Inversion of geophysical data using an approximate inverse mapping

Abstract

SummaryRealistic geologic features are 3-D and inverse techniques which rely upon linearization and computation of a sensitivity matrix to show how a change in the model affects a particular datum, can require prohibitive amounts of computation. Even 104 data collected over an earth parametrized into 102× 102× 102 elements has a sensitivity matrix which is 104× 106. The generation of that matrix requires the solution of many 3-D forward problems and its solution is also computationally intensive. In this paper we formulate a general technique for solving large-scale inverse problems which does not involve full linearization and which can obviate the need to solve a large system of equations. The method uses accurate forward modelling to compute responses, but only uses an approximate inverse mapping to map data back to model space. The approximate inverse mapping is chosen with emphasis on the physics of the problem and on computational expediency. There are two ways to implement the AIM (Approximate Inverse Mapping) inversion. At any iteration step, AIM-MS applies the approximate inverse mapping to forward modelled data and also applies the same mapping to the observations; the model perturbation is taken as the difference between the resulting functions. In AIM-DS, an alteration to the data is sought, such that the approximate inverse mapping applied to the altered data yields a model which adequately satisfies the observations. The approximate mapping inversion is illustrated with a simple parametric inverse problem and with the inversion of magnetotelluric (MT) data to recover a 1-D conductivity model. To illustrate the technique in a realistically complicated problem we invert MT data acquired from a line of stations over a 2-D conductivity structure. TE and TM mode data are inverted individually and as determinant averages. As a final example we invert 900 data, with and without noise, to recover a model that is parametrized by 1500 cells of unknown conductivity. The inversion is found to be computationally efficient and robust.