In the previous chapter, we saw how geostatistics can be used to generate 3D heterogeneity models satisfying a number of input statistical parameters, such as the vari-ogram, or equivalently, the spectral density. Relationships between simulated parameters and seismic-derived information were statistical, usually coded as a linear correlation between the seismic parameter and the simulated variable. This was the “nprimary- versus secondary-variable” approach. We also saw earlier that the geostatistical paradigm, based on the concept of trend and covariance, can be considered to be an approach for coding the a priori model that is often used in Bayesian terminology.
In parallel with geostatistical developments, inversion technology made significant progress in the last three decades (Fig. 5-1). Thanks to the Bayesian formalism promoted by authors such as Tarantola (1987) and Duijndam (1988), the standard optimization-based deterministic approach (minimization of an objective function with a regu-larization term) was improved, and it became possible to produce an estimate of uncertainty together with the inverted acoustic-impedance block.
Logically, the idea emerged in the early 1990s to apply the conditional simulation approach to acoustic-impedance inversion to produce multiple 3D realizations, all constrained by seismic data. This resulted in geostatistical-inversion methodology. In the following, we will consider that geostatistical and stochastic inversion are exactly the same thing.
The method discussed here is that presented in the papers of Bortoli et al. (1992) and Haas and Dubrule (1994). Geostatistical inversion (GI) consists of generating 3D acoustic-impedance realizations, all