The continuation inverse problem revisited |
| |
| Authors: | Stephen P Huestis |
| |
| Institution: | Department of Earth and Planetary Sciences, University of New Mexico, Albuquerque, NM 87131, USA. E-mail: |
| |
| Abstract: | The non-uniqueness of the continuation of a finite collection of harmonic potential field data to a level surface in the source-free region forces its treatment as an inverse problem. A formalism is proposed for the construction of continuation functions which are extremal by various measures. The problem is cast in such a form that the inverse problem solution is the potential function on the lowest horizontal surface above all sources, serving as the boundary function for the Dirichlet problem in the upper half-plane. The desired continuation, at the higher level of interest, must then be in the range of the upward continuation operator acting on this boundary function, rather than being allowed the full freedom of itself being part of a Dirichlet problem boundary function. Extremal solutions minimize non-linear functionals of the continuation function, which are re-expressed as different functionals of the boundary function. A crux of the method is that there is no essential distinction between the upward and downward continuation inverse problems to levels above or below data locations. Casting the optimization as a Lagrange multiplier problem leads to an integral equation for the boundary function, which is readily solved in the Fourier domain for a certain class of functionals. The desired extremal continuation is then given by upward continuation. It is found that for some functionals, application of the Lagrange multiplier theorem requires a further restriction on the set of allowable boundary functions: bandlimitedness is a natural choice for the continuation problem. With this imposition, the theory is developed in detail for semi-norm functionals penalizing departure from a constant potential, in the 2-norm and Sobelev norm senses, and illustrated by application for a small synthetic Deep Tow magnetic field data set. |
| |
| Keywords: | Fourier transforms inverse problem magnetic anomalies |
|
|
