Successive Substitution Methodology (SSM)

A simple substitution method was proposed by Sahin and Sen (1998) for dynamic estimation of Angstrom’s coefficients which play significant role in relation of the global solar irradiation to the sun shine duration through a linear model. Their math­ematical estimation procedures are presented on the basis of successive global ra­diation and sun shine duration record substitutions into the model. This procedure yields a series of parameter estimations and their arithmetic averages are closely related to the classical regression method estimates. The series of model parameter estimations provide an ability to assess these parameters statistically. Consequently, such a dynamic parameter estimation procedure evaluates and enables one to make interpretations with their normal and extreme values. Additionally, necessary rel­ative frequency distribution functions of these parameters appear in the form of Beta probability distribution function. Routinely recorded daily and monthly global irradiation and sunshine duration values are used by the regression technique for determining the coefficients in Angstrom equation. The use of such a deterministic model provides linear unique predictions of global solar irradiation given the sun­shine duration. In order to consider effects of unexplained part, it is necessary to estimate coefficients from the successive data pairs “locally” rather than “globally” as in the classical regression approach (Fig. 6.3).

Parameter b represents variation and relation of ratios d(H/H0) and d(S/S0) which corresponds to the slope of the linear relationship defined as,

d(H/Ho) d(S/So)

This first-order differential equation can be written in terms of backward finite dif­ference method as,

Herein, n is the number of records and b’ is the rate of global solar irradiation change with the sunshine duration between time instances i — 1 and i. For daily data, these are successive daily rates or in the case of monthly records, they are monthly rates. Arrangement from Eq (6.10) by considering Eq. (6.17) leads to the successive time estimates of a’i as,

a’ =( HQ — ^(iQ. (i = 2,3,4,…….. ,n) (6.18)

The application of these last equations to actual relevant data yields (n — 1) coef­ficient estimations. Each pair of the coefficient estimate (a!, b’) explains the whole information for successive pairs of global solar irradiation and corresponding sun­shine duration records. Comparison of Eq. (6.17) with Eqs. (6.11), (6.18) and (6.12) indicates that regression technique estimation does not allow any randomness in the coefficient calculations. However, the proposed finite differences method coefficient estimations assume the regression technique estimations and it provides flexibility in the parameter calculations.

Furthermore, it is possible to obtain the relative frequency distribution of a’’s and b’ds. In addition to any statistical parameter such as variance or standard deviation. Confidence limits can also be stated at a certain significance level as 5% or 10%. Extreme values of a’ and b’i also become observable by finite difference method solution.

Taking the average values of both sides in Eq. (6.18) leads to finite difference averages of coefficients as,


The difference of this expression from Eq. (6.12) results in,


since ab < ab, this expression can be written in the form of an inequality as,


As a result, it is shown that by SSM, a dynamic behavior can be given to Angstrom equation.

Updated: August 2, 2015 — 10:26 pm