Semivariogram analysis

The theoretical basis of the kriging method is the so-called intrinsic hypothesis. A random function Z(x) accomplishes this hypothesis if two assumptions are fulfilled  

Under these suppositions, the application of linear geostatistical methods, like point kriging, is possible on the basis of the semivariogram. 

Usually the computation of the semivariance is the first step in geostatistical analysis. The semivariance y(l) is expressed by  

An important advantage of geostatistical methods is that the sampling points do not necessarily have to be regularly distributed. For environmental investigation, this point is of particular importance because sometimes certain locations in the area cannot be sampled. 

If the sampling points are not distributed regularly, the locations of the sampling points have to be transformed into polar coordinates (direction P A P, distance of the points l+AI) for the computation of the semivariogram. 

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Similar models have been applied in geological exploration and environmental studies. There semivariogram analysis (Akin and Siemens [1988], Einax et al. [1997]) plays a comparable role than autocorrelation analysis for the characterization of stochastic processes. 

Objects which are internally correlated for example volumes sampled from rivers, soils, or ambient air, can be treated by autocorrelation analysis or semivariogram analysis. The range up to a critical level of error probability is an expression of the critical spatial or temporal distance between sampling points. 

If the purpose of sampling is the detailed description of the composition of an object, the character of the internal correlation has to be investigated. The methods of autocorrelation and/or semivariogram analysis, as described in Sections 6.6 and 4.4.2, may be useful for clarification of the internal spatial and/or temporal relationships existing within the parent population to be sampled. Geostatistical methods, e.g. kriging, enable undistorted estimation of the composition of unsampled locations in the area of investigation. 

Olea, R. A. Measuring spatial dependence with semivariograms , Kansas Geological Survey Series on Spatial Analysis , no. 3, University Kansas, Lawrence... 

The required distance has to be chosen from the empirical semivariogram or (if the first sampling was done equidistantly) by autocorrelation analysis also (see example for soil sampling in Sections 9.1 and 9.4). Clearly, the required distance depends on the relationship between nugget effect and sill. The length determination is described in detail by YFANTIS et al. [1987],... 

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