Geostatistical methods were developed to create mathematical models of spatial correlation structures with a variogram as the quantitative measure of spatial correlation. The variogram is commonly used in geostatistics and the interpolation technique, known as kriging, provides the best, unbiased, linear estimate of a regionalized variable in a no sampled location, where best is defined in a least-squares sense. The emphasis is set on local accuracy, i.e. closeness of the estimate to the actual, but unknown, value without any regard for the global statistical properties of the estimates. The kriging estimation variances are independent of the value being estimated and are related only to the spatial arrangement of the sample data and to the model variogram. One development of geostatistics, that has become more popular in the last decade, is the stochastic simulation which represents an alternative modeling technique, particularly suited to applications where global statistics are more important than local accuracy (Webster and Oliver 2008). Application of geostatistics techniques in hydrological sciences is a useful approach to avoid some errors and increase of calculation accuracy as well. In classic statistic samples taken from a population are lack of spatial properties. Therefore the calculated values of a parameter in a homogeny sample do not include any information of the same parameter in another sample with a defined distance. Geostatistics consider the value as well as location of the sample. Then it is possible to analyze value and location of the samples together. To achieve this purpose it is necessary to relate spatial properties (distance, direction) of different samples using mathematical formula called spatial structure (Babakhani et al. 2016).
Natural resources and environmental concerns, including groundwater, have benefited greatly from the use of Geographic Information System. One can use GIS to integrate spatial data with other information. GIS is used for analyzing and presenting the spatial information, which facilitates the environmental protection, and resource planning. GIS also help in finding the close relationship between water quality and various natural or anthropogenic activities (Dhanasekarapandian et al. 2016). ArcGIS geostatistical analyst effectively bridges the gap between geostatistics and GIS analysis (Kumar et al. 2007; Nas 2009). Geostatistical analysis has been useful to determine water variables in space and time. Many studies have successfully used interpolation techniques with and without the use of the ArcGIS Geostatistical tool (Nas 2009).
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Obviously geostatistical methods use various variables and then produce different results. In many causes a method which is selected to reach the prediction process but it is vital to find out the most appropriate interpolation technique for precipitation prediction. The accuracy of interpolation methods for spatially predicting soil and water properties has been analyzed in several studies. Safari (2002) used kriging method to estimate spatial prediction of Groundwater in Chamchamal plain in west of Iran. Results showed that suitable method of geostatistics to estimate one variable depends on variables type and regional factors which influence this and any selected method for given region cannot be generalized to others. Nazarizadeh et al. (2005), used geostatistics method to study spatial variability of Groundwater quality in Balarood plain. Their results showed spherical model is the best model for fitting on experimental variogram of EC variable. Ahmed (2002) used kriging method to estimate total dissolution solid in Groundwater and demonstrated accuracy of this method to prediction of total dissolution solid.
Kriging is one of the most popular and robust interpolation techniques among other techniques. It integrates both the spatial correlation and the dependence in the prediction of a known variable. Estimations of nearly all spatial interpolation methods can be represented as weighted averages of the sampled data (Bodrud-Doza et al. 2016). The presence of a spatial structure where observations close to each other are more alike than those that are far apart (spatial autocorrelation) is a prerequisite to the application of geostatistics. The experimental variogram measures the average degree of dissimilarity between no sampled values and a nearby data value, and thus can depict autocorrelation at various distances. The value of the experimental variogram for a separation distance of h (referred to as the lag) is half the average squared difference between the value at Z(x i ) and the value at Z(x i + h) (Eq. 2).
The validation and the sufficiency of the developed model variogram can be tested via a technique called cross validation. Cross validation estimation is obtained by leaving one sample out and using the remaining data. This test allows assessing the goodness of fitting of the variogram model, the appropriateness of neighborhood and type of kriging used. The interpolation values are compared to the real values and then the least square error models are selected for regional estimation (Leuangthong et al. 2004). Cross validation result of predicted and measured EC data in ordinary kriging method is presented in Fig. 3 and this graph show well evaluation of estimation by using this method.
Electrical conductivity is a parameter related to total dissolved solids (TDS). EC is actually a measure of solution in terms of its capacity to transmit current. The importance of EC and TDS lies in their effect on the corrosion of a water sample and in their effect on the solubility of slightly soluble compounds such as CaCO3. In general, as TDS and EC increase, the corrosion of the water increases. Therefore, study of spatial variation of groundwater EC is necessary for optimized management of groundwater resources. According to evaluation criteria, the accuracy of geostatistical methods in the estimation of groundwater EC is assessed as very good. The results showed that ordinary kriging can be applied as an appropriate tool to estimate the EC of groundwater in areas with data restriction. In this study, interpolation methods especially OK showed amount of groundwater EC was high in coastal land and lowlands than the other regions, therefore effective actions should carry out to prevent EC increasing. Generally, results of this research indicated that geostatistics models are suitable for estimation of groundwater quality.
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