Quadratic Variation of a with b

Present author and his co-worker (Akinoglu and Ecevit 1990) make use of the vari­ation of the Angstrom coefficients a with b to find a quadratic form. They only used the published values of a and b of 100 locations all over the World to derive a quadratic form and did not use any measured irradiation and sunshine hours.

Quadratic form dictates the variation of Angstrom coefficients with respect to n/N values. In other words for a specific value of n/N, the result that would be obtained from a + b(n/N) must be equal to that obtained from the quadratic form. Therefore, slope of the quadratic form is the Angstrom coefficient b and the intercept of the line having this slope is the Angstrom coefficient a. These are:

b = a1 + 2a2(n/N)

a = a0 — a2(n/N )2 (5.30)

which has similar dependence as found by Rietveld, using the coefficients obtained from measured data, for the variation of a and b with respect to n/N (Rietveld 1978). That is, b decreases with n/N while a increases. Note that the third coefficient a2 is negative and that is why b decreases with n/N while a increases. In fact, the coefficient a2 always comes with a negative sign, not only in the fits to the real data as in Eq. (5.27) of Ogelman et al. but also in the physical formalism developed and presented in section 4.3 for the monthly values (see Table 5.2 last columns). Note also that monthly a and b values obtained using the formalism presented in section 4.3 but without considering the first reflection cycle (Akinoglu 1993) (which results in a linear relation in n/N) give exactly the similar variations of the monthly values of a and b with n/N as Eq. (5.30).

Another natural outcome of the quadratic relation between H/H0 and n/N is the quadratic nature of the dependence of a to b, which was validated using the
published values of a and b all over the world (Ogelman 1984; Akinoglu and Ecevit 1990). If the quadratic nature of the relation between H/H0 and n/N has a global implication, then the linear Angstrom-Prescott forms are the family of straight lines which will be various chords on the quadratic curve, Eq. (5.25).

Consequently, the coefficients of the quadratic form Eq. (5.25) can be obtained in terms of the coefficients of the quadratic relation between a and b. Equating the slopes of the tangents of Eq. (5.25) to b, namely the first expression in Eq. (5.30), the specific value of n/N can be extracted as n/N =(b — a1)/2a2. At that specific value of n/N, Angstrom-Prescott equation written with the slope and intercept of the tangents to the quadratic curve, that is [a + b(b — a1)/2a2] and the quadratic form itself, that is {a0 + a1(b — a1)/2a2 + a2[(b — a1)/2a2]2} must give the same results. This equality then gives a relation between a and b of the form:

(5.31)

as given in Akinoglu and Ecevit (1990). Hence, a quadratic fit to the curve a versus b can be used to obtain the coefficients of the quadratic relation, that is a0, a1 and a2 of Eq. (5.25), using this fit and Eq. (5.31). Such a regression fit can be attained using the published Angstrom coefficients a and b that are obtained by regression analysis between the measured values of global solar irradiation and bright sunshine hours. Therefore, a quadratic relation between H/H0 and n/N can be obtained without any measured H and n values directly, but by using the relation between a and b that can be found in the literature for different locations, as given in Akinoglu and Ecevit (1990). Such a relation should have global applicability as it will be derived only from the set of a and b values for the locations with different latitudes and climates all over the World. Figure 5.2 shows this relation between a and b values obtained from hundred locations all over the World presented also in Akinoglu and Ecevit (1990), which has the same type of variation as implied by the quadratic relation between H/H0 and n/N.

Hence, a quadratic curve is fitted to the curve a versus b in Fig. 5.2 using a computer program prepared in Cern Computer Centre (James and Roos 1977). The result obtained was (Akinoglu and Ecevit 1990):

a = 0.783 — 1.509b + 0.892b2.

The coefficients a0, a1 and a2 of a quadratic expression between H/H0 and n/N were obtained in terms of the coefficients of Eq. (5.32) using Eq. (5.31) (Akinoglu and Ecevit 1990). The expression was:

and as mentioned and outlined above it was obtained only by using 100 published values of Angstrom coefficients a and b, without using any measured data of H and 5 of any location. Comparisons with 13 other correlations were carried out

Fig. 5.2 Variation of a with respect to b for 100 locations

and quadratic correlations (Eq. (5.33) and Eq. (5.29)) were observed to have better performances (Akinoglu 1991, 1990a, 1990b, 1993; Tasdemiroglu and Sever 1989; Badescu 1999).

A self-explanatory excel worksheet which calculates the monthly-average daily values using Eq. (5.33) can be found in the CD, namely ‘monthly-mean-daily- comparison-workbook. xls’. In this worksheet, calculations for statistical errors MBE and RMSE – which will be explained in the following section-, are also included for comparisons. Of course, any expression (model, correlation etc.) can be written in the cell that calculates H (column J, J10-J21), so that comparisons can be carried out. The data is the measured monthly average bright sunshine hours and monthly average global solar irradiation which must be imported or directly written to the cells H10-I21. The latitude of the location under consideration is of course the main input.

Updated: August 2, 2015 — 6:14 am