Angstrom Equation

Solar irradiation and sunshine duration records depend on the combined effects of astronomical and meteorological events. The first relationships occurred in the form of a linear expression as suggested by Angstrom (1924). His formula has been used in practical applications for many years to estimate the daily, monthly and annual global solar irradiation, H, from the comparatively simple measurements of sun­shine duration, S, according to the following expression


— = a + b – (6.10)

H0 S0

where H and S are the daily global irradiation received on a horizontal surface at ground level and sunshine duration, respectively, and a and b are model parameters. As explained above, although H and S vary temporally in a random manner, H0 and S0 have fixed values that are given by deterministic expressions and the question is whether the model parameters a and b also vary temporally and randomly at a given station. In most applications so far in the literature, a and b are considered as con­stants for the time period used in the application of Eq. (6.10). For instance, if daily values are used then a straight-line is matched through the scatter of solar irradiation versus sunshine duration plots which minimizes the sum of square deviation from this line. On the other hand, estimation of Angstrom coefficients by the application of regression technique yields constant values as


Angstrom linear model relates global irradiation to sunshine duration by ignor­ing other meteorological factors such as the rainfall, relative humidity, maximum temperature, air quality, elevation above mean sea level, etc. The effects of other me­teorological variables appear as deviations from the straight-line fit to the scatter di­agram. In order to cover these errors to a certain extent, it is necessary to assume that the model coefficients are not constants, but random variables that change with me­teorological conditions (Sahin and §en 1998). On the other hand, many researchers have considered additional meteorological factors in order to increase the accu­racy of estimations (Prescott 1940; Swartman and Ogunlade 1967; Rietveld 1978; Soler 1990; Sen et al. 2001). However, a common point to all these studies is that parameter estimates are obtained by the least squares method with minimum but remaining error. Many researchers (Sabbagh et al. 1977; Dogniaux, Lemonie 1983; Gopinathan 1988; Jain 1990; Akinoglu and Ecevit 1990; Lewis 1989; Samuel 1991; Wahab 1993; Hinrichsen 1994) have considered additional parameters increasing the estimation accuracy. For instance, Ogelman et al. (1984) incorporated the sun­shine duration standard deviation for better model parameter estimations.

Soler (1986, 1990) has shown that monthly variations of (a + b) are mete­orologically sound and similar for different locations. It has been shown by Hinrichsen (1994) that, physically a > 0. Furthermore, Gueymard et al. (1995) showed that a corresponds to the relative diffuse radiation on an overcast day, whereas (a+b) corresponds to the relative cloudless-sky global irradiation. Depend­ing on weather and atmospheric conditions H and S vary temporally and spatially in a random manner but H0 and S0 have fixed values. The question is whether the model parameters, a and b also vary temporally at a given station. In most applications, a and b are considered as constants for the time periods used in practice. However, it is shown by Sahin and Sen (1998) that a and b also change temporally and spatially.

A detailed historical evolution of Angstrom equation is explained by Martinez – Lozano et al. (1984) with further criticisms are presented by Gueymard et al. (1995) and accordingly some authors suggested alternative methods (Suehrcke 2000; Sen 2001; Sen and Sahin 2001; Sahin et al. 2001).

Updated: August 2, 2015 — 10:28 am