Estimation of accumulation parameters for urban runoff quality modeling

WATER RESOURCESRESEARCH, VOL. 17, NO. 6, PAGES 1657-1664,DECEMBER 1981

Estimation of Accumulation Parameters for Urban Runoff Quality Modeling WILLIAM

M.

ALLEY

U.S. Geological Survey, Reston, Virginia 22092

PETER E.

SMITH

U.S. Geological Survey, Bay St. Louis, Mississippi 39529

Many recentlydevelopedwatershedmodelsutilize accumulationand washoffequationsto simulate the qualityof runofffromurbanimperviousareas.Thesemodelsoftenhavebeencalibratedby trial and error and with little understanding of model sensitivityto the variousparameters.Methodologiesfor estimating best fit values of the washoff parameters commonly used in these models have been presentedpreviously.In this paper, parameteridentificationtechniquesfor estimatingthe accumulation parametersfrom measuredrunoffqualitydata are presentedalongwith a sensitivityanalysisof the parameters.Resultsfrom applicationof the techniquesand the sensitivityanalysissuggesta need for data quantifyingthe magnitudeand identifying the shape of constituentaccumulationcurves. An exponentialaccumulationcurve is shownto be more generalthan the linear accumulationcurvesused in most urban runoff quality models. When determiningaccumulationrates, attention needs to be given to the effectsof residualamountsof constituentsremainingafter the previousperiod of storm runoff or street sweeping. INTRODUCTION

run. The alternative

of continuous

simulation was assumed

to representa better approachand the modelwas developed as a variable time step, continuoussimulationmodel. The model provides detailed simulationof the quality of stormnot most of these models utilize some variation of the runoff during days for which short-time interval (1 min to accumulation and washoff algorithmsfor impervious area runoffthat were incorporatedin the first versionof the well- hourly) data are input to the program. Between these storm known storm water management model, abbreviated event days, daily precipitation data are used to provide a SWMM [Metcalf and Eddy, Inc., et al., 1971]. Models daily accounting of constituent accumulation. The model containingmany of SWMM's algorithmsinclude STORM consistsof two main components:constituentaccumulation and constituent washoff. [U.S. Army Corps of Engineers,1976]and the NPS model [Donigian and Crawford, 1976].These modelshave often been calibratedby trial and error and with little understandCONSTITUENT ACCUMULATION ing of model sensitivityto the variousparameters.In fact, 'default values,' recommendedin model user's manuals,are Most urban runoff quality modelsassumethat the accumuoften arbitrarily used. In an earlier paper, Alley [1981] lation of water quality constituentson impervioussurfaces A number of modelshave been developedduring the last decadewhich simulatethe quality of urban runoff. Many if

presentedmethodologies for estimatingbestfit valuesof the washoffparameterscommonlyused in these models.The purposeof thispaperis to exploremethodsof estimatingthe accumulationparametersusedin thesemodelsand to investigate the sensitivityof model output to changesin these parameter values.

occurs at a constant rate. However, several studies have

suggested that the rate of accumulationis nonlinearandthat there is a limit to the amount of constituents that can

accumulatebetween storms,regardlessof the lengthof dry

period.Data collectedby Sartorand Boyd [1972]and Pitt [1979] suggestthat the accumulationrate is largest for severaldaysafter a periodof streetcleaningor rainfall.Then

To perform the study, accumulationand washoff algorithmswere formulatedinto a lumpedparametermodel. The the rate decreasesand eventually approacheszero, apparqualifter'lumped parameter'refers to the fact that spatial entlybecauseconstituents are resuspended by windandland variationsin modelparametersare not accountedfor. Thusa use activities such as vehicles moving along a highway. The singleset of valuesfor the accumulationand washoffparam- amountof constituents depositedon perviousandnoneffecetersis assumedto applyfor a particularwatershed,andthe tive impervious areas is less available for transport by model uses runoff values at the outlet from the watershed. It rainfallrunoff.This phenomenon canbe modeled[Shaheen, is assumedthat the runoff load originatesentirely from the 1975] as

effectiveimperviousarea of the watershed.(The effective imperviousarea refers to the area of impervioussurfaces that are directly connectedto the drainagesystem.Impervi-

dL

•

dt

= K-

K2L

(1)

ous surfaces such as roofs that drain to lawns are considered

noneffective.)

where L is the amount of constituenton effective impervious

Many runoff quality modelsare single-eventmodelsthat simulate only one individual rainfall period during a given

areas in kilograms,K is a constantrate of constituent depositionin kilogramsper day, K2 is a rate constantfor

This paperis not subjectto U.S. copyright.Publishedin 1981by the American Geophysical Union.

Papernumber1W1383.

constituentremovalin day-l, and t is time in days.The parameterK2 can accountfor lossesdue to wind and vehiclesas well as the biologicalandchemicaldecayof some

1657

1658

ALLEY AND SMITH: ESTIMATING PARAMETERSFOR URBAN RUNOFF where WSHOFF

is the amount of constituent removed from

the effectiveimperviousarea duringa time stepin kilograms, L0 is the amount of constituenton the effective impervious area at the beginningof the time step in kilograms,K3 is the

washoffcoefficientin mm-•; R is the runoffrate in mm/h (millimeters of runoff refers to the volume of runoff in

millimetersdistributedover the effectiveimperviousarea of the watershed), and At is the time step in hours. Constituent concentrations are determined by dividing WSHOFF by AR At whereA is the effectiveimperviousarea of the watershed. The primary assumptionof (6) is that the rate of constituentwashoffis proportionalto the amountof

00

te

• t

constituent remainingonthelandsurface.Because theexponential term contains the product of runoff intensity and duration, the amount of constituentwashed off during a

TIMESINCELASTCLEANING, EITHER MECHANICAL ORBY STORMRUNOFF

Fig. 1. Constituent accumulation.

storm is a direct function of the total volume of storm runoff.

constituents.Integration of (1) yields L = K•[1 - exp (-K2T)]

(2)

where K• = K/K2 is the maximum amount of constituenton effective impervious areas in kilogramsand T is time since the last period of street sweepingor storm runoff (accumulation time), in days. Traditionally, (2) has been derived with the assumption that urban impervious surfaceswere completelywashedby the last period of runoff or street sweeping. In order to eliminate this assumption,T in (2) can be redefinedas the equivalent accumulation time' T = t + te

(3)

where t is the time sincethe last street sweepingor runoff and te is the time required for a land surface load to accumulateequalto that at the end of the lastperiodof street sweepingor runoff, assuminginitially clean urban impervious surfaces.The variable te is computedas

t•-

K:

In 1-

(4)

During simulationon a daily basis, the washoffof constituents can be simulatedusing a At of 1 day and a runoff rate equal to daily precipitationminusimperviousretention. Limitationsof the exponentialwashoffequationhavebeen

describedby Alley [ 1981].If the exponentialwashoffequation is appliedon a distributedbasis,the routingof constituents througha channelnetwork and the mixingof contributions from different subwatershedscan result in many different distributions of simulated constituent concentra-

tions over a storm hydrograph.However, applied on a lumpedparameterbasis,the exponentialwashoffequation predicts decreasingconcentrationsof a given constituent with increasingtime since the start of a storm. Many urban runoff quality modelsutilize an 'availability factor' to adjust washoff predicted by the exponential washoffequationfor the effectsof runoff intensity.However, the resultsof a previous study by Alley [1981] suggests that the advantagesof includingthe additionalavailability parameterswill often be more than offset by the advantages of having only the singlewashoffparameter.

It shouldalsobe notedthat the qualityof the precipitation input to a watershedwill affectrunoffquality.Accounting for precipitationqualitywould affectthe bestfit valuesof the accumulationand washoffparameters.This factor was not

whereL• is the land surfaceloadin kilogramsat the end of includedin the presentstudy,but the mechanics of doingso the last periodof streetsweepingor runoff. are not difficult (assuming the substancesin rainfall are Figure 1 illustratesthe exponentialaccumulation equaconservative).One couldsimplysubtractout the precipitations. The upper curve accountsfor a residualamount of

tion quality before fitting the accumulationand washoff constituentremainingon effectiveimpervioussurfacesat the parameters.The main difficultywould be in estimatingthe end of the last periodof streetsweepingor stormrunoff, quality of the precipitation,particularlyits time variability. whereas the lower curve assumes no residual.

In order to approximatelinear accumulation, (1) can be rewritten

as

OF MODEL

PARAMETERS

Application of this model requires estimation of three

dL dt

ESTIMATION

= K•K2 - K2L

(5) parameters:K•, K2, and K3. The parametersK• andK2 could

Assuming a very small value for K2, L would be much smaller than K• for normal accumulationtimes. Thus K2L would be negligible, and accumulationwould be at a linear rate equal to the product of K• and K2. CONSTITUENT WASHOFF

Constituent washoff from effective impervious areas can be simulatedusing an exponentialwashoff equation:

WSHOFF = L0[1 - exp (-K3R At)]

(6)

be determinedfrom samplescollectedfrom a street surface sampling program [Sartor and Boyd, 1972; Pitt, 1979]. However, the values of K• and K2 determined in this manner might only reflect constituentloads at a few sampledpoints rather than on all the effectiveimpervioussurfaces.Also, the sampled material removed by sweeping, vacuuming, or flushing techniquesmay not be related to the amount of constituent available for transport by storm runoff. An alternate approach is to estimate these parametersusing runoff quality data collected at the watershedoutlet. Care should be exercised in using this approachbecausethe outlet data represent the combined effects of accumulation, _

ALLEY AND SMITH: ESTIMATING PARAMETERS FOR URBAN RUNOFF

,-, 0.25 z •

I

I

I

I

I

I

I

0.20

-

a: 0.15

-

o O.lO

-

_z 0.05

-

-

o 250

1659

both the magnitude of simulated concentrations and the distribution of simulatedconcentrationsduring a storm. The effects of changesin Ls and K3 are shown in Figure 2. Alley [1981] presented an optimization schemefor determining the best fit value of K3 for a given constituentand set of storm events. The procedurecouldbe usedto determinea separate value of K3 for each storm, or a 'multistorm' value for a set of stormscould be determined.The remainingparts of this paper assume that the latter option of selecting a particular K3 which applies to all storms (for a particular water quality constituent)would be the approachmostlikely taken for multievent

simulation.

After selectinga value for K3, one approach to determining best fit values of K• and K2 would be to use a direct search optimization technique such as the one devised by

200

K3=0.10 mm-: 150

Rosenbrock [1960]. Rosenbrock's method revises the model --

100

•

50

--

0

parameter magnitudes and recomputes the objective function using the revised set of model parameter values. If the result is an improvement over previous runs, the revised set is accepted; if not, the method returns to the previous best set of parameters.This procedurewas investigatedwith the objective function

Kg

z 250•!• y..I I -I I I I I I o

10.20•

a: "'

min Z = •

Ls=50Kg

z

150

100

50 -0.0••,,, ....

[S(i)- M(i)]2

(7)

i=1

-

where i is the storm number, N is the number of storms in

the objectivefunction,and S and M are the naturallogarithms of the simulated and measured values of storm runoff

load, respectively.Becausechangesin all concentrations for storm i are proportionalto changesin Ls(i), replacingloads O0 20406080100120 140160180 200 in (7) with concentrations wouldresultin nearlyan equivaTIME, IN MINUTES lent optimizationproblem. Fig. 2. Model sensitivityto Ls and K3. Rosenbrock'smethodof optimizationproceedsby stages. Duringthe first stage,eachparameterrepresentsoneaxisin washoff,routing,and contributionsfrom not only effective an orthogonalset of searchdirectionsuntil end of stage imperviousareasbut perviousareasand rainfall as well. In criteriaare satisfied.At the end of eachstagea new set of addition,effectiveimperviousareasmay includedifferent orthogonaldirectionsis computed,basedon the experience land usetypeswith differentaccumulation rates.The proce- of parametermovementduringthe precedingstage.The duresdescribedhereinare thereforeintendedonlyfor appli- majorfeatureof thisprocedureis thatafterthefirststageone axis is alignedin a directionreflectingthe net parameter cation to small watersheds of uniform land use and for storm eventswith little perviousarea runoff. When theseassump- movementexperiencedduringthe previousstage. In general,the Rosenbrocktechniqueis mostapplicableif tions are not met, best fit parametervalues may differ the initial estimatesof modelparametervaluesare closeto substantially from 'true' values. A characteristicof stormwaterqualitysamplingprograms their optimumvalues,if the objectivefunctionis sensitiveto is that field or automaticsamplinglogisticsresultin the first changesin the values of the model parameters,and if sample of storm runoff being collectedsometimeafter the parameterinteractionsare minimal. These aspectsare disstart of the runoff and the last samplecollectedbeforethe cussedin the followingparagraphs. end of the runoff period. Often a considerablepart of the INITIAL ESTIMATES total stormrunoffmay not be includedin runoffoccurring between the first and last sample.Thus measuredrunoff For single-eventsimulationa value for Ls which will result loadsoften are basedonly on the runoffoccurringbetween the firstandlastsample.Thisproblemis accounted for in the in a simulatedstorm runoff load equal to the measuredload can be estimatedby first determining following procedure. For the assumptionof exponentialwashoff,the distribu-

tion of constituentconcentrations duringa periodof storm Lsp= (8) runoffis independent of the amountof a constituent (Ls)on [1 - exp (-K3(Vb - Va))] theeffectiveimpervious areasat thestartof thestorm[Alley, 1981].That is, changesin Ls affectthe magnitudeof simulat- where Lsp is the amountof the constituenton effective ed concentrations by a commonmultiplier.For example,if impervious surfaces at thestartof thesampling period,AL is Ls for a storm is doubled, then all simulatedconcentrations the measuredrunoffloadoccurringbetweenthe firstandlast would be doubled.On the otherhand,changesin K3 affect sample,K3 is determinedusingthe methodology presented

1660

ALLEY AND SMITH'

ESTIMATING PARAMETERS FOR URBAN RUNOFF

0.50

where Rs(KO is the relative sensitivityof a simulatedload to K• and S is the simulated load. But

0.45

S = K•[1 - exp (-K2T)] [1 - exp (-K3V)]

(13)

0.40

where

V is the volume

of storm runoff

in millimeters.

Equation (13) is derived by letting L = L0 in (2) and substituting (2) into (6). EvaluatingOS/OK1 and substitutingthis expression,aswell as (13), into (12), one arrives at

0.35

0.30

-o0.25 •

R•(KO = 1.0

.

(14)

0.20

Equation (14) statesthat simulatedloads are proportional to K1. Thus if K1 is doubled,then so are the simulatedloads. Actually, simulatedloadswill not be exactly proportionalto K1 because(14) is derived assumingcomplete independence of events. If K1 is increased,then t• in (3) and thus T alsowill be increased.However, this effect usually will be very small. Similarly,

0.15

0.10

0.05

0.00 0 2'.0 5.0 7.5 10.012.5 15.017.5 20.0 22.525.0 EOUIVALENT ACCUMULATION TIME,IN DAYS

Fig. 3. FractionF of maximumloadoneffectiveimpervious areas as a functionof K2 and equivalentaccumulationtime.

by Alley [1981], Vb is the cumulative storm runoff at the time of the last sample in millimeters, and Va is the cumulative storm runoff at the time of the first samplein millimeters. By

lettingWSHOFF = Ls - Lsp,(6) can be rearrangedas Ls =

Lsp exp (-K3Va)

exp (-K3Va)[1 - exp (-K3(V0 - Va))]

K2T exp (-K2T) [1 - exp (-K2T)]

(15)

where Rs(K2) is the relative sensitivityof a simulatedload to K2. The relationshipof Rs(K2)to equivalent accumulation time and K2 is shownin Figure 4. The relative sensitivityof

K2 approaches1.0 for very small valuesof K2 and short (9)

Substituting(8) into (9) resultsin Ls =

Rs(K2) =

(10)

For multievent simulation using the accumulationequa-

tionspreviouslydescribed, Lsis determined asa functionof K1 and K2 describedby (2). The parameterK1 is the

equivalent accumulationtimes but rapidly decreasesas K2 and T increase.Again, Rs(K2)may be slightlydifferentthan the values shownin Figure 4 becauseT is not a constantbut varies with K2 due to effects on previous storms. Finally, note that the relative sensitivity of K3 has the same algebraicform as Rs(K2)and is obtainedby replacing K2 with K3 and T with V in (15). The relative sensitivityof K3 decreasesfor larger values of K3 and larger storm runoff volumes. For many stormrunof[ eventsthe relative sensitivity of simulatedloadsto K3 would be muchlessthan 1.0. For

maximumvalueof Ls that can be predictedusing(2). Thusa example,for a K3of 0.18mm-1anda stormrunoffvolumeof reasonableinitial estimateof K1 couldbe obtainedby solving 25 mm the relative sensitivity of K3 would be about 0.05. (10) for a valueof Ls for eachstormand settingK• to the Again, on a multistormbasis,the actual relative sensitivity maximumvalueof Ls. The parameterK2 is a measureof the would be somewhat different due to effects on previous net rate of constituentaccumulationon effective impervious

surfaces.Figure3 showsthefractionof the maximumloadF accumulated as a function of K2 and the equivalent accumulation time T:

F = 1 - exp (-K2T)

(11)

The simulated rate of constituent accumulation increases as

the value of K2 increases;thus for a given equivalent accumulation time the simulated value of F increasesas the value of K2 increases.

simulated events of changesin K3. PARAMETER

INTERACTIONS

All predictedloads are affectedabout equally by changes in K1, but predictedloadsare most sensitiveto K2 for storms with short equivalent accumulationtimes. The parameterK2 tends to be a measure of the relative effects of antecedent conditions on runoff loads. Parameter interactions should be

minor for values of K2 greater than about 0.10 and a set of storms having a range of antecedentconditions.However, as the value of K2 approacheszero, the correlationbetween MODEL SENSITIVITY TO PARAMETERS the values of K1 and K2 would approach 1.0, and thus the estimatedvaluesof theseparameterswould be lessreliable. Becauseof the simplicityof the accumulation model, However, as shown by (5), the product of K1 and K2 direct differentiationcan be used in determiningmodel representsan approximatelylinear accumulationrate for sensitivity. To providea validmeans forcomparison, param- small values of K2, and thus individual estimatesof K1 and eter sensitivity will be expressed as relativesensitivityK2 are less important than their product. [McCuen, 1973]' Jewell et al. [1980], in a similar studyto identify accumulation and washoffparameters,attemptedto determinebestfit aS K1 as(KO(12) values of K1, K2, and K3 simultaneouslyfor many waterOK1 S shedsand storms.They reportedfindingunrealisticallylow

ALLEY AND SMITH:

ESTIMATING

PARAMETERS FOR URBAN RUNOFF

1.0

1661

lead, and 71% for suspendedsolids.The resultsfor all trials are shown in Table

0.9

1.

Often, land surface loads are assumedto approach their maximum values after dry periods of the order of 10 days. However, the values of K: were consistentlyof the order of 0.03 to 0.1 and thus appear to be smaller than commonly assumed. For example, the equivalent accumulationtime required for 90% of the maximum land surface load to accumulate is about 50 days for a K: of 0.05. The fact that different initial estimates of K: produced similar best fit values of K: was encouraging.However, additionaloptimization runs with startingvaluesof K: of 0.05 resultedin even smaller final values. Thus the best fit value of K: appearedto be asymptoticallyapproachingzero. Further investigationof this phenomenonled to an alternative optimizationscheme.

0.8

0.7

0.6

0.5-

0.4-

0.3-

0.2-

ALTERNATIVE

OPTIMIZATION

APPROACH

0.1

Because(for a given K3) simulatedstorm-runoffloads are proportional to the estimatedinitial amountof constituenton •.00 0.05 0.40 0.45 0.20 0.25 0.30 0.35 0:40 0.•,5 030 the effectiveimpervioussurfaces,the exponentialaccumulaKz,IN DAYS -• Fig. 4. Relativesensitivityof K2 asa functionof K2 andequivalent tion equation itself can be fit. This is similar to the problem accumulation time T. of finding ultimate BOD (biochemicaloxygen demand) and the first-order

rate

coefficient

for the BOD

reaction.

A

values ofK: andK3.Atthese ve• lowvalues therelativenumber of procedures exist for fitting such a curve (for sensitivity of simulatedloads to the accumulationand wash- example, see Thomas [1950]). The method selectedwas a off parameterswould be approximatelyequal for all parame- nonlinear least squares technique similar to that recently reported by Barnwell [1980]. The method involves solving ters and all storms and thus individual estimates of these parameters would be meaningless.By recognizingthat the the following unconstrainedoptimization problem: accumulation and washoff parameters can be estimated separately, this problem can be eliminated. min Z = • [K•[1- exp(-K:T(/))] - Ls(/)]: (16) i=1

APPLICATION

TO A FLORIDA

WATERSHED

The method of solutionof (16) would be to estimate values

The model was applied to a 5.95-ha urban watershed for K• and K:. The model can then be run with these located near Miami, Florida. The land use of the watershed

consistslargelyof multifamilyresidentialdwellings.Approximately 4.21 ha of this drainagebasin are covered by impervioussurfacesof which 2.63 ha are effectiveimpervious area. The remaining 1.58 ha are.impervious areas drainingonto perviousground.The soilsof the watershed are very permeableand runoff from the basin originates predominantlyfrom the effectiveimpervioussurfacesof the basin. Therefore the runoff contributingarea of the basin

estimatesof K• and K: to determine T(i), i = 1, ß ß ß , N. The values for L•(i) can be determined from (10). A necessary conditionfor an optimum solutionwould then be oz

oz

= • = o OK• OK:

(•7)

Values of K• and K: for which (17) holds could be determined by taking partial derivatives of (16) and using the was considered to be 2.63 ha. Rainfall, runoff, and water Newton-Raphson method to solve the generatedpair of quality data for the basin are reportedby Hardee et al. nonlinear equations. The model can then be run usingthe new estimatesfor K• [1979]. and K: to generatenew estimatesfor equivalentaccumulaThe model was run using total nitrogen, total lead, and tion times (T(i), i = 1, ß ß ß , N). The procedure is repeated suspendedsolids data for eight periods of storm runoff. Values for K3 were 0.063 for total nitrogen, 0.098 for total until the changes in equivalent accumulation times from lead, and 0.13 for suspendedsolids.These are the 'multi- model run to model run are lessthan a specifiedconvergence storm' valuespresentedby Alley [1981]for the eightperiods criterion. This procedurewas performed usingthe Florida data and found to have difficulty in convergingto specific of storm runoff. The model was calibratedfor theseeight stormsusingthe values for K• and K:. However, at each iteration the measureddata and the Rosenbrockalgorithm. Values for Ls procedure gave values for K: that were very small. This for each storm and constituent were computed using (10). suggestedthat linear accumulationmay be a reasonable Initial estimates of K1 were determined as the maximum approximationfor this exampleand that the instabilityof the value of Ls for the constituentof interest. Three different procedurewas due to the stronginteractionsbetween painitial estimates of K2 were tried for each water quality rameter values of K• and K:. A modification of the proceconstituent:a low value (0.10), a mediumvalue (0.25), and a dure which only used one of the partial differentialequations high value (0.50). For each trial the model was run for 50 was tried, namely, Rosenbrock iterations. Root mean square errors of load OZ iv = 2 • A(i)[K•A(i) - L•(i)] = 0 (18) estimates(as percentof the meanvalue)for the best trial for o/•1 i•l each constituent were 29% for total nitrogen, 53% for total

1662

ALLEY AND SMITH: ESTIMATING

TABLE

PARAMETERS FOR URBAN RUNOFF

1.

Model Results

K•, kg

K2, day-1

RMSE, % of

Trial

Initial

Final

Initial

Final

Mean

Number

Value

Value

Value

Value

Load

2.4 2.4 2.4 0.55 0.55 0.55 530 530 530

3.2 2.6 3.5 0.82 0.76 0.45 560 200 29O

0.10 0.25 0.50 0.10 0.25 0.50 0.10 0.25 0.50

0.051 0.066 0.085 0.040 0.047 0.092 0.027 0.10 0.057

29 32 32 53 55 55 71 76 73

Water Quality Constituent

I 2 3 I 2 3 I 2 3

Total nitrogen

Total lead

Suspended solids

where

explains the very low values of K2. Similar plots were obtained for suspendedsolidsand total lead. Simple linear regressionanalysiswas usedto fit a relationship of the form

A(i) = 1 - exp (-K2T(i)) Equation (18) can be solved for K• as K• =

Zi=lN A(i) Ls(i) N Ei=I [A(i)]2

Ls(i)= •o + •,X(i)

(19)

(20)

where• and• aretheestimated regression coefficients and

X(0 is the antecedentconditionsparameter. Three antecedent conditionsparameterswcrc used for each of the three constituents.These included equivalent accumulationtime T, time sincelast rainfall, t, and number of days prior to the storm duringwhich the accumulatedrainfall was lessthan 25 mm, DD. This last parameteris one commonlyusedin other runoff quality models, such as SWMM and STORM. The changed.For all threeconstituents the valueof the objective time since last rainfall, t, is the antecedent conditions function decreasedas the value of K2 was decreased.The parameterusuallyusedwith exponentialaccumulationequaminimum objective function occurred ata K2of0.001day-1, tions. in this particular example the previousrainfall was which was the smallest value tried. always greaterthan 5 mm. Of the nine regressionequations A plot of equivalentaccumulation time T(i) versusinitial (3 constituents times 3 antecedent conditions parameters) coefficient •o wasnotsignificantly ditfcrcnt land surfaceload Ls(i) is shownfor total nitrogenin Figure5. theregression This suggests that the datafit a linearaccumulation function from zero at the a = 0.05 level for any of the nine Thus•o wasdropped from(20).Table2 shows better than an exponentialaccumulationfunctionand thus regressions.

The final valuesfor K• and K2 reportedfor trim number 1 in Table 1 wcrc used in the model to generateestimatesof equivalentaccumulationtime for each runoff period (i.e., T(i), i = 1, ß ß ß , N). Thesewcrc usedin (19) to estimatebest fit valuesof K• for variousvaluesof K2. For eachpair of K• and K2 values the objective function (16) was computed assumingthat the equivalentaccumulationtimes were un-

2.5

1.0

03

o

0

I

2

I

4

I

6

I

8

I

10

i

12

I

14

!

16

T(i),IN DAYS

Fig. 5. Relationshipof initial land surfaceload of total nitrogen,Ls(i), to equivalentaccumulationtime T(i).

ALLEY AND SMITH: ESTIMATING PARAMETERSFOR URBAN RUNOFF

TABLE 2.

1663

Results of RegressionEquations of Initial Land Surface Load Versus Antecedent Conditions

Parameters Coefficient of

Antecedent

WaterQuality Constituent Total nitrogen

Total lead

Suspendedsolids

Conditions Parameter

•,

Determination,

kg d- •

r2

Coefficient

of Variation,%

T

0.14

0.92

34

t DD

0.19 0.12

0.70 0.90

65 37

T

0.035

0.83

53

t DD

0.038 0.027

0.59 0.85

82 49

T

22

0.72

77

t DD

26 16

0.58 0.66

94 84

the results of regressionequationsof the form

Ls(i)= •,X(i) (21) Thevalues for • areestimated linearaccumulation rates and are closeto the productof the final valuesof K• and K: shownin Table 1. In general,the linear regressionequations are best with T as the independent parameter and worst usingt as the independentparameter.Thusfor this example, accountingfor residualloadsgivesbetter model resultsthan usingthe commonassumptionof no residual.As expected,

been presented is a rational alternative to the common practice of trial and error estimates of these parameters. Because(for a given K3 and the assumptionof exponential washoff) simulated storm runoff loads are proportionalto the estimated

initial

amount

of constituent

on the

effective

impervious surfaces, it was shown that the accumulation parameters could be estimated by fitting the exponential accumulation curve to computed values of initial land surface load Ls(i) and equivalent accumulation time T(i). The results can be plotted (as in Figure 5) to partially reveal the best fit accumulation rates are smaller using T as the adequacy of the model. If a plot of L•(i) versus T(i) appears independentvariable than those obtained using t as the to represent an exponential accumulation curve or linear independentvariable. For this particularcasethe valuesof accumulation curve, then values of K• and K2 can be the parametersDD and T were similar;thusthe two parame- estimatedby the methodscontainedin this paper. Otherwise ters gave similarresults.However, there is not necessarilya a different model shouldbe tried. This may include usingthe universal relationshipbetween the two. same relationships for impervious area accumulation and These results are based on a small site specific data set. washoff but using additional algorithmsto account for other They are also dependent on the assumptionsabout K3, a sourcessuch as pervious area washoff. parameterthat hasbeendemonstratedto showconsiderable Results from application of the model suggesta need for storm to storm variability [Alley, 1981]. Other runs with data quantifying the magnitude and identifying the shape of different assumedvalues for K3 resulted in different values constituent accumulation curves. An exponential accumulafor• butstillindicated thatlinearaccumulation wasa good tion curve is shown to be more general than the linear assumption.The nonlinearcharacteristicsof accumulation accumulation curves usually used in runoff quality models. may be more evidentin drier (and/orwindier) climateswith More attention needs to be given to the effects of residual their associatedlongeraccumulationtimes. In any event the amounts of constituentsremaining after the previous period inherent flexibility of the model to simulate linear or expo- of storm runoff or street sweeping. nential accumulation was demonstrated. The results suggest REFERENCES a needfor data quantifyingthe magnitudeand identifyingthe shapeof constituentaccumulationcurves. Attention needs Alley, W. M., Estimation of impervious-area washoff parameters,

to be directed toward the effects of residual amounts of

Water Resour. Res., •7(4), 1161-1166, 1981. Barnwell, T. O., Least squaresestimates of BOD parameters, J. Environ. Eng. Div. Am. Soc. Civ. Eng., 106(EE6), 1197-1202,

constituentsremaining after the previous period of storm runoff or street sweeping.Effects of precipitationquality on 1980. the estimatedparametervalues is also worthy of investiga- Donigian, A. S., and N.H.

Crawford, Modeling nonpoint source pollution from the land surface, U.S. Environ. Prot. Agen. Off. Res. Der. Rep., EPA-600/3-76-083, 1979. Hardee, J., R. A. Miller, and H. C. Mattraw, Jr., Stormwater-runoff SUMMARY AND CONCLUSIONS data for a multifamily residential area, Dade County, Florida, Approachesfor estimatingimpervious area accumulation U.S. Geol. Surv. Open-File Rep., 79-1295, 1979. parametersalongwith an investigationof parameter sensitiv- Jewell, T. K., D. D. Adrian, and D. W. Hosmer, Analysis of stormwater pollutant washoff estimation techniques, paper preity have been presented. Limitations of the methodology sented at the International Symposiumon Urban Storm Runoff, include the requirement for runoff quality data and the Univ. of Kentucky, Lexington, July 28-31, 1980. assumptionthat effective impervious surfacesare the pre- McCuen, R. H., The role of sensitivity analysis in hydrologic dominant source of storm runoff loads. This methodologyis modeling, J. Hydrol., 18(1), 37-53, 1973. useful only when runoff quality is characterized by decreas- Metcalf and Eddy, Inc., University of Florida, and Water Resources Engineers,Inc., Storm Water ManagementModel, U.S. Environ. ing concentrations over a storm hydrograph which can be Prot. Agen. Rep., EPA-11024DOC 07/71, 1971. described by the exponential washoff equation. However, Pitt, R., Demonstrationof nonpoint pollution abatementthrough many measuredurban runoff quality data exhibit this characimproved street cleaning practices, U.S. Environ. Prot. Agen. Off. Res. Dev. Rep., EPA-600/2-79-161,1979. teristic, and the analytical optimization approachwhich has

tion.

1664

ALLEY AND SMITH:

ESTIMATING

Rosenbrock,H. H., An automaticmethodof findingthe greatestor least value of a function, Camput. J., 3, 175-184, 1960. Sartor, J. D., and G. B. Boyd, Water pollution aspectsof street surfacecontaminants,U.S. Environ. Prat. Agen. Rep., EPA-R2-

72-081,1972. Shaheen, D. G., Contributionsof urban roadway usageto water pollution, U.S. Environ. Prat. Agen. Off. Res. Dev. Rep., EPA600/2-75-004, 1975.

PARAMETERS FOR URBAN RUNOFF

Thomas,H. A., Graphicaldetermination of BOD curveconstants, Water Sewage Works, 97, 123, 1950.

U.S. Army Corps of Engineers,Storage,treatment, overflow, runoff model (STORM), report, Hydrol. Eng. Center, Davis, Calif., 1976.

(ReceivedApril 7, 1981; revised August 13, 1981; acceptedAugust20, 1981.)