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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/23598575 A clear-sky spectral solar radiation model for snow-covered mountainous terrain Article in Water Resources Research · September 1980 DOI: 10.1029/WR016i004p00709 · Source: NTRS CITATIONS READS 173 114 1 author: Jeff Dozier University of California, Santa Barbara 370 PUBLICATIONS 12,266 CITATIONS SEE PROFILE All content following this page was uploaded by Jeff Dozier on 22 May 2014. The user has requested enhancement of the downloaded file. WATER RESOURCES RESEARCH, VOL. 16, NO. 4, PAGES 709-718, AUGUST 1980 A Clear-Sky SpectralSolar Radiation Model for Snow-Covered Mountainous Terrain JEFF DOZIER Departmentof Geography,Universityof California,Santa Barbara, California93106 A dear-sky spectralsolarradiationmodel for direct and diffusefluxes,combinedwith topographiccalculationsfrom digital terrain data, computeseither incident,net, or reflectedsolarradiation at any point on a snowsurfacein mountainousterrain. The radiation may be integratedover any wavelengthrange from 250 to 5000 nm, or over any time step.Atmosphericattenuationparametersare ozone,water vapor, theAngstrom turbiditycoefficient andexponent, andtheabsorptance to reflectance ratioof the atmosphericaerosols.The model derivesthese,from measurementswhich may contain both systematicand random errors,by finding the leastsquaressolutionto an overdeterminedsetof nonlinear equations.For calculationsover a specifiedarea, it employstable look-up procedures,sothat computationspeedfor the spectralmodel approachesthat for a lumped model. Thus it may be usefulas part of a snowsurfaceenergy budget calculation over a drainage basin. INTRODUCTION typical, but not universal, size distribution of the particles. The wavelength range consideredis 250 to 5000 nm. AbsorpIn this paper I describea methodby which incidentor net tion bands outside this range are ignored. spectralsolarradiationunder clear skiesmay be calculated Monochromatic direct radiation at wavelength X (at the over a ruggedor mountainoussnow-coveredsurfacefrom a earth's surfaceon a plane perpendicular to the sun'srays) is sparsesetof measurements. The modelis particularto snow only in the specificationof surfacereflectanceand its variation with wavelengthand illumination angle. It could be applied to othersurfacesif appropriatelymodified. Q$[Xl -- Q0[Xlr-2 ro[Xlr•[Xl r•[Xl r^[X] *miX] (1) For tabulatedvaluesfor the solar constantQo[X],the model usesdata from Makarova and Kharitinov [1972], with adjust1. It is spectral,henceit can be comparedwith measure- ments from Willson[1978]. All of the transmissivitiesare funcmentsoverspecificwavelengthranges(e.g.,from satellites),or tions of wavelength. Except for absorptionby water vapor and miscellaneous gases, they follow the Beer-Bourget-Lambert it can be used to calculate net solar radiation for a surface material, suchas snow,whosereflectancevarieswith wavelength. law [Gates and Harrop, 1963; Robinson, 1966]: 2. Topographiccalculationsincludeeffectsattributableto (2) to[X]= exp [-ko[h] mo(O3)[zlll Some useful attributes of the model are: altitude, slope,exposure,horizon, and reflectionfrom adjacent terrain, utlizing the Digital Terrain Tapes available from the U.S. GeologicalSurvey.In mountainousterrain thesecan all vary significantly,and a radiationmodelwhichignoresany of them will be inaccurate for some locations. Forest canopy calculationsinclude beam and diffuse shading functions, us- ing photographsfrom a wide-anglecamera. 3. Values for atmospheric turbidity and water vapor are calculatedfrom global solar radiation measurementswith instrumentswhich can be either easilycarried in the field or can be operatedat remote, unattendedinstallations. 4. Table look-up procedures,usedwhen radiation is calculatedfor a terrain grid, prevent the model'sspectralattribute from degradingthe computationspeed.Therefore the radiation calculationsover an area are fast enoughto be usedin conjunctionwith calculationsof other componentsof the snow surfaceenergy balance. *w[h]= exp [-kw[h] (roww[zll)'/:l (3) ß•[h] = exp [-o•[X] maP[z]/P[O]] (4) *A[X] = exp [--OA[h]ma] (5) *m[X]----1 -- km[X](maP[zI/P[O])'/2 (6) Except for the aerosol attenuation coefficient, all of the absorptionand scatteringcoefficientsneededin the above equations are available as experimentally determined values (see Table 1). The aerosolattenuation coefficient usedis /•ngstrom's [ 1961, 1964] turbidity function: o^[X]= fi[zl X-• (7) This parameterization depends upon a Junge distribution of the particulatesover the range of sizesthat contribute significantly [Paltridge and Platt, 1976], and it is probably not applicable for heavy concentrationsfrom maritime or fire sources. DESCRIPTION OF THE MODEL A normally acceptedvalue for a is 1.3 or 1.5, with a maximal Direct and DiffuseFluxes range of 0.8 to 2.0 [Leckner, 1978].If •, is measuredin nm, valThe solar radiation model is simplified by using a single- ues for fi range from 0 to about 12,500.The advantageof this layer atmosphere,whereby any variations with altitude are parameterizationover a more precisesolution of the radiative expressedas analytic functions. The attenuation attributable transfer equation [e.g., Herman and Browning,1965] is a conto aerosolsis specifiedby a parameterization which assumesa siderable decrease in computation time. Formulas for path lengthsfor ozone [Lacis and Hansen, 1974],water vapor, and • Presentaddress:NOAA National EnvironmentalSatelliteService, air mass [Kasten, 1966] are available, as are altitude correcWorld Weather Building, Washington, D.C. 20233. tionsfor ozone[Kreugerand Minzer, 1974;Giorgis,1977],water vapor [Yamamoto,1949],and fi [Robinson,1966].Within Copyright¸ 1980 by the American GeophysicalUnion. Paper number 80W0216. 0043-1397/80/080W-0216501.00 709 710 DOZIER: SOLAR RADIATION TABLE 1. Sources for Attenuation Coefficient available for backscattering.Becauseof the non-Lambertian nature of snow (and many other surfaces),the reflectance computations are complicated.Three kindsof reflectanceare Coefficients Source ko[X,]ozone k,•[X], water vapor oR[X], Rayleigh km[X],miscellaneous Inn and Tanaka [1953] Leighton[1961] Gates[ 1960] Gatesand Harrop [ 1963] Penndorf[1957] Leighton[1961] Gatesand Harrop [ 1963] considered: specularreflectance from the beamaa[A], diffuse reflectancefrom the beam a'a[h], and reflectanceof diffuse radiation aq[h].Dunkle and Bevans[1956] and O'Brien and Munis[1975]givevaluesfor diffusereflectance aq[h]for snow the visiblepart of the spectrum,the main attenuationparameters are Rayleigh scatteringand aerosols.Beyond the visible, in the near infrared, water vapor absorptionis the major attenuation process. On a horizontal or slopingsurface,the direct radiation is Q'•[Xl= Q•[Xlcosz'(1 - I•e) MODEL (8) of variousgrain sizesand ages.The generalshapeof the reflectance versus wavelength curve shows that reflectanceis high in the visible portion of the spectrumbut decays very rapidly above about 1000 nm. O'Brien and Munis point out that the relative decaywith ageis about the samefor all wavelengths.Petzold[1977] has derivedtypical decayfunctionsfor accumulationand ablation seasons.Such functionsmay give reliable results for regional values, but for local values in mountainousterrain, they are only approximations.My approachis to assignan exponentialagingparameter'tt, which is basedupon field or satellitemeasurements [Frew, 1980] and which is independentof wavelength,without specificallyrelating this parameterto the actual age of the snow: z', the solar angle measuredfrom normal to the surface,may be calculated by standard methods [Sellers, 1965; Robinson, 1966;Paltridgeand Platt, 1976],and three-termFourier series provide accurate approximationsfor the solar declination, exp [-•] -- (aq[h]for actualsnow)/(aq[h]for new snow) (12) earth-sunradiusvector,and equationof time [Dozierand Outcalt, 1979].Va is a beamradiationshadingfactorattributable For reflectance from direct radiation, Paltridge and Platt to the forest canopy and is a function of Z'. Its effect is to averagethe portionsof the surfacein the shadewith thosein the sun, and its measurement is discussed in a later section on [1976]givean empiricalrelation: a'a[h] + aa[h] = aq[h]q- (1 - aq[h])exp[(18/•r)(•r/2- Z)] (13) Correctionsfor Terrain and Vegetation. According to data presentedby Middleton and Mungall [1952],a reasonablyaccurateapproximationfor the portion flected upward from the surfaceand subsequentlybackscat- which is specularlyreflectedfrom snowmay be obtainedby tered toward the earth. The total amount of radiation at waveaveragingthe Fresnelreflectances for normalandparallelpolarizationfor an ice surface(whoseindexof refractionis 1.31). length 3, that is scatteredout of the beam is Diffuse radiation comes from two sources: radiation that is scattered downward out of the beam, and that which is re- The limited data in Dirrnhirn and Eaton [1975], however, in- qo[Jk] -- Qo[Jk]r-: {1 - exp[-ma (on[X] + (1 + a/Re) + oa[X]P[zl/P[OI)I} dicatethat this approximationmay not be as valid for new (9) snow. The total amount of reflected radiation available for back- Someof this is absorbedafter scattering.I assumethat, on the scatteringis average,the scatteringtakes place from the level P[z]/2, QI[X] -- Q$[Jk] cosZ (a'e[h] + aa[h])+ qh•[X]aq[X] whereP[z] is surfacepressure. From the hydrostaticequation, (14) the equationof state,and empiricalequationswhichdescribe In this equationthe albedosrefer to an averageof the surthe vertical distributionof ozone, water vapor, and aerosols, roundingarea rather than to a particularpoint, becausethe one can calculate the amounts of these absorbingsubstances regionallyreflectedradiationwill be reflectedin many differbetween this level and the surface. This altitude is denoted ent directions.The backscatteredportion is assumedto be and the transmission function for the scattered radiation is Rayleighscatteredfrom the altitudeof half the surfacepressure. Aerosol scattering is generally forward peaked and ßsiX]-- exp {-1.9 [ko[hl(O3)[z:•l + w[Xlw[zll thereforeis not consideredin the backscatteringcalculations. + •[zl- fi[z:l)X-"/(1 + 1/(a/Re))l} (10) Some of the scatteredradiation is absorbedby ozone, water sothe portionof Q•'[2qwhichis actually Generally a/Re, the absorptance/reflectance ratio of the aero- vapor,and aerosols, backscattered is: sols,is independentof wavelength[Paltridgeand Platt, 1976]. A typical value is 0.5. If a/Re -- 0, the secondline of (10) is B[X] = 0.5 ,•[X] {1 - exp [-1.9 ,a[X] P[z]/P[O]]} (15) omitted. The factor 1.9 [Kondratyev,1969]is usedto integrate the scattered radiation over a hemisphere(a value of 2.0 Total backscattered radiation on an unobscured horizontal would be used if the diffuse radiation were perfectly iso- surfaceincludesmultiple reflectionsbetweenthe groundand but the infinite seriesconverges [Hay, 1976]: tropic). The downward-scattered radiationon an unobscured the atmosphere, horizontal surfaceis [Giorgis,1977] qb•[Jk] = QI[X]B[Jk]/(I- B[Jk] aq[•,]) (16) qh[•] ----CZCsqO[X]•'s[X] coSZ (11) The correctionfactorsCz -- 0.5 cos•/3 Z and c• -- 1 + cos:Z' sin3 Z accountfor the portion of the radiation scatteredtoward the surface[Robinson,1966]and for brighteningof the sky in the vicinity of the sun [Tempsand ½oulson,1977]. All of the radiation reflectedfrom the surfaceis potentially Total diffuseradiation on a slopeis then the sumof the backscatteredand diffuseradiationfrom the sky,correctedfor the portionof the sky seenand for brighteningnear the sun: q'•[X]-- (1 - Vt LJVq)(czc•qo[Xl,s[X] cosZ + qt,•[X]) (17) The combinedtopographic/forestview factor Vt LJ Vq repre- DOZIER:SOLARRADIATIONMODEL 711 I havechosena procedure for losentstheportionof theskyobscured by thesurrounding ter- After someexperimentation, derivedby Akima [1970]to producecubic rain,or thetreecanopy, or bothß Thederivation of Vt and Vq cal interpolation whichare thenusedfor the actualinterare discussed in a later sectionon Correctionsfor Terrain and splinecoefficients, of thiscurveis continuous, but Vegetation. Temps andCoulson [1977] introduce anadditional polation.The firstderivative factorto accountfor brightening of theskynearthe horizon, the secondderivativeis not. The methodseemssuperiorto an but for mountainousterrain this correction is usually inappropriate. Radiationthat is reflectedto a slopefrom adjacentterrain canbe dividedinto threecategories: reflectance of diffuseradiation,diffusereflectance of directradiation,andspecular reflectanceof directradiation.Of these,specularreflectanceof earlierattemptthat usedcubicsplineswhichhad continuous secondderivativesbut which causedsevereovershootprob- lemsin the 'windows'in the watervaporabsorptionportionof the spectrum. Two typesof integrationare neededfor the model:overa specificwavelengthrangeat a singletime or over a wave- overwavelength, directradiationfromadjacentterrainis ignored.Whileit oc- lengthandsometimestepAt.To integrate the modelusesAkima's[1970]interpolationroutineto evalcasionally occurs at particular combinations of solarandslope betweeneachwavelength valuethat angles, it is tooinfrequent to beimportant in theradiation uatesplinecoefficients corresponds to a wavelength in oneof thetabulatedfunctions. budget.Reflected diffuseradiationis The integralis then calculatedfrom the splinecoefficients q•[•] -- Vt(l - Vq)aq[•](q•[•]+ qb,•[•]) (18) [Ahlberg et al., 1967].For integration overtime,an adaptive quadrature method [Forsythe et al., 1977] workssatisfactorily, In this equationthe albedoaq[•] is the albedoof the surbecausethe dependence on time is typicallya smoothfunc- rounding terrain.To calculate reflected directradiation, it is tion. Where there is interference from the local horizon, the necessary to makesomegeneralizations aboutthesurround- function is not so smooth,and at theselocationsthe adaptive ingterrain. If, in anydirection, weconsider theterrain tobea methodrequiresmorefunctionevaluations. constant slopeto thehorizon, thenS', theanglebetween the planeofthepointandtheplanetothehorizon in a givendirection, canbereadilycalculated fromthedirection angles of CORRECTIONS FOR TERRAIN AND VEGETATION Terrain informationis availablein digital form on 'Digital thelinesnormalto the planes.For anyslope$ with exposure TerrainTapes'fromthe NationalCartographic Information E, thecosines of thedirection angles withrespect to thex, y, Center,U.S. GeologicalSurvey.From theseone can calcuandz axesarecosE sinS, sinE sinS, andcosS, respectively. lateslope,exposure, andhorizoninformation, subjectto some Hence S' is given by errorswhich are introducedby the resolutionof the terrain data[DozierandOutcalt,1979].Vegetation shades thesurface. It blocksout a portionof the sky,thusinfluencing diffusera+ sin E sin E' + cosS cosH) (19) diation,andit shades thebeam,especially at largesolarzenith functions varywiththe depth H isthehorizonanglein thedirection -(•r - E') if E' isposi- angles.Moreover,theshading sinS' -- sinS sinH (cosE cosE' tive,or (•r+ E') if E' isnegative. Theestimate of thetotalreflecteddirectradiationis obtainedby averaging the reflection of the snow. The method I use for the terrain calculations has been Only two rowsof the terrain fromthesurrounding slopes. WhereS' wouldbenegative, i.e., adaptedto a smallcomputer. grid need be in main memory at anyonetime,but asa conwhereIE - E'[ • •r andS • H, S' is setto zero.If thesursequence the methodrequiresa largenumberof random roundingterrainis dividedinto N segments: access input/outputoperations andthusdepends uponan operatingsystemthat can handletheseefficiently.The most time-consuming portionof the terrainanalysisis the calcu- q•[X] --(l/N) IQ&[X] (1- gq)] ß • cosZ"[/1a'Q[h,Z"l•l [1- cos 2(S')I/2)] p=o lation of the horizonvectors,but we have recentlydeveloped (20) a veryfastmethod[Dozieretal., 1979],sothatthistaskisnow computationally reasonable, evenfor largeterraingrids.The Asin (14),a'o[X]arethediffuse albedos of directradiation for programoutputconsists of an elevation, slope,aspect, view thesurrounding slopes andarefunctions of wavelength and factor,and horizonanglevectorfor eachpointin the terrain solarangle.The Z" arethesolarangleson theseslopes. grid.For storage space economy andfor dataportability beFinally,globalsolarradiation at wavelength X ontheslope tweendifferenttypesof computers, the dataarestoredasbiis: G[x] = + + + qA[X] and net monochromatic solar radiation is: Qn[X] = Q•&[x]- {Q'&[x] (a'o[X]+ ao[X]) + (q'•[X]+ q•[X] + q•[X])aq[X]} nary fractionswithin specifiedranges. Informationabout the vegetationcanopyis derivedfrom (21) skywardphotographs taken with a wide-anglecamera(a 'Widelux')with a pivotinglensthat sweeps througha 140ø field of view. From thesephotographs, it is possibleto mea- surea total hemispheric-shading portion(Vq) and a beam- (22) shadingfanction(Vo) whichvarieswith the solarangleZ. The corrections are thus statistical in nature and could not be wherethe albedosreferto the pointin question. For satellite usedto predictthe solarradiationundera forestcanopyat a radiometry purposes, thediffusely reflected solarradiation is: precise locationat a specific time.Theycanbeused,however, qT[X]--a'o[X]Q'l[X] + aq[X](q'•[X] + qa•[X]+ q•[X]) (23) to calculatea solarradiation value that is integratedover time [e.g.,EvansandCoombe,1959;Clark,1961;Anderson, 1964] or averagedover an area. Where the canopyis dense,the Interpolationroutinesare necessary to calculatevaluesfor transmittedsolarradiationis very smalland may generallybe of the surfaceenergybudget[ReifsnyQo[X], 'ko[X],o•[X], kw[X],and km[X]from tabulatedvalues. ignoredin calculation Interpolation andIntegrationMethods 712 DOZIER: SOLAR I•DIATION MODEL der and Lull, 1965]. However, in areas that are only sparsely content, and for snow the reflectancein thesewavelengthsis forested, solar radiation is the dominant term because the very high anyway. Clearly the instrumentsthat are usedmust trees may block the wind, even if providing little shade,and includeportionsof the attenuationbandsof the unknown pathe sensibleand latent heat fluxes are considerablyreduced rameters; for example, one must have measurementswhich [Priceand Dunne, 1976].Two problemsinhibit the application include wavelengthsgreater than 855 nm in order to estimate of these vegetation correctionsover drainage basins: one is precipitablewater vapor. The essenceof the procedureis simply described:measurethat the shadingfunctionsvary with depth of snow (and with season,in areas of deciduousvegetation);the other is the ne- ments of global radiation, or of somecombinationof global, cessityfor a large number of sampleswithin an area. At pres- direct, and diffuseradiation, preferably in at least two waveent, work is underway to derive approximatecanopy informa- lengthintervals,are taken at differenttimesof day and thus at tion from satellite data. differentatmosphericpath lengths,and valuesfor the parameters (unknowns)are selectedwhich best reconcilethe set of DETERMINATION OF THE ATTENUATION PARAMETERS measurements.Becauseof measurementerror (both random The solar radiation model has independentvariableswhich and systematic)it is generally not possibleto exactly match the measurements,but it is possibleto make more measuremay be classifiedinto four groups. mentsthan there are unknowns.The only restrictionis the asGroup Variable sumptionthat the parametersthemselvesdo not changedurradiative wavelengthrange ing the measurement period, so in general, morning and solar constant astronomic/temporal radius vector solar declination topographic/geographic equationof time solarangle view factor horizons shadingfactors altitude surface albedo atmospheric/geometric path length air pressure ozone precipitablewater vapor Angstromturbidity coefficient Angstromturbidity exponent absorptance/refiectance ratio The variables in the last group are thosewhich actually cause atmospheric attenuation of solar radiation under clear-sky conditions,and five of them cannot be easily determined from normally available information. These potential unknowns are: the ozone and water vapor contentof the atmosphere,the afternoon data should not be combined. Statedformally, we have a set of n unknowns(x) and a set of rn functions, each of which compares a measured value with its correspondingvalue that is calculatedfrom the model: f•[x]= log {Q•[measl/Qi[calc]} (24) Other comparisonscould be used(e.g.,differencebetweencalculated and model values), but the form of Beer'slaw makes this one preferable. The solution to this overdeterminedsystem of nonlinear equationsis x such that m--I E {ffx]} j----O is minimized. Dennis [1973] providesa useful review of methods of solvingsystemsof equationsin the least squaressense. For the solar radiation model I have used the finite difference Levenberg-Marquardt method [Levenberg,1944; Marquardt, 1963;Brown and Dennis, 1972]. Numerical convergenceproblems are reduced considerablyif the original constrainedunfi anda parameters of the/•ngstromturbidityfunction,and knownsare mappedinto unconstrainedvariables[Box, 1966]. An alternativemethod of determiningthe attenuationparamthe a/Re ratio of the aerosols. . etersis to simply treat the sum of squaresas a function of one In this sectionI proposea flexible method whereby two of or more variablesand to use a global or local minimization altheseparameters (watervaporandthe/•ngstrom fi value)can gorithm [see Brent, 1973]. This method generafly requires be determinedby a setof measurementsthat can be made easmore iterationsbut appearsto be lesssensitiveto random erily with either portable field equipment or unattended in- ror. struments at a micrometeorologicalstation. In contrast to standard methods [e.g.,Jngstrom, 1961,1964],it usesglobal Usually errorsin radiation measurementsare biasedin one direction. Wherever the error can be parameterized,an addivalues and broad wave bands, has lessstringent accuracy retional unknown can be added. A very simpleexamplewould quirements,and can deal with a variety of instrumentorientabe where all of the measurementsmade with a given intions. Furthermore, dependingon the wavelengthbands over strumentare wrongby a constantmultiplicativevalue,suchas which radiation is measured, additional parameters can be might be causedby miscalibration.The addedunknowns,septreated as unknowns. For example, if narrow-band wavearate for each instrument,would be the valuesby which readlength measurementsof both diffuse and direct radiation are available,themethodcouldbe usedto estimate theAngstrom ings must be multiplied to obtain the true radiationvalues. More complicatedbias functions,for exampleto compensate a exponentand the absorptance/reflectance ratio of the aerofor instrumentdeviationfrom cosineresponse,couldbe develsols,much in the way that King [1979] determined the comoped. plex index of refraction for a more completesolution to the radiative transfer equation in a wavelengthrange that did not include water vapor absorption.In our work in the southern Testswith SyntheticMeasurements Sierra Nevada, we have assumed a = 1.3 and a/Re = 0.5, and Initial tests of the method described above were made by have used a season/latitude/longitude approximation for generatinga set of 'synthetic'measurements for assumedvalozone [Fan Heuklon, 1979].To accuratelyestimateozone, one ues of the attenuation parameters,perturbing thesewith sysinstrument,restrictedto the shorterwavelengths(<400 nm), tematic and random errorsof known magnitudeand then uswould be preferred. The model is not very sensitiveto ozone ing the procedure described above to try to solve for the DOZIER: SOLAR RADIATION MODEL attenuationparametersand systematicerror corrections.Tests were carried out with systematicerrorsof up to +_5%with 2 instruments, and with random errors of 2%, 5%, and 10% 713 280-2800 nm and one for 700-2800 nm. All 115 time periods were thus of either 3- or 4-hoursduration, to try to avoid any problemscausedby increasein turbidity duringthe measurement period. Becauseof the broad wavelengthbands of the (standard deviation of Gaussian distribution). Under these circumstances the model performedadequatelyand was able instruments, onlywatervaporandtheAngstrom fl valuewere to calculate values which were closer to the 'true' values than treated as unknowns. the synthesized'measurements'were. However, random error of 10%,combinedwith systematicerror of 5%, was enoughto swamp the Brown and Dennis [1972] method. Even with 5% random and 5% systematicerror, the method required data over a large range of atmosphericpath lengths(7 A.M. to 12 noon) in order to accurately determine attenuation coefficients. The Brent [1973] algorithm for global minimization could copewith 10% random error, but it is computationally For the data set, atmosphericwater vapor content apparently varied from 0.2 to 30 mm (but wasgeneraBylessthan 10 mm), and • varied from 2 to 5000. The rangefor fl appearsto conform to the general seasonalranges,from 200 to 2000 (after conversionfrom/•m to nm), determinedby R6ssler[1979] at 1560m on the Rauschbergin Bavaria. Also in conformance with other observationswas an increasein the fl parameter from morning to afternoon. In the 40 observationdays, there unreasonable for more than three unknowns. These tests with were 17 instancesin which we measureda fl value of lessthan synthetic measurementsindicated that the method demands 500. All but one of thesewas in the morning.With the range lessaccuracy in itsinputdatathanthemethods of/[ngstrom of values in the field experiment, the results,summarized in [1961, 1964]or King and Herman [1979],which require mea- Figure 1 and in the table below, indicate that the method is surements accurate to 1% or 2%. reliable enoughto be useful. Tests with Field Measurements Location For the 1979 and 1980 snow seasons we have installed a sat- ellite data collectionplatform in the southernSierra Nevada. For 1979it was locatedin the OwensRiver drainageat an elevation of 3049 m. Included in the array of instrumentsare two Eppley Precision Spectral Pyrnaometers,one with a clear dome(wavelengthrange:280-2800nm) and the otherwith an RG-8 filter (wavelengthrange:700-2800 rim). Instrumentoutput voltagesin 1979were recordedhourly and transmittedto the SMS/Goes sateBiteevery 6 hours. The station was operated from mid-March to the end of the snowseasonin May of 1979, and it has been operating, since mid-October of the 1979-1980 season,in the Kings River drainage. Between March 19 and May 16, 1979, 115 setsof measurements from the 40 clear days on which we have data were used for analysis.Each set consistedof three or four pairs of measurements,each pair consistingof a measurement for HorseshoeMeadow, Owens River Drainage 36.5øN, 118.2øW elevation, 3049 m Instrument wavelength ranges 1. 280-2800 2. 700-2800 Dates nm nm March 22 to May 11 (40 clear days) Times 7:39 A.M. to 4:39 P.M. Results with instrument 1 correlation coefficient 0.993 0.981 19.8 slopeof regression intercept Results with instrument correlation coefficient 2 0.978 0.964 -8.3 slopeof regression intercept Results with both instruments correlation coefficient combined 0.994 0.999* slopeof regression intercept - I I I * Not significantlydifferent from 1.0 at 0.50, 0.95, or 0.99 con-. fidence limits •-Not significantlydifferent from 0.0 at 0.50, 0.95, or 0.99 con- 1000 - fidence limits CALCULATION • OF SOLAR RADIATION OVER RUGGED TERRAIN 750 - < 0.2t Given estimatesof the atmosphericattenuation parameters, as determined by methods describedin the previoussection, the solar radiation model can be applied to every point in an elevation-slope-aspect-horizonfile to produce maps of incomingor net radiation in any wavelengthband. Such results 5o0 , would be useful for calculations of snowmelt runoff, satellite 25O , / 0 0 I I I 250 500 750 1000 -2 CALCULATED, Wm Fig. 1. Scatter diagram of calculatedversusmeasuredvalues at data collectionplatform. Measurementswith pyranometerwith clear dome (280-2800 nm) and with RG-8 filter (700-2800 nm) are included. Analysis of correlation is in text. __ determination of snow surface albedo, or integration with other types of surface heat exchange calculations.Running the model for every point in such a grid is computationally time consuming,however.For example,a 7.5-min quadrangle at latitude 36ø, digitized at 100-m intervals, has over 15,000 points, and while the model is efficient,this large number of iterations requirestoo much time for most applications. Fortunately there are some considerable computational economies.The geometriceffectsattributable to terrain are all independentof wavelength,hence equationssimilar to those 714 DOZIER: SOLAR RADIATION MODEL for monochromaticvaluescan be applied to integratedvalues as well, namely: Q'• = Q• cosz' (1 - %) (25) q'• ---(1 - Vt U Vq)(qh•Cs[Z']/Cs[Z] + qb•) (26) qr•---- Vt (1- Vq)ad,r[O](qh• + qb•) (27) N = [O, - Fq)/]v] Y, cosz" ß[l - cos:(s'/M (:8) Qg• = Q'• + q'• + qr• + qR• (29) In order to apply theseequationsat any time of day, the model must preparelook-up tablesfor (a) integratedbeam, :.. -}..•:, .- .:.:•. .... study area, and (b) integrated albedosfor diffuseand total reflectanceat a range of incidenceanglesfrom 0 to •r/2. Then (25-29) can be applied by interpolation.Once the look-up tables exist, calculation speed is just as rapid as for a non- creasesin proportion to the total time. . .. -'" zontal surfaceat a rangeof elevationsspanningthosein the spectralmodel,and asthe numberof pointsin the terrain grid increases,the time requiredto preparethe look-up tablesde- .. .. ........ diffuse, and backscattered radiation for an unobscured hori- ...;.•. ?•'"i ........ '.............. •:::•: " ' ..-'........... - ''"".":•:* *•-%::,. ......... ?,. ..;:.::..:.•.-: ........... '.............. ß ß Figures 2 and 3 illustrate sample model output over a 7.5min quadrangle in the Bullfrog Lake area in the southern Sierra Nevada, digitized at 100-m spacing. ' :... :.... - ......... ..x:--.:.<;:. ...:.;..;?...;•...... -.,..s:, "':•'::f ' -.'.--:"" •".'""i":': ' "' ' ,.•:• ;*?;•:: •!-:::-,.:.:{•:•i•: '•':i':""": ......... •:' .! ......... s..,•...... .... ::::..;;.. ............ ..... Example of Model Output ..... ;::'"' :,;.'.. ""•" '.......... •*'%.:i';;" ..•,,-:" ............ ,.•:,;;;.:',...,.-.:,,,..•.,;....:<.::, ';;:•' *;::":;;'; ........ ?..: "%....... ,:.;:.:, .;.';•,'ß '....................... "i:,: *%;:': :....................................... Fig. 3. Density map of incomingsolarradiation in Bullfrog Lake area at 10:19A.M., February 25, for spectralrange 250 to 2500 nm. Atmosphericparametersare: (O3), 3.15 mm; w, 18 mm; fl, 1000;a, 1.3; a/Re, 0.5. Regional snow age parameteris 0.8. The values portrayed rangedfrom65 W m-2 (darkest)to 1079W m-2 (lightest).Scaleisthe sameas in Figure 2. Errors Attributable to Calculation Methods The error introduced by these table look-up methodsis acceptably small. The model was run in both the slow, detailed mode and the fast, table look-up mode for the area in Figure 1 and for the entire model spectral limits (250-5000 nm) for 10:19 A.M., February 25, 1977. For an area this size (15,820 points)the computationtime requiredfor the model to run in the slow mode was almost 100 times that of the fast mode. The RMS error between the rapid and slow methodswas 1.7 W m-:, or only 0.3% of the mean value. The maximum error was 3 W m -:. DISCUSSION Under clear-sky conditions solar radiation measurements can be usedto determine atmosphericattenuation parameters. Once these are measured, the solar radiation model can be combined with the topographic calculations to estimate the spatial distributionof incoming,net, or reflectedsolar radiation valuesin any wavelengthband over a designatedterrain area (suchas a drainagebasin).Within modellimits the radiation may be integratedover any time and wavelengthinterval. Such estimates could be used in combination with snowmelt runoff models,or for comparisonwith satellitespaceradiance o I 1 I 2I 3i 4• • measurements, to determine snow albedo. Fig. 2. Topographicmap of 7.5-min quadranglein Bullfrog Lake area. Contour interval is 100 m. Latitude-longitude boundaries are The refinementsof this model over other slope radiation models[e.g., Garnier and Ohmura, 1968; Williamset al., 1972] are: (a) it is spectral;(b) by using a physically based calcu- 36o42'30" to 36o50'00" N, 118o20'00" to 118o27'30" W. lation km of radiation attenuation it accounts for effects caused DOZIER: SOLAR RADIATION MODEL 715 by varying elevationwithin the area of interest;(c) it storesa file of horizon vectorsthat do not need to be recomputed for another time or date; (d) it includescalculationsfor reflection from adjacentterrain. However, theseattributesall make this model more complicated and, therefore, more expensive to run. The questionis, then, are they justified? Clearly the spectralattribute is necessaryfor albedo estima- Lack of an altitude dependencein radiation calculations can also lead to significant errors. In the southern Sierra Nevada in winter, snow-coveredarea ranges in elevation from 1200 to 4400 m. By using exponentialdecay functionsfor atmospheric water vapor and aerosol content, and by calculating air pressurefor Rayleigh scattering,the model is able to use measurementsat a single elevation within this altitude tion from satellites that measure radiation rangeto estimatetransmissivities throughoutit. Under typical conditions(water vapor 10 mm, fl 2000, a 1.3, a/Re 0.5) the incoming solar radiation at 1200m is 25% lessthat at 4400 m. Figure 5, a graph of the diurnal variation in incomingradi- in narrow wave- length bands.For evaluationof the net solar radiation componentin the snowsurfaceenergybudget,the spectralmodel provides a small but definite improvement over a lumped model. Figure 4 shows sample model output for a site in ation for three dates at a site between Vidette Meadow mountainous terrain, the frozen snow-covered surface of Bull- and Junction Meadow on Bubbs Creek, demonstratesthe imporfrog Lake, in the Kings River drainageof the Sierra Nevada, tance of calculating horizons in mountainous terrain. At all at 10:19 A.M. on February 25. times of year the horizonsreducethe effectiveday length by Depending on whether attenuationis by aerosolsor water interceptingdirect beam radiation at low sun angles [Charvapor, the reduction in incomingradiation occursin different bonneauet al., 1979], and in winter this interceptionmay not parts of the spectrum,and this differenceis convalved with be restrictedto the early morning and late afternoon. Our rethe spectral snow reflectancecurve. Table 2 also illustrates cently developedhorizon algorithm [Dozier et al., 1979]is, for this point. For a high-altitudehorizontal surfacewhich is un- a grid the size of Figures 1 and 2, about 5000 times faster than obstructedby surrounding terrain during the winter, the in- the methodsof Williarnset al. [1972], Lacornrne-Lahourguette tegratedsnowalbedo is a functionof whether or not attenua- [1973], or Dozier and Outcalt [1979]. The importance of terrain-reflectedradiation varies spation of solar radiation is by aerosolsor water vapor. From the base case of a relatively clean and dry atmosphere, adding tially, but in someareasit can be significant.A singleanalysis enoughwater vapor to reducethe incoming solar radiation by of the topographicarea coveredin Figures 2 and 3 indicated 7% causesa reduction in net solar radiation by 14%, whereas that for a condition in which total incoming.solarradiation the same 7% reduction in solar radiation causedby aerosols varied from 65 to 1079 W m-2, with a mean of 549, terrainresults in only a 5% reduction in net solar radiation. To reflectedradiation varied from 0 to 100 W m-2, with a mean achievethe same 14%reduction in net solar radiation by aero- of 54. As shownin Figure 6, the portion of the total incoming sol attenuation, it is necessaryto reduceincoming solar radia- solarradiation that was terrain reflectedaveraged17%,with a maximum of 66%. tion by 21%. solar constant I.T5 E 1.25 N 'E - 1.00 • .75 .25 net oo 5oo iaaa 15oo 2ooo 25oo 5ooo wavelength, am Fig. 4. Samplemodeloutput,10:19A.M., February25, at BullfrogLake,KingsRiver Drainage,36.8øN,118.4øW,ele- vation3240m. Terrainviewfactoris0.19.Atmospheric parameters aresameasin Figure3. Regionalsnowageparameter is 0 8 (to allow for darkeningcausedby exposedrock or vegetation);point snowageparameteris 0.2. The dottedlines showthe effectof increasing fl to 3000;the dashedfinesshowthe effectof increasingw to 40 mm. 716 DOZIER: SOLAR RADIATION MODEL TABLE 2. Sensitivityof Incomingand Net SolarRadiation to Attenuationby Water Vapor and Aerosols Parameters Parameter Values Problem Parameters Location 36.5øN, 118.5øW Altitude 2500 m Time 10A.M., February25 Snow age parameters,• 0.22 Constant Attenuation ozone, mm 3.15 1.3 a/Re Attenuation . .•......•:•:-. •; :• •.--.,,.. ........•:-....;-½;...•, .. ',-...•., •;; ....... '½ ;•,,.....:•:•;;:.:::.":: ...... .....;;% .;.-.:.:•:: ..,.::.:• •. L ,•:.: :::::-•- :' .... '"• -.-:-":' ...::•:,,: ... :: '.... ? :'""" ?"'•:': .;• :.: .: ..;:;•";':? •.. *;; ..... ;;:: ::%: ..:...:. "-........ ;•:.,.,::,'":"•......•;;;• .... ::.:•.•.... %•>.'-".:,-.',•::•;%.: ..... •..., .... .?•..?:,.,,.-..?-'-"•:• •::);:.... '*•"•;• :%.". 0.5 Variable .•..::. ,,--..---:.,:...--.-----------,,, ............. .?% :.......• i•:"'":,•-•:.:..,:.:.;; •"....?';;....: .;•:: ..,......•,:...;• Parameters a .... .• Parameters . • water vapor . ..•;:•:::' .... . .... ... . ........ . ........ . .. .... ........ .;:...........•. :... •.... ..... ........... ½ ..... .. ;;-..•...-:.•, .:..:•:..; ...... .,'•&..,.:..:...:.::•'•:...... . •;:: •:. .. Integrated Solar Radiation Values,W m-2 Ultraviolet and Visible Incoming Net Effective 35 132 108 Incoming 127 3. .... . ...... . - 6.7 -14.4 26 117 - 6.7 - 4.8 -20.7 143 albedo -14.4 ;:•.'-:;:..,.•;:. •L' :4;' "%::?:;• ? -.... , - ........•:.. .•:; :..... ""-"? y.:... .. ":......;:"'•':'•:'/'"•-':" •'"- .. ";:' ' ' ..... . -.... • .;:-.:'...:;;...... .. ':::;? •.:.; -:: .., ½.•.::......•.½..,• .... , . -.... ......•:.. ..... -½ -- ,.:.?:::..... ¾:....• .....•... •:... .:•.. Fig. 6. Map of percentageof total simulatediffadiance,which is reflectedfrom adjacenttefrain, for •e area in Figures2 and 3. Atmoseefic and surfaceparametersare sameas in Figure 3. T•e is 10:19 A.M.; date is Feb•aw 25. The mean percentageis 17%;the range is 0% (darkest)to 66% (lietest). Terrain-reflectedradiation clearly not be ignored in mountainousareas. 0.71 *Base case. CONCLUSION Under clear-sky conditions the solar radiation model provides a detailed . ,:, ..,.:;?•;•:; ",:•.,::• "•.:•::.:; ....•..... :::•::: .• .......... .•...:.:, ...... ":.-. ---"-%.:... ½• •.,-.. ':;-•',•".:'....' ...... ..... ½½ .... ... .... :- .,.,.---:. ,• ...... ......:....?'.:.: ;:•;;;.•:.:... ....--. ,:. ----•,.• • Water Vapor, lOrnre;fl, 4600 223 264 487 Net - ..:: .....--: ............... ?-'-.::•::;: . -; 159 0.72 albedo Incoming ... - -..•:...... Water Vapor, 10 ram; fl, 2010 280 293 573 32 . ....... •;::.,.. '"-"' -.--'.,,'"•;::":•:•-::•-";:;':.•'.": ...... ........ :?..?:•' ,...... . 143 0.75 albedo .,.•.:•..:...•;::•:..•:•.;:: ..... ¾, .......;.;....;...• .... ...... •:., ....... ....:.:.:.............;.. ...... .½;: ...:::...::.?•,:..•:,:.:.:•;.'7.½;.,, ........ ½.: .'•: ;:::• ,::...,...•:,•?-,.... ..... ....... ......... :½.•::'½&: •½..:.:.:;.......... --•. ..½. .:.. Water Vapor, 40 ram; fl, 1000 308 265 573 35 2. Effective ... Case, % 167 0.73 albedo 1. Net Effective Total ..,.•: ....... ..•:.•::.... .:?:::.:-. .... Water Vapor, 10 ram; fl, 1000' 308 306 614 Incoming Net Effective Deviation from Base Near Infrared calculation of incident irradiance over a ter- albedosare known. It is thus a step toward calculation of the surface energy budget over rugged snow-coveredterrain, either as direct input into energy budget equationsor in comparison with area-wide satellite measurementsof upwelling radiation to determine rain grid; it may also provide net radiant flux density if point surface albedo. NOTATION [ ] function of, e.g., w[z•] is water vapor above altiiooo tude z. _ I May I March ation. oa 750 ••/•• IJanuary i a'o[3•] monochromatic diffusereflectance of beamradia- E E tion. 5OO aair integrated effectivediffuse reflectance. a,t integrated effectivetotal reflectance. a/Re absorptanceto reflectance ratio for atmospheric ._o ½r a•[3•] monochromatic diffusereflectance. ao[•] monochromatic specularreflectance of beamradi- 250 aerosols. B[3•] portion of monochromatic reflected radiation 6 8 I0 Time, 12 14 16 18 20 hours Fig. 5. Simulateddiurnal variation of incomingsolarradiation on Bubbs Creek, Kings River Drainage, between Junctio• Meadow and Vidette Meadow. Site coordinatesare latitude 36o45'54" N, longitude 118ø26'12" W, elevation 2986 m. Terrain view factor is 0.36. The three datesshown are January 1, March 1, and May 1, and the daily totalsare7.07, 15.92,and 19.46MJ m-2, respectively. Atmospheric attenuation parametersare sameas in Figure 3. The suddenchangesare due to the sun'semergencefrom or disappearancebehind a local horizon. which is backscattered. Cs factor to account for increased diffuse radiation in vicinity of sun. Cz portion of radiation scatteredtoward surface. E aspect(exposure)of slopefrom south. E' aspectof slope with respectto opposingslope. H horizon angle (from horizontal). km[3t] absorptioncoefficientfor miscellaneousgases. ko[3•] absorptioncoefficientfor ozone. kw[3•] absorptioncoefficientfor water vapor. DOZIER:SOLAR RADIATION MODEL ma mo mw N (03) P[z] relativepath lengthfor air mass. relativepath lengthfor ozone. relativepath lengthfor water vapor. numberof segments for surrounding terrain. atmospheric ozonecontent(mm). air pressure at altitudez (Pa). Forest Service and National Park Service,and the Los Angeles De- partment ofWaterandPowerhavebeengenerous in cooperation with our measurement program. REFERENCES Ahlberg,J., E. Nilson,andJ. Walsh,TheTheoryof Splines andtheir Applications, pp. 42-52,Academic, New York, 1967. Akima,H., A newmethodof interpolation and smoothcurvefitting basedon local procedures, J. Ass. Comput.Mach., 17, 589-602, q'•[•] monochromatic* diffuseradiation. qb•[•,] monochromatic backscattered radiation. 1970. qh•[•'] monochromatic diffuseradiation on unobscuredAnderson,M. horizontal surface. qr•[•,]monochromatic diffuse radiation reflected toslope Sphere, Tellus, 13,214-223, 1961. from surroundingterrain. Q•[•,] monochromatic beamradiation onplanenormal to sun. Q'•[X] monochromatic beamradiation onslope. Qs•[•,] monochromatic globalradiation onslope. Qn[•,] monochromatic netradiationon slope. qo[•,] monochromatic radiation scattered frombeam. Qo[•,] monochromatic solarconstant. q•'[•,] monochromatic diffusely reflected radiation. Q•'[•,] monochromatic reflected radiation. Qg•,Q, integrated globalornetradiation (W m-2). r earth-sun radius vector. R angleof reflection. S slopefrom horizontal. S' slopewith respect to opposing slope. t time. Vq portionof skyobscured by forest. Va portionof beamobscured by forest. V, portionof skyobscured by terrain. w precipitablewater vapor,mm. x vector of unknowns. z elevation, m. Z solar zenith angle. Z' solaranglemeasured fromnormalto slope. Z" solaranglefromnormalon opposing slope. a exponent in Jmgstrom turbidity equation. fi coefficient in J•ngstrom turbidity equation. X wavelength(nm). OA[•] aerosolattenuationcoefficient. oR[•,] Rayleighscattering coefficient. •A[•'] aerosoltransmission function. ßm[•,] transmission functionfor miscellaneous gases. ßo[•,] ozonetransmission function. •w[•,] watervaportransmission function. •R[•,] Rayleightransmission function. •[•,] transmission functionfor scattered radiation. •[•,] watervaportransmission function. •t, exponentialagingparameter. Units for all monochromatic radiationvaluesare W m-2 nm -l. C., Studiesof woodlandlight climate, 1, The photo- graphiccomputation of lightconditions, J. Ecol.,50,27-41,1964. Jmgstrom, A., Techniques fordetermining theturbidityof theatmo- qR•[•,]monochromatic beamradiation diffusely reflected to slopefromsurrounding terrain. 717 gstrom,A., Parameters of atmospheric turbidity,Tellus,16,64-75, 1964. Box, M. J., A comparison of severalcurrentoptimizationproblems, and the useof transformationsin constrainedproblems,Comput.J., 9, 67-77, 1966. Brent,R. P., Algorithms for MinimizationwithoutDerivatives, pp. 61167,Prentice-Hall,EnglewoodCliffs,N.J. 1973. Brown,K. M., andJ. E. Dennis,Jr., Derivative-free analogues of the Levenberg-Marquardt and Gaussalgorithms for nonlinearleast squares approximation, Numer.Math.,18,289-297,1972. Charbonneau,R., J. -P. Lardeau,and C. Obled, Comparisonof deterministicmodelsfor a high mountainous watershedwith domi- nantsnowyields:Analysisof sensitivity to modelstructure and to inputdataandtheirspatialextrapolation, J. Hydrol,in press,1980. Clark,F. G., A hemispherical photocanol•ymeter, J. Forest.,59, 103105, 1961. Dennis, J. E., Jr., Somecomputationaltechniquesfor the nonlinear leastsquaresproblem,in NumericalSolutionof Systems of Non- linearAlgebraicEquations, editedby G. D. Byrneand C. A. Hall, pp. 157-183,Academic,New York, 1973. Dirmhirn, I., and F. D. Eaton, Somecharacteristics of the albedoof snow,J. Appl.Meteorol.,14, 375-379, 1975. Dozier,J., and S. I. Outcalt,An approachtowardenergybalancesimulationoverruggedterrain,Geogr.Anal., 11, 65-85, 1979. Dozier, J., J. Bruno,and P. Downey,A fastersolutionto the horizon problem,Rep.TR-CSL-7907,Comput.Syst.Lab.,Univ. of Calif., Santa Barbara, 1979. Dunkle, R. V., and J. T. Bevans,An approximateanalysisof the solar reflectance and transmittance of a snow cover, J. Meteorol., 13, 212-216, 1956. Evans,G. C., and D. E. Coombe,Hemisphericaland woodlandcan- opyphotography andthelightclimate,J. Ecol.,47, 103-111,1959. Forsythe,G. E., M. A. Malcolm,andC. B. Moler,Computer Methods for Mathematical Computations, pp. 84-109,Prentice-Hall, Englewood Cliffs, N.J., 1977. Frew, J. E., Remote sensingof snow surfacealbedo, M.A. thesis, 82 pp., Univ. of Calif., SantaBarbara, 1980. Garnier, B. J., and A. Ohmura, A method of calculatingthe direct shortwaveradiation incomeof slopes,J. Appl. Meteorol., 7, 796800, 1968. Gates,D. M., Near-infraredatmospheric transmission to solarradiation, J. Opt. Soc.Amer.,50, 1299-1304,1960. Gates, D. M., and W. J. Harrop, Infrared transmissionof the atmosphereto solarradiation,Appl.Opt.,2, 887-898, 1963. Giorgis,R. E., Jr.,A simplesolarradiationmodelfor computing directanddiffusespectralfluxes,M.S. thesis,100pp.,Univ. of Calif., Davis, 1977. Hay, J. E., A revisedmethodfor determining the directand diffuse components of thetotalshort-wave radiation.Atmosphere, 14,278287, 1976. Herman, B. M., and S. R. Browning,A numericalsolutionto the equationof radiativetransfer,J. Atmos.Sci.,22, 559-566,1965. Acknowledgments. Theworkwassupported by theNationalAeronauticsandSpaceAdministration: grantsNSG-5155and NSG-5262; Inn, E. C. Y., and Y. Tanaka,Absorptioncoefficientof ozonein the ultraviolet and visible regions,J. Opt. Soc. Amer., 43, 870-873, the National Oceanicand AtmosphericAdministration:grant 04-81953. MO; and the Universityof CaliforniaWater Resources Center:grant Kasten, F., A new table and approximationformulafor the relative UCAL-W-546. Computations werecarriedout at the ComputerSysopticalair mass, Arch.Meteorol.Geophys. Bioklimatol. Ser.B, 14, tems Laboratory,University of California, Santa Barbara, on a 206-223, 1966. PDP11/45with Unix operatingsystem.I am indebtedto William St. of thegroundalbedoandtheindexof Lawrence,Robert Fraser,ThomasDunne, John Hay, LeonardMy- King,M.D., Determination absorption of atmospheric particulates byremotesensing, 2, Applirup,JamesFrew,WebbMiller,CharlesObled,DavidSimonett, and cation,J. Atmos.Sci., 36, 1072-1083, 1979. the editorsand reviewersof thisjournal for helpfuldiscussion or cor- respondence. The MammothMountainSki Corporation, the U.S. King,M.D., andB. M. Herman,Determination of thegroundalbedo 718 DOZIER: SOLAR RADIATION MODEL and the index of absorptionof atmosphericparticulatesby remote sensing,1, Theory, J. Atmos. Sci., 36, 163-173, 1979. Kondratyev, K. Ya., Radiationin the Atmosphere,pp. 161-216, Academic, New York, 1969. Kreuger, A. J., and R. A. Minzer, A mid-latitude ozone model for the U.S. standard atmosphere, 1975, Rep. X-912-74-291, NASAGSFC, 1974. Rayleigh scatteringcoefficientfor the spectralregion between 0.2 and 20.0 microns and their application to atmosphericoptics, J. Opt. Soc.Amer., 47, 176-182, 1957. Petzold, D. E., An estimation technique for snow surface albedo, McGill Univ. Climatol. Bull., 21, 1-11, 1977. Price, A. G., and T. Dunne, Energy balancecomputationsof snowmelt in a subarctic area, Water Resour. Res., 12, 686-694, 1976. Lacis,A. A., and J. E. Hansen,A parameterizationfor the absorption Reifsnyder,W. E., and H. W. Lull, Radiant energyin relation to forests,Tech.Bull. 1344, 111 pp., U.S. Dep. Agr., Washington,D.C., of solar radiation in the earth'satmosphere,J. Atmos.Sci., 31, 118133, 1974. Lacomme-Lahourguette,A., Etude de l'ensoleillementnumerique, Bull. d'inform.23, Inst. Geogr. Nat., Paris, 1973. Leckner, B., The spectraldistribution of solar radiation at the earth's surface--Elements of a model, Solar Energy, 20, 143-150, 1978. Leighton,P. A., Photochemistry of Air Pollution,pp. 6-103, Academic, New York, 1961. Levenberg,K., A method for the solutionof certain nonlinear problems in least squares,Quart. Appl. Math., 2, 164-168, 1944. Makarova, Ye. A., and A. V. Kharitinov, Distribution of energy in the solarspectrumand the solarconstant,NASA TT F-803, 245 pp., 1972. Marquardt, D. W., An algorithm for least squaresestimationof nonlinear parameters,SIAM J. Appl Math., 11, 431-441, 1963. Middleton, W. E. K., and A. G. Mungall, The luminous directional reflectanceof snow,J. Opt. Soc.Amer., 42, 571-579, 1952. O'Brien, H. W., and R. H. Munis, Red and near-infraredspectralrefiectanceof snow, in Workshopon OperationalApplicationsof Satellite SnowcoverObservations,edited by A. Rango, pp. 319-334, 1965. Robinson,N., Ed., Solar Radiation,pp. 29-160, Elsevier,New York, 1966. R6ssler, F., Trfibungsmessungen im Gebirge, Arch. Meteorol. Geophys.Bioklimatol., Ser. B, 27, 69-74, 1979. Sellers,W. D., PhysicalClimatology,pp. 11-39, Universityof Chicago Press,Chicago,II1., 1965. Temps, R. C., and K. C. Coulson, Solar radiation incident upon slopesof differentorientations,Solar Energy,19, 179-184, 1977. Van Heuklon, T. K., Estimatingatmosphericozonefor solar radiation models,Solar Energy,22, 63-68, 1979. Williams, L. D., R. G. Barry, and J. T. Andrews,Applicationof computedglobal radiationto areasof high relief, J. Appl. Meteorol.,11, 526-533, 1972. Willson, R. C., Accurate solar 'constant' determinations by cavity pyrheliometers,J. Geophys.Res.,83, 4003-4007, 1978. Yamamoto, G., Average vertical distributionof water vapour in the atmosphere,Sci. Rep. TohokuImp. Univ., Ser. 5, 1, 76-79, 1949. NASA SP-391, 1975. Paltridge, G. W., and C. M. R. Platt, Radiative Processesin Meteorologyand Climatology,pp. 89-141, Elsevier,New York, 1976. Penndorf, R., Tables of the refractive index for standard air and the View publication stats (ReceivedAugust31, 1979; revisedFebruary 6, 1980; acceptedFebruary 13, 1980.)