Kriging is an advanced geostatistical procedure that generates an estimated surface from a scattered set of points with z-values. Using geostatistical techniques, you can create surfaces incorporating the statistical properties of the measured data. Kriging is based on statistics. These techniques produce not only prediction surfaces but also error or uncertainty surfaces, giving you an indication of how good the predictions are. More so than other interpolation methods, a thorough investigation of the spatial behavior of the phenomenon represented by the z-values should be done before you select the best estimation method for generating the output surface.
Many kriging methods are associated with geostatistics, but they are all in the kriging family. Ordinary, simple, universal, probability, indicator, and disjunctive kriging, along with their counterparts in cokriging, are all available in the Geostatistical Analyst. Not only do these kriging methods create predictions and error surfaces, but they can also produce probability and quantile output maps depending on user needs.
Let’s focus more on ordinary kriging (OK) interpolation in this section. This form of kriging usually involves four steps:
- Removing any spatial trend in the data (if present).
- Computing the experimental variogram, γγ, which is a measure of spatial autocorrelation.
- Defining an experimental variogram model that best characterizes the spatial autocorrelation in the data.
- Interpolating the surface using the experimental variogram.
- Adding the kriged interpolated surface to the trend interpolated surface to produce the final output.
Kriging is the estimation procedure using known values and a semi-variogram to determine unknown values. The procedures involved in kriging incorporate measures of error and uncertainty when determining estimations. Based on the semi-variogram used, optimal weights are assigned to unknown values in order to calculate the unknown ones. Since the variogram changes with distance, the weights depend on the known sample distribution.
The basic equation used in ordinary kriging is as follows:
- d is the distance between known points,
- n is the number of pairs of sample separated by d;
- Z is the attribute value (elevation of known points).
The equation indicates that the semi-variance is expected to increase as d increases. One of the most popular approaches is the ordinary kriging, which will be applied in this study. Ordinary kriging assumes the model:
Z(s) = μ + ε(s),
where μ is an unknown constant.
One of the main issues concerning ordinary kriging is whether the assumption of a constant mean is reasonable. Sometimes there are good scientific reasons to reject this assumption. However, as a simple prediction method, it has remarkable flexibility.