Reprinted from the Journal of Environmental QualityVol. 4, no. 1, Jan.-Mar. 1975, Copyright © 1975, ASA, CSSA, SSSA

677 South Segoe Rd., Madison, Wis. 53711 USA

The Watershed Approach to Understanding our Environment

David A. Woolhiser

Purchased byUSDA Agric. Research Servitor Official Use.. _ ica

The Watershed Approach to Understanding our Environment1

David A. Woolhiser2

ABSTRACT

Current approaches used in modeling a particular part of thehuman environment—the agricultural watershed—are reviewed andquestions are raised regarding the objectives, approaches, and interpretation of agricultural water quality models. Models-eithersymbolic (mathematical) or material—are essential to understanding and predicting environmental phenomena on agricultural watersheds. Models describing the transport of water, sediment, andchemicals through a watershed system can become very complicated and frequently must be simplified. Simplifications inevitablyinvolve distortion and may make interpretation of model parameters difficult. The use of material models may assist in interpreting the parameters of mathematical models.

Additional Index Words: modeling, simulation, water quality,hydrology.

What do we mean when we speak of understanding ourenvironment? Hempel (1963) suggests that we understand anevent or a regularity if we can give a scientific explanation of it. If we accept this notion it isobvious that"understanding our environment" is one of the commonly accepted goals of all science, and that when we speakof "the watershedapproach," we are considering a limitedportion of the environment.

In common usage, the term environmental problemsincludes air pollution, water pollution, soil erosion, anddestruction of scenic beauty, and therefore refers to certainparts of our surroundings that are strongly influencedby man. Environmental problems involve economic externalities—situations where the provision of goods orservices for an individual or group affects the welfare ofothers.

We have long recognized that erosion of agriculturalsoils has substantial effects on people downstream. Thisproblem has been studied for a long time, no doubt because soil erosion adversely affects farmers as well. Associety has become more aware of the quality of surfaceand ground water and of adverse effects of certain substances on the health of man and other organisms and onesthetic values, questions have been raised about the effect of fertilizers and pesticides on downstream waterquality. Because nutrients or pesticides may be trans-

1Contribution from Agricultural Research Service, USDA, in cooperation with the Colorado State University Agr. Exp. Sta. Received 12 Feb. 1974.

2Research Hydraulic Engineer, USDA, Fort Collins, CO 80523.

ported by running water or may be adsorbed by sedimenttransported by runoff, these questions can best beanswered through the use of hydrologic models. Thewatershed is an appropriate area! element to consider forhydrologic models because all uncontrolled surface waterflux out of the system is zero except at the stream draining it.

The purpose of this paper is to briefly review currentapproaches used in modelingthe particular part of humanenvironment known as the agricultural watershed and toraise some questions regarding the objectives, the approaches, and the interpretation of agricultural waterquality models.

THE USE OF MODELS IN SCIENCE

Manydiscussions of modeling in general and hydrologicmodeling in particular have appeared in the literature inthe last decade. However, I feel that the rationale formodel building expressed by Rosenblueth and Wiener(1945, p. 316) is superior to more recent statements

No substantial part of the universe is so simple thatit can be grasped and controlled without abstraction.Abstraction consists in replacing the part of the universe under consideration by a model of similar butsimpler structure. Models, formal or intellectual onthe one hand, or material on the other, are thus acentral necessity of scientific procedure.

A formal model is a symbolic, usually mathematical,representation of an idealized situation which has the important structural properties of the real system. A material model is a physical representation of a complex systemby a model system that is assumed to be simpler and isalso assumed to have some properties similar to those ofthe prototype system. One can kick a material model; aformal model, being an abstract entity, cannot be kicked.

Models assist us in understanding or explaining naturalphenomena and may also allow predictions to be madeunder certain circumstances. But what do we mean whenwe say we have explained something? The essence ofHempel's definition of an explanation follows: Supposewe have a statement E which describes some phenomenonto be explained. Then if E can be inferred from a set LuL2 . . . Ln of general laws or theoretical principles and aset Cu C2 . . . Cm of statements of empirical circum-

J. Environ. Qua!., Vol. 4, no. 1, 1975 17

stances, we say that the phenomenon has been explained.Explanation can be either deductive-nomological or inductive-probabilistic (Hempel, 1963). A deductive-nomological explanation is based on deterministic laws,whereas an inductive-probabilistic explanation involves atleast one statistical law. The laws and statements of empirical circumstance required as part of scientific explanation are the formal or symbolic models of Rosenbluethand Wiener.

Formal models also allow certain predictive statementsto be made. A deductive-nomological (deterministic)model allows only predictions with certainty; an inductive-probabilistic (stochastic) model allows only predictions in the form of probability statements.

A distinction is sometimes made between theoretical

and empirical models. A theoretical model includes botha set of general laws and a set of statements of empiricalcircumstances; an empirical model allows predictionsbased only on a set of particular data. This distinctionbecomes less clear when we consider a model that includessome but not all of the necessary general laws.

A GENERAL DISTRIBUTED WATERSHED MODEL

For purposes of discussion, let us visualize a watershedsystem as shown in Fig. 1. This open system is boundedby impervious rock on the bottom, by the surface 5 onthe sides and by the surface A (which may or may not coincide with the soil surface) on top. If wc knew the fluxof water in liquid and vapor form, and the volumetricmoisture content at all points within this volume as afunction of time, wc could answer Penman's (1961) question and definition of hvdro'rtOV "Wliai !»»r»op»»* tn thp

rain?" Vertical flux through the surface A consists ofprecipitation (positive) and evapotranspiration (negative)

Water

Table

V*.y.*." ^

Fig. 1—Schematic drawing of watershed as a distributed system(From Woolhiser, 1971).

18 J. Environ. Qual., Vol. 4, no. 1, 1975

and is designated as %(x,y,t). This flux can best be described as a stochastic process.. The saturated and unsaturated flow of water in a direction normal to the surface 5 is designated r\(x,y,z,t). Surface streamflow fromthe system is more concentrated than the other fluxes soit will be considered as the point process $x(t) and themass rate of sediment transport will be designated f 2(t).Imported water, always a possibility, will be representedas the process oi(t). The volumetric moisture contentwithin the soil or porous rock is d(x,y,z,t). In some circumstances it may be necessary to include the temperature within the system as a fundamental dependent variable. We assume that one of the fundamental lawsgoverning this system is the conservation of mass; therefore, we can integrate the fluxes over the appropriatedomains and equate the net flux to the time rate ofchange of integrated storage to obtain the continuityequation.

Finally, consider some substance i distributed over asubarea a{ at a rate of Ty per unit area. The total mass ofthis substance contained in a unit spatial volume isv{(x,y,z,t). If this material can be adsorbed by soil particles or ingested by living organisms, the total quantityper unit volume must be subdivided into appropriateparts. Chemical reactions must also be accounted for.The mass rate of transport of substance i in the stream is

U(t) = pci(t) U(t) + 4> [U(tJ\; • - 3,4 ... [ i ]

where p is the density of water, c{(t) is the concentrationof substance i in streamflow and ^[fjftj] represents theamount of material carried by sediment. Figure 1 and thesubsequent definitions suggest that water transport canbe described by considering the porous medium as a continuum and utilizing the partial differential equationsrepresenting three-dimensional saturated-unsaturated flowcoupled with equations of unsteady free-surface flow.Freeze (1972a, 1972b) has developed a computer program to solve these equations by finite difference techniques for three-dimensional porous media flow and one-dimensional open-channel flow. The movement of chemicals through porous media by water and the chemical reactions that occur during movement can also be describedby partial differential equations. (See Boast, 1973, for arecent review.) These equations can also be solved byfinite difference techniques in conjunction with solutionsfor unsteady flow of water. (Reddell and Sunada, 1970.)Similar methods can be used to describe the transport ofdissolved substances by unsteady, free-surface flow.(Bella and Dobbins, 1968; Domhclm and Woolhiser,1968; Dresnack and Dobbins, 1968; Harleman, Lee, andHall, 1968; and Fischer, 1969.) The mathematical description of detachment, transport, and deposition of soilparticles has not been as fully developed as those forwater and chemical transport because it is an extremelycomplex phenomenon. However, Meyer and Wischmeier(1969) have developed mathematical models that accountfor many of the important aspects of the erosion process.Models of unsteady sediment transport in rivers andstreams are being developed.

All of the models cited deal with the physics of fluidmotion and chemical dispersion and read ion, but all

either ignore or treat indirectly the effects of living organisms. The amount of water transpired by plants isusually the largest output of the watershed system andobviously must be included in a comprehensive transportmodel. Furthermore, plants and animals have importantinfluences on the transport of chemicals within the watershed system and must be included in models of the system. Models presently used for estimating evapotranspira-tion consist of a continuity relationship, a method ofcomputing potential evapotranspiration,£0,and a methodof computing actual evapotranspiration as a function ofE0, soil moisture content, and stage of vegetative growth.If we consider the fact that the computer program for thesubsurface portion of Freeze's (1972a) model strains thecapacity of the largest present generation of computers, itappears that a numerical model incorporating evapo-transpiration, chemical reaction, and transport mechanisms, plus biological mechanisms at this degree of detailwould be completely intractable. Parameter estimationand determination of initial conditions would also be prohibitively expensive.

It is evident that if we wish to develop a mathematicalmodel to describe transport of water, sediment, and chemicals within and out of an agricultural watershed, we mustsimplify our approach. To use H. T. Odum's (1971) picturesque terminology, we must look at the system througha macroscope—z. device that eliminates detail but retainsthe essential structure. But how do we design such amacroscope? What is the essential structure? Do we commit some sort of scientific sin if we deviate from the"rigorous" methods?

OBJECTIVES OF WATERSHED MODELS

At this point we must consider the purpose of ourmodeling activity. Our purpose is not to develop evermore complex models of the watershed system. Rather,our models (which may be interpreted to include predictions made by them) represent our descriptions of howwater, sediment, and chemicals move on watersheds underexisting or proposed circumstances. Individuals or groupsthen evaluate possible courses of action whose consequences are described by our models and decide upon the"best" option. Evaluation is possible only after someonehas described the process. If the description is in error,the evaluation is sure to be wrong. However, scientificexpertise does not enable one to make any "ought to" or"should" statements. This doesn't mean that scientists

shouldn't take part in public decisionmaking; it merelymeans that they are not uniquely qualified to do so.Scientists are certainly uniquely qualified to evaluate themodels themselves.

If models are too complicated, they may provide muchmore information than is necessary for a decision. Decision models include components derived from the socialscience of economics and usually involve problems of optimization. The objective function (which is to be maximized by an appropriate choice of design variables) isstated in economic terms and usually requires condensedinformation about the outputs from a system.

Consider two sample functions of an output processthat might be obtained by simulation from a watershed

C2(»)C'(t>

-£jU) Before Treotment

C'(t) After Treatment

Fig. 2—Sample functions of sediment transport rates with and without watershed modification (From Woolhiser. 1371).

transport model. Sample functions of the processes$i(t) and £'2(t) are shown in Fig. 2 where J2f£) is the massrate of sediment transport out of the watershed underpresent conditions and $2(t) 's tne sediment transport rateunder a proposed condition. Now a decision-maker mightsay, "I can't possibly use information of that detail. All Ican use is the expected annual sediment yield under bothconditions." The modeler must than decide if he can ob

tain estimates of the mean sediment yields with a simplermodel.

If the decision-makers ask, "If substance i is spread ona particular field within the watershed, will some of it getto the stream?", the modeler may be able to answer "Yes"without any further calculations. However, he might alsouse model results to point out that the question may betrivial. Almost anything we do on the watershed will affect streamflow quantity or quality. The pertinent question is "How much?"

MODEL SIMPLIFICATION

Models consisting of sets of partial differential equations can be simplified by reducing their dimensionality,i.e., by treating the streamflow as a problem in one-spacedimension, or by ignoring terms that are relatively unimportant. Examples include linearizing nonlinear equations(Dooge and Harley, 1967), using the kinematic approximation for overland flow (Woolhiser and Liggett, 1967),or neglecting molecular diffusion and hydrodynamic dispersion in transport of chemicals through porous media(Nelson and Eliason, 1966).

Another simplification can be made by recognizing thetremendous difference in response time between surfacewater systems and flow through porous media. In manycases this means that ground-water flow can be treated asa steady-state problem, or as a series of steady-state problems.

Perhaps the greatest simplification is to eliminate spatialvariability by using "lumped" or compartment models.The Stanford Model (Crawford and Linsley, 1962) is anexcellent example of this approach in hydrology. Thepartial differential equations of the "distributed" systemsare replaced by ordinary differential equations and themathematical difficulty as well as the computer storagerequirements and computation time are greatly reduced.

J. Environ. Qual., Vol. 4, no. 1,1975 19

Lumped models can be considered as abstractions of thedistributed model that hopefully retain some importantcharacteristics. As a further abstraction, we might represent the watershed as a "general system" where the output (runoff, sediment transport, chemical transport) is assumed to be related to the input (precipitation and inputsof chemicals), but where no explicit assumptions havebeen made regarding the internal structure of the system.Nonlinear systems theory was first applied in hydrologyby Amorocho and Orlob (1961), and several other hy-drologists have worked on this problem in recent years[Amorocho (1963), (1967); Jacoby (1966); Bidwell(1971); Amorocho and Brandstetter (1971)]. In thegeneral systems approach (linear or nonlinear), the systemparameters are estimated by minimizing some function ofthe observed and computed output sequences, given theinput sequence. Obviously this approach cannot be usedif we propose to change the system in some way. Thephysical significance of the parameters in lumpedsystemmodels also tends to be obscure; therefore, unless reliablecorrelations have been found between parameters andmeasurable entities in the real world system, it may bemisleading to use this type of model for predicting the effects of changes in the system.

In this discussion I have used the words distributed,lumped, and general system as if any model can be easilyplaced in one of these categories. It is more accurate toconsider these classifications as successively higher orderabstractions of the real system. Although the equationswe start with are continuous, we usually resort to discretemethods to solve them. Therefore, the distributed modelsinvolve lumped parameters. The lumped models andgeneral system models are much more general than afinite-difference model of overland flow, for example.This generality is obtained at a cost of a less detailed description of the system.

MATERIAL MODELS

So far I have considered the application of symbolic(mathematical models) to environmental problems of agricultural watersheds. Do material models have any application?

Material models may be subdivided into iconic or "lookalike;" models, and analog models. The validity of ananalog model depends on the existence of identicalmathematical relationships describing both the real systemand its analog, and so depends on symbolic models.Iconic models are simplified versions of real-world systemsthat are more convenient to work with. Laboratory setups can be considered as iconic models, and have playedan important role in modeling the transport of environmental pollutants by water.

To be useful, material models must be easier to workwith than the real system and must provide some information that is not already incorporated in mathematicalmodels. One of the fundamental problems involved inmodeling agricultural watersheds is the question of interpretation of parameters in lumped system models. It appears to me that large material models with simple geometry as compared to real watersheds but involving three-dimensional flow in a porous medium as well as surface

20 J. Environ. Qual., Vol. 4, no. 1,1975

runoff, could give us a great deal of insight into the problem of interpretation. Nonhomogeneities or spatial variability could also be included in these models. Successively simpler mathematical models describing this systemcould be formulated. These models would first includespatial variability and then would be modified so thatsuch variability would be averaged in some sense. Ifmodel parameters were shown to be ambiguous for asimple system, one might suspect that they would not bereliable in more complex systems.

DISCUSSION

Environmental problems invariably involve economicexternalities—situations where the provision of goods forone group of individuals affects the welfare of anothergroup. In evaluating the external costs or benefits ofsome agricultural practice, we must ask the questions:What? How much? When? and Where?3 To answer thesequestions we must utilize a mathematical model of thewatershed if water is an important transporting medium.Rather complex models may be required to understandthe important processes involved. Models used in decision-making will include components derived from thesocial sciences as well as the physical, chemical, and biological sciences. Therefore, they will include only themost important of the subprocesses involved or they willbecome unduly cumbersome. We must be very carefulwhen a model that was developed for one purpose is usedin another application. For example, a watershedmodeldeveloped for predicting water quantity may performvery well for that purpose but may be misleading if usedas the basis for a chemical transport model. In the firstapplication, the water flow path may be the subject ofheated arguments (i.e., interflow vs. surface runoff) butdoesn't really affect predictive capability; in the secondapplication, the flow path is of crucial importance.

As modeling efforts proceed, we must continuallyquestion our procedures if we are to make real progress.Can we say in any objective way that model B is betterthan model A? Has our simplification of the model structure resulted in a complicated curve-fitting device withlittle structural relationship to the real world system? Canmaterial models be useful in interpreting the parametersof mathematical models? The answers to these questionsare not obvious. Hopefully, research will answer some ofthem or will raise other, more important questions.

LITERATURE CITED

1. Amorocho, J. 1963. Measures of the linearity of hydrologicsystems. J. Geophys. Res. 68(8):2237-2249.

2. Amorocho, J. 1967. Nonlinear prediction problem in thestudy of the runoff cycle. Water Resour. Res. 3:861-880.

3. Amorocho, J., and A. Brandstetter. 1971. Determination ofnonlinear functional response functions in rainfall-runoff processes. Water Resour. Res. 7:1087-1101.

4. Amorocho, J., and G. T. Orlob. 1961. Nonlinear analysis ofhydrologic systems. Water Resour. Center Contrib. 40. Univ.of Calif., Los Angeles.

3R. William Nelson (unpublished manuscript) hassuggested thatthese questions should be answered in environmental impact state-

5. Bella, D. A., and W. E. Dobbins. 1968. Difference modelingof stream pollution. Proc, J. Sanit. Eng. Div. ASCE. 94(5):995-1016.

6. Bidwcll, V. J. 1971. Regression analysis of nonlinear catchment systems. Water Resour. Res. 7:1118-1126.

7. Boast, C. W. 1973. Modeling the movement of chemicals insoils by water. Soil Sci. 115:224-230.

8. Crawford, N. H., and R. K. Linsley. 1962. Synthesis of continuous streamflow hydrographs on a digital computer. Stanford University Dept. Civil Eng. Tech. Rep. 12. 121 p.

9. Dooge, J. C. I., and B. M. Harley. 1967. Linear routing inuniform open channels. Proc. Int. Hydrol. Symp. Ft. Collins,Colorado. 1:57-63.

10. Dornhelm, R. B., and D. A. Woolhiser. 1968. Digital simulation of estuarine water quality. Water Resour. Res. 4:1317-1327.

11. Dresnack, R., and W. E. Dobbins. 1968. Numerical analysisof BOD and DO profiles. Proc. ASCE, J. Sanit. Eng. Div.p. 789-807.

12. Fischer, H. B. 1969. A Lagrangian method for predictingpollutant dispersion in Bolinas Lagoon, California. USGSOpen-File Report. Menlo, Calif.

13. Freeze, R. A. 1972a. Role of subsurface flow in generatingsurface runoff. I. Base flow contributions to channel flow.Water Resour. Res. 8:609-623.

14. Freeze, R. A. 1972b. Role of subsurface flow in generatingsurface runoff 2. Upstream source areas. Water Resour. Res.8:1272-1283.

15. Harleman, D. R. F., Chok-Hung Lee, and L. C. Hall. 1968.

Numerical studies of unsteady dispersion in estuaries. Proc.ASCE, J. Sanit. Eng. Div. p. 897-911.

16. Hempel, C. G. 1963. Explanation and prediction by coveringlaws. p. 107-133. In B. Baumrin (ed.) Philosophy of science:The Delaware Seminar. Intersciehce. John Wiley & Sons,New York &: London.

17. Jacoby, S. L. S. 1966. A mathematical model for nonlinearhydrologic systems. J. Geophys. Res. 71(20):4811-4824.

18. Meyer, L. D., and W. H. Wischmeier. 1969. Mathematicalsimulation of the process of soil erosion by water. Trans.ASAE 12(6):754-758.

19. Nelson, R. W., and J. R. Eliason. 1966. Prediction of watermovement through soils—a first step in waste transport analysis. Proc. 21st Industrial Waste Conf., Purdue Univ. Eng.Ext. Ser. 121, pt. 2. p. 744-758.

20. Odum, H. T. 1971. Environment, power and society. Inter-science. John Wiley 8c Sons, New York. 331 p.

21. Penman, H. L. 1961. Weather, plant and soil factors inhydrology. Weather 16:207-219.

22. Reddell, D. L., and D. K. Sunada. 1970. Numerical simulation of dispersion in groundwater aquifers. Hydrology Paper41, Colorado State Univ'. 79 p.

23. Rosenblueth, A., and N. Wiener. 1945. Role of models inscience. Phil. Sci. XII(4):316-321.

24. Woolhiser, D. A. 1973. Hydrologic and watershed modeling-State of the art. Trans. ASAE 16(3):553-559.

25. Woolhiser, D. A., and J. A. Liggett. 1967. Unsteady, one-dimensional flow over a plane—The rising hydrograph. WaterResour. Res. 3:753-771.

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