Modeling Solar Radiation at the Earth’s Surface

Distributions of 1-minute кb Values Conditioned by Optical Air Mass

Figure 3.16 shows the kb frequency distributions for different optical air masses for Armilla (Spain). All the curves present a bimodal appearance, with two well de­fined maxima. The first maximum, located in the interval (0.00, 0.02), corresponds to conditions associated with direct irradiance close to zero. These conditions corre­spond to overcast or partially-cloudy skies, more frequent for high optical air mass. For greater zenith angles, scattered clouds hide proportionally greater areas on the surface of the Earth and horizontal cloud layers have large effective thickness than for lower zenith angles. Therefore, for higher values of optical air mass, the block­ing effect of clouds is more efficient. The probability of that solar global irradiance

mainly consists of diffuse irradiance increases with the optical air mass. Skartveit and Olseth (1992) have found similar results. The experimental probability for the (0.00,0.02) interval (Figs. 3.16, 3.18) can be represented by an exponential function, which depends on the relative optical air mass:

f1(ma) = 17.34 - 36.75 • exp(-0.975ma), withR2 = 0.98. (3.38)

The second maximum of the distribution, located on the right, covers a range be­tween 0.7 for m= 1.0 and 0.1 for m=3.0. Intermediate values between both maxima present a low probability. When optical air mass increases, the second maximum shifts towards lower values of kb. This is a result of the enhancement of direct irradi­ance extinction. The shape of the distribution around this second maximum presents a marked asymmetry towards the left side of the distribution.

In order to model the bimodality that characterises the distribution of the data, the sum of two functions can be used (Fig. 3.17). The first corresponds to the first kb interval, expressed as a Dirac delta multiplied by a factor depending on the optical air mass. The second corresponds to the remaining intervals, and can be adjusted by means of a function that reproduces the observed asymmetry. To this end, the same kind of functions used in a previous work (Tovar et al. 1998) to model the frequency distribution functions of 1-minute kt values, can be used. I have modified this function including an additional parameter p, that accounts for the asymmetry of the function. This modified equation is:

t e(kb-kb0)X

f2 (kb) = A - —-------------- -

[1 + e(kb-kbG)(X+P)]

and satisfies the normalisation condition:

f(kb) dkb = f1(kb) dkb + f2(kb) dkb = 1. (3.40)

0 0 0

The degree of asymmetry depends on the ratio of the parameters P and t. The sign of P determines whether the asymmetry goes towards the right or the left side.

Fig. 3.18 The amplitudes of instantaneous kb distribution in the interval (0.00, 0.02) and its fitting using Eq. (3.38)

The kb0 parameter is related to the position of the maximum in the PDF. The product A • X depends on the size of the frequency distribution maximum.

The use of this equation to describe the statistic behaviour of kb is interesting since, in this way, a formal coherence with the functions used for the kt distributions can be maintained (Tovar et al. 1998). The equation modified by the parameter в accounts properly for the experimental values.

The parameter kb0 locates the maximum of the distribution function. Note that kb0 shifts towards lower kb values as the optical air mass increases. The tendency to a decrease of the asymmetry with the optical air mass is modelled by the decrement of the ratio в to X. Figure 3.16 shows the PDFs and their respectives CDFs. Figure 3.17 shows the division of the PDF into two functions: the first corresponding to the (0.00,0.02) interval and the second corresponding to the rest of the intervals. We can appreciate in the CDFs of the Fig. 3.16 that the initial value for these curves as the optical air mass varies. For the lowest optical mass, the interval (0.02,1.00) includes 72% of the cases. This percentage diminishes with the optical air mass, mainly due to the increase in the direct beam extinction for greater optical air masses.

The parameter that rules the distributions presents a dependence on the optical air mass. After a linear multiple regression adjust, we obtain for the Armilla data the following results:

A = 0.9984 - 0.01686ma + 0.00171m2, withR2 = 0.986, (3.41)

kb0 = 0.7862 -0.0546ma -0.00543m2,withR2 = 0.99, (3.42)

X = 8.855 - 0.2666ma - 0.3229m2, withR2 = 0.935, (3.43)

в = 83.74 - 34.059ma - 6.1846m2, withR2 = 0.975. (3.44)

The R-squared between the experimental values and the modelled ones each air mass is close to 0.98; similar results are obtained for other optical air masses.