Hydrologic and Watershed Modeling— State of the Art

David A. Woolhiser ASSOC. MEMBER ASAE

material model is the representation of further as lumped or distributed, sto P environmentalRESENT concernqualityof requiressociety confor the real system by another system that chastic or deterministic. In general, a sideration of water as a transporting is assumed to have similar properties but lumped model can be represented by an medium for pollutants. Symbolic hydro is not as complicated or difficult to ordinary differential equation or a series logic models provide a quantitative, work with. A symbolic model is a 0f linked ordinary differential equa mathematical description of the trans mathematical description of an idealized tions. A distributed model includes spa port processes within a watershed. A situation that shares some of the struc tial variations in the inputs, parameters conceptual model of a watershed as a tural properties of the real system. and dependent variables and consists of continuous system in three space dimen Material models include the iconic or a partial differenital equation or linked sions is presented. Examples are given of “look alike” models and analog models. partial differential equations. mathematical formulations of this mod For example, lysimeters or rainfall simu Stochastic models describe processes el as a distributed system (partial differ lators can be classified as material mod occurring in time governed by certain ential equations) and as a lumped sys els. Symbolic or mathematical models probability laWs. A model is determinis tem (ordinary differential equations). are sometimes subdivided into theoreti tic if when the initial conditions, bound The structure 0f several currently used cal models and empirical models. This is ary conditions and inputs are specified, watershed models is examined briefly. a rather arbitrary subdivision because the output is known with certainty. Penman (1961) proposed a concise one man’s empiricism may be another The purpose of this paper is to definition of in the form of man’s theory. However, the point can briefly review currently used watershed the question: “What happens to the be made that an empirical model merely models and to examine how they might ?” This is the general question we presents the facts—it is a representation be used in understanding and predicting are trying to answer through the use of of the data. If conditions change, it has transport of pesticides, plant nutrients material or symbolic watershed models. no predictive capabilities. The theoreti or other substances that might affect As society becomes more aware of the cal model, on the other hand, has a water quality. Other important aspects environmental problems that may result logical structure similar to the real of the environment such as scenic from man’s activities on a watershed, we world system and may be helpful under beauty are not considered because must direct our activities toward an changed circumstances. hydrologic modeling does not seem to swering the questions: “What happens All theoretical models simplify and be directly involved in their evaluation. to the fertilizer?” or “What happens to therefore are more or less wrong. All the pesticides?” Because nutrients or empirical relationships have some A GENERAL DISTRIBUTED pesticides may be carried by running chance of being fortuitous and in prin WATERSHED MODEL water or may be adsorbed by sediments ciple should not be applied outside the transported by runoff, the last two range of data from which they were Consider the general distributed questions can only be answered through obtained. Both types of models are model of a watershed shown in Fig. 1. the use of hydrologic models. useful, but in somewhat different cir A watershed is an extremely compli cumstances. a cated natural system that we cannot If anyone wishes to develop a model hope to understand in all detail. There to aid in understanding a process, they fore abstraction is necessary if we are to should choose the theoretical model. understand or control some aspects of However, if they wish to make a deci watershed behavior. Abstraction con sion based upon answers obtained by sists in replacing the watershed under using the model, the choice is not consideration with a model of similar necessarily obvious. For example, engi but simpler structure. There are two neering models contain components de classes of models: material and symbol rived from the social science of econom Water ic. (R.osenblueth and Wiener, 1945). A ics as as physically based compo Table nents. Because the objective of engineer ing design is stated in economic terms, Article was submitted for publication on March 16, 1912; reviewed and approved for the physical fidelity of the model com publication by the Soil and Water Division ponents is irrelevant. Net benefits of of ASAK on February 19. 1913. Presented as ASAE Paper No. 11.747. any project are a function of design Paper is a contribution from the Northern costs. Therefore if an empirical compo Plains Branch, ARS, USDA in cooperation with the Colorado State University Experi nent gave equal accuracy at a lower ment Station. cost, it would be preferred to a theoreti ‘rho author Is: DAVID A. WOOLHISER, Research Hydraulic Engineer. ARS, USDA, cal model. FIG. 1 Schematic drawing of watershed as a Colorado State University, Fort Collins. Symbolic models may be classified distributed 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

S C ds a PT+wT_ET_QT_~(T)= I_ aOi ‘(II a 1’]

0, WSeSflId where 000”d,’y

= f ~ (~yj;=~fl~444 = precipi A ration rate (vol/time)

EvopoIronip~ rol a, = ~(x,y,t = T)dA FIG. 2 Schematic representation of sample 4 fA distribution of surfuce flux (x,y) over a rate vol/ watershed time)

This open system is bounded by imper = f ~)(x,y,z,t) = T)dS = net FIG. 3 Sample function of vertical flux at a point x, y. vious rock on the bottom, by the subsurface outflow rate (vol/ imaginary surface S on the sides and by time made in the system. If we have long the imaginary surface A on the top. If periods of record, we can estimate the we knew the flux of water in liquid and = f 0 x,y,z,t = T dv total parameters of the unmodified processes vapor form and the volumetric moisture V from historical data. If records are content at all points within this volume, storage. volume short, it may be necessary to use a we could answer the question “What model in conjunction with longer rec happens to the rain?” Vertical flux Finally consider some substance i ords of precipitation to estimate the through the surface A consists of precip added to an area a1 at a rate of lj per parameters of the existing process. It itation (positive) and evapotranspiration unit area. If this material is added as will always be necessary to use a model (negative) and is designated as ~(x,y,t). part of an agricultural operation, the This flux through the surface may be area, the amount and the intervals be to estimate the parameters of the modi considered as a stochastic process be tween applications may be deterministic fled processes cause even though we know its value at or nearly so. It may also be possible to some instant to, we can only make treat the input as instantaneous. The ALTERNATE MATHEMATICAL probabilistic statements regarding its total mass per unit spatial volume is FORMULATIONS OF THE value at t0 + At.t Sample functions of denoted as Pj(x,y,z,t). After some time CONCEPTUAL MODEL this process at a time t T and at a lag, the substance added will be trans Although the conceptual model pre point (x,y) are shown in Figs. 2 and 3. ported past the mouth of the watershed sented in the previous section is an The flux of water in a direction and will appear in streamflow at a abstraction of reality, it presents a very normal to the surface S is designated concentration c1(t). The mass rate of general description of the behavior of a n(x,y,z,t). This function includes both transport of substance i out 0f the watershed. In fact it is too general to be (saturated) flow and unsa watershed is of much use in any particular situation. turated flow. Where the groundwater = pc1 t ~1(t) + ~‘j~2(t)] ;i = 3,4, To develop a more detailed formulation, catchment corresponds exactly with the it is quite natural to begin with the surface water catchment ?7(x,y,z,t) = 0, [2] partial differential equation of continu if the position of zero gradient is time ity for flow of water in porous media or invariant. The surface streamfiow from where p is density of water and for free surface flow in streams. These the system is more concentrated than *[~2(t)] represents the amount of ma equations can be derived quite readily the other fluxes and so will be consi terial carried by sediment. If we are and, along with Darcy’s law for satur dered as the point process ~1(t). Stream- primarily concerned with the outputs of ated flow in porous media and the flow includes both and a watershed, we will confine our atten momentum equation for free surface water contributed to the stream from tion to the multidimensional stochastic flow, constitute a rather rigorous mathe the saturated zone. Let the mass rate of process ~1(t); i = 1,2,..N which repre matical description of transport of wa sediment transport out of the watershed sents the instantaneous rate of transport ter within the system. Conceptually, be t2(t). Imported water, which is of water, sediment, and other sub these equations along with initial and always a possibility if not a fact, will be stances out of the watershed. In some boundary conditions would enable one represented as the process a(t). The situations we may be interested in the to solve for the streamflow process in a volumetric moisture content at any processes t~(t); in others we may be deterministic manner if the rainfall and point in the system is O(x,y,z,t). The more concerned with the total transport factors affecting evapotranspiration are continuity equation or the water bal during a time interval (0,t) given. These partial differential equa t tions are a distributed model of the ‘It could be argued that this process can be reduced to a deterministic one If a detailed t f ~1(s)ds [3] system. Unfortunately these equations meteorological model is Included. Considering 0 either do not have analytical solutions present skiff In weather forecasting utilizing numerical models, the surface flux model win To evaluate alternate courses of ac or can be solved only for very simple remain stochastic for At on the order of days tion, we need to have some information geometries or boundary conditions. The or hours. As At decreases to the order of minutes, we might consider the process to be about the historical processes ~1(t) or equations may be written in finite- deterministic, although a realization of the X~°~t) and also the processes t and process will retain an apparently random difference form and solved numerically, component. X0) (t) after some changes have been or a more abstract model may be postdated. The more abstract model from rainfall, q1, and the output from obtaining general relationships for pa would be simpler than the distributed the upstream storage, Q1~1. This system rameter estimation through numerical model but hopefully would have a can be described by the following series experiments on distributed models and similar structure and would retain most of linked ordinary differential equa lumped system approximations to them. of its important characteristics. tions: As a further abstraction we might As an example of this process of represent overland flow as a general abstraction, let us consider various mod dS1 system where the output (runoff) is + bS = q~ (t) assumed to be related to the input els for the description of unsteady flow dt over a plane—the “overland flow” prob (precipitation) but where no explicit lem. If we make the assumption that a assumptions have to be made regarding one-dimensional formulation of the ~ + bS~ = q2(t) + bS~ the internal structure of the system. problem is adequate, we can describe dt j This approach was first applied to unsteady, free surface flow on a plane ~.. [7] hydrology by Amorocho and Orlob by the continuity equation (1961) and several investigators have dS. ah a~h —‘ + bS~ = q1(t) + bS1 worked on this problem in the following + = q(x,t) {4] dt decade: (Amorocho 1963, 1967; Jacoby at ax 1966; Bidwell 1971; Amorocho and

and the momentum equation: + bS~ = t + bS~ 1 theBrandstettergeneral system1971).approach,A specialthecasetheoryof 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 of u = local mean velocity in x an obvious similarity between the series the linear assumption were clearly direction of nonlinear storages and the kinematic stated. h = local depth model. During the rising stage due to muiationAs we ofconsidersuccessivelythis examplemore abstractof for q(x,t) = lateral inflow rate per un- uniform lateral inflow, the solution it area within the zone of determinacy for ~ < models, it is obvious that many of the

g = acceleration of gravity for the kinematic model is given indistinct.model classificationsA certain amounttend toofbelumpingrather

s0 = bed slope S~ = friction slope of inputs and parameters is necessary and x and t are space and time coordi. h = qt for the distributed model because we nates respectively. or can only obtain discrete data and be. The above model distorts reality in cause we must use discrete mathemati many ways and its application to flow Q = a(qt~’ [8] cal methods to solve the partial differen over 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 ap flow 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 use hydrologic circumstances the above element can be written as the partial differential equation model equations can be very accurately ap- to predict the runoff response if the proximated by the kinematic wave N a watershed surface were changed pro- equation (Woolhiser and Liggett 1967). ~ = a [ SN] [9] vided one had independent information L In the kinematic wave equation, the a on the friction relationship for the momentum 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 the S=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 lumped in 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 can include the effects of slope, roughness plane is represented by a single storage all be looked upon as successively more and the regime of flow (laminar or element, however, this equivalence dis.. abstract models of the real physical turbulent). 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 distor propose that overland flow be approxi- and the ratio of the parameters for the Hon. This distortion or the lack of mated by a cascade of N non linear kinematic model and the lumped-system physical significance of model parame storages, each receiving inputs model. This suggests the possibility of ters is the price we pay for simplicity and reduced data requirements. 4cIN STATE OF THE ART WATERSHED MODELS

I______- - ‘~ 5N The Stanford Model (Crawford and 1 2-- ————N Linsley — 1962, 1966) is certainly the ters, particularly for the componeiit. Stanford Watershed Model IV Flowchart Dawdy et al (1970) have recently developed and tested a lumped system - Ke~ model describing surface runoff from

~ (S~#O.’j!!!S small watersheds. The model structure is similar to parts of the Stanford Model and it can be described by equation [101; however, there are differences in the functional relationships. A distributed watershed model in cluding overland flow, porous media flow and open-channel flow is not yet available. Freeze and Harlan (1969) I— —— ,,,ofl. — — — — proposed a three-dimensional approach which has not been completely imple mented as yet. (Fig. 5). Freeze (1971) has presented a three-dimensional model for unsteady, saturated or unsaturated flow in a groundwater basin. His model 0-I !.I”.’ is physically based in that it has a —in, structure based upon the theory of flow ~ led in a porous medium. Models that in ‘ji’!91!LflQa. clude a part of the hydrologic system have been presented by others (Kibler FIG. 4 Flowchart of the Stanford Watershed Model IV (from Crawford and Linsley 1966). and Woolhiser 1970; Brakensiek 1967; Henderson and Wooding 1964; best known general watershed model in shed. The Stanford Model has been used Machmeier and Larson 1967; Morgali current use. A flow chart ior this model by many investigators and many differ and Linsley 1965; Schaake 1965; Harley is shown in Fig. 4. This is basically a ent versions are available. However, the et al 1970; Amisial eta1 1969; Huggins lumped-parameter model, although basic structure of the model is essential and Monke 1970; and Smith and ly the same in all versions. Because of large, heterogeneous watersheds can be Woolhiser 1971). the extensive experience with this mod divided into subwatersheds if sufficient The parameters in the lumped system el, it is quite appropriate to compare data are available to estimate the param models have little direct physical signifi any new model structure with it to eters for the subwatersheds. The essen cance and, therefore, can be estimated ascertain if there has been an improve tial structure of the Stanford Model and only by using concurrent rainfall and ment. lumped parameter models in general can runoff data. In principle, the parameters be expressed in the following concise Holtan and Lopez (1971) have de in the distributed model have some notation: scribed the USDAHL-70 model of wa physical significance and the possibility tershed hydrology. Although their ob dS. N exists that they can be evaluated by —I-- + y~ ÷ ZY..=X ;i1,2,...N jective is to develop a distributed water 13 1 independent measurements. However, in dt shed model, the present model is pri practice the partial differential equa 1*1 madly lumped although, like the [10] tions describing distributed systems Stanford Model, a watershed can be must be solved by finite difference where S~ is the amount of storage (state broken down into smaller homogeneous methods. This representation requires variable) in the it~ storage element; Y1 is areas. Spatial variations in soil proper the specification of parameter values at output from the ~ element; ‘V11 is the ties and slopes are accounted for by a finite number of points. If the number flow from storage i to storage j; X1 is the dividing the watershed into land capabil of points is large, the cost of measure exogenous input to the ~ element. The ity classes which correspond to uplands, ments of the parameters becomes pro will usually be functions of the S~ hillslopes and bottom lands. The per hibitive. If the number of points is and Si and may be dependent on certain centage of outflow from each capability reduced drastically, the measured pa threshold parameters. The inputs ~ are class that is contributed to the other rameter values will not necessarily give independent of the state of the system, classes or the channel is estimated from good predictions of watershed behavior but the outputs, ‘Y1, depend upon the topographic maps. Overland flow is sim because of the distortions introduced by state of the system and may also depend dated with a lumped, nonlinear storage the order of approximation. upon current values of some inputs. For model. Because the structure of equation example, evapotranspiration rates may Although there are many similarities [10j is general enough to include an be a function of incoming solar radia between components of the Stanford extremely large number of different tion as well as soil moisture storage. The Watershed Model and the USDAHL-70 models, the question arises “How do I above system of equations are solved by model, the USDHAL-70 model attempts choose the best model for my particular finite-difference techniques on a digital to incorporate some aspects of spatial application?” Dawdy and Lichty (1968) computer. The methods by which infil variability by dividing the watershed suggest four criteria of choice that tration and evapotranspiration are com into land capability classes with speci might be used: (a) accuracy of predic puted infer spatial variability in these fied spatial relationships. The tion, (b) simplicity of the model, (c) fluxes but they are considered to be USDAHL-70 model also puts more em consistency of parameter estimates and independent of position on the water- phasis on a prjori estimation of parame (d) sensitivity of results to changes in Percipitotion -P (I) Orlob and Woods (1964) studied channel water quality aspects of an irrigated area Overland Flow by computer modeling of the water transport system and concurrent trans Eva port of conservative pollutants. They Tab e used a lumped-system model which could be described by equation (10]. They used a time step of one month. Runof 1-0 (I) Their investitagion demonstrated the Flow Lines possible effects of over- and (Soil Moisture 8 Groundwater) recirculation of waters on con Equipolentiol Lines centration of pollutants. (Soil Moisture & Groundwater) Another model using the lumped system approach to describe transport of a substance from the point of deposi tion on a watershed to the outlet of the watershed is that developed by Huff (Huff and Kruger 1967 (a, b)). Jn his initial work at Stanford University, Huff used the Stanford Watershed Model to describe the movement of water through a basin and expanded it to include the transport 0f radioactive aerosols (Sr90). Subsequently, he and FIG. S Schematic diagram of (a) watershed and (b) three-dimensional, his associates at the University of Wis discrete model of the watershed (from Freeze and Harlan). consin (Huff et al 1970) are attempting to modify the radionuclide transport model to include the transport of nutri parameter values. Although it is impossi (1956), Dobbins (1964). ents (Watts et al 1970). The initial ble to obtain an unambiguous ranking These refinements include the effects attempt in this area is the modeling of with multiple criteria, these criteria are of photosynthesis, sedimentation, bot the transport of nitrogen in an effort to obviously related to a criterion which tom scour and surface runoff on the discover mechanisms by which nutrient might be to maximize net benefits dissolved oxygen balance. More recently enrichment of lakes and streams is through use of a model in a design O’Connor (1967) developed a model related to human activity. situation. including, in addition to the above, Bloom et al (1970) used a very artificial aeration, photosynthetic pro WATER QUALITY COMPONENTS elementary lumped storage representa duction of oxygen, temporal and spatial OF WATERSHED MODELS tion of soil water and surface water as variations in flow rates, and diurnal elements of a mathematical model to There have been many investigations variation in photosynthetic activity. evaluate the transport and accumulation concerning water quality in certain re Goodman and Tucker (1969) utilized a of radionuclides. Their model was very stricted segments of the conceptual wa time-varying model in studying effec comprehensive in that it attempted to tershed discussed previously. Perhaps tiveness of treatment plants. include the transport of radionuclides the earliest and most widely used mathe All of the previously cited works from deposition on plants and soil, matical model describing water quality have either considered steady-state situ transport by water, ingestion by fish is that proposed by Streeter and Phelps ations or have considered time-varying and finally ingestion by man. This com (1925) to describe the dissolved oxygen cases as a series of steady states. This plex system was subdivided into ten relationship in a stream. The Streeter approach is probably adequate for non- “compartments.” From continuity con Phelps model considers a particular tidal rivers where streamfiow and siderations, the behavior of the system reach of a stream with an initial bio oxygen-demanding inputs do not vary was described by the system of equa chemical oxygen demand (BOD) and rapidly with time. tions: dissolved oxygen content at the upper Recognizing the possible inadequacy boundary. Under the assumption that of the series of steady states or the dy. N biochemical oxidation and reaeration by quasi-steady state case used in earlier ~D Aj~y~,i1,2,...N; dt n1 absorption of atmospheric oxygen are water quality work on esturaries, n#i first-order processes and the streamfiow Dresnack and Dobbins (1968), Bella and [11] process is pure translation at constant Dobbins (1968), Harleman et al (1968) velocity, the Streeter-Phelps model en and Dornhelm and Woolhiser (1968) where ables the prediction of oxygen deficit at devleoped methods for predicting water the amount of radionucide any point within the reach. The quality based upon the partial differen in the ith compartment Streeter-Phelps model considers only tial equation of unsteady free surface y,, = the amount 0f radionuclide the first stage of biochemical oxidation flow and longitudinal dispersion for the in the nth compartment and ignores the second or nitrification one-dimensional case. Solutions were the transfer rate coefficient stage which may also utilize significant obtained using finite-difference or finite A11 the elimination-rate coeffici amounts of oxygen. Several more com element methods. Unsteady, two-dimen ent plete models have been proposed—Camp sional cases have since been considered N the total number of com (1963), Churchill and Buckingham by Fischer (1969). partments in the system. Equation [11J is obviously a special system and involve the numerical solu unambiguous answers to specific re case of equation [10]. tion 0f partial differential equations. search questions. The authors contend that this simple, Obviously, any model describing the Empirical, lumped-system models linear, lumped-system model is justified dissolved oxygen content can also de will probably be useful tools in predict because sufficient experimental data are scribe transport of conservative sub ing transport of substances which are not available to estimate parameters of stances. naturally present in the environment more complex models. The Stanford Watershed Model has and which are currently being moni Two models that use a distributed been used to describe the hydrologic tored. However, if we must evaluate the system approach toward describing wa transport of radioactive aerosols environmental impact of new substances ter quality in an urban environment (Strontium 90) with some success and is before they are released or if the trans have been reported recently (Metcalf being modified to handle transport of port system itself may be changed, and Eddy, University of Florida and nutrients. The nutrient transport prob theoretical models appear to be the only Water Resources Engineers 1971, Uni lem appears to be less suitable for a possible approach. This means that we versity of Cincinnati 1970). lumped-system approach because the must develop objective techniques of a Hyatt et al (1970) developed a simu inputs are not spatially uniform. The priori estimation of model parameters. lation model of the transport of salts in transport of nitrogen as an important Effective research on transport of the upper Colorado River Basin. The nutrient seems to be especially difficult environmental contaminants by water hydrologic transport model used was because it can exist in six major forms will inevitably involve several disciplines similar in structure to the Stanford including organic compounds, all 0f because of the variety of problems in model but was solved by an analog which must be induded in the model. volved in constructing adequate models. computer. The salt transport model Added to the hydrologic complexity ignored ionic exchange and chemical which includes transport and storage of References liquid water and its transformation into precipitation phenomena within the soil 1 AmIsial, Roger A., .3. Paul Riley, and therefore is only applicable to solid or gas are the chemical or biologi K. 0. Renard, E. K. Israelsen. 1969. Analog computer solution of the unsteady flow equa steady-state situations. The basic time cal transformations of the various nitro tions and its use In modeling the surface unit was one month and the smallest gen forms. While many 0f these trans runoff process. Research Progress Report to ARS, USDA, Utah Water Research Labora spatial unit was a sub-basin of the formations can be described quantita tory, Utah State University. tively in the laboratory, little can be 2 Amorocho, J. and 0. T. Orlob. 1961. Colorado River. in further work at Utah Nonlinear analysis of hydrologic systems. State (Thomas et al 1971) a salt trans said regarding the extent to which they Water Resources Center Contrib. 40. UnIver take place under field conditions sity of California, Los Angeles. port model has been developed that 3 Amorocho, J. 1963. Measures of the includes reactions occurring in the soil (Stanford et al 1970). linearity of hydrologic systems. J. Geophys. Another problem which appears to Res. 68(8):2237-2249. such as ionic exchange and chemical 4 Amorocho, J. 1967. The nonlinear be significant is that of estimating initial prediction problem in the study of the runoff precipitation of gypsum and line. Both cycle. Water Resources Res. 3(3):861-880. of these models are intended to aid conditions. In hydrologic modeling, the 5 Amorocho, S. and A. Brandstetter initial conditions may not be particular 1971. Determination of nonlinear functional management decisions with regard to response functions in rainfall-runoff pro water quality effects of irrigation. ly important because the system has a cesses. Water Resources Res. 7(5):1087-1 101. relatively short memory—the state of 6 Bela, David A., and William B. If we consider only the lumped Dobbins. 1968. Difference modeling of parameter storage models, adding a wa the system after a few years is not very stream pollution. Journ. Sanitary Engr. Divi sensitive to the initial condition. How sion, Proc. ASCE. p. 995-1016. ter-transported pollutant requires a par 7 Bidwell, V. J. 1971. Regression anal ever, nutrients present in soil can be allel system of flows and storages similar ysis of nonlinear catchment systems. Water very large compared to those added by Resources Res. 7(5):1118-1126. in structure to that describing the flow 8 Bloom. S. D., A. A. Lain, W. E. fertilizer and will affect the system for a Martin and 0. B. Raines. 1070. Mathematical of water. The parameters of the quality long time. methods for evaluating the transport and model include the hydrologic model accumulation of radionuclides. Battelle Mem Considering the urgency of the prob orial Institute, Columbus. Ohio. NITS Report parameters plus certain reaction rate BMI-171-030. 39 p. lems associated with water quality, it is parameters to describe transformation 9 Brakenslek, P. L, 1967. A simulated certainly desirable to construct mathe watershed flow system for hydrograph predic of the substance under consideration. tion: A kinematic application. Proc. interna matical models describing these phe tional Hydrology Symposium, Fort Collins, Certain elements of the hydrologic mod nomena. However, we must give more Cob. el may be more important in predicting 10 Camp, T. R. 1963. Water and its serious attention to the question of impurities. Reinhold Press, New York. water quality than they were in predict 11 Churchill, M, A. and R. A. Bucking- model verification. Can we say in any ham. 1956. StatIstical methods for analysis of ing streamfiow rates. meaningful way that model A is better stream purification capacity. Sewage and In dustrial Wastes 28(4):517-537. than model B? if we cannot do this, we 12 Crawford, N. H. and R. K. Linsley. SUMMARY AND DISCUSSION are unable to determine if we are 1962. The synthesis of continuous streaniflow hydrographs on a digital computer. Stanford In this paper, I have briefly discussed making progress in our research a very University Dept. of Civil Engr. Tech. Report the structure of current general water unsatisfactory situation. it appears to 12, 121 p. 13 Crawford, N. H. and R. K. Linsley. shed models or those that include im me that research into environmental 1966. Digital simulation in hydrology: Stan ford Watershed Model IV. Tech. Report No. portant parts of the hydrologic cycle aspects of hydrology should involve two 39 Dept of Civil Engr., Stanford Univ. and have been used or could be used to types of modeling: (a) models of rela 14 Dawdy, D. R. and R. W. Lichty. describe the transport of pesticides, tively complex systems and (b)models 1968. Methodology of hydrologic model building. In the use of analog and digitel nutrients, radioactive nuclides or other of very simple systems. in constructing computers in hydrology. tnt. Ass Sd. Hydrol. Symp. Proc. Tucson, Arizona. II, No. substances. The objective of most water models of complex systems, we will find MATHS. p. 347-3 55. quality models has been to predict out what we need to know and we may 15 Dawdy, David R., Robert W. Lichty dissolved oxygen concentrations in seg develop operationally useful interim and James M. Bergmann. 1970. A rainfall-run off simulation model for estimation of flood ments of a stream or an estuary. The models. However, only by carefully peaks for small drainage basins—A progress more advanced models treat the river as constructed, controlled experimentation report. USGS Professional Paper 50GB. 28 p. 16 Dobbins, W. B. 1964. .BOD and oxy. a one- or two-dimensional unsteady on a more limited scale can we obtain gen relationships in streams. Proc. ASCE, Journ. of the Sanitary Engr. Div. 90. No. The chemical and physical parameters In a 40 O’Connor, Donald J. 1967. The tem SA3, P. 53-78. hydrologic transport model for radioactive poral and spatial distribution of dissolved 17 Dooge, J. C. I. 1959. A general aerosols. Proc. Int’l Hydrology Symp. oxygen in streams. Water Resources Res. theory of the unit hydrograph. Journ. 1:128-135, Fort Collins, Cob. 3(1):6 5-7 9. beophys. Res. 64(2):241-256. 30 Huff, Dale D. and Paul Kruger. 41 Orlob, Gerald T. and Philip C. 18 Dooge, J. C. 1. 1967. Hydrologic 196Th. A numerical model for the hydrologic Woods. 1964. Lost river system a water systems with uniform non-linearity. Mimeo transport of radioactive aerosols from precipi quality management study. Proc. ASCE, graphed Report, Univ. College, Dept. of Civil tation to water supplies. Isotope Techniques Journ. Hydraulics Dlv. 90(2):1-22. Engr., Cork, Ireland. in the Hydrologic Cycle. Am. Geophys. Un 42 Penman, H. L. 1961. Weather, plant 19 Dornhelm, Richard B. and David A. ion, Geophysical Monograph Series No. 11. p. and soil factors in hydrology. Weather Wo olhiser. 1968. DigItal simulation of 85-96. 16: 207-21 9. estuarine water quality. Water Resources Res. 31 Huff, D. D., D. G. Watts, 0. L. 43 Roseablueth, Arturo and Norbert 4(6):1317-1327 Loucks and M. Teraguchi. 1970. A study of Wiener. 1945. The role of models in science. 20 Dresnack, Robert and William E. nutrient transport with the Stanford Water’ Philosophy of Science XIl(4):316-321. Dobbins. 1968. Numerical analysis of BOD shed Model. Int’l Bio. Program, Deciduous 44 Schaake, J. a, Jr. 1965. SynthesIs of and DO profiles. Proc. ASCE, Joum. Sanitary Forest Biome Memo Report 70-1. Univ. of the inlet hydrograph. Tech. Report 3, Storm Engr. Div. p. 789-807. Wisconsin. Madison. project, Dept. of Sanitary 21 FIscher, Hugo B. 1969. A lagrangian 32 Huggins, L. F. and S. J. Monke Engineering and Water Resources, The Johns method for predicting pollutant dispersion in 1970. Mathematical simulation of hydrologic Hopkins Univ., Baltimore, Md. Bolinas Lagoon, California. USGS Open-File events of ungaged watersheds. Tech. Report 45 Sherman, L. K. 1932. Stream flow Report. Menlo Park, Calif. 14. Purdue Univ. Water Res. Res. Center, from rainfall by unit graph method. Engr. 22 Freeze, R. Allan. 1971. Three- Lafayette, Indiana. News Record 108:501. dimensional, transient, saturated-unsaturated 33 Hyatt, M. Leon, S. Paul Riley. M. 46 Smith, Roger S. and D. A. Woolhiser. flow In a groundwater basin. Water Resources Lynn McKee and Eugene K. Israelsen. 1970. 1971. overland flow on an infiltrating sir- Res. 7(2):347-366. Computer simulation of the hydrologic-salin face. Water Resources Res. 7(4):899-913. 23 Freeze, it Allen and R. L. Harlan ity flow system within the Upper Colorado 47 Stanford, G., D. B. England and 1969. Blueprint for a physically-based, digital River Basin. Report No. PRWGS4-1, Utah A. W. Taylor. 1970. FertiLizer use and water ly-simulated hydrologic response model. Water Rn. Lab., Utah State Univ.. Logan. quality. USDA-ARS 41-168. Journ. of Hydrology 9:237-258. 34 Jacoby, S. L. 5. 1966. A mathemati 24 Goodman, Alvin S. and Richard J. cal model for nonlinear hydrologic systems. J. 48 Streeter, H. W. and S. B. Phelps. Tucker. 1969. Use of mathematical models in Geophys. Res. 71(20):4811-4824. 1925. A study of the pollution and natural water quality studies. Report to Federal 35 KibIer, David F. 1968. A kinematic purification of the Ohio River. U. S. Public Water Pollution Control Administration, Na overland flow model and its optimizatiolt Health Bulletin, No. 146. tional Technical Information Service PR Ph.D. Dissertation, Colorado State Univ., Fort 49 Thomas, Jimmie, L., S. Paul Riley 188494. Collins. and Eugene K. Israelsen. 1971. A computer 25 Harleman, Donald R. L, Chok’Hung 36 Kibler, David F. and D. A. Woolhlser. model of the quantity and chemical quality of Lee and Lawrence C. Hall. 1968. Numerical 1970. The kinematic cascade as a hydrologic return flow. Report No. PRWG77’l, Utah studies of unsteady dispersion in esturaries. model. Colorado State Univ. Hydrology Papet Water Res. Lab., Utah State Univ., Logan. Proc. ASCE, Joum. Sanitary Engr. Div. p. No. 39. 50 Univ. of Cincinnati. 1970. Urban 897-911. 37 Machmeier, a. S. and C. L. Larson. runoff characteristics. Water Pollution Con 26 Harley, Brendan M., Frank E. Perkins 1967. A mathematical watershed routing trol Series 11024 DQU 10/70 EnvIronmental and Peter S. Eagleson. 1970. A modular model. Proc. Int’l Hydrology Symp., Fort Protection Agency, U.S. Govt. Printing Of distributed model of catchment dynamics. Collins, Cob. p. 64-71. fice. MIT Dept. of Civil Engr., Hydrodynamics 38 Morgafl, J. R. and Ray K. Linsley. 51 Watts, D. G, D. D. Huff, 0. L. Lab. Report No. 133. 1965. Computer analysis of overland flow. Loucks, M. Teraguchi. 1970. Models for 27 Henderson, F. M. and R. A. Woodiag. Proc. ASCE, Journ. Hydraulics Div. 91, No. systems studies -of nutrient flows In lakes and 1964. Overland flow and HY3, p. 81-100. streams. Int’l Bio., Program. Deciduous Forest from a steady rainfall of finite duration. 39 Metcalf and Eddy, Univ. of Florida Biome, Memo Report 70-4, Univ. of Wiscon Journ. Geophys. Res. 69(8):1531-1540. and Water Resources Engineers. 1971. Storm sin, Madison. 28 Holtan, H. N. and N. C. Lopez. 1971. water management model. Vol. I-IV Water 52 Woolhiser, D. A. and J. A. Liggett. USDAHL-70 mode! of watershed hydrology. Pollution Control Research Series 11024 DOC 1967. Unsteady, one-dimensional flow over a USDA Tech. Bull. No. 1435. 84 p. 10/71 Environmental Protection Agency, U.S. plane—the rising hydrograph. Water Resources 29 Huff, D. D. and P. Kruger. 1967a. Government Printing Office. Res. 3(3):753.771. ~urchase4 by USDA Agric. Research Sen-joe for °~~~Cja1 Use.