Development and Application of Geostatistical Models with Translation Variant Variograms

The kriging process begins by the selection of an appropriate variogram. A variogram is often deemed appropriate after being compared favorably with an experimental variogram. One of the caveats with this approach is that in order for the experimental variogram to make sense, the increments of the stochastic process in question must have a translation invariant second moment. We make the case that processes with nonstationary increments (which have a translation variant second moment) are likely to be found in geophysical settings. A method for kriging in the presence of translation variant variograms is also presented. The method consists of introducing nonstationarity into common, translation invariant variograms by employing a nonlinear spatial or temporal transformation. Analytical model selection criteria are then used to help inform the selection of an appropriate variogram, including whether nonstationarity is present or stationary approaches are sufficient for a given problem.