Generalized Riccati equations for 1-D magnetotelluric impedances over anisotropic conductors Part I: Plane wave field model

Abstract

In the 1-D magnetotelluric theory, a Riccati equation governs the depth variation of the impedance, or admittance, for a given distribution of the electrical conductivity. This equation can be used to compute the surface magnetotelluric functions for generally piecewise continuous conductivity profiles. In case of a simple layered medium, it provides the classical formulae for recalculating recursively the impedances between the individual layer boundaries. We present an extended version of the Riccati differential equations for generally anisotropic 1-D structures for the case of a plane wave incident field. Relation between the standard matrix propagation procedure for a layered medium and the Riccati equation approach, as a limiting case of the former, is demonstrated. In the anisotropic case, all elements of the 2 × 2 impedance tensor are present and, consequently, a system of four coupled Riccati equations has to be considered. Standard methods for the numerical solution of systems of ordinary differential equations are applied to the Riccati system, which gives an efficient alternative to the current matrix propagation procedures for the numerical evaluation of 1-D magnetotelluric impedances in anisotropic media. As an application, a synthetic study on the influence of a depth-variable regional strike on magnetotelluric decomposition results is presented, with the variable strike simulated by a variable anisotropy within the 1-D section.

Canonical decomposition of magnetotelluric responses: Experiment on 1D anisotropic structures

“…However, it's notable that geophysical models resolved by different independent or combinations of impedance elements can indicate a substantial contrast to each other; although various modes of response have distinct sensitivities to the same structural features, and the reverse models are affected by various factors, such as data noises, errors and insufficient assumptions of 2D distribution, anisotropy should still be taken into account; moreover, field survey itself in tensor frame provides much convenience for us to research into anisotropy. In addition, there are some similar features between model responses of 1D anisotropy and 2D isotropy, as under some circumstances, the trace of impedance tensor equals to zero, diagonal elements are zero while offdiagonal elements are not zero when the tensor is rotated to electrical principal directions, and when galvanic distortion is considered, sometimes column elements have the same phases and shifted magnitudes (Kováciková and Pek, 2002;Santos and Mendes-Victor, 2000); thus it's nearly impossible to distinguish them by rotation analyses. Nevertheless, Jones (2012) has reviewed several methods that attempt to recognize different properties of 1D anisotropy and 2D isotropy; Heise et al (2006) made a comparison of phase split and phase tensor of several simple anisotropic models, which pointed out that these two phenomena are not direct evidences of the existence of anisotropy.…”

Generalized Riccati equations for 1-D magnetotelluric impedances over anisotropic conductors Part II: Non-uniform source field model

“…cited by a non-uniform source field To find the solution of (4) for the field components within a 1-D anisotropic medium that consists of a stack of homogeneous layers underlain by a homogeneous anisotropic half-space, we could almost exactly duplicate the procedure for solving (1) with the uniform source field (see Kováčiková and Pek, 2002, section 2.2).…”

“…To find the solution to (4) for the field vectors e and h for a particular pair of the spatial frequencies (ξ, η), we can again adopt the matrix propagation procedure used earlier for the plane wave model (Kováčiková and Pek, 2002, section 2.2). The principal difference with respect to the uniform field case is the general presence of both the vertical magnetic fields and vertical currents within the structure for non-zero ξ and η.…”

“…In the first part of this paper (Kováčiková and Pek, 2002), we have demonstrated that the Riccati equation approach to studying magnetotelluric (MT) impedances in 1-D anisotropic conductors is a simple and effective alternative to the matrix and impedance propagation methods generally used for this purpose in the plane wave model. We will summarize the main theoretical results of that previous study here for reference purposes:…”

“…In this paper we will show that the Riccati equation approach from Kováčiková and Pek (2002) can be readily extended to the spectral impedances in generally anisotropic 1-D conductors.…”