Intersections, ideals, and inversion

• Main Author: Vasco, D. W.
• Other Authors: LBNL Director's Research and Development
• Format: Report
• Language: English
• Published: Lawrence Berkeley National Laboratory 1998
• Subjects: Numerical Solution; Integral Equations; Algebra; 99 Mathematics; Computers; Information Science; Management; Law; Miscellaneous; 58 Geosciences; Partial Differential Equations; Antarctica; Magnetotelluric Surveys; Antarc*
• Online Access: https://doi.org/10.2172/6448; https://digital.library.unt.edu/ark:/67531/metadc690439/

Techniques from computational algebra provide a framework for treating large classes of inverse problems. In particular, the discretization of many types of integral equations and of partial differential equations with undetermined coefficients lead to systems of polynomial equations. The structure of the solution set of such equations may be examined using algebraic techniques. For example, the existence and dimensionality of the solution set may be determined. Furthermore, it is possible to bound the total number of solutions. The approach is illustrated by a numerical application to the inverse problem associated with the Helmholtz equation. The algebraic methods are used in the inversion of a set of transverse electric (TE) mode magnetotelluric data from Antarctica. The existence of solutions is demonstrated and the number of solutions is found to be finite, bounded from above at 50. The best fitting structure is dominantly onedimensional with a low crustal resistivity of about 2 ohm-m. Such a low value is compatible with studies suggesting lower surface wave velocities than found in typical stable cratons.