Semivariogram

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. 

Fitting a model equation to an experimental semivariogram is a trial-and-error process, usually done by eye. Clark describes and gives examples of the manual process, while 01ea > provides a program which computes a model. 

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

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. 

Geostatistical methods are also termed random field sampling as opposed to random sampling [BORGMAN and QUIMBY, 1988], The spatial dependence of data and their mutual correlation can be analyzed by use of semivariograms. Statements on the anisotropy of the spatial distribution are also possible. Kriging, a geostatistical method of... 

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

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. 

According to Eq. 4-28 the theoretical semivariogram function has the value y = 0 for / = 0. Semivariograms obtained from experimental data often have a positive value of intersection with the y(/)-axis expressed by C0 (see Fig. 4-5). The point of intersection is named nugget effect or nugget variance. This term was coined in the mining industry and indicates an unexplained random variance which characterizes the microinho-... 

Fig. 4-5. Example of an experimental semivariogram with nugget effect Cc, sill C, and range L...

An experimental semivariogram can be modeled by fitting a simple function to the data points. Linear, spherical, or exponential models are often used [KATEMAN, 1987 AKIN and SIEMES.1988]. 

At this point it should be noted that the experimental semivariogram may be influenced by many errors. One way to minimize these errors is the computation of the indicator semivariogram I(x,Zcrit) [JOURNEL, 1983]. The function Z(x) has to be transformed according to ... 

A comparison between an autocorrelogram and a semivariogram, as exemplified by the design of a sampling plan for aquatic sediments, is given by WEHRENS et al. [1993]. 

If the sampling locations are not distributed in a regular raster, the following methods are available to determine the semivariogram [AKIN and SIEMES, 1988] ... 

The sampling points can be projected to lines of different orientation. The semivariogram can then be computed along these lines with the distance l Al. 

The semivariogram as a variance function can also be used to estimate the value and the variance for new points not sampled in the investigated area. The method applied for this purpose is termed kriging. Kriging is a special regression method for interpolation of spatially or temporally correlated data with minimization of variance. The normal distribution of the data is an important condition. If the original data are not normally distributed, which is often the case for trace components in environmental compartments, the logarithm of the data or otherwise transformed data have to be applied to obtain a normal distribution of the data (see also Section 9.4). 

The quality of a theoretical semivariogram model can be assessed by means of point kriging and cross-validation [MYERS, 1991]. Each point of the data set is deleted one after another and then newly estimated according to Eqs. 4-33 to 4-35 by means of its neighbors. Additionally, the kriging variance is calculated for each estimated point (Eq. 4-36). 

The goodness of fit for experimental data to a theoretical semivariogram model can be tested by means of cross-validation. The best model is that with the smallest deviation from the mean zero and with the smallest standard deviation. 

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. 

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],... 

Semivariograms shall be applied for the characterization of the spatial structure of the data. First, the data have to be tested for normal distribution. The results of the test hypothesis are acceptable for potassium only, in all other cases the hypothesis has to be rejected. After logarithmic transformation of the original data, however, a normal distribution can be obtained. Thus, the following calculations must be performed on the logarithms of the data - only for potassium can the untransformed values be used. 

Tab. 9-9. Results of semivariogram estimation for some selected elements...

Fig. 9-21. Directional semivariogram for iron ( A north-south direction, east-west direction)...

The state of pollution of the soil can be more objectively described by means of geostatistical methods. The computation of semivariograms and the use of kriging uncover spatial structures which are not discernible by means of simple univariate statistical tests. 

The semivariogram gives information about both the spatial structure of the element distribution and the anisotropy, i.e. the dependence of this distribution on direction. 

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