Articulo 1984

Solar Energy, Vol. 32, No. 4, pp. 523 535, 1984 00384)92X/84 $3.00 + .00 Printed in Great Britain. Pergamon Press Ltd.

ESTIMATION OF DAILY AND MONTHLY DIRECT, D IFFUSE AND GLOBAL SOLAR RADIATION FROM SUNSHINE

DURATION MEASUREMENTS

R. B. BENSON, M. V. PARIS, J. E. SHERRY and C. G. JUSTUS School of Geophysical Sciences, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

(Received 9 December 1981; accepted 4 April 1983)

Abstract--An accurate 200Wire 2. threshold pyreheliometer instrument for measuring the duration of bright sunshine has been used to derive daily and monthly regressions for direct, diffuse, and global solar radiation component vs sunshine duration. Daily regression for diffuse/global are linear in sunshine duration, while quadratic regression forms are employed for direct normal, direct horizontal, and global/extraterrestrial components. Only the daily direct normal component had regression values which depend on season while all of the monthly regressions depend on season. Linear regression relations for monthly direct normal, diffuse[global and global/extraterrestrial are employed, with a quadratic form being used for direct horizontal. Effects of rainfall, especially in overcast conditions, and of atmospheric turbidity and precipitable water, especially under clear-sky conditions, are observed and documented.

1. INTRODUCTION

(a) Diffuse and direct components from global radiation measurements

Thermal analyses for energy conservation or solar energy applications require knowledge of the solar radiation incident on buildings or collector surfaces which are typically inclined with respect to the horizontal. In the absence of solar radiation measurements on an ap- propriately tilted surface, this radiation may be estimated by one of several methods which usually require values of diffuse and direct components on a horizontal surface. Other applications, such as focusing or concen- trating solar collectors, require values for the direct- beam radiation, which, in the absence of direct beam measurements, may be estimated directly from the global component or indirectly from the difference between global and estimated diffuse components.

Results from methods for estimating hourly diffuse solar radiation from hourly observed global radiation by Liu and Jordan[l], Bues[2], Bugler[3] and Orgill and Hollands[4] have been analyzed recently by Spencer[5]. Erbs et a/.[6] have recently examined these and other methods for use with hourly, daily[7-11], and monthly [%14] radiation averages. Most of these methods involve a regression of the ratio of diffuse-to-global vs the ratio of global-to-extraterrestrial radiation. Some involve a regression of diffuse-to-global vs the ratio of global-to- clear-sky radiation.

Most of the methods for estimating direct-beam radiation from global radiation measurements first estimate the diffuse to global ratio and then derive the direct from the diffuse and global values[l, 7-11]. The method of Randall and Whitson[15] uses a statistical algorithm to estimate direct beam for measured (or cloud-estimated) global radiation, with the diffuse component being com- puted by differencing the global and direct components.

There are three advantages for estimation of diffuse or direct solar radiation components from sunshine duration rather than from measured global radiation. First, there are more sites available which measure sunshine dura-

tion than those which take quantitative global radiation measurements. Hence, unless actual on-site radiation data are available, it is more likely that a nearby sunshine-duration site could be available than a near-by solar radiation site. (Some caveats on the use of Foster sunshine switch data are presented later which should be observed when using near-by sunshine duration site data.) Secondly, the observed diffuse/global ratios vs measured global show considerable scatter (see Fig. 2 of Spencer[5]). Plots of diffuse/global ratio vs sunshine duration might actually exhibit less scatter than those versus measured global (see Fig. 5 of Erbs et al.[6]). Thirdly, estimates of direct-beam radiation from sunshine duration are also likely to be more accurate than those done indirectly by first estimating diffuse and then deriving direct from the global-diffuse difference. Better accuracy is also likely for estimates of direct beam from sunshine duration than for the estimates for direct vs

JUNE 79 RMS=I S. :) RV=' 9 . / RF 2~.~

moo L . . . . . . . . . . . . . . '~ ~J

w 60 + + ++ + ~ i

o ~- 50 . +,

~ 40 ~ + + -~ /+ + + + +

cc 20 + + +

[- - - / I

!

-I0 _ i .~ : IO [3 IrO 210 310 410 510 610 710 80 90 fO@ II r NiP ~SS

Fig. 1. Hourly average plot of CS vs NIP per cent Sunshine for June 1979, shows a 15.9 per cent r.m.s., which is typical of the

hourly average data over a month.

523

524 R.B. BENSON et al.

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; ,'o 2'o ~'o ;o 5'o o'o 7'o 8'o do 40 o NIP ZS5

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RMS=I8 . ~ AV=58.B RE= 30,92

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+ ++ + ++ ++ +

i

+ + 4++

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/

0 - /0 10 30 40 510

NIP %SS

60 ~0 8'0 90 10% IlO

Fig. 2. Daily average plot of CS vs NIP per cent sunshine for winter, 1981, shows almost a 50 per cent drop in r.m.s from hourly average to 8.5 per cent r.m.s., which is typical of the daily

average data over a season.

Fig. 4. Daily average plot of Foster vs NIP sunshine for Spring, 1979, shows an 18.2 per cent r.m.s, which is typical of the daily average data over a month. Bias error in the Foster data is

evident from comparison to the one-to-one regression line.

S

+--. co

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CE ( J

MONTHLY OS.VS.NIP

II13 i ~

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80

70

613

50

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,'o 2'o ~'o ,0 s'o ~'o 7'o NIP 255

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Fig. 3. Monthly average plot of CS vs. NIP per cent sunshine for the two years, April 1979 to March 1981, shows a reduction in r.m.s, by a factor of about two from the daily averages to 4.1 per

cent r.m.s.

global regression, such as the Randall-Whitson technique[15], which exhibit considerable scatter about the mean regression relation.

(b) Diffuse and direct components from cloud measurements

Many studies of solar radiation components vs cloud type have been limited to overcast-only conditions (e.g. Kasten and Czeplak[16]) or have examined only global radiation vs cloud amount (e.g. Flocas[17]). Other studies of cloud effects have not even attempted to relate solar radiation to cloud amount, but have limited them- selves to estimates of sunshine duration vs cloud amount (e.g. Reddy[18] and Barbaro et al.[19]).

MDNTNLY FOS.VS.N IP

110 l ~

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80

70

60

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+

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++ + , + ++ "~

+ +/ /

+ + ++ ~"

, / /

+ //

. /

20 30 40 510 610 710

NIP 7.55

[

J I

f

8(] 90 I00 I~O

Fig. 5. Monthly average plot of Foster vs. NIP per cent sunshine for the two years, April 1979 to March 1981, shows a 12.4 per cent r.m.s, deviation between Foster and NIP data. Bias error in the Foster data is evident from the comparison to the one-to-one

regression line.

In the SOLMET[20] derived data base, hour!y global horizontal radiation H is estimated from a non-linear regression relation with observed opaque cloud amount OPQ and precipitation indicator variable R(R = 0 or 1)

H = Hc x [Co+ c,)(OPQ)+ c2(OpQ)2+ c3(OPQ)3+ c4R], (1)

where Hc is the hourly clear-sky solar radiation on a horizontal surface, given as a function of zenith angle Z by the relation

H~=ao+alCOSZ+a2cos2Z+a3cos3Z. (2)

Estimation of daily and monthly direct, diffuse and global solar radiation

Tables of ao-a3 and Co-C4 are given in the SOLMET report for 26 sites in the old (pre-1977) NOAA/NWS solar radiation network, The ao-a3 coefficients have different values during morning and afternoon periods, and the ao values also vary with month. Values of direct normal radiation are not estimated directly from cloud cover data in the SOLMET derived data base, but are estimated from the Randalt-Whitson[15] direct-vs-global regressions by using the cloud-estimated global radiation.

An advantage to the use of sunshine duration instead of cloud cover for the estimation of radiation components is that sunshine duration measurements can more readily be automated. Cloud amount or cloud type data require the regular presence of a trained observer.

(c) Solar radiation components from sunshine duration measurements

One method first suggested by Kimball [21] and refined by ~,ngstrom[22] for estimating daily global horizontal radiation H is

= Hc[a + (1 - a)(~//Q)] (3)

where Hc is the daily clear-sky radiation and a is an empirically determined coefficient and a//q is the daily sunshine duration determined from ri, the observed daily minutes of sunshine, divided by N, the maximum possible daily minutes of sunshine for the date. Because/4c varies from site to site and varies with season and atmospheric transmission at a given site, it is usual, following Prescott[23], to use

=/4o[a + b(a//%] (4)

where Ho is the daily total extraterrestrial radiation on a horizontally oriented surface.

Both McQuigg and Decker[24] and Harris[25] have found that non-linear relations, such as

H = Ho(a + b(a/~)+ c(a//q) 21 (5)

or

H = Ho[a + b(alN) + c log (a//%] (6)

give slightly better results, especially at low values of ti//Q. Swartman and Oglunde[26] also introduced a relative humidity parameter into linear and non-linear regression of /-I vs a[/q. Whereas many investigators ([24, 27-32]) have found that the linear relation (4) works well, and explains roughly 60 to 90% of the variations in /4, Baker and Haines[34] observed that correlations were not uniformly high at 19 Alaskan and north central United States sites which they investigated. Among other possible causes for the observed discrepancies, Baker and Harris identified problems in that the times near sunrise and sunset in which direct radiation is too weak to activate the sunshine recorder is a variable which depends on atmospheric turbidity and water vapor con- tent, and that there may also be "undetected instrumen- tal or observational errors".

525

Coulson[35] and Schulze[36] have reviewed the applications of sunshine duration regressions to estimate solar radiation, especially the global component. While most work has emphasized the estimation of the daily global radiation component, some work has used sunshine duration to estimate hourly values [20] or monthly totals [37, 38]. The SOLMET hourly values for global radiation H are given by regression relations similar to those for opaque clouds [Eqn (1)]

H = H~[bo + b,(nlN)+ b2R] (7)

where (n/N) is the minutes of bright sunshine in the hour divided by the total number of possible minutes. Sears et a/.[39] have used sunshine duration to estimate the diffuse and direct component (daily total) as well as the global. Their relation for the direct component S, resolved onto a horizontal surface, is

= Ho[a + b(a//q)], (8)

while the diffuse component (D) regression is vs global /~, rather than versus the extraterrestrial/4o:

/5 =/-]'[a + b(~/N)]. (9)

(d) Methods for sunshine duration measurements There are three different types of instruments dis-

cussed in this paper, which are designed to measure sunshine duration. The Campbell-Stokes (CS) recorder (the World Meteorological Organization standard[40] in- "strumen0, the normal-incident pyrheliometer (NIP) and the Foster[41] Sunshine switch (the National Weather Services' standard instrument).

The CS recorder uses a glass ball as a focusing lens. A cardboard chart with special timing marks is inserted into a bowl under the ball. The cardboard chart burns when the direct radiation of the sun is sufficiently intense (above about 210 W/m2). If the relative humidity is high or rain has occured, the chart may become damp and more intense radiation is needed to burn it than usual. At any given thme the CS threshold for response can vary from as little as 70 W/m 2 to as much as 300 W/m 2 (WMO, [40]).

The Foster sunshine switch makes use of two selenium barrier layer photovoltaic cells. They are mounted so that one cell is blocked from the direct radiation of the sun, but both cells receive diffuse sky radiation. The diffuse signals are balanced so no electrical signal is generated if a strong beam is absent. In the presence of a strong direct beam, the cell not blocked from the sun triggers a relay and activates a recorder.

The NIP is composed of a multijunction thermopile attached to the base of a brass tube that is chromed on the outside and blackened on the inside. The NIP is airtight, which enables it to remain accurate in wet weather. For application as a sunshine duration meter, the NIP recording circuitry was designed to record only the duration of direct-beam radiation above a fixed threshold of 200 W/m 2.

The NIP sunshine measurement system requires

526 R. B. BENSON et al.

automatic tracking which must be checked regularly. This instrument should be considered a research instrument only. Both the Foster sunshine switch and the Campbell-Stokes instrument are of the non-tracking type, requiring only periodic adjustment to correct for changing solar declination angle. However, the need to change the Campbell-Stokes charts during each night time period represents a limitation on its use, especially for remote-site sunshine monitoring. Newer sunshine instruments which employ a single photocell with a rotary light chopper (which produces a modulating signal in the presence of bright sunshine) will not be addressed here.

2. ANALYSIS OF ERROR IN SUNSHINE MEASUREMENTS

To assess the relative accuracies of sunshine measurement techniques, a comparison was undertaken of sunshine duration measurements from a Foster sunshine switch [41] at one National Weather Service (NWS) site to measurements taken at two different instruments in the same city. Results of this comparison calls into question radiation estimates based on Foster sunshine switch data.

Measurements were taken at the Atlanta Airport with a Foster sunshine switch and at the Georgia Institute of Technology with an Eppley model NIP and a Campbell- Stokes (CS) recorder[40]. The CS and NIP data agree well, but the Foster sunshine switch gives consistently and significantly different results.

The NIP and CS sunshine recorders used in this study are located on the roof of the Civil Engineering Building of Georgia Tech. In comparing the data from these instruments, the NIP was used as the reference instrument since it is believed to be most accurate. The CS readings vary with humidity, and human judgement has to be taken into consideration when interpreting the burns on the chart.

Comparisons of the CS and NIP were made with three different data bases: hourly averages, daily averages, and monthly averages. The hourly averages were plotted monthly over the two year period between April 1979 and March 1981. The daily averages were plotted seasonally and yearly over the same two-year period. The monthly averages were plotted over the entire two years. (See Figs. 1-3). All data were analyzed in terms of per cent of possible sunshine for the hour, day or month. Differences in measurement by the three techniques are compared as r.m.s, differences in.per cent possible sunshine. For the hourly CS-vs-NIP comparisons (e.g. Fig. 1),

data points falling significantly below the line (i.e. low CS vs high NIP sunshine) were usually at low sun angles when turbidity, haze, or horizon clouds apparently affected the CS sunshine threshold. Data points falling significantly above the line (i.e. high CS values vs low NIP readings) usually resulted during partly cloudy (especially scattered cumulus) conditions when the CS burn strip appeared more continuous than the occurence of NIP readings above 200 W/m 2 should warrant.

For the hourly averages, there was 15.5 per cent average r.m.s, difference between the CS and NIP (see

Table 1. r.m.s. Values of Campbell-Stokes vs NIP using hourly averages

RMS FOR CS % SUNSHINE MONTH VS. NIP % SUNSHINE*

APR 79 11.9 MAY 79 18.3 JUN 79 15.9 JUL 79 19.3 AUG 79 17.4 SEP 79 16.0 OCT 79 14.0 NOV 79 11.3 DEC 79 9.9 JAN 80 15.4 FEB 80 8.8 MAR 80 13.8 APR 80 17.9 MAY 80 19.4 JUN 80 21.3 JUL 80 24.4 AUG 80 22.6 SEP 80 17.8 OCT 80 14.6 NOV 80 8.7 DEC 80 12.4 JAN 81 10.6 FEB 81 12.8 MAR 81 17.3

(15.5)

*To avoid large errors which occur at low sun angles, only hours with solar elevations above 15 were considered.

Table 1). The r.m.s, difference for the daily average dropped by almost one-half to 8.5 per cent average r.m.s. The monthly average r.m.s, difference of 4.1 per cent expresses about another 50 per cent drop in r.m.s, from the daily average r.m.s, differe'nce (see Table 2).

For the comparison of the Foster and the NIP, the only available data were daily and monthly averages of per cent sunshine. These averages were plotted in the same fashion as the CS-vs-NIP plots for the same data bases (see Figs. 4 and 5).

For the daily averages there was an 18.2 per cent average r.m.s, difference between the Foster and NIP. The monthly average plots revealed a 12.4 per cent r.m.s. between the two instruments (see Table 2).

Because of random errors, there should be a significant drop in r.m.s, difference from a shorter time-span data base to a longer time-span data base. The CS-vs-NIP comparison reveals this drop by decreasing by approximately a factor of 2 from one data base to another. The Foster vs-NIP plots do not reveal the same type of significant drop in r.m.s, from one data base to another. This behavior is indicative of a non-random bias error between the two instruments. For the data available, the Foster sunshine switch at Atlanta Airport appears to read consistently about 12 per cent higher, on average, than the CS and the NIP. The consistency between the NIP and CS readings would support the argument that it is the Foster readings which are high. This is also sup- ported by comparison with sunshine estimates from sunrise-to-sunset average cloud-cover readings at the Atlanta Airport (see Table 3).

Estimation of daily ahd monthly direct, diffuse and global solar radiation

Table 2. Comparison of seasonal and yearly daily averages and total average of the r.m.s, monthly averages of sunshine duration

527

DATA TIME SPAN

RMS FOR CAMPBELL-STOKES

% SUNSHINE VS. NIP % SUNSHINE.

RMS FOR FOSTER % SUNSHINE VS.

NIP % SUNSHINE

DAILY AVG. APR 79-JUN 79 9.0 18.2 DAILY AVG. JUL 79-SEP 79 7.8 21.6 DAILY AVG. OCT 79-DEC 79 7.9 12.4 DAILY AVG. JAN 80-MAR 80 10.2 14.7 DAILY AVG. APR 80-JUN 80 7.7 20.1 DAILY AVG. JUL 80-SEP 80 8.8 23.9 DAILY AVG. OCT 80-DEC 80 8.1 15.4 DAILY AVG. JAN 81-MAR 81 8.5 16.5 DAILY AVG. APR 79-MAR 80 8.8 17.1 DAILY AVG. APR 80-MAR 81 8.3 19.3 MONTHLY AVG. APR 79-MAR 81 4.1 12.4

Table 3~ Comparison of Monthly averages of available sunshine measured by three methods

Atlanta, Georgia Tech Site Atlanta Airport Site Observed Sunshine

Observed % Sunshine Observed Inferred % Sunshine (Campbell- % Sunshine from

(NIP) Stokes) Deviation (Foster) Deviation Cloud Cover Deviation

Apr 79 49 49 0 57 +8 37 -12 MAY 79 40 46 +6 53 +13 32 -8 JUN 79 49 48 -1 66 +17 39 -10 JUL 79 35 36 +1 51 +16 23 -12 AUG 79 51 51 0 68 +17 47 -4 SEP 79 33 30 -3 42 +9 27 -6 OCT 79 68 62 -6 74 +6 59 -9 NOV 79 58 57 -1 62 +4 49 -9 DEC 79 54 50 -4 59 +5 41 -13 JAN 80 29 29 -0 35 +6 19 -16 FEB 80 58 51 -7 66 +8 43 -15 MAR 80 44 38 -6 52 +8 25 -19 APR 80 53 54 +1 64 +11 45 -8 MAY 80 44 40 -4 60 +16 35 -9 JUN 80 53 47 -6 68 +15 47 -6 JUL 80 63 57 -6 83 +20 59 -4 AUG 80 55 50 -5 76 +21 53 -2 SEP 80 49 47 -2 66 +17 41 -8 OCT 80 58 56 -2 70 +12 51 -7

NOV 80 54 50 -4 60 +6 46 -8 DEC 80 56 57 +1 65 +9 48 -8 JAN 81 68 63 -5 78 +10 58 -8 FEB 81 51 48 -3 62 + 11 32 -19 MAR 81 64 58 -6 74 + 10 46 - 18

52 ~ 4 R---MS 63 12 RMS 42 11 RMS (-3) (+ 11) (- 10)

.In addition to consistent bias errors, the cloud cover and Foster sunshine data in Table 3 show revealing seasonal variation in their deviations from the NIP sunshine values. No such seasonal variations exist in the CS-NIP differences, again tending to confirm the accuracy of these instruments. The observed seasonal variation in the cloud-inferred sunshine estimates is easily explained: Not all clouds are completely opaque, hence there is a small negative bias. In the summertime, clouds are predominaltly cumulus type and opaque, with win- tertime clouds being more frequently composed of partially transparent cloud types. The sunshine estimates

based on average cloud, not average opaque cloud, would therefore tend to be most accurate in the summer and least accurate in the winter with a consistent negative bias, as observed.

The seasonal variations of the Foster sunshine deviations from the NIP per cent sunshine, shown in Table 3, indicates a possible source of error. If a sinple, seasonally-invariant threshold error is assumed for the Foster switch, its daily sunshine readings would be expected to be least accurate in the shorter winter days and most accurate in the longer summer days. In fact, the observed errors in the Foster data have just the opposite pattern,


with an average error of about + 9 per cent in the winter 1~9 1.0

months and about + 17 per cent in the summer months. In the time of sunshine duration, these represent about 1 hr 90 extra indicated sunshine in winter and about 2 hr extra ~:

f___ B0

indicated sunshine in summer. The observed Foster errors are consistent with readings relative to a higher threshold in = J 70 winter, at low sun angles, and to a lower threshold in summer, at high sun angles, with the threshold being lower ~ \ ~o than 200 Wire ~ (the NIP threshold used) for all times of the ~ so year. This behavior suggests a "cosine-correction" type of ~ ~0 error in the Foster measurements

From discussions with meteorologists at several air- -~ ~ port sites at which Foster sunshine switches are used, ~ ~o this level of inaccuracy is not surprising, and although the Atlanta instrument gave consistently high readings, ~o these meteorologists would not be surprised to find con- . oo sistently low Foster readings at other sites The errors are typically due to lack of quality control and calibration during regular operation Because of such in- consistencies in the Foster switch, the bo and b~ coefficients of the SOLMET[20] sunshine-regression equation (eqn 7), which were determined by use of a Foster switch, should not be used at sites other than the ones for which they were derived Seasonal discrepancies similar to those discussed above might also be expected from the SOLMET sunshine-regression equa- tions, even at the site for which the coefficients were derived

ta_

3. SUNSHINE ESTIMATES OF DAILY SOLAR RADIATION >_

Figures 6-11 present results of comparisons of daily solar radiation components with daily sunshine duration for the period April 1979-March 1982 for the Atlanta, Georgia Tech site. Sunshine duration is measured with the NIP system with a 200W/m: threshold and is expressed as per cent of daily possible sunshine The solar radiation data is divided into "summer" (April- September) and "winter" (January-March, October- December) periods to examine seasonal differences The

1979 1982 HINTER 8ERSON 8 : ,566E-O2R: .17~. SY : 069

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-10 0 10 ~0 30 40 50 60 70 80 90 lO0 110

DRILY NIP PERCENT SUNSHINE

Fig. 6. Daily global horizontal/horizontal extraterrestrial vs sunshine duration in per cent for the winter season at Atlanta, Georgia Tech site. Solid line is eqn (10). Dashed line is the Sears et a/.[39]

regression.

1981 SUMMER 5ERSON 8 : ,470E-OZQ= .276 8Y : .054

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+ 4."++. + + * + 4 -+++ +++

++"

+

- I0 0 I0 20 30 40 50 60 70 80 90 I00 I i0


Fig. 7. As in Fig. 6 for the summer season.

1979 1982 WINTER SERSON B=- .914E 02R= t . OO SY= .101 1 .0 ~ , . . . . . . . . . . /1" " " '1 . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . 1 . . . . . . . . . ] . . . . . . . . . I . . . . . . .

xN~ .911

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-10 O JO 20 30 40 50 60 70 80 90 IO0 l lO


Fig. 8. Daily diffuse horizontal/global horizontal vs sunshine duration in per cent for the winter season at Atlanta, Georgia Tech site. Solid line is eqn (11). Dashed line as the Sears et

al. [39] regression.

IX3 C3

C~

CE

I979 1981 SUMMER SEASON B= _763E-OSP: .952 SY= .082

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- I0 0 10 20 30 40 50 60 70 80 90 100 110



L~

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w CK

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1979 198~ WINIER SEASON

L~

T~ L] cE

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8= .54++E-Oelq=-.Ie2E 01 SY= .O~9 1.0 . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . u ' " " " l . . . . . . . . . I . . . . . . . . . I . . . . . . .

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-io o ~o 2o 3o 4o so 6o 70 8o 9o 1-oo ~io

DAILY NIP PERCENT SUNSHINE

Fig. 10. Daily direct normal/extraterrestrial normal vs sunshine duration in per cent for the winter season at Atlanta, Georgia

Tech site. Solid line is equation (12).

1979 1981 SUMMER SEflSON B= .LfOE~O2Q=-.I64E-OI SY= 1 , 0 ~H'" , , , I . . . . . . . . . ] . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . ~ . . . . . . . . . ~ . . . . . . . . . I . . . . . . . . . ] . . . . . . . . . ~HHIHT

.90

.80

.70

.60 +

++

+ + s0 +~++ .40 +

+

. DO . . . . . . I . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . t . . . . . . . . . I . . . . . . . . . I . . . . .

- I0 0 10 ~0 30 aO SO 60 70 80 90 100 110

DAILY NIP PERCENT SUNSHINE


044

daily total solar radiation components are normalized as global/extraterrestrial (horizontal), diffuse/global, and direct normal/extraterrestrial (normal).

Figures 6, 8 and 10 show that. winter radiation data consists predominantly of clear (near 100 per cent sunshine) or overcast (nearly 0 per cent sunshine) conditions, with relatively less partly cloudy (10 per cent ~i[/~


solid line in Figs. 8 and 9. Values of the seasonal average Sears et al. [39] regression results [eqn (9)] are plotted as the dashed lines [Average winter values are a = 1.00, b=-0 .86 ; average summer values are c= 1.07, b= - 0.961.

The restriction that direct beam radiation is zero at zero sunshine duration, and the observation of significant concave-upward curvature in Figs. 10 and 11 lead to the relation

S.lSo = co(filEt) + c,(a/Et) 2 (12)

for the daily total direct normal component S, relative to the extraterrestrial normal So. Equation (12) is plotted as the solid lines in Figs. 10 and 11, with winter values co=0.31, co=0.26 and summer values c0=0.26, c~ = 0.23.

A regression relation for daily direct horizontal radiation S may be derived from the relations for diffuse/) and global H, since S = H - D, or

S/#o = (~/~o) [1 -/5/_0]. (13)

Substitution of eqns (10) and (11) into (13) yields

SIflo = aobo(hlEt) + a,bo(hlEt) 2 + a2bo(hlEt) 3. (14)

The a and b values determined above yield coefficient values for (14) of aobo = 0.22, a,bo = 0.64, a2bo = -0.23, valid for both summer and winter seasons. This relation is plotted as the solid line in Fig. 12. The Sears et a/.[39] regression for S[Ho [eqn 8] are shown in Fig. 12 as the dashed line for winter and the dotted line for summer.

Figure 9 and Fig. 12 show unrealistic zero-intercept values for the Sears et al. diffuse and direct regressions in summer because at the site they studied (Davis, CA) clear skies (sunshine = 100 per cent) strongly dominate in "the summer season. Individual monthly regressions of Sears et al. had zero intercepts ranging from +0.31 to

CD i

E F-

t J CL

g

1.0

.90

.80

.70

.60 I

.50 i

.40

.30

.10

.00

' I

I ~ l I I I -~o o ~o ~o 3o 4o so 6o 7o 8o 90 ~oo tto

AVFI ILRBLE 5UNSI~ INE, %

Fig. 12. Daily direct horizontal/extraterrestrial horizontal vs sunshine duration in per cent. Solid line is from eqn (14). Dashed line is Sears et a/.[39] regression for winter; dotted line is Sears

et al. regression for summer.

-0.70 (instead of the necessary zero value) for direct horizontal/extraterrestrial and from 0.42 to 1.59 (instead of the necessary 1.00 value) for diffuse/global. If the summer regression results of Sears et al. are taken to apply only to large sunshine duration values (fi/N)~> 50 per cent), then good correspondence is found between the present results from Atlanta and the Sears et al. results from Davis, CA, especially when the wide variation in climate, turbidity, and atmospheric water vapor levels at these two sites are considered.

CE

L.J

Z: c~ ~J z

r--

c~

>-

c1_ c~

4. THE EIF~ECTS OF TURBIDITY AND PRECIP1TABLE WATER

Effects of aerosols, e.g. as measured by the base-e turbidity at 500 nm wavelength, and of water vapor, e.g. as measured by total column precipitable water, are expected in the solar radiation ratios, especially under clear-sky (sunshine near 100 per cent) conditions. Both direct beam and global ratios with respect to extraterrestrial should decrease with increasing turbidity or precipitable water, while the diffuse/global ratio should increase (because of increased scattering) with increasing turbidity. The turbidity effect on global radiation is expected to be smaller in magnitude than for direct and diffuse, because of the partially compensating effect of the opposite trends of the turbidity effect on the diffuse and direct components.

Figure 13 shows observed trends of clear sky (sunshine > 90 per cent solar radiation ratio with base-e 500 nm turbidity, measured by an automated direct beam sensor with 500 nm (-+ 10 nm bandwidth) filter, located at the Atlanta Georgia Tech site. The observed trends [downward for direct/ETR and global/ETR in Figs. 13(a) and 13(c), and upward for diffuse/global in Fig. 13b] are as expected. The slope of the least squares straight line fit to the global data in 13(c) is only about 1/3 the slope for the direct/ETR data in Fig. 13(a), as expected due to the compensating direct and diffuse effect on global radiation.

tO IBO 3 /82 SS > gO B : - .352E~O2A= .585 SY .039 1 .0 . . . . ~ . . . . I . . . . I . . . . i . . . . I . . . . I . . . . i ' ' '~

.90

.80 1

7o ~

60 E ~

1 5O A " ' - .o} o l

~ ~

,o~ 00 i l i I L l l l l l l I I I I I I I ' ' ] . . . . I . . . . i . . . . [ ,

0 5 I0 15 20 25 30 35 aO

5OO NM TURBID ITY . PERCENT

Fig. 13(a). The daily ratio of: (a) direct/extraterrestr ial normal. (b) diffuse/global, (c) global/extraterrestrial horizontal, (d) diffuse/direct normal vs base-e turbidity at 500 nm wavelength for clear skies (daily sunshine > 90 per cent). Solid lines show

least squares straight line fit.


to /8o - 382 B= .353E-021q: .791E-01 SY=

,30 I f . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . . I . . . .

o o

125 ]

g o

.80 . . -d

o o o "

t~_ o o o ~- .15 ~ o o

o o o o

.10 ~ o o o o O&o

. 050

.00

1

. . . . i . . . . i . . . . ~ . . . . i . . . . i . . . . i . . . . i . . . .

5 10 15 20 25 30 35 z~o

500 NM TUR~IDITY , PERCENT

10/80 - 3 /82 SS > 90

"90

N .80 I

.70 cc E ~ .60 E

~ .8 !

,~ .4@

5 .20

.10

034

Fig. 13(b).

.09

E E

E E , ,

D

8 . l t3E 020- .707 SY= .03t

I . . . . I . . . . ~ . . . . I . . . . t . . . . I . . . . I . . . . 1

x >1< x xx

x __y x ,..~x,x,,{ x x x x x,X~ ~- -~ , ;/~ ~ ,~ . . . . . . . .

, : . . . . L . . . . i . . . . i . . . . i . . . . ~ . . . . I ,h J ,~

5 10 I5 20 25 30 35 40

500 NM TURBID ITy , PERCENT

Fig. 13(c).

tO /BC 3 /82 B : .377E-02R= .104E Ot 8Y=

~ .20 !

oq / o , "

m L o "

>- .1o ~ o o o

E ooOO oo

~ ~ O~@o o o .058 ~- o O ~ O o o o o I o7 / o -

|

.8@ I , i , i r l l l ~ l . . . . J . . . . i . . . . i . . . . i . . . . I , , k ,

D

. o18

5 i0 15 20 25 30 35 aO

500 NH TURBID ITY , PERCENT

Fig. 13(d).

531

Recently Peterson and Dirmhirn[42] have noted that the ratio of diffuse horizontal to direct normal radiation remains essentially constant throughout the day, provided that the turbidity remains relatively uniform throughout the day and large zenith angles (> 75 degrees) are not considered[43, 44]. The results of Fig. 13(d) indicate that the daily diffuse/direct normal ratio is related to the daily average 500 nm turbidity. Hence the hourly diffuse/direct ratio would remain constant throughout a given day only if the turbidity remains relatively unchanged. Hourly values of diffuse/direct normal ratio vs effective broad band turbidity, shown in Fig. 14, confirm a close relationship between direct-beam turbidity and diffuse/direct ratio. Note that the four data points in Fig. 13(b) and 13(d) with diffuse/global values ~> 0.2 [or diffuse/direct ~> 0.13] are probably significantly affected by cloud cover (see Fig. 8).

The observed effect of precipitable water on the daily solar radiation ratios are shown in Fig. 15(a-c). Similar trends with precipitable water are seen as were observed for turbidity, although the global/extraterrestrial vs precipitable water slope (Fig. 15c) is almost ~' that for direct/extraterrestrial vs precipitable water (Fig. 15a), compared to about a 1/3 relationship for those slopes for turbidity effects.

If the effects of water vapor on solar radiation were limited to selective water vapor absorption only, then the water vapor effect in diffuse radiation would be significantly different than the aerosol effects on diffuse. Comparison of Figs. 13(b) and 15(b) indicate that such is not the case, however. Indeed the effect of non-selective absorption or scattering by hygroscopic aerosols [45, 46] are observed to occur in the water-vapor absorption "window" regions of the spectrum. As evidenced by Fig. 16, there is a considerable degi'ee of dependence of the 500nm (presumably aerosol-related) turbidity and the precipitable water.

The conclusions from these results are that, although one could use data from Fig. 13 to apply an aerosol

OCT-DEC 80 CLR SKY AM- I.--

.SO

.15

l-- . lO

.050

l ~ I l i

o ~o/~O o o

o o

o o o / oOO o

.08 I O0 . 050 .ll 0 .+l 8 .120 .188 . 30

DIFFUSE/DIRECT RATIO

Fig. 14. Hourly broad-band base-e turbitity vs diffuse/direct ratio for clear skies (100 per cent shine) and high solar elevation

(relative air mass < 2.0).


1.0

.90 c~ ~ .80

~E .70

C3 z .~0

u .50 LtJ

cz~ . ~ o

.30

g

1/80 - 12/80 B: .48t~E-OIA: .567 5Y=

+ ++ + + - - - _ _

'I , O0 I I I I I I I I I l l l I L l J l l l l l l I l ~ l l l r I r i i ,

O0 , 50 110 1,5 2 .0 2 .5 3 .0 3 .5

PRECZPITABLE WAFER, CH

Fig. 15(a). The daily ratio of:(a) direct/extraterrestrial normal, (b) diffuse/direct and (c) global/extraterrestrial horizontal, vs precipitable water for clear skies (daily sunshine > 90 per cent).

Solid lines show least squares straight line fit.

r j LJ Q:

?

L L~_

Q3

EE 0

t~/79 12180 SS > 90 B= .491E-OIR= .41BE-OI SY=

1 o .35 .30

.25 I

o .20 a

o o , - (3

.15 o ooo o

o 8

,0 o ~,~_~o "so o

" O0 . 0 I . I I 0 .501 . . . . i . ol . . . . I.sl . . . . 2. 0 I . . . . Z ~ * 5 . . . . 3~. " ` r I O

PRECIP ITABLE WATER, CM

Fig. 15(b).

1/80 l.O

.90

F-J ' .80

Q:

~- .7D QZ

L~ .60

--J

ISO Q~ 0 J r~ . z~O

.30

A 20

. i 0

.00

.047

3.5

- 12 /80 SS > 90

i . . . .

V

E P

r l , i i i , , , , i , i i , i i i . . . . I , .OO .50 1 .0 1 .5 2 .0 2 .5

PRECIP ITRSLE NATER, CH

8=- . 221E-O IR : .712 SY:

I . . . . I . . . . I . . . . g . . . . I . . . . ] . . . .

.Dt~l

3 .0 3 .5

Fig. 15(c).

.035

OCT-DEC 80 CLR SKY RM < 2 B: .169 A= .173E-01 SY=

.50 I I I I I

.35 >- }--

.30 173

.25

I--- .20

o .15 o Co

. tO

.OSO

~x A

a

a a a

a

.080

.DO I I I I I

.00 .50 1 .0 1 .5 2 .0 2 .5 3 .0

PRECIP ITRBLE WATER, CM

Fig. 16. Hourly 500nm turbidity vs precipitable water for clear skies (100 per cent sunshine) near solar noon (relative air mass

< 2.0).

correction to the observed regression, and one could use the data from Fig. 15 to apply a water vapor correction, the interdependence of aerosol and water vapor effects, probably means that both of these corrections should not be applied linearly. Further amplification of this must await a more complete multiple regression analysis of the radiative components against sunshine, turbidity and water vapor variables.

s. StmSmNE K~rIMATES OF MONTHLY RAn/AT/ON

Available sunshine amount can effectively be used to estimate the monthly average solar radiation component, as evidenced by the 'regression results in Fig. 17(a-c). As has been found in some regression studies of diffuse/global via global/extraterrestrial horizontal[I,6, 12], there is a seasonal dependence on the monthly

F~ W

LO CS

t~

4 /79

60

55

50

aS

4D

35

30

25

20

.15

.10

.050

.00

3/B2 B: .5SIC 02R: .275E-01 BY:

i , ....... t ......... I ......... I ......... I ......... I ......... h,l,,,,,, I ......... I ......... I ........ 0 10 20 30 z~O 5O 60 70 BO 90 100

AVAILABLE SUNSHINE, %

.028

Fig. 17(a). The monthly ratio of: (a) direct/extraterrestrial normal, (b) diffuse/global, (c) global/extraterrestrial horizontal vs monthly available sunshine. Solid line = winter season; dashed

line = summer season.

c~ 0

I L

c3c

LLI

J CE

Z 0

cc 0

c~

J

4/79

1.0

.90

.80

.70

.60

.80

.z.O

.30

,20

. tO

.DO

4/79

l .O

.90

.80

70

50

40

30

80

. tO

.OO


3/82 8=- . 694E-028- .766 SY= .077

4

"" \ 1 ".A. \

? ?

. . . . . . . . I . . . . . . . . . I . . . . . . . . . [ . . . . . . . . . I . . . . . . . . . I . . . . . . . . . l . . . . . . . . . I . . . . . . . . . I . . . . . . . . . J . . . . . . . ~ 10 20 30 z~O 50 60 70 80 90 100

IqVA I LRBLE SUNSHINE, Z

Fig. 17(b).

3/82 B= ,561E-028= .210 SY= .024

. . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . .

i : o ~ i ' - " ' "

*- o I~ d

. . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . J . . . . . . . . . L . . . . . . .

10 20 30 40 SO 60 70 80 90 100

~tVRI LRBLE SUNSHINE, Z

Fig. 17(c).

533

normal (S,) vs monthly available sunshine 0i)/(N) are expressed by

(g.)l(go) = ao(a)l(/q) (15)

with values ao=0.545 for winter and ao = 0.445 for summer.

The monthly average diffuse/global regression in Fig. 17(b) are expressed by

(D)I(H) = bo + b,(R)l(IQ) (16)

with values bo=0.66, b1=-0.61 for winter and bo = 0.88, bl =-0 .80 for siammer. These values apply only between the limits of approximately 0.2 ~< (~)/(/~)~< 0.7, since the physical constraint (/3)1(/~)= 1 at (ri)/(/~)= 0 should apply (i.e. the more complete relation must be non-linear in such a way as to approach (D)/(H)= 1 as (ti)l(/V) approaches 0).

The monthly average global regression in Fig. 17(c) are expressed by

(H)l(l-lo) = Co + c,(~)l(l~) (17)

with values co = 0.18, c, = 0.60 for winter and Co = 0.24, cl = 0.53 for summer.

The monthly average direct radiation on a horizontal surface (S), relative to the extraterrestrial horizontal (Ho) can be found by a combination of eqns (16) and (17)

(g ) / (#o) = ( f l ) l ( f I o ) - (D)I(Ho)

= ( (#) / (#) ] [1 - (D) / (#) ]

= do + d,(a)l(bl) + d2[(ti)/(/Q)] 2 (18)

where do = c(1 - bo), = 0.06 (winter) = 0.03 (summer), d, = c1(1 - bo)- cob, = 0.31 (winter) = 0.26 (summer) and d2 = blc, = 0.36 (winter)= 0.42 (summer). These results are plotted as the solid (winter) and dashed (summer) lines in Fig. 18, along with the regression of lqbal[38],

regressions vs sunshine. Summer [April-September] regression values (dashed line) for direct/extraterrestrial are lower than winter [January-March; October- December] values (Solid line) in Fig. 17(a). These seasonal effects on diffuse/global are reversed for direct beam regressions because of the opposite physical effects on direct beam and on diffuse processes. A partial compensation effect on the global component is seen in Fig. 17(c), in that the seasonal difference is much smaller for the global component than for the direct or diffuse components.

The monthly global regression results for Atlanta used by Lof[28, 35], shown as the dotted line in Fig. 17(c), appear to overestimate the global radiation at low sunshine and underestimate it at high sunshine values. The monthly diffuse regression results of Iqbal[38], shown as the dotted line in Fig. 17(b), agree fairly well with the present summer results but not the winter values.

The regression lines for monthly average direct normal radiation (S,) relative to monthly average extraterrestrial

1.O

.90

N .80 C/:_ C3 : I .70

Q:. q.~ .6[J

~-- .511

~ .4 [ ) C~

>~ 38

~ .88

I E

.18

I ~ I I I I ~ ! I J - T - ,

4i !

>t

/ ] .DO i i, I i I ~ - -

-10 0 t8 88 310 410 58 610 710 818 90 180 118

RVRI LFIBLE SUNSHINE,

Fig. 18. Monthly ratio of direct/extraterrestrial horizontal derived from present regressions (winter = solid line, summer = dashed

line) and that of Iqbal[38] (dotted line)


who derived values given by the dotted line in this figure. The results of Iqbal are seen to be realistic only between the approximate limits of 20-60 per cent sunshine, where they agree quite well with the present results. Negative values, which result for sunshine values below about 15 per cent and decreasing values with increasing sunshine, as results for sunshine values above 65 per cent are both unrealistic features of the Iqbal regression.

6. CONCLUSIONS

Based on the preceding discussion, the following conclusions are offered:

(I) Because of calibration and maintenance problems significant errors may exist in sunshine duration values measured with the Foster sunshine switch, and solar radiation regression derived from this instrument should be applied, if at all, only at the station for which the coefficients were derived.

(2) Problems of underestimation at low sun angles and overestimation during periods of bright sunlight between thick (e.g. cumulus) clouds means that errors of about -+15 per cent exist in Campbell-Stokes measured hourly sunshine duration. Errors are reduced to about + 8 per cent daily average sunshine and -+ 4 per cent for monthly average sunshine measured by the Campbell- Stokes instrument.

(3) When the daily stlnshine duration (ti//~) is accurately measured, relations (10)-(14) may be used to provide reasonable estimates of the daily global (/~'),. diffuse (/)), direct normal (S,) and direct horizontal (S) components relative to the daily extraterrestrial direct- beam (So) or extraterrestrial horizontal (Ho).

(4) Rainfall conditions significantly decrease the daily global radiation at zero sunshine, below an already somewhat decreased overcast value relative to partly cloudy values (eqn 10 with ao ~ ao and ao dependent on rainfall).

(5) Aerosol effects, as measured by base-e 500nm turbidity coefficient, decrease the direct and global component at given sunshine values and increase the diffuse component. Water vapor, as measured by the column precipitable water, has a similar effect, although the aerosol and water vapor effects are not independent, perhaps indicating an important influence of hygroscopic aerosol scattering process.

(6) When the monthly average sunshine duration ((fi)/(N)) is accurately measured, relations (15)-(18) may be used to provide reasonable estimates of the monthly average global ((/~)), diffuse ((/3)), direct normal ((/~'o)) and direct horizontal ((So)) components of solar radiation, relative to monthly average extraterrestrial normal (So) or extraterrestrial horizontal ((Ho)).

(7) The regression values presented here provide some improvements in the physical consistency and reason- ableness of regression values over those which have been proposed by others who have examined daily or monthly sunshine regression relations.

Acknowledgments--Support for the data collection and analysis of these results was provided by the DOE Solar Energy Meterological Research and Training Site program under DOE grant DE-FGO5-77ET20153.

/-/, fl, (fl)

no, ~0,

Estimation of daily and monthly direct, diffuse and global solar radiation 535

17. A. A. Flocas, Estimation and prediction of global solar radiation over Greece. Solar Energy 24, 63 (1980).

18. S. J. Reddy, An empirical method for estimating sunshine from total cloud amount. Solar Energy 15, 281 (1974).

19. S. Barbaro, G. Cannata, S. Coppolino, C. Leone and E. Sinagra, Correlation between relative sunshine and state of the sky. Solar Energy 26, 537 (1981).

20. SOLMET, Volume 2-Final Report. Hourly solar radiation surface meteorological observations, TD-9724 (1979).

21. H. H. Kimball, Variations in the total and luminous solar radiation with geographical position in the United States. Mon. Weather Rev. 47, 769 (1979).

22. A. K. ,~ngstron, Solar and terrestrial radiation. Quart, J. R. Meterol. Soc. 50, 121 (1924).

23. J. A. Prescott, Evaporation from a water surface in relation to solar radiation. Trans. R. Soc. So. Augst. 64, 114 (1940).

24. J. D. McQuigg and W. L. Decker, Solar energy-A summary of records at Columbia, Missouri. Mort. Agr. Exp. Sta. Res. Bull. 671 (1958).

25. A. R. Harris, Solar Radiation Reception and Its Correlation with Sunshine. Masters Thesis, Soil Sci. Dept., Univ. of Minnesota, Minneapolis (1966).

26. R. K. Swartman and O. Oglunde, Solar radiation estimates from common parameters. Solar Energy 2, 107 (1967).

27. N. J. Rosenberg, Solar energy and sunshine in Nebraska. Neb. Agr. Exp. Sta. Bull. No. 213 (1964).

28. G. O. G. Lof, J. A. Duffle and C. O. Smith, World distribution of solar radiation. Solar Energy 10, 27 (1966).

29. J. N. Black, C. W. Bonython and J. A. Prescott, Solar radiation and the duration of sunshine. Quart. J. R. Meteorol. Soc. 80, 231 (1954).

30. I. Bennett, Correlation of daily insolation with daily total sky cover, opaque sky cover and percentage of possible sunshine. Solar Energy 12, 391 (1969).

31. S. J. Reddy, An empirical method for the estimation of total solar radiation. Solar Energy 13, 289 (1971).

32. S. Barbaro, S. Coppolino, C. Leone and E. Sinagra, Global solar radiation in Italy. Solar Energy 20, 431 (1978).

33. V. Modi and S. P. Sukhatme, Estimation of daily total and diffuse insolation in India From Weather data. Solar Energy 22, 407 (1979).

34. D. C. Baker and D. A. Haines, Solar radiation and sunshine duration relationships in the north-central region and Alaska. Tech. Bull 262, Agr. Exp. Sta., Univ. of Minnesota, Min- neapolis (1969).

35. K. L. Coulson, Solar and Terrestrial Radiation Methods and Measurements. Academic Press, New York (1975).

36. R. C. Schulze, A physically based method of estimating solar radiation from suncards. Agr. Meteor. 16, 85 (1976).

37. G. O. G. Lof, J. A. Duffle and C. O. Smith, World distribution of solar radiation, Rep. No. 21, Engr. Expt. Sta., Madison (1966).

38. M. Iqbal, Correlation of average diffuse and beam radiation with hour of bright sunshine. Solar Energy 27, 169 (1979).

39. R. D. Sears, R. G. Flocchini and J. L. Hatfield, Correlation of total, diffuse and direct solar radiation with the percentage of possible sunshine for Davis, California. Solar Energy 27, 357 (1981).

40. World Meteorological Organization, Measurement of radiation and sunshine. Guide to meteorological instrument and observing practices. 2nd edn, WMO 8, TP3 (1965).

41. N. B. Foster and L. W. Foskett, A photoelectric sunshine recorder. Bull. Amer. Meteorol. Soc. 34, 212 (1953).

42. W. A. Stevenson and I. Dirmhirn, The ratio of diffuse to direct solar irradiance (perpendicular to the sun's rays) with clear skies: A conserved quantity throughout the day. J. Appl. Meteorol. 20, 826 (1981).

43. M. R. Weber and C. B. Baker, Comments on The ratio of diffuse to direct solar irradiance (perpendicular to the sun's rays) with clear skies: A conserved quantity throughout the day. J. Appl. Meteorol. 21, 883 (1982).

44. T. A. Cerni, Comments on The ratio of diffuse to direct solar irradiance (perpendicular to the sun's rays) with clear skies: A conserved quantity throughout the day. J. Appl. Meteorol. 21, 886 (1982).

45. R. S. Fraser, Degree of interdependence among atmospheric optical techniques in spectral bands between 0.36 and 2.4~m. J. Appl. Meteorol. 14, 1187 (1975).

46. J. R. Humeml, Anamolous water-vapor absorption: Im- plications for radiative-convective models. J. Atmos. Sci. 39, 879 (1982).

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