Proceedings—Research on Coniferous Forest Ecosystems—A symposium.Eellii4ham, Washington—March 23-24, 1972 I6&2

Hydrologic modeling in theConiferous Forest Biome

George W. BrownChairman, Hydrology SubcommitteeOregon State UniversityRobert H. BurgyHydrology Project LeaderUniversity of California, DavisR. Dennis HarrHydrology Project LeaderOregon State UniversityJ. Paul RileyHydrology Project LeaderUtah State University

AbstractThe objective of the hydrology program is to prepare a model which will provide predictions of the hydrologic

Mate of a coniferous watershed at any desired time and in any desired place, where state is defined by the inputweds of the other submodels or systems, particularly the producer and biogeochemical processes. Subsurface flows the dominant runoff mechanism in coniferous watersheds and one of the least understood processes inrydrology. Research projects in hydrology seek to understand this process using three different techniques. Onewoject relates subsurface flow to soil properties using direct measurement techniques. Another project approacheshe problem using simulation techniques. The third project utilizes systems analysis and statistical decomposition#f runoff events to make inferences about subsurface flow. These studies of hydrologic processes will bencorporated into a hydrologic model and linked to studies of other systems. The first step in linking our model9ith those of other groups is watershed stratification, a problem now solved by our modeling efforts.

IntroductionWater is an essential component of any eco-

ystem. In the Pacific Northwest, water is aominant element. The coniferous forests ofits region are noted as some of the best-.atered terrestrial ecosystems in the Unitedrates; water is the major linkage which tiesie terrestrial portion of the coniferous eco-'stem to the aquatic portion.Water performs several functions which

foster this linkage. Water must be viewed as acarrier. It carries organic and inorganic nutri-ents between the several compartments of theterrestrial portion of the ecosystem and fromthe terrestrial to the aquatic portion. Wateralso carries sediment from the terrestrial tothe aquatic portion of the system.

Water must also be viewed as a nutrientitself. It is an essential component of mostbiologic processes. The availability of water inthe soil governs both the initiation and termi-

49

nation of any process as well as the rate atwhich it proceeds.

Objective of the Hydrology Program

The objective of our program is to preparea model which will provide predictions of thehydrologic state of a coniferous watershed atany desired time and in any desired place,where state is defined by the input needs ofthe other submodels or systems, particularlythe producer and biogeochemical processes.

Structure of the Hydrology Modeling Effort

Our modeling effort is organized to payparticular attention to the many functions ofwater in the forest ecosystem. A generalizedmodel for water flow through a forest systemwas conceptualized long ago. This model isoften called the hydrologic cycle. Rothacheret al. (1967) showed for the H. J. AndrewsExperimental Forest that water movement inthe forest soil and evapotranspiration are themost significant processes governing waterflow.

Our studies focus upon subsurface watermovement. One project measures subsurfaceflow directly in the study watershed. Anotherstudy is a computer simulation of a water-shed. This approach will provide yet anotheravenue for assessing soil moisture and the sub-surface flow of water. The simulation modelwill be calibrated using 14 years of record onwatershed 2. Then the model will be verifiedwith the data available on watershed 10. Athird project seeks to develop techniques forpredicting the subsurface flow componentusing a systems analysis technique for statisti-cal decomposition of the hydrograph into itscomponents. Other studies will contributesubmodels to the simulation of the hydrologicsystem. The work of the Primary Producergroup on a transpiration model as well as thework of the Meteorology group on an evapo-transpiration model will aid our modelingeffort significantly.

We are concurrently working toward amore spatially refined model, the character of

which is determined as much by biologic ashydrologic constraints. Our latest efforts havefocused upon devising a system for stratifica-tion of watershed 10 which is amenable toboth hydrologic and biologic models.

Other hydrology projects are included inthe biome effort. We hope to provide hydro-logic measurements as a part of the work atFindley Lake and the evapotranspirationstudy at the Thompson site. We shall soonbegin to solicit and organize available datafrom coniferous watersheds throughout theWest in anticipation of extrapolating ourmodel to other ecosystems.

This has provided a broad overall view ofthe Coniferous Biome Hydrology Program—itsstructure to achieve a better understanding ofwater flow through the ecosystem and itsinteraction with other system components. Amore detailed description of the hydrologyefforts follows. Each project focuses uponevaluating water flow in a forested watershed.Analytical techniques vary between projects,but the ultimate goal of modeling subsurfaceflow mechanisms and watershed responselinks all projects together.

Subsurface Movementof Water on

Steep, Forested Slopes1With the exception of stream channel inter-

ception, the hydrographs of watersheds in theforested, steep topography of western Oregonreflect overall subsurface movement of water.Watershed response is rapid but withoutsurface runoff (Barnett 1963, Rothacher et al.1967). Although subsurface flow is by far themajor component of the hydrograph, virtuallynothing is known about the process on steepslopes. The objective in our study is to char-acterize the subsurface movement of water insteeply forested topography.

l Authored by R. Dennis Harr, Assistant Professor,Oregon State University, Corvallis.

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50

The study areas are located at several of thelower watersheds in the H. J. Andrews Experi-mental Forest near Blue River, Oregon. Vege-tation is typical of the low-elevation Douglas-fir forest. Study slopes average about 75 per-cent. Soil depth is variable with maximumdepths in excess of 5 meters. Because of highporosities (70-80 percent) and large propor-tions of macropores, these soils drain rapidly.Permeabilities of 5,000 and 900 mm per hourhave been noted on nearby watersheds forsurface soil and subsoil, respectively (Dyrness1969).

Methods

Initial investigations are being directedtoward describing the physical properties ofthe porous medium through which watermoves on its way to a stream. These field andlaboratory investigations will indicate wherewater movement most likely occurs.

Drilling with a portable power drill willfollow a grid pattern over a small stream-to-ridge portion of slope. At each grid point soildepth, depth and thickness of saprolite, anddepth to unweathered bedrock will be deter-mined. Additional drilling between initial gridpoints will indicate in more detail the surfacecontour of the impermeable parent material.Aluminum tubing placed in each hole willprovide access for measurement of ground-water level or soil moisture content.

In the laboratory, undisturbed soil corestaken from various depths in soil pits locatedover the study area are being analyzed. Suchproperties as porosity, pore-size distribution,stone content, permeability, and moisture re-tention characteristics are being evaluated.

The type and amount of measurements tobe made during and following winter stormevents in 1972-73 will depend on the in-formation gathered during initial field andlaboratory investigations now underway.Anticipated measurements include soil mois-ture content, vertical and lateral extent ofsaturated flow, soil moisture tension, precipi-tation, and water outflow from the base ofthe slope. Drilling and tracer studies willattempt to define the source area for this

water.

Preliminary Results

Although the study has just recently begun,certain observations have provided qualitativeinformation concerning the subsurface flowprocess on steep slopes. Precipitation movesdownward under the influence of gravity untilthis movement is obstructed. In some parts ofthe study area this obstruction may be causedb y rock fragments which cause shallow,localized saturation as evidenced in severalsoil pits during a period of heavy rain. Whererock fragments are not present, downwardmovement of water continues until the rela-tively impermeable parent material is reached.Here saturation occurs, flow acquires a hori-zontal component, and water begins movingtoward the stream.

At some point on the slope this saturatedflow is concentrated into pipelike subsurfacechannels. The cause of this concentration isunknown but could conceivably result fromthe microrelief of the impermeable material,from bedrock fractures, or from decayed rootchannels. At the toe of the slope the channelsare spaced about 1-6 meters apart. They lie onthe bedrock surface and appear associatedwith surface micro-relief. Shapes of their crosssections range from circular to flat rectan-gular. Width is also variable, ranging from 1centimeter to about a meter. Where thesechannels discharge into the stream channel,they are separated by soil which may containa shallow saturated lower layer from whichseepage occurs. Water velocity of the seepageappears to be several orders of magnitudelower than that of the subsurface channels.The latter accounts for the greatest portion ofstormflow.

The subsurface channels evident at the toeof the study slopes may be outlets of a sub-surface drainage system much like that de-scribed for other humid areas (Jones 1971).Water can be observed discharging from suchchannels in roadcuts and recent soil slumps atvarious slope positions in the vicinity of thestudy area. Such a subsurface drainage systemcould account for the rapid hydrOlogicresponse of these steep slopes.

51

Computer Simulationof Forest

Watershed Hydrology 2A hydrologist is often faced with the need

to predict system responses under variouspossible management alternatives. One ap-proach to this problem is to apply the tech-nique of computer simulation, whereby aquantitative mathematical model is developedfor investigating and predicting the behaviorof the system. In this study, a computermodel is being developed to simulate thehydrologic responses of a forest watershed,emphasizing the measurable variables relatedto the plant communities and soil types of thewatershed. The model represents the inter-related processes of the system by functionswhich describe the different components ofphysical and biological phenomena in awatershed.

Scope and Objectives of theSimulation Study

In the first phase of the study, the scope isbeing limited to the formulation of a funda-mental model of watershed hydrology whichtakes precipitation as the basic input andevapotranspiration and streamflow as outputsof the system. The various component proc-esses within the system are linked by the con-servation of mass principle. Depending uponenergy levels, water can vary among its solid,liquid, and vapor forms; hence, the energybudget is used as an auxiliary tool for main-taining the water balance. That aspect of thesystem involving water as a carrier of nutri-ents and sediments will be examined in asubsequent phase of the study. Under thisnext phase a water quality submodel will beformulated and added to the quantity modelnow being developed.

2 Authored by J. Paul Riley, Professor, Utah WaterResearch Laboratory, Utah State University, Logan;George B. Shih, Research Engineer, Utah WaterResearch Laboratory, Utah State University, Logan;and George E. Hart, Associate Professor, ForestScience, Utah State University, Logan, Utah.

The specific objectives of the current phaseof the study are stated as follows:

To develop and verify (calibrate andtest) a hydrologic simulation model for asmall forested subwatershed on the H. J.Andrews Experimental Forest.To estimate through model sensitivitystudies the relative importance of variousprocesses within the hydrologic systemof the model, with particular emphasison evaluating the soil moisture and inter-flow components.

Hydrologic System Models

Several hydrologic simulation models arecurrently available. Examples which might becited include Crawford and Linsley (1964),Sittner et al. (1969), and Riley et al .(1966).However, in order to meet the needs of thisstudy all existing models require some modifi-cations and further development. Therefore,on the basis of previous work at Utah StateUniversity a computer model is being devel-oped to simulate the hydrologic behavior offorest watersheds. The model will be appli-cable to a wide variety of geographical areasand management problems. In this study, datafrom watershed 2 on the H. J. AndrewsExperimental Forest will be used to demon-strate the utility of the model. Figure 1 sum-marizes the geophysical features of the studyarea (Rothacher et a. 1967). Data require-ments include air temperature, precipitation,runoff hydrographs, and characteristics of thewatershed (average slope, degree and aspect,vegetative cover, density of vegetative cover,soil moisture holding characteristics, anddrainage density). Other observed records,such as snow depth, soil moisture content willbe used to check the performance of themodel in simulating various component proc-esses of the system.

Figure 2 illustrates the various componentprocesses represented in this model, with theboxes representing storage locations and thelines transfer functions. Under this study, thehydrology of the drainage area will be syn-thesized first as a lumped parameter model inwhich the entire watershed area is consideredas a single space unit. On this basis, a dis-

52

1!ilIM

114•P‘404.1(Womul)

KmFigure 1. Watershed 2, H. J. Andrews Experimental Forest, Oregon. Area: 60.3 hectares; aspect NW;

average slope (percent): 61.1; elevation min.: 526 m; elevation max.: 1,078 m; main channel length:1,108 m; drainage density: 4.3 km/km 2 ; precipitation (1952-62): 2,400 mm/yr; runoff (1953-62):1,560 mm/yr; average evapotranspiration (1959-62): 540 mm/yr.

tributed parameter model will be developed inwhich the watershed will be divided into fourspace units, roughly corresponding to sub-watersheds within the area. The model willcompute continuous daily streamflow foreach subarea and route the contribution ofeach down the streams to the gaging station,where the computed and observed dischargeswill be compared. Other important outputfunctions from the model will include soilmoisture, actual evapotranspiration, and snowdepth. Several of the component processeswhich are illustrated by figure 2 are discussedin the following sections.

Interception

Interception is the part of precipitation

that is caught temporarily by forest canopiesand then redistributed either to the atmos-phere by evaporation or sublimation or to theforest floor. The amount of interceptiondepends upon storm size and intensity, andcanopy type and density. A report byRothacher (1963) showed that throughfall inthe study area was related to storm size bythe equation:Throughfall = 0.8311 x (gross precipitation) — 0.117 (1)

In equation 1 throughfall is, of course,bounded by the condition that it must begreater than or equal to zero. Although largeramounts of snow may be temporarily inter-cepted than rain, there is strong evidence thatmost intercepted snow ultimately falls and be-comes part of the snowpack.

53

CLOUDS

I

EVAPORATIONSUBLIMATION

INTERCEPTION

EVAPOTRANSPIRATION

SNOW SNOWA

0

CI)V

SNOW

STORAGEz01-1E-4

1-1

Cf)

E-4

C°

Y

SNOWMELTWATER STORED ONLAND SURFACE

quic

z0

OVERLAND FLOWSTREAM12

>s•E-11-1 INTERFLOW CHANNEL

r=4 STORAGE

SURFACEOUTFLOW

EVAPOTRANSPIRATION Y

SOIL MOISTURE

GROUNDWATERIN FLOW

z01-4H

aw ■-.1w 0A (-)

C443Pov

GROUNDWATERSTORAGE

EFFLUENT FLOW—Om"'INFLUENT FLOW-E--

GROUNDWATER OUTFLOW

Figure 2. A flow diagram of the hydrologic system within a typical watershed area.

54

100

s 0 0.5 1.0 2.0 3.0 4.0 Tr-1.5 -1.0 -0.5

90I)U

f:4rzUO

O

z

1:4

z0-1

30

20 >-c.)ELI

O'1 0

0

1 V 1

.111,

80

70

60

50

40

An alternative way of considering inter-ception quantities in a model is to expressinterception rate as a decaying function oftime limited by an average interception stor-age capacity for the watershed canopy. Thisapproach was incorporated into a watershedsimulation model by Riley et al. (1966).

Snow Storage and Melt

Forms of Precipitation

Only two forms of precipitation, rain andsnow, are considered in this study, with asurface air temperature criterion being appliedto establish the occurrence of these twoforms. Figure 3 (U.S. Army Corps of Engi-neers 1956) shows that at a temperature of1.5°C there is a 50-percent chance that theprecipitation will be in the form of snow. Astraight-line fit to figure 3 is used to deter-

mine the portion of rain in a given day ac-cording to the following equation.

Ta -R = P T (2)

r TTs

sin which

R = estimated portion of total dailyprecipitation occurring as rain

P = total daily precipitationTa= mean daily surface air temperatureTs= mean daily air temperature below

which all precipitation is assumedto occur as snow

Tr= mean daily air temperature abovewhich all precipitation is assumedto occur as rain

Precipitation falling as snow will be accu-mulated on the watershed until air tempera-tures rise sufficiently above the freezing pointto initiate snowmelt.

0

10cncla

20ct

30O

O40

zUoL

50

-a60

0En 700

80zO

90

100

SURFACE AIR TEMPERATURE, °C

Figure 3. Frequency distribution of precipitation in rain and snow forms.

55

PRECIPITATIONAT LAND SURFACE

r- - -- TEMP. —

NO YES

CONDENSATIONAND

EVAPORATION

SNOWPACKT >0°C YESL NO

YMELT

LIQUID WATER-HOLDINGCAPACITY OF SNOWPACK

GROUNDME LT

AVAILABLE WATERAT THE SOIL SURFACE

Figure 4. A flow chart of the snow accumulation and ablation processes.

56

a constant of proportionalityvegetation transmission coefficientfor radiationradiation index for a horizontal sur-face at the same latitude as theparticular watershed or zone understudyradiation index for a particularwatershed zone possessing a knowndegree and aspect of slopesurface air temperature in °Calbedo, or reflectivity, of the snow-pack surfaceprecipitation reaching the snow sur-face in the form of rain, incentimeters

in whichkm =kv =

RIh =

Rls =

Ta =A=

Prg =

Snowmen

A flow chart of the snow accumulation andablation processes is shown by figure 4. Rateof snowmelt depends primarily upon the rateof energy input to the snowpack. However,both the complex nature of snowmelt anddata limitations prevent a strictly analyticalapproach to the simulation of this process,and air temperatures are frequently applied asan index of available energy. Examples ofresearchers who have used this approach arePysklywec et al. (1968), Anderson and Craw-ford ( 1964 ) , Amorocho and Espildora(1966), and Eggleston et al. (1971). Becausetemperature data are the only indicators ofenergy levels available on watershed 2, adegree-day approach based upon the work ofEggleston et al. (1971) will be used in themodel of this study to represent the snowmeltprocess at the surface of the snowpack. Thiscomponent submodel includes mathematicalrelationships for various phenomena involvedin the snowmelt process. The submodel isapplicable to any geographic location by de-termining appropriate constants for certain re-lationships through a verification procedure.The relationship for surface melt rate isexpressed as follows:

Mrs = kmkviiTh Ta (1 - A) + Ta 80 (3)

Infiltration

Rates of water supply on the ground sur-face, whether in the form of rainfall minusinterception or snowmelt, must exceed infil-tration rates before any surface runoff occurs.The infiltration rate depends on the physicaland moisture characteristics of the soil, aswell as the surface organic conditions, and itis often expressed in the form of Horton'sexponential equation. However, the soils ofwatershed 2 are very porous and no overlandflow has been observed. Thus, all precipita-tion reaching the ground surface is assumed toinfiltrate into the soil and to move to thestream channels as subsurface flow.

Soil Moisture

Soils on the study watershed are relativelydeep and have a high porosity. Data on thephysical properties of the watershed soils areavailable (Rothacher et al. 1967), and thisinformation will be used to determine the soilmoisture holding characteristics.

The computer model allows infiltratingwater to satisfy first the available moistureholding capacity of the soil within the rootzone of the forest canopy. When the availablesoil moisture holding capacity is reached,additional infiltration is assumed to percolateby gravitation either somewhat laterally with-in the root zone or downward to deeper soilzones. Water which moves laterally usuallyreaches a surface channel within a relativelyshort period of time, whereas deep percola-tion moves from the watershed more slowlyand sustains streamflow during dry seasons.From preliminary studies (Rothacher et al.1967), approximately 87 percent of the totalannual precipitation reaches the ground sur-face, and about 75 percent of this quantitybecomes surface runoff. From an analysis ofstreamflow hydrographs it is estimated thatabout 10 percent of the runoff comes frombaseflow, which is contributed from deeppercolation. Thus, of the average annualprecipitation of 2,400 mm which falls on thewatershed approximately 2,100 mm enter thesoil, 440 mm are abstracted by evapotran-spiration, and the remaining 1,660 trim leave

RIs prg

57

the watershed as surface runoff, with 170 mmof this quantity occurring as baseflow. Thesoil moisture content computed by the modelwill be checked with observed data.

Evapotranspiration

Factors affecting evapotranspiration includetemperature, solar radiation, wind, humidity,and consumptive use by plants. However,only temperature and humidity data areavailable for the watershed. Among the com-m only used evapotranspiration equations(Veihmeyer 1964), the Penman equation isperhaps the most rational, but the data re-quirements are extensive. For this reason, themodified Hargreaves (Veihmeyer 1964), willbe used in this study. The equation is statedas follows.

U = ZKd (0.38 - 0.0038h) Ta(4)

in whichU= the daily potential evapotranspira-

tion in centimetersd= the daily daytime coefficient de-

pendent upon latitudeh= the mean daily relative humidity at

noonTa = the mean daily surface air tempera-

ture in °Ca monthly consumptive use coeffi-K=cient which is dependent uponplant related characteristics, such asspecies, growth stage, and densityon the watershed

The influence of soil water on evapotran-spiration has been the subject of much re-search and discussion. It is now generallyrecognized that there is some reduction inevapotranspiration rate as the quantity ofwater within the root zone decreases. In thisstudy it will be assumed that evapotranspira-tion occurs at the potential rate through acertain range of the available soil moisture. Acritical moisture level is then reached at whichactual transpiration begins to lag behind thepotential rate. Within this range of the avail-able soil moisture the relationship betweenavailable water content and transpiration ratewill be assumed to be virtually linear. Thus,

E = U, [Mes < M s(t) < Mcs]

(5)

in whichE = daily evapotranspiration adjusted

for the influence of soil moisturelevels

Ms= quantity of water stored within theroot zone and available for plantuse at any time, t

NIes = limiting root zone available mois-ture content below which soil mois-ture tensions reduce evapotranspira-tion ratesroot zone storage capacity of waterMcs =available to plants

Ms (t)E = U, [Mes > Ma (t) > Oj (6)Mes

Considering the pressure effect, the total rateof gravity water storage depletion throughboth interflow and deep percolation isassumed to be directly proportional to thequantity of water in this form of storage re-maining in the soil profile at any particulartime. The interflow portion of this depletion,Nr, will be expressed as follows:

1 ddtGsNr (t) = K • - K- (Kg Gs (t)) (7)

in whichinterflow= depletion coefficient

Kg= gravity water depletion coefficient

That is, Nr = KiKgGs(0)e -Kgt

, in which grav-ity storage at time, t = 0 is represented byGs(0) and no input to Gs is assumed to occurbetween t = 0 and any other time, t. It is esti-mated that on watershed 2 about 90 percentof the gravity water storage leaves the area asinterflow, in which case KiKg = 0.9.

Groundwater

Water enters groundwater storage as deeppercolation from the overlying plant rootzone. The rate of deep percolation, Gr, isnumerically equal to the total rate of gravitywater depletion within the root zone less theinterflow rate. Thus,

58

G - r Qrg dtd Gw

(12)

Gr = (1 - K.)1 dt

By integrating equation 9 over a specific timeperiod the accumulated inflow to the ground-water basin, Gw, is estimated for this timeperiod.

GcGov f t (1 - K.1 dt) d dt

If the groundwater basin is considered as alinear reservoir, the outflow rate is given bythe expression

Qrg = Kb Gw (11)

in whichKb = a coefficient which is estimated

from dry season streamflow hydro-graphs

Qrg = the outflow rate from the ground-water reservoir

By combining equations 9 and 11, the netrate of storage change within the groundwaterbasin is derived as

By substituting equation 11 into equation 12and rearranging terms, the following relation-ship is obtained.

d dtQrg - Kb [Gr (t) - Cirg ( t)] (13)

The rate of discharge from the groundwaterbasin as baseflow is obtained by solving equa-tion 13 for Qrg.

Runoff

The possible sources of streamflow at anyreach within a channel are overland flow (sur-face runoff), interflow, groundwater, and up-stream input. Manning's equation is usuallyapplied to compute overland and channel flowrates at any point. Under conditions on water-shed 2, however, surface runoff does not occur,and channel routing on a daily time increment

is not significant. Therefore, runoff rates at thestream gage are given by summing the inter-flow and groundwater discharge rates..

Model Verification

Model verification includes calibration ofthe model parameters to a particular area,testing the sufficiency of processes defined inthe model, and examining the prediction per-formance of the model. A self-calibration sub-routine will be included in the model wherebythe program will search for optimal modelparameter values. Under this procedure eachwater year is used as a unit for optimizationand the objective function is to minimize thevariance between observed and computedstreamflow (Shih 1971). The sufficiency ofprocesses defined in the model is reflected inthe dispersion of parameter values resultingfrom each year of calibration. After themodel is calibrated, those years of data whichwere not used for calibration are used toexamine the confidence level of predictionsby the model. A flow diagram of the modelverification procedure is shown by figure 5.

Model Parameters

Model parameters are the coefficients usedin defining the processes which have not beenaccurately measured or which cannot bedirectly measured. By establishing the valuesof these coefficients the general model isfitted to the hydrologic system of a specificwatershed. Depending upon the resolution ofthe model and the availability of data, thenumber of parameters to be calibrated mayvary. In order to avoid using a large numberof degrees of freedom in the calibrationprocess and to save computation time, thenumber of model parameters should be keptas few as possible. In this study, the prelimi-nary model parameters to be calibrated areinterception storage capacity (Si), snowmeltcoefficient (Ks), soil moisture retentioncapacity (Mcs), gravity water depletion coeffi-cient (Kg), and groundwater recession coeffi-cient ( Kb).

d Gs(9)

(10)

59

I

CHECK ACCURACY ANDPARAMETER VALUE DISPERSION

YES

SENSITIVITYANALYSIS

APPLICATION

SUBWATERSHEDMODEL

'AVAILABLE DATA CALCULATE WATERSHED COEFFICIENTS

A ESTIMATE THE INITIAL VALUE ANDLIMIT OF MODEL PARAMETERS SET doi WATERSHED HYDROLOGIC

SIMULATION MODELWATER YEARHYDROLOGIC UP PARAMETER VALUE CONSTRAINTS

DATA

ENTIRE WATERSHED AS SINGLE UNIT CALIBRATE THEMODEL PARAMETERS

IMPROVEiMODEL

DAILY TIME INCREMENT

I CONVERT MODEL COEFFICIENTSAND PARAMETERS FOR DAILYTIME INCREMENT

Y

NO

CALIBRATION, RESULTS STUDYAND COMPARISON WITH

MONTHLY MODEL

NO

NO

CALCULATE AND ESTIMATESUBWATERSHED COEFFICIENTS

AND PARAMETERS

IMPROVEDEFINITION

OF PROCESSES

Figure 5. Flow diagram for verification procedure.

60

Calibration of Parameters

In general, it is anticipated that realisticammeter values are established through theiilibration procedure. However, when stream-

:-.ow is the only available component for:ecking the model, it is possible that severaliimbinations of parameter values will yieldtt isfactory agreement between observed andimputed outflow hydrographs. The problem

of establishing unique parameter values isvproached on the basis of hydrologic judg-ment and by using "interior" observations,uch as snow depth and soil moisture, ashock points on model performance. Other

.Says of testing the model include the timeiii.tribution of output quantities, such asstrea m flow , and known (or estimated)monthly or annual quantities. For example,or watershed 2 it is estimated that intercep-

. .on storage is about 0.5 cm, and total inter-ept ion amounts to approximately 17 percent.1 the annual precipitation.

'ilder the self-calibration technique modelammeter values are altered or purturbed in a

:,:ldorn sequence and the resulting changes in:hi• objective function are examined (Shih

11. A computer flow chart for the calibra-,•n subroutine is shown by figure 6. The.;;re program model, including the calibra-

. .11 subroutine, will be synthesized on avbrid computer.

Sensitivity and Management Studies

Sensitivity

\ sensitivity analysis is performed byii;inging one system variable while holdingho remaining variables constant and noting

changes in the model output functions. If-mall changes in a particular system parameter

large changes in the output or response...inct ion, the system is said to be sensitive to

parameter. Thus, through sensitivityi!:,Ilyses it is possible to establish the relativen i!)ortance with respect to system response

various system processes and input func-ii,,ns. This kind of information is useful from

standpoint of system management, system: • o ileling, and the assignment of priorities in.0 collection of field data. Under this study,

the verified model will be used to performvarious sensitivity analyses for the hydrologicsystem of watershed 2.

Management

Opportunities for management of a forestwatershed are widely varied, and range fromchanges in logging practices to forms of soiltreatment. Actual implementation of amanagement scheme depends upon benefitsgained as compared with possible disbenefits.The simulation model developed under thisstudy will not make direct comparisons ofbenefits and disadvantages, but will predictchanges in the system output associated withgiven management alternatives. Under thisstudy the capability of the model will bedemonstrated for rapidly testing manypossible management alternatives.

Much of the work discussed by this paper isbased upon past developments in watershedsimulation at Utah State University. Hydro-logic modeling of the H. J. Andrews Experi-mental Forest for Coniferous Forest Biomehas just begun, and the preceding discussionhas been influenced by a consideration ofparticular conditions in the study area. How-ever, the model will be fundamental inconcept, and therefore generally applicable ina geographic sense. Whenever feasible, themodel will use basic equations which are validfor short time intervals to define the variousprocesses in the model. The output will thenbe summed for application to daily or longertime increments. For example, Horton's infil-tration equation is applicable in minute units,but by summing these quantities, the model iscapable of calculating equivalent daily infiltra-tion rates. For the area under this study, how-ever, it is assumed that water supply rates atthe ground surface do not exceed infiltrationcapacities, so that surface runoff does notoccur, thus considerably simplifying themodel calibration process.

System functions which are important toforest management and other aspects of thetotal project include evapotranspiration andsoil moisture. These functions, along withstreamflow, will be estimated by the modelfor use in other parts of the total system

61

RESULTIMPROVEDIMPROVEMENT

SIGNIFICANT

PARAMETERVALUE AT EXTREMES

OF LIMIT ? RESULTIMPROVED

9

SET BACK TO ORIGINALPARAMETER VALUE

RESULTSFROM THIS SE

OF PARAMETER VALUESATISFY THECONSTRAINTS

YES

EQUALSL NO. OF

PARAMETERS

NOYES

ASSIGN OPPOSITE DIRECTIONOF PERTURBATION INNEXT CYCLE

YES

31,1 OBTAIN INITIA L MODEL ERROR

SET UP PARAMETER OPTIMIZINGORDER IN RANDOM SEQUENCE

PERTURBATE A PARAMETER

No

CHANGE THE PERTURBATION DIRECTIONWITHIN GIVEN PARAMETER LIMIT

NO

PARAMETERS TESTEDWITHOUT SIGNIFICANT

IMPROVEMENTL = L + 1

ADJUST THEINITIAL VALUES A,

ALL

PARAMETER PERTURBED9

NO NO

RETURN I

Figure 6. Flow chart for the model calibration.

62

INPUT(KNOWN)

SYSTEM

(KNOWN)

OUTPUT

(UNKNOWN)

(A) SYSTEM SYNTHESIS

INPUT(KNOWN)

SYSTEM

(UNKNOWN)OUTPUT(KNOWN)

(B) SYSTEM ANALYSIS

model being developed under the ConiferousBiome Program. The important underlyingfeature throughout the entire study will bethat all of the separately described hydrologicprocesses and phenomena are interlinked intoa total system. Thus, from the model, hope-fully, it will be possible to evaluate the rela-tive importance of the various items, explorecritical areas where data and perhaps theoryare lacking, and finally establish guidelines forthe improved management of forest water-sheds.

Hydrologic Systems Analysis 3

The purpose of this research is to devise atechnique for statistical decomposition of ahydrologic event such that system processessuch as precipitation, subsurface flow, andevapotranspiration, which contribute to theobserved streamfiow can be separated anddescribed. This technique will therefore pro-vide one more avenue for determination ofthe subsurface flow process on forest soils.The technique chosen for this research is aform of systems analysis.

Systems, Definitions and Basic PrinciplesSystem may be defined as an aggregate of

physical parts that do not change with time,operating on an input to produce an output,both being functions of time. The simplifiedrepresentation of a watershed, given in figure2, can be considered as a system whose inputis precipitation and runoff its output. System-synthesis" is a technique employed when the,y,;tem is known in terms of a mathematicaloquation; the objective is to determine thenature of the output for any class of inputfig. 7). In system "analysis" a system

response function or kernel which best de-st •ribes a given input-output pair is derivedfig. 7). The term "best" implies that the

derived kernels are not unique. Combinationso f both techniques can be used for the solu-tion of hydrologic problems. A system can

Authored by Z. G. Papazafiriou, Research\ ,ciate, and R. H. Burgy, Professor, University of

t l ornia, Davis.

(C) MULTI-INPUT/OUTPUT SYSTEM

Figure 7. Illustration of systems.

have one input and one output or many in-puts and outputs (fig. 7). A system is"lumped parameter" if input and output arefunctions of a single variable. Otherwise, thesystem is of the "distributed parameter" type.If the system response at any time, due to agiven input, is uniquely determined, thesystem is said to be "deterministic." If thesystem response is subject to uncertain influ-ences, the system is "stochastic orprobabilistic."

A quantity z is a "functional" for the func-tion x(t) in the interval (a,b), if it dependsupon all values taken by x(t), when t varies inthe interval (a,b). An illustration of a func-tional is given in figure 8. The output of asystem is a functional of the input and, forthe same reason, runoff is a functional ofprecipitation. A system is "time invariant" ifit does not change with time. Such systemscan be represented by functionals. "Physicallyrealizable" is a system whose output at time tdepends only upon past values of the input.

63

Figure 8. Demonstration of a functional

Hydrologic systems are physically realizablesince their outputs (runoff) at time t dependonly on the past values of their inputs (pre-cipitation). The "memory" of a system is thetime period between some past time and thepresent for which the output depends onlyupon the input. If the output depends onlyon the present value of the input, the systemis said to be a "no-memory" system. If theoutput of a time invariant system is analyticabout zero input at some time to, the systemis "analytic". Analyticity is very important,since if a system is analytic, its output can beexpanded in Volterra series. Hydrologicsystems are assumed to be analytic.

A deterministic system H is said to be"linear," if given the inputs X i (t) and X2 (t)such that

can be almost linear, but there is no linearsystem which can be almost nonlinear. Ingeneral, linearity is a limiting case of non-linearity. Therefore, any theory or techniqueadequate for a general nonlinear system isequally adequate for linear systems.

Deterministic Linear Hydrologic Systems

The theory behind most linear methods canbe generalized in the following manner. Sup-pose that s and a are continuous variablesrepresenting position in space, and t and rdefine position in time. Consider the linearP. D. E. of the general form

L[g(s,t)] = f(s,t) (17)where L is linear P. D. operator of arbitraryorder, and g(s,t) some function which satisfiesequation 17 within a certain region R. Giventhe appropriate homogeneous boundary con-ditions along R, the solution of equation 17can be written according to Hildebrand(1958) as

g(s,t) = f f G(s,t; a,r) f (a,r) da dr (18)It

If

f(s,t) = fs (s) ft (t). (19)equation 18 can be written in the form

g(s,t) = f f(r) [ f G(s,t; a,r) dal dr (20)

If fs (s) is spatially invariant, we may writey / (t) = H [AX / (t)]

Y2 (t) = H [BX 2 (t))

(14) rg(s,t) = j G (s,t; r) f (r) dr

(15)(21)

implies thaty i ( t) + Y2 CO Y[X(t)]

= 11 [AX/ + BX 2 (t)]

= ALI [Xi (t)] + BH [X2 (t)]

where

(16) f (r) = f(s,t), and G(s,t; r)

= fG(s,t; a,r) da

(22)

that is, in a linear system, each member of asequence of input values influences the out-put independently of every other. This is thewell known principle of superposition. If asystem does not satisfy the above condition itis said to be "nonlinear." A nonlinear system

We can write equation 21 in differentialequation form as

dng(s,t)dtn4

+ Ao(s,t) g(s,t) = f(s,t) (23)

An(s,t)dtn

n-1+ An_i(s,t) d g(s,t)

64

EFFECT WERAINFALL

LOSSES

Usually, we are interested in the output vari-able at some particular point in space (aparticular gaging station), in which case equa-tion 23 becomes

dg(t) do-1g(t)An(t) + An-1(t)dtndtn-1

+ Ao(t) g(t) = f(t) (24)which describes a spatially lumped parameter,time-varying linear system. If we assume thatthe parameters in equation 24 are time in-variant, we obtain

dng(t)A + A -An dtnn-1

Aog(t) = f(t) (25)which describes a time invariant, lumpedparameter linear system. If we assume thatthe system is completely at rest at t=0, we canwrite equation 25 in the form of the con-volution equation

g(t) = f h(r) f(t-r) dr (26)0

which is the basis of the unit hydrographtheory and many other hydrologic tech-niques. In equation 26, g(t) represents runoff,f(t) rainfall, and h(t) is the kernel, or in thiscase, the unit hydrograph.

In summary, application of equation 26implies that the watershed behaves as a linearsystem, it is time invariant, the rainfall isuniformly distributed over the watershed

area, and the watershed is completely at restat the beginning of the rain. The "effective"precipitation is used as an input to thesystem. This implies that we know somemethod for the separation of the runoffhydrograph into base flow and direct runoff(fig. 9). This approach in fact is a combina-tion, using system synthesis for the estimationof the effective precipitation, and systemanalysis for the estimation of runoff. Methodsusing this technique have been developed bySnyder (1955), Eagleson et al. (1966), Nash(19 5 7, 1960), O'Donnell (1960), Dooge(1965), and others.

Deterministic Nonlinear Hydrologic Systems

As it was noted in a previous section, atime invariant analytic system can be ex-panded in Volterra series. Such an expansioncan be written in the form

Oa

y(t) = ho + f h i (r i ) x(t-r i ) dri

ff h2 (T1 ,T2 ) X (t-T Xa-T2 dT2...OP ...Oa

GO

hn (r Irn) x(t-r )

x(t-rn) drn

n-d1 g(t)

dtn4

(27)

EFFECTIVE RAINFALL DIRECT RUNOFF

X(T)DIRECTRUNOFF

III■ xxt..tBASE FLOWY (T)

Figure 9. Estimation of effective rainfall through hydrograph separation.

65

where hi are the kernels of the system andh0=0 unless a source or sink is present. If thesystem is physically realizable, and has finitememory (as it happens with hydrologicsystems), equation 27 can be written in theform u

y(t) = f h i ( r 1 ) x(t-r 1 ) dr10

U

f f h2 (T 1 I,T2 X(t-T 1 ) X(t-T2 WTI dT2

0 0

U

ff hn(Ti Tn) x(t-T1)0 0x(t-rn)dr idrn

(28)

subject to the condition

hi(t) = 0 for all T < 0

and where u is the length of the memory. If,instead of continuous functions, discrete setsof data are used, equation 28 can be writtenin the form

U

y(T) = E H i (S i ) X(T-S1)S1=0

U U

+ E E H2 (Si ,S2 ) X(T-S 1 )S 1 =0 S2=0 X(T-S2)

U U+ Hn(Si Sn) X(T-S1)S 1 =0 Sn=0 x(r_sn) (29)

The first term in equation 28 or 29 repre-sents a linear system, that is an ordinary con:volution integral. The other terms are ageneralization of the convolution integral. Ingeneral, the ith term represents a pure sub-system of order i. Therefore our system y(t) iscomposed of the summation of a linear and a

series of nonlinear subsystems. If we representthe successive terms of the system by H i (t),}1 2 (t), . . ., respectively, the sys-tem can be written in the form

y(t) = H i (t) + H2 (t) .... Hn(t) + (30)

An illustration of a nonlinear system is givenin figure 10.

A system is identified whenever its kernelshn(t) are calculated. This evaluation is basedon the past behavior of the system. Theoreti-cally, after knowing all the kernels, theresponse of the system can be calculated forany given set of input values.

Equation 28 (in the form given) is verygeneralized and its usage is limited and timeconsuming. It is desirable to reduce the opera-tions involved. For example, given a sequenceof inputs, sometimes sequential values arehighly correlated, and the system itself maysmooth out rapid fluctuations. Hydrologicsystems demonstrate both of these charac-teristics. Thus, we may look for ways whichcan describe the sequence by a smallernumber of parameters. If the sequence x i ....xj....xn represents observations at timesti....tj....tn, respectively, they can be approxi-mated by a polynomial of the form

Xk (t) = ao + t + + aktk k

amen(31)m-O

where k<n, and such that

ej = IXk(tj ) - xi! , (j = 1.... n) (32)

is sufficiently small. Using such a polynomialapproximation a number of difficulties arise.The most important is that the method be-comes very sensitive to error when k exceeds7 or 8 (Forsythe 1957). The use of ortho-gonal polynomials solves these difficulties.The normal equations of the least square datafitting completely decouple. The evaluationof the standard deviation employing thenull hypothesis becomes much less time con-suming. Fourier series expansion of a functionfalls into this category. Another approach isthat of weighted polynomial expansion withinan interval (the length of memory). If onetakes as many polynomials as the length of

66

3rd ORDERY a (T)

LINEAR

Y1 (T)

2nd ORDER

Y2 (T)

>- OUTPUTY (T))it

••■•••• ••■•••

INPUTX (T) •

1=11■11• ■•■••••■ •■•

nth ORDER

Yi (T)

Figure 10. Illustration of nonlinear system representation by Volterra Series_

the memory, an exact match could be madeto each of the input values. However, noeconomy in the description would have beenaffected. Instead, fewer polynomials can beused. This introduces an error, but due to thenature of the operation, it will be the leasterror possible.

Methods using the above procedure havebeen introduced by Jacobi (1966), Harderand Zand (1969), and Brandstetter andAmorocho (1970). It is this method which isbeing considered for use in this study. Thereare certain distinct advantages associated withthe process. The method is quite general andcan handle a variety of problems such as run-off, chemical quality of runoff waters, andsuspended sediment predictions, when inputvalues are properly weighted by functionsdescribing the physical processes involved in

each case. Once the response functions orkernels of the system are evaluated, they maybe used for predictions given any sequence ofinputs. It requires a minimum amount ofdata, possibly 1 to 2 years of good records.Systems can be used in cascade, like

predict predictprecip.--)-runoff-o-quality or sediment

It can be used for quantitative evaluations ofthe changes created by any type of watershedmanagement procedure by evaluating thekernels of the original and the managedhydrologic system. Emphasis will be given toestablishing weighted functions for the bestdescription of the physical process, and onderiving tools for the greatest economizationof the procedure.

67

2100BOUNDARY

- COMPARTMENTSUBCOMPARTMENT

WATERSHED-100 200 4001'1 I FEET

CONTOUR INTERVAL = 25 FEET

Watershed Stratification:A Problem on Watershed 104

We have also begun to structure thehydrologic model for watershed 10 concur-rently with the simulation study underwayfor watershed 2 and the watershed systemsanalysis. Our objective here is to prepare afine-resolution hydrologic model which in-corporates the transpiration model from thePrimary Producers, the evapotranspirationmodel from Meteorology, and provides soilmoisture and subsurface water flow for thePrimary Producer and Bio-Geochemical Proc-esses groups. The first step in structuring sucha model is system stratification.

One key to the analysis of complex systemsis the compartmentalization of the systeminto homogeneous subsystems which can then

4 Authored by George W. Brown, Associate Pro-fessor, Oregon State University, Corvallis.

be isolated for study. This should be done insuch a way that the linkage between compart-ments is simple and direct.

Another key is placement of compartmentboundaries in such a way that the cells areeasily uncoupled, or are coupled as a simplelinear cascade. To do otherwise would compli-cate the modeling considerably. A hydrologicmodel that consists of a series of compart-ments arranged as a branching cascade isextremely difficult to manage. Water flowfrom an upper compartment must be some-how divided between lower compartments inthe cascade. The basis for such division isgenerally obscure and usually arbitrary.

In our attempt to structure the hydrologicmodel for watershed 10 at the next level ofresolution, we began by setting compartmentboundaries along stream courses. This auto-matically decoupled the compartments, sincewater does not cross the channel (fig. 11).

Next, it was necessary to consider the ar-rangement of the plant communities within

19" :°1/4WM46""":

2000

17001 00

ov,°%47414%ftwoula...

-

alaW1 "(4S

!Ar t 1707 or4

Figure 11. Initial and secondary stratification of watershed 10, H. J. Andrews Experimental Forest, Oregon.

68

the watershed. The hydrologic model and theprimary producer model are obviously linked.The principal coupling variable betweenmodels is the soil moisture profile. Soil mois-ture is that portion of the "hydrologic state"of the watershed which closely regulates plantgrowth. Plants, in turn, influence soil mois-ture by transpiration. Thus, superimpositionof the primary producer's vegetative structureupon the structure of the hydrologic system isessential. This structure was combined withthe initial hydrologic stratification and de-fined the subcompartment boundaries. Sub-compartment boundaries approximate thevegetative type-map boundaries and are ar-ranged into riparian, midslope and ridgetopzones. These zones undoubtedly reflect thechanges in soil moisture regime within thewatershed. Also, this stratification allows usto consider flow between subcompartments assimple linear cascades.

This final stratification for watershed 10will provide the basis for sampling schemes tocharacterize soil moisture, water flow andother hydrologic and biologic processes neces-sary for the next round of model construc-tion. It is essential to note that this stratifica-tion is compatible for modeling hydrologicprocesses and is also compatible for linkingthe hydrologic model with that of the pri-mary producers. It is the major achievementof our initial modeling effort and sets thestage for continued progress.

AcknowledgmentsThe work reported in this paper was sup-

ported in part by National Science Founda-tion Grant No. GB-20963 to the ConiferousForest Biome, U.S. Analysis of Ecosystems,International Biological Program. This is Con-tribution No. 24 to the Coniferous ForestBiome.

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