Large-scale magnetic inversion using differential equations and ocTrees

Abstract

The inversion of large-scale magnetic data sets has historically been successfully achieved through integral transforms of the large, dense sensitivity matrix. Two well-known transforms, the discrete Fourier and multi-dimensional wavelet, reduce the required storage and ultimately speed of the inversion by storing only the necessary transform coefficients without losing accuracy. The main drawback of the approaches is the required calculation of the entire dense sensitivity matrix prior to the transform. This process can be much more costly than the inversion itself. We solve the magnetostatic Maxwell's equation using a finite volume technique on an ocTree-based mesh. The ocTree mesh greatly reduces the time required for the inversion process. When working in the differential equation domain it is not necessary to explicitly form the sensitivity matrix; this decreases the storage requirement of the problem and increases the overall speed of the inversion. The principal mesh is broken up into sub-domain ocTree grids to further enable parallelization of the forward problem. These grids extend the entire domain of the principal mesh to include large regional features that may influence the data. We present the discretization of the equations and verify the accuracy of the modelling both with the principal mesh and with multiple sub-domains. We show a synthetic example and a large field example consisting of over 4 million data and 5 million model cells that was inverted on a desktop computer.