Estimation of solar radiation from Australian meteorological observations

Solar Energy Vol. 47, No. 3, pp. 209-218, 1991 0038-092X/91 $3.00 + .00 Printed in the U.S.A. Copyright © 1991 Pergamon Press plc

ESTIMATION OF SOLAR RADIATION FROM AUSTRALIAN METEOROLOGICAL OBSERVATIONS

W. W. MORIARTY Moriarty Meteorological Services, Sunbury, Victoria 3429, Australia

Abstract--A carefully prepared set of Australian radiation and meteorological data was used to develop a system for estimating hourly or instantaneous broad band direct, diffuse and global radiation from meteorological observations. For clear sky conditions relationships developed elsewhere were adapted to Australian data. For cloudy conditions the clouds were divided into two groups, high clouds and opaque (middle and low) clouds, and corrections were made to compensate for the bias due to reporting practices for almost clear and almost overcast skies. Careful consideration was given to the decrease of visible sky toward the horizon caused by the vertical extent of opaque clouds. Equations relating cloud and other meteorological observations to the direct and diffuse radiation contained four unknown quantities, functions of cloud amount and of solar elevation, which were estimated from the data. These were the proportions of incident solar radiation passed on as direct and as diffuse radiation by high clouds, and as diffuse radiation by opaque clouds, and a factor to describe the elevation dependance of the fraction of sky not obscured by opaque clouds. When the resulting relationships were used to estimate global, direct and diffuse radiation on a horizontal surface, the results were good, especially for global radiation. Some discrepancies between estimates and measurements of diffuse and direct radiation were probably due to erroneously high measurements of diffuse radiation.

1. I N T R O D U C T I O N

Information about the climatology of broad band solar radiation is required for a range of purposes, and for many (e.g. building design and air conditioning, surface water and energy budgets and snow melt calculations in hilly terrain, solar power installations with tilted or moving collectors) a division of the global radiation into direct (or beam) and diffuse components is needed.

Sometimes there are measurements available close enough to the site of interest to provide the necessary information. However, in general the existing measuring instruments even for global radiation are too widely spaced to capture the variations. Major causes of the spatial variability are the variation of cloudiness typically associated with varying topography, and the variations of aerosol type and concentration associated especially with distance from major industrial centres. Measurements of diffuse and/or beam radiation are even more widely spaced. Thus many workers wishing to describe the climatology of solar radiation have tried to develop techniques for making estimations [ 1]. Be- cause of their wide availability, methods using surface meteorological observations have received much at- tention [ 2-5 ].

In Australia there has been a comparatively good network of radiation observations maintained over many years[6 ], but there is still a call for radiation information in places where there have been few or no measurements. Starting with the available radiation data, a method applicable anywhere Jn Australia has been developed for estimating hourly or instantaneous broad band radiation, direct, diffuse, and global, from routine surface meteorological observations. The main emphasis of the project was on the effects of clouds, which were treated by using regression techniques to fit functions of cloud cover to observed radiation values, hut as a first step estimation techniques for clear

sky Australian conditions were derived. This paper de- scribes the project, and gives some results from using the method.

2. DATA

The radiation data came from five years of measurements ( 1981 to 1985) from twelve stations maintained by the Australian Bureau of Meteorology (Fig. 1 ). Each of the stations measured global and diffuse solar radiation received on a horizontal surface, using Epply pyronometers for the global measurements and similar instruments with occulting discs attached for the diffuse measurements. The data used were half- hourly totals archived in units of 10 kJ m -2, beginning at midnight local apparent time. They had been subjected to quality controls and corrections, including a correction to the diffuse measurement (+3%) to allow for the amount of sky obscured by the occulting disc. Possibilities of error arose from the instruments them- selves, and also from the calculations and correction procedures applied to the raw data. A recent internal Bureau study (P. Novotny) of a randomly selected set of the observations has estimated that the RMS errors for half-hourly totals of global and diffuse radiation may be as high as 8%-10% and 10%-12%, respectively, close to mid-day, and 10%-15% and 12%-17% respectively for low solar elevations, provided the instruments have been well maintained. Occasional fail- ures of maintenance procedures, and in particular fail- ure to keep the instruments level or the occulting discs aligned, may have resulted in larger errors.

For the present study these data were further edited by removing all occasions with either global or diffuse radiation measurements missing, and some other occasions with obviously suspect observations. These were cases with values of direct radiation (found by

209

210 W.W. MORIARTY

~f ILLIS ISLAND

; ,ALICE S RP INGS XLON(3REACH~-~ CKHAMPTON

• ~ for data set A(Basic set) MT. (3AMBIER

o Stations for data set B(Temperate zone comparison set) 0 x Stations for data set (3(Tropical zone comparison set) ~HC""~IOBART

Fig. 1. Locations of the Australian radiation measuring stations used in this study.

subtracting the observed diffuse from the observed global radiation) less than - 2 units, and clear sky occasions when a large exceeder of the measured diffuse radiation over a rough estimate suggested that the occulting disc was misaligned. (Large exceeders, greater than 2.5 standard deviations, greatly outnumbered large deficits.) When either type of error occurred on a number of successive days, or with only a few inter- vening days, persisting instrumental faults were sus- pected, and all the observations in the period concerned were deleted.

There was a possibility that the shadows of trees or buildings or hills could interfere with the radiation measurements close to sunrise or sunset, so the observations for solar elevations less than 5 ° were deleted from the data set as well. (This also removed any half- hours which included time before sunrise or after sunset.)

Surface meteorological observations at the stations involved in this study are made at 0300 hours local time and at three hourly intervals through the day. As local apparent time is used in measuring radiation there is no synchronisation between the two sets of observations. For the present study, half-hourly radiation totals at each station were selected for those half-hours which included the times of the meteorological observations. It was assumed that the meteorological ele- ments changed sufficiently slowly for their observed values to be representative of the whole half-hour of the radiation measurement (the "radiation half- hour")[7]. Four radiation half-hours per day were chosen, those corresponding to the meteorological observations for 0900, 1200, 1500, and 1800 hours.

The surface observations used were those routinely

observed, quality checked, and recorded in the archives of the Australian Bureau of Meteorology. Observations of surface pressure, temperature, dew point, visibility, total cloud cover, total low cloud, middle cloud, and high cloud cover were used. If any of these observations were unavailable, the occasion in question was deleted from the data set.

The data set finally obtained and used contained 7461 observations with clear skies, and 52,605 observations all told.

3. DIRECT RADIATION WITH CLEAR SKIES

When skies are clear the broad band transmissivity of the atmosphere to solar radiation may be expressed approximately as

r = r a r r r o r g r w ( 1 )

where ra, rr, ro, rg, and rw are, respectively, the partial transmissivities due to aerosols, Raleigh scattering, ozone, uniformly mixed gases, and water vapour.

For this study expressions for %, %, and rw given by Bird and Hulstrom[ 8,9 ] and quoted by Iqbal (his model C)[1 ] were used. An average value of 3 cm was adopted throughout for ozone thickness. For ra, rr, and the precipitable water (w), needed in the calculation of r~, expressions from the literature were adapted to Australian conditions using the basic data set. For ra the initial expression was one developed by Machler and quoted by Iqbal and based on the horizontal visibility (V), which was the only relevant pa- rameter routinely measured in Australia. For r, the initial expression was one given by Bird and Hulstrom,

Solar radiation from Australian

and for w it was a modification of an expression based on surface dew point in degrees C (Ta) given by Atwater and BaU[10]. The functional forms of the initial expressions were assumed correct, and the numerical constants were recalculated successively, in the order r~, w, r~, to give the best fit to the Australian data. (SAS procedures were used for fitting throughout this project[11].) The initial expression for ra was valid only for visibilities down to 5 km, but the revised one for Australian conditions was useable for visibilities down to 1 kin.

The expressions finally adopted for r~, r~, and w were

r , = 0.917(0.993 - 2.265 g-l'558)mb

where

mb = m a 1"632 (20 km ~< V ~< 180 km) (2a)

and m~ is the optical mass of the atmosphere.

r . = 0.917(0.959 - 0.347 V-L°26)~C

where

me= m~ °81 (1 km~< V < 2 0 k m ) (2b)

r, = exp[-0.086m°824(1.0 + m~ - m~°°6)] (2c)

w = exp(0.1 Td)" ( P / 1 0 1 3 . 2 5 ) ° 7 5 ( 2 7 3 / T ) °'5 (2d)

p and T being station level pressure and temperature. This expression for w included a correction to the water vapour path length to allow for the temperature and pressure dependance of water vapour absorptivity [ 12 ]. The station level temperature and pressure were used to approximate the correction because the distribution of water vapour with height was in general not known, and because the use of surface dew point to estimate w had already implied the assumption that most of the precipitable water was in the lowest layers.

When the revised model, as against the original one, was used to estimate r for the basic clear sky data set, the standard deviation of the error was reduced from 0.050 to 0.040, and the mean error from 0.076 to 0.001. The mean value of r was 0.632.

4. DIFFUSE RADIATION WITH CLEAR SKIES

In this study the diffuse radiation on a horizontal surface and its components (and also the global and direct radiation) were expressed as relative values, that is the absolute values divided by the radiation on a horizontal surface at the top of the atmosphere.

The diffuse radiation under clear skies (D~) is the sum of diffuse radiation due to Rayleigh scattering (Dr), diffuse radiation due to scattering by aerosol particles (D~), and diffuse radiation due to multiple reflections between the earth and atmosphere (Din). In general the expressions which have been given for D,, Da, and

meteorological observations 211

D,, depend on the partial transmissivities of the atmosphere, and also on certain physical parameters whose values may vary with the prevailing conditions, viz. the absorbed fraction (/7,) of the radiation incident on aerosol particles, the downward scattered fraction (Fc) of aerosol scattered radiation, and the albedo (Ag) of the ground. Three models were tried, all of which assumed isotropy, to see which would best fit average Australian conditions. They were fitted to the observations by adjusting the values of the above mentioned physical parameters, and also allowing for the possibility that multiple scattering might cause the downward fraction (Fr) of the Rayleigh scattered radiation to differ from the upward.

The best model was found to be one given by Iqbal ( model C) [ 1 ] based on the considerations of Bird and Huistrom[ 9 ]. F~ was taken to be quadratic in the solar elevation (×), following the conclusions of Robin- son[13]. It was found that Fa = 0.08, Ag = 0.223, and F~ = 0.456. The final form of the model is given below:

Dc = (Dr + D ~ + 0 .223rA, ) / (1 - 0.223A,) (3)

where

D, = 0.79rorgrr. O.456ra,(1 - r r ) / ( l - m ~ + m ~ "02)

(3a)

Da = 0.79rorgrr 'Fc(ra~ - r a ) / ( l - m a + m ~ °2)

(3b)

A, = 0.0685 + (1 - Fc) . ( l - r J % a ) (3c)

ra, = 1 - 0 . 0 8 . ( 1 - m a + m ~ ° 6 ) - ( 1 - r a ) (3d)

F~ = 0.895 - 1.059 sin x + 0.528 sin2X. (3e)

The expression found for Fc implied a decrease with increasing solar elevation. This strange seeming result was checked by finding estimates of F~ for a wide variety of data sets and diffuse radiation models. The tests indicated that the result was not due to an accident associated with the particular data set used; to the editing of the data set; to any effects varying with locality in Australia; to the modification of partial transmissivities for Australian conditions; to the use of relative as against absolute radiation values; nor to the particular diffuse radiation model used. However, the possibility remained that it might reflect an inadequacy in the solar elevation dependence of the expressions used for ra, or (less likely) rw or r,. In this case Fc would have to be regarded as including a correction factor. On the other hand, the effect might reflect some kind of diurnal change in the average scattering prop- erties of aerosol particles, perhaps associated with changing relative humidity.

When the modified model was applied to estimating relative diffuse radiation for the original data set, there was a reduction in the magnitude of the mean error (as against the results for Iqbal's model C using Aus- tralian partial transmissivities but otherwise unmodi- fied) from -0.020 to 0.000; and a reduction in the

212 W.W. MORIAR/Y

standard deviation of the error from 0.0194 to 0.0191. The mean value o f De was 0.097.

5. DIRECT RADIATION WITH CLOUDY SKIES

In general the direct radiation is affected by low and middle clouds which block it out, and by high clouds which diminish it. In a given case it is not possible (unless the sky is clear or overcast) to tell from the cloud observations whether the sunlight is blocked, reduced, or arrives unaffected by clouds. However, it is possible to calculate the average effect of a given cloudiness condition if it is assumed that the average distribution of clear and cloudy patches across the sky is symmetrical about the zenith (but see [14]).

Before considering such a calculation, it was necessary to deal with two difficulties arising from Aus- tralian observing practice. One was that the reported cover of middle and high clouds is not the cover visible to an observer on the ground, but the estimated cover which would be visible in the absence of lower clouds. There is the relationship T,. = ( L + M + H ) , where To, L, M, and H are, respectively, the fractions of sky obscured by all clouds and by visible low, middle and high clouds, but if low, middle, and high cloud layers are all present only T c and L are given, and it is not possible to find values for M and H. However, if only two of these layers are reported, the reported cover by the lower one is the visible cover, and the visible cover by the higher one may be deduced, e.g., if there is no middle cloud, H = Tc - L. In finding relationships between radiation and cloud cover, then, only occasions with two or less of these cloud layers were used. There were 48,623 such occasions in the basic data set.

The second difficulty arose from the convention for reporting almost clear and almost overcast skies. All cases of cloud cover between 0 and ~ are reported as

(cloud cover is reported in okras); and all cases of cover between 7 and ~ are reported as 7. Thus the figures

and 7 are not true averages of the cloud cover for the cases they represent. The effect of this with respect to opaque clouds was examined first, by taking all cases with zero reported high cloud and finding the average relative direct radiation for each value of opaque cloud cover; and then calculating corrected values for ~ and 7 so as to make all the averages fall on or near to a smooth curve. It was found that ~ should be replaced by 0. l, and 7 by 0.936. These substitutions were made throughout the data set, with the amount of high cloud cover being modified appropriately. A similar procedure for high clouds, taking cases with no opaque cloud, suggested that no modification was needed to the reported figures.

When cloud cover is observed from the ground, the vertical extent of low and middle level clouds causes the proportion of sky they obscure to increase toward the horizon. Hence, the proportion of sky available for patches of clear sky and visible high clouds decreases toward the horizon. (This matter is considered in more detail in a companion paper [ 15 ]). When cloud reports are used to deduce the amount of sky given over to

clear and high cloud patches, the result is an all sky average, which must therefore be modified by an elevation dependant factor when the probability of sunlight reaching the ground is being considered. We call this factor g. We suppose also that the proportion of direct radiation transmitted by high cloud (n) is de- pendent on solar elevation.

The plan was to find typical or average values ofg and n by using the observations and an equation relating averages of direct radiation to cloud cover. This would require grouping of the observations according to X, L, M, and H, and finding appropriate radiation averages for each group. However, with such a division of the available observations many groups would be too small to have reasonably stable averages. Middle and low clouds have similar effects in blocking the direct solar beam, and roughly similar effects in causing an increase in the fraction of sky obscured toward the horizon, so it was decided to consider their cover to- gether as the "opaque cloud cover" L,,,(Lm = L + M ) .

If we consider all the observations with the same X, H, and Lm, and take averages, we find

Average [Ih/~-] =g{1 - ( 1 - n ) H - L m } (4)

where Ih is the relative direct radiation on a horizontal surface,

In (4) the factor g would need to be l when Lm = 1, since the vertical extent of clouds cannot modify the fraction of visible clear sky towards the horizon when the sky is overcast. Similar reasoning shows that g = 1 when Lm = 0. After some trials with the data a function was posited for g of the form

g = 1 + aLto + b L ~ + c L ~ - (a + b + c ) L 4 (5)

a, b, and c being functions of X to be found from the data.

Examination of the data for cases with no opaque clouds suggested that n, too, was a function of cloud cover as well as of solar elevation. This was consistent with the observation that there is a tendency for high clouds to be thicker when there is a larger expanse of them, and this observation also suggested that total high cloud cover (Ht) would be relevant rather than visible high clouds. After some trials with the data for zero opaque clouds, the functional form assumed for n was

n = d + eHt + f H 2 (6)

where d, e, and f were functions of x to be fitted. The data were grouped according to X, H, and L,,,

and the averages of the left hand side (LHS) of (4), and also o f l i t (for use in determining n), were found for each group. For purposes of this grouping ten ranges of solar elevation were used, the first covering 5 degrees and the remainder 10 degrees. They were centred on 7.5 °, 10 °, 15 ° , 20 °, 30 °, 40 °, 50 °, 60 °, 70 °, and 80 ° . The coefficients in the functions g and n were then fitted for each solar elevation range separately, and the

Solar radiation from Australian meteorological observations 213

values found for the coefficients were fitted to polynomials in ×. The results were

a = 2.97 - 16.55X + 41.92X 2

- 38.60X 3 + 11.85x 4 (7a)

b = -31 .3 + 202.49( - 468.0x 2

+ 418.8x 3 - 127.1x 4 (7b)

c = 58.1 - 402.6x + 934.3x 2

- 834.4x 3 + 252.7x 4 (7c)

d = 0.91 - 0.13x (7d)

e = -2 .97 + 10.52x - 11.00x 2 + 3.47x 3 (7e)

f = 2.71 - 12.249( + 13.52× 2 - 4.38× 3 (7f)

with × in radians. Sometimes during a radiation half-hour with a re-

ported opaque cloud cover of 88-, there might be breaks in the cloud which allowed the sun to shine through briefly (though not at the t ime of the cloud observa- t ion). There would not then be the expected zero observation of direct radiation. Thus, (4) could be used for cases of overcast opaque cloud only when considering instantaneous radiation values. For half-hourly (or hourly) radiation totals the RHS of (4) was replaced by a polynomial in × chosen to fit the observed average values of the LHS for the different elevation ranges, and (4) became

Average [Ih/r] = 0.126 -- 0.591X

+ 1.155X 2 -- 0.946X 3 + 0.275X 4

Similar considerations regarding seasonal and latitude effects applied to g. Also, the g values for cloud cover 0.1 and 0.9 and low solar elevation were higher than expected. This was probably due to the lack of any elevation dependence in the corrections applied to the i and 7 cloud cover values.

6. DIFFUSE RADIATION WITH CLOUDY SKIES

For reasons similar to those discussed in relation to direct radiation, we treat middle and low clouds as a single layer (opaque cloud). Also, we neglect the average effects of anisotropies in the diffuse radiation (but see [ 4,16,17 ] ). Suppose that the proportions of the total incident radiation (on a horizontal surface) which are passed on as diffuse radiation by high and opaque clouds are h and m respectively. The factors h and m will be functions of solar elevation and perhaps of cloud amount (due to reflection effects and relationships between cloud amount and cloud thickness).

The diffuse radiation at the ground consists of diffuse radiation from clear sky patches, from high clouds, from opaque clouds, and from multiple reflections. In the case of the opaque cloud layer, the incident radiation consists of direct and diffuse radiation from clear sky patches, and from high clouds. The relative diffuse radiation on a horizontal surface at the ground is then approximately

D = [ h H + {1 - Ht(1 - n - h ) } - m L m

+ (Da + Dr ) ' (1 - T c ) + l h R ] / ( l - R ) (9)

where

with X in radians R = 0.223[0.2H + 0.7Lm + As(1 - To)] (9a)

(for Lm = 1 ). (8)

If, then, it was assumed that for a particular solar elevation and cloudiness condit ion ( I j z ) could be approximated by (Th/z) , eqns ( 4 ) - ( 8 ) could be used to estimate Ih, the half-hourly (or hourly) relative direct radiation on a horizontal surface. To estimate instantaneous values of direct radiation ( 4 ) - ( 7 ) could be used, with (4) instead o f (8) applying to the case of Lm = 1. In either case the values found would be averages for all occasions with the given solar elevation and cloudiness.

Figure 2 shows the variations of g with elevation above the horizon, and of n with solar elevation, for various amounts of the relevant cloud cover. It was found that n did not increase monotonically as solar elevation increased from 0 ° to 90 °. This would be because the values found for n were averages for all sea- sons and for many Australian locations, with corresponding variations in the thickness and type of the predominant high clouds. The values o fn for the higher solar elevations would reflect an increasing influence o f summer t ime and low latitude cloud types.

and where the direct radiation incident on the opaque clouds has been taken as an average for the given Ht and Lm. In the reflection factor R we have taken the albedos of high and opaque clouds to be 0.2 and 0.712,18], and 0.223 is the albedo of the ground for average Australian conditions. As, Da, and Dr, the clear sky albedo, diffuse radiation due to aerosols, and diffuse radiation due to Rayleigh scattering, are given by (3c), (3b), and (3a).

Rearranging, averaging, and using (4), we find

Average [( 1 - R ) D - (Da + Dr). ( 1 - To)

- ~ - g R { l - ( 1 - n ) H - L m } ]

= [ h n + mLm{l - / 4 , ( 1 - n - h)}] . (lO)

In this equation, the averages should be taken over sets of observations grouped according to X, H, Lm, and Ht. In the current study it was found that such sets would be too small to have stable averages, so the observations were grouped simply by ×, H, and Lm. When applying (10) a mean value of Ht for each observation group was used. Since n also depended on kit this im-

214 W.W. MORIARTY

1.6 ' ' ' ' ' (J

1o4] , - ~ ~ "

.... y ... --:_"_2

| .4 Lrn:O'1-- . . . . | / / Lrr, : 0"3 . . . . . . . n,,~-] / / t m - - 0 " 5 - -

" ' 1 / / Lr~ : 0"7 . . . . . 0.J / Lm:o.9 . . . . . . . .

0 - 2 0

ELEVATION (X)

(a)

0"3 -

~0 .2.

0.1-

r r l

\ \

o lbO 2bo 3'0 o t,'0o B? 6? 70 ° 80 ° SOLAR ELEVATION x

(c)

0.9

0.6.

0.5.

¢

0-3-

n

/ ~ . ~'~

/

0,1"

SOLAR ELEVATION x

(b)

0"/,1 = , , I i a

o.~C_ " ................... h

. . . . . . .

0-I \ ...... " f - ~

ol o ° Ib ° io ° 3'0 ° 4b ° s'o ° 6'0 ° 7'o ° 8'0 ° 9

SOLAR ELEVATION x (d)

Fig. 2. The parameters g, n, m, and h, derived from the data, shown as functions of elevation above the horizon for various values of cloud cover. The legend in (a) indicating the amounts of cloud cover applies also to the other diagrams. In (a) and (c) it refers to the amount of opaque cloud cover, as indicated, but

in (b) to the visible amount of high cloud, and in (d) to the total amount of high cloud.

plied that, for given x, H , and L,, , Ht was nearly constant.

In ( 1 0 ) observed values were used for D , and the basic data set was used to find the best fitting values for h and m. Both parameters were assumed to be functions o f × , h to be a linear function of Hi, and m to be a quadratic function o f L,, . The coefficients in the expressions for h and m were first found for each range o f x, and then these coefficients were fitted to polynomials in ×. The resulting expressions were (with X in radians)

h = ho + h~" Ht ( 1 1 )

where

ho = 0.39 - 1 .195x + 1 .146x 2 - 0 . 3 1 5 x 3 (1 la )

h~ = - 0 . 1 1 6 + 1 . 3 3 6 x - 1 .322x2 + 0 . 3 6 2 x 3 ( l i b )

m = mo + m,L,,, + m2L2m ( 1 2 )

where

mo = 0.18 + 0.49X - 2.01X 2 + 2.07X 3 - 0.67X 4

( 1 2 a )

m~ = - 0 . 0 6 + 1.50X + 1.41X 2 - 3.50X 3 + 1.47X 4

( 1 2 b )

m2 = 0.03 - 2.19X + 1.40X 2 + 0.69X 3 - 0.58X 4.

( 1 2 c )

If, then, it was assumed that Ht and the clear sky estimates for ro and rw (and hence z, Do, Dr, and R ) were roughly constant for given X, H , and Lm, ( 1 0 ) - ( 12 ) could beused to find an estimate o f D, the relative diffuse radiation at the ground. The value found would be an average for all cases with the given solar elevation and sky condit ions.

Solar radiation from Australian meteorological observations 215

Figures 2 (c) and (d) show the variation of h and m with solar elevation for various values of cloud cover. For high solar elevations there were again seasonal and latitudinal effects.

7. THE ACCURACY OF THE RADIATION

ESTIMATION TECHNIQUE

The methods derived in the foregoing sections could be used for estimating direct and diffuse radiation on a horizontal surface when there were two or less layers of clouds (out of low, middle and high layers) present. Global radiation could be estimated by adding the estimates for direct and diffuse radiation. The results were extended to cases with three layers of clouds by assum- ing that the difference between total and low cloud cover was equally divided between visible middle and high clouds, and then making corrections if the resulting estimates of visible cover for either layer exceeded the layer's reported total cover. Table 1 shows that this led to reasonably good results.

The accuracy of the overall estimation procedure was tested against three data sets. The first was the basic data set, including cases with three cloud layers (data set A). The second and third were independant sets containing observations for five times daily (0600, 0900, 1200, 1500, 1800 hours) during the years 1983 to 1986 from three temperate zone stations (data set B) and four tropical stations (data set G). The locations of all the observing stations are shown in Fig. 1. Data set B had observations of both global and diffuse radiation, but data set G had only global measurements. Neither was subjected to the special editing procedures used for the basic set. To facilitate the examination of variations with solar elevation the observations in each data set were formed into the solar elevation groups described in Section 5 and recombined to form a modified data set.

Mean errors in the estimated radiation values (ex- ceedance of the measured over the estimated values) were then calculated with the data sets subdivided in various ways. The proportion of absolute errors greater than 0.1 MJ m -2 in the half-hourly totals of diffuse radiation was also calculated, as giving an indication of the usefulness of the results for some applications [ 19 ]. For global and direct radiation this proportion would not be meaningful, as there would be many large errors (positive and negative) due to the technique's inability to distinguish cases when the direct solar beam penetrated through gaps in the clouds from cases when it was blocked. Tables 2 and 3 give the results by observing station and by solar elevation group.

The mean errors in relative global radiation were quite small for all the data sets. There were no stations in the basic data set more than about 2 ° inside the tropics, so the results for the tropical data set (G) were surprisingly good.

The mean errors for estimated relative diffuse (and direct) radiation were quite small in data set A, though slightly larger than for global radiation. For data set B (unedited) the mean errors in relative diffuse radiation for all observing stations and all solar elevations were positive, corresponding to measurements larger than the estimates, while the errors in direct radiation were mostly negative. Since the global estimates, formed by adding the direct and diffuse estimates, were better than either of them, it seems likely that some of the discrepancies in the diffuse radiation (and hence in the direct) were a result of erroneously high observed values. This could be due to an equatorward tilt of the instrument, misalignment of the shade disc, and/or some other instrumental fault. This suggestion is sup- ported by a finding of the Bureau study of the radiation data mentioned in Section 2, that for Canberra the daily totals of diffuse radiation exceeded the global totals on over 10% of days.

Table I. Mean errors (observation-estimation) in estimated relative radiation on a horizontal surface for cases in the basic data set (A) with three cloud layers and with two or less cloud layers, by elevation group. The mean errors are expressed as percentages of the corresponding mean relative radiation. The relative

radiation is the radiation at the ground expressed as a proportion of the radiation at the top of the atmosphere

S o l a r elevation Global radn. Direct radn. Diffuse radn. (degrees) 2 lyrs. 3 lyrs. 2 lyrs. 3 lyrs. 2 lyrs. 3 lyrs.

.................................................................

5-10 1.8 2.4 2.6 3.9 1.0 1.7 5-15 1.5 2.8 1.8 3.3 1.2 2.5

10-20 1.9 0.2 2.4 -3.3 1.2 3.0 15-25 2.2 -3.3 3.4 -12.7 0.4 4.5 25-35 1.7 -3.5 2.3 -6.6 0.2 0.i 35-45 1.9 -1.8 3.5 -2.6 -1.9 -0.6 45-55 1.3 -0.8 3.2 -0.6 -4.2 -1.2 55-65 0.4 -1.5 1.8 -4.4 -3.5 2.7 65-75 0.3 -4.2 1.4 -9.2 -2.9 3.7 75-85 0.9 1.8 2.6 i.I -5.0 3.0

All 1.5 -1.3 2.7 -4.0 -0.9 1.6

216 W. W. MORIARTY

Table 2. Errors (observation-estimation) in estimated radiation on a horizontal surface by observing station, for the basic data set (data set A ), a smaller independant data set of three temperate zone stations (data set B), and another independant set of four tropical stations (data set G). The table shows the percentage mean errors (%ME) in relative radiation, and for diffuse radiation the percentage frequency of large absolute errors (%LAE), i.e. errors greater than 0.1 MJ m -2, in the estimated half-hourly totals of radiation. The relative

radiation is the radiation at the ground expressed as a proportion of the radiation at the top of the atmosphere

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Observing No. of Global radn. Direct radn. Diffuse radn. station obs. Rel. radn. Rel. radn. Rel. I/2 hr.

% ME % ME radn. total %ME %LAE

.................................................................

Data set A. Albany 3771 -i.0 -7.4 5.6 23.8 Guildford 5982 -0.8 -1.5 0.8 15.5 Kalgoorlie 4851 2.2 4.9 - 4.6 19.6 Geraldton 5104 4.4 4.1 5.1 18.4 Port Hedland 5006 3.6 5.9 - 4.6 15.0 Mt. Gambler 4821 4.9 5.5 4.2 24.0 Alice Springs 5726 -0.3 3.0 -11.3 15.7 Hobart 5072 -0.6 1.6 - 3.4 19.7 Laverton 4877 2.0 2.7 I.I 24.9 Mildura 5461 0.i -i.0 2.5 17.7 Wagga Wagga 5412 1.4 1.4 1.5 21.1 Rockhampton 4353 -0.9 3.6 -I0.i 33.4 All stations 60436 1.3 2.3 - 0 . 7 20.3

Data set B. Adelaide 3458 5.5 5.9 5.0 24.2 Brisbane 3485 -2.1 -4.3 i.i 22.3 Canberra 1479 2.7 -7.1 20.0 29.6 All stations 8422 2.0 -0.6 6.3 24.4

Data set G. Hall's Creek 7508 2.7 Darwin 7818 1.7 Longreach 5707 2.0 Willis Island 6584 4.5 All stations 27617 2.7

.................................................................

8. CONCLUSION

A system developed for estimating hourly or instantaneous ground level global, direct, or diffuse solar radiation from Australian surface meteorological observations was found to give quite good results. The estimates of global radiation especially were found to be good, for both the original data set and two unedited comparison sets. The estimates for diffuse and direct radiation were also found to be good for the original data set. For the comparison sets the estimates did not agree so well with the measurements, but the nature of the differences suggested that they were due to erroneously large measurements of diffuse radiation.

It should be noted, however, that these good results were estimates of average radiation for given solar elevation and cloudiness conditions. The method cannot distinguish between cases when the sun shines through gaps in the clouds and when the direct radiation is blocked out. Hence on individual occasions there might be quite large errors in the estimates, particularly of direct and global radiation.

The development of the system involved deriving four unknown quantities in the theoretical relationships between radiation and cloud amount. These quantities were functions of solar elevation and of cloud amount,

and their determination involved grouping the observations according to solar elevation and the amount of the different cloud types, and then finding radiation averages for each group. To ensure that there were enough observations in each group to give stable averages, it was necessary to treat low and middle level clouds as a single type, "opaque cloud." About 500,000 observations would be needed to overcome this re- striction. Ideally, each group should have been subdivided further, so that each sub-group had the same amount of total high clouds, and similar partial transmissivities due to aerosols and water vapour. To allow this several million observations would be needed. In view of the uncertainties in the system, especially those due to the use of single cloud cover figures to sum- marise the effects of the many kinds of low, middle, and high clouds, it is doubtful whether such further refinements are warranted.

The convention for reporting cloud cover with almost clear and almost overcast skies gave rise to a . problem in using the data. It was necessary to find more representative averages to replace the reported opaque cloud cover values of ~ and I, and this was done using average values of measured direct radiation. However, the corrected values did not allow for any variation with solar elevation, and this was the probable

Solar radiation from Australian meteorological observations

Table 3. Errors (observation-estimation) in estimated radiation on a horizontal surface by solar elevation group, for the same data sets as in Table 2. The table shows the percentage mean errors (%ME) in relative

radiation, and for diffuse radiation the percentage frequency of large absolute errors (%LAE) in the estimated half-hourly totals, in the same way as Table 2.

Solar No. of Global radn. Direct radn. Diffuse radn. elevation obs. Rel. radn. Rel.radn. Rel. i/2 hr. group % ME % ME radn. total (degrees) %ME %LAE

Data set A. 5-I0 1894 1.8 2.7 i.i 0.2 5-15 4190 1.7 1.9 1.4 0.8

1 0 - 2 0 6 0 4 5 1 . 7 2 . 0 1 . 4 3 . 7 15-25 8668 1.8 2.6 0.7 8.3 25-35 12664 1.4 1.9 0.2 19.3 35-45 9668 1.6 3.1 -1.8 27.4 45-55 7260 i.I 2.9 -3,8 31.5 55-65 5168 0.2 1.4 -2.8 38.8 65-75 3005 0.0 0.9 -2.3 38.7 75-85 1874 1.0 2.5 -4.0 41.1

Data set 8. 5-10 378 2.2 -4. i 5.8 0.8 5-15 574 0.7 -6.3 5.5 I .6

10-20 550 1.1 -6.5 8.5 5.5 15-25 1140 0.7 -6.4 9.4 10.8 25-35 1931 i.i -1.6 6.0 21.3 35-45 1397 0.5 -2.2 5.3 31.8 45-55 791 1.4 -0.6 4.8 39.2 55-65 874 5.6 5.8 5.0 39.1 6 5 - 7 5 5 4 2 6 . 6 5 . 9 8 . 3 4 7 . 0 7 5 - 8 5 2 4 5 2 . 2 2 . 6 1 . 4 4 9 . 8

Data set G. 5-10 2337 6.6 5-15 3458 3.7

10-20 1831 -4.0 15-25 1652 I.I 25-35 3473 3.6 35-45 4440 3.4 45-55 5062 3.0 55-65 2053 1.4 65-75 1288 0.9 75-85 2023 1.9

217

cause of irregularities in some of the funct ions derived from the data. A new calculat ion of the correction, making al lowance for the change of effective cloud cover with elevat ion above the horizon, might result in an improved per formance of the overall system.

Acknowledgments.--The work on which this paper is based began while the author was working in the Australian Bureau of Meteorology, and was completed in cooperation with the Bureau. The author profited from various discussions with his former colleagues, especially Ms. C. Skinner and Messrs. P. Novotny, R. Atkinson, and F. Callus. A reviewer also made some helpful comments.

NOMENCLATURE

Ag Albedo of the ground As Albedo of the cloudless atmosphere

a, b, c Coefficients in the expression for g D Relative diffuse radiation on a horizontal surface

Dc Clear sky relative diffuse radiation on a horizontal surface

Do, Din, Dr Clear sky relative diffuse radiation due respectively to aerosol, multiple, and Rayleigh scattering

d, e, f Coefficients in the expression for n

Fo Fraction of incident radiation absorbed by aerosol particles

Fc Downward fraction of aerosol scattered radiation

F, Downward fraction of Rayleigh scattered radiation

g Factor giving elevation dependence of sky cover by clear and high cloud patches

H Fraction of sky visibly covered by high clouds H, Fraction of sky covered by total high clouds

h Proportion of incident radiation passed downward as diffuse radiation by high clouds

ho, h~ Coefficients in the expression for h lh Relative direct radiation on a horizontal surface L Fraction of sky covered by low clouds

L,, Fraction of sky covered by opaque (low and middle) clouds

M Fraction of sky visibly covered by middle clouds m Proportion of incident radiation passed down-

ward as diffuse radiation by opaque clouds mo, m~, m2 Coefficients in the expression for rn

ma Optical mass of the atmosphere rnb = rn~rn; mc = m~ "°sl

n Fraction of direct radiation transmitted by high clouds

p Station level pressure T Surface temperature

Tc Fraction of sky covered by all clouds

218 W.W. MORIARTY

Ta Surface dew point, in degrees C V Horizontal visibility in km w Precipitable water in cm x Solar elevation z Transmissivity of the atmosphere to solar ra-

diation Partial transmissivities of the atmosphere due respectively to aerosols, absorption by aerosols, Rayleigh scattering, ozone, uniformly mixed gases, and water vapour

Ta , Taa~ Tr~

TO, Tg~ T w

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Estimation of solar radiation from Australian meteorological observations