Magnetotelluric tensors, electromagnetic field scattering and distortion in three‐dimensional environments

Abstract

This paper describes how subsurface resistivity distributions can be estimated directly from the magnetotelluric (MT) tensor relationship between electric and magnetic fields observed on a three‐dimensional (3‐D) half‐space. It presents an inhomogeneous plane wave analogy where relationships between horizontal electric and magnetic fields, and an apparent current density define an apparent resistivity tensor constructed from a quadratic function of the MT tensor. An extended‐Born relationship allows the electric field to be normalized with respect to an apparent background current density. The model is generalized by including the vertical magnetic field in a 3 by 3 MT response tensor. A complex apparent wave number tensor, constructed from this tensor, has eigenvalues which, using the plane wave analogy, are the vertical wave numbers associated with the eigenpolarizations of propagating waves in the model half space. The elements associated with the vertical magnetic field transfer function define the horizontal wave numbers. An extended 3 by 3 phase tensor contains four elements of the conventional 2 by 2 phase tensor and two elements associated with the vertical magnetic transfer function. The extended phase tensor and a single real distortion tensor with six independent elements can be used to quantify static electric and magnetic field distortions. The approach provides a theoretical basis for visualization and migration of MT data, in comparison with results from other electrical and EM techniques, a starting point for constrained 3‐D inversions, and an assessment of results with other geophysical and geological data.

Magnetotelluric Apparent Resistivity Tensors for Improved Interpretations and 3‐D Inversions

“…The relationship E h = ZB h between the horizontal electric field and magnetic induction, E h (V/m) and B h (T) respectively, at the surface of the Earth, contains information about the subsurface electrical conductivity in the period‐dependent 2 by 2 MT response tensor, Z (m/s). Brown (2016) used an analogy for the propagation of plane EM waves through a homogeneous, dissipative, and anisotropic conductivity model half‐space to construct a second rank 3 by 3 tensor γ (m −2 ) consisting of quadratic functions of complex wave numbers associated with the EM waves in the model. The approach has similarities with the c‐response (Schmucker, 1987), which relates a 1‐D resistivity distribution to the first vertical derivative of the horizontal electric field.…”

“…The approach has similarities with the c‐response (Schmucker, 1987), which relates a 1‐D resistivity distribution to the first vertical derivative of the horizontal electric field. In the following, the formulations of Brown (2016) are restricted to the horizontal EM fields and all tensors ( Z, γ, σ, σ a , ρ a ) have a dimension of 2 by 2.…”

“…The apparent conductivity tensor is a quadratic function of Z given by (Brown, 2016) where μ is the free space magnetic permeability, det represents the determinant of a matrix, the superscript T represents the transpose, and Y = Z −1 (s/m) is the inverse of the MT response tensor. The complex apparent resistivity tensor, ρ a (Ω m), is therefore defined as …”

“…If local conductive heterogeneities are very different from regional structures, the condition may not be a good approximation so the regional horizontal magnetic field may be changed. Brown (2016, equation 28) assumed that similar to the electric field, the distorted magnetic field is also a linear superposition of the regional magnetic field and a scattered magnetic field in‐phase with the regional magnetic field. In these circumstances, a single 2 × 2 tensor C is sufficient to parameterize the distortion of both EM fields.…”

“…Several studies have introduced apparent resistivity tensors in different contexts: Caldwell and Bibby (1998) used them for long‐offset time domain electric field data, Caldwell et al (2002) used them to present controlled‐source MT data, and Weckmann et al (2003) showed examples for natural source field MT data. Brown (2016) developed the theory to define an extended‐Born MT scattering tensor and apparent background electric and magnetic fields from a complex apparent conductivity tensor. In this paper we use the inverse of this apparent conductivity tensor, the complex apparent resistivity tensor, and decompose it into a real and imaginary tensor.…”