Spatial solar radiation estimation
In the previous sections, the modeling of solar radiation is discussed on a given site. However, in practical solar energy assessment studies, it is also necessary to have spatial (multiple sites) solar energy estimation procedures. The main purpose of this section is to develop a regional procedure for estimating the solar irradiation value at any point from sites where measurements of solar global irradiation already exist. The spatial weights are deduced through the regionalized variables theory [67,81], semivar - iogram (SV) and the cumulative SV (CSV) approach . The SV and CSV help to find the change of spatial variability with distance from a set of given solar irradiation data. It is then employed in the estimation of solar irradiation value at any desired site through a weighted average procedure, which takes into account a certain number of adjacent sites with the least error. The validity of the methodology is first checked with the cross-validation technique and then applied for the spatial irradiation estimations.
Spatial variability is the main feature of regionalized variables, which are very common in the physical sciences . In practice, the spatial variation rates of the phenomenon concerned are of great significance in fields such as solar engineering, agriculture, remote sensing and other earth and planetary sciences. A set of measurement stations during a fixed time interval (hour, day, month, etc.) provides records of the regionalized variable at irregular sites, and there are few methodologies to deal with this type of geographically scattered data. There are various difficulties in making spatial estimations due to not only the regionalized random behavior of the solar irradiation, but also from the irregular site configuration. Hence, the basic questions are
1. how to transfer the influence of each neighboring
measurement station to the estimation point, and
2. how to combine them in order to make reliable
regional estimations of solar irradiation.
Based on empirical work by Krige  to estimate ore grades in gold mines, the regionalized variable theory was developed by Matheron . It is also known as geostatistics, which has been used to quantify the spatial variability through the Kriging technique. The basic idea in geostatistics is that for many natural phenomena, such as solar irradiation, close samples have higher probability of being similar in magnitude than samples further apart. This implies spatial correlation structure in the phenomena. Especially, in earth sciences, considerable effort has been directed towards the application of the statistical techniques leading to convenient regional interpolation and extrapolation methodologies [26,85].
The spatial solar irradiation estimation problem has been addressed by Dooley and Hay , and Hay . They tried to evaluate the errors using solar irradiance data at a number of sites in Canada. The basis of their approach was the optimal interpolation techniques as suggested by Gandin  in the meteorology literature. The main interest was to estimate the long-term average of all the sites considered for each month irrespective of any particular year. Systematic interpolation evaluations have been carried out in solar irradiation networks by different authors [55,149,150].
It is possible to prepare solar irradiation maps of a region based on a set of measurements at different sites by using basic geostatistical techniques such as SVs and then the Kriging methodology . The success of Kriging maps is dependent on the suitability of the theoretical SV with the data at hand. In fact, SVs are the fundamental ingredients in Kriging procedures, because they represent the spatial correlation structure of any phenomenon. There are however, practical difficulties in the identification of SVs from available data [124,125]. Empirical CSVs are adopted as spatial correlation structure representatives of irradiation data. They are transformed into standard weighting functions (SWFs), which show the change of weighting factor with dimensionless distance values. As the dimensionless distance value increases the effect of weighting decreases.