Hydrologic and Watershed Modeling—State of the Art

David A. WoolhiserASSOC. MEMBER ASAE

P RESENT concern of society forenvironmental quality requires consideration of water as a transportingmedium for pollutants. Symbolic hydrologic models provide a quantitative,mathematical description of the transport processes within a watershed. Aconceptual model of a watershed as acontinuous system in three space dimensions is presented. Examples are given ofmathematical formulations of this model as a distributed system (partial differential equations) and as a lumped system (ordinary differential equations).The structure 0f several currently usedwatershed models is examined briefly.

Penman (1961) proposed a concisedefinition of hydrology in the form ofthe question: “What happens to therain?” This is the general question weare trying to answer through the use ofmaterial or symbolic watershed models.As society becomes more aware of theenvironmental problems that may resultfrom man’s activities on a watershed, wemust direct our activities toward answering the questions: “What happensto the fertilizer?” or “What happens tothe pesticides?” Because nutrients orpesticides may be carried by runningwater or may be adsorbed by sedimentstransported by runoff, the last twoquestions can only be answered throughthe use of hydrologic models.

A watershed is an extremely complicated natural system that we cannothope to understand in all detail. Therefore abstraction is necessary if we are tounderstand or control some aspects ofwatershed behavior. Abstraction consists in replacing the watershed underconsideration with a model of similarbut simpler structure. There are twoclasses of models: material and symbolic. (R.osenblueth and Wiener, 1945). A

material model is the representation ofthe real system by another system thatis assumed to have similar properties butis not as complicated or difficult towork with. A symbolic model is amathematical description of an idealizedsituation that shares some of the structural properties of the real system.

Material models include the iconic or“look alike” models and analog models.For example, lysimeters or rainfall simulators can be classified as material models. Symbolic or mathematical modelsare sometimes subdivided into theoretical models and empirical models. This isa rather arbitrary subdivision becauseone man’s empiricism may be anotherman’s theory. However, the point canbe made that an empirical model merelypresents the facts—it is a representationof the data. If conditions change, it hasno predictive capabilities. The theoretical model, on the other hand, has alogical structure similar to the realworld system and may be helpful underchanged circumstances.

All theoretical models simplify andtherefore are more or less wrong. Allempirical relationships have somechance of being fortuitous and in principle should not be applied outside therange of data from which they wereobtained. Both types of models areuseful, but in somewhat different circumstances.

If anyone wishes to develop a modelto aid in understanding a process, theyshould choose the theoretical model.However, if they wish to make a decision based upon answers obtained byusing the model, the choice is notnecessarily obvious. For example, engineering models contain components derived from the social science of economics as well as physically based components. Because the objective of engineering design is stated in economic terms,the physical fidelity of the model components is irrelevant. Net benefits ofany project are a function of designcosts. Therefore if an empirical component gave equal accuracy at a lowercost, it would be preferred to a theoretical model.

Symbolic models may be classified

further as lumped or distributed, stochastic or deterministic. In general, alumped model can be represented by anordinary differential equation or a series0f linked ordinary differential equations. A distributed model includes spatial variations in the inputs, parametersand dependent variables and consists ofa partial differenital equation or linkedpartial differential equations.

Stochastic models describe processesoccurring in time governed by certainprobability laWs. A model is deterministic if when the initial conditions, boundary conditions and inputs are specified,the output is known with certainty.

The purpose of this paper is tobriefly review currently used watershedmodels and to examine how they mightbe used in understanding and predictingtransport of pesticides, plant nutrientsor other substances that might affectwater quality. Other important aspectsof the environment such as scenicbeauty are not considered becausehydrologic modeling does not seem tobe directly involved in their evaluation.

A GENERAL DISTRIBUTEDWATERSHED MODEL

Consider the general distributedmodel of a watershed shown in Fig. 1.

a

WaterTable

Article was submitted for publication onMarch 16, 1912; reviewed and approved forpublication by the Soil and Water Divisionof ASAK on February 19. 1913. Presented asASAE Paper No. 11.747.

Paper is a contribution from the NorthernPlains Branch, ARS, USDA in cooperationwith the Colorado State University Experiment Station.

‘rho author Is: DAVID A. WOOLHISER,Research Hydraulic Engineer. ARS, USDA,Colorado State University, Fort Collins.

FIG. 1 Schematic drawing of watershed as adistributed system.

This article is reprinted from the TRANSACTIONS of the ASAE (Vol. 16, No.3, pp. 553, 554, 555, 556, 557, 558, 559, 1973)Published by the American Society of Agricultural Engineers, St. Joseph, Michigan

C I TI ance for this system can be expressed as

This open system is bounded by impervious rock on the bottom, by theimaginary surface S on the sides and bythe imaginary surface A on the top. Ifwe knew the flux of water in liquid andvapor form and the volumetric moisturecontent at all points within this volume,we could answer the question “Whathappens to the rain?” Vertical fluxthrough the surface A consists of precipitation (positive) and evapotranspiration(negative) and is designated as ~(x,y,t).This flux through the surface may beconsidered as a stochastic process because even though we know its value atsome instant to, we can only makeprobabilistic statements regarding itsvalue at t0 + At.t Sample functions ofthis process at a time t T and at apoint (x,y) are shown in Figs. 2 and 3.

The flux of water in a directionnormal to the surface S is designatedn(x,y,z,t). This function includes bothgroundwater (saturated) flow and unsaturated flow. Where the groundwatercatchment corresponds exactly with thesurface water catchment ?7(x,y,z,t) = 0,if the position of zero gradient is timeinvariant. The surface streamfiow fromthe system is more concentrated thanthe other fluxes and so will be considered as the point process ~1(t). Stream-flow includes both surface runoff andwater contributed to the stream fromthe saturated zone. Let the mass rate ofsediment transport out of the watershedbe t2(t). Imported water, which isalways a possibility if not a fact, will berepresented as the process a(t). Thevolumetric moisture content at anypoint in the system is O(x,y,z,t). Thecontinuity equation or the water bal

‘It could be argued that this process canbe reduced to a deterministic one If a detailedmeteorological model is Included. Consideringpresent skiff In weather forecasting utilizingnumerical models, the surface flux model winremain stochastic for At on the order of daysor hours. As At decreases to the order ofminutes, we might consider the process to bedeterministic, although a realization of theprocess will retain an apparently randomcomponent.

dsPT+wT_ET_QT_~(T)= I_

1’]where

= f ~ (~yj;=~fl~444 = precipiAration rate (vol/time)

4 = f ~(x,y,t = T)dAAevapotranspiration rate vol/time)

= f ~)(x,y,z,t) = T)dS = net

subsurface outflow rate (vol/time

= f 0 x,y,z,t = T dv totalVstorage. volume

Finally consider some substance iadded to an area a1 at a rate of lj perunit area. If this material is added aspart of an agricultural operation, thearea, the amount and the intervals between applications may be deterministicor nearly so. It may also be possible totreat the input as instantaneous. Thetotal mass per unit spatial volume isdenoted as Pj(x,y,z,t). After some timelag, the substance added will be transported past the mouth of the watershedand will appear in streamflow at aconcentration c1(t). The mass rate oftransport of substance i out 0f thewatershed is

= pc1 t ~1(t) + ~‘j~2(t)] ;i = 3,4,

[2]

where p is density of water and*[~2(t)] represents the amount of material carried by sediment. If we areprimarily concerned with the outputs ofa watershed, we will confine our attention to the multidimensional stochasticprocess ~1(t); i = 1,2,..N which represents the instantaneous rate of transportof water, sediment, and other substances out of the watershed. In somesituations we may be interested in theprocesses t~(t); in others we may bemore concerned with the total transportduring a time interval (0,t)

tt f ~1(s)ds [3]

0

To evaluate alternate courses of action, we need to have some informationabout the historical processes ~1(t) or

X~°~t) and also the processes t andX0) (t) after some changes have been

made in the system. If we have longperiods of record, we can estimate theparameters of the unmodified processesfrom historical data. If records areshort, it may be necessary to use amodel in conjunction with longer records of precipitation to estimate theparameters of the existing process. Itwill always be necessary to use a modelto estimate the parameters of the modifled processes

ALTERNATE MATHEMATICALFORMULATIONS OF THE

CONCEPTUAL MODEL

Although the conceptual model presented in the previous section is anabstraction of reality, it presents a verygeneral description of the behavior of awatershed. In fact it is too general to beof much use in any particular situation.To develop a more detailed formulation,it is quite natural to begin with thepartial differential equation of continuity for flow of water in porous media orfor free surface flow in streams. Theseequations can be derived quite readilyand, along with Darcy’s law for saturated flow in porous media and themomentum equation for free surfaceflow, constitute a rather rigorous mathematical description of transport of water within the system. Conceptually,these equations along with initial andboundary conditions would enable oneto solve for the streamflow process in adeterministic manner if the rainfall andfactors affecting evapotranspiration aregiven. These partial differential equations are a distributed model of thesystem. Unfortunately these equationseither do not have analytical solutionsor can be solved only for very simplegeometries or boundary conditions. Theequations may be written in finite-difference form and solved numerically,or a more abstract model may be

SC

aOiaa

0, WSeSflId

000”d,’y

‘(II

FIG. 2 Schematic representation of sampledistribution of surfuce flux (x,y) over awatershed

EvopoIronip~ rol a,

FIG. 3 Sample function of vertical flux at apoint x, y.

from rainfall, q1, and the output fromthe upstream storage, Q1~1. This systemcan be described by the following seriesof linked ordinary differential equations:

postdated. The more abstract modelwould be simpler than the distributedmodel but hopefully would have asimilar structure and would retain mostof its important characteristics.

As an example of this process ofabstraction, let us consider various models for the description of unsteady flowover a plane—the “overland flow” problem. If we make the assumption that aone-dimensional formulation of theproblem is adequate, we can describeunsteady, free surface flow on a planeby the continuity equation

dS1

dt

ah a~h

obtaining general relationships for parameter estimation through numericalexperiments on distributed models andlumped system approximations to them.

As a further abstraction we mightrepresent overland flow as a generalsystem where the output (runoff) is

+ bS = q~ (t) assumed to be related to the input

(precipitation) but where no explicit

~ + bS~ = q2(t) + bS~ the internal structure of the system.assumptions have to be made regarding

dt j This approach was first applied to

~.. [7] hydrology by Amorocho and Orlob(1961) and several investigators havedS.—‘ + bS~ = q1(t) + bS1 worked on this problem in the following

at ax 1966; Bidwell 1971; Amorocho and+ = q(x,t) {4] dt decade: (Amorocho 1963, 1967; Jacoby

Brandstetter 1971). A special case ofand the momentum equation: + bS~ = t + bS~ 1 the general system approach, the theory

au uau gah qu dt of linear systems, has found hydrologic+ + —

at ax ax g S~ S~ 1~ where bS1c represents the outflow, Q1, application since Sherman’s (1932) unit

[5] from the it~ storage element. Dooge hydrograph theory was proposed. It was(1967) used equation [7] to represent a not until Dooge’s classic paper (Dooge

where: “uniformly nonlinear system.” There is 1959) that most of the implications ofu = local mean velocity in x an obvious similarity between the series the linear assumption were clearly

direction of nonlinear storages and the kinematic stated.As we consider this example of forh = local depth model. During the rising stage due to muiation of successively more abstract

q(x,t) = lateral inflow rate per un- uniform lateral inflow, the solutionit area within the zone of determinacy for ~ < models, it is obvious that many of the

model classifications tend to be ratherg = acceleration of gravity for the kinematic model is given indistinct. A certain amount of lumpings0 = bed slopeS~ = friction slope of inputs and parameters is necessary

and x and t are space and time coordi. h = qt for the distributed model because wenates respectively. or can only obtain discrete data and be.

The above model distorts reality in cause we must use discrete mathematimany ways and its application to flow Q = a(qt~’ [8] cal methods to solve the partial differenover a rough, natural surface is subject Kibler (1968) has shown that if a tial equation. It is readily apparent,to greater doubt than its application to storage element is considered to be however, that the general systems apflow on a parking lot, analogous to a length of plane equal to proach is valid only for time-invarient

It has been found that in most LQ/N that the discharge from the ~h systems while in principle one might usehydrologic circumstances the above element can be written as the partial differential equation modelequations can be very accurately ap- to predict the runoff response if theproximated by the kinematic wave N a watershed surface were changed pro-equation (Woolhiser and Liggett 1967). ~ = a [ SN] [9] vided one had independent information

LIn the kinematic wave equation, the a on the friction relationship for themomentum equation is simplified to as N becomes large. This expression is changed surface condition,

similar to equation (8} and is a very This example demonstrates that theS=S I‘ ° I good approximation for N> 10. If N is partial differential equations, the finite-or [6] large, the parameters in the storage difference equations, the ordinary dif

u = ah”1 model have a direct equivalence to those ferential equations describing lumpedin the uniform flow relationship for the systems with decreasing N, and the

where a and n are parameters which kinematic model. If overland flow on a general linear and nonlinear systems caninclude the effects of slope, roughness plane is represented by a single storage all be looked upon as successively moreand the regime of flow (laminar or element, however, this equivalence dis.. abstract models of the real physicalturbulent). appears although there is likely to be a system. They share some common prop-

As a further abstraction, we might relation between the number of storages erties and all inevitably involve distorpropose that overland flow be approxi- and the ratio of the parameters for the Hon. This distortion or the lack ofmated by a cascade of N non linear kinematic model and the lumped-system physical significance of model paramestorages, each receiving inputs model. This suggests the possibility of ters is the price we pay for simplicity

and reduced data requirements.

4cIN

I____ ____ - - ‘~ 5N

STATE OF THE ARTWATERSHED MODELS

The Stanford Model (Crawford andLinsley — 1962, 1966) is certainly the1 2-- ————N

I—

best known general watershed model incurrent use. A flow chart ior this modelis shown in Fig. 4. This is basically alumped-parameter model, althoughlarge, heterogeneous watersheds can bedivided into subwatersheds if sufficientdata are available to estimate the parameters for the subwatersheds. The essential structure of the Stanford Model andlumped parameter models in general canbe expressed in the following concisenotation:

NZY..=X ;i1,2,...N

13 11*1

[10]

where S~ is the amount of storage (statevariable) in the it~ storage element; Y1 isoutput from the ~ element; ‘V11 is theflow from storage i to storage j; X1 is theexogenous input to the ~ element. The

will usually be functions of the S~and Si and may be dependent on certainthreshold parameters. The inputs ~ areindependent of the state of the system,but the outputs, ‘Y1, depend upon thestate of the system and may also dependupon current values of some inputs. Forexample, evapotranspiration rates maybe a function of incoming solar radiation as well as soil moisture storage. The

above system of equations are solved byfinite-difference techniques on a digitalcomputer. The methods by which infiltration and evapotranspiration are computed infer spatial variability in thesefluxes but they are considered to beindependent of position on the water-

shed. The Stanford Model has been usedby many investigators and many different versions are available. However, thebasic structure of the model is essentially the same in all versions. Because ofthe extensive experience with this model, it is quite appropriate to compareany new model structure with it toascertain if there has been an improvement.

Holtan and Lopez (1971) have described the USDAHL-70 model of watershed hydrology. Although their objective is to develop a distributed watershed model, the present model is primadly lumped although, like theStanford Model, a watershed can bebroken down into smaller homogeneousareas. Spatial variations in soil properties and slopes are accounted for bydividing the watershed into land capability classes which correspond to uplands,hillslopes and bottom lands. The percentage of outflow from each capabilityclass that is contributed to the otherclasses or the channel is estimated fromtopographic maps. Overland flow is simdated with a lumped, nonlinear storagemodel.

Although there are many similaritiesbetween components of the StanfordWatershed Model and the USDAHL-70model, the USDHAL-70 model attemptsto incorporate some aspects of spatialvariability by dividing the watershedinto land capability classes with specified spatial relationships. TheUSDAHL-70 model also puts more emphasis on a prjori estimation of parame

ters, particularly for the infiltrationcomponeiit.

Dawdy et al (1970) have recentlydeveloped and tested a lumped systemmodel describing surface runoff fromsmall watersheds. The model structure issimilar to parts of the Stanford Modeland it can be described by equation[101; however, there are differences inthe functional relationships.

A distributed watershed model including overland flow, porous mediaflow and open-channel flow is not yetavailable. Freeze and Harlan (1969)proposed a three-dimensional approachwhich has not been completely implemented as yet. (Fig. 5). Freeze (1971)has presented a three-dimensional modelfor unsteady, saturated or unsaturatedflow in a groundwater basin. His modelis physically based in that it has astructure based upon the theory of flowin a porous medium. Models that include a part of the hydrologic systemhave been presented by others (Kiblerand Woolhiser 1970; Brakensiek 1967;Henderson and Wooding 1964;Machmeier and Larson 1967; Morgaliand Linsley 1965; Schaake 1965; Harleyet al 1970; Amisial eta1 1969; Hugginsand Monke 1970; and Smith andWoolhiser 1971).

The parameters in the lumped systemmodels have little direct physical significance and, therefore, can be estimatedonly by using concurrent rainfall andrunoff data. In principle, the parametersin the distributed model have somephysical significance and the possibilityexists that they can be evaluated byindependent measurements. However, inpractice the partial differential equations describing distributed systemsmust be solved by finite differencemethods. This representation requiresthe specification of parameter values ata finite number of points. If the numberof points is large, the cost of measurements of the parameters becomes prohibitive. If the number of points isreduced drastically, the measured parameter values will not necessarily givegood predictions of watershed behaviorbecause of the distortions introduced bythe order of approximation.

Because the structure of equation[10j is general enough to include anextremely large number of differentmodels, the question arises “How do Ichoose the best model for my particularapplication?” Dawdy and Lichty (1968)suggest four criteria of choice thatmight be used: (a) accuracy of prediction, (b) simplicity of the model, (c)consistency of parameter estimates and(d) sensitivity of results to changes in

-

—— ,,,ofl. — — — —

Stanford Watershed Model IVFlowchart

Ke~

~ (S~#O.’j!!!S

0-I!.I”.’

—in,

~led‘ji’!91!LflQa.

FIG. 4 Flowchart of the Stanford Watershed Model IV (from Crawford and Linsley 1966).

dS.—I-- + y~ ÷dt

parameter values. Although it is impossible to obtain an unambiguous rankingwith multiple criteria, these criteria areobviously related to a criterion whichmight be to maximize net benefitsthrough use of a model in a designsituation.

WATER QUALITY COMPONENTSOF WATERSHED MODELS

There have been many investigationsconcerning water quality in certain restricted segments of the conceptual watershed discussed previously. Perhapsthe earliest and most widely used mathematical model describing water qualityis that proposed by Streeter and Phelps(1925) to describe the dissolved oxygenrelationship in a stream. The StreeterPhelps model considers a particularreach of a stream with an initial biochemical oxygen demand (BOD) anddissolved oxygen content at the upperboundary. Under the assumption thatbiochemical oxidation and reaeration byabsorption of atmospheric oxygen arefirst-order processes and the streamfiowprocess is pure translation at constantvelocity, the Streeter-Phelps model enables the prediction of oxygen deficit atany point within the reach. TheStreeter-Phelps model considers onlythe first stage of biochemical oxidationand ignores the second or nitrificationstage which may also utilize significantamounts of oxygen. Several more complete models have been proposed—Camp(1963), Churchill and Buckingham

(1956), Dobbins (1964).These refinements include the effects

of photosynthesis, sedimentation, bottom scour and surface runoff on thedissolved oxygen balance. More recentlyO’Connor (1967) developed a modelincluding, in addition to the above,artificial aeration, photosynthetic production of oxygen, temporal and spatialvariations in flow rates, and diurnalvariation in photosynthetic activity.Goodman and Tucker (1969) utilized atime-varying model in studying effectiveness of treatment plants.

All of the previously cited workshave either considered steady-state situations or have considered time-varyingcases as a series of steady states. Thisapproach is probably adequate for non-tidal rivers where streamfiow andoxygen-demanding inputs do not varyrapidly with time.

Recognizing the possible inadequacyof the series of steady states or thequasi-steady state case used in earlierwater quality work on esturaries,Dresnack and Dobbins (1968), Bella andDobbins (1968), Harleman et al (1968)and Dornhelm and Woolhiser (1968)devleoped methods for predicting waterquality based upon the partial differential equation of unsteady free surfaceflow and longitudinal dispersion for theone-dimensional case. Solutions wereobtained using finite-difference or finiteelement methods. Unsteady, two-dimensional cases have since been consideredby Fischer (1969).

Orlob and Woods (1964) studiedwater quality aspects of an irrigated areaby computer modeling of the watertransport system and concurrent transport of conservative pollutants. Theyused a lumped-system model whichcould be described by equation (10].They used a time step of one month.Their investitagion demonstrated thepossible effects of over-irrigation andrecirculation of drainage waters on concentration of pollutants.

Another model using the lumpedsystem approach to describe transportof a substance from the point of deposition on a watershed to the outlet of thewatershed is that developed by Huff(Huff and Kruger 1967 (a, b)). Jn hisinitial work at Stanford University, Huffused the Stanford Watershed Model todescribe the movement of waterthrough a basin and expanded it toinclude the transport 0f radioactiveaerosols (Sr90). Subsequently, he andhis associates at the University of Wisconsin (Huff et al 1970) are attemptingto modify the radionuclide transportmodel to include the transport of nutrients (Watts et al 1970). The initialattempt in this area is the modeling ofthe transport of nitrogen in an effort todiscover mechanisms by which nutrientenrichment of lakes and streams isrelated to human activity.

Bloom et al (1970) used a veryelementary lumped storage representation of soil water and surface water aselements of a mathematical model toevaluate the transport and accumulationof radionuclides. Their model was verycomprehensive in that it attempted toinclude the transport of radionuclidesfrom deposition on plants and soil,transport by water, ingestion by fishand finally ingestion by man. This complex system was subdivided into ten“compartments.” From continuity considerations, the behavior of the systemwas described by the system of equations:

dy. N~D Aj~y~,i1,2,...N;

dt n1n#i

[11]

wherethe amount of radionucidein the ith compartment

y,, = the amount 0f radionuclidein the nth compartmentthe transfer rate coefficient

A11 the elimination-rate coeffici

N the total number of compartments in the system.

channelOverland Flow

Eva

Percipitotion -P (I)

Runof 1-0 (I)

Equipolentiol Lines(Soil Moisture & Groundwater)

Tab e

Flow Lines(Soil Moisture 8 Groundwater)

FIG. S Schematic diagram of (a) watershed and (b) three-dimensional,discrete model of the watershed (from Freeze and Harlan).

ent

Equation [11J is obviously a specialcase of equation [10].

The authors contend that this simple,linear, lumped-system model is justifiedbecause sufficient experimental data arenot available to estimate parameters ofmore complex models.

Two models that use a distributedsystem approach toward describing water quality in an urban environmenthave been reported recently (Metcalfand Eddy, University of Florida andWater Resources Engineers 1971, University of Cincinnati 1970).

Hyatt et al (1970) developed a simulation model of the transport of salts inthe upper Colorado River Basin. Thehydrologic transport model used wassimilar in structure to the Stanfordmodel but was solved by an analogcomputer. The salt transport modelignored ionic exchange and chemicalprecipitation phenomena within the soiland therefore is only applicable tosteady-state situations. The basic timeunit was one month and the smallestspatial unit was a sub-basin of theColorado River. in further work at UtahState (Thomas et al 1971) a salt transport model has been developed thatincludes reactions occurring in the soilsuch as ionic exchange and chemicalprecipitation of gypsum and line. Bothof these models are intended to aidmanagement decisions with regard towater quality effects of irrigation.

If we consider only the lumpedparameter storage models, adding a water-transported pollutant requires a parallel system of flows and storages similarin structure to that describing the flowof water. The parameters of the qualitymodel include the hydrologic modelparameters plus certain reaction rateparameters to describe transformationof the substance under consideration.Certain elements of the hydrologic model may be more important in predictingwater quality than they were in predicting streamfiow rates.

SUMMARY AND DISCUSSION

In this paper, I have briefly discussedthe structure of current general watershed models or those that include important parts of the hydrologic cycleand have been used or could be used todescribe the transport of pesticides,nutrients, radioactive nuclides or othersubstances. The objective of most waterquality models has been to predictdissolved oxygen concentrations in segments of a stream or an estuary. Themore advanced models treat the river asa one- or two-dimensional unsteady

system and involve the numerical solution 0f partial differential equations.Obviously, any model describing thedissolved oxygen content can also describe transport of conservative substances.

The Stanford Watershed Model hasbeen used to describe the hydrologictransport of radioactive aerosols(Strontium 90) with some success and isbeing modified to handle transport ofnutrients. The nutrient transport problem appears to be less suitable for alumped-system approach because theinputs are not spatially uniform. Thetransport of nitrogen as an importantnutrient seems to be especially difficultbecause it can exist in six major formsincluding organic compounds, all 0fwhich must be induded in the model.Added to the hydrologic complexitywhich includes transport and storage ofliquid water and its transformation intosolid or gas are the chemical or biological transformations of the various nitrogen forms. While many 0f these transformations can be described quantitatively in the laboratory, little can besaid regarding the extent to which theytake place under field conditions(Stanford et al 1970).

Another problem which appears tobe significant is that of estimating initialconditions. In hydrologic modeling, theinitial conditions may not be particularly important because the system has arelatively short memory—the state ofthe system after a few years is not verysensitive to the initial condition. However, nutrients present in soil can bevery large compared to those added byfertilizer and will affect the system for along time.

Considering the urgency of the problems associated with water quality, it iscertainly desirable to construct mathematical models describing these phenomena. However, we must give moreserious attention to the question ofmodel verification. Can we say in anymeaningful way that model A is betterthan model B? if we cannot do this, weare unable to determine if we aremaking progress in our research a veryunsatisfactory situation. it appears tome that research into environmentalaspects of hydrology should involve twotypes of modeling: (a) models of relatively complex systems and (b)modelsof very simple systems. in constructingmodels of complex systems, we will findout what we need to know and we maydevelop operationally useful interimmodels. However, only by carefullyconstructed, controlled experimentationon a more limited scale can we obtain

unambiguous answers to specific research questions.

Empirical, lumped-system modelswill probably be useful tools in predicting transport of substances which arenaturally present in the environmentand which are currently being monitored. However, if we must evaluate theenvironmental impact of new substancesbefore they are released or if the transport system itself may be changed,theoretical models appear to be the onlypossible approach. This means that wemust develop objective techniques of apriori estimation of model parameters.

Effective research on transport ofenvironmental contaminants by waterwill inevitably involve several disciplinesbecause of the variety of problems involved in constructing adequate models.

References1 AmIsial, Roger A., .3. Paul Riley,

K. 0. Renard, E. K. Israelsen. 1969. Analogcomputer solution of the unsteady flow equations and its use In modeling the surfacerunoff process. Research Progress Report toARS, USDA, Utah Water Research Laboratory, Utah State University.

2 Amorocho, J. and 0. T. Orlob. 1961.Nonlinear analysis of hydrologic systems.Water Resources Center Contrib. 40. UnIversity of California, Los Angeles.

3 Amorocho, J. 1963. Measures of thelinearity of hydrologic systems. J. Geophys.Res. 68(8):2237-2249.

4 Amorocho, J. 1967. The nonlinearprediction problem in the study of the runoffcycle. Water Resources Res. 3(3):861-880.

5 Amorocho, S. and A. Brandstetter1971. Determination of nonlinear functionalresponse functions in rainfall-runoff processes. Water Resources Res. 7(5):1087-1 101.

6 Bela, David A., and William B.Dobbins. 1968. Difference modeling ofstream pollution. Journ. Sanitary Engr. Division, Proc. ASCE. p. 995-1016.

7 Bidwell, V. J. 1971. Regression analysis of nonlinear catchment systems. WaterResources Res. 7(5):1118-1126.

8 Bloom. S. D., A. A. Lain, W. E.Martin and 0. B. Raines. 1070. Mathematicalmethods for evaluating the transport andaccumulation of radionuclides. Battelle Memorial Institute, Columbus. Ohio. NITS ReportBMI-171-030. 39 p.

9 Brakenslek, P. L, 1967. A simulatedwatershed flow system for hydrograph prediction: A kinematic application. Proc. international Hydrology Symposium, Fort Collins,Cob.

10 Camp, T. R. 1963. Water and itsimpurities. Reinhold Press, New York.

11 Churchill, M, A. and R. A. Bucking-ham. 1956. StatIstical methods for analysis ofstream purification capacity. Sewage and Industrial Wastes 28(4):517-537.

12 Crawford, N. H. and R. K. Linsley.1962. The synthesis of continuous streaniflowhydrographs on a digital computer. StanfordUniversity Dept. of Civil Engr. Tech. Report12, 121 p.

13 Crawford, N. H. and R. K. Linsley.1966. Digital simulation in hydrology: Stanford Watershed Model IV. Tech. Report No.39 Dept of Civil Engr., Stanford Univ.

14 Dawdy, D. R. and R. W. Lichty.1968. Methodology of hydrologic modelbuilding. In the use of analog and digitelcomputers in hydrology. tnt. Ass Sd.Hydrol. Symp. Proc. Tucson, Arizona. II, No.MATHS. p. 347-3 55.

15 Dawdy, David R., Robert W. Lichtyand James M. Bergmann. 1970. A rainfall-runoff simulation model for estimation of floodpeaks for small drainage basins—A progressreport. USGS Professional Paper 50GB. 28 p.

16 Dobbins, W. B. 1964. .BOD and oxy.gen relationships in streams. Proc. ASCE,

Journ. of the Sanitary Engr. Div. 90. No.SA3, P. 53-78.

17 Dooge, J. C. I. 1959. A generaltheory of the unit hydrograph. Journ.beophys. Res. 64(2):241-256.

18 Dooge, J. C. 1. 1967. Hydrologicsystems with uniform non-linearity. Mimeographed Report, Univ. College, Dept. of CivilEngr., Cork, Ireland.

19 Dornhelm, Richard B. and David A.Wo olhiser. 1968. DigItal simulation ofestuarine water quality. Water Resources Res.4(6):1317-1327

20 Dresnack, Robert and William E.Dobbins. 1968. Numerical analysis of BODand DO profiles. Proc. ASCE, Joum. SanitaryEngr. Div. p. 789-807.

21 FIscher, Hugo B. 1969. A lagrangianmethod for predicting pollutant dispersion inBolinas Lagoon, California. USGS Open-FileReport. Menlo Park, Calif.

22 Freeze, R. Allan. 1971. Three-dimensional, transient, saturated-unsaturatedflow In a groundwater basin. Water ResourcesRes. 7(2):347-366.

23 Freeze, it Allen and R. L. Harlan1969. Blueprint for a physically-based, digitally-simulated hydrologic response model.Journ. of Hydrology 9:237-258.

24 Goodman, Alvin S. and Richard J.Tucker. 1969. Use of mathematical models inwater quality studies. Report to FederalWater Pollution Control Administration, National Technical Information Service PR188494.

25 Harleman, Donald R. L, Chok’HungLee and Lawrence C. Hall. 1968. Numericalstudies of unsteady dispersion in esturaries.Proc. ASCE, Joum. Sanitary Engr. Div. p.897-911.

26 Harley, Brendan M., Frank E. Perkinsand Peter S. Eagleson. 1970. A modulardistributed model of catchment dynamics.MIT Dept. of Civil Engr., HydrodynamicsLab. Report No. 133.

27 Henderson, F. M. and R. A. Woodiag.1964. Overland flow and groundwater flowfrom a steady rainfall of finite duration.Journ. Geophys. Res. 69(8):1531-1540.

28 Holtan, H. N. and N. C. Lopez. 1971.USDAHL-70 mode! of watershed hydrology.USDA Tech. Bull. No. 1435. 84 p.

29 Huff, D. D. and P. Kruger. 1967a.

The chemical and physical parameters In ahydrologic transport model for radioactiveaerosols. Proc. Int’l Hydrology Symp.1:128-135, Fort Collins, Cob.

30 Huff, Dale D. and Paul Kruger.196Th. A numerical model for the hydrologictransport of radioactive aerosols from precipitation to water supplies. Isotope Techniquesin the Hydrologic Cycle. Am. Geophys. Union, Geophysical Monograph Series No. 11. p.85-96.

31 Huff, D. D., D. G. Watts, 0. L.Loucks and M. Teraguchi. 1970. A study ofnutrient transport with the Stanford Water’shed Model. Int’l Bio. Program, DeciduousForest Biome Memo Report 70-1. Univ. ofWisconsin. Madison.

32 Huggins, L. F. and S. J. Monke1970. Mathematical simulation of hydrologicevents of ungaged watersheds. Tech. Report14. Purdue Univ. Water Res. Res. Center,Lafayette, Indiana.

33 Hyatt, M. Leon, S. Paul Riley. M.Lynn McKee and Eugene K. Israelsen. 1970.Computer simulation of the hydrologic-salinity flow system within the Upper ColoradoRiver Basin. Report No. PRWGS4-1, UtahWater Rn. Lab., Utah State Univ.. Logan.

34 Jacoby, S. L. 5. 1966. A mathematical model for nonlinear hydrologic systems. J.Geophys. Res. 71(20):4811-4824.

35 KibIer, David F. 1968. A kinematicoverland flow model and its optimizatioltPh.D. Dissertation, Colorado State Univ., FortCollins.

36 Kibler, David F. and D. A. Woolhlser.1970. The kinematic cascade as a hydrologicmodel. Colorado State Univ. Hydrology PapetNo. 39.

37 Machmeier, a. S. and C. L. Larson.1967. A mathematical watershed routingmodel. Proc. Int’l Hydrology Symp., FortCollins, Cob. p. 64-71.

38 Morgafl, J. R. and Ray K. Linsley.1965. Computer analysis of overland flow.Proc. ASCE, Journ. Hydraulics Div. 91, No.HY3, p. 81-100.

39 Metcalf and Eddy, Univ. of Floridaand Water Resources Engineers. 1971. Stormwater management model. Vol. I-IV WaterPollution Control Research Series 11024 DOC10/71 Environmental Protection Agency, U.S.Government Printing Office.

40 O’Connor, Donald J. 1967. The temporal and spatial distribution of dissolvedoxygen in streams. Water Resources Res.3(1):6 5-7 9.

41 Orlob, Gerald T. and Philip C.Woods. 1964. Lost river system a waterquality management study. Proc. ASCE,Journ. Hydraulics Dlv. 90(2):1-22.

42 Penman, H. L. 1961. Weather, plantand soil factors in hydrology. Weather16: 207-21 9.

43 Roseablueth, Arturo and NorbertWiener. 1945. The role of models in science.Philosophy of Science XIl(4):316-321.

44 Schaake, J. a, Jr. 1965. SynthesIs ofthe inlet hydrograph. Tech. Report 3, Stormdrainage research project, Dept. of SanitaryEngineering and Water Resources, The JohnsHopkins Univ., Baltimore, Md.

45 Sherman, L. K. 1932. Stream flowfrom rainfall by unit graph method. Engr.News Record 108:501.

46 Smith, Roger S. and D. A. Woolhiser.1971. overland flow on an infiltrating sir-face. Water Resources Res. 7(4):899-913.

47 Stanford, G., D. B. England andA. W. Taylor. 1970. FertiLizer use and waterquality. USDA-ARS 41-168.

48 Streeter, H. W. and S. B. Phelps.1925. A study of the pollution and naturalpurification of the Ohio River. U. S. PublicHealth Bulletin, No. 146.

49 Thomas, Jimmie, L., S. Paul Rileyand Eugene K. Israelsen. 1971. A computermodel of the quantity and chemical quality ofreturn flow. Report No. PRWG77’l, UtahWater Res. Lab., Utah State Univ., Logan.

50 Univ. of Cincinnati. 1970. Urbanrunoff characteristics. Water Pollution Control Series 11024 DQU 10/70 EnvIronmentalProtection Agency, U.S. Govt. Printing Office.

51 Watts, D. G, D. D. Huff, 0. L.Loucks, M. Teraguchi. 1970. Models forsystems studies -of nutrient flows In lakes andstreams. Int’l Bio., Program. Deciduous ForestBiome, Memo Report 70-4, Univ. of Wisconsin, Madison.

52 Woolhiser, D. A. and J. A. Liggett.1967. Unsteady, one-dimensional flow over aplane—the rising hydrograph. Water ResourcesRes. 3(3):753.771.

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