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A simple solar radiation model for computing direct and diffuse spectral fluxes
SolarEncrgyVol.27,No. 4, pp. 323-329,1981 Printedin GreatBritain.
0038/092X/81/I~)323-07502.0(H0 PergamonPressLtd.
A SIMPLE SOLAR RADIATION MODEL FOR COMPUTING DIRECT AND DIFFUSE SPECTRAL FLUXESt J. L. HATFIELDR. B. GIORGIS,JR. and R. G. FLOCCHINI:~ Department of Land, Air and Water Resources, University of California, Davis, CA 95616, U.S.A. (Received 2 December 1980; revision accepted 1 April 1981) Abstract--A solar spectral model that describes the solar radiation flux on a clear day at any given location was developed and tested. The model computes spectral fluxes of global, global photon, direct and diffuse solar radiation incident at the surface. Input parameters describe location and atmospheric characteristics. Location is described by latitude, altitude, slope orientation and surface albedo. Atmospheric characteristics described are turbidity, precipitable water vapor and total ozone content. The model was constructed using a one-layer homogeneous atmosphere with refinements, which are: (I) Use of climatological data to predict the total ozone content, if it is not known. (2) A more advanced treatment of infrared solar radiation (0.8-4.5 ~m) absorption. (3) A more complex scheme for predicting diffuse radiation. (4) The capability of handling a spectral albedo. (5) Inclusion of albedo dependence on zenith angle. Input parameters are minimized and several simplifying features are incorporated for ease of handling variables not routinely measured. Turbidity and total ozone content are treated as climatological estimates if specific location measurements are not available. Procipitable water vapor can be predicted using surface vapor pressure, since the sounding network is not dense. These features allow researchers outside the field of solar radiation to use the model. Because complete measurements with needed location and atmospheric characteristics could not be found, the validity of the model was tested by comparing it with a more complex, multilayered atmosphere model by Dave et al. [3]. Calculated fluxes of total direct, diffuse, and global radiation from the model presented were 11.5, 20.1 and 13.2 per cent lower, respectively, when they were adjusted for differences in extra-terrestrial solar radiation fluxes. Direct spectral fluxes closely agreed in spectral composition, with slight exception in the 0.8-0.95 #m region. Diffuse spectral flluxes were slightly higher in the UV region lower in the rest of the spectrum than were those in the multilayered atmosphere model. A sensitivity analysis of the model was also conducted: the most influential inputs were found to be latitude, slope orientation and turbidity, and the least influential was total ozone content. The hourly integrated values for the model compared very well with measured values for clear days at Davis. Discrepancies between predicted and measured values were due to lack of local turbidity coefficients. 1. INTRODUCTION
Dav~ [5]. These models are difficult to use by researchers outside the physical sciences because of mathematical and computational complexity. Robinson [6] presented a model that is adequate for most general purposes, but it does not have a complete treatment of radiation absorption by atmospheric gases. Szeicz[7] presented a solar radiation model that is specialized for use in plant growth. Goldberg and Klein [8] present a spectral model that works reasonably well for broad spectral regions. The goal of this research was to develop a relatively simple, versatile solar spectral model that is sufficiently accurate over the solar spectrum from 0.29 to 4.50 ~m. Inputs are minimized for ease of use and define needed location and atmospheric characteristics. Output parameters of the model include direct, diffuse, global and global photon spectral fluxes,
Many solar radiation models exist in the scientific literature, but few have been constructed to simulate spectral fluxes of direct and diffuse radiation. It appears that the solar spectral models available could be used more widely in such fields as solar energy, plant ecology, agriculture and environmental photochemistry. Part of the problem in acceptance of solar spectral models is due to the complexities of solar radiation and difficulties in their application. There are several spectral models available in the literature. However, a simple model that is versatile and consistently accurate over the entire solar spectrum does not exist. Leighton [1] and Green et al. [2] presented models that deal mainly with the UV portion of the solar spectrum. Some models use a multilayered atmosphere, such as that of Day6 et al. [3], Day6 and Braslau [4] and
2. THEORETICALBASIS
tContribution from the University of California Agricultural Experiment Station. Research supported by NSF Grant No. DEB-22390 and DOE Grant No. DE-FEO3--79ET20187. :~Biometeorologist; Former graduate research assistant, presently Research Meteorologist, Air Resources Control Board, State of California, Sacramento;and SolarEnergyLecturer,Dept. of Land, Air and WaterResources.Universityof California,Davis, CA 95616, U.S.A. 323
The model is set up to calculate the monochromatic fluxes of direct and diffuse radiation at the earth's surface. The diffuse flux is then related to the radiation scattered from the direct beam by a scheme that considers the many factors influencing the scattering process. Extraterrestrial solar radiation measurements for mean
324
J. L. HAT~ELDet
sun--earth distance have been made[9]. These measurements have been summarized in a table[10]. Values from that table were added to the model using an Aitken interpolation scheme[l 1], which allowed intensities to be calculated at wavelengths not given. The basic information for the framework of the model can be found in Sellers [12]. We will present the information here that is unique to this model and extensions to basic theory will provide the reader with a basis for the model. The decadic optical depth of the atmosphere is defined in terms of four components
at..
Unlike absorption, scattering must be considered to be two subprocesses: Rayleigh and Mie scattering. Robinson [6] provided a method for calculating Rayleigh scattering. The decadic monochromatic optical depth for Rayleigh scattering was calculated as a function of wavelength. The optical depth due to aerosol extinction, which is almost totally due to scattering (less than 5 per cent is from absorption by aerosols), is determined by methods introduced by Angstr6m. The equation he proposed can be modified as: r,(X) = B~-* 1
r(X) = ~-o(X)+ ~-8(~0+ r,(X) + rd(X)
(1)
in which to(A) is due to ultraviolet and visible absorption by ozone, rR(A) is due to gaseous IR absorption, r,(X) is due to Rayleigh scattering and ~',~(A) is due to Mie scattering. UV absorption by ozone mainly occurs in the HartleyHuggins band. Leighton [1] pointed out that the decadic absorption coefficient, Ko(,~) in mm -t, for this band (0.29--0.35 #m can be described approximately by the following relation: log [Ko(A)] = 16.58 - 0.00564A.
(2)
For absorption by ozone in the visible range, Inn and Tanaka [13] provide Chappuis band absorptiou coefficients. Coefficient values lying between the given wavelengths may be found using the previously mentioned interpolation scheme. Other ozone bands above 0.8 ~m are considered in the scheme for IR absorption. To obtain the optical depth due to absorption by ozone, the absorption coefficients must be multiplied by the total ozone content. The total ozone content of the atmosphere varies with latitude and time of year. An adjustment for altitude was made by applying a height correction of 2.5x 10-Smm 03m -~, derived from data given by Krueger and Minzner [14]. The decadic optical depth due to UV and visible absorption was calculated using the total ozone content and the absorption coefficient. Many gases, such as water vapor, carbon dioxide, oxygen, ozone and methane, absorb solar radiation at wavelengths above 0.8 t~m. Of these gases, water vapor is the predominant absorber; carbon dioxide is the only other significant absorber. Monochromatic absorption coefficients for IR (>0.8t~m) are given by Gates [15], and Gates and Harrop [16]. These coefficients are multiplied by the square root of the total precipitable water vapor in a vertical column, w (ram), to obtain the decadic, monochromatic optical depth of gases absorbing in the infrared region. It was assumed that the total precipitable water vapor does not vary within any given horizontal layer of the atmosphere. If atmospheric soundings are not available for estimating precipitable water vapor, the may be calculated using an approximation given by Leighton [1]. Reber and Swope [17] discuss problems of using this approximation scheme.
(3)
in which B is the turbidity coefficient and al is the wavelength exponent. A generally accepted value for at is 1.5. Robinson [6] gives average values of B as a function of latitude and height based on long-term measurements at several locations. Otherwise, turbidity coefficients can be measured by a Voltz sun photometer. If the optical depths due to absorption and scattering are known, the transmissivity of the atmosphere (To(,~)) can be found using the formula:
The spectral flux of direct radiation incident on a given surface at the bottom of the atmosphere can then be given using the formula: F(A)AA = (~/r) 2 cos (0,)
~
X2
Fo(A) To(A)d,~
(5)
I
in which At and a2 are increment boundaries. Diffuse radiation results from scattering and reflection of solar radiation. There are two components of diffuse radiation: the radiation scattered downward from out of the direct beam to the surface, and radiation reflected off the surface, then backscattered down to the surface. The amount of radiation scattered out of the direct beam is equal to that incident at the top of the atmosphere times the fraction scattered out of the direct beam. The fraction that undergoes Rayleigh scattering is easily calculated. The amount of Mie scattered radiation, the Mie scattered radiation must be assumed to be about equal to the amount of direct beam radiation undergoing aerosol extinction. The amount of scattered direct radiation, F~(A), is then determined by the formula: Fd(A) = Fo(A)(1 - I0-"~>'-'~<~"').
(6)
Some of this scattered radiation is absorbed. Assuming that, on the average, radiation is scattered from the 500 mb pressure level of the atmosphere, it travels isotropically through 3.5 per cent of the total atmospheric ozone. This percentage was calculated from data presented by Krueger and Minzer[14] for mid-latitude vertical ozone distribution. Absorption by gases other than ozone are ignored, since their vertical distributions
A simple solar radiation model for computing direct and diffuse spectral fluxes can be strongly variable and their absorption bands occur in low energy regions of the diffuse spectrum. Therefore, the transmis.sivity of the atmosphere to incoming scattered radiation is given as:
To(A) = 10-°'°7'~)
(7)
in which To(A) is the average monochromatic transrnissivity due to absorption of diffuse radiation scattered out of the direct beam. The exponent of eqn. (24) includes a factor of 2.0, so that the oblique path of isotropic radiation is considered. The fraction of radiation scattered down toward the surface, k~, is given [6] as: k~ = 0.5 coP'~(O).
(8)
Downward scattered direct radiation incident at the surface is then given by
D~(A) = ktk,.F~(A)cos(O)lO-°'°7~
transmitted to space. Since Mie scattering is mostly forward, it is not considered in the backscattering process. Half of the surface-reflected radiation that undergoes Rayleigh scattering is assumed to be scattered back toward the ground, since Rayleigh scattering is symmetric. The backscattered radiation is assumed to be isotropic and, on the average, scattered from the 500mbar pressure level of the atmosphere. The backscattered diffuse radiation, Db(A), is then given by Db(A) = 0.5Fb(A)(I - 10-2°"A))10 -°'°~'~)
in which Db(,~) is the diffuse radiation incident on a horizontal surface. If the surface is tilted at some angle i, part of the sky is blocked out and some radiation is reflected onto the receiving surface from the horizontal surface that is assumed to be at the base of slope. The fraction of the sky that is not blocked out, [', is given by using the formula: f' = cos2(i]2)
(|0)
where 0, is the angle between the sun and normal to the plane. Surface properties play an important role in the amount of hack-scattered radiation received at the surface. For instance, the higher the albedo, the greater the amount of radiation available for back-scattering. The albedo for direct radiation is a function of zenith angle. Paltridge and Platt [9] provide the following formula for the albedo to be used in calculating the amount of direct radiation reflected from the surface: a,(0) = a + (1 - a) exp (- k2(90° - 0))
(13)
(9)
in which Dd(A) is the monochromatic diffuse flux at the surface contributed by scattering from the direct beam and k., is a correction factor for the brightening of the sky in the vicinity of the sun[18]. Temps and Coulson[18] provide the following as an equation to quantify the sky brightening. = 1 ,de.COS2(0s) Sin3(0)
325
(14)
Temps and Colson [15] provide a correction factor for horizon brightening, f": f"=
l+sin3(il2)
(15)
Monochromatic diffuse radiation that has not been reflected directly from other surfaces, D,(A), is then determined from D,,(A)= [Ds(A) + Db(A)](f')(.f").
(16)
Reflected monochromatic radiation falling on the receiving surface, D,(A), is given [15] as: D,(A) = IF(A) + D,(A)]a(I -/311 + sin2(0/2)][cos (a)] (17] where a is the azimuth angle. The monochromatic diffuse flux is then given using
(11) D(A) = D,,(A)+ D,(A)
(18)
in which a is the surface albedo at low zenith angles, k2 is a constant, equal to about 0.1 degree -l, and a,(#) is the zenith-angle-dependent albedo for direct radiation. By assuming isotropi.c diffuse radiation and by integrating eqn (30) over a hemisphere, the surface albedo for diffuse radiation is found to be about equal to a. The total amount of radiation available for backscattering is then given by
Calculation of global radiation Once the direct and diffuse solar fluxes are known, the global radiation can be calculated easily. The monochromatic global radiation is simply the sum of the direct and diffusemonochromatic fluxes:
Fb(A) = FO:)a,(0) + De(A)a
G(A) = F(X) + D(A)
(12)
in which Fb(A) is the monochromatic solar radiation flux back into the atmosphere from the surface. In this instance, higher-order surface backscattering is ignored. As radiation reenters the atmosphere from the surface, it is assumed that it is isotroic and that it will be absorbed by ozone, scattered by Rayleigh particles, or
in which D(A) is the monochromatic diffuse radiation.
(19)
in which G(A) is the monochromatic flux of solar radiation at the surface. The monochromatic global photon flux is then found by dividing the monochromatic global solar radiation by the energy per photon: P(x) =
G(A)xlhc.
(20)
J. L. HATFIELDet aL
326
Table 1. Input and output parametersfor the spectralmodel and the necessaryunits Input Parameter
Unit
Wavelength i n t e r v a l
~
Wavelength l i m i t s
I.m
Output P a r u s e t e r Exterrestrial
Time
Unit
f l u x , Honocbroaatic
WI'2s'l
Monochromatic s u r f a c e fluxes
length and l i m i t s
hour
~'2s'l
direct, diffuse, total
Station description Latitude
degrees
Nonocbrolstic g l o b a l photon
flux
~-2s-I
m
Elevation Slope
degrees
Slope azimuth
degrees
Albedo
unitless
Surface vapor pressure
cm
Surface a i r p r e s s u r e
mb
Ozone ~/
I
T u r b i d i t y ,~/
I
T o t a l g l o b a l flux
~-2s-1
H o n o c h r o ~ ' t i c global flux
WB-2s'I
~/ Estimated c l i m a t o l o g i c a l l y i f no s ~ a t i o n l o c a t i o n information is a v a i l a b l e .
Finally, the total global flux is ob~ned by integrating over 0.05 p.m intervals the direct and diffuse monochromatic fluxes:
GT = f:
[ F ( A ) + D ( A ) ] d~
(21)
in which GT iS the total global flux.
case, the site was at sea level at a latitude of 40°N, and the tLme was solar noon of the spring equinox. Turbidity in the atmosphere was assumed to be 0.09 and total ozone content to be 3.318mm. Surface vapor pressure was assumed to be 12 mbars. A level surface was used that had an albedo of 0.25. Plots were generated for comparison using the output from the base case and variations from it.
3. METHOD
4. RESULTS AND D/SCUSS/ON
' The spectral model developed from the theory included the input and output parameters given in Table I. The ozone content and turbidity coefficient are estimated cl/matologically by model if they are not given. Two interpolation schemes were used to obtain values of parameters for which data were not provided in continuous form [1 l]. A listing of the model will be provided by the senior author upon request. The reliability of the proposed model was tested by comparing calculations against spectral calculations made by Dave et al. [3], who used multilayered atmosphere model. Their calculations were made for a sealevel site at mid-latitude during a clear day in the summer for which the zenith angle was 60°. Total precipitable water vapor was 2.925 cm, and total ozone content was 3.18 ram. Since their model atmosphere was of low turbidity, the value used for the turbidity coefficient in the model presented had to be reduced from the climatological value to 0.05. The albcdo was 0, a perfectly absorbing surface. The two models were compared using the spectral and total fluxes of extraterrestrial, direct, and diffuse solar radiation. A sensitivity analysis was conducted to assess the response of the model to variations in input parameters as compared with an arbitrary base case. In the base
Figure I ~ves results of the companson with the multilayered model of Dave et aL [3]. The two top curves are solar fluxes onto a horizontal surface at the top of the atmosphere. The top curve is predicted using the multilayered model. Discrepancies between the two spectra are due to the differeing sources of extraterrestrial flux data for the models. Values used in the multilayered model were obtained from Howard et ol. [20]; values used in the presented model are from more recent measurements [9]. Since the extraterrestrial values used ,s,~
16 --
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Fig. 1. Comparison of extraterrestrial (upper curves) direct (middle curves) and diffuse (lower curves) spectral fluxes as calculated by the multilayered Dave et al. [3] model and the model.
A simple solar radiation model for computingdirect and diffuse spectral fluxes by the multilayered model are larger (approx. 6.4 per cent), the predicted direct and diffuse values are also higher. Therefore, direct and diffuse values from the two models show more lack of agreement than actually exists. When the extraterrestrial flux differences are taken into account, the direct spectra appear to agree very well from 0.29 until 0.8 ~m, whereas the presented model neglects to properly simulate the region from 0.8 to 0.95/zm because of poor oxygen and water vapor absorption data used in the present model. The direct spectra agree very well in the waveband from 0.95 to 2.5 ~m. When corrections were made for extraterrestrial flux differences, the total direct flux of the presented model was found to be 11.5 per cent lower than that of the multilayered model. The diffuse spectra do not agree as well as the direct. The presented model appears to generate higher values at wavelengths greater than 0.5/zm. Also, since absorption by water vapor is not included in the diffuse parameterization of the presented model, its diffuse spectrum lacks absorption bands in the IR. The last important disagreement is that the total diffuse flux of the presented model is 20.1 per cent lower than that of the muitilayered atmosphere model. Combining this difference with that of the total direct gives a difference in the global values of 13.2 per cent; the presented model has the lower total global flux.
Sensitivity analysis The first sensitivity variation used was that of latitude. Three variations in latitude were considered: 0°, 40°N (base case) and 70°N. All other input parameters remained at base case values, except for total ozone content and turbidity, which were allowed to vary climatologicaUy with latitude. Figure 2 compares direct values. The highest direct spectral fluxes were received at the equator. Direct radiation at 40 and 70°N received about 73 and 24 per cent of that received at the equator, respectively. There also appears to be a red shift in the direct and diffuse spectra with latitude (Fig. 3). Sinclair and Lemon [21] pointed out this red shift with greater zenith angle and its significance to plant phytochrome systems. When the optical depths due to different attenuating mechanisms were inspected, Rayleigh and Mie scattering appeared to be the most responsible for the differential attenuation causing the red shift. Variations in total ozone content had only a small influence on the solar radiation spectrum. Direct spec-
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Fig. 3. Latitude effects on the diffuse spectral flux as simulated by the one-layermodel trum changes were found in the ultraviolet and visible range, whereas the only detectable changes in the diffuse
occurred in the UV. When the total ozone content was changed from 0.0 to 6.0 ram, the total direct and diffuse fluxes were decreased by 2.4 and 2.8 per cent, respectively. Gases absorbed the most in the infrared region. The surface vapor pressures used for calculating infrared absorption were 0, 6, 12 (base case) and 30 mbars. In turn, the precipitable water vapor estimates were 0, 1.04, 2.08 and 5.19cm. Since absorption is directly proportional to the square root of the precipitable water vapor, the condition of no IR absorption was simulated by setting the surface vapor pressure equal to zero. Without infrared absorption, calculated total direct flux was 24.4 per cent higher than the base case value. Increasing the surface vapor pressure from base case value to 30 mbars decreased absorption by only 5.6 per cent, indicating that absorption bands in the IR (which are mostly water vapor bands) tend to saturate very fast. Since the diffuse parameterization did not include IR absorption, the diffuse spectrum did not change as surface vapor pressure varied. Turbidity affected both direct and diffuse spectra very strongly. Increasing turbidity attenuated the direct beam but enhanced the diffuse. Increasing the turbidity coefficient of sky values from clear (B = 0.05) to turbid (B = 0.30) decreased the total direct by 37.4 per cent but
increased the total diffuse by 118 per cent. A red shift occurs in both the direct and diffuse spectra, as is shown by Figs. 4 and 5. But because of the compensating changes in the direct and diffuse spectra, the global radiation does not undergo such strong changes (13 per "¢--
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Fig. 4. Turbidity effects on the direct spectral flux as calculated by the one-layermodel.
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determine how slope azimuth affects the total amount of global radiation received throughout the day. All slopes used in this analysis were set at 30°, except in the case of the horizon~ surface. Five azimuths were used in the analysis: south, southwest, west, northwest and north. The south slope received almost twice as much radiation as the north slope. The west slope received almost the same amount of radiation as a horizontal surface. One of the problems with these calculations is that the slopes were assumed to have a horizontal surface at their bases. Therefore, the spectra given may be somewhat dissimilar to those received on the same slopes in mountainous terrain. In the structure of the model, it was possible to integrate over the solar spectrum to achieve a tom
irradiance. These results for clear skies at Davis are given in Figs. 7(a--d). The results for all days are quite good and show that this model predicts all components with an accuracy of better than 5 per cent. There were only two points at which the model did not perform satisfactorily. On day 80, the global amount was underpredicted because the model did not account for a large DAVIS 38N
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Fig. 6. Daily global spectral fluxes incident on the surfaces of various the surfaces of various slope orientations at the spring equinox as calculated by the one-layer model.
cent for the same change). These results indicate that the turbidity must be known to accurately calculate direct and diffuse fluxes. Sensitivity to albedo was tested by examining the behavior of surfaces with constant and spectral albedos. Change in albedo does not affect the direct spectrum, and in~uences only the amount of backscattered diffuse. Three constant albedos were used in the analysis: 0.0, 0.25 (base case) and 1.0. Decreasing the albedo from 0.25 to 0.0 caused a decrease of 10.0 per cent in the total diffuse, and increasing it to 1.0 caused an increase of 30.0 per cent. The effect of slope on the direct and diffuse spectrum was calculated with atl slopes south facing at 30, 45 and 90~, except for the base case, which was a horizontal surface. The sun's azimuth and zenith angles were 0 and 40', respectively, so a south-facing 40~ slope would receive maximum direct radiation. For this reason, the 30 and 450 slopes show the highest reception of direct radiation, and the vertical slope shows the lowest. Diffuse spectra show a red shift as slope increases, because of the blocking out of a portion of the sky and the interception of radiation reflected off of the nearby horizontal surface, which is at the base of the slope. Part of the sensitivity analysis was conducted to
I000
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(~,m)
n o
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Fig. 7. Comparison of simulated and measured direct, diffuse and global solar radiation on a horizontal surface for a clear Julian day 80 (a), 172 (b), 264 (¢) and 355 (d) at Davis, California.
A simple solar radiation model for computing direct and diffuse spectral fluxes amount of dust particles in the air. On day 172, June 20, there was an increase in diffuse at midday due to agricultural burning near the measurement site. Overall, this model produces acceptable results and should be useful for a number of locations around the world. It incorporates a large number of local parameters, yet remains easy to use and can be run on a small minicomputer quite efficiently. REFERENCES
1. P. A. Leighton, Photochemistry of Air Pollution, p. 300. Academic Press, New York, (1961). 2. A. E. S. Green, T. Sawada and E. P. Shettle, The middle ultraviolet reaching the ground. Photochem. and Photobiol. 19, 251 (1974). 3. J. V. Dave, P. Halpern and N. Braslau, Spectral distribution of the direct and diffuse solar energy received at sea-level of a model atmosphere. Rep. G320-3332,p. 20. IBM Palo Alto Scientific Center (1975). 4. J. V. Dav~, and N. Braslau, Effect of cloudiness on the transfer of solar energy through realistic model atmosphere. J. Appl. Meteor. 14, 388 (1975). 5. J. V. Dave,, Validity of the isotropic distribution approximation in solar energy estimation. Solar Energy 19, 331 (1977). 6. N. Robinson, Solar Radiation, p. 347. Elsevier, Amsterdam (1966). 7. G. Szeicz, Solar radiation for plant growth. J. Appl. Ecology 11,617 (1974). 8. B. Goldberg and W. H. Klein, A model for determining the spectral quality of daylight on a horizontal surface at any geographical location. Solar Energy 24, 351 (1980).
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9. M. P. Thekaekara and A. J. Drummond, Standard values of the solar constant and its spectral components. Nature: Phy. Sci. 229, (1971). 10. K. L. Coulson, Solar and Terrestrial Radiation: Methods and Measurements, p. 322. Academic Press, New York (1975). I 1. B. Caruahan, H. A. Luther and J. 0. Wilkes,Applied Numerical Methods, p. 604. Wiley, New York (1969). 12. W. D. Sellers, Physical Climatology, p. 272. University of Chicago Press, Chicago (1965). 13. E. C. Y. Inn and Y. Tanaka, Absorption coefficientof ozone in the ultraviolet and visible regions. J. Opt. Soc. Am. 43, 870 (1953). 14. A. J. Krueger,and Raymond A. Minzner, A mid-latitade ozone model for the U.S. standard atmpsphere, 1975. Rep. X-912-74-291, p. 17, NASA--Goddard Space Flight Center (1974). 15. D. M. Gates, Near infrared atmospheric transmission to solar radiation. J. Opt. Soc. Am. $0, 1299(1960). 16. D. M. Gates and W. J. Harrop, Infrared transmission of the atmosphere to solar radiation. Appl. Opt. 2, 887 (1963). 17. E. E. Reber and J. R. Swope, On the correlation of the precipitable water in a vertical column and absolute humidity at the surface. J. Appl. Meteor. 11, 1322 (1972). 18. R. C. Temps and K. L. Coulson, Solar radiation incident upon slopes of different orientations. Solar Energy 19, 179 (1977). 19. C. W. Paltridge and C. M. R. Plait, Radiative Processes in Meteorology and Climatology, p. 318. Elsevier, Amsterdam, (1976). 20. J. N. Howard, J. I. F. King and PI R. Gast, Thermal radiation. In, Handbook of Geophysics, revised Edn., Chap. 16 MacMillan, New York, (1961). 21. T. R. Sinclair and E. R. Lemon, The distribution of 660 and 730nm radiation in corn canopies. Solar Energy l-g, 89
(1973).
0038/092X/81/I~)323-07502.0(H0 PergamonPressLtd.
A SIMPLE SOLAR RADIATION MODEL FOR COMPUTING DIRECT AND DIFFUSE SPECTRAL FLUXESt J. L. HATFIELDR. B. GIORGIS,JR. and R. G. FLOCCHINI:~ Department of Land, Air and Water Resources, University of California, Davis, CA 95616, U.S.A. (Received 2 December 1980; revision accepted 1 April 1981) Abstract--A solar spectral model that describes the solar radiation flux on a clear day at any given location was developed and tested. The model computes spectral fluxes of global, global photon, direct and diffuse solar radiation incident at the surface. Input parameters describe location and atmospheric characteristics. Location is described by latitude, altitude, slope orientation and surface albedo. Atmospheric characteristics described are turbidity, precipitable water vapor and total ozone content. The model was constructed using a one-layer homogeneous atmosphere with refinements, which are: (I) Use of climatological data to predict the total ozone content, if it is not known. (2) A more advanced treatment of infrared solar radiation (0.8-4.5 ~m) absorption. (3) A more complex scheme for predicting diffuse radiation. (4) The capability of handling a spectral albedo. (5) Inclusion of albedo dependence on zenith angle. Input parameters are minimized and several simplifying features are incorporated for ease of handling variables not routinely measured. Turbidity and total ozone content are treated as climatological estimates if specific location measurements are not available. Procipitable water vapor can be predicted using surface vapor pressure, since the sounding network is not dense. These features allow researchers outside the field of solar radiation to use the model. Because complete measurements with needed location and atmospheric characteristics could not be found, the validity of the model was tested by comparing it with a more complex, multilayered atmosphere model by Dave et al. [3]. Calculated fluxes of total direct, diffuse, and global radiation from the model presented were 11.5, 20.1 and 13.2 per cent lower, respectively, when they were adjusted for differences in extra-terrestrial solar radiation fluxes. Direct spectral fluxes closely agreed in spectral composition, with slight exception in the 0.8-0.95 #m region. Diffuse spectral flluxes were slightly higher in the UV region lower in the rest of the spectrum than were those in the multilayered atmosphere model. A sensitivity analysis of the model was also conducted: the most influential inputs were found to be latitude, slope orientation and turbidity, and the least influential was total ozone content. The hourly integrated values for the model compared very well with measured values for clear days at Davis. Discrepancies between predicted and measured values were due to lack of local turbidity coefficients. 1. INTRODUCTION
Dav~ [5]. These models are difficult to use by researchers outside the physical sciences because of mathematical and computational complexity. Robinson [6] presented a model that is adequate for most general purposes, but it does not have a complete treatment of radiation absorption by atmospheric gases. Szeicz[7] presented a solar radiation model that is specialized for use in plant growth. Goldberg and Klein [8] present a spectral model that works reasonably well for broad spectral regions. The goal of this research was to develop a relatively simple, versatile solar spectral model that is sufficiently accurate over the solar spectrum from 0.29 to 4.50 ~m. Inputs are minimized for ease of use and define needed location and atmospheric characteristics. Output parameters of the model include direct, diffuse, global and global photon spectral fluxes,
Many solar radiation models exist in the scientific literature, but few have been constructed to simulate spectral fluxes of direct and diffuse radiation. It appears that the solar spectral models available could be used more widely in such fields as solar energy, plant ecology, agriculture and environmental photochemistry. Part of the problem in acceptance of solar spectral models is due to the complexities of solar radiation and difficulties in their application. There are several spectral models available in the literature. However, a simple model that is versatile and consistently accurate over the entire solar spectrum does not exist. Leighton [1] and Green et al. [2] presented models that deal mainly with the UV portion of the solar spectrum. Some models use a multilayered atmosphere, such as that of Day6 et al. [3], Day6 and Braslau [4] and
2. THEORETICALBASIS
tContribution from the University of California Agricultural Experiment Station. Research supported by NSF Grant No. DEB-22390 and DOE Grant No. DE-FEO3--79ET20187. :~Biometeorologist; Former graduate research assistant, presently Research Meteorologist, Air Resources Control Board, State of California, Sacramento;and SolarEnergyLecturer,Dept. of Land, Air and WaterResources.Universityof California,Davis, CA 95616, U.S.A. 323
The model is set up to calculate the monochromatic fluxes of direct and diffuse radiation at the earth's surface. The diffuse flux is then related to the radiation scattered from the direct beam by a scheme that considers the many factors influencing the scattering process. Extraterrestrial solar radiation measurements for mean
324
J. L. HAT~ELDet
sun--earth distance have been made[9]. These measurements have been summarized in a table[10]. Values from that table were added to the model using an Aitken interpolation scheme[l 1], which allowed intensities to be calculated at wavelengths not given. The basic information for the framework of the model can be found in Sellers [12]. We will present the information here that is unique to this model and extensions to basic theory will provide the reader with a basis for the model. The decadic optical depth of the atmosphere is defined in terms of four components
at..
Unlike absorption, scattering must be considered to be two subprocesses: Rayleigh and Mie scattering. Robinson [6] provided a method for calculating Rayleigh scattering. The decadic monochromatic optical depth for Rayleigh scattering was calculated as a function of wavelength. The optical depth due to aerosol extinction, which is almost totally due to scattering (less than 5 per cent is from absorption by aerosols), is determined by methods introduced by Angstr6m. The equation he proposed can be modified as: r,(X) = B~-* 1
r(X) = ~-o(X)+ ~-8(~0+ r,(X) + rd(X)
(1)
in which to(A) is due to ultraviolet and visible absorption by ozone, rR(A) is due to gaseous IR absorption, r,(X) is due to Rayleigh scattering and ~',~(A) is due to Mie scattering. UV absorption by ozone mainly occurs in the HartleyHuggins band. Leighton [1] pointed out that the decadic absorption coefficient, Ko(,~) in mm -t, for this band (0.29--0.35 #m can be described approximately by the following relation: log [Ko(A)] = 16.58 - 0.00564A.
(2)
For absorption by ozone in the visible range, Inn and Tanaka [13] provide Chappuis band absorptiou coefficients. Coefficient values lying between the given wavelengths may be found using the previously mentioned interpolation scheme. Other ozone bands above 0.8 ~m are considered in the scheme for IR absorption. To obtain the optical depth due to absorption by ozone, the absorption coefficients must be multiplied by the total ozone content. The total ozone content of the atmosphere varies with latitude and time of year. An adjustment for altitude was made by applying a height correction of 2.5x 10-Smm 03m -~, derived from data given by Krueger and Minzner [14]. The decadic optical depth due to UV and visible absorption was calculated using the total ozone content and the absorption coefficient. Many gases, such as water vapor, carbon dioxide, oxygen, ozone and methane, absorb solar radiation at wavelengths above 0.8 t~m. Of these gases, water vapor is the predominant absorber; carbon dioxide is the only other significant absorber. Monochromatic absorption coefficients for IR (>0.8t~m) are given by Gates [15], and Gates and Harrop [16]. These coefficients are multiplied by the square root of the total precipitable water vapor in a vertical column, w (ram), to obtain the decadic, monochromatic optical depth of gases absorbing in the infrared region. It was assumed that the total precipitable water vapor does not vary within any given horizontal layer of the atmosphere. If atmospheric soundings are not available for estimating precipitable water vapor, the may be calculated using an approximation given by Leighton [1]. Reber and Swope [17] discuss problems of using this approximation scheme.
(3)
in which B is the turbidity coefficient and al is the wavelength exponent. A generally accepted value for at is 1.5. Robinson [6] gives average values of B as a function of latitude and height based on long-term measurements at several locations. Otherwise, turbidity coefficients can be measured by a Voltz sun photometer. If the optical depths due to absorption and scattering are known, the transmissivity of the atmosphere (To(,~)) can be found using the formula:
The spectral flux of direct radiation incident on a given surface at the bottom of the atmosphere can then be given using the formula: F(A)AA = (~/r) 2 cos (0,)
~
X2
Fo(A) To(A)d,~
(5)
I
in which At and a2 are increment boundaries. Diffuse radiation results from scattering and reflection of solar radiation. There are two components of diffuse radiation: the radiation scattered downward from out of the direct beam to the surface, and radiation reflected off the surface, then backscattered down to the surface. The amount of radiation scattered out of the direct beam is equal to that incident at the top of the atmosphere times the fraction scattered out of the direct beam. The fraction that undergoes Rayleigh scattering is easily calculated. The amount of Mie scattered radiation, the Mie scattered radiation must be assumed to be about equal to the amount of direct beam radiation undergoing aerosol extinction. The amount of scattered direct radiation, F~(A), is then determined by the formula: Fd(A) = Fo(A)(1 - I0-"~>'-'~<~"').
(6)
Some of this scattered radiation is absorbed. Assuming that, on the average, radiation is scattered from the 500 mb pressure level of the atmosphere, it travels isotropically through 3.5 per cent of the total atmospheric ozone. This percentage was calculated from data presented by Krueger and Minzer[14] for mid-latitude vertical ozone distribution. Absorption by gases other than ozone are ignored, since their vertical distributions
A simple solar radiation model for computing direct and diffuse spectral fluxes can be strongly variable and their absorption bands occur in low energy regions of the diffuse spectrum. Therefore, the transmis.sivity of the atmosphere to incoming scattered radiation is given as:
To(A) = 10-°'°7'~)
(7)
in which To(A) is the average monochromatic transrnissivity due to absorption of diffuse radiation scattered out of the direct beam. The exponent of eqn. (24) includes a factor of 2.0, so that the oblique path of isotropic radiation is considered. The fraction of radiation scattered down toward the surface, k~, is given [6] as: k~ = 0.5 coP'~(O).
(8)
Downward scattered direct radiation incident at the surface is then given by
D~(A) = ktk,.F~(A)cos(O)lO-°'°7~
transmitted to space. Since Mie scattering is mostly forward, it is not considered in the backscattering process. Half of the surface-reflected radiation that undergoes Rayleigh scattering is assumed to be scattered back toward the ground, since Rayleigh scattering is symmetric. The backscattered radiation is assumed to be isotropic and, on the average, scattered from the 500mbar pressure level of the atmosphere. The backscattered diffuse radiation, Db(A), is then given by Db(A) = 0.5Fb(A)(I - 10-2°"A))10 -°'°~'~)
in which Db(,~) is the diffuse radiation incident on a horizontal surface. If the surface is tilted at some angle i, part of the sky is blocked out and some radiation is reflected onto the receiving surface from the horizontal surface that is assumed to be at the base of slope. The fraction of the sky that is not blocked out, [', is given by using the formula: f' = cos2(i]2)
(|0)
where 0, is the angle between the sun and normal to the plane. Surface properties play an important role in the amount of hack-scattered radiation received at the surface. For instance, the higher the albedo, the greater the amount of radiation available for back-scattering. The albedo for direct radiation is a function of zenith angle. Paltridge and Platt [9] provide the following formula for the albedo to be used in calculating the amount of direct radiation reflected from the surface: a,(0) = a + (1 - a) exp (- k2(90° - 0))
(13)
(9)
in which Dd(A) is the monochromatic diffuse flux at the surface contributed by scattering from the direct beam and k., is a correction factor for the brightening of the sky in the vicinity of the sun[18]. Temps and Coulson[18] provide the following as an equation to quantify the sky brightening. = 1 ,de.COS2(0s) Sin3(0)
325
(14)
Temps and Colson [15] provide a correction factor for horizon brightening, f": f"=
l+sin3(il2)
(15)
Monochromatic diffuse radiation that has not been reflected directly from other surfaces, D,(A), is then determined from D,,(A)= [Ds(A) + Db(A)](f')(.f").
(16)
Reflected monochromatic radiation falling on the receiving surface, D,(A), is given [15] as: D,(A) = IF(A) + D,(A)]a(I -/311 + sin2(0/2)][cos (a)] (17] where a is the azimuth angle. The monochromatic diffuse flux is then given using
(11) D(A) = D,,(A)+ D,(A)
(18)
in which a is the surface albedo at low zenith angles, k2 is a constant, equal to about 0.1 degree -l, and a,(#) is the zenith-angle-dependent albedo for direct radiation. By assuming isotropi.c diffuse radiation and by integrating eqn (30) over a hemisphere, the surface albedo for diffuse radiation is found to be about equal to a. The total amount of radiation available for backscattering is then given by
Calculation of global radiation Once the direct and diffuse solar fluxes are known, the global radiation can be calculated easily. The monochromatic global radiation is simply the sum of the direct and diffusemonochromatic fluxes:
Fb(A) = FO:)a,(0) + De(A)a
G(A) = F(X) + D(A)
(12)
in which Fb(A) is the monochromatic solar radiation flux back into the atmosphere from the surface. In this instance, higher-order surface backscattering is ignored. As radiation reenters the atmosphere from the surface, it is assumed that it is isotroic and that it will be absorbed by ozone, scattered by Rayleigh particles, or
in which D(A) is the monochromatic diffuse radiation.
(19)
in which G(A) is the monochromatic flux of solar radiation at the surface. The monochromatic global photon flux is then found by dividing the monochromatic global solar radiation by the energy per photon: P(x) =
G(A)xlhc.
(20)
J. L. HATFIELDet aL
326
Table 1. Input and output parametersfor the spectralmodel and the necessaryunits Input Parameter
Unit
Wavelength i n t e r v a l
~
Wavelength l i m i t s
I.m
Output P a r u s e t e r Exterrestrial
Time
Unit
f l u x , Honocbroaatic
WI'2s'l
Monochromatic s u r f a c e fluxes
length and l i m i t s
hour
~'2s'l
direct, diffuse, total
Station description Latitude
degrees
Nonocbrolstic g l o b a l photon
flux
~-2s-I
m
Elevation Slope
degrees
Slope azimuth
degrees
Albedo
unitless
Surface vapor pressure
cm
Surface a i r p r e s s u r e
mb
Ozone ~/
I
T u r b i d i t y ,~/
I
T o t a l g l o b a l flux
~-2s-1
H o n o c h r o ~ ' t i c global flux
WB-2s'I
~/ Estimated c l i m a t o l o g i c a l l y i f no s ~ a t i o n l o c a t i o n information is a v a i l a b l e .
Finally, the total global flux is ob~ned by integrating over 0.05 p.m intervals the direct and diffuse monochromatic fluxes:
GT = f:
[ F ( A ) + D ( A ) ] d~
(21)
in which GT iS the total global flux.
case, the site was at sea level at a latitude of 40°N, and the tLme was solar noon of the spring equinox. Turbidity in the atmosphere was assumed to be 0.09 and total ozone content to be 3.318mm. Surface vapor pressure was assumed to be 12 mbars. A level surface was used that had an albedo of 0.25. Plots were generated for comparison using the output from the base case and variations from it.
3. METHOD
4. RESULTS AND D/SCUSS/ON
' The spectral model developed from the theory included the input and output parameters given in Table I. The ozone content and turbidity coefficient are estimated cl/matologically by model if they are not given. Two interpolation schemes were used to obtain values of parameters for which data were not provided in continuous form [1 l]. A listing of the model will be provided by the senior author upon request. The reliability of the proposed model was tested by comparing calculations against spectral calculations made by Dave et al. [3], who used multilayered atmosphere model. Their calculations were made for a sealevel site at mid-latitude during a clear day in the summer for which the zenith angle was 60°. Total precipitable water vapor was 2.925 cm, and total ozone content was 3.18 ram. Since their model atmosphere was of low turbidity, the value used for the turbidity coefficient in the model presented had to be reduced from the climatological value to 0.05. The albcdo was 0, a perfectly absorbing surface. The two models were compared using the spectral and total fluxes of extraterrestrial, direct, and diffuse solar radiation. A sensitivity analysis was conducted to assess the response of the model to variations in input parameters as compared with an arbitrary base case. In the base
Figure I ~ves results of the companson with the multilayered model of Dave et aL [3]. The two top curves are solar fluxes onto a horizontal surface at the top of the atmosphere. The top curve is predicted using the multilayered model. Discrepancies between the two spectra are due to the differeing sources of extraterrestrial flux data for the models. Values used in the multilayered model were obtained from Howard et ol. [20]; values used in the presented model are from more recent measurements [9]. Since the extraterrestrial values used ,s,~
16 --
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Fig. 1. Comparison of extraterrestrial (upper curves) direct (middle curves) and diffuse (lower curves) spectral fluxes as calculated by the multilayered Dave et al. [3] model and the model.
A simple solar radiation model for computingdirect and diffuse spectral fluxes by the multilayered model are larger (approx. 6.4 per cent), the predicted direct and diffuse values are also higher. Therefore, direct and diffuse values from the two models show more lack of agreement than actually exists. When the extraterrestrial flux differences are taken into account, the direct spectra appear to agree very well from 0.29 until 0.8 ~m, whereas the presented model neglects to properly simulate the region from 0.8 to 0.95/zm because of poor oxygen and water vapor absorption data used in the present model. The direct spectra agree very well in the waveband from 0.95 to 2.5 ~m. When corrections were made for extraterrestrial flux differences, the total direct flux of the presented model was found to be 11.5 per cent lower than that of the multilayered model. The diffuse spectra do not agree as well as the direct. The presented model appears to generate higher values at wavelengths greater than 0.5/zm. Also, since absorption by water vapor is not included in the diffuse parameterization of the presented model, its diffuse spectrum lacks absorption bands in the IR. The last important disagreement is that the total diffuse flux of the presented model is 20.1 per cent lower than that of the muitilayered atmosphere model. Combining this difference with that of the total direct gives a difference in the global values of 13.2 per cent; the presented model has the lower total global flux.
Sensitivity analysis The first sensitivity variation used was that of latitude. Three variations in latitude were considered: 0°, 40°N (base case) and 70°N. All other input parameters remained at base case values, except for total ozone content and turbidity, which were allowed to vary climatologicaUy with latitude. Figure 2 compares direct values. The highest direct spectral fluxes were received at the equator. Direct radiation at 40 and 70°N received about 73 and 24 per cent of that received at the equator, respectively. There also appears to be a red shift in the direct and diffuse spectra with latitude (Fig. 3). Sinclair and Lemon [21] pointed out this red shift with greater zenith angle and its significance to plant phytochrome systems. When the optical depths due to different attenuating mechanisms were inspected, Rayleigh and Mie scattering appeared to be the most responsible for the differential attenuation causing the red shift. Variations in total ozone content had only a small influence on the solar radiation spectrum. Direct spec-
T 20 "r E
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Fig. 3. Latitude effects on the diffuse spectral flux as simulated by the one-layermodel trum changes were found in the ultraviolet and visible range, whereas the only detectable changes in the diffuse
occurred in the UV. When the total ozone content was changed from 0.0 to 6.0 ram, the total direct and diffuse fluxes were decreased by 2.4 and 2.8 per cent, respectively. Gases absorbed the most in the infrared region. The surface vapor pressures used for calculating infrared absorption were 0, 6, 12 (base case) and 30 mbars. In turn, the precipitable water vapor estimates were 0, 1.04, 2.08 and 5.19cm. Since absorption is directly proportional to the square root of the precipitable water vapor, the condition of no IR absorption was simulated by setting the surface vapor pressure equal to zero. Without infrared absorption, calculated total direct flux was 24.4 per cent higher than the base case value. Increasing the surface vapor pressure from base case value to 30 mbars decreased absorption by only 5.6 per cent, indicating that absorption bands in the IR (which are mostly water vapor bands) tend to saturate very fast. Since the diffuse parameterization did not include IR absorption, the diffuse spectrum did not change as surface vapor pressure varied. Turbidity affected both direct and diffuse spectra very strongly. Increasing turbidity attenuated the direct beam but enhanced the diffuse. Increasing the turbidity coefficient of sky values from clear (B = 0.05) to turbid (B = 0.30) decreased the total direct by 37.4 per cent but
increased the total diffuse by 118 per cent. A red shift occurs in both the direct and diffuse spectra, as is shown by Figs. 4 and 5. But because of the compensating changes in the direct and diffuse spectra, the global radiation does not undergo such strong changes (13 per "¢--
'_o
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0.00
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Fig. 4. Turbidity effects on the direct spectral flux as calculated by the one-layermodel.
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et aL
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o
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t.2
WAVELENGTH
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determine how slope azimuth affects the total amount of global radiation received throughout the day. All slopes used in this analysis were set at 30°, except in the case of the horizon~ surface. Five azimuths were used in the analysis: south, southwest, west, northwest and north. The south slope received almost twice as much radiation as the north slope. The west slope received almost the same amount of radiation as a horizontal surface. One of the problems with these calculations is that the slopes were assumed to have a horizontal surface at their bases. Therefore, the spectra given may be somewhat dissimilar to those received on the same slopes in mountainous terrain. In the structure of the model, it was possible to integrate over the solar spectrum to achieve a tom
irradiance. These results for clear skies at Davis are given in Figs. 7(a--d). The results for all days are quite good and show that this model predicts all components with an accuracy of better than 5 per cent. There were only two points at which the model did not perform satisfactorily. On day 80, the global amount was underpredicted because the model did not account for a large DAVIS 38N
OAY 80
o
meosure4 . . . . . comoufed
800
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600
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Fig. 6. Daily global spectral fluxes incident on the surfaces of various the surfaces of various slope orientations at the spring equinox as calculated by the one-layer model.
cent for the same change). These results indicate that the turbidity must be known to accurately calculate direct and diffuse fluxes. Sensitivity to albedo was tested by examining the behavior of surfaces with constant and spectral albedos. Change in albedo does not affect the direct spectrum, and in~uences only the amount of backscattered diffuse. Three constant albedos were used in the analysis: 0.0, 0.25 (base case) and 1.0. Decreasing the albedo from 0.25 to 0.0 caused a decrease of 10.0 per cent in the total diffuse, and increasing it to 1.0 caused an increase of 30.0 per cent. The effect of slope on the direct and diffuse spectrum was calculated with atl slopes south facing at 30, 45 and 90~, except for the base case, which was a horizontal surface. The sun's azimuth and zenith angles were 0 and 40', respectively, so a south-facing 40~ slope would receive maximum direct radiation. For this reason, the 30 and 450 slopes show the highest reception of direct radiation, and the vertical slope shows the lowest. Diffuse spectra show a red shift as slope increases, because of the blocking out of a portion of the sky and the interception of radiation reflected off of the nearby horizontal surface, which is at the base of the slope. Part of the sensitivity analysis was conducted to
I000
1.8
(~,m)
n o
Z( HOUR
6
8
Fig. 7. Comparison of simulated and measured direct, diffuse and global solar radiation on a horizontal surface for a clear Julian day 80 (a), 172 (b), 264 (¢) and 355 (d) at Davis, California.
A simple solar radiation model for computing direct and diffuse spectral fluxes amount of dust particles in the air. On day 172, June 20, there was an increase in diffuse at midday due to agricultural burning near the measurement site. Overall, this model produces acceptable results and should be useful for a number of locations around the world. It incorporates a large number of local parameters, yet remains easy to use and can be run on a small minicomputer quite efficiently. REFERENCES
1. P. A. Leighton, Photochemistry of Air Pollution, p. 300. Academic Press, New York, (1961). 2. A. E. S. Green, T. Sawada and E. P. Shettle, The middle ultraviolet reaching the ground. Photochem. and Photobiol. 19, 251 (1974). 3. J. V. Dave, P. Halpern and N. Braslau, Spectral distribution of the direct and diffuse solar energy received at sea-level of a model atmosphere. Rep. G320-3332,p. 20. IBM Palo Alto Scientific Center (1975). 4. J. V. Dav~, and N. Braslau, Effect of cloudiness on the transfer of solar energy through realistic model atmosphere. J. Appl. Meteor. 14, 388 (1975). 5. J. V. Dave,, Validity of the isotropic distribution approximation in solar energy estimation. Solar Energy 19, 331 (1977). 6. N. Robinson, Solar Radiation, p. 347. Elsevier, Amsterdam (1966). 7. G. Szeicz, Solar radiation for plant growth. J. Appl. Ecology 11,617 (1974). 8. B. Goldberg and W. H. Klein, A model for determining the spectral quality of daylight on a horizontal surface at any geographical location. Solar Energy 24, 351 (1980).
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