ON THE GEOSTATISTICAL APPROACH TO THE INVERSE PROBLEM

The geostatistical approach to the inverse problem is discussed with e mphasis on the importance of structural analysis. Although the geostat istical approach is occasionally misconstrued as mere cokriging, in fa ct it consists of two steps: estimation of statistical parameters (''s tructural analysis'') followed by estimation of the distributed parame ter conditional on the observations (''cokriging'' or ''weighted least squares''). It is argued that in inverse problems, which are algebrai cally undetermined, the challenge is not so much to reproduce the data as to select an algorithm with the prospect of giving good estimates where there are no observations. The essence of the geostatistical app roach is that instead of adjusting a grid-dependent and potentially la rge number of block conductivities (or other distributed parameters), a small number of structural parameters are fitted to the data. Once t his fitting is accomplished, the estimation of block conductivities en sues in a predetermined fashion without fitting of additional paramete rs. Also, the methodology is compared with a straightforward maximum a posteriori probability estimation method. It is shown that the fundam ental differences between the two approaches are: (a) they use differe nt principles to separate the estimation of covariance parameters from the estimation of the spatial variable; (b) the method for covariance parameter estimation in the geostatistical approach produces statisti cally unbiased estimates of the parameters that are not strongly depen dent on the discretization, while the other method is biased and its b ias becomes worse by refining the discretization into zones with diffe rent conductivity. Copyright (C) 1996 Elsevier Science Ltd