A Simple Energy Budget Algorithm for the Snowmelt Runoff Model

WATERRESOURCES RESEARCH, VOL. 30,NO. 5, PAGES1515-!527, MAY !994

A simpleenergy budget algorithm for the snowmelt runoff model

WilliamP. Kustas and Albert Rango HydrologyLaboratory, Agricultural Research Service, U.S. Departmentof Agriculture,Beltsville, Maryland

Remko Uijlenhoet Departmentof Water Resources, Agricultural University, Wageningen, Netherlands

Abstract.The snowmeltrunoff model (SRM) uses a degree-dayapproach for melting snowin a basin.A simpleradiation component was combined with the degree-day approach(restricted degree-day method) in an effortto improveestimates of snowmelt andreduce the needto adjustthe meltfactor over the ablationseason. A dailyenergy balancemodel was formulated that requiresnot onlythe inputof radiationbut also measurementsof daily wind speed,air temperature,and relative humidity. The three approachesfor computingsnowmelt, namely, the degree-day,restricted degree-day, anddaily energybalance model were testedat the local scaleby comparingmelt rates with lysimeter outflow measurements.Because radiation measurementsare not often available,a simplemodel for simulatingshortwave and longwavecomponents of the radiationbalance that requiresminimal irrformation (i.e., daily cloud cover estimates, airtemperature, and relative humidity) was developed.It was found that cloudsand theireffects on daily insolationat the surfacecan producesignificant differences betweenmeasured and model estimates.In the comparisonsof snowmeltestimates withthe lysimeter outflow, the restricted degree-daymethod yielded melt rates that werein better agreement with the observedoutflow than the degree-daymethod and werepractically the same as estimatesgiven by the energybalance model. A sensitivityanalysis of runoff generatedwith SRM using as input the local snowmelt computationsgiven by the three modelsand measuredoutflow from the lysimeter was performedfor a basin. A comparisonof the synthetichydrographs for the basin suggeststhat a radiation-basedsnowmelt factor may improveranoff predictionsat the basin scale.

Introduction versus!5%). This causesa large differencein energyabsorp- tion and subsequentthermal heating. Additionally, the low Theoccurrence of precipitationin the solidform (snow)as thermal conductivityof snow greatly reducesheat exchange opposedto the liquidform (rain)typically causes a changein betweenthe groundand the atmosphereso that snowserves drainagebasin responseto the input of water becausesnow as an insulatingblanket for the underlyingsurface. isstored in a basinfor an extendedperiod of time beforeit Energy balance models can account for many of the entersinto the ranoff process.At the end of the winter physicalprocesses that affect snowmelt,but the inputdata to accumulationperiod the seasonalsnow cover melts in sev- mn the models are rarely available. As a result, the use of eralweeks to severalmonths depending on the amountof averagetemperature above freezing as an index of energy snowand location.When precipitationis snowrather than available for snowmelt is common. rain,evaporation losses during the runoffperiod are usually The degree-daymethod for snowme!t-runoffcalculations reduced. has been in use in different ways for almost 60 years [e.g., Snowmeltranoff is especiallyimportant in the mountain- Collins, 1934]. In general, the degree-day method involves ouswestern United States, where water suppliesare scarce computingthe daily snowmelt depth by multiplying the andwater demand is high.Most mountainous basins where number of degree-daysby the degree-dayfactor (see equa- snowoccurs generally have at least 50% of their water tion (!)). Linsleyand Franzini [ 1979]point out that the factor supplyproduced by snowmelt.In certainlocalities, 95% of generally increasesas the melting period progresses,as thewater supply is derivedfrom snowmelt in thespring and observedearlier by Anderson [1968, 1976]. earlysummer [Shafer et al., 1982]. An advantageof the method is that it is easy to use Snowon a drainagebasin can cause significant changes in operationallybecause a limitedamount of data are required, theenergy exchange. Because of its high albedo,snow usually only daily maximum and minimum temperatures. reflectsa much higher percentage of incomingsolar short- The methodhas been shown many times to produceaccurate waveradiation than snow-free surfaces (of the orderof 80% ranoff predictionsfor alpine and subalpinedrainage basins Copyright1994 by theAmerican Geophysical Union. [World MeteorologicalOrganization, 1986]. Becausetem- Papernumber 94WR00152. peratureis oneof the key climatevariables to be affectedby 0043.1397/94/94WR-00152505.00 climatechange, the degree-dayapproach is easilyadaptable 15!5 1516 KUSTAS ET AL.' ALGORITHM FOR SNOWMELT RUNOFF MODEL to evaluationof variousclimate change scenarios associated Grangerand Male, 1978;Kuusisto, 1980; Moussavi etal., with a temperaturechange [e.g., Rango and van Katw(jk, 1989]. 1990a; Martinec and Rango, 1989]. However,because (1) implicitlyaccounts for all termsof The snowmelt runoff model (SRM) improves upon the the energy budget that affect the mass balanceof the traditionaldegree-day approach by usingremotely sensed snowpack,a hashigh spatial and temporal variability. Under observations of the snow-covered area. The basin is divided most circumstancesthe range in the degree-dayfactor is into elevation zones, and the remote observationsare used between0.35 and 0.60 cm øC-• d-•. Attemptshave been to specify the portion of each elevation zone where the made to adjust the value of a using physicalparameters degree-dayapproach should be appliedto melt the existing which affectthe melt processand that can be easilyrhea. snowpack.As the snowpackis depleted, the degree-day suredor estimated.These include snow density [Martinec, approachis appliedto only thoseportions of the basinwhere 1960;Kuusisto, 1980], average daily air temperaturerange snow remains. The areal distribution employed in SRM [Moussaviet al., 1989],and meanwind speed[Martinec, providesa more physicallybased application of the degree- 1960]. day approach. A major drawbackwith this approachis that althoughthe Restricted Degree-Day Radiation Balance Approach degree-dayfactor is a bulk melt factor that can provide a In the current study, net radiation was chosen becauseit reasonablygood measureof an averageenergy flux, other appearsto explain most of the variation in snowmelt[Zuzet importantmelting factors like solarradiation, albedo, topog- and Cox, 1975;Granger and Male, 1978;Marks andDozier, raphy, and the turbulent energy exchangeprocesses are not 1992].The combinationof the surfaceradiation budget and a specificallytaken in to account.Therefore the degree-day temperatureindex has been proposedby severalinvestiga- approach will be unreliable under conditionswhen these tors [Martinec and de Quervain, 1975; Arebach, 1988;Mar- other factors may largely dominate the melt process[Ander- tinec, 1989] and appears to satisfy the requirement for an son, 1968].There is also high temporaland spatialvariability operational yet more physically based snowmelt model: in melt rates associated with these factors, which are not reflected in the degree-day approach. Therefore any im- M = art d q-m QRn (2) provementsthat can be added to the physical basis of the wherea r is a restricteddegree-day factor, m Q is thecon- degree-day method will likely improve its use in both snow- versionfactor for energy flux density to snowmeltdepth (cm melt runoff forecasting and evaluation of the effects of d-1 (W m-2)-•), andRn is netradiation (watts per square climate change. meter).The value of mQ is approximately0.026, so each 1W The objective of the present work is to develop a more m-2 of dailyaverage energy input results in a dailysnowmelt physically based snowmelt factor for SRM by combining a depth of about 0.03 cm water equivalent. simplified radiation budget approach with the degree-day Preliminary findings by Martinec [1989] showed valuesof method. In doingso it may be possibleto restrict variation of' the restricted degree-day factor ranging between 0.20 and the degree-day factor during the melt season. The study 0.25 cm øC-1 and exhibitingsignificantly less variability provides a comparisonof snowmelt measurementsat a point throughout the ablation period than the original degree-day (lysimeter) with three approachesfor calculating snowmelt, factor. Lower values of a r were generally found on days namely, the normal degree-day method, a restricted degree- with low wind speeds, reducing sensible heat transfer,or day and radiation balance approach, and a total energy with low humidity, which increased latent heat loss dueto balance approach. Further evaluation is carded out by evaporation.The resultsof Martinec [1989]indicate that the comparing snowmelt hydrographs from a watershed using restricted degree-day factor relates more to the physical SRM with the different snowmelt calculation methods. processesit supposedlyparameterizes, namely turbulent energy exchanges.

Theory and Model Formulations Surface Energy Balance Method A statisticalanalysis by Zuzel and Cox [1975]indicated Degree-Day Method that daily snowmeltestimates improved by incorporatingnet The degree-daymethod is a temperature index approach radiation,vapor pressure,and wind in modelformulations that equatesthe total daily decreaseof the water equivalent rather than air temperaturealone. In other words, improve- of the snowpack to a temperature difference between the ment in snowmeltpredictions for a muchwider range of daily averageair temperatureand a base temperature(usu- conditionswould result from computingthe turbulentfluxes ally 0øC).The coefficientmultiplying this temperaturediffer- of latent and sensibleheat [Anderson, 1976]. Hencethe ence is the degree-dayfactor [e.g., Westerstrom,1982]: equationfor estimatingthe energyavailable for snowmelt becomes M = a(Ta - Tb) (1) AQ = Rn + H + LE (3) where a is the degree-dayfactor (centimetersper degrees Celsiusper day), Ta is averagedaily temperature(degrees and is converted to snowmelt depth by Celsius), Tb is the base temperature(degrees Celsius), and m = mQ• Q (4.t, M is the snowmeltrate (centimetersper day). The temperature differencein (1) is representedby Ta, the degree-day whereA Q is theenergy available for melt,H thesensible temperature.When T a < Tb, then Ta = 0, and no melt will heat flux, and LE the latent heat flux, all in watts per sq• occur. Equation (1) has been widely used for short-term meter.Equation (3) is valid when the snowpack isisothermal prediction of snowmelt [Martinec, 1960; Anderson, 1968; at 0øCand its liquidwater content is at the irreducible KUSTASET AL.:ALGORITHM FOR SNOWMELT RUNOFF MODEL 1517 saturationpoint. This is assumed tobe a reasonableapprox- integration as a function of solar zenith angle c% was imationduring the ablation season. performedto obtaindaily totalsfor Kdiroand K difo,which In orderto minimizethe data and computational require- has been shown to be reasonably accurate [Garnier and ments,the simplifiedThornthwaite-Holzman bulk transfer Ohmura, 1968, 1970:Olyphant, 1986b]. approachfor parameterizingthe turbulent transfer of heat, The expressionfor the direct radiationreaching the sur- momentum,and water vapor was adopted, together with face under clear skies is stabilitycriteria based on the bulk Richardsonnumber Ri [Morris,1989]: Kdiro= Eo[(1 - Ao)tR - Aw]tadt•d (7) H = pCpChk2[ln(Z/Zo)]-2U(Oa- Os) (5a) wheretaa is the transmissionafter absorptionby dust,tsd is the transmissionafter scatteringby dust, t R is the transmis- LE = pLCek2[ln(Z/Zo)]-2u(qa- qs) (5b) sion after Rayleigh scattering,E0 is the extraterrestrial radiation,A•, is the absorptionby water vapor (absorptiv- where ity), and A0 is the absorptivityof ozone. For water vapor, Ch = (1 -- 58 Ri) 0.25for Ri < 0 expressionsfrom Wang [1976] for the absorptivity, using Kasten [1966] for computing the relative path length, were Ch= ( l + 7 Ri)-ø'• forRi>0 employed along with the model developed by Brutsaert [1975] for estimatingthe zenith path water vapor content. Ce = 0.5 Ch For ozonethe parameterizationsof Lacis and Hansen [1974] for absorptivityand Rodgers [1967] for the relative path andRi= gz(O a - Os)(OaU2)-1, /9is the density ofair, cp is lengthwere used. The quantity of ozone was estimatedfrom the specificheat of air at constantpressure, k is yon a formuladeveloped by Van Heuklon [1979]. Values of t aa, Karman'sconstant (0.4), z0 is the roughnesslength for t sa, and t R were obtained from expressionsreferenced by momentumand scalars,L is the latentheat of vaporization, Munro and Young [1982]. For diffuse radiation the formula 0is the potential temperature, q is the specifichumidity, u is from Munro and Young [ 1982] reads thewind speedat elevationz abovethe surface,t7 is the gravitationalacceleration, and the subscriptsa and s repre- Kdifo= E0{0.5(1 - A0)(1 - tr) sent values at z and at the surface. The bulk transferapproach is basedon somesimplifying + 0.8[(1 - Ao)tR - Aw]tad(1- tsd)} (8) assumptionswhich includeusing similar roughness lengths formomentum, heat, and water vaporand approximationsto with the values0.5 and 0.8 representingthe scatteringratios thestability corrections via Ch and Ce. However,because for Rayleighand Mie scatteringadopted from Davies and Idso [ 1979]. dailyinputs were used,the simplificationsleading to (5) were considered reasonable. Moreover, similar schemes have In order to have a truly operational model the important beenused with successin energy budget modelsfor the effects of clouds on the daily radiation balance need to be predictionof snowmelt in different environments[Dozier simply parameterized.The approachdeveloped by Munro andOutcatt, 1979;Granger and Male, 1978;Williams, 1988]. and Young [1982] was adopted where the direct radiation under cloudy skies was calculated with

Simulatingthe Radiation Balanceat an Kdir = Kdiro( 1 - m c) (9) Unobstructed Point at the Surface where mr . is the fractionalcloud cover. In order to use the restricted degree-dayfactor or the Diffuseradiation under cloudyskies, Kdi f, was computed approachoutlined in (3) and (4), the daily net radiation by adding the contribution of diffuse radiation from the balance at the surface must be determined. When there are fraction of the sky which is cloud-free and that passing no radiation measurements, this requires a modeling ap- throughthe cloudy portion, K difs, tO the contributionof proachwhereby the shortwaveand longwavecomponents of diffuse radiation by multiple reflection between the cloud Rn can be simulated with minimal ancillary meteorological base and surface, K difc.Thus data.This is discussedbriefly below. Global and Net Solar Radiation Kdif = Kdifs+ Kdifc (10) Underclear sky conditionsat a point on an unobstructed where horizontalsurface, the total shortwaveradiative flux re- ceivedper unit area (K, globalradiation) is the sumof the Kdifs= Kdifo(l -- rrtc) + mcKdir(tmc- Otct) (11) directinsolation and the diffusesky radiation[e.g., Kondra- tyev, 1973]: K difc=(K dir + K difs)rn ca ,.6 a dif(1 - a ct, a dif')- 1 ( 12) K o = K diro3' Kdifo (6) The symbolact is the cloudtop albedo,act, is the cloudbase whereK o is the globalradiation for clearskies, K dirois the albedo, adif is the surfacealbedo for diffuse radiation, and directsolar radiation under cloudless conditions, and Kdifo is tmcis the cloudtransmission. Values of ac0 (0.6) and tmc thediffuse component for clearskies. Note that in (6) andin (0.82) were taken from Munro and Young [!982], and the whatfollows the subscripto will alwaysindicate cloudless equationfrom Fritz [!954], modifiedby Munro and Young conditions.The direct and diffusecomponents under cloud- [1982],was adoptedfor estimatingOtct. Values of O•difwere lessconditions were determinedwith an operationalap- determined with a snow grain size algohthm discussed proachdescribed by Munroand Young[1982]. Numerical below. 1518 KUSTAS ET AL.' ALGORITHM FOR SNOWMELT RUNOFF MODEL

During the snowmelt season the fraction of incoming Warren,1982]. Furthermore, because it is assumedthat radiation reflected from the surface representedby the snowmeltoccurs when there is a net energyinput after the broadbandalbedo, asfc, decreases between snowfalls from snowpackbecomes isothermal at 0øC,Ts was takenas a around 0.9 to about 0.5 as a result of grain growth, contam- constant(i.e., 273 K) throughoutthe melt season.This ination, and shallow depth. Typical snow albedo decay yieldeda meanvalue for L sfcof the orderof 300W m-2. functions for the ablation season have been derived [e.g., The downwellinglongwave radiation from the atmo- Anderson, 1968, 1976; PetzoId, 1977]. However, to deter- sphere,Lsky, cannot be describedanalytically. Although mineastc requires estimates of the diffusesurface reflectivity radiationmodels with varyingdegrees of complexityfor for direct radiation,adir, and diffuseradiation, adi f. For the computingLsky have beendeveloped and tested[e.g., presentstudy an approachby Williams[1988] was adopted Lutheret al., 1988],they are stilltoo complicatedfor usein which uses a physicallybased broadband parameterization the presentmodel. Approaches involving the useof screen for the snow albedo. This schemeimplicitly accountsfor level air temperatureor vapor pressureor both for estimat- someof the distinctspectral properties of snowreflection via ing the atmosphericemission are much more applicable separate determination of the reflectivity for direct and [e.g.,Brunt, 1932; Swinbank, 1963; Idso and Jackson, !969; diffuse radiation under all sky conditions: Brutsaert, 1975; Unsworth and Monteith, 1975]. On the other hand, most of these expressionsare empiricaland adir'-O•diri- (0.083 + 0.23EI/2) cos 1/2 a s (13) hence contain "constants"which vary significantlywith locality. a dif'-- a difi-- 0.21E1/2 (!4) A noteworthyexception is the modelof Brutsaert[1975], where who took a more physically based approach. Brutsaert's analysisyielded the effectiveatmospheric emissivity under E = (E•3 + 0.252Ds)•/3 (15) clearskies, œskyo, aS a functionof both screenlevel vapor pressuree a (pascals)andair temperatureTa (kelvin): whereE is the meangrain radius (millimeters), a s is the solar zenith angle, E i is the mean grain radius before melting eskyo= 0.642(ea/Ta)1/7 (18) occurs,usually between 0.1 and 0.4 mm, D s is the numberof days sincelast snowfall, adiriis the diffusesurface reflectiv- Aase and Idso [1978]found that underfreezing conditions, ity for directradiation at sunriseor sunset(•0.965), and adifi (18) generallyunderestimated the atmosphericemissivity. is the surface reflectivity of diffuse radiation of fresh, dry Satterlund[1979] derived an exponentialformula containing snow (•-0.96). The above formulation implicitly assumes both vapor pressureand air temperaturethat improvedthe that the surfaceof a melting snowpackis wet during the agreement with measurements under freezing conditions. entire day and that each new snow accumulationconsists of However, Brutsaert's equation seemspreferable becauseof dry uncontaminatedsnow with the mean grain radius of new its theoretical foundation and the fact that his derivation snow. Clearly, both the fraction of a day that the snow allows adjusting for a decreasing amount of atmospheric surface is wet and the mean grain radius are related to the water vapor with increasing altitude [U•jlenhoet, 1992]. amount of radiation absorbed and other feedback mecha- Furthermore, daily average air temperaturesduring the melt nismsimplicit in the energybudget of the snowpack[Dozier seasongenerally are not significantly below 0øC; hence(18) et al., 1989]. Fortunately, (13) and (14) are not very sensitive wouldnot yield significant errors. Thus the value of Lskyois to changesin the empiricalfactor usedin (15) for estimating evaluated with the following formula: the changesin snow grain radius due to melting. The above parameterizationssummarized in (6)-(!5) yield Lskyo= eskyoO'ra4 (!9) an estimateof the daily net shortwaveradiation balance K n on an unobstructed horizontal surface: By combining(17) and (19) the net longwaveradiation under clear skies,Lno, for an unobstructedhorizontal surface can K n = (1 - a dir)Kdir+ (1 -- a dif)Kdif= (1 -- • sfc)K (16) be written as follows' Longwave Radiation Lno '- Eo'(• •kyoT• 4- T• 4) (20) Longwave radiation in the terrestrial environmentorigi- For cloudy conditionsthe atmosphericemission increases, nates mainly from two sources: emission from the atmo- mainly as a result of the increased water vapor content. sphereand emissionfrom the Earth's surface.The longwave Someinvestigators [e.g., Brutsaert,1982; Unsworth and emissionfrom the surfaceis proportionalto the fourthpower Monteith, 1975]have presentedcorrection factors for Ln; of its absolutetemperature according to Stefan-Boltzmann's but this assumesL n is negative,which typically is notthe relation [e.g., Liou, 1980] and can be expressedby the casefor snow-coveredsurfaces during the meltseason. Thus following formula: anadjustment to theeffective atmospheric emissivity would Lsfc= errrs 4 (17) seemmore appropriatefor radiation modelingover snow. For operationalapproaches in the nearfuture it islikely where e is the surface emissivity, rr is Stefan-Boltzmann's that daily cloudcover observations will be the onlyinforma- constant(•-5.67 x 10-8 W m-2 K-4), andT, isthe absolute tion routinely available. Therefore in the presentstudy the surfacetemperature (kelvin). For snow the emissivityis atmosphericemissivity for clearskies was corrected for the between 0.98 and 0.99 [Dozier and Warren, 1982;Kondra- effect of cloudiness via a nonlinear function of the mean tyev et al., 1982]and is not significantlyaffected by changes fractional cloud cover [Brutsaert, 1982]: in snow cover propertiesresulting from grain growth or contaminationduring the snowmelt season [Dozier and sky= Esky o( 1 q-cm c2) (21) KUSTAS ET AL.: ALGORITHM FOR SNOWMELT RUNOFF MODEL 1519 wherem c is the fractionalcloud cover and c is an empirical Evaluation of Snowmelt Algorithms coefficient.Although the coefficient c in (21)is dependent on ExperimentalData Set cloudtype, Brutsaert [1982] suggestedthat c •- 0.22 is a satisfactorymean value. This is supportedby theexperimen- The performancesof the degree-day,restricted degree- talfindings of Kimball et al. [1982].Substitution of (21) into day, and the bulk transfer/energybalance approaches for (20)yields a dailyestimate of the netlongwave flux for an estimatingdaily snowmelt were assessedusing outflow mea- unobstructedhorizontal surface: surementsfrom a snowlysimeter at the test siteof the Swiss Federal Institute for Snow and Avalanche Research at 2 4 4 Ln = I•o'(t•skyo(1q-Cmc)T a -- r s) (22) Weissfluhjoch/Davos,Switzerland (46.8øN, 9.8øE). The snowlysimeter, which has a surfacearea of 5 m2, is situated Bycombining (16) for the net dailyshortwave flux Kn with in a horizontal snow field at an altitude of 2540 m above (22)for the net longwaveflux Ln, the expressionfor com- mean sea level. The snow lysimeteris an advancedtype putingthe dailynet radiation for a horizontalunobstructed featuring high side walls (60 cm) to preventlateral inflows surfaceis obtained: andimproved subsurface drainage on the experimentalplot compatiblewith the surroundinglandscape. Its outflowis Rn= Kn + Ln-- (1 - a dir)Kdir+ (1 -- a dif)Kdif interceptedby a steel vesseland recordedcontinuously usinga tippingbucket gauge. Due to resistancein the + ea(e skyr• 4- r•4) (23) unsaturatedand saturated snow layers and in the pipe It should also be mentioned that satellite remote sensing leadingfrom the vessel to thegauge, the transformationof a for assessingcloud properties is showing some promise snowmeltdepth resulting from a positiveenergy budget at [Rossow,1989] and may one day provide the necessary the snow surfaceinto a outflow hydrographfrom the entire informationfor more physically basedmodeling. Moreover, snowpackin the lysimeterexhibits a certaintime lag and attenuation.However, it was found that, apart from daily estimationof surface insolation and the net shortwave radi- fluctuationsassociated with daytime variationsin snowmelt ationbalance [e.g., Chou, 1989;Darnell et al., 1988;Dedieu and nighflyrefreezing, the liquid water contentof the et al., 1987] as well as net radiation[Pinker and Tarplcy, snowpackdid not increase any more after the dayon which 1988]from satellite remote sensingis another potential the lysimeteroutflow started. These features of the snow sourceof input datafor modelcalibration and verification of lysimeterand concurrent checks of the hydrologicalbalance catchmentand regional scale computations(for a recent at the experimentalsite indicate that the lysimetermeasure- reviewsee Sellers et al. [ 1990]). Nevertheless, in the present mentsare representativeof the snowmeltprocesses in the modelformulation such approaches are not yet suitablefrom area [Martinec, 1987]. Consequently, one can assume on a anoperational standpoint. dailybasis that the changein snowmeltdepth (water equivalent)of the site essentiallyequals lysimeter outflow [Mar- ModelingRadiation Over ComplexTerrain tinec, 1989]• Difficultiesin modelingthe radiationbalance in mountain- The snowlysimeter data set usedfor comparisonwith ousterrain are mainly associatedwith topographiceffects model simulationswas collected during the 1985 ablation whichmodify the incidentradiation via obstruction,reflec- seasonby the FederalInstitute for Snow and Avalanche tion,and emissionby adjacentsurfaces. These factors are Research.It consistedof daily averagesof air pressure(p), moreprevalent in alpinewatersheds where most of the larger air temperature(Ta), relativehumidity (RH), wind speed snow-coveredareas have slopesof 10ø to 30ø and signifi- (u), fractionalcloud cover (me), sunshineduration (n), canflyinfluence the radiationbalance [Olyphant, 1986a]. globalradiation (K), precipitationoccurring as rain (Pr), For the presentstudy, no attempthas beenmade to precipitationoccurring as snow (Ps), andlysimeter outflow includethe effects of adjacentterrain and slope/aspect onthe (Ql). The entire1985 snowmelt season lasted from May 9 radiationbalance at a point.Models for solar[e.g., Dozier, (start of snowmelt)to July 15 (last day with lysimeter outflow). During the first week, however, no lysimeter 1980;Olyphant, 1984] and thermal [e.g., Marks and Dozier, outflow occurred because the entire snowmelt depth was 1979;Olyphant, 1986a] radiation of suchcomplexity have usedto increasethe liquid water contentof the snowpack beendeveloped and implemented with digital elevation data. gradually.Hence ripe conditions only occurred on the 58 However,at presentthese algorithms are difficult to include daysbetween May 16 (startof lysimeteroutflow) and July inan operational mode due to the amount and nature of input !2, whichtherefore were takeninto consideration. data(e.g., cloud cover) and the computational requirements. Thedata pertaining to the mass(i.e., water)balance of the Onthe other hand, studies improving the efficiency inspatial snowlysimeter were collected at a differentlocation than the integrationalgorithms [Dozier and Frew, 1989] and extrap- meteorologicaldata. These data includedp, which was olationof inputdata [Running et al., 1987]will inevitably measured at an altitudeof 2667 m, and Ta, RH, u, m c, and allowmore detailedtreatment of the radiationbalance in K, whichwere evaluated at 2693m abovemean sea level (at complexterrain. the automaticmeteorological station of the SwissMeteoro- Asa meansof incorporatingsome of themajor terrain logicalOffice). Hence the meteorologicaldata are from 127 effectsonradiation, Uijlenhoet [1992] developed a submodel and 153 m above the snow field containing the lysimeter. forcomputing the radiation budget for aninfinitely long Measurementsof p and Ta were extrapolateddownward to V-shapedvalley. Thus only gross features ofa watershed,the snowfield using standard atmosphere profiles. The value suchas basin orientation and average slope and elevation, of RH was assumedto be constantover this altitudediffer- arerequired. The radiation budget for a watershedusing this ence[Marks andDozier, 1979],which allowed easy compu- idealizedcase will be testedin a future study. tationof the vaporpressure at the lysimeterand is basically 1520 KUSTAS ET AL.: ALGORITHM FOR SNOWMELT RUNOFF MODEL

Table 1. Minima, Maxima, Averages,and StandardDeviations of Some of the Daily AverageInput VariablesCollected at the WeissfluhjochTest Site During the 1985 Ablation Period

10•, Pa Ta,K ea, Pa mu, s-1 mc WK, m-2 cmPr,d -• cmQl, d-1 Minima 739 270 326 0.5 0.13 57 0 0.05 Maxima 757 285 818 6.9 1.00 458 3.9 5.98 Averages 749 276 605 3.0 0.76 270 0.24 1.70 Standard deviations 4 3 104 1.5 0.29 87 0.66 1.49

See text for notation of symbols.

1 the sameas assumingan exponentialvertical vapor pressure riseto sunsetusing Simpson's ] rule with a temporalincre- profile [Brutsaert,1975]. Although u will likely be overesti- ment of 1 hour. The resulting daily averagesof global matedwhen applieddirectly to the lysimeterbecause it was radiation represented the measured values rather well on a measured at a mountain summit, there was no alternative seasonal average basis, as is indicated by a MBE of 2.35%. due to a lack of more appropriatedata. No correctionfor On a daily basis there were larger differences between elevation differenceswas made to rnc either, which is measuredand modeled values, as is indicated by a RMSE of supportedby the findingsof Olyphant[ 1984].Finally, K will 21%. A linear regression analysis of the simulated versusthe be overestimatedwhen applied directly to the lysimeter, measuredvalues yielded 0.92 for the slope,29 W m-2 forthe becauseit was measuredat a ridge top where the effectsof intercept,and 57 W m-2 for thestandard error. The value of obstructionby surroundingterrain are negligible.However, r 2 = 0.56 indicatedsignificant scatter between simulated no correction was made to K because the effects of obstruc- and measured radiation (Figure 1). As can be seen from tion by neighboringsurfaces to the amountof solarradiation Figure 2, the daily simulated values seem to follow the received by the snow lysimeter could not be estimateddue to overall trends in the measuredvalues but generallyoverp•- a lack of appropriate topographic data. dict high values and underpredict low values. The measured The minima, maxima, averages, and standard deviations global radiation on days when the mean fractional cloud of some of the corrected input variables are listed in Table 1. coverequaled 1 rangedfrom 57 to 365W m-2, varyingby From the table, note that the average vapor pressure of the more than a factor of 6. This result plus the comparisonin air (ca = 605 Pa) duringthe ablationperiod is only slightly Figure 2 illustrate the problemsassociated with modelingthe lower than the saturated vapor pressure over melting snow variability of global radiation due to cloud cover effectson a (Ca = 611 Pa). The average is significantlyhigher than daily average basis without taking into account the diurnal observations collected in other midaltitude alpine basins variations or the cloud type. [Marks et al., 1992]. This may indicate that the humidity The simulated daily average surface albedo decreased sensor was losing calibration and as a result overestimating from about 0.85 for each new accumulation of fresh dry humidity (D. Marks, personal communication, 1993). Also, snow (with a mean grain radius of 0.2 mm) to 0.59 for there is significantvariability in the magnitudeof the mean saturated and contaminated snow at the end of the ablafion fractional cloud cover (me) and, consequently,in the range period(Figure 3). The net shortwaveradiation simulated by in values of the daily average global radiation (K). The total the model is quite sensitiveto variation in the albedodecay !ysimeter outflow (•Ql) during the 58 nearly equilibrium days of the 1985 snowmelt season at the test site at Weiss- duringthe snowmeltseason. Furthermore, variability in the fiuhjoch amounted to 98.6 cm, of which 13.6 cm can be net longwaveradiation can greatly influencethe surface attributedto dischargeresulting from rainfall (EPr) and the radiation budget. In the model the magnitudeof the net remaining 85.0 cm, consequently, to actual snowmelt (EM). longwaveradiation is primarily affectedby the typeof formulafor determiningthe effectiveatmospheric emissiv- Evaluating Model Performance ity. A seasonalaverage net radiation of + 19W m-2 was To assessthe simulation performance of the three previ- obtained,composed of +76 W m-2 for the net shortwave ously describedpoint snowmeltpri•diction methods, several radiationand -56 W m -2 for the net longwaveradiation. different statistics were employed. The mean bias error Thus the net shortwaveradiation simulated by the model (MBE) and the root mean square error (RMSE) [Willmort, makesup nearly58% of the radiationfluxes, leaving 42% 1982] were computed and expressedas a percentageof the contributedby the longwavebalance. The minimumand measuredmean. The coefficientof determination(r 2) and maximumdaily average net radiation were found to be-40 the slope and the intercept (and the associated standard and +96 W m-2, respectively. error) resultingfrom a linear regressionanalysis between the The total simulatedsnowmelt occurring as a resultof a simulated and measured values were also evaluated. positiveradiation budget at the snowsurface accounted for 54% of the lysimeteroutflow during the entireablation period.This is slightlysmaller than 60% obtained by Marti- Results nec [1989] using measuredradiation data for the same Instantaneousvalues of global radiation were generated snowmeltseason. Because the average energy flux density accordingto the parameterizationpresented in (6)-(16). The thatis associatedwith cooling of precipitationreceived by instantaneousvalues were integratednumerically from sun- thesnowpack is lessthan 0.5 W m-2 (witha totalmeltwater KUSTASET AL.' ALGORITHMFOR SNOWMELT RUNOFF MODEL 152!

500

1: ! line

400

0 ß•-- 300

_o 200

• lOO

o 0 100 200 300 400 5oo MeasuredGlobal Radiation (W m'•) Figure1. Relationshipbetween simulated versus measured daily average global radiation for the 1985 snowmeltseason at Weissfluhjochtest site. equivalentof only 0.7 cm), it is reasonableto concludethat proportionalitybetween the two [Zuzel and Cox, 1975; theremaining 46% of the cumulativelysimeter outflow is the Olyphant, 19841. result of net turbulent heat transfer between the snow The seasonal averages of the simulated flux densities surfaceand the atmosphericboundary layer. The value of r 2 associatedwith the input of sensible heat and the loss of betweenthe measured lysimeter outflows and the simulated latentheat were 13 and -0.8 W m-2, respectively.The latter radiativemelt was 0.78. The RMSE was found to be 60%, is the result of a net loss of 1.3 cm of water equivalentfrom andthe slope,intercept, and standard error of theperformed the lysimeter due to evaporation of meltwater, which is linearregression analysis were 0.64, -0.13, and 0.43 cm negligiblein the massbalance of the snow !ysimeterin this d-1, respectively.These statistics confirm the findings of particular case. For the same site and ab!ation period, severalauthors, who argue that althoughnet radiation ex- Martinec [1989] estimated4.8! cm of evaporationof melt- plainsmost of the variationin snowmelt,there is no simple water from the computation of hourly melt rates. This

500 -o----'MeaSUred Simulated 4o0

300

lOO

00 10 2•0 ß ' 3¸ .... 40...... 50 - 60 Days Since Start of Snowmelt Season Figure2. Simulatedversus measured daily average global radiation throughout the !985 snowmelt season(May !6 to July12) at Weissfluhjochtest site, plotted as a functionof time. 1522 KUSTAS ET AL.' ALGORITHM FOR SNOWMELT RUNOFF MODEL

0.9

----. 0.8

o •- 0.7

'• 0.6 E

0.5

0 10 20 30 40 50 60 Days Since Start of Snowmelt Season Figure 3. A time plot of the simulateddaily averagebroadband albedo for the 1985snowmelt season at the Weissfluhjoch test site.

translatesinto an average loss of latentheat of-3.1 W m-2 In Figure 4 the seasonal variation of the energy budget for the snowmelt season. According to the atmospheric terms in (3) are illustrated. Note the relatively small latent stability criterion (see equation (5)) stable (near neutral) heat flux component and the fact that almost an equal conditions prevailed throughout the snowmelt season (i.e., number of positive and negative values exist. This behavior 43 of the total of 58 days). The correction factor to account in LE is not consistentwith estimates in a small alpinebasin for departuresfrom neutral conditionsnever departed much by Marks and Dozier [1992]. One cause may be instrumen- from unity. The average restricted degree-day factor (ar) tal. As discussedabove, consistently high relative humidities assessedto fit the simulated daily average turbulent transfer measured by the sensor may indicate that it was losingits throughoutthe snowmeltseason amounts to 0.18cm øC-• calibration. With (Sb) such high humidities will significantly d-1, whichis closeto the valueof 0.20 cm øC-• d -• that reduce the computed values of daily LE. Another reason Martinec [1989] assessedfor the same ablation period using may be the use of daily data in (5a) and (Sb) insteadof measurementsof global radiation instead of simulations. meteorologicalobservations available at a higher frequency.

lOO -' Net RadiatiOn O Sensible Heat Flux & LatentHeat Flu

-50 .... I I I 0 10 20 30 40 50 60 Days Since Start of Snowmelt Season Figure 4. Simulateddaily averageenergy balance components, net radiation, and sensibleand latent heat fluxes during the 1985 snowmeltseason at the Weissfluhjochtest site. KUSTAS ET AL' ALGORITHM FOR SNOWMELT RUNOFF MODEL 1523

Table 2. SummaryStatistics Comparing Lysimeter Outflow Measurements With Simulationsof the Daily SnowmeltAccording to the Original Degree-DayMethod a, the RestrictedDegree-Day Method a,, andthe TurbulentTransfer/_+ Energy Balance Method AQ

Standard Error of the Estimated Water Water Equivalent MBE, RMSE, Intercept, Equivalent, (Snowmelt) % % r 2 Slope cmd-• cm d -•

a - 1.8 65 0.54 0.82 0.23 0.94 a r -5 49 0.73 0.96 -0.02 0.72 AQ -7.4 50 0.74 0.99 -0.09 0.73

The input variables were collectedat the test site at Weissfluhjochduring the 1985 ablation period.

The use of daily meteorological data as input does not agreementwith measuredsnowmelt from the lysimeter.The accountfor the significant temporal variability in turbulent relative reduction in RMSE is about 20%, and r- •ncreases fluxesand gradients. by nearly40%. Overall,there is little differencebetween the The simulation capabilitiesof the three presentedmethods restricteddegree-day and the bulk transfer/energybalance forthe predictionof the point snowmeltwere comparedin a methodin simulatingthe snowmelt.This result is encourag- statisticalanalysis with the melt and flow rates measuredby ing becausemore physicallybased models like the bulk the snow lysimeter for the 1985 ab!ation period. For the transfer/energybalance approach outlined in (3)-(5) require restricteddegree-day factor approacha constantvalue of 0.2 atmosphericdata (i.e., screenheight, wind speed,and vapor cmøC -• d-• was usedthroughout the snowmeltseason, pressure)which are not commonly available in alpinebasins. whereasin the original degree-dayfactor methodthe value Apart from direct comparisonbetween the simulated wasgradually increased from 0.48 cm øC -• d-1 in Mayto snowmeltdepths and the measuredlysimeter outflows dur- 0.50cm øC-• d -• in Juneand 0.52 cm øC-• d-• in July ingthe 1985snowmelt season at the Weissfluhjochtest site, [Martinec,1989]. The resultingstatistics are summarizedin a brief sensitivityanalysis was carried out through the Table2, and the daily snowmeltindicated from the lysimeter intercomparisonof hydrographsgenerated for a complete outflowdata versusthe three modelpredictions are shownin watershed.The pointsnowmelt depths simulated according Figure5. to the three methods and the outflows measured at the snow The results in Table 2 suggestthat relative to the degree- lysimeterwere transformed into the respectiverunoffs that daymethod both the restricteddegree-day and the bulk would occurfrom the nearby Dischmabasin, providedthat transfer/energybalance method significantlyimprove the theinputs were representative for the wholebasin. In other

--m- Lysimeter _ o EnergyBudget •, RestrictedDegree Day x Degree Day

0 10 20 30 40 50 60 DaysSince Start of SnowmeltSeason Figure5. Dailymeltwater equivalent measured bythe lysimeter andsimulated bythe original degree- daymethod, therestricted degree-day method, andthe bulk transfer/energy balancemethod through the 1985snowmelt season at theWeissfluhjoch test site. 1524 KUSTAS ET AL.: ALGORITHM FOR SNOWMELT RUNOFF MODEL

Table 3. SummaryStatistics Comparing the Simulationof Daily Discharge (Expressedas Water Equivalent)Using SRM for the DischmaBasin With Daily SnowmeltGiven by the LysimeterVersus the Original Degree-Day Method a, the RestrictedDegree-Day Method a r, and the Turbulent Transfer/Energy Balance ApproachAQ for EstimatingMelt

Standard Error of the Estimated Water Water Equivalent MBE, RMSE, Intercept, Equivalent, (Flow) % % r 2 Slope cm d- • cm d -1

a -1.6 35 0.82 0.72 0.19 0.19 a r 4.0 25 0.91 0.97 0.05 0.18 AQ -3.8 25 0.91 0.94 0.02 0.17

words, the point melt rates given by the three models and ing snowmelt compared to the lysimeter outflow measure- measured by the lysimeter were assumedrepresentative of mentsare summarizedin Table 3. Note that the discharge the melt rate for the entire Dischma basin. These values resultingfrom (24a) and (24b) is converted to equivalent were the estimatesof the snowmelt rate M required in (24a). waterdepths for convenience.It can be seenby comparing Although the use of point inputs for representingarea inputs the results from Tables 2 and 3 that the snowmelt-runoff obviouslyis a grosssimplification [e.g., Bl6schtet al., 1991], transformationgiven by (24a) and (24b) yields a relative it will provide someinsight into the sensitivity of the applied reduction in the RMSE of all three methodsby nearly50% snowmelt-runoff transformation model to this data. The and increasesr 2 or the proportionof the varianceof the simplest form of the snowmelt runoff model (SRM), where measured water equivalents that is explained by the simu- the basin is not subdivided into different elevation zones, lated values by nearly 25%. Furthermore, the standard was used for this purpose [Martinec et al., 1983]: errors of the estimated water equivalent values are reduced by an order of 4. The synthetic hydrographsproduced by Qn+I = cn(MnSn+ Pn)A( 1 - kn+l) + Qnkn+l (24a) (24) are illustrated in Figure 6. Note that the largest differences occur during the peak runoff period, where melt rates kn+1 = xQ•-y (24b) are generally the highest (see Figure 5). This is a common where Q is the daily discharge(cubic meters per day), c is a result since the most dynamic period of the melt seasonis runoff coefficient (•-0.9), S is the ratio of snow-covered area the most difficult to simulate. In evaluating the snowmelt- to the basin area, P is the precipitationcontributing to runoff runoff process at the basin scale, sensitivity of simulated (metersper day), A is the basinarea (squaremeters), k is the dischargeto the three methods of computing snowmeltwas recessioncoefficient, x is the recessionfactor (•0.85), y is reduced. Nevertheless, the findings summarized in Tables2 the recessionexponent (•-0.086), and the subscriptdenotes and 3 indicate that including a radiation componentin SRM the sequence of days during the discharge computation to predict daily snowmeltgenerally improves model predic- period. Note that the valuesused for the parametersin (24a) tion. Similar conclusions have been reached in earlier stud- and (24b) are typical for the alpine Dischma basin, Switzer- ies [e.g., Tarboron et al., 1991] that combine radiationwith land, and are not applicableto other basins. temperature index approaches. Representativeparameter values for the Dischmabasin in 1985 were obtainedfrom a set of parameter values estab- Conclusions lished using 10 years of SRM simulations on the Dischma basin[Martinec and Rango, 1986].Although their publica- The resultof this studyshowed that combininga radiation tion showsthat runoffcoefficients for snowmeltand precip- componentwith the temperature-basedapproach for esti- itation can differ markedly from each other, one value is mating snowmelt generally improves results comparedto appliedfor the current purposebecause the contributionof usingsolely a temperature-basedsnowmelt factor and is in rainfall will be small as comparedto that of snowmelt. In good agreementwith a more complicatedenergy balance SRM, new snowfallis just incorporatedinto the existing model.The comparisonwas carriedout at the localscale snowpack. The relative snow-covered area of the basin is with snowmeltestimates compared to lysimeteroutflow assumedto decreaselinearly from 1 to 0 duringthe snowmelt measurements. Furthermore, runoff for a basin was gener- season, although analyses of data obtained from aircraft atedwith SRM usinglocal snowmelt computations from the photographyand satellite imagery show that areal snow threemodels and using the outflowmeasurements from the coverdepletion generally follows an S curve[Rango and van lysimeter. These melt rates were assumedto be representa- Katwijk, 1990b].It canbe seenfrom (24a) that during periods tivevalues for thewhole basin. Comparison of thesynthetic of true recession,kn+ 1 = Qn+l/Qn. In SRM this recession hydrographsassociated with the three modelsversus the coefficientis not assumedto be a constamas usual(leading synthetichydrograph using the lysimeter data suggests that a to an exponential recession)but rather to be a function of the morephysically based snowmelt factor may improve predic- dischargeon the day before, accordingto (24b). tions at the basin scale. The dischargeresults using the three methodsfor simulat- Theeffects of cloudson the daily insolation at thesurface KUSTASET AL.: ALGORITHMFOR SNOWMELT RUNOFF MODEL 1525

2.5 ,,,

Energy Budget RestrictedLysimeterDegree Day DegreeDay

--- 0.5

S 0 0 10 2; 30 40 50 60 Days Since Start of Snowmelt Season

Figure6. Synthetichydrographs for the Dischmabasin (43.3 km 2, 1670-3150m abovemean sea level) for the 1985snowmelt season converted to equivalentwater depths and generated with the following assumptions:recession coefficient k = 0.85Q-0.086 and snow-covered area decreasing linearly to zero. Estimatesof a basinaverage daily snowmelt were obtained from the lysimeteroutflows and simulated usingthe originaldegree-day method, the restricted degree-day method, and the bulktransfer/energy balance approach.

are shown to produce significant differencesbetween mea- cover, NOAA Tech. Rep. NWS 19, 134 pp., Silver Spring, Md., surements and model estimates. While this scatter to some 1976. degreeis due to the simplifiedapproach used in the present B16schl,G., The influenceof uncertaintyin air temperatureand albedo on snowmelt. Nord. Hydrol., 22, 95-I08, 1991. formulation,errors associatedwith the groundobservation B16schl,G., D. Gutknecht, and R. Kirnbauer, Distributed snowmelt of daily cloud cover also contributedto the disagreement simulationsin an alpine catchment,2, Parameterstudy and model [McGuffieand Henderson-Sellers, 1989]. Nevertheless,the prediction, Water Resour. Res., 27, 3181-3188, 1991. radiationmodel in its present form cannot adjust daily Brunt, D., Notes on radiation in the atmosphere, !, Q. J. R. insolationvalues without information on sky conditions. Meteorol. Soc., 58, 389-420, 1932. Computationsof snowmeltby radiationare alsosensitive to Brutsaert,W., On a derivable formula for long-wave radiation from clear skies, Water Resour. Res., 11,742-744, 1975. theestimated albedo values [e.g., Bl6schl, 1991].Therefore Brutsaert, W., Evaporation Into the Atmosphere, Theo•, History. it maybe more suitableto usea radiation-basedapproach for and Applications, 299 pp,, D. Reidel, Norwell, Mass., !982. runoff simulations than for real time runoff forecasts because Chou, M.D., On the estimationof surfaceradiation using satellite it is difficult to forecast cloudiness,albedo changes,etc. data, Theor. Appl. Climatol., 39, 2657-2666, 1989. However, use of satellite remote sensingfor estimating Collins,E. H., Relationshipof degree-daysabove freezing to runoff, cloudcover, albedo, and insolationis another avenuewhich Eos Trans. AGU, 15, 624-629, 1934. Darnell, W. L., W. Staylor, S. K. Gupta, and F. M. Denn, mayone day have operationalcapabilities for providingreal Estimation of surface insolation using sun-synchroroussatellite timedata [e.g., Ewingand Pinker, 1988;Sa!by et al., 1991]. data, J. Clim., 1,820--835, 1988. Davies, J. A., and S. B. Idso, Estimating the surface radiation balanceand its components,in Modification of the Aerial Envi- .Acknowledgments.Wewould like to thankJ. Martinecfor supply- ronmentof Plants, ASAE Monogr., edited by J. F. Gerber and mgthe basic lysimeterand climate data for the study.Also, many B. J. Barfield, pp. 183-210, American Society of Agricultural valuablecomments on an early draft of the paperwere provided by Engineers, St. Joseph, Mich., 1979. D. Marks,R. Davis, D. DeWal!e,and J. Martinec.Cooperation and Dedieu, G., P. Y. Deschamps,and Y. H. Kerr, Satellite estimation supportof the Departmentof AgriculturalEngineering at the Uni- of solar irradiance at the surface of the earth and of surface albedo versityof Marylandis appreciated. usinga physicalmodel applied to Meteosatdata, J. C!im. Appl. Meteorol., 26, 79--87, 1987. Dozier, J., A clear-sky spectral solar radiation model for snow- References covered mountainousterrain, Water Resour. Res., 16, 709-718, Aase,J. K., andS. B. Idso,A comparisonof twoformula types for 1980. calculatinglong-wave radiation from the atmosphere, Water Re- Dozier, J., and J. Frew, Rapid calculationof terrain parametersfor sour. Res., 14, 623-625, 1978. radiationmodeling from digital elevationdata, paperpresented at Arnbach,W., Interpretationof the positive-degree-daysfactor by !989 InternationalGeosciences and RemoteSensing Symposium heat balance characteristics--West Greenland, Nord. Hydrol., (!GARSS), Int. Council of Sci. Unions, Vancouver, B.C., I9, 217-224, 1988. Canada, July !0-!4, 1989. Anderson,E. A., Developmentand testing of snowpack energy Dozier, J., and S. I. Outcalt, An approachtoward energybalance balanceequations, Water Resour. Res,, 4, 19-37,1.968. simulationover ruggedterrain, Geogr. Anal., 11, 65-85, 1979. Anderson,E. A., A pointenergy and mass balance model of a snow Dozier, J., and S. G. Warren, Effectof viewingarqgle on the infrared 1526 KUSTAS ET AL.: ALGORITHM FOR SNOWMELT RUNOFF MODEL

brightnesstemperature of snow, Water Resour. Res., 18, 1424- Martinec,J., andA. Rango,Parameter values for snowmeltrunoff 1434, 1982. modelling, J. Hydrol., 84, 197-219, 1986. Dozier, J., R. E. Davis, and A. W. Nolin, Reflectance and trans- Martinec,J., andA. Rango,Effects of climatechange on snowmelt mittance of snow at high spectral resolution, paper presentedat runoffpatterns, in RemoteSensing and Large-Scale Global Pro. 1989 International Geoscienceand Remote SensingSymposium cess,Proceedings ofthe Baltimore Symposium, IAHS Publ. 186, (IGARSS) Int. Council of Sci. Unions, Vancouver, B.C., Can- 31-38, 1989. ada, July 10-14, 1989. Martinec,J., A. Rango,and E. Major,The snowmelt-runoffmodel Ewing, J. A., and R. T. Pinker, Sensitivityof surfacesolar fluxes to (SRM) user'smanual, NASA Ref. Publ. 1100,110 pp., 1983. cloud parameterization, J. Atmos. Sci., 45, 881-884, 1988. McGuffie,K., and A. Henderson-Sellers,Almost a centuryof Fritz, S., Scatteringof solar energy by clouds of large drops, J. "imaging"clouds over the whole-sky dome, Bull. Am. Meteorol. Meteorol., 11,291-300, 1954. Soc., 70, 1243-1253, 1989. Garnier, B. J., and A. Ohmura, A method of calculatingthe direct Morris,E. M., Turbulenttransfer over snow and ice, J. Hydrol., shortwave radiation income of slopes, J. Appl. Meteorol., 7, 105, 205-223, 1989. 796-800, 1968. Moussavi,M., G. Wyseure,and J. Feyen,Estimation of meltrate in Garnier, B. J., and A. Ohmura, The evaluation of surfacevariations seasonallysnow-covered mountainous areas, Hydrol. Sci. J., 34, in solar radiationincome, Sol. Energy, 13, 21-34, 1970. 249-263, 1989. Granger, R. J., and D. H. Male, Melting of a prairie snowpack,J. Munro, D. S., and G. J. Young, An operationalnet shortwave Appl. Meteorol., 17, 1833-1842, 1978. radiationmodel for glacierbasins, Water Resour.Res., 18, Idso, S. B., and R. D. Jackson, Thermal radiation from the 220-230, 1982. atmosphere, J. Geophys. Res., 74, 5397-5403, 1969. Olyphant, G. A., Insolationtopoclimates and potentialablation in Kasten, F., A new table and approximationformula for the relative alpine snow accumulationbasins: Front Range, Colorado,Water optical airmass,Arch. Metorol. Geophys. Bioklimatol., Ser. B, Resour. Res., 20, 491-498, 1984. 14, 206-223, 1966. Olyphant, G. A., Longwave radiation in mountainousareas and its Kimball, B. A., S. B. Idso, and J. K. Aase, A model for thereal influence on the energy balance of alpine snowfields,Water radiation from partly cloudy and overcast skies, Water Resour. Resour. Res., 22, 62-66, 1986a. Res., 18, 931-936, 1982. Olyphant, G. A., The componentsof incomingradiation withina Kondratyev, K. Ya. (Ed.), Radiation Characteristics of the Atmo- midlatitudealpine watershedduring the snowmeltseason, Arct, sphere and the Earth's Surface, translated from Russian by V. Alp. Res., 18, 163-169, !986b. Pandit, pp. 311-463, Amerind, New Delhi, 1973. Petzold, D. E., An estimationtechnique for snow surfacealbedo, Kondratyev, K. Ya., V. I. Korzov, V. V. Mukenberg, and L. N. Climatol. Bull. 21, pp. 1-11, McGill Univ., Montreal, Que., 1977. Dyachenko, The shortwave albedo and the surfaceemissivity, in Pinker, R. T., and J. D. Tarp!ey, The relationshipbetween the Papers Presented at the JSC Study Conference on Land Surface planetaryand surfacenet radiation: An update,J. Appl. Meteo- Processes in Atmospheric General Circulation Models, Green- rol., 27, 957-969, 1988. belt, Maryland, January 1982, edited by P.S. Eagelson, pp. Rango, A., and V. van Katwijk, Climate change effects on the 463-514, Cambridge University Press, New York, 1982. snowmelt hydrology of western North American mountainbasins, Kuusisto, E., On the values and variability of degree-daymelting IEEE Trans. Geosci. Remote Sens., GE-28(5), 970-974, 1990a. factor in Finland, Nord. Hydrol., 11,235-242, 1980. Rango, A., and V. F. van Katwijk, Development and testingof a Lacis, A. A., and J. E. Hansen, A parameterization for the snowmelt-runoff forecasting technique, Water Resour. Bull., 26, absorptionof solar radiation in the earth's atmosphere,J. Atmos. 135-143, 1990b. Sci., 31,118-133, 1974. Rodgers, C. D., The radiative heat budget of the troposphereand Linsley, R. K., and J. B. Franzini, Water ResourcesEngineering, p. lower stratosphere, Rep. A2, 99 pp., Planetary Circulations 51, McGraw-Hi!l, New York, 1979. Project, Dep. of Meteorol., Mass. Inst. of Technol., Cambridge, Liou, K. N., An Introduction to Atmospheric Radiation, Int. Mass., 1967. Geophys. Ser., vol. 26, 392 pp., Academic, San Diego, Calif., Rossow, W. B., Measuring cloud properties from space:A review, 1980. J. Clim. , 2, 201-213, 1989. Luther, F. M., R. G. Ellingson, Y. Fouquart, S. B. Fels, N. A. Running, S. W., R. R. Nemani, and R. D. Hungerford, Extrapola- Scott, and W. J. Wiscombe,Intercomparison of radiationcodes in tion of synopticmeteorological data in mountainousterrain and its climate models(ICRCCM): Longwaveclear-sky results, A work- use for simulatingforest evapotranspirationand photosynthesis, shop summary, Bull. Am. Meteorol. Soc., 69, 40-48, 1988. Can. J. For. Res., 17, 472-483, 1987. Marks, D., and J. Dozier, A clear-skylongwave radiation model for Salby, M. L., H. H. Hendon, K. Woodberry, and K. Tanaka, remote alpine areas,Arch. Meteorol. Geophys.Bioklimatol., Ser. Analysisof global cloud imageryfrom multiplesatellites, Bull. Am. Meteorol. Soc., 72,467-480, 1991. B, 27, 159--187, 1979. Sattefiund, D. R., An improved equation for estimatinglong-wave Marks, D., and J. Dozier, Climateand energyexchange at the snow radiation from the atmosphere,Water Resour. Res., 15, 1649- surfacein the alpine region of the Sierra Nevada, 2, Snow cover 1650, 1979. energy balance, Water Resour. Res., 28, 3043-3054, 1992. Sellers, P. J., S. I. Rasool, and H-J. Bolle, A review of satellitedata Marks, D., J. Dozier, andR. E. Davis, Climateand energy exchange algorithmsfor studiesof the land surface,Bull. Am. Meteor& at the snow surfacein the alpine region of the Sierra Nevada, 1, Soc., 7I, 1429-1447, 1990. Meteorological measurementsand monitoring, Water Resour. Shafer, B. A., E. B. Jones,and D. M. Frick, Snowmeltrunoff Res., 28, 3029-3042, 1992. modellingin simulationand forecasting models with theMartinec- Martinec,J., The degree-dayfactor for snowmelt-runoffforecasting, Rangomodel, Rep. 170542,NASA GoddardSpace Flight Cent., in "Surface Waters", Proceedingsof the General Assemblyof Greenbelt, Md., 1982. Helsinki, pp. 468-477, International Association of Scientific Swinbank,W. C., Long-waveradiation from clearskies, Q. J. R. Hydrology, Gentbrugge,Belgium, 1960. Meteorol. Soc., 89, 339-348, 1963. Martinec,J., Meltwaterpercolation through an alpinesnowpack, in Tarboton,D. G., M. J. AI-Adhami,and D. S. Bowles,A preliminary AvalancheFormation, Movementsand Effects Symposium,Pro- comparisonof snowmelt models for erosionprediction, Proceed- ceedingsof the Dayos Symposium,IAHS Publ., 162, 255-264, ings of the 59th Annual Snow Conference,pp. 79-90, Juneau, !987. Alaska, April 12-15, 199!. Martinec, J., Hour-to-hour snowmeltrates and lysimeteroutflow Uijlenhoet,R., A simplesurface radiation budget model for a point during an entire ablation period, in Snow Cover and Glacier in snowcovered mountainous terrain, MSc thesis,125 pp., Dep. Variations,Proceedings of the BaltimoreSymposium, Maryland, of WaterResour., Agric. Univ., Wageningen, Netherlands, 1992. May 1989, edited by A. Rango, IAHS Publ., I83, 19-28, 1989. Unsworth,M. H., andJ. L. Monteith,Long-wave radiation atthe Martinec, J., and M. R. de Quervain,The effectof snowdisplace- ground,I, Angulardistribution of incomingradiation, Q. J. R. ment by avalanches on snow melt and runoff, in Snow and Ice Meteorol. Soc., 101, 13-24, 1975. Symposium,Proceedings of the Moscow Symposium,August Van Heuklon,T. K., Estimatingatmospheric ozone for solar 1971, IAHS Publ., 104, 364-377, 1975. radiation models, Sol. Energy, 22, 63-68, 1979. KUSTAS ET AL.: ALGORITHM FOR SNOWMELT RUNOFF MODEL 1527

Wang,W. C., A parameterizationfor the absorptionof solar Zuzel,J. F., andL. M. Cox, Relativeimportance of meteorological radiationby water vapor in the earth'satmosphere, J. Appl. variablesin snowmelt, Water Resour. Res., 11, 174-176, 1975. Meteorol., 15, 21-27, 1976. Westerstrom,G., Estimatingsnow cover runoff by the degree-day W. P. Kustasand A. Rango,Hydrology Laboratory, Agricultural approach,Vannet Norden, 3, 47-53, 1982. ResearchService, U.S. Departmentof Agriculture,Beltsville, MD Williams,L. D., Estimationof frequencyof occurrenceof dry 20705. snowcoverin Germany, report, 41 pp., R. Signalsand Radar R. Uijlenhoet,Department of Water Resources,Agricultural Estab., Malvern, England, 1988. University, Wageningen,Netherlands. Willmott, C. J., Some comments on the evaluation of model performance,Bull. Am. Meteorol.Soc., 63, !309-1313,1982. WorldMeteorological Organization, Intercomparison of modelsof (ReceivedAugust 2, 1993;revised January 12, 1994; snowmeltrunoff, Oper. Hydrol. Rep. 23, Geneva, Switz., !986. acceptedJanuary 12, !994.)