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Geostatistics originate from the work of French mathematician G. Matheron and South African mining engineer D.G. Krige, who worked on developing an optimal interpolation procedure based on regionalized variables, for application in mining. Consequently, geostatistical techniques are also known as ‘kriging’. Geostatistical methods are founded in the principles of statistical spatial autocorrelation. They are suited to situations when the variation within or density of sample attribute data are too great to be adequately modeled by simpler methods. Another advantage is that these methods contain a probabilistic estimate of the interpolation quality itself (Burrough and McDonnell, 1998).
When data sampling is sparse, it is crucial to adequately evaluate the validity of the assumptions that are made in regards to the underlying variation amongst the data. Geostatistical interpolation techniques are based upon the acknowledgement that modeling by means of a simple, smooth mathematical function is often inappropriate for spatially continuous variable because the spatial variation is too irregular. Geostatistical methods divide spatial variation into three components: “(a) deterministic variation (different levels of trends) that can be treated as useful, soft information, (b) spatially auto correlated, but physically difficult to explain variations, and finally (c) uncorrelated noise” (Burrough and McDonnell, 1998).
Geostatistical methods of spatial estimation are an improvement over the simpler Theissen method because the use of information from surrounding samples produces estimates of greater precision and the estimation variance, which provides an uncertainty measurement for the estimates, is minimized (Pardo-Iguzquiza, 1998). Geostatistical procedures are also an advancement over other interpolation methods because their weights are chosen a priori such that the interpolated surface will be optimized, providing a Best Linear Unbiased Estimate (BLUE). Earlier interpolation methods did not provide any means of determining whether or not the best weighting values had been chosen. As a result of these improvements, geostatistical approaches have become the standard tool for interpolation (Martinez-Cob, 1996; Pardo-Iguzquiza, 1998; Goovaerts, 2000).
Many specific geostatistical techniques exist. ESRI (2002) provides the following definitions of the three most common forms of kriging:
Ordinary Kriging produces interpolation values by assuming a constant but unknown mean value, allowing local influences due to nearby neighbouring values. Because the mean is unknown, there are few assumptions. This makes Ordinary Kriging particularly flexible, but perhaps less powerful than other methods.
Simple kriging produces interpolation values by assuming a constant but known mean value, allowing local influences due to nearby neighbouring values. Because the mean is known, it is slightly more powerful than Ordinary Kriging, but in many situations the selection of a mean value is not obvious.
Universal Kriging produces interpolation values by assuming a trend surface with unknown coefficients, but allowing local influences due to nearby neighbouring values. It is possible to over fit the trend surface, which does not leave enough variation in the random errors to properly reflect uncertainty in the model. When used properly, Universal Kriging is more powerful than Ordinary Kriging because it explains much of the variation in the data through the nonrandom trend surface.”
Other forms include Indicator Kriging, Probability Kriging and Disjunctive Kriging. Technical details on all these methods can be found in Burrough and McDonnell (1998). Cokriging is a multivariate form that uses information on more than one co-regionalized variable. It can be especially useful when there is a second attribute that is more densely measured or cheap to measure. However, the second attribute must be measured at a greater number of points; otherwise Cokriging will not result in any improvement over univariate kriging methods. Furthermore, there must exist a high degree of correlation between the two variables (Eastman, 1999; Goovaerts, 2000). Cokriging can have the same forms as the above mentioned kriging forms.
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