Finding the shape of a local heterogeneity by means of a structural inversion with constraints

Various aspects of structural inversion are considered. The aim of the inversion is limited to finding the shape of an isolated 2D homogeneous body, although the technique may be generalized to the case of interfaces with steep fragments, faults or overhangs. The unknown parameters are shifts of border points. The shift directions can be normal to the initial heterogeneity configuration or to another contour. Medium properties (seismic velocities, densities, etc.) within the heterogeneity are assumed to be known. The optimal shape is determined iteratively, a quadratic objective function being minimized at each iteration with the conjugate-gradient method. Special attention is paid to preventing self-intersections, for which purpose each unknown parameter is forced to lie within a certain predetermined interval. In order to achieve this, the classical conjugate-gradient method has been modified accordingly. Three numerical examples are considered. These illustrate how the developed technique can be applied to different practical problems. The first example is devoted to monitoring an oil/steam interface by gravity gradiometry measurements. In the second example, a cross-hole seismic experiment is simulated. It is shown that a structural inversion can restore the configuration of a local body much more accurately than traditional seismic tomography. In the third example, the shape of a salt dome is reconstructed by joint inversion of refracted traveltimes and gravity measurements. This example demonstrates how different kinds of data, used simultaneously in a structural inversion, can complement each other.