. Management Objectives—What questions does the m

have to answer, and with what degree of precision?2. Site-Specific Characteristics—What system character

are relevant to the management objectives?3. Available Resources—What resources are available to

port the modeling?Aspects of these three important considerations are illustratFig. 3.

Consideration of Management ObjectivesClear delineation of the management objectives to be addr

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The model must be selected to fully answer the managementions posed by the TMDL. At first glance, the management otives associated with modeling in the TMDL process may sobvious. The objectives need to be specified, however, at aof detail beyond simply stating the pollutants of concern andrequired level to achieve through the TMDL process if they ainfluence the model selection process. Stating the managobjectives should also specify the error tolerance, and if posthe spatial and temporal resolution with which the pollutant tawill be assessed.

For example, the need to specify the spatial and temscales of the problem is essential~i.e., how, when, and where wwater quality standard compliance be assessed!. Different modelsprovide steady-state, temporal average, or hourly outputcoarse or refined spatial definition in one, two, or three dimsions. For example, localized hourly assessments reflectingnal variations are often needed to address dissolved oxygsues, whereas coarse-scale seasonal or peak conditions aradequate for algal/nutrient considerations. The model needscompatible with the problem and site-specific spatial and teral scales of interest.

Another important aspect of defining management objecis to specify whether NPS loads are important to the problemif so, what types and how they will be estimated. Will NPS lobe measured, specified from historical data or literature, oreled? All models cannot address all types of NPS loads antypes of proposed controls. Furthermore, the importance ofsources and how they will be specified will determine the neea watershed loading model linked to a receiving water bmodel for the analysis.

It is also important to attempt to couch the objectives ofmodeling in terms of the maximum level of uncertainty thatbe tolerated in defining the load–response relationships.specification of desired accuracy will drive the data needsassociated model complexity. The proper use of models forsion making requires not only a consideration of the output vitself, but also a firm understanding of the confidence level~un-certainty! associated with the prediction. Both consideratmust be incorporated when selecting the right model. The inity to provide TMDL results that meet desired accuracy mayquire consideration of adaptive or iterative model approa~Limno-Tech, Inc. et al. 2002!.

Other management objectives relevant to model selectioninclude the levels of review and acceptance that the modereceive. This may also include regulatory, public, and peeview. The model selected may also need to be consistent orpatible with other efforts in the watershed. Finally, the cost omodeling software and the public availability of the model an

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Consideration of Site-Specific CharacteristicsEvery watershed and receiving water is unique; the physicalacteristics and the important natural processes will differ forTMDL assessment. The unique features of the watershed aceiving water should, therefore, influence model selection. Mels are neither universally applicable to all watersheds andbodies, nor capable of describing all processes.

The impairments and potential sources affecting waterlinked in the real world through a series of cause-and-emechanisms. The intent of the watershed/receiving water moto describe the important relationships in mathematical teThis aspect of model selection requires documenting the conents and processes of concern. The recommended approadocumenting these linkages can be conducted in a three-stecess:~1! Develop detailed conceptual model;~2! estimate magntude of component processes; and~3! remove/combine minor processes.

The first step is to define all of the potential processes cposing the linkages between causes and effects. This documtion can be in the form of a simple list, or a more formal band-arrow process diagram. The above step consists of deall potentially important processes linking watershed causesreceiving water body effects.

The next two steps are designed to eliminate those procthat do not play a significant role. This is done by first estimathe order of magnitude of each of the component processescan be accomplished using available data and simple ‘‘bacthe-envelope’’ calculations. The final step, once the magnitudthe component processes has been identified, is to deleteprocesses from consideration that play an insignificant role icause–effect linkage. Spatial and temporal scales determinthe problem and system characteristics also come into playdeciding on the necessary processes for inclusion. This reconceptual model needs to be compared with the computamodel selected.

Developing this conceptual model upfront, before modelection, is important because models do not have universal acability. For example, individual watershed models are oftenter suited to urban or rural applications, with varying levelsflexibility and capabilities that need to be matched to the probSimilarly, individual receiving water models have characteristhat may map well for a river but not for a lake or estuary. Invidual process details can differ widely even in simple dissooxygen models which address algal and periphyton. It is essthat the conceptual model be developed first and then usedtool to select the best-suited computational model.

For the receiving water body, the conceptual model musdress temporal and spatial resolution, plus important variableprocesses. The watershed model must consider pollutant soland uses, and most importantly management/control optionsavailable model inventories referenced previously providereader a more detailed list of these features and capabilities

Consideration of Resource ConstraintsTMDL model applications require a variety of resources, whcan be grouped into data, time, and human resources. Site-sdata are essential to any model application, as models are ogood as the data upon which they are based. Time~schedule! isanother important consideration, as sufficient time must be mavailable to collect the necessary data, develop the modeconduct predictive scenarios. Human resources are a third imtant component, both in terms of quality and quantity. Suffic
staff time must be made available for model development and

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application. The proper application of TMDL models alsoquires significant training and judgment, making the qualitstaff an important consideration. One additional resource isget, as this can dictate the amount of data to be collected aas the quality and quantity of staff resources.

The large number of TMDLs that need to be developed ovrelatively short period of time will result in many TMDLs beiconducted with fewer resources than desired. Constraints iof the available resources will lead to limitations in the predicaccuracy of the TMDL model. There will often be an incompibility between the desired accuracy defined by the manageobjectives and the accuracy that can be delivered becausesource constraints. Methods to deal with incompatibility aredressed in the following guiding principle, as well as in the ging principles related to model application.

Questions Drive Model Complexity, but DataConstrain It

When conceptually evaluating model frameworks for TMDLis easy to list dozens of complicating factors to consider inmodel. Questions about spatial or temporal resolution, linkbetween different water quality parameters and processes, cquences of changes in watershed or environmental condiand other factors, however, can push models toward incrcomplexity. Modelers must resist this temptation, and focuthe core management issues, not merely scientific curiosity.pering the whole issue of complexity, however, is a pracconstraint—the extent of data available to support the mMathematical models and associated site-specific data hsymbiotic relationship that must be achieved in any TMDL meling application. Correct interpretation of site-specific dataefits from a modeling framework to assist in that interpretaand a site-specific model can only achieve acceptance for umanagement applications through a comparison with system

While no universal criterion for the minimum quantity of dis needed for a successful TMDL, it is safe to say that the amof data needed to achieve a desired level of reliability is lardetermined by the complexity of the model being applied. Mcomplexity is a function of three factors that comprise the deopment and application of a model: Kinetic complexity, sparesolution, and temporal resolution. A model that combinestiple individual fate processes into a single lumped loss rasimpler than a model that describes each individual procemodel that predicts lake-wide average concentrations is simthan a model that predicts how concentrations vary over sFinally, a steady-state model is less complex than a time-vamodel. In general, the more complex the model, the morenecessary for its successful application. In turn, the modelplexity is driven by the problem definition.

One essential concept in assessing whether the data arecient for the model application is that a model is only as goothe data available to support it. This is often characterized badage ‘‘garbage in, garbage out.’’ Data support refers to datheoretical understanding, process description and parametion, and field observations for calibration, confirmation, andplication. This concept is illustrated in Fig. 4, which showsincreased data are needed to support models as they becomcomplex. It is desirable to stay on the ‘‘reliable’’ side, so thathas at least as much data as needed, if not more. Often tsources~time, money, or both! available for data collectionsupport modeling are limited for TMDLs. In this case, the av
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fully applied; under conditions of limited data budget, one mask, ‘‘Can that level of model complexity answer the managemquestions being posed?’’ Of course, if there were unlimitedsources, enough data could be collected to support even thesophisticated model. That does not necessarily mean, howthat the answer would be better~as discussed in the next sectio!.

Fig. 4 appears to indicate that as long as there are enoughone can continue to increase model complexity and therebytinue to increase model reliability~as defined by increased acracy!. Experience has shown, however, that even with virtuunlimited resources, one can produce a model in which utilityreliability do not continue to increase—and may even decrwith increased complexity. This occurs when limits to the abto mathematically represent the complexity of nature are enctered. As most modelers realize, environmental models aresimplifications of the actual ecosystem being addressedtherefore, true validation of the models of natural systems ispossible~Reckhow and Chapra 1983a; Oreskes et al. 1994!. Also,budgets are rarely sufficient to support highly complex modebe used for management purposes.

Fig. 5 presents a graph that depicts the qualitative relationbetween model complexity and model utility~or reliability!.Model reliability can be thought of as the inverse of modelcertainty.

The challenge in balancing model reliability against mocomplexity is selecting the simplest model that can answemanagement questions being posed: In other words, reachioptimum model reliability on a curve that represents the avairesources. The ‘‘optimum’’ point is driven by both the manament questions being asked and the resources available to dand apply the model. As indicated, if one has more resourcescan generally answer more complex questions with greaterability. This may be necessary if the relevant management

Fig. 4. Relationship between available data and model comple

Fig. 5. Relationship between model reliability, model complexand available resources/data

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tions require the degree of reliability indicated by point B in F5. If the management questions require the degree of reliaindicated by point A, however, it does not make sense to useresources and increase model complexity to get to point Beither case, there is an optimum point on the curve~the ‘‘knee’’!where increasing model complexity adds little in terms of reliaity or utility.

Data richness and quality govern the level of model compity that can be applied to a given system. The model should nbe more complex than the data allow. Although the approplevel of model complexity is difficult to quantify and generarequires experienced professional judgment, it is always desto begin with a simple model and increase complexity only adata, and other project constraints, allow and the problem dtion ~water quality endpoints and tolerable uncertainty! demandSimple models can provide an insightful diagnostic understanof the system. Also, in applying a model, one should alwquantify the uncertainty in predictions. In general, quantifyuncertainty is easier with simple models, which is another reto begin with a simple framework.

Select the Simplest Model that Adequately AddressesManagement Objectives

TMDL models are being developed to answer managementtions and to establish a TMDL. The selection and adequacytherefore be judged by how it meets these needs, not by theof sophistication with which it describes the system. When galternative models that can address these objectives, the mshould apply ‘‘Okham’s razor’’; in the 15th Century, Okh~paraphrasing! said: ‘‘Given two theories that both explainobservation, the simpler theory is better’’; or in other wo‘‘keep it simple’’ whenever possible.

First and foremost, one should select a model with a comity ~spatial, temporal, and kinetic resolution! that is compatiblwith the available resources, and no more complex than is neto address management objectives and site-specific charatics. The best practice is to start with a simple modeling repretation of the system and increase complexity only as the pro~defined by management question, desired level of accuracsystem characteristics! demands and the available resouallow.

If one starts with the basic tenet to ‘‘start simple and incrcomplexity of the analysis only as required by managemenjectives ~including desired level of certainty! and permitted bavailable data and resources,’’ then one should end up witmost compatible model with the correct level of complexityother words, the relationship between problem definition, mcomplexity, and available data must be continually examthrough the modeling process.

Keeping the analysis and modeling simple has several atages over more complex approaches:

• Resources: Simple models consume fewer resources, wheunnecessarily complex models waste valuable time ansources if they do not provide any better information for ming the required decision.

• Understanding: Simple models provide an easier frameworunderstand the system response to assumptions. Comodels often prevent insightful and intuitive understandinthe system because of the complex interactions.

• Communication: The TMDL and its basis must be communi-


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• Parameterization: Simple models generally require fewerputs and specification of parameters. Complex modelsrequire large lists of inputs and parameters for which linformation or data are available. This can lead to more untainty, not more accuracy.

• Uncertainty: Simple models generally incorporate moreproximations than complex models. Ideally, the sophisticaof complex models~when properly supported! provides moreaccuracy. It is generally easier to define and fully understhe uncertainty of simple models, however, whereas commodels often have many assumptions with undefined coquences. As discussed later, understanding the uncertacentrally important to good decision making.

Basic Principles for Model Development

Once a specific model framework has been selected, the nexis to develop the model application to best describe site-spconditions. This section provides four guiding principles relato model development:1. Start simple and build complexity only as justified;2. Model credibility requires comparison to data;3. Model calibration is a scientific process, not a mathema

exercise; and4. Model confirmation with data should be judged qualitativ

and quantitatively.

Start Simple and Increase Complexity Only as Neededand Supported

It is almost always valuable to begin a TMDL analysis wit‘‘simple’’ model to gain a general understanding of the sysbehavior. The modeling process should include a concemodel and back-of-the-envelope calculations at the earliestof the TMDL process, and then proceed to the preliminary acation of the model selected to support the development oTMDL. These early stages of simple modeling may not be scient to conduct a TMDL; however, for many situations the inanalysis will identify uncertainties, data gaps, or inadequate slation of system behavior. The latter situation may require ational data collection or, only if necessary data are availablethe problem requires it, increased model complexity.

The benefits of starting simpler and building complexity willustrated previously in Fig. 5. It is better to begin model deopment on the left-hand side of the shown curve, moving toright-hand side with increased complexity and improreliability/utility. Starting at the right-hand side of the curve, areducing complexity to try to reach the optimum can oftenfrustrating; this is because the extreme complexity can madifficult to interpret model output and to prioritize processelump to reduce spatial or temporal resolution. A general ruthat it is better to have a robust application of a simpler mthan a weak application of a complex model. In the former,fully understands the utility and reliability; in the latter case,liability is often unknown or large, and, therefore, utility is lo

Model Credibility Requires Comparison to Data

The review of existing TMDL studies conducted as part of
research found that many, if not most, of the models applied were

done so in the absence of any formal model calibration proModel inputs were defined and predictions made for futureditions with no demonstration that the model was capable ocurately predicting water quality.

To develop a scientifically credible TMDL model applicatione must have sufficient quality and quantity of data to demstrate its ability to simulate actual observed conditions. Becacomplete theory of terrestrial and aquatic system behavior isponse to external perturbations does not yet exist, a modemechanistically captures all processes relevant to a given procannot be developed. It is important, however, to use a modeis consistent with current scientific theory. Whether this modsimple or complex, it will still have a certain degree of empcism that must be calibrated on a site-specific basis. Modelbration consists of comparing model predictions to observedand adjusting model coefficients as necessary to best deobserved conditions. The understanding of the relevant procis such that the use of literature values for model coefficientsresult in predictions that range over a wide magnitude, depeupon which literature coefficients are used. Without a demontion that model inputs are appropriate for a given site, the utainty in model predictions will be large.

The degree of demonstration necessary depends on the nof model inputs that have to be estimated. A river dilution etion or an empirical lake model may have no coefficientsneed to be calibrated, and will require a much lower levedemonstration than a more complex model. But these simodels often will have much less predictive power~i.e., management utility! than more complex models.

This principle covers the most remedial aspect of modelbration by requiring some type of reality check to demonsthat the model is capable of describing the system observaThe following principles focus more on the calibration procitself.

Model Calibration is a Scientific Process, Not aMathematical Exercise

Model calibration involves adjusting process formulations andefficients until a model is able to reproduce site-specific obstions. Chapra~1997! provides a schematic of the model calibtion process as depicted in Fig. 6. As indicated in this diagonce a model has been selected, a coordinated data collprogram is undertaken to obtain a calibration data set. Thisdata collection program should be designed to simultaneoustain site-specific physical data, forcing function data, neceprocess-related data, and spatial and temporal profilesmodel state variables~dependent variables!. The calibration datset should be as similar as possible to design conditions foproblem being studied. For example, for a dissolved oxTMDL, the design condition might be for summer low flow inriver. Consequently, the calibration data set should be collduring warm low-flow conditions. Also, a statistical critershould be established for the acceptable difference betweemodel computation and the measurement.

While the design of a calibration data set depends on thequality problem and parameters of concern, some general caries of data should be included in all calibration data sets. Tare, by category:

• Physical data to set up the model configurations,• External loading/environmental data to drive the model

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• Ambient system response data on loadings and concentrto compare against model output.With a coherent complete calibration data set that is cons

with the resolution of the model, the focus of the calibration tbecomes the adjustment of the coefficients~within an acceptablrange dictated by the field and laboratory literature values! untilthe difference between model computations and measuredvariables is within acceptable tolerance.

Modelers are strongly discouraged from arbitrary adjustmof coefficients to improve a point-by-point fit and relative statical performance. Without scientific justification, this becommerely a process in curve fitting. The result is a model withappearance of high precision but actually unknown predictivliability. It is better to use a logical and consistent set of cocients and accept model-to-data discrepancies. These disccies should be quantified and used as a basis to quantifdegree of model uncertainty/precision.

The process of adjusting the model coefficients to calibradata is an art with scientific foundations. A few consideratthat should be kept in mind are as follows:• Define as many coefficients independently as possible, an

to leave these fixed while other coefficients are adjusted.• Always a priori establish an expected range for each co

cient that is to be adjusted in calibration. The range shoubased on scientific experience from laboratory and otherstudies. Avoid deviating from the range.

• To the extent possible, justify coefficient adjustments baselogical scientific observations or theory, qualitative ifquantitative.

• Do not arbitrarily adjust coefficients spatially and/or temrally merely to improve the model performance without local justification. This is curve fitting, not calibration.The following principle provides guidance on evaluating

Fig. 6. Schematic of the mode

quality of a site-specific calibration.

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Model Confirmation with Data Should be JudgedQualitatively and Quantitatively

Model reliability is best judged by comparison to data. Thioften termed model confirmation or validation. The terms mcalibration or verification are also used in this context, referrina process of adjusting model parameters and then a comparimodel results to data. Whatever terminology is used, it is essthat model credibility be supported by some comparison to adata. Most complex models are underdetermined; they comore coefficients than state variables. Multiple sets of calibrparameters can therefore provide results that are roughly ctent with the observed data. It is often possible to follow a sctific calibration approach as described above, and still be fwith difficult decisions on how to judge the quality of the cbration and how to select which specific set of parameters tfor the final calibration.

The traditional approach to assess the validity of a calibrwas to present a graphical comparison of model predictionsrely on the judgment of the modeler to select the set of cocients that ‘‘best’’ described the observed data. This approacthe limitation of being primarily qualitative, and not providingquantitative assessment of model performance. Furthersimple graphical comparisons are often not feasible when dewith multidimensional time-variable models. Thomann~1982!and Reckhow and Chapra~1983a,b! called for the use of moquantitative measures to assess model performance, and deseveral statistical techniques that could be used to assessperformance. These include measures such as regression aor calculation of relative error. Parameter optimization routhave been developed that can define parameter values thavide the best comparison to a given quantitative performmeasure.

While the calibration process should be done in a quantitand systematic fashion, there is often considerable profes

ration process~from Chapra 1997!
l calib
judgment involved in obtaining a satisfactory model calibration.


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There is as much ‘‘art’’ as ‘‘science’’ in model calibration. Aumated error-minimization techniques can provide some guidbut they almost never derive the best calibration data set beit is often more important to describe some aspects of theserved data than others. It is difficult to define a priori, oquantitative terms, the specific importance of capturing varaspects of the observed data.

As a result, model practitioners must avoid ‘‘curve fitting’’ afocus on observing if the model performs consistent with theand scientific understanding. Some considerations which arportant to examine include:• Does the model respond consistently with the theory and• Are the spatial trends in the model and data consistent?• Are the temporal variations in the model and data consis• Are the timing and magnitude of peaks and valleys in

model and data similar?Anomalous data, and high environmental variability, can o

confuse the model calibration process if governed by statisassessments alone or a point-by-point focus. Thus, consistemagnitude and trends is equally as important as the statiresiduals. In the end, the adequacy of the calibration mujudged by a weight of evidence approach considering quantitmeasures as well as qualitative characterizations and explan~Reckhow et al. 1980!.

As a closing note on model calibration, we have contenbefore that models are only approximations of nature; they rreflect the natural process with complete accuracy and anever fully reflect the significant variability that exists in natuSo do not be discouraged by isolated discrepancies or uncasmall-scale variability. Paraphrasing Aristotle: ‘‘The mark ofeducated mind is to be satisfied with an approximate answerthat is all that the nature of the problem permits.’’

Ideally, before a calibrated model is applied to address TMmanagement questions, it undergoes a confirmation proModel confirmation is conducted by applying the calibramodel to a new data set for the system of interest~or ideally, morethan one data set collected under a range of conditions! that wasideally collected under a set of forcing conditions~i.e., loadsflows! as different as possible from those used for the calibraFor example, one will have more confidence in applying a mfor making management decisions if it simulates field obsetions well under both low and high flow conditions withoutjusting model coefficients. To accomplish this, the model simtion for the confirmation period is compared against the inobservations made during this period,without changing anmodel coefficients from those determined in the calibrationcess. If the error associated with the model simulation ofconfirmation data set is no larger than that for the calibraperiod, then the model has been ‘‘confirmed’’ as an effectivediction tool for the range of conditions observed. In fact,uncertainty analysis on calibration and confirmation data setrecommended means of determining a margin of safety foTMDL process. If the confirmation process yields significalarger error than for the calibration data set, it may be neceto iterate it again through the model building/selection ormodel calibration process.

It is recognized that model confirmation is almost never din the TMDL process because of cost and time constraints.being the case, the TMDL practitioner should acknowledgethe uncertainty determined using the calibration data set maderestimate the uncertainty associated with application o
model to a different set of loads that might be used in determining

.

assimilative capacity or in evaluating alternative load reducscenarios.

Principles for Management Application of Models

Once the model has been developed, the next step is to appdefine the TMDL. Four guiding principles are provided inregard:1. Model scenarios for TMDLs must incorporate realistic c

ditions, not idealized assumptions;2. Model sensitivity and diagnostics are essential to confi

model use;3. Using model predictions without a defined uncertainty i

responsible; and4. When resource constraints are an issue, consider an ad

approach.

Model Scenarios for Total Maximum Daily Loads MustIncorporate Realistic Conditions, Not IdealizedAssumptions

The TMDL program is designed to establish loading reducand management conditions that will lead to compliancewater quality standards. It is therefore essential that the minputs be selected to represent realistic conditions, assumfor background loadings, and realistic expectations for futurand NPS loads.

It is not uncommon for inexperienced practioners to mquick and overly conservative choices for critical conditiocompounding the effects of various individual critical assutions. The selection of overly critical conditions and/or baground loadings can result in an unfair and/or unattainTMDL, while the opposite can lead to a TMDL that will nachieve water quality standard attainment. The responsibletitioner selects a critical condition for the TMDL model simution that matches well with the duration and frequency attribof the standard, as well as providing a realistic representatithe watershed. The same needs to be held true for backgconditions. Irresponsible or poor choices for backgroundcritical conditions can easily undermine and discredit modevelopment efforts.

Modeling projects often wrongly focus more on developmthan on applications. The forecasts can often be conductedquick and arbitrary assumptions, when faced with a presschedule delayed by prolonged efforts to develop a more remodel. It is important, however, to reserve significant schetime and budget costs for these final TMDL forecasts. As mthought needs to go into these simulations as went into the mdevelopment. Unreliable assumptions in these forecasts cadermine the credibility of even an exceptional model.

Model Sensitivity and Diagnostics Are Essential toConfident Model Use

While the primary use of the model is to define the totalrequired to attain water quality standards, it may also be usprovide insight on important environmental processes and thtimal means to reduce loads. Thorough model sensitivity anaand additional model diagnostic simulations are essentialthat a modeler must undertake to ensure they have confidetheir results and recommendations. Through sensitivity anaand component analyses, the modeler can gain insight to the
tive importance of different model assumptions and environmen-
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tal factors. This provides insights into one’s confidence inmodel results and which topic areas may need additional inor investigation.

The model can also be used to generate a load–responserepresenting different levels of PS and NPS controls. Througanalysis of these simulations, the modeler can establish thetive effectiveness of various proposed controls. An exampleload–response plot for a steady-state linear model is showFig. 7~a!. From this plot, one can then determine the maximallowable load to achieve a given ambient water quality targedetermining the load on the plot corresponding to the targetbient concentration. Fig. 7~b! illustrates how the load responmight look if only PS loads are adjusted while holding the Nloads constant. As Fig. 7 illustrates, there may be a point ominishing returns~a ‘‘knee-of-the-curve’’! below which furthereductions of the PS load will not accomplish water qualityprovements in the system. The point of diminishing returns itrated in Fig. 7~b! can be caused by the dominance of unconlable sources, background loads, irreversible alterations, ornatural stresses as point sources are reduced to low levels.

The TMDL practitioner must be careful in using these moderived load–response plots. First, the plots presented in Frepresent a steady-state view of the response of the systethere is, for example, a significant reservoir of the pollutanconcern in the sediments of the system, there could be ensediment feedback to delay the approach of the system to sstate for a long time~years to decades, in some cases!. In thiscase, one should consider developing a series of load–rescurves that reflect how the system responds to external loaductions over time. Because of sediment feedback, there mvery little response to a point source load reduction in theseveral years after implementation. At the very least, this isable information in planning follow-up monitoring.

The results of the sensitivity and diagnostic analyses, suthose discussed above, need to be summarized and reportethe model findings to allow the decision maker a full understing of the responsiveness of the model and the sensitivity oresults to different assumptions or choices.

Using Model Predictions without Defining theirUncertainty is Irresponsible

Models are inherently approximations of nature. In this regais irresponsible to use model results as absolute to make retory or watershed management decisions. Since the mode

Fig. 7. Example of steady-state load–response curve for

approximations, they have an inherent range of uncertainty that

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must be defined and considered when calculating the TMDLmaking recommendations for controls. The NRC Expert TMPanel~NRC/NAS 2001! was explicit in citing the need for uncetainty analyses, stating that uncertainty analyses need to bestandard components in TMDL modeling studies.

The issue of model uncertainty cannot be overemphasizea minimum, model uncertainty can be estimated from statistests employed in establishing model calibration and throughsitivity analysis. More rigorous methods are available and pticed as discussed elsewhere. The bottom line is that any aption of a model must not only provide a forecasted result,must also include a characterization of the degree of uncert

When Resource Constraints are an Issue, Consider anAdaptive Approach

As described previously, model reliability and accuracy oftehand in hand with the need for more resources, more datapossibly more model complexity. Unfortunately, many TMprojects do not have the adequate resources to achieve thThe result may be a model that has significant uncertainty.

The most common problem is a lack of adequate data. Tmodel development requires significantly more data than thtial 303~d! listing process. Defining impairment requires only don water conditions, whereas model development requires da wide range of factors including loading, environmental cotions, and numerous other important impacts. One telling exaof these problems was highlighted in a study by the Generacounting Office GAO~2000! that found that only three states fthey had sufficient NPS data to support TMDL developmWithout data, models can be impotent and dangerous inpotential inaccuracy.

Faced with limited resources, the TMDL analyst has a chof a simple model well done or a complex model poorly sported. The recommendation as stated previously is that it ister to start simple and later build complexity as data and resobecome more available. A robust application of a simple modmuch better than the weak application of a complex model.

Simple models may have limited accuracy because of sor temporal averaging and simplifications of system procebut they still offer advantages. First, it is often easy to definerange of uncertainty in these models, which can be used todecision making. Further, the models typically provide sufficinsight to select priority controls, estimate a TMDL, and gu

tment total load~a! and for adjustment of point source load only~b!
adjus
needs for additional data and model complexity. Full TMDL


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model development then proceeds on an adaptive or iterativecess.

This is consistent with the concept ofiterative modelingre-cently recommended by the National Research Council~NRC/NAS 2001!. In this approach, one begins with a simple screemodel that is less complex than the optimum. Then, with ational data collection, one can impose the more obvious conand continue to increase model complexity and reliability topoint where the TMDL can be completed.

The iterative modelingprocess does not necessarily requicomplete progression to the most complex model. In many cthe initial controls implemented by this process may be sufficto attain water quality standards and to achieve the TMDL wout using a more complex model. This adaptive approachreduces some of the concerns related to model uncertainty amargin of safety. By incrementally increasing the level of magement controls, the observed water quality response willvide a direct indication of when sufficient load reductions hbeen implemented~i.e., reduce loads incrementally until waquality standards are achieved!.

Iterative or adaptive modelingis necessary if a given modapplication does not fully address the load allocation questionthe system or produces an application that demands a marsafety that is too high relative to the objectives set forth inproblem definition. There are several points to which oneiterate in this process. First, one can return to model selectioincrease model complexity to reduce uncertainty and adhigher-resolution questions. This would, of course, require rlibration and most likely additional higher-resolution data to cduct that recalibration. It is also possible to strategically coadditional data to reduce the MOS with the existing simmodel framework. If project constraints preclude the collectioadditional data and/or increasing model complexity, then itbe necessary to apply the existing framework with the exislevel of uncertainty~margin of safety! to initiate the most obvioulevel of controls. This should, however, be followed by moniing the response of the system to these initial controls withgoal of postauditing the model to reduce uncertainty and, thethe necessary margin of safety. In summary, TMDL modelingoften be viewed as an evolutionary/adaptive process, not aone-shot effort.

Full Disclosure in Total Maximum Daily LoadModelingA final guiding principle that covers the entire TMDL modelprocess is that the full disclosure of modeling details leadbetter credibility and utility. Modeling combines both art andence, and typically involves many assumptions. It is thereessential that model development and use be fully transpthrough comprehensive documentation of all steps in the ming process. ‘‘Black box’’ modeling, wherein only selected inpand results are provided, is inappropriate for TMDL applicatModel credibility is best enhanced by fully communicatingnature and limitations of the modeling. This is best facilitatedcomprehensive documentation, including clear descriptionsaspects of the TMDL modeling process:• Problem definition and conceptual model development,• Data syntheses,• Model selection and justification,• Model theoretical formulations,• Model site-specific configuration and key assumptions,
• Model calibration and confirmation strategy and results,

• Model application approach and results,• Model sensitivity and uncertainty analysis, and• Limitations and recommended improvements.

This will allow stakeholders and decision makers to fullyderstand the utility and limitations of the model. TMDLs ofinvolve costly or imposing recommendations. These decisionbest made when one fully understands the reliability and utainty of modeling tools used to support the decision. The rtion of model results is often more a result of poor communtion and limited disclosure than poor analysis.

Model documentation should be simple enough to conveyassumptions and important limitations to the nonmodelerssupplemented with enough detail so that model reviewerscritically evaluate the effort or, if not, duplicate it. A good gufor the type of information sought by peer reviewers of envimental regulatory models is provided by EPA on their SciePolicy Council web page ~http://www.epa.gov/osp/spmodelpr.htm!.

Acknowledgments

The work described in this paper was part of a larger reseeffort funded by the Water Environment Research Found~WERF! entitled ‘‘Navigating the TMDL Process Evaluation aImprovements’’ Limno-Tech, Incorporated et al.~2003!. The ef-forts of scientists for the WERF document were also aideexpert members of a Project Steering Committee identified iWERF report who invested significant effort on this topicprovided ideas, information, and critical review.

References

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Models Quantify the Total Maximum Daily Load Process

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