Extinction of solar radiation due to the clouds is more important than that due to any other atmospheric constituents. The majority of solar irradiation models take into account the extinction of radiation in relation to cloud cover via the Angstrom-Prescott equation (Angstrom 1924; Prescott 1940). Customary derivation (Jain 1990) leads to relative sunshine duration as a natural parameter in this type of correlation. Over time, in order to increase accuracy, the original Angstrom’s equation has been modified and related to other surface meteorological parameters. The fractional cloud amount N is often used instead of relative sunshine duration (Haurvitz 1945, Kasten and Czeplak 1980).
Many previous modeling efforts have been conducted to include daily extreme of air temperature tmax,, tmin besides daily mean of cloudiness N in empirical solar irradiation estimation. Taking air temperature and cloudiness in computation is motivated by the usual availability of both meteorological parameters. Embedding the air temperature in models is meant to increase prediction quality, having the practical experience that accuracy decays with increasing cloudiness. The cause of increasing error with increasing N mainly derives from the definition of cloudiness, which for a given N does not take into account whether the sun is shinning or is behind the clouds. But, the drawback of including air temperature in the fitting process leads to a closer connection of the model to parental geographical location.
The equation from Supit and Van Kappel (1998):
H(N, At) = He (a (tmax – tmin)1/2 + b (1 – N)1/2) + c (7.1)
is a typical model which linearly relate daily global solar irradiation at ground level H to its extraterrestrial value He. Practically, Eq. (7.1) combines the square root of temperature amplitude dependence (Hargreaves et al. 1985) and a non-linear cloudiness dependence of the daily solar irradiation (Kasten and Czeplak 1980). The coefficients a, b and c are provided by Supit and Van Kappel (1998) for 95 various location in Europe from Finland to Spain exhibiting large dispersions that reveals the model local specificity. For example the temperature coefficient a takes values between 0.028 at Goteborg, Sweden (latitude 57.7°N; longitude 12.0°E; altitude 2m) and 0.115 at Murcia, Spain (38.9°N; 1.23°W; 62m), but it is also irregular with latitude – a = 0.086 at Lund, Sweden (55.7°N; 13.2° W; 73m).
The three models from (El-Metwally 2004) derived with data coming from northern Africa use other expressions for the correlation. Apart from Supit and Van Kappel model, it incorporates separately tmax and tmin either in a linear relation, such as:
H(N, tmax, ^min) = aHe + btmax + ctmin + dN + e (7.2)
or in an unfamiliar exponential one, as long as He is under the exponent. The regression coefficients in Eq. (7.2) given for seven Egyptian locations are also distinctly different.
To explain the air temperature role in this type of equations, we computed the second order polynomial regression coefficients with data (daily tmax, tmin, N, H recorded in 1998-2000) from Timisoara, Romania (45.76°N; 21.25°E; 85m), using the least square method, for the following two equations:
H(N) = -0.01He2 +(—0.42N + 0.9)He — 3.948N2 + 4.577N – 1.685 (7.3) H(N, At) = — 0.014H2 + (0.025At — 0.256N + 0.56) He — 2.64N2+
+ (0.06At + 1.804)N — 0.05At — 4.723 • 10—3At2 + 0.1 (7.4)
The difference between these two equations consists of the presence of daily temperature amplitude At = tmax — tmin in the Eq. (7.4).
The effect of taking temperature into consideration in the H model is depicted in Fig. 7.1 where the Eqs. (7.3) and (7.4) are plotted in respect to cloudiness. The H(N) curve from Eq. (7.3) is shifted by Eq. (7.4) in a band H(N, At), which can be interpreted that for a given N, At acts as a refinement according to weather conditions. It makes sense if we bear in mind that the air temperature amplitude is lower in the cloudy days than in the sunny days.
All of these models natively have been designed to avoid the task of solar irra- diance computation, being focused on direct computation of daily solar irradiation. However solar irradiation basically represents a sum over time of the irradiance. Therefore it is possible to include air temperature in a solar irradiance model.