The geometrical background of inversion

Abstract

The results of model calculation (direct problem solutions above model parameter space) determine an embedded continuous and differentiable surface in the Euclidean space of measurements. This multidimensional subspace contains the possible expected values of measurement vectors according to the assumed rock model as a projection of measurement points (expressing the model and real rock equivalences). The model parameters are the natural coordinates of this subspace, determining a contravariant curvilinear coordinate system (“flat world” for the inversion). The local curvature of this surface is very important factor of covariance matrices and the possible bias of estimated parameters. In this article the role of curvature is discussed and the shortage of conventional (first order) inversion is demonstrated by simple example and the possibility of bias correction.