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UBC Theses and Dissertations

Forward modelling and inversion of geophysical magnetic data Lelièvre, Peter George


The ultimate goal of this research was to invert geophysical magnetic data to recover three dimensional distributions of subsurface magnetic susceptibility of any possible magnitude and geometric complexity. Magnetic data collected over bodies of high susceptibility contain significant self-demagnetization effects. Self-demagnetization causes magnetizations to rotate away from the external inducing field and causes the amplitude of the magnetic response to scale nonlinearly with susceptibility. These effects are highly dependent on the shape of the object and they complicate interpretation. Examples where self-demagnetization is important include surveys for detection and discrimination of unexploded military ordnance (UXO) and mineral exploration surveys over highly mineralized banded iron formations and nickel deposits. Current modelling methods that account for self-demagnetization effects are limited to simple bodies, such as ellipsoids, where the geometry of the body is represented by a few parameters. Standard forward modelling methods for general susceptibility distributions (i.e. methods that can deal with complicated bodies) neglect the effects of self-demagnetization and can produce inaccurate results and subsequent deterioration in performance of the inverse solution. Here, a full solution to Maxwell's equations for source-free magnetostatics was developed using a finite volume discretization. The Earth region of interest is discretized into many prismatic cells, each with constant susceptibility, which allows for models of arbitrary geometric complexity. The finite volume forward modelling method is valid for any linear medium and is appropriate for modelling the response of highly magnetic objects. The forward modelling method was refined and the code was tested against analytical solutions for simple bodies and against a slower, more memory intensive full solution for general distributions formulated in the integral equation domain. All tests showed the forward modelling method to be sound within expected error tolerances. The finite volume modelling method formed the foundation for a subsequent inversion algorithm. In the discretization, many more model cells are used than there are data. As such, the inverse problem is underdetermined. The inverse problem was formulated as an unconstrained optimization problem in which an objective function is minimized. The objective function was designed so that the data are fit to an acceptable degree and the recovered model has desired spatial characteristics. The resulting optimization problem was nonlinear and required an iterative solution, for which a Gauss-Newton approach was used. Testing for the inversion code included inversion of synthetic data for simple bodies and inversion of survey data collected over a planted UXO target. All tests showed positive results.

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