## 2 Materials and Methods  
The solar irradiance model (dubbed SOLAR) described here is semi-empirical in nature in that it consists of a blend of basic theoretical concepts and empirical results. The input requirements of the model include Julian day, time, geographical latitude and atmospheric pressure. Knowledge of cloud cover is also needed but if this is not available, then it can be estimated from one of two options, namely, the daily global solar flux or daily sunshine hours. The latter provision is made since measurement of daily duration of sunshine is more common amongst meteorological stations as it is easier to measure and requires less accurate instruments [11]. The instantaneous direct solar irradiance at the earth's surface (\(S_{dir}^{c}\)), in the presence of a turbid atmosphere but clear sky (i.e., no clouds), can be calculated using the following equation given in textbooks on solar radiation theory (see, for example, [14; 15]),  
\[S_{dir}^{c}=S_{o}\beta\cos\psi\tau^{m} \tag{1}\]  
where \(S_{o}\) is the solar constant (1373 W m\({}^{-2}\)), which is defined as the solar irradiance on a surface perpendicular to the sun's rays just outside the earth's atmosphere and at the mean distance of the earth from the sun; \(\beta\) is the square of the ratio of the length of the semi-major axis of the earth's orbit about the sun to the earth-to-sun separation, given by Equation (2); \(\psi\) is the zenith angle, defined as the angle between the direction of the sun and a vertical axis (to the surface of the earth at the place of observation), given by Equation (3); \(\tau\) is the turbidity coefficient (or atmospheric transmissivity) which quantifies the attenuation of solar radiation by the atmosphere, assumed to be equal to 0.6, which is a reasonable value for cloud-free atmospheres [14] of urban areas, especially those in the East Midlands, United Kingdom [16]; \(m\) is the optical air mass defined as the ratio of the path length in the direction of the sun at a given zenith angle to the path length in the vertical direction, i.e., atmospheric depth, given by Equations (4) and (5). The square of the ratio of the length of the semi-major axis of the earth's orbit about the sun to the earth-to-sun separation, \(\beta\), is given by  
\[\beta\approx 1+2\epsilon\cos\left(\frac{2\pi}{365}d\right) \tag{2}\]  
where \(\epsilon\) is the eccentricity of the earth's orbit about the sun (0.01675); \(d\) is the Julian day. Furthermore, we have the following:\[\cos\psi=\sin\varphi\sin\delta+\cos\varphi\cos\delta\cos h_{s} \tag{3}\]  
where \(\varphi\) is the geographical latitude (in degrees); \(\delta\) is the solar declination angle, defined as the angle between the orbital (earth round the sun) and earth's equatorial planes, given by Equation (6) (in degrees); \(h_{s}\) is the hour angle, given by Equation (7) (in degrees). Note that whenever angles are independent variables of sinusoidal functions, then they can be either in radians or degrees; otherwise, they are in radians unless stated. Derivation of Equation (3) is given in standard textbooks on solar radiation theory, but practically all of the derivations are based on the geometrical approach, which makes them cumbersome and algebraically complicated. An easier approach is to use vector analysis by considering the dot (or scalar) product of two vectors corresponding to two (parallel) sun rays, one which impinges on any arbitrary position on the earth's surface and the other such that it is radial to the earth's sphere. When the dot product is equated to zero (due to vectors being parallel), Equation (3) is obtained.