Existence and uniqueness properties for the one-dimensional magnetotellurics inversion problem

Abstract

The one-dimensional magnetotelluric (MT) inversion problem is well known to be ill posed and nonlinear. This paper seeks an understanding of the mappings underlying the nonlinear relationships. These properties are used to study the validity of some exploration aspects of the problem that are essential for the practical use of MT as an exploration tool. This study has two major segments—results in response space and results in geologic space. In response space, the existence of optimal admittance curves has been proven. In the case of continuous data, uniqueness has been established. For discrete data, a relationship between alternative optimal solutions has been derived. In geologic space, the results are extremely significant to MT’s applicability to exploration. A class of conductivity functions Y, which contains all possible natural geological conditions, has been used as a framework for the study. Over this class Y, the one-dimensional magnetotellurics problem has been proven to be uniquely invertible from the admittance curves. This precludes the possibility of two different geologic models yielding the same complete data. As a result, the path is now clear to generate a description of a unimodal statistical distribution of feasible inversions to the real-world exploration problem with finitely sampled noisy data.