Conditional Simulation for Heterogeneity Modeling and Uncertainty Quantification
In the previous chapter, kriging proved to be an interpolation technique that was flexible enough to filter correlated or uncorrelated noise from the data or to combine seismic and well information. Thanks to the flexibility in the choice of the trend and the covariance model, kriging is closely related with splines, multiquadrics, and trend-surface analysis. Kriging, when it is based on the trend and covariance models fitted to the data, also provides an estimate of the uncertainty at every location of the map. However, kriging remains a deterministic approach that provides a very smooth image of geological variables that are, in most cases, very erratic.
As an example, Fig. 4-1 shows in red the kriged surface already used in Fig. 3-24 (Hohn data set). This surface is very smooth. A variogram calculated on the points of this surface would be extremely different from the spherical variogram fitted on the data and used as input to kriging. Is there not a contradiction here? Should not the vari-ogram of the kriged surface be the same as the spherical model used as input to kriging? The answer is definitely no, because the goal of kriging is not to generate a surface that mimics the actual variations of the interpolated variable, but to provide, at each location, an estimate that is as close as possible — on average — to the unknown value.
A simpler way to understand this is to use the example of a random variable, Z, taking