Forward modeling and inversion of potential field data using partial differential equations

The aim of geophysical inversion is to provide a feasible subsurface structure or distribution of physical properties of the Earth. There must be a forward modeling to predict the observations given a distribution of such properties. Standard forward modeling methods for potential fields are based on the integral equations. For large-scale or high-resolution data, however, forward models based on partial differential equations (PDE) are more efficient. These methods have been proven to be faster and less memory intensive than the standard approach. In the present work, I extend the PDE based modeling methods for both gravity and magnetics.;The gravity field from a density distribution can be computed through an indirect formulation by first solving Poisson's equation for the gravitational potential, and then numerically differentiating the potential to obtain the field. Alternatively, given the gradient of a density distribution one may directly solve the Poisson's equation of the gravity field, which does not require differentiation of the potential. I investigate this formulation and study its relative advantages. The direct formulation has the same degree of accuracy as the indirect formulation. Based on this result, the direct formulation can be used in inversion algorithms to recover derivatives of the density distribution as a mean to image density boundaries.;Like gravity, magnetic fields are described by Poisson's equation. I formulate the PDE solution of the magnetic problem by including anisotropy in the forward calculation of the magnetic field. I compare the forward modeling algorithm against analytical solutions of simple bodies and against the integral equation domain solution. I find that the PDE-based forward model can accurately predict the magnetic signal of anisotropic materials.;To conclude the study, I use PDE based forward methods to invert gravity and magnetic data collected over a copper-lead zinc deposit in northern New Brunswick, Canada. The solutions are consistent with other inversion algorithms that use the standard integral equation approach. The inversion of the magnetic field is for isotropic susceptibility only. Inversion which takes into account susceptibility anisotropy requires further development of the inversion algorithm which is beyond the scope of the present work.