An eigenstate formulation of the magnetotelluric impedance tensor

Abstract

An important step in the interpretation of magnetotelluric (MT) data is the extraction of scalar parameters from the impedance tensor Z, the transfer function which relates the observed horizontal magnetic and electric fields. The conventional approach defines parameters in terms of elements of a coordinate‐rotated tensor. The rotation angle is chosen such that Z′(θ) approximates in some sense the form for a two‐dimensional (2-D) subsurface conductivity distribution, with zero elements on the diagonal. There are two major problems with this approach. (1) Apparent resistivities, defined from the off‐diagonal elements of the rotated tensor, are independent of the trace of Z. It is problematic that apparent resistivities, the parameters for which we have physical analogs and which are most heavily used in interpretation, are insensitive to the addition of an arbitrary constant on the diagonal of Z. (2) The conventional parameter set is incomplete; there are two degrees of freedom in Z which are transparent to all parameters. Through a variation of the classical eigenstate formulation of a matrix, it is shown that in general there exist two, and only two, polarization states for which the electric and magnetic fields have the same polarization at perpendicular orientations. For each eigenstate the magnetic and electric fields are related by a scalar, the eigenvalue for that state. This scalar relationship between fields is of identical form to the solution for transverse electromagnetic (TEM) waves in a homogeneous medium and thus provides a physically more satisfactory basis for defining apparent resistivity than the conventional approach using the off‐diagonal elements of the coordinate‐rotated impedance tensor. The eigenstate and coordinate‐rotation methods yield identical results in the limited cases of 1-D and 2-D subsurface conductivity distributions. The eigenstates provide the basis for new definitions of parameters as concise, closed expressions which are complete and more amenable to interpretational insight. The polarization ellipses defined by the eigenstates provide a concise display in real space of all the information contained in the impedance tensor.

Application of structural geology to mineral and energy resources of the Central and Western United States

“…The data points were subjectively weighted based on their quality. If the data are considered equally, the resulting curve is equivalent to using the geometric mean sounding curve, which for two-dimensional structures is an invariant curve even for tensor (phase-amplitude measurements) data (Berdichevsky and Dimitriev, 1976;Eggers, 1982;Park and Livelybrooks, 1989). Nevertheless, direct information about the dimensionality of structures is lost in this approach.…”

The distortion tensor of magnetotellurics: a tutorial on some properties

“…Eigenanalysis and singular value decomposition (SVD) are two methods of linear algebra which it is natural to assess for applications to such studies (Eggers, 1982;LaTorraca et al, 1986;Yee and Paulson, 1987). If applied to the distortion tensor, as in this paper, eigenanalysis and SVD will individually determine, with a 90 ambiguity, the geologic strike direction for the most simple 2D case.…”

Magnetotelluric analysis using Mohr circles

“…The tensor quoted by Eggers (1982) is thus shown in Figure 2, with real and quadrature circles. What may now be sought, for each circle, are two directions for which the electric and magnetic fields are at perpendicular orientations.…”

“…The last decade has seen a development of interest in methods for analysing a magnetotelluric impedance tensor for this sort of information. General analysis has been advanced in a series of papers by Eggers (1982), Spitz (1985), LaTorraca et al (1986), and Yee and Paulson (1987). Another line of inquiry has been the development of particular models, involving a combination of local and regional effects.…”

“…The method is explored by seeking directions for the Eggers (1982) and LaTorraca et al (1986) analyses, when the real and quadrature parts of a tensor are examined separately. Another traditionally important concept in magnetotelluric studies has been that of invariance under the rotation of axes, and a number of traditional magnetotelluric invariants are discussed, again taking the real and quadrature parts of a tensor separately.…”