Ambiguity and Depth Resolution in Potential Field Inversion

We study the inversion of potential fields and evaluate the degree of depth resolution achievable for a given problem. To this end, we introduce a powerful new tool: the depth-resolution plot (DRP). The DRP allows a theoretical study of how much the depth resolution in a potential-field inversion is influenced by the way the problem is discretized and regularized. The DRP also allows a careful study of the influence of various kinds of ambiguities, such as those from data errors or of a purely algebraic nature. The achievable depth resolution is related to the given discretization, regularization, and data noise level. We compute DRP by means of singular-value decomposition (SVD) or its generalization (GSVD), depending on the particular regularization method chosen. To illustrate the use of the DRP, we assume a source volume of specified depth and horizontal extent in which the solution is piecewise constant within a 3D grid of blocks. We consider various linear regularization terms in a Tikhonov (damped least-squares) formulation, some based on using higher-order derivatives in the objective function. DRPs are illustrated for both synthetic and real data. Our analysis shows that if the algebraic ambiguity is not too large and a suitable smoothing norm is used, some depth resolution can be obtained without resorting to any subjective choice of depth weighting.