Calculating solar radiation for horizontal surfaces—I. Theoretically based approaches

Rene**able Ener.qy Vol. 3, No. 4/5, pp. 357 364, 1993

0960 1481/93 $6.00+.00 Pergamon Press Ltd

Printed in Great Britain.

CALCULATING SOLAR RADIATION FOR HORIZONTAL SURFACES--I. THEORETICALLY BASED APPROACHES JOHN E. HAY Environmental Science, University of Auckland, Auckland, New Zealand Abstrac~This paper reviews the theoretical basis of relatively simple models that use widely available meteorological data to determine the solar irradiation at the earth's surface. Spectral and non-spectral models for clear and cloudy sky conditions are considered. A sample of validation results is presented. These indicate that most of the models studied are capable of estimating the global irradiation with an acceptable accuracy, especially for time intervals of a month of more.

INTRODUCTION

Quantitative information on the amount of solar energy reaching the earth's surface is often unavailable due to the low density of the observing network and the strong spatial variations in solar radiation. Such data are required in many applications of importance in Africa--crop growth models, water balance and soil moisture estimates and the designing of photovoltaic systems all need solar radiation data. So do the development of solar crop-drying systems and building design. The prospect of global warming may well add urgency to the construction of buildings with lower solar thermal gains and increase the use of solar cooking in tropical countries. The amount of solar radiation reaching the earth's surface (the "solar irradiation") is influenced by astronomical factors, gaseous absorption, gaseous and aerosol scattering and by scattering and absorption by both clouds and the underlying surface. Previous contributions in this volume will have shown that, in principle, many of these influences are reasonably well understood. Problems arise when the physical atmospheric conditions are poorly described in both the spatial and temporal dimensions. The most difficult processes to include are those related to clouds, since clouds are highly variable in both time and space. Even for a particular instant at a specific location it is impossible to characterize the cloud microphysical and geometric properties in sufficient detail to simulate the radiative transfer processes with great accuracy. Thus, solar radiation modelling is always a compromise between the desire for accuracy and the practical constraints imposed by data availability and corn-

putational effort. Various balances are struck, sometimes as a consequence of the fundamental requirements of the investigation but often for more practical reasons. The following discussion will illustrate the range of approaches that are followed and it will indicate the accuracies that are achieved as a result. Part I of this paper will focus on models that have a strong theoretical basis while Part II will review models which are based on empirical (statistical) relationships that often bear little resemblance to the physical processes operating in the atmosphere. THE THEORETICAL BASIS

Figure 1 provides an indication of the range of factors influencing the intensity of solar radiation at the earth's surface and their relative importance. The seasonal and diurnal variations in the extraterrestrial (i.e. top of atmosphere) radiation are, of course, dominant, except in the equatorial region for the former and in polar regions for the latter. Fortunately these influences can be modelled exactly, apart from the smaller variations in the solar output (the "solar constant"). Satellite measurements are now confirming and quantifying the shorter term variations in the solar constant [1], but in most applications a value of (around) 1367 W m 2 is accepted [2]. The effects of s u n ~ a r t h geometry on the extraterrestrial radiation for a horizontal surface are given by Eo = S o ( R o / R ) 2 [ s i n ~b sin 6+cos 0 cos 6 cos h] = S o ( R o / R ) 2 cos z

where 357

(l)

358

J . E . HAY

Zenith angle varies with Latitude, Lime, and season

Sun-earth distance seasonaLLy variable

= I. 32 to 1.42 kW/m

Scattering %

Absorption %

it

-.17/

0.5 t o 5 Ozone

/ - - 0 . 5 to 5 - - 0 . 5 to 5

Upper dust Layer 15-25 km

/ [

/ ,/ Dry air

to 5 \ ~

"~ W.V. 31:o9 tower dust 0.5 to 5

[ ~

0-30 kin

]

Sky cover

Dry air 6to 8

~

l

and Lower dust \Layer O - 3 km

0 . 4 to I0

ION

O. I t o I

"% sunshine

0.5 to 5

Fig. 1. Indicative values of clear sky absorption and scattering of solar radiation in the earth's atmosphere. Values are representative of unit optical air mass (i.e. overhead sun). From Randall and Bird [7]. E0 = extraterrestrial global radiation for a horizontal surface So = solar constant R0 = mean sun-earth distance R = actual sun-earth distance q9 = latitude of location 3 = solar declination h -- solar hour angle z = solar zenith angle. Most texts and other studies providing more than a cursory treatment of solar radiation will include definitions of the above variables and practical methods for their computation. Examples include Iqbal [3], Hulstrom [4], Frohlich and L o n d o n [2], Davies and Hay [5] and Paltridge and Platt [6]. Calculation of solar radiation at the earth's surface involves some approximation of the scattering and absorbing processes in the atmosphere. These are treated explicitly in the equation of radiative transfer #d!~ d'c~ where

L~+ 4nn ~o0 fo l~(r,~, y')p(L~, y', y) do9

(2)

1;, = spectral radiation intensity at wavelength 2 # = cosine of the solar zenith angle (z) z~ = atmospheric optical depth p ( r ~ , y ' , y ) = scattering distribution or phase function from direction y' into the direction y 0% = single scattering albedo co = solid angle. Once solved, eq. (2) will give the total solar irradiance at the earth's surface after integration over azimuth and zenith angles (yielding the spectral irradiance) and then over the wavelengths of the solar spectrum. While such exact solutions are computationally intensive and are inconsistent with the uncertainties associated with specifying the optical properties of the atmosphere, they do provide stateof-the-art computations for model atmospheres and hence serve as a baseline for the evaluation of simpler, more practical parameterizations of the physical processes. The usual approach is to simplify the radiative transfer equation and to differentiate between cloudless and cloudy sky conditions.

359

Calculating solar radiation for horizontal surfaces--I (A) Chmdless s k y conditions Following Davies and Hay [5], eq. (2) can be readily solved for direct radiation by ignoring the scattering phase function. Integrating between the ground and the top of the atmosphere (where z;~ = 0), the spectral transmittance at normal incidence is expressed by Beer's law l , ( t ) / L ( O ) = exp ( - trip).

(3a)

This formulation assumes that the atmosphere is homogeneous over the entire path length. If t~. varies along the path eq. (3a) should be written as I; (z) f~t 1~(0)• - exp-- 2 z;~ ds

f;

L(0) exp ( - r ~ / # ) d2.

(5)

Numerical evaluation of eq. (5) requires detailed knowledge of spectral optical properties. Those for aerosol are particularly difficult to define though Mie scattering theory can be applied to give aerosol extinction (and scattering and absorption) efficiencies and thence the associated spectral optical depth. For situations where it is important to distinguish between the

(6)

where S is the direct solar radiant intensity at normal incidence. The exponential function defines a total transmittance, the component terms of which are usually combined multiplicatively so that : S = SopTo(uom)Tw(u,,m)TR(m)T,,(m)

(7)

where To, Tw, Tr and T~L are appropriate transmittances after absorption by ozone and water vapour, Rayleigh scatter and extinction by aerosol, Uo and uw are optical path lengths through ozone and water vapour and m is the optical air mass. The multiplicative assumption is not strictly correct for Tw as water vapour absorbs at longer wavelengths than ozone. Its absorptance (a,,.) should therefore be subtracted, yielding S = S~,p[To(uom)TR(m)-aw(u,,m)]T~(m).

(8)

The various attenuations are shown schematically in Fig. 2. A similar treatment may be provided for the diffuse radiation if polarization is neglected, it is assumed that all scattering is isotropic (i.e. non-directional) and that water vapour absorption attenuates only the direct beam. The two components (one from Rayleigh scattering DR and one from scattering by aerosols DA) of the diffuse radiation stream (Fig. 2) are then given by

(4)

where r ..... rw~, TR/. and r~,. are the optical depths due to absorption by ozone and water vapour, scattering by dry air molecules (i.e. Rayleigh scattering) and absorption and scattering by aerosols, respectively. Integrating eq. (3a) over wavelength gives the following expression I=

S = So exp ( - r / # )

(3b)

where s is the path of the radiation. Equation (3) strictly applies only to monochromatic radiation but without too much error it can be used to model finite spectral bandwidths in those spectral regions where rapid changes caused by molecular absorption do not occur. In molecular absorption bands with structure, models with higher spectral resolution should be used. "Band models" were developed to reduce the computational and data burdens. One of the most widely used is the LOWTRAN computer code which now includes the extraterrestrial solar spectrum (as does its derivative SOLTRAN), permitting computation of the direct solar radiation. Recent versions of L O W T R A N have progressed from including single scattering into the direct beam (appropriate only for computing diffuse radiation under very clear conditions) to more complex twostream multiple scattering schemes. The principle scatters and absorbers are included in z, such that r~ = r,,~ + r w ; + t R ~ + G ;

relative contributions of aerosols (including clouds) to absorption and scattering such calculations are desirable, but more often spectrally integrated values of the radiant fluxes are expressed in the same form as eq. (5) such that

D R = I(O)#T,,(Uom)[l -- TR(m)]T.,(m)/2

(9)

and DA

=

l(O)p[To(uorn) T R (m) -- aw(uwm)]

× [1-- T~,(m)]c')0&~a (10) where B~, is the ratio of forward to total scatter for aerosols. In eq. (9) the isotropic assumption results in half of the Rayleigh scattered radiation reaching the ground while in eq. (10) the single scattering albedo allocates to scattering the fraction of the radiation that is attenuated by aerosol. Interactions between the incoming radiation and the ground surface result in multiple reflections between the ground and the atmosphere. The diffuse component arising from multiple reflections is Ds = ~ h ( S t ~ + D R + D . a ) / ( l - - ~ X ~ b )

(l 1)

360

J.E. HAY

Extraterrestriat

Irmdiance I(o)/~

+ Absorption by ozone

TO(uoml: I-Ooluoml Rayleigh scattering TR(m)

~

,

I bsorption by water- [ vapour aw (uw m I

/ , ~

Attenuotionbyoerosot To (TI

I

l++=er+ l component [ I.TR(ml]/2

Attenuotion byaerosol TO(m)

"Downward sCattered-

component[1-To(ml]=oB~ J

1 Fig. 2. Schematic of steps in calculating solar irradiance and its direct and diffuse components. From Davies and Hay [5].

where ~b is the reflectivity of the atmosphere for surface reflected radiation and a+ the reflectivity of the surface. Hence, for cloudless skies, the total diffuse radiation reaching the earth's surface (Ed~) is Ed$ = DR + DA + Ds.

(12)

Implementation of these equations requires parameterizations of the attenuation processes. Again most comprehensive texts [3, 5, 6] include the relevant equations, and they will not be repeated here. For a given location usually only the amounts of precipitable water and ozone need be measured or estimated. Other terms are either known functions of location and time or are typically unknown and must be estimated. The parameters needed to specify aerosol attenuation usually generate the greatest uncertainty. (B) Cloud efJects Despite the apparent difficulties, calculation of solar radiation under cloudless conditions can be performed with relative ease and with resaonable accuracy. Clouds drastically degrade the accuracy due not only to the paucity of observed data on cloud amount and type but also to the failure of such observations to define the pertinent microphysical and geometric properties and their temporal and spatial variations

and the assumptions necessary to provide a tractable computational algorithm. The literature contains numerous examples of analytical studies based on idealized, though in some instances geometrically complex, clouds or cloud fields [8]. While these studies serve to highlight the role of cloud properties in modifying radiative flows through the atmosphere, they do little to solve the practical dilemmas facing the modeller as a consequence of data paucity and computational complexity. In some approaches, clouds are treated as a layer of aerosols of such thickness that the radiation reaching the ground is all diffuse. However, even in such cases, the distribution of radiation over the sky hemisphere may not be uniform (i.e. isotropic) due to variations in cloud thickness or the absence of clouds in some sectors. Also, in the case of partial cloud cover, cloud edge effects can cause momentary increases in solar irradiance to values greater than the solar constant. Highly simplified approaches tend to be used for most climatological applications, with some consequential deterioration in accuracy. The approach with the strongest physical basis is to treat the cloud as occurring in distinctive layers, each with its characteristic transmissivity. Following Davies and McKay

361

Calculating solar radiation for horizontal surfaces--I

[9], cloud layer models have the general form n

F.~$ = F.o.[ [ I ( l - C i

+t,C,)(1-~,,)

'

C~ : "1

(13)

I

C3 = "5

where Eo,b is the cloudless sky irradiance, C, is cloud C2 = ' 5

a m o u n t and t~ is the transmissivity of an individual layer. Usually values of the cloud transmittances are calculated from tj = a exp ( - b m )

C 2 : '6

(14)

where a and b are parameters that depend on cloud type [10]. Commonly used values are based on mean data from a single site in the U.S.A. this compounds other errors and certainly ensures that reliable estimates are probably limited to time-averaged calculations. Equation (13) shows that transmittances are applied to a theoretical cloudless sky irradiance (E0,,), other atmospheric properties are assumed unchanged by the presence of cloud and allowance is made for multiple reflections between the cloudy atmosphere and the underlying surface. Typically, cloud a m o u n t is assumed to be uniformly distributed over the sky hemisphere and to remain that way over the period represented by the cloud observation. In eq. (I 3) the term (1 -C~) defines the transmittance of the cloudless portion of the cloud field through which incident radiation passes unimpeded while tiC~ is the transmittance of the clouded portion. Total cloud transmittance is the product of the layer transmittances. Cloud layer models require estimates of the fraction of the sky at each level i that is occupied by cloud. The sum of the cloud amounts for all layers may thus exceed unity, a situation which is inconsistent with the reporting of observed cloud amounts. In the latter case, individual layer amounts are expressed as a fraction of the total cloud amount, One solution [ll] is to correct observed cloud amounts above the lowest level due to some of this cloud

A j

A

C I = "5

Cv=-9 Fig. 3. Diagrammatic representation of the correction for reported cloud layer amounts due to layers being partially obscured by lower cloud. From Suckling and Hay [12].

being obscured from the observer. The corrected cloud a m o u n t in the ith layer (i counted downwards) is obtained from

C, =

1-

C;

where C: is the observed cloud a m o u n t and the summation is for the observed cloud amounts in the layers beneath. This correction procedure is shown schematically in Fig. 3. DESCRIPTION OF SELECTED OPERATIONAL MODELS

(A) Spectral so&r radiation models.fbr clear skies As described above, a rigorous model of radiative transfer through the atmosphere uses numerical methods to solve the integral form of the radiation transfer equation [eq. (2)]. One approach is the Monte Carlo or "ray tracing" method in which the atmosphere is divided into several layers and key atmo-

500 /ModeLed m~rA~asured

E :a. , 2 o o

'&

6CC

05

,,

I

I

05

07

I ~ 09

I

~

(15)

i=/t

t

5

15.

f"

I 9

L 21

25

WoveLengt,h, p_m Fig. 4. C o m p a r i s o n o f spectral solar r a d i a t i o n measurcd at G o l d e n , CO, U.S.A., and modelled values using

the SERI rigorous spectral solar radiation model (BRITE). F r o m Bird [14].

362

J . E . HAY Carlo radiative transfer model [13]. The massive computer code a n d c o m p u t i n g requirements m a k e such a model time-consuming, exacting a n d expensive to run. However, it shows excellent agreement with values determined using a spectroradiometer (Fig. 4). A relatively recent d e v e l o p m e n t has been to use models such as the B R I T E code to evaluate the use of a single, h o m o g e n e o u s , atmospheric layer and deterministic as opposed to statistical m e t h o d s to solve the radiative transfer e q u a t i o n [15]. Only fluxes at the top a n d b o t t o m o f the a t m o s p h e r e can be determined. C o m p a r i s o n s o f the results provided by the rigorous a n d simpler codes show excellent agreement even for zenith angles as large as 80 U, but only in the absence o f strong molecular a b s o r p t i o n . Even in the presence of strong a b s o r p t i o n b a n d s nearly identical multilayer a n d singlelayer results are o b t a i n e d by d e t e r m i n i n g equivalent t e m p e r a t u r e s a n d pressures for the a b s o r b i n g gases from the b a n d a b s o r p t i o n

Table 1. Essential measured variables for three models. (After Davies and McKay [9].) Total Cloud Cloud Surface cloud layer layer humidity amount amount type Davies and McKay [19] Josefsson [17] Monteith [18]

x × ×

× x ×

× ×

× ×

spheric p a r a m e t e r s are defined for each layer. These n o r m a l l y include temperature, pressure, the density o f i m p o r t a n t molecular species a n d the aerosol density. Mie scattering theory is used to model the effects o f aerosols, Rayleigh scattering theory the molecular scattering effects a n d a b a n d model for molecular a b s o r p t i o n . One such rigorous model is the B R I T E M o n t e

Table 2. Summary performance statistics for selected models. (After Davies and McKay [9].) Global radiation Measured = 10.99 MJ m 2 day ~and 910 kJ m 2 h Daily

Davies and McKay

Josefsson

Monteith

BEST

1.27 -0.01 1.76

1.20 -0.24 1.67

1.58 0.30 2.15

1.04 -0.00 1.42

Josefsson

Monteith

172.6 - 19.0 266.4

197.2 25.8 286.6

MAB MBE RMSE

Davies Hourly and McKay MAB MBE RMSE

177.5 0.4 270.7

Diffuse radiation Measured = 5.45 MJ m 2 day ~and 443 kJ m 2 h Daily MAB MBE RMSE

Davies and McKay

Josefsson

1.41 0.06 1.93

1.43 0.13 1.93

Davies Hourly and McKay MAB MBE RMSE

Josefsson

169.8 1.8 256.6

174.3 7.8 261.6

Direct radiation Measured = 9.49 MJ m 2 day J and 731 kJ m 2 h- i Daily MAB MBE RMSE

Davies and McKay

Josefsson

1.42 0.15 2.09

1.42 --0.16 2.07

Davies Hourly and McKay MAB MBE RMSE

182. l 13.1 298.3

Josefsson 180.9 --13.0 296.1

363

Calculating solar radiation for horizontal surfaces--I model. Such comparative studies have resulted in a new model [16]. The inputs required for this model are atmospheric ozone, water vapour, turbidity at 0.5 #m, solar zenith angle, surface pressure and ground albedo.

40

2500,

Alice Springs 20001 I t 1500

50

~

[] 2O

000

(B) Operational non-spectral modelsfor cloudy skies Davies and McKay [9] describe an evaluation of several non-spectral models that treat the atmosphere as plane parallel and assume single scattering except for multiple reflections between ground and atmosphere. The models examined include those described by Josefsson [17], Monteith [18] and Davies and McKay [19]. The latter model uses eqs (13) (15) to simulate the attenuation by clouds. In Josefsson's model observer overestimation of total cloud amount and amounts in the lowest two layers are corrected using C=

( C ' ) '6

500

% . . . .

o

transmission is reduced by 30% if precipitation occurred during the hour in question, and by 20% if it ended within that hour. Monteith's model uses total cloud amount and consequently eq. (13) is used for only one layer. Data requirements for the three models are given in Table 1. Davies and McKay used data from 15 locations in North America, Europe and Australia to evaluate the models. Model performance was assessed primarily from the mean bias error (MBE), which measures systematic error, the root mean square error (RMSE), which measures shortterm nonsystematic errors and the mean absolute bias error (MAB) which is used to reveal any significant positive or negative biases which would not be indicated by the MBE. To provide an estimate of the best performance any model can attain Davies and McKay regressed atmospheric transmission and percent possible sunshine on a month by month basis for each station and subsequently used the regression coefficients for a given month to compute the daily global irradiation for that month. These estimates are called "BEST" in Table 2. The summary of performance statistics (Table 2) indicate that the two cloud layer models outperform the Monteith model, with Josefsson usually being superior. Both come very close to optimum model performance as indicated by BEST. Model performance as a function of cloudiness was also investigated by Davies and McKay. Both layer models perform well for the most frequently observed cloud amounts (Fig. 5) but they tend to underestimate global radiation for the less frequently observed cloud

I0

.... ,

L__~

r---,

L

L ' M-~[

I

2

3

4

5

_J

"

-I

I

I l-t'

6

7

8

10

9

Cloud (]mount (tenths) 2500

40

~'~ E

ALbuquerque

2

3o ~5oo

~. 2o

%1

o-

~0 500

la-

. . . . . . . .

(16)

w h e r e C' is the o b s e r v e d c l o u d a m o u n t . C l o u d transm i s s i v i t y does n o t v a r y w i t h air mass a n d c l o u d field

r--q

q

I

1

1

i

2

3

I 4

' 5

1 6

1

I

7

8

9

',0

CLoud amount (tenths) 2500 I

2000 I--

r-

l

j- 30

/

w

,oool/

5oo I - 'r---~k___ o

40

Zurich

I_ I

L

~

2C

%

,---J

['%, Io N

]

I "-I--'-I-=]--I

2

-~

5

4

5

6

I

7

•

Io

8

CLoud amount (Oktes) Fig. 5. Comparison of measured (solid line) and modelled global radiation for three selected stations. Estimates based

on the Davies and McKay model are shown by squares, those of Josefsson by dots. Cloud frequency distributions are shown by dashed lines. (From Davies and McKay [9].)

amounts between two and seven tenths. The influence of cloud reporting procedures was examined. Little error is introduced when 3-h observations are used while results for 6-h cloud observations suggest that even in such cases useful estimates may be obtained. Such a conclusion has favourable implications for estimates of cloud amount and solar radiation based on a single satellite pass. CONCLUSIONS Solar irradiation models based on radiative transfer theory have evolved substantially over the past few decades as numerical and computational capabilities have undergone substantial improvement. Such devel-

364

J. E. HAY

opments have laid the basis for improvements in the less rigorous but computationally more efficient codes which are still capable to reproducing the longer-term characteristics of the radiation climate. Since all models rely on statistical descriptions of average relationships they cannot provide instantaneous radiation estimates that compare favourably with the radiation measurements themselves. On the basis of their comprehensive evaluation, Davies and M c K a y [9] conclude that the differences between the best and worst performing models they assessed for global irradiation estimates may not be significant for solar energy or any other applications. They also concluded that models may well be close to the limit of prediction given that the layer models performed nearly as well as a statistical model tuned to the measured data. M a j o r error sources may now be associated with surface meteorological measurements and observations rather than with the models. For the direct and diffuse components they showed that estimates of 30-day means using the layer models had errors comparable to those for statistical models which simply apportion the global irradiance (see Part II). While data on short-term variability in the direct and diffuse components are required for many applications the competitive performance of layer models in the longer term is an important finding.

REFERENCES

1. R. C. Willson and H. S. Hudson, Solar luminosity variations in Solar Cycle 21. Nature 332, 810 812 (1988). 2. C. Frohlich and J. London (Eds), Revised Instruction Manual on Radiation Instruments and Measurements. World Climate Research Programme, Publications Series No. 7, WMO/TD-No. 149, World Meteorological Organization, Geneva, 140 pp. (1986). 3. M. lqbal, An Introduction to Solar Radiation. Academic Press, Don Mills, 390 pp. (1983). 4. R. Hulstrom (Ed.), Solar Resources. MIT Press, London, 408 pp. (1989).

5. J. A. Davies and J. E. Hay, Calculation of the solar radiation incident on a horizontal surface. In Proceedings First Canadian Solar Radiation Data Workshop

6.

7. 8. 9. 10. 11. 12. 13. 14. 15.

16.

17. 18. 19.

(Edited by J. E. Hay and T. K. Won), pp. 32--58. Canadian Atmospheric Environment Service, Downsview, Canada (1980). G. W. Paltridge and C. M. R. Platt, Radiative Processes in Meteorology and Climatology. Developments in Atmospheric Science, 5, Elsevier, New York, 318 pp. (1976). C. Randall and R. Bird, Insolation models and algorithms. In Solar Resources (Edited by R. L. Hulstrom), pp. 61-142. MIT Press, London (1989). J. M. Davis, S. K. Cox and T. B. McKee, Vertical and horizontal distributions of solar absorption in clouds. J. Atmos. Sci. 36, 1976-1984 (1979). J. A. Davies and D. C. McKay, Evaluation of selected models for estimating solar radiation on horizontal surfaces. Solar Ener,qy 43(3), 153-168 (1989). B. Haurwitz, Insolation in relation to cloud type. J. Meteorol. 5, II0 113 (1948). J. A. Davies, W. Schertzer and M. Nunez, Estimating global solar radiation. Boundary-Layer Meteorol. 9, 3352 (1975). P. W. Suckling and J. E. Hay, A cloud layer--sunshine model for estimating direct, diffuse and total solar radiation. Atmosphere 15, 194-207 (1977). R. E. Bird, Terrestrial solar spectral modelling. Solar CellsV, 107 111 (1982). R. E. Bird, Spectral terrestrial solar radiation. In Solar Resources (Edited by R. L. Hulstrom), pp. 309 333. MIT Press, London (1989). K. Stamnes and R. A. Swanson, A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. J. Atmos. Sci. 3, 387-399 (1981). R. E. Bird and C. Riordan, Simple solar spectral model for direct and diffuse irradiance on horizontal and tilted planes at the Earth's surface for cloudless atmospheres. J. Climate Appl. Meteorol. 25(I), 87 97 (1986). W. Josefsson, Modelling direct and global radiation from hourly synoptic observations. Unpublished manuscript (1985). J. L. Monteith, Attenuation of solar radiation: a climatological study. Quart. J. R. Meteorol. Soc. 87, 171 179 (1962). J. A. Davies and D. C. McKay, Estimating solar irradiance and components. Solar Enerqy 29, 55 64 (1982).