Modeling Solar Radiation at the Earth’s Surface

One-Minute kd Values Distributions Conditioned by the Optical Air Mass

The analysis of the experimental distributions of 1-minute horizontal diffuse irradi­ation has been carried out following the same approach as for kt and kb. The diffuse component has been analysed by means of the coefficient kd. Figure 3.19 shows the 1-minute kd distributions for every optical air mass (k^ma) using the data of Armilla. The curves are unimodal with a maximum value similar for all the dis­tributions. Nevertheless, there is a slight shift towards higher kj values when the optical air mass increases. This shift is minor than that encountered in kt (Tovar et al. 1998) and kb analyses. This index ranges from 0 to about 0.5. The maximum of the distribution, for all the optical air masses considered, is located between 0.05 and 0.15, presenting a slight displacement toward higher values as the optical air mass increases. However, the optical air mass influence on the shape of the distribu­tion is minor. The CDFs are presented in Fig. 3.19; note their similarity. The greatest differences between the curves correspond to the kd values in the range 0.1 to 0.2.

The experimental distribution has been fitted using the same modified Boltzmann functions previously used:

Xe(kd-kdd0)X

f (kd) = A - . (3.45)

[1 + e(kd-kd0)(X+e)] 2

The parameters present a dependence on optical air mass that could be adjusted by multiple linear regression; we have obtained the following results:

A = 0.07062 - 0.07609ma + 0.02989m2, withR2 = 0.98, (3.46)

kd0 = 0.0248 + 0.0222ma, withR2 = 0.98, (3.47)

X = 538.69 - 152.34ma, with R2 = 0.996, (3.48)

в = -305.27 + 122.154ma - 11.468m2,withR2 = 0.998. (3.49)

It must be pointed out that all the distributions show similar kd0 values. Addition­ally, the parameter A also presents similar values for all distributions. The similarity of both parameters, governing the position and amplitude of the maximum of the distribution, is related to the evident similarity of the experimental distributions. These results could be explained as follows. The scattering process increases with the optical air mass, leading to an enhancement of solar diffuse irradiance. On the other hand, the extinction process by both scattering and absorption increases as the optical air mass increases, thus diminishing the solar radiation reaching the Earth’s surface. In this way, regarding the slight difference among the kd distribution as­sociated with different optical air mass, it seems that there is a compensation of these two effects. Therefore, there is a relative independence of the kd distribution function with the optical air mass.

However, X and в present a more marked difference for the different distributions. Note that X decreases as the optical air mass increases, evidencing an increase of the

dispersion in the experimental distributions. Nevertheless, the ratio в to X slightly changes with the optical air mass, reflecting the fact that the asymmetry is rather similar for all distributions.