Solar Radiation Model for Snow-Covered Terrain

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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
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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.
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andtheir
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Akima,H., A newmethodof interpolation
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J. Ass. Comput.Mach., 17, 589-602,
q'•[•] monochromatic*
diffuseradiation.
qb•[•,] monochromatic
backscattered
radiation.
1970.
qh•[•'] monochromatic
diffuseradiation
on unobscuredAnderson,M.
horizontal
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onplanenormal
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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.
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reflected
radiation.
Qg•,Q, integrated
globalornetradiation
(W m-2).
r
earth-sun radius vector.
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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.
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radiationvaluesare W m-2
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A., Techniques
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theturbidityof theatmo-
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to slopefromsurrounding
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717
gstrom,A., Parameters
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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
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F., A new table and approximationformulafor the relative
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(ReceivedAugust31, 1979;
revisedFebruary 6, 1980;
acceptedFebruary 13, 1980.)