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Geochemists examine the spatial patterns of geochemical elements to understand the mechanisms and that control their spatial distribution, therefore, an understanding of the spatial variability of geochemical elements has important implications. It has been argued that all spatial data fulfill the generalization that values from samples near to one another tend to be more similar than those that are further apart (Liebhold & Sharov, 1998; Epperson, 2000). This tendency is termed spatial autocorrelation or spatial association (Cliff & Ord, 1981). Consequently, there has been an increasing interest in the use of variogram, spatial correlograms and covariance functions for describing patterns of spatial variability and measuring degree of spatial autocorrelation (Frogbrook & Oliver, 2001; Zhang & McGrath, 2004; Frutos et al., 2007; Borcard & Legendre, 2012; Legendre & Legendre, 2012).
The classical regional variogram estimator proposed by Matheron (1963) was first used to spatial variability of spatial data. However, according to Cressie and Hawkins (1980), the sample variogram can give a poor estimate of the regional variogram if there are outliers in the data. The sample mean is not stable estimator of theoretical mean. Genton (1998) shows that one single outlier can destroy this estimator completely. For that reason, several types of estimators based on robust estimation of scale and quantiles have been proposed, such as typical robust variogram proposed by Dowd (1984) is the median of the magnitude of increments, variants proposed include the quantile variogram (Armstrong & Delfiner, 1980), the jack-knifing (Chung, 1984) and a variogram estimator based on a highly robust estimator of scale (Genton, 1998). Especially the variogram of order ½ proposed by Cressie and Hawkins (1980) has been widely used. The robust estimation of the Cressie variogram using of a fourth-root transformation was proposed when the distribution is normal-like in the central region but heavier than normal in the tails. The usual product moment covariogram estimator of a Gaussian process can have bias. In order to decrease the bias, the sample mean in the estimator is replaced with the sample median (Cressie & Hawkins, 1980). Variograms decompose the spatial variability of observed variables among distance classes (Legendre & Legendre, 2012), thus they have been widely used in modeling and interpreting ecological spatial dependence and spatial structure (Legendre & Legendre, 2012; Saraux et al., 2014; Roy et al., 2015), soil science (Iqbal et al., 2004; Tripathi et al., 2015), studies of spatial patterns of sill physico-chemical variables (Abu et al., 2011; Jiménez et al., 2011), quantifying the distribution of spatial patterns and changes in soil organic carbon in environmental science (Frogbrook & Oliver, 2001; Zhang & McGrath, 2004).