Re parameter of each month at each station are calculated and it is seen that some Re’s are negative. As can be concluded from Eq. (6.33) when extraterrestrial ratio is higher than terrestrial ratio, a positive Re value occurs, otherwise a negative Re value
is obtained. Generally, in Adana station terrestrial ratios are higher than extraterrestrial ratios except for few months. At this station, sunshine duration values are high but solar irradiations are not as high as expected. As a result, terrestrial ratios are higher than extraterrestrial ratios.
In Ankara station, most of the characteristic Re’s have negative values. These ratios also show atmospheric effect to extraterrestrial ratio. In some months, extraterrestrial ratio reduced by 60% and 40% is received by horizontal surface. Under all circumstances, Re values represent atmospheric effect irrespective of their signs. Positive and negative values must be considered for comparison between terrestrial and extraterrestrial ratios. Monthly mean values indicate that in the first and the last two months of the year, terrestrial ratios are higher than extraterrestrial ratios. On the contrary, in other months terrestrial ratios are smaller than extra terrestrial ratios. Hence, negative Re values occur during eight months in this station.
In contrast to Adana and Ankara stations, majority of Re values at Istanbul are positive (Fig. 6.9). Four months atmospheric effects to extraterrestrial ratios are higher than 0.6. It is estimated that average monthly terrestrial ratio values are higher than extraterrestrial ratios during seven months (Sen and Sahin 2001). In this figure polynomial connections occur below the straight line of extraterrestrial ratios. In the CD accompanying this book the reader will find details of this figure.
By using Re values, measured terrestrial variables (sunshine duration and solar irradiation) can be estimated with Eqs. (6.35) and (6.36). Although Angstrom parameters, a and b, are constants, in the proposed method there are different Re values for each month with a sequence of Re values and hence it is possible to make probabilistic estimations, which provide an opportunity for the temporal prediction of Re values and solar irradiation reductions. If Re value of each month is used for estimation then there might be very little error. For optimal usage, constant Re value must be considered with minimum estimation error. Herein, positive and negative Re values are estimated. Hence, average positive Rep value is calculated from positive
Fig. 6.9 Re values for Istanbul station 
Rep values and average negative Ren is calculated from negative Re values. Three stations have Rep and Ren values in addition to AgstrOm parameters are presented in Table 6.3. Other variables are taken into account directly.
If Re value is positive then Rep is taken into account for terrestrial sunshine duration and solar irradiation estimations from Eqs. (6.35) and (6.36), otherwise Ren is used.
Estimations for Istanbul by both methods are compared and it is seen that R2 value for Angstrom method estimation is 0.97. This means that correlation coefficient of this representation is 0.98 which is a very good result. However, the proposed method has R2 value as 0.87 which is also a good representation, but not better than Angstrom approach. In the case of time series graphics, regression technique gives some misleading information. All maximum values of Angstrom estimations are higher than measurements. In other words, overestimations occur in H values in this station (Fig. 6.10a). Sometimes physically impossible values are estimated by Angstrom equation. Finally, measured sunshine duration, S values are estimated by both methods. It is observed that the proposed method gives better results than the Angstrom approach (see Table 6.3). In this station, some sunshine duration estimations have negative values (Fig. 6.10b). In addition, details of this figure could be maximized in the accompanying CD.
Adana station H estimations with proposed method are compared by measurements through the regression technique on the basis of the coefficient of determination (R2). The same procedure is also used for terrestrial variables estimation through Angstrom equation. Proposed and Angstrom method estimations of H with high R2 are given in Table 6.3. It is observed that Angstrom equation estimations are better than proposed method in Adana station. On the other hand, if time series comparison is considered then generally measured data are represented better except at maximum and minimum values. At maximum values overestimations occur, but at minimum values underestimations exist in Adana station. The same procedure is applied for sunshine duration estimation in Adana, and it is seen that proposed method estimation of sunshine duration is better than Angstrom equation. Especially, sunshine duration estimation is not meaningfully represented by Angstrom equation. In other words, R2 attached with the Angstrom equation is not meaningful.
Table 6.3 Parameter and variables estimation of classical regression and new equations

Fig. 6.10a Comparison of measured and estimated solar irradiation H values by both methods in Istanbul 
In time series comparison Angstrom equation appears weak for terrestrial sunshine duration estimation. In one of the months, Angstrom equation estimates sunshine duration as zero which is physically impossible. In other words, during one month absolutely closed conditions could not be observed at these latitudes especially in this station.
Ankara station estimations by both methods indicate that terrestrial solar irradiation by Angstrom equation estimation is better than the proposed method. Like Adana, both methods have R2 values higher than 0.94 and at maximum values in some months overestimations occur by proposed method. It should be
Fig. 6.10b Comparison of measured and estimated sunshine duration S values by both methods in Istanbul
remembered that, R2 value of the proposed method is 0.94 which means that correlation coefficient between measured and estimated value is 0.96. This result is very representative for estimation purposes. In addition to measured global solar irradiation H and sunshine durations, S values are also estimated by both methods. It is understood that better estimations are possible through the proposed method where R2 values are higher than Angstrom equation results (Table 6.3). Time series approach shows that the proposed method is more representative than Angstrom approach. Similar to Adana station in Ankara one month has physically impossible result by Angstrom equation estimation.
For the accuracy of the proposed and classical Angstrom models mean bias error (MBE), root mean square error (RMSE) and relative error (RE) are used and compared (Table 6.4).
It is not easy to see differences between estimated and measured values by using MBE except for Istanbul. In Adana and Ankara MBE values are approximately equal to each other, but in Istanbul MBE of H values estimated from Angstrom equation, is 42.86% that is very high for engineering approaches. When one looks at RMSE values, it is seen that as a result of summation of square differences, these errors are higher than MBE and generally bigger than 10% except errors of estimated H values for Adana and Ankara by Angstrom equation. Other RMSE values are higher for H and S than estimations by Angstrom equation. It is clearly seen that Angstrom equation is not a good approach for these parameters. Especially, in Istanbul unacceptable errors are estimated by classical approach. One of the other comparison methods is the relative error (RE) approach that is a very useful tool for engineering calculation. It is seen that for all station, RE values of Angstrom equation are higher than errors of proposed method except H values in Adana and Ankara that are estimated by Angstrom equation (Table 6.4).
Table 6.4 Different estimation errors for classical regression approach and proposed method

In this chapter, different new methods are applied to the Angstrom equation and a new alternative methodology is proposed to see dynamic behavior of this equation and solar irradiation variables. A dynamic model estimation procedure as the successive substitution method (SSM) is proposed, which leads to a sequence of parameters and hence it is possible to look at the frequency distribution function (probability distribution function, PDF) of the model parameters and decide whether the arithmetic average of the parameters or the mode (the most frequently occurring parameter value) should be used in further solar irradiation estimations. It is shown on the basis of some solar irradiation and sunshine duration data measurements on different locations in Turkey that the model parameter estimations abide by the Beta PDF. Besides, it is also possible to find the relationship between the model parameters at a single station by using SSM, which shows temporal parameter variations. In addition, it is easy and practical to do statistical analysis of Angstrom equation parameters and variables with SSM.
Apart from the dynamic model parameter estimation procedure, an unrestricted model (UM) for solar irradiation parameter estimation procedure, is also presented which considers the conservation of the model input and output variables’ arithmetic mean and the standard deviations only, without the use of least squares technique. Assumptions in the restrictive (Angstrom) model cause overestimations in the solar irradiance amounts as suggested by Angstrom for small (smaller than the arithmetic average) sunshine duration and underestimations for large sunshine duration values. Around the average values solar irradiation and sunshine duration values are close to each other for both models, however, the UM approach alleviates these biasedestimation situations. Additionally, the UM includes some features of nonlinearity in the solar energy data scatter diagram by ignoring consideration of crosscorrelation coefficient.
Finally, an alternative formulation to Angstrom equation is proposed for sunshine duration and solar irradiation variables estimation. According to the suggested formulation, extraterrestrial variable ratio S0/H0 is assumed to have a reduction amount, Re, due to cloud cover, dust, humidity, etc. Such reductions in sunshine duration and solar irradiation are measured on the horizontal surface. There is a relation between extraterrestrial and terrestrial ratios due to atmospheric effects. This reduction amount, Re represents atmospheric effect to extraterrestrial solar irradiation. Given the astronomical calculations of H0 and S0 together with measurements of the H and S, Re can be calculated easily from proposed formulation. This methodology and Angstrom equation procedure are compared and it is shown that there are some physical problems with classical Angstrom approach.