As mentioned in section 4.1, if the spectral averages of some of the physical properties of the atmosphere can be defined instantaneously as given in Eq. (5.8), then the only requirement is to consider the time variation of these quantities. Consequently, an expression of the form as Eq. (5.9) can be utilized. Instead of Go, extraterrestrial instantaneous solar irradiation outside the atmosphere on a perpendicular surface, I0n can be used. Then, for the direct component of the solar radiation on a horizontal surface at the bottom of the atmosphere can be written as;
I = I0n¥Cosz. (5.15)
In this expression z is the zenith angle and ¥ is a term that accounts various absorption and/or transmission mechanisms. Monteith (1962) for example used ¥ =( 1 – ф’)( 1 – p’) for the direct component where ф’ stand for Rayleigh, Fowle, and dust scattering while p’ is for the absorption coefficient of water vapor and dust. In his model, Monteith used the fractional cloudiness c of two monthly mean of daily observations.
One can start with the fractional cloud amount c for the cloudy sky as discussed by Davies et al. (1984), and then taking into account the multiple reflection cycles the solar irradiation can be written as:
G = G0(1 – c + Tcc)(1 – aft)1 (5.16)
where Tc is the cloud transmittance a is the ground albedo and в is a total atmospheric backscattering coefficient. G0 is theoretical cloudless sky global irradiation. In this expression 1c is for the solar irradiation on the surface coming from the cloudless part of the sky while Tcc is for the portion coming after transmitted by the cloudy part. Such approach of course introduce a new question on the time integration of Eq. (5.16) as one also needs the time interval that the sky has the fractional cloud amount c.
Instead of c one can also start with bright sunshine fraction within an infinitesimally small time interval щ (Akinoglu 1992b). In such a consideration, infinitesimally small means a small enough time interval so that numerical integration on time gives acceptable approximate results for the average parameters. Firstly, by considering the direct component Id which comes during the bright sunshine period within the specified time interval as:
Id = І0ЩТ. (5.17)
I0 in this equation is the extraterrestrial solar irradiance at the site of interest on a horizontal surface above the atmosphere coming within the specified time interval and t is transmittance of the atmosphere during the clearsky period, an average value for all wavelengths. Diffuse component during the same bright sunshine period can be written as:
Id1 = І0Щ(1 – т)в’ (5.18)
where в’ is the atmospheric forward scattering coefficient. Finally, diffuse component during cloudy sky period is:
Id2 = І0(1 – щ)ттс. (5.19)
In this expression, t is also included as the sun rays must pass through the whole atmosphere also in the presence of clouds (Akinoglu 1992b). Including the first reflection cycle between the ground and the atmosphere, namely the first term in the binomial series, total global solar irradiation on the surface may be written as:
І = І0ІПіТ + щ( 1 – т)в’ + (1 – щ)TTc]( 1 + а в) (5.20)
where ав is for the irradiation reflected back by the atmosphere due to all its components. Note that this expression now contains three diffuse parts, two coming from Eqs. (5.18) and (5.19) and one coming because the first multiple reflection cycle between the ground and the atmosphere is considered. The third component can be deduced from Eq. (5.20) as:
Id3 = І0 [пт + Пі(1 – т)в + (1 – Пі) TTc] ав. (5.21)
As defined above, в is the total atmospheric backscattering coefficient and can be defined with two components: backscattering from the clear atmosphere with a coefficient aR, and from the cloud base with the coefficient ac. This consideration yields:
в = ацп + ас(1 – пі) (5.22)
similar to that explained by Davies and McKay (1982), but they used the cloud amount c. An aerosol scattering term may be introduced as written in the article of Davies and McKay (1982), however this would not change the nature of the relationship that will be obtained below, between the global solar irradiation and bright sunshine hours. This form is quadratic in ni, and given as:
I = Іо[щт + Пі(1 – т)в + (1 – ni)TTc][1 + а(ацПі + ас (1 – пі))] (5.23)
Assuming that the form of the equations does not change after daily integrations and monthly averaging, analog equation for the monthly mean daily values can be written. But then of course all the parameters should be replaced by their monthly effective counterparts. Also ni must be replaced with the monthly average values of n/N. Thus one can obtain a quadratic relation as:
H = H0 [(n/N)(Te + (1 – ге)в) + ( 1 – n/N) TeTce]
x[1 + ae(aRen/N + ace(1 – n/N))] (5.24)
where index e stands to indicate that the parameters are monthly effective parameters. Last equation has a quadratic form as:
H П (П2
H0 = + », n + n) (5’25)
where the coefficients a0, a1 and a2 can be written in terms of the effective parameters in Eq. (5.24). As these parameters have some physical interpretations, so we may conclude that a0, a1 and a2 and the quadratic relation (5.25) have physical base as they are written in terms of them. It should be noted that, although these effective parameters are newly defined, approximate values of some of them can be obtained from the literature. Some of them can be left free to be calculated for each month as a0, a1 and a2, by using the measured diffuse and total component of the monthly average daily solar irradiation at any site on the Earth surface (Akinoglu 1992b). Expressions for the three coefficients in terms of the parameters in Eq. (5.24) are:
a0 — Te Tce (1 + ae ace )
a1 — Te( 1 + aeace )(1 fte) + fte( 1 + aeace) + TeTce(aReae 2aeace 1)
a2 — Teae(aRe ace)(1 fte)+ fie ae(aRe ace) TeTceae(aRe ace) (5.26)
For some further description of the nature of a possible quadratic relation between solar irradiation and bright sunshine hours here several results on the use of Eqs. (5.24) and (5.25) will be given. Some of these results already appeared in the literature (Akinoglu 1992b; Akinoglu 1993).
The value of ground albedo a varies from 0.1 for forest and grass up to 0.7 for fresh snow (Davies et al. 1984). However, for semiurban and cultivation site 0.2 is a rational value as measured by Ineichen et al. (1990). Although a seasonal variation exists, Ineichen et al. concluded that a unique average ground albedo can give satisfactory results in the calculation of the ground reflected radiation. Cloud reflectance depends on the type, height and amount of clouds but an average value can be assigned as proposed by Fritz (1949), the value he obtained was 0.5. Later, Houghton (1954) and Monteith (1962) determined the same value using different approaches. Using the results obtained by Houghton (1954) a value of around 0.250.40 can be derived for the average forward scattering coefficient of the clear atmosphere (Houghton 1954). Finally, for aR, a value of 0.0685 can be used as Davies and McKay (1982) which was proposed by Lacis and Hansen (1974). In the highlight of these values, one can assign and change within an appropriate interval the values of the effective parameters in Eq. (5.24), namely ae, aec, aRe and в’є – The parameters Te and Tec can be left free to be calculated for each month.
Using measured monthly averages of daily global and diffuse irradiation together with monthly average of bright sunshine hours, monthly values of the coefficients a0, a1 and a2 of the quadratic correlation Eq. (5.25) and the monthly values of the parameters Te and Tec are calculated for four stations (Table 5.1) from different latitudes. Table 5.1 also includes the climate types of the locations as given in Trewartha (1968). The data of three of these stations are those used in the work of Jain (1990) who constructed a similar formalism to write the global solar irradiation in the linear form and applied it to find the Angstrom coefficients a and b of these three locations. This work of Jain is summarized at the end of this section. Linear form of the outlined formalism above (that is, if the first reflection cycle is not accounted) was applied to a location in Turkey before (Akinoglu 1992b) and the results for this station will also be presented here.
Table 5.2 gives the calculated monthly values of the parameters Te and Tec and the coefficients a0, a1 and a2 of the quadratic correlation. In the calculations, 0.2 is used as the ground reflectance, 0.35 is used as the forward scattering coefficient and 0.5 is used for the cloud albedo. The values of the ground albedo varied between the
Table 5.1 Some characteristics of the locations

Table 5.2 Values of the parameters Te and Tec, and the coefficients a0, a1 and a2 Bulawayo Salisbury

values 0.154 and 0.220, which are reflecting the range for the semiurban and cultivation sites (Ineichen et al. 1990), to determine the validity of the assumption for the value 0.2 assigned for this parameter. No considerable changes in the calculated monthly values of the parameters and the coefficients were observed with the values 0.154 and 0.220. However use of extreme values for this parameter affects the results especially the second and third coefficient. One can state that inclusion of first multiple reflection cycle effect in the models may be an indication of natural quadratic relation between the solar irradiation and bright sunshine hours or at least this may be one of the reasons of a slight curvature observed by Ogelman et al. (1984). It may be one of the causes for relatively better performances of quadratic correlations (Akinoglu and Ecevit 1990a, 1990b, 1993; Tasdemiroglu and Sever 1989; Badescu 1999) or the correlation of Rietveld who expressed the Angstrom coefficients as a function of n/N (Rietveld 1978).
In Table 5.2, mean values of the parameters Te and Tec and the coefficients a0, a1 and a2 are also presented in the last rows. The monthly values of the parameters do not have large deviations for a location which may be thought of as the typical effective value, different for the locations with different climate type and latitude as can be observed from Table 5.2. Two locations, Salisbury and Bulawayo, which are having close latitudes and also similar climates seem to give rather close values for the parameters and also for the coefficients. In fact, climate type is a quite valuable starting point for the accurate solar radiation estimations. A recent study of the present author showed that climate type alone can be used to estimate the annual profile of monthly average daily global solar irradiation on horizontal surface with acceptable accuracy, without any measured input data, at least for the locations in USA (Akinoglu 2004).
Variation of the coefficients with respect to n/N is given inFig. 5.1, for two locations one from south latitudes and the other north and with different climates to scan a wider range of n/N values. The coefficients a0, a1 and a2 seem to span relatively smaller range of values compared to the wide ranges of values of the Angstrom coefficients. However, the formalism should be applied to different locations from all over the world to reach any further conclusions about the physical base of quadratic correlation and to talk about the universal superiority of it over linear relation.
In fact, a linear form for the relation between the monthly average solar irradiation and bright sunshine hours can be obtained if the multiple reflection effect is not accounted in the above outlined formalism (Akinoglu 1993). Then, the Angstrom coefficient of the linearform are obtained as: a = TeTec, b = Te(1 — Tec )+ве(1 — Te) and a+b = Te + Д(1 — Te) which seem indeed appropriate for the physical meanings attributed to these coefficients by many authors.
Jain (1990) used a similar formalism to obtain a linear relation. He has written Angstrom coefficients as a = and b = (ym + am — j3m) where subscript m stands to indicate the monthly average, Дп is similar to Te defined above and ym and am are the monthly average transmittance of atmosphere for the diffuse component during
U 
■ 
■ 

0.6 
■ 
■ ■ ■ ■ ■ 
1 
■■ 
■ 
■ ■ ■ □ 

■ 

0.4 
ao 

ф 
X* ♦ 
♦# ♦ 
♦♦♦ 
♦ 
t 
al 

0.2 
♦4 
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0.8 
0.0 
0.3 
ДА a ▲▲ 0.5 
АД Д A A li ‘ 0.7 0.9 
0.2
n/N
Fig. 5.1 Variations of the coefficients with n/N for Bulawayo and Macereta
cloudy and clear times, respectively. Both formalisms can be used to write various relations between the direct, diffuse and global solar irradiation and the bright sunshine hours. Such relationships were given for his formalism by Jain (1990) and only for the diffuse component by Akinoglu (1993). For example, the coefficients a’ and b’ in the linear relation between the monthly average diffuse component and bright sunshine hours: D/H0 = a’ + b’n/N, can be obtained. Both in the formalisms of Jain and in the above outlined formalism but without multiple reflection effect (Akinoglu 1993; Jain 1990), coefficient a of Angstrom equation equals to a’. Table 5.3 gives the coefficients a, b and b’ obtained by Jain’s formalism and for comparison mean values of those derived monthly from the above formalism without multiple reflection effect is also presented.
Ogelman et al. (1984) noticed that quadratic fits to the daily data of two locations in Turkey had better correlation coefficient than the linear fits. They also observed that a single quadratic curve can represent two locations having quite different climatology. They obtained the quadratic fit for the daily data of these two locations as:
— = 0.204 + 0.758 – 0.250 f) . (5.27)
H0 N NJ
For the monthly average values one needs to take the average of the square of n/N, and the mean of a square of any value is related to its standard deviation a as < (n/N)2 >=< n/N >2 +a2. Then, the quadratic form for the monthly average values can be written as:
They obtained an empirical quadratic correlation between a2 and n/N using the same daily data set of two locations in Turkey. Then, by inserting this empirical correlation into Eq. (5.28), they have written:
— = 0.195 + 0.676n – 0.142 f. (5.29)
H0 N NJ ’
They proposed that this expression can be used for the estimations of monthly average daily global solar irradiation for the locations without measured data. Later
Table 5.3 Calculated values of the Angstrom coefficients and a and b and also the coefficient b’ by Jain and by the present formalism

similar procedure as followed by Ogelman et al. (1984) is used in another research for the data of six locations in Turkey and another quadratic form was obtained (Aksoy 1997).
Correlation derived by Ogelman et al. (1984) Eq. (5.29), and the one given in the following section are compared with the estimations of different models and found to be the best (Akinoglu and Ecevit 1990a, 1990b, 1993; Tasdemiroglu and Sever 1989; Badescu 1999).
An excel worksheet is given in the CD which calculates the monthly values of the parameters and the coefficients (Eqs. 5.26) of the model presented in this section, namely ‘monthlymeandailyquadparametersworkbook. xls’. Input is monthly mean bright sunshine hours, monthly mean global solar irradiation and monthly mean diffuse irradiation of the location of interest. This input can either be imported to the cells H10J21 or directly written. The main input of course is the latitude of the location under consideration.