Then, the horizontal irradiance is equal to the integral of the contribution of each solid angle, and can be written as
D(0) = L(dZ, f) cos dZ dfl (22.30)
Figure 22.13 Angular coordinates 0Z and ф of an elemental solid angle of the sky
where the integral is extended to the whole sky, that is, 0 < 0Z < n/2 and 0 < ф < 2n. When we are dealing with an inclined surface, a similar reasoning leads to
D(fi, a) = L($Z, f) cos 4 dfl (22.31)
where $’Z is the incident angle from the solid angle element to the inclined surface, and a means that the integral is extended to the non-obstructed sky. The general solution of this equation is difficult because, under realistic skies, the radiance is not uniform and varies with the sky condition. For example, the form, brightness and position of clouds strongly affect the directional properties of the radiance.
The distribution of radiance over the sky is not measured routinely. Nevertheless, a number of authors [22-24] have developed instruments to measure it and have presented results for different sky conditions. Some general patterns may be discerned from these.
With clear skies, the maximum diffuse radiance comes from the parts of the sky close to the sun and to the horizon. The minimum radiances come from a region at an angle of 90° to the solar zenith (Figure 22.14). The diffuse radiation coming from the region close to the sun is called circumsolar radiation and is mainly due to the dispersion by aerosols. The angular extent of the sun’s aureole depends mainly on the turbidity of the atmosphere and on the zenith angle of the sun. The increase in diffuse radiance near the horizon is due to the albedo radiation of the Earth and is called horizon brightening.
The radiance distributions associated with overcast skies are very well described by Kondratyev: ‘for dense non-transparent cloudiness, the azimuthal dependence of diffuse radiation intensity is very weak. There is a slight monotonic increase of the radiance from the horizon upward towards the zenith’ .
Some models can be derived from these general ideas. The simplest model makes use of the assumption that the sky radiance is isotropic, that is, every point of the celestial sphere emits light with equal radiance, L(0Z, ф) = constant. The solution of Equations (22.30 and 22.31) leads to
1 + cos в
D(fi, a) = – D(O)— ^ (22.32)
Because of its simplicity, this model has achieved great popularity, despite the fact that it systematically underestimates diffuse irradiance on surfaces tilted to the equator.
The opposite approach assumes that all the diffuse radiation is circumsolar, that is, from the sun. This is really a case of treating diffuse radiation as though it were direct, and leads to
D(f), a) = D(0) max(0, cos 6>s) (22.33)
This model also has the advantage of being very simple to use, but in general it overestimates diffuse irradiances.
In general, better results are obtained with so called anisotropic models. Hay and Davies  proposed considering the diffuse radiation as composed of a circumsolar component coming directly from the direction of the sun, and an isotropic component coming from the entire celestial hemisphere. Both components are weighted according to the so-called anisotropy index ki, defined as
Dc(fi, ot)= —:——— max(o, cosds)
respectively, define the contribution of the isotropic and of the circumsolar components.
Note than k1 is just the ratio between a pyrheliometer’s reading and the solar constant, once corrected by the eccentricity due to the ecliptic orbit of the Earth around the sun. In this way, ki can be understood as a measure of the instantaneous atmospheric transmittance for direct irradiance. When the sky is completely clouded over, k1 = 0 and this equation becomes the same as that for the simple isotropic model. This anisotropic model is an excellent compromise between simplicity and precision. It has been well validated against measurements performed at different worldwide locations, and has been extensively used, for example, for the elaboration of the European Atlas of the Solar Radiation .
Also, very commonly employed, in particular with digital machines, is the model that has been put forward by Perez [27, 28]. It divides the sky into three zones acting as diffuse radiation sources: a circumsolar region, a horizontal band and the rest of the celestial hemisphere. The relative contribution of each component is modulated by means of empirical factors determined from the study of data from 18 measurement stations at 15 sites in North America and Europe. The Perez model used to perform slightly better than others , because the larger number of modulating factors allows for the consideration of a larger number of different sky conditions.