Statistical Investigation of the Clearness Index
7.1.1 Bimodal Character of the Probability Density Functions
The measurements of instantaneous solar radiation values allow to considerer the effect of optical air mass. As a consequence, the distributions conditioned by the optical air mass not only describe how the instantaneous radiation values are distributed for a given mean value, but also how the instantaneous solar radiation varies with air mass and the time of the day (Suehrcke and McCormick 1988a).
Prior to studying the 1-minute distributions, the influence of the temporal integration interval on the shape of the density distribution function has been analysed. Notice that the size of the integration interval has an important influence on the bimodal character of the distributions. Figure 3.7 shows the comparison between 1-minute frequency distribution f(kt|ma) and the hourly frequency distributions f(kH|ma) for the same value of the optical air mass (ma = 3.0).
The distribution curve corresponding to 1-minute kt values presents a more marked bimodality than the corresponding to the hourly clearness index values. This behaviour was first noticed by Suehrcke and McCormick (1988a), who analysed the effect of internal averaging on the CDFs (Fig. 3.8) and confirmed later by Jurado et al. (1995) and by Gansler et al. (1995). Suehrcke and McCormick (1988a) suggested that, for averaging periods longer than 60 minutes, there is no evidence of bimodality in the kt distributions. Thus, they obtained similar distributions to those derived by Bendt et al. (1981) for daily averages. Nevertheless, some degree of bimodality persists for the hourly distribution corresponding to higher values of optical air mass (Fig. 3.7).
Figure 3.9 shows some degree of bimodality in all the distributions. This feature increases with increasing optical air mass. The present finding is in line with the results reported in Suehrcke and McCormick (1988a), Skartveit and Olseth (1992) and Jurado et al. (1995). However, Gansler et al. (1995) found a different behaviour in three U. S. locations.
The distribution densities in Fig. 3.9 show that the probability for values of clearness index in the intermediate range is low. The low probability associated with intermediate kt values indicates that it is possible to relate the curves of the distribution with two levels of irradiation in the atmosphere, for each optical air mass. The major peak in the density function corresponds to high values of kt, associated with cloudless conditions, and the secondary maximum corresponds to low values of kt, associated to cloudy conditions. An increase in optical air mass implies a decrease in the intensity of the first maximum and a subsequent increase in the secondary maximum.
Furthermore, by increasing optical air mass, the principal maximum is shifted towards lower kt values. The decrease in probability density for the principal maximum implies an increase in the probability density of kt in the lower range
Instantaneous kt values (0/0.02/1) Hourly kt values (0/0.02/1)
Fig. 3.7 Comparison between 1-minute (instantaneous) frequency distribution and hourly frequency distributions for the same value of the optical air mass (ma = 3.0). The distribution corresponding to 1-minute kt values presents a more marked bimodality than the corresponding to the hourly clearness index values
that leads to the enhancement of the second maximum in order for the area under the curve to remain constant. When the optical air mass tends to higher values, there is an increase in the probability density for the lower kt range. This result can be associated with the fact that, for small zenith angles, the clouds shade a smaller Earth surface area than for larger angles (Fig. 3.11). On the other hand, for horizontal layers of clouds, the effective thickness of the clouds is also larger for high zenith angles; hence, the clouds are less transparent. Thus, for higher values of optical air mass, the effect of the clouds is stronger, and the bimodality suffers an increase. On the other hand, the probability density of the intermediate states of kt does not vary considerably with the optical air mass. The increase of the bimodal character when the optical air mass increases is explained by the fact that the largest kt values and their frequency tends to decrease, therefore increasing the lowest partitions.
In Fig. 3.9 also we can appreciate that kt reach values close to unity, specially at low optical air mass due to multiple cloud reflections of solar radiation (Fig. 3.10).
Suehrcke and McCormick (1988a) proposed a model, based on the Boltzmann statistics, to explain the bimodal character of the distributions by using three functions associated with three different irradiation levels. Particularly, two of the functions were associated with the extreme conditions related to the cloudless and cloudy conditions, i. e. higher and lower kt values, and a third function was associated with the intermediate kt range. Figure 3.12 shows the clearness index distributions for Armilla (Granada, Spain) and Perth (Australia) in Suehrcke and McCormick’s work.
Fig. 3.10 The clearness index, kt, at point P can reach values close to unity, especially, at low optical air mass. These high kt values are associated with an enhancement due to cloud reflections (Suehrcke and McCormick 1998a)
A similarity in the shape of the distributions can be observed. The figure also depicts the adjusted curve proposed by the authors from their data.
There are several other approaches to model bimodality. Many of them use the sum of two functions, each of them describing the behaviour around one of the two maxima. Intermediate values can be obtained as the sum of the “tails” of the distributions between the maxima. For instance, Jurado et al. (1995) proposed the use of two Gaussian distributions.
In order to properly describe the shape of the distribution we should use functions, which meet certain criteria:
• The bimodal character may be expressed as the sum of two functions corresponding to two discernible atmospheric conditions: clear and overcast skies.
• The function must be as simple as possible, such as we do not have to implicitly assess any parameter.
• The parameters governing the function should be interpreted in terms of the climatic and atmospheric variables involve in the process: average clearness index, optical mass, climatology, etc.
• The function should be versatile enough to adapt to any kind of distribution.
• The function may be used to model the direct and diffuse radiation distributions.
• All the function parameters may be formulated by means of the optical air mass, the determinant variable in the process.
This function is symmetrical, centred at x0 and its width is determined by the parameter X. The introduction of a parameter в in the denominator of the exponential allows to get asymmetrical distributions, as Fig. 3.13 reveals.
7.1.2 Modelling with Boltzmann Distribution
As mentioned early, the experimental distributions are described by the sum of two functions:
f(kt |ma)= f1(kt)+f2(kt), (3.27)
subject to the normalisation condition:
/ft |ma) dkt = 1. (3.28)
The f1 and f2 functions are obtained from the Boltzmann statistic:
This function yields unimodal symmetrical curves around kt0i, where the function reaches its maximum. A; determines the function height and X; is related to the width of the distribution function. This can be integrated to:
and this can be analytically inverted:
These characteristics allow the generation of synthetic data of instantaneous values from the CDFs by methods of inferential statistics. Also it is possible to obtain explicitly the kt coefficients.
The coefficients A1 and A2 must satisfy the normalisation condition:
When modelling this dependence for the data of Armilla (Granada, Spain), we found for the maxima distributions, kt01 and kt02, the following expressions:
kt01 = 0.763 -0.0152ma -0.012m2, withR2 = 0.996, (3.33)
kt02 = 0.469 -0.0954ma + 0.01m2, withR2 = 0.992, (3.34)
where R2 is understood as the proportion of response variation “explained” by the parameters in the model.
The position of the principal maximum, kt01, shifts towards lower values as the optical air mass increases. The same trend occurs for the value of kt02, corresponding to the second maximum of the distribution. However, the shift is smaller than that associated with the principal maximum kt01, as it can be concluded by comparing the coefficients of the optical air mass terms in each equation. This implies that, when the optical air mass increases, the two maxima tend to be closer.
The values of the width parameters, X1 and X2, can also be expressed in terms of the optical air mass:
X1 = 91.375 -40.092ma + 6.489m2, withR2 = 0.999, (3.35)
X2 = 6.737 + 1.248m + 0.4246m2, withR2 = 0.975. (3.36)
The coefficient A1 has been fitted using the following expression:
A1 = 0.699 + 0.1217m-21416, withR2 = 0.994. (3.37)
Considering the A1 and A2 dependence (A1 + A2 = 1, because of the normalisation condition), it is obvious that while A1 decreases with air mass, A2 shows the opposite trend. The ratio between the intensity of the two peaks depends on ma. This ratio decreases when ma increases, that is a decrease in ma implies an enhancement of the first maximum relative to the second one.
Figure 3.14 shows the fitting curves using both, the Suehrcke and McCormick’s model and the Tovar’s model based on the Boltzmann statistics. Figure 3.14a shows the case of the best adjustment provided by the Suehrcke-McCormick model for data collected in Armilla (Granada), adapting conveniently the parameters to fit the maxima of the distribution. Figure 3.14b shows the Tovar model adjustment for the same set. Figure 3.14c shows the results by applying the Tovar model to the
data collected in COrdoba. It can be observed that the Boltzmann model provides a reasonable adjustment (Tovar et al. 1998a; Varo et al. 2006). Nevertheless, the maxima of the bimodal distribution depend on the location and its climatic features. Therefore, the fitting parameters inEqs. (3.33-3.37) will also depend on the location and its climatic features. However, there are some common characteristics for all the functions used to fit the distributions.