Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona ...... Walnut Gulch Experimental Watershed, Tombstone, Arizona,.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. D24, PAGES 33,421-33,433,DECEMBER 27, 2001
Multiobjective calibration and sensitivityof a distributed land surface water and energy balance model Paul R. Houser,1HoshinV. Gupta,andW. JamesShuttleworth Departmentof HydrologyandWaterResources, Universityof Arizona,Tucson,Arizona
JamesS. Famiglietti Departmentof Earth SystemScience,Universityof California,Irvine, California
Abstract. The feasibilityof usingspatiallydistributedinformationto improvethe predictive ability of a spatiallydistributedlandsurfacewater andenergybalancemodel(LSM) was exploredat the U.S. Departmentof AgricultureAgriculturalResearchService(USDA-ARS) WalnutGulchExperimentalWatershedin southeastern Arizona. The inclusionof spatially variablesoil andvegetationinformationproducedunrealisticsimulationsthat were inconsistent with observations, whichwaslikely an artifactof bothdiscretelyassigninga singlesetof parameters to a givenareaandinadequateknowledgeof spatiallyvarying parametervalues. Becausesomeof the modelparameterswere not measuredor are abstract quantitiesa multiobjectiveleastsquaresstrategywas usedto find catchmentaveraged parametervaluesthat minimizethe predictionerrorof latentheatflux, soil heatflux, and surfacesoil moisture. This resultedin a substantialimprovementin the model'sspatially distributedperformanceandyieldedvaluableinsightsinto the interactionandoptimal selectionof modelparameters. 1. Introduction
Many aspectsof LSM calibrationhave been exploredin other investigations. For example, Sellers et al. [1989] employedmanualcalibrationof nineparameters in the simple biosphere(SiB) modelto improveits comparison with field observations. Franks and Beven [1997] use a generalized likelihooduncertainty estimationapproach to reducetheoutput uncertaintyfor the TOPUP model using First International SatelliteLand SurfaceClimatologyProjectField Experiment (FIFE) and Anglo-Brazilian Climate Observation Study (ABRACOS) observations.Lettenmaieret al. [1996] reported that in an intercomparisonof various LSMs, those whose
A land surfacewater and energy-balancemodel (LSM) [Famigliettiand Wood, 1994] was used to simulatethe spatiallydistributedbehaviorof the U.S. Departmentof AgricultureAgricultural ResearchService (USDA-ARS) WalnutGulchExperimental Watershed(WGEW). This type of modelgenerallyhas a large numberof parameters that describesurfacevegetationand soilsthat mustbe correctly specifiedto produceaccuratewater and energy balance simulations.Many of theseparameters arephysicallyrealistic and observable;that is, they are dimensionsor capacities parameterswere calibratedperformedbetterthanthosewithout which can be measuredreliably. For these parameters, parametercalibration.For a reviewof conceptualhydrologic observedvaluescan be usedwhen available. Other parameters modelcalibrationto observeddata,seeGuptaet al. [ 1998]. are not observableor are not physicallyrealistic. Such Automatic spatially distributedLSM calibrationhas been parameters maywell be conceptual representations of abstract severelylimited by the absenceof adequatespatiallyvariable watershedcharacteristics or empiricalconstantsthat can be hydrologiccalibrationdata. Automaticcalibrationmethods usedto optimizethe simulationresultson a trial-and-error adjustLSM parameters to obtaina bestfit of LSM predictions basis to match the simulations to observations. with observations. Therefore,to automatically obtainspatially Two aspectsadd complexityto the problemof parameter distributedLSM parameters,spatiallydistributedobservations specificationin this study. First, becausethe surface of LSM predictions,suchas soil moisture,surfaceheatfluxes, conditions are spatiallyvariable,the parameters mightalsobe and runoff, must be available. Currently, distributed expectedto vary spatially. Se•nd, because the performance observationsof LSM predictions are scarce, and when of the model is to be judged in terms of its ability to available, have relatively high uncertainty [Franks et al., simultaneously predictthe soil moisturestatesand the latent, 1998]. Therefore spatially distributedLSM calibrationhas sensible, andsoilheatfluxes,the specification of parameters is generallybeenlimitedto (1) specification of spatiallyvariable a multiobjective calibrationandevaluationprocess. parameters from soil, vegetation,or topographic mapsderived from field surveysor remotesensingor (2) lumpedor area1NowatHydrological Sciences Branch, NASA/GSFC, Greenbelt,averagedcalibration,where parametersare assumedto be Maryland.
Copyright 2001bytheAmerican Geophysical Union. Papernumber2000JD900803. 0148-0227/01/2 000JD900803 $09.00
constant over an area and are adjusted coincidentallyto minimize error in streamflow[e.g. Storcket al. 1998]. An exceptionis a studyby Frankset al. [1998], wheredistributed model parameterizations were first conditionedon lumped discharges and then further conditioned on distributed 33,421
33,422
HOUSER ET AL.: MULTIOBJECTIVE
CALIBRATION
estimatesof saturatedareasestimatedfrom remotesensingand energy fluxesatthelandsurface andlocalverticalrecharge to topographyusingthe weightingof estimatesin a fuzzy set the watertable. It incorporates simplerepresentations of framework.
This paperconsistsof six sections.Section2 describesthe studysite andthe dataset. Section3 providesan overviewof the LSM. The fourthsectionexploresthe hypothesis that the useof spatiallydistributedsoilsandvegetationinformationcan help to improvethe predictiveability of the LSM. Section5 explores the hypothesisthat a multiobjectiveparameter calibrationmethodcan be used to improvethe estimatesof nonobserved modelparameters.The last sectionpresentsthe
atmosphericforcing, vertical soil moisturetransport, plant-controlledtranspiration,interception,evaporation, infiltration,surfacerunoff, and sensibleand groundheat fluxes. The LSM incorporates a diurnalcycleandis driven withstandard meteorological datawith an hourlytimestep, thisbeingconsidered sufficient to resolve thedynamics of the land-atmosphere interaction.
conclusions and some final remarks.
4. Use of Spatially Distributed Soilsand Vegetation Information
2. Walnut Gulch and Monsoon '90 Observations
This sectionexaminesthe hypothesisthat the use of spatiallydistributed soilsandvegetation information canhelp
The WalnutGulchExperimental Watershed (WGEW) is to improvethe predictiveability of the LSM. The LSM was thelandsurface waterandenergydynamics in operated by the Southwest Watershed Research Center usedto simulate (SWRC) of the Agricultural Research Service, U.S. a spatiallydistributedmannerfor the entire WGEW at a 40 m
resolution. Predictions were madehourly from July 22 to August15, or day of year (DOY) 204 to 228, of 1990. LSM parameter estimates were first specified by using catchment-averaged parametervalues based largely on groundwater variesfrom 45 m at the lower endto 145 m in the observations. Next, spatially distributedvalues for the centerof thewatershed.Soiltypesrangefromclaysandsilts parameterswere derived from geographic information to well-cemented boulderconglomerates [Kustas et al., 1992], system's(GIS) maps. withthe surfacesoiltextures beinggravellyandsandyloams containing, on average,30% rock and little organicmatter. 4.1. Spatially ConstantParameters Themixedgrass-brush rangeland vegetation ranges from20 to Spatiallyconstantparametersfor the LSM were specified 60% in coverage.Thisrangeland regionreceives250 to 500 primarily on the basisof observations made duringMonsoon mm of precipitationannually,typicallytwo thirdsof it as '90 [Daughtryet al., 1991; Kustaset al., 1992, 1994a, 1994b, convectiveprecipitation duringthe summermonsoonseason 1994c; Stannard et al., 1994; Moran et al., 1994a, 1994b;
Department of Agriculture.Thestudycatchment is a heavily instrumented areacomprising theupper148km: of theWalnut Gulchdrainagebasinin an alluvialfan portionof the San PedroRiver watershed in southeastern Arizona. Depthto
[Renardet al., 1993].
Humes et al., 1994a, 1994b; Weltz et al., 1994; Menenti and
Eighty-fiverecordingrain gauges,11 primarywatershed Ritchie,1994;Hippset al., 1994;Spangler,1969]. The albedo runoff-measuring flumes, and micrometeorologicalof dry soilwas specifiedby adoptingthe parameters suggested
observations make the WGEW
a valuable research location.
by Dicla'nsonet al. [1993]. Soil profile temperatures measured
Duringthe Monsoon '90 experiment (July23 throughAugust in three trenchesat the Lucky Hills and Kendall siteswere 10, 1990),extensiveremote-sensing observations weremade, analyzedand modeledto determinethe temperaturediurnal while eight micrometeorological energy flux (Metflux) dampingdepth. The saturationsoil moisturewas assumedto
instruments provided continuous measurementof local be close to the porosity calculated from bulk density
meteorological conditionsand the surfaceenergybalance measurements.The surfacesaturatedhydraulicconductivity [Kustasand Goodrich,1994].
wasbasedon porosityusingthe Kozeny-Carmen equationwith Precipitation is the mostimportantspatialforcingvariable a coarsefraction correction,following RawIs et al. [1993]. in semiaridregionsbecause of its highlyvariable,convective The depth of the surfacezone was specifiedequal to the nature;thusmucheffort was devotedto derivingspatially maximum reported L band microwave penetrationdepth distributedprecipitationdata sets for the Monsoon '90 [Jackson,1993] to aid compatibilitywith remotesensingdata. experiment. A multiquadric-biharmonic interpolation Becauseof the deep water table at Walnut Gulch, surface algorithm[Syed, 1994] was used to producespatially processesshow no sensitivity to the LSM water table distributedprecipitationvaluesfor the entire model domain parameters,so thesewere assignedto arbitraryvalues. The fromthe availableraingaugedata. All othermeteorologicalvegetationstresssoil moistureparameterwas assumedto be forcingwasassumed to be spatiallyconstant andderivedfrom halfway between the saturatedand residual soil moisture averaging observations at the eightMetfluxstationsin place values,while the wiltingpointwasassumed to be equalto the during theexperiment. A spatial averaged soilmoisture profile valueof residualsoilmoisturebecausedesertvegetationrarely was derived from several in situ profile observationswilts. The soil moistureat which evapotranspiration reaches [Schmugge et al., 1994]andusedto initializetheLSM onJuly thepotentialratewasassumed to be 10% belowsaturation, this 22 (Table 1). beingthe approximate ratio suggested by Sellerset al. [ 1986]. Values for albedoof dry vegetationfor semiaridareaswere obtainedfrom Dicla'nsonet al. [1993], and the albedoof wet vegetationwas assumedto be 5% lower than that for dry The land surfacewater and energybalancemodel(LSM) vegetation.The stomatalresistance, root activity factor,root [Famiglietti and Wood, 1994] simulatesland surfacerunoff, density,root resistivity,and criticalleaf water potentialwere energyfluxes,andsoilmoisturedynamicsin threelayers.This not measured; indeed,becausemanyof theseparameters were LSM is designedto predictdiurnaldynamicsof the water and notobservable quantities, estimates wereused. 3. Land
Surface
Model
HOUSER ET AL.: MULTIOBJECTIVE CALIBRATION
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Table1. Catchment-Averaged LSMParameters forWalnut Gulch, Based onObservations, GISCoverage, and MultiobjectiveCalibration a Parameter
Source
Vegetationparameters vegetationheight,m leaf area index
minimumstomatal resistance, s m-• initial waterstoragein canopy,m unstressed transpiration soilmoisture,% wiltingpointsoilmoisture,% vegetationfraction albedo,wet vegetation albedo,dry vegetation albedo,bare soil
root activityfactor
rootdensity, m m'3 rootresistivity, s m4 criticalleaf waterpotential,m Soil parameters surfacezonedepth,m initial surfacesoil moisture, %
root depth,m initial root zone soil moisture, %
maximum rateof capillary rise,m s'• initial transmissionzone soil moisture, %
percentsand,% percentclay, %
bulkdensity, gcm'3 residual soil moisture, % saturated soil moisture, %
saturated hydraulic conductivity, m s'• baresoilroughness length,m baresoil zeroplanedisplacement, m Water tableparameters averagetopographic index K sexponentialdecayparameter initial watertabledepth,m Energybalanceparameters soil moisturefor calculationof PET, %
diurnalheatpenetration,m temperature of deepsoillayer,øK
Observed
GIS
Calibrated
Humeset al. [ 1994a] Daughtryet al. [1991] Chowet al. [ 1988] assumed dry at startof simulation
(0.22) 1.275 40 (0)
(1.31) 267
(574)
assumed halfwaybetween0sand0r assumed 0r (cactirarelywilt) Kustaset al. [1994a] Dickinsonet al. [ 1993] Dickinsonet al. [1993] Dickinsonet al. [1993] Famiglietti[ 1992] Famiglietti[1992] Fatniglietti[ 1992] Fatniglietti[ 1992]
(20) (1) (0.42) (0.2) (0.25) (0.33) 10,000 1 1e9 -210
Basedon PBMR sensitivitydepth Schmugge et al. [ 1994] Schrnugge et al. [1994] Schrnugge et al. [ 1994] Famiglietti[1992] Schmugge et al. [ 1994] Kustasand Goodrich[1994] Kustasand Goodrich[ 1994] Kustasand Goodrich[ 1994] Rawlset al. [1993] Kustasand Goodrich[1994] Rawlset al., [ 1993] assumption assumption
(0.1) (10.0) 1.5 (17.0) (0.1) (17.0) 70.9 8.5 (1.6) 1 38.0 6.9e-5 (0.001) (0.0)
calculationfromA(tan(B))image Famiglietti[ 1992] Gilbert[ 1996]
(8.314) (7.0) (100.0)
Sellerset al. [1986] from observations [Houser,1996] fromobservations [Houser,1996]
28
0.5 297.0
(348025) (86.5) (4.8e 11) (-500)
(0.8)
63.1 22.1
(14.4) (8.•)
5.1 33.6 4.7 e-6
(2.2) (30) (8.7e-6)
(47) (0.33) (287.6)
Values inparentheses produce the"best" Monsoon '90 simulation. Readl e9as1fro.
4.2. Spatially Variable Parameters
for eachsoil seriesin the soil survey. Bulk density,B, was
estimated usingan empiricalformula[Rawlset al., 1993]: Spatialdistributions of the soil andvegetation parameters wereestimated usingGIS mapsof WalnutGulchvegetation B = 1.51+0.0025(S)- 0.0013(S)(O)- 0.0006(C)(O) andsoils. The potentialrootingdepthsfor eachsoil series were taken from a recent Walnut Gulch soil survey (D. J.
Breckenfeld, unpublished document,1993). Valuesof the effectiveporosity,•be, residualsoil moisture,0r, and the saturatedhydraulicconductivity, Ks, were determinedby Mayeux[1995]usingvaluessuggested by Rawlset al. [1993] andBouwer[1966]. J. Berglund,unpublished report,(1995) calculatedsaturatedsoil moisture,0e, for each soil classas follows:
Oe--•e-{-Or.
- 0.0048(C)(E).
(2)
whereS is percent sand,C is percent clay,O is percentorganic matter, and E is the cation exchangecapacity of clay normalizedby percentclay. Measurements at the eight Monsoon'90 Metflux sites [Daughtryet al., 1991] and
parameter rangesgiven by Jones[1983] were usedin conjunction withtheWalnutGulchvegetation GIS coverage to estimate the spatialdistribution of leaf areaindex(LAI) and
minimum stomatalresistance. The topographicindex was fromdigitalelevation models(DEMs)derivedfrom Percentsandandclay werecalculated by J. Berglund(1995) calculated 23 1:5000 stereo aerial photos [Matthews,1992]. DEM fromvaluesof percentmaterialpassing throughcertainsieves
33,424
HOUSER ET AL.' MULTIOBJECTIVE CALIBRATION a) Topographic Index
b)1August 1990 Rainfall
ß '••
20
.•
..?
'•'• '•'•' •--' '" Omm
c) Leaf Area Index
d) Saturated Soil Moisture ...,--
.
.::.. :
:•::
o
0%
e) 7 August 1990 LSM Surface Soil Moisture(%)
f) 7 August 1990 LSM Surface Soil Moisture(%)
using Spatlally Constant Parameters ...•.• .•..• using Spatially Variable Parameters 't
• ....
g) 5 August 1990 PBMR Soil Moisture
½;½• '• •:•5•?"• .....
•:½:::,½;•...;•t•½ •
•"
h) 9 August 1990 PBMR Soil Moisture
4O
Scale(km •'• • •Scale(kr•• • 0%
o
5
0%
Figure 1. SpatiallydistributedWalnutGulchExperimentalWatershed:(a) topography,(b) precipitation, (c) vegetation, and(d) soils. LSM spatialpredictions of surfacesoilmoistureat 1200LT on August7 (DOY 219) using(e) spatiallyconstantand (f) spatiallyvariablesoilsand vegetationparameters (all simulations using spatiallyvariabletopographyand precipitation). Pushbroommicrowaveradiometer(PBMR)'derived soil moisturefor (g) August5, 1990 (DOY 217) and (h) August9, 1990 (DOY 221) [Houseret al., 1998]. The additionof spatiallyvariablesoilsandvegetation produces unrealisticpolygonartifactsin the simulation.
overlapswere averaged,then a smoothingalgorithm was appliedto reducenoise,and isolatedrunoffsinkswere filled. The watershedwas delineatedand the topographicindex was derived using the standardmethodologydetailedby Beven [ 1995]. A selectionof representative spatiallyvariableWalnut Gulchparameters is shownin Figure1a-1d, andthe watershed averages of theseparameters arereportedin Table 1. 4.3. Results and Discussion
1e'1fi It isevident thatthespatially variable parameters havea largeimpacton the spatialpatternsof the simulation.Further, the soil moisturepatternsderivedusingthe spatiallyvariable parametersappearto be unrealisticand do not comparewell with observedpush broom microwaveradiometer(PBMR) surfacesoil moisture(Figures1g- 1h). A seriesof simulations, whereonly one spatiallyvariable parametersetwasusedat a time (theotherparameters were set to theirestimated catchment-averaged values),wereperformed to determinewhich subsetof spatialparameterscontribute
Thesimulated spatial patterns of surface soilmoisture for mostto thesepatterns(the topographic index and the August 7, 1990(DOY219),using boththespatially constantprecipitation fieldswere spatiallyvariablein all these andthespatially variable parameters, areshown in Figuressimulations).The simulations with spatiallyvariable
HOUSER ET AL.: MULTIOBJECTIVE
vegetationparametersperform similarly to the control run, indicatingthat spatialvariationin theseparametershas little effect on predictions. All of the simulationsusing spatially variablesoil parametersshowdistinctpolygonpatterns. The parameterspecifyingsaturatedsoil moisturehas the most influence on simulatedspatial patterns,while those which specify the percentagesof sand and clay, the saturated hydraulicconductivity,and the residualsoil moisturehave a more
moderate
influence.
The
soil
characteristics
are
extremelyimportant,and the vegetationcharacteristics are less important in regulating soil moisture variability and other surfacewaterand energycomponents in this watershed.The resultsof this sensitivityalso showthat the spatiallyvariable topographicindex hasvery little influenceon the simulations; thisresultis a consequence of the largedepthto the watertable
CALIBRATION
33,425
Gupta et al. [1999], this situationinvolvesthe problemof finding parameterestimateswhich can simultaneously minimize several noncommensurable criteria.
This section
examinesthe hypothesisthat a multiobjectiveparameter calibrationmethodcanbe usedto improvethe estimates of the nonobserved modelparameters of the LSM model. 5.1. Data Available
for Calibration
The primary observationsused to calibrate the LSM parametersin this study were hourly time series of catchment-averaged, water balancecorrected,latent heat flux during unstableatmosphericconditions,soil heat flux, and gravimetric calibrated near-surfacesoil moisture measured
with resistance sensors (Table 1). Duringthe Monsoon'90 study period, three replicate gravimetricsurfacesoil moisture at Walnut Gulch. samples were collected each day at the eight Metflux sites The pattern of enhanced spatial soil and vegetation [Schrnuggeet al., 1994]. Resistancesensorscollected polygonsapparentin the simulationsis an artifact of both continuous time series of soil moisture at 2.5 cm and 5 cm assigninglumped values of the parametersto discretely below thesurfaceat all eightMetflux sites[Arneret al., 1994]. partitionedareasas well as the misspecification of parameter Becausesuchsensorsare generallydifficult to calibrateand values. A moreappropriatespecification of spatialparameters would reflecttheir continuous variabilityin space,as obtained tend to drift, they were recalibratedeach day against measurements for the purposeof this study. The with remotesensing. It is also clearthat the specificationof gravimetric
energy balance was determined from measurementsof net the spatialparametervaluesusingscatteredobservations, table radiation, soil heat flux, and estimatesof sensibleand latent lookup,andphysicalrelationships (as describedin section4.2) variance result in inadequateparameterspecification. These results heat flux by either eddy covarianceor temperature methods [Kustas et al., 1994a]. A detailed water balance study indicatethat the available WGEW parameterobservations simply have insufficientaccuracyand spatial frequencyto showedthat during Monsoon'90 the sensibleheat flux was becauseof eddycorrelation propellerstalls, realistically constrainthe LSM. Becausethe simulations underestimated source area mismatch, or horizontal flux divergence in hilly using spatially constant vegetation and soil parameters terrain [Houser et al., 1997; William& 1996; Keefer et al., comparewell to the PBMR patterns,the subsequent parameter calibration studies reported in section 5 assume spatially 1997]. Therefore,for this study the observedlatent and with the constantsoil and vegetationparametersacrossthe catchment, sensibleheat fluxeswere adjustedto be consistent water balance, and unreliable observations during stable leaving only topographicindex and precipitationas spatially varyingentities. The use of catchment-averaged parametersin this study is Table2: Calibration Parameter Ranges andObjective Functions a supported by Whiteet al. [ 1997],who foundthatthe difference betweenthe calculatedarea-averagedsurfaceenergy fluxes Calibration Parameter Possible Range givenfor the WGEW by a singlepointlandsurfacemodelwith 1 to 1e6 Root activityfactor aggregateparametersand that given by a distributedarray of 1 to 100 Rootdensity,m m-3 land surface models was small.
This conclusion
was tested
here; althoughthere was more variability than reportedby Whiteet al. [1997] becauseof the finer spatialresolution,there was overall agreement. The relevanceof this finding to the presentstudyis that it supportsthe useof catchment-averaged parametersand forcingfor the WGEW if catchment-averaged surfacefluxes are required. This conclusionis only valid when catchment-averaged fluxes are of interest; for cases wheredownslopeflows or water availabilityin valley bottoms is of interest,thenspatiallyvariableparameters and forcingare required. In this study, the spatially distributedLSM is requiredto allow predictionof spatialpatternsof soil moisture at the resolutionof the availableremotelysensedobservations.
Rootresistivity,s m4 Criticalleafwaterpotential,m Potentialevaporation soil moisture,% Penetration of diurnalheating,m Temperature of deepsoillayer,K Minimumstomatalresistance, s m'l
le6 to le12 - 1 to - 1000 10 to 50
0.1 toO.7
285 to 310 0 to 700
Percentsand,%
lto
Percentclay, % Residualsoil moisture,% Saturatedsoil moisture,%
1 to 100
100
1 to20
20 to 50
Surface saturated hydraulic conductivity, m s4 5e-5to 5e-7 ObjectiveFunction
5. Multiobjective Calibration of Nonobserved Model
Parameters
The purposeof this study was to obtain estimatesof the nonobserved modelparametersthat would result in improved simulations of the overall behavioral responsesof the watershed.In particular,the desirewasto selectvaluesfor the parameters that minimizedthe error in predictedsoil moisture and sensible,latent,and groundheat fluxes. As discussed by
sensible heatflux(eightsites, waterbalance corrected, daytime average fromtemperature variance)
latentheatflux(eightsites, waterbalance corrected, daytime averagefromtemperature variance)
soilheatflux(eightsites average fromsoilheatfluxplates) surface soilmoisture (eightsites, gravimetric corrected average fromresistance sensors) aThe sensible heatflux objectivefunctionwasincludedasa diagnostic;it was not used in the determinationof the Pareto set. Read 1e6 as 10'6.
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atmospheric conditions wereexcluded fromobjective functionFigure2a illustrates howeachof the meansquare error calculations.
functions f•(0) andf2(0) mightvarywith 19.Notethat the parameter value19= 02minimizes the function f• buthasa relatively poorvalueforf2, whilea differentvalue19= 02 minimizes thefunction f2 buthasa relatively poorvalueforf•.
5.2. Parameters Calibrated
The 13 parameters listedin Table2 wereidentified for In otherwords,the individualsolution19= 0• that givesthe calibration.Table2 alsoliststhetypicalrangeof theirvalues; best match to the first model flux might give rather poor
theserangeswerespecified largerthanmightnormallybe performance in termsof matching thesecond modelfluxand expected in orderto allow the calibration procedure some vice versa.However,in general,we wish to havea model leewayin accounting for modeland dataerror. Moreover, solutionthatsimultaneously givesacceptable "good"matching becausesome of the parametersare nonphysical, the of both fluxes. specification of theirranges is somewhat arbitrary.Notethat Tothisendwenotethatthereexists a region ofvaluesOv= thesoilheatfluxparameters (temperature of thedeepsoillayer { 01< 0< 02} consisting of thoseparameter values forwhichit andpenetration of diumalheating) wereincluded in thislist is possible to obtainimprovement in f2(0 ) (by varying0 ) eventhoughtheyweremeasured with confidence.This is whilesimultaneously allowingsomedeterioration inf2(19);this because the soil heatflux was poorlysimulated by the LSM regionis calledthe "Pareto,""non-inferior," or "trade-off' owingto the simplicity of its soilheatflux submodel; this solution.Similarly, all valuesof 19q•O•arecalled"inferior" suggests thatnonphysical valuesfortheseparameters maybe solutions. Figure2b shows theproblem plottedin thefunction necessary forthemodelto performadequately. Notealsothat space withf• (19)andf2(0) ontheaxes,andthecurverepresents the five soil water retentionparameters,which can be
the trajectoryobtainedby varying19overits feasiblerange. Clearly,theobjective isto find 0 suchthatwe comeascloseto calibration, because this greatlyimprovedthe simulation of theorigin• ,f2}={0,0}aspossible.Notethatthesegment of soilmoisture andyieldedinsightintotheuseof observed soil the curve betweenB and C represents the trade-offregion, parameters. whileotherportions of the curverepresent inferiorsolutions. Any one of the trade-off solutions can be selected by theuser 5.3. TheoreticalBackgroundto the Multiobjective as "best," depending on which trade-off in matching of model Calibration Approach performance is considered mostacceptable by theuser. A theoreticaland practicalbasisfor the calibration of models withmultipleandnoncommensurable outputfluxeshas 5.4. ObjectiveFunction
estimated from observations,were also included in the
beenpresented by Guptaet al. [1998,1999],Bastidas et al. Threeobjectivefunctions wereusedto calibrate the model [1999],andMeixneret al. [1999]. Because of errorsin the parameters: thehourlymeansquare errorin matching of latent model(structure) andin the datait is impossible to find a heat flux duringunstableatmospheric conditions, soil heat single"best" parameter setthatsimultaneously minimizes the flux, andgravimetric calibrated near-surface soilmoisture.In errorsin matching all of theobservable modeloutputs.This eachcasethe meansquareerror objectivefunction,F, was leadsto a multiobjective optimization problemfor whichthe computed usingthenormalized sumof thesquared difference "solution" is a regionof the parameter spacethat reflects betweenthe catchment-averaged, water balancecorrected differenttrade-offsin the matchingof the outputs. An observations, Oi, and the predictions,Pi, wheneverthe
acceptable solution can beselected bytheuser from this region observation was available; thus of trade-offsolutions usingsubjective judgment.
To clarify,a simpleone-parameter example is presented in
Figure 2;we assume that data are available for two model output fluxes thatmust bematched, andweletf•(O)and f2(0)
F=n1y,• (O•P•) i=•
(3)
represent the meansquareerror(MSE) in matching these fluxesfor selected valuesof a specificmodelparameter 19. wherei isthetimestepandn isthenumber of observations. (b)
(a)
)
f•
f1(02 )
B
f•(O•)
01 '- 02
f2(02 ) f2(01 )
Figure2. A simpleone-parameter, two-objective modelcalibration example: (a) howeachof themean square errorfunctions, f•(0) andf2(0), mightvarywithmodel parameter 19,and(b) problem plotted in the function space withthecurverepresenting thetrajectory obtained byvarying 19overitsfeasible range.
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5.5.MultiobjectiveCalibration Methodology
CALIBRATION
33,427
shows thecorresponding trade-off among theparameters.
Inspeclion of theseresults shows thatthefive solutions giving
Theprocedure adopted forthemultiobjective calibration of bestmatching to soil moisture alsogive relativelygood the LSM parameters at WalnutGulchwasto conduct a matching of theotherfourfluxes.In addition, it wasdecided Monte-Carlo search of thefeasible parameter space indicatedthatthe soilmoisture objective wasmostimportant for the inTable2. A totalof 200,000 parameter setrealizations were purposes of thisstudy. Therefore the parameter setthat randomly generated, andthethreeobjective function valuesproduced the lowestsoil moisture objective was chosen werecomputed at eachparameter set. Fromthese,the (bolded linein Figure4). Theselected parameter values are "noninferior"trade-off parameterregion was estimatedby shown in Table 1. findingthosepointsfor which theredid not existat leastone It is interesting to notethatthecalibrated parameter values othermemberof the original200,000realizationsfor whichall arereasonably closeto the defaultvalues,with the exception three objective function values were better. Only 125 of the rootparameters, andare in generalmorerealistic. A parameter set realizationswere found to belong to this rootdensity of 86m m'3ismuchmorerealistic than1 m m'3, estimatednoninferiorregion, indicatinga rather substantial for instance, while the high minimumstomatalresistance is reductionin overallparameteruncertainty.In addition,at each realisticfor water-conserving semiaridplants. The default noninferiorpoint,the meansquareerror in matchingof hourly percentsandand clay parameters that were derivedfrom a sensibleheatflux wasalsocomputedfor diagnosticpurposes. numberof soil analyses(J. Berglund,unpublishedreport, The estimatednoninferiorregion is displayedin Figure 3 1995)were likely too highbecause the largecoarsefraction usingthe formatsintroducedby Sorooshianet al. [ 1993] and (D. J. Breckenfeld, unpublished document, 1993)wasremoved
Gupta etal. [1998,1999].Figure 3 (le•) shows thetrade-offpriorto analysis.Basically, if the sand,silt, andclay amongthe objectives, whileFigure3 (right)showsthe percentages (whichin thepublished soilanalysis addupto corresponding trade-off among theparameters. Notethatthe 100%)areadjusted sothattheyaddupto 100%minusthe objectivefunctionvalueshave been normalized onto a coarsefraction, thentheyget verycloseto the parameters zero-to-one scale, where zerocorresponds tothesmallest error found using calibration. These insights leadtotheinclusion of andonecorresponds to thelargest errorin thefeasible region. a coarsefractioncorrection in the modelphysics[Houser, Similarly,the parameter setsare alsonormalized ontoa 1996].Thelargedifferences in soilheatfluxparameters are zero-to-one scale, withzerocorresponding to theminimumprobably anartifact of theLSMhaving a simplified soilheat valueof theparameter andonecorresponding tothemaximumfluxsubmodel. Finally,thehighvalues of criticalleafwater valuein theirdefined ranges.Theseplotsenable easein potential andsoilmoisture for thecalculation of potential identifying whichparameters arepoorly orwellidentified and evapotranspiration (PET)reflect thefactthattheplants donot alsohow variousparameter valuesaffectprediction errors. wilt and that the environment rarelyexperiences potential
Theestimated noninferior solution region shows relatively low evaporation. values formostoftheobjectives, withthelargest spread being Thereis a clearcompromise in the calibration between in the matching of soilmoisture.Someof the calibratedlatentheatfluxprediction andsoilmoisture prediction (Figure (noninferior) parameters ranges arequitetight(e.g.,deepsoil 5). Therange of prediction shows thatlatentheatfluxcanbe temperature andpercent clay),showing thattheseparameters calculated wellbytheLSMbutattheexpense of soilmoisture. areidentified well. Thenoninferior parameter values alsoare By choosing to predict soilmoisture mostaccurately, rather generally confined to moderate ranges of theparameter space. thanlatentheatflux, we obtain"flat tops"or "hats"in the However,quitea lot of parameterinteractionis observed,with simulatedlatentheat flux time series. Accordingto Carlson two clustersof parametersevident(i.e., seeminimumstomatal [ 1991,p. 353],"thereis considerable observational evidence of resistance, potentialevaporationsoil moisture,diurnalheating [such]behaviorin the field." This LSM prediction of latent penetration, residualsoilmoisture,etc.). heatflux may not be entirelyincorrect, but it is certainlynot Despite the relatively low objective function values supported by theMonsoon '90 observations. It is difficultto producedby the noninferiorparameterrangethe corresponding drawanydefinitiveconclusions herebecause the accuracy of total rangeof LSM flux predictionremainshigh. The eight these latent heat flux observationshas been questioned Metflux site averagerangesof predictionobtainedfrom the [Williams,1996;Keeferet al., 1997;Houseret al., 1997]. Paretoparametersetsare shownin Figure 5. There is still a large amountof predictionvariability even within the Pareto 6. Conclusions set,andhencethe final parametersetmustbe chosencarefully. High-quality land surface water and energy balance All of the noninferiorparametersets are good but with different compromises as to which objective is more fully simulationsare essentialfor the predictionof drought,floods, minimized. To selecta single"best"parameterset from the agriculturalproduction,land surfaceinputsto the atmosphere, of landsurfacesto climatechange. However, 125 noninferiorsolutions,a necessarilysubjectiveprocedure andthe response was followed. For each of the three objectives,the five the land surfacewater and energybalancemodelsthat make generallyutilize largenumbersof parameters, noninferior parameter sets giving the best value of that thesepredictions objectivefunctionwere selected(a total of 15 solutions);in manyof which are not regularlymeasuredand someof which addition,the five membersof the noninferiorregiongivingthe describe abstract land surface characteristics that cannot be bestvalueof hourlymeansquareerrorin matchingof sensible measured. The inherent heterogeneityof land surfaces heat flux also were selectedas shownin Figure 4. Note that requires spatially distributed LSM predictions, which the sensibleheat flux objectivefunctionwas not usedin the complicatesthe opli•num selectionof parameters. Further, estimationof the noninferior region. Each row of plots LSMs are expectedto predictmultipleland surfacestatesand corresponds to a differentoptimizedobjectivefunction,while fluxes reasonably well, which requires the use of a Figure4b showsthe trade-offamongobjectivesand Figure4a multiobjective calibration procedure to achieve optimal
33,428
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a) Best5 Sensible HeatFluxObjective Parameter Sets
CALIBRATION
1.0
•_1.0 b Best 5Sensible Heat Flux Objective Parameter Sets
0.5
a' 05•
0.0
z 0.0
13
ß
0
1.0 C Best 5Latent Heat Flux Objective Parameter Sets ,_
0.5 [
•
d) Best5 Latent HeatFluxObjective Parameter Sets
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0.5
.__N_ o
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1.0
,_ 1.
o.s
,•
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O f)Best 5Soil Heat Flux Objective Parameter Sets
o.s
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z 0.0
1.0 'q)Best 5Soil Moisture Objective Parameter Sets
._ 1.0
h) Best 5SoilMoisture Objective Parameter Sets
0.5
0.5
o
z z
Figure 3. Paretosetnormalizedobjectivefunctionandparametervaluesfor the 13-parameter, threeobjective function Monte-Carlo calibration. The best five parametersets and objective function values for each objectivefunctionare shown,with the subjectivelychosen"best"parametersetshownin bold.Parameterand objectivefunctiondefinitionsare shownin Table 2.
performance. Therefore, thisstudyhasexplored thefeasibility distributed LSM should improve localpredictions, thisstudy of usingspatiallydistributed soilsandvegetation information showsthat the polygonnatureof thesedata setsresultsin anda multiobjective parameter calibration methodto improve unrealistic simulations which were inconsistent with the predictiveability of a spatiallydistributedLSM at the observations. A parameter sensitivity studyrevealedthat WGEW. spatialvariations in vegetation parameters havelittle effecton Although it seems reasonable thattheinclusion of spatially predictions, butthatsoilparameters havea largeeffect. This variablesoil and vegetationinformationin a spatially problem is bothanartifactof discretely assigning a singleset
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1.0a) 0.8
0.6
0ø4 0.2
0.0
b)
0.8
0.6
0.4
0.2
Figure4. Pareto setnormalized (a) objective function and(b) parameter values forthe13-parameter, three objective function calibration. Parameter andobjective function definitions areshown inTable2. of parametersto large areas of the catchmentand a However, the unrealisticvery strong surface soil spatial misspecification of spatialparameter fieldsdueto inadequatepatternspredictedon August7 (Figure 1) would likely be enhanced givena spatiallyvariablesoilmoistureinitialization measurement of the requiredparameters. or longer termmodelspinup.It is possible thatthe spatially ß It is alsorecognized thatthechoiceof spatiallyconstant soil moistureinitializationon July 22, 1990 and the shortmodel distributedinitial conditionscouldbe manipulatedto producea prediction.However,we had no spinupinfluencesthe spatiallydistributedpredictionsand betterspatiallydistributed influences thus the calibration of the distributed model. spatiallydistributed observations on July 22, for validating
33,430
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a)
-100
' 204
• 206
•
I 208
•
I 210
•
I 212
•
I
,
214
CALIBRATION
Heat Flux
I
216
I
I
218
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•
I
,
220
,
I
i
224
1
226
228
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I
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Observed Calibrated LSM
Heat Flux
2OO
-4OO
204
206
208
210
212
214
216
218
Day of Year
220
222
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...................... LSMPareto SetRange Observed :-- ;-' Calibrated LSM
Figure5. Catchment-averaged timeseriesof (a) latent,(b) sensible, (c) soilheatflux, and(d) near-surface soilmoisture andtheirassociated prediction rangesfor the 13-parameter, threeobjective function Paretoset. Thesubjectively chosen calibrated parameter setprediction andarea-averaged observations arealsoindicated.
such manipulations,and we seriouslydoubt that such only eight surfaceflux observationpoints. Given that the manipulation wouldchangeour conclusions giventhe very availablespatial informationis insufficientto accurately strongspatialpatterns observed in thesubsequent predictions. specifythe LSM spatialparameterdistributions, we choseto There is doubtthat the processof derivingthe spatially calibratethecatchment average, parameters to providethebest variable parameterfields from a few scatteredobservations possiblesimulation. Becauseprecipitationis the dominant usingestablished physicalrelationships andtablelookupis driverof spatialvariabilityin soilmoistureat theWGEW, this sufficientfor realistically constraining a complexdistributed approach resultsin reasonable LSM spatialpredictions. land surfacemodel. The spatiallyvariableparameter fields It is acknowledged that the simulationshouldimproveif were derived from a variety of sources,which leads to therewere sufficientinformationto truly calibratethe LSM inconsistencies in the resultingparameterset and degraded spatially. It may have beenpossibleto use the spatially LSM performance. Ideally,the modelshouldbe calibrated distributed PBMR soil moistureremote-sensing estimates to for eachpointin thedomainto produce consistent parameter calibratethe spatiallydistributed parameters.However,this sets that accountfor the spatial variability of surface was not done becauseour intentionin this studywas to characteristics.For the WGEW this is a poorly posed performa multiobjective calibration that wouldimprovethe problem,beingthat thereare 90,000modelgrid pointsand predictionof a numberof quantities.If the LSM had been
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Soil Heat Flux
• t
....
210
212
•
214
•.•
216
•
•
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•
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•.
•
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•
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35
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SoilMoisture
25
20 o. 15 lO
o
204
206
208
21o
212
214
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........ LSM Pareto SetRange
• Observed (gravimetric) Observed (resistance)
.......
Calibrated LSM
Figure 5. (continued)
calibrated spatially using only remotely sensed soilmoisture, werederived by optimization. A multiobjective LSM thenthesurface energy fluxprediction skillmayhavecalibration technique was used tocalibrate these parameters by degraded. estimating thenoninferior parameter region based ona large Theuseofcatchment-averaged parameters inthisstudy is number of Monte-Carlo parameter setrealizations, then also supported bythesmall difference between thecalculated selecting the"best" parameter setin a semisubjective way. surface energy fluxes given bya single LSMwithaggregate Thechosen parameter setcontained physically reasonable parameters andthatgiven bya distributed array ofLSMs.values which, inthecase ofthesoilwater retention parameters, to measured values (because of theremoval of This findingalso suggests that a spatiallydistributed LSM weresuperior appliedat theWGEWwiththeresolution usedin thisstudyis thecoarsefractionpriorto measurement). Severaladditionalinsightsinto multiobjective calibration probably notneededif catchment-averaged surfacefluxesare asa resultof additional experiments notreported required.However,a spatially distributed LSM isrequired for weregained andforwhichadditional investigation isrequired. prediction of spatialpatterns of soilmoisture at theresolutioninthispaper Theseincludethefollowing:(1) Someparameters tendto have of the available observations. LSMs have many parameters which must be carefully similarvaluesfor minimumobjectivesfrom severalsites, averageparameters may be specified fortheiroptimalimplementation. Severalparametersindicatingthat watershed However, thespatial patterns present inthePBMR were directlyobservable at Walnut Gulch,and their values acceptable. thatimproved distributed representations maybe couldbe readilyspecified,but otherswere nonphysical and datasuggest
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possible,and thereforea lumpedrepresentation may not be Chow, V. T., D. R. Maidment,andL. W. Mays, AppliedHydrology, acceptable for all the parameters. (2) Paretooptimalparameter McGraw-Hill, New York, 1988. Daughtry,C. S. T., M. A. Weltz, E. M. Perry, and W. P. Dulaney, sets often produce objective functionswith large error Direct andindirectestimatesof leaf areaindex,paperpresentedat indicating that some model componentsmay require Tenth Conferenceon Biometeorologyand Aerobiology,Special Sessionon Hydrometeorology,Am. Meteorol. Soc., Salt Lake improvement (the soil heatflux component for example).(3) City, Utah, 1991. Someobjectives thatarelinkedthroughmodelphysics,suchas Dickinson, R. E., A. Henderson-Sellers,and P. J. Kennedy, Biosphere-atmosphere transfer scheme(BATS) Version l e as make the multiobjectivecalibrationmore identifiable,only coupledto the NCAR CommunityClimateModel, NCAR Tech.
sensibleand latent heat fluxes, containsimilar information. To
objective functions with significantly different model performance informationshouldbe usedin the calibration
Note 387+STR, Natl. Cent. For Atmos. Res., Boulder, Colo.,
1993.
process. (4)Theinclusion ofsoilparameters inthecalibration Famiglietti, J. S.,Aggregation andScaling of spatially-variable hydrologicalprocesses:local, catchment-scale and macroscale results in significantly betterLSMperformance becausemodels ofwater and energy balance, Ph.D. dissertation, 207pp. measurements of these parameters werelikelybiased owingto
Dep.ofCiv.-Eng. andOper. Res., Princeton Univ., Princeton, N.
the removalof the coarsefraction(i.e., rocksand gravel), the
J., 1992.
presence ofmacropores, etc.(5)A large number ofobjectiveFamiglietti, J.S.,andE.F.Wood, Application ofmultiscale water
functions leads toalarge Pareto setsize, making identification and energy balance models onatallgrass prairie, Water Resour. Res.,30(1 i), 3061-3078, 1994.
of a preferred parameter setimpossible. Therefore, when Franks, S.W.,mad K.J.Beven, Bayesian estimation ofuncertainty in implementing a multiobjective calibration, it is criticalto select a minimumnumberof mostrelevantobjectives.(6) Because
landsurface-atmosphere fluxpredictions, J. Geophys. Res.,102, 23,991-23, 999,1997.
multiple paran•eter setscan be Pareto, multiobjective Franks, S.W.,P.Gineste, K.J.Beven, and P.Merot, Onconstraining the predictionsof a distributedmodel:The incorporation of fuzzy calibrationproblemsare inherentlysubjective.Procedures for estimatesof saturatedareasinto the calibrationprocess,[Vater findingthe Paretoparametersetsare relativelyautomaticand Resour.Res.,34(4), 787-797, 1998. will help to identify parametersets that minimize all the Gupta, H. ¾., S. Sorooshian,and P.O. Yapo, Toward improved calibration of hydrologic models: Multiple and objectivesto some extent. However, to choosea single noncommensurablemeasuresof information, Water Resour. Res., parameterset from thosethat are Pareto,a setof rulesabout 34(4), 751-763, 1998. whichobjective(or trade-offamongobjectives) is mostcritical Gupta,H. V., L. Bastidas,S. Sorooshian,W. J. Shuttleworthand Z. L. Yang, Parameterestimationof a land surfaceschemeusing to minimizemustbe adopted. Alternatively,in this context, multicrilefiamethods,d. Geophys.Res., 104, 19, 491-19, 503, the concept of a preferred parameter set may not be 1999. meaningful. It might be betterto understand the rangeof Hipps, L. E., E. Swiatek, and W. P. Kustas, Interactionsbetween model behaviorsgiven by the Pareto set. The range and regionalsurfacefluxesandthe atmospheric boundarylayerover a characterof the Pareto set may be an indicationof model heterogeneous watershed,Water Resour.Res., 30(5), 1387-1392, physicaldeficiencies, aswhenthe observations lie outsidethe 1994.
pareto setrange. However, a large Pareto setparameter rangeHouser, P.R.,Remote-sensing soilmoisture using four-dimensional dataassimilation,Ph.D. dissertation, 391pp.,Dep. of Hydrol. and may also result from a poorly posedparameterestimation problem.
Water Res., Univ. of Ariz., Tucson,Ariz., 1996. l-louser,P. R., C. Harlow, W. J. Shuttleworth, T. O. Keefer, W. E.
Emmerich, and D.C. Goodrich, Evaluation of multiple flux measurementtechniquesusing water balance informationat a Acknowledgments. This work was supported,fostered,and semiaridsite,paperpresentedat 13th Conferenceon Hydrology, carriedout to completionat the Universityof Arizona,Departmentof Am. Meteorol.Soc.,LongBeach,Calif., 1997. Hydrologyand Water Resources,with financialsupportprovided Houser,P. R., W. J. Shuttleworth, J. S. Famiglietti,H. V. Gupta,K. throughNASA grantsNAGW-4165andNAGW-4087andthroughthe H. Syed,and D.C. Goodrich,1998: Integrationof soil moisture awardof a NASA GlobalChangeFellowship.The studywouldnot remotesensingand hydrologicmodelingusingdata assimilation, havebeenpossiblewithoutthe dataprovidedby theUSDA-Southwest WaterResour.Res.,34(12), 3405-3420, 1998. WatershedResearchCenterandwithouttheir assistance in interpreting Humes,K. S., W. P. Kustas,andM. S. Moran, Useof remotesensing and reference site measurements to estimate instantaneous surface thesedata from the Walnut Gulch ExperimentalWatershed. We are energybalancecomponents over a semiaridrangelandwatershed, pleasedto thankCorrieThiesfor typographic suggestions andseveral WaterResour.Res.,30(5), 1363-1373, 1994a. anonymous reviewersfor very constructive suggestions. Humes, K. S., W. P. Kustas,M. S. Moran, W. D. Nichols, and M. A.
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(ReceivedApril 22, 1999;revisedSeptember22, 2000; acceptedOctober5, 2000.)
