We have presented the model used by geostatistics for 2D and 3D petroleum applications. This quantified geological model assumes that the variable of interest is the sum of a deterministic trend plus a residual characterized by its variogram or its covariance. When there are enough data, the variogram model can be fitted to the experimental function. In other cases, assumptions based on geological knowledge, combined with a variogram analysis of seismic data, will be used to define the variogram model. Variograms can be related to fractal models and to a priori information used by geo-physicists in seismic inversion or in Fourier analysis (Fig. 7-1).
The geostatistical model can address the problem of deterministic interpolation through kriging. A degree of smoothing can be applied to kriging through the error-cokriging approach, which allows the filtering of random noise, whereas the factorial kriging approach allows the filtering of short-range — or high-frequency — terms due (for instance) to seismic-acquisition artifacts. Kriging based on well data can also incorporate extra information coming from seismic data, through the external drift or the collocated cokriging approach. Kriging is closely related to other interpolation techniques, such as splines or radial-basis functions. Specifying a kriging model amounts to specifying the regularization term of energy-based inversion techniques.
Kriging and all its family of associated techniques remain a deterministic method. The production of a minimum variance estimate results in an interpolation that is very smooth away from the data points. In practice, however, geology has no reason to