R. W. Groom

There are many occasions on which the magnetotelluric impedance tensor is affected by local galvanic distortion (channelling) of electric currents arising from induction in a conductive structure which is approximately two‐dimensional (2‐D) on a regional scale. Even though the inductive behavior is 2‐D, the resulting impedance tensor can be shown to have three‐dimensional (3‐D) behavior. Conventional procedures for rotating the impedance tensor such as minimizing the mean square modulus of the diagonal elements do not in general recover the principal axes of induction and thus do not recover the correct principal impedances but rather linear combinations of them. This paper presents a decomposition of the impedance tensor which separates the effects of 3‐D channeling from those of 2‐D induction. Where the impedance tensor is actually the result of regional 1‐D or 2‐D induction coupled with local frequency independent telluric distortion, the method correctly recovers the principal axes of induction and, except for a static shift (multiplication by a frequency independent real constant), the two principal impedances. Also obtained are two parameters (twist and shear), which partially describe the effects of telluric distortion. It is shown that the tensor operator which describes the telluric distortions can always be factored into the product of three tensor sub‐operators (twist, shear, local anisotropy) and a scalar. This product factorization allows assimilation of local anisotropy, if present, into the regional anisotropy. The method of decomposition is given in the paper along with a discussion of the improvements obtained over the conventional method and an example with real data.

The Born and Rytov approximations, widely used for solving scattering problems, are of limited utility for low‐frequency electromagnetic scattering in geophysical applications where conductivity can vary over many orders of magnitude. We present four new, relatively simple nonlinear estimators that can be used for rapid electromagnetic modeling. The first, termed the static localized nonlinear approximation, is designed specifically to correct the magnitude of the electric field internal to the scatterer. The second, termed the localized nonlinear approximation, improves the estimate of the phase of the scattered field and includes some of the cross‐polarization effects due to full wave scattering. Two further new estimators, based on the Rytov transformation (the localized nonlinear Rytov and the static localized nonlinear Rytov approximations) are designed to further improve the estimation of the phase of the scattered field, especially at high frequency and for larger size scatterers. Although these approximations are nonlinear functions in conductivity, they are generally much faster to compute than the full forward problem, and are almost as efficient as the Born or Rytov approximations. Moreover, the enhanced accuracy of the new estimators has made us optimistic about their application to low‐frequency three‐dimensional inverse problems in electromagnetics. The approximations developed in this paper will also be applicable to fields such as quantum mechanics, optics, ultrasonics, and seismology.

Vertical spatial sensitivity and effective depth of exploration ( d e ) of low‐induction‐number (LIN) instruments over a layered soil were evaluated using a complete numerical solution to Maxwell's equations. Previous studies using approximate mathematical solutions predicted a vertical spatial sensitivity for instruments operating under LIN conditions that, for a given transmitter–receiver coil separation ( s ), coil orientation, and transmitter frequency, should depend solely on depth below the land surface. When not operating under LIN conditions, vertical spatial sensitivity and d e also depend on apparent soil electrical conductivity (σ a ) and therefore the induction number (β). In this new evaluation, we determined the range of σ a and β values for which the LIN conditions hold and how d e changes when they do not. Two‐layer soil models were simulated with both horizontal (HCP) and vertical (VCP) coplanar coil orientations. Soil layers were given electrical conductivity values ranging from 0.1 to 200 mS m −1 As expected, d e decreased as σ a increased. Only the least electrically conductive soil produced the d e expected when operating under LIN conditions. For the VCP orientation, this was 1.6 s , decreasing to 0.8 s in the most electrically conductive soil. For the HCP orientation, d e decreased from 0.76 s to 0.51 s Differences between this and previous studies are attributed to inadequate representation of skin‐depth effect and scattering at interfaces between layers. When using LIN instruments to identify depth to water tables, interfaces between soil layers, and variations in salt or moisture content, it is important to consider the dependence of d e on σ a

Abstract An outcropping hemispherical inhomogeneity embedded in a two-dimensional (2-D) earth is used to model the effects of three-dimensional (3-D) near-surface electromagnetic (EM) 'static' distortion. Analytical solutions are first derived for the galvanic electric and magnetic scattering operators of the heterogeneity. To represent the local distortion by 3-D structures of fields which were produced by a large-scale 2-D structure, these 3-D scattering operators are applied to 2-D electric and magnetic fields derived by numerical modeling to synthesize an MT data set. Synthetic noise is also included in the data.These synthetic data are used to study the parameters recovered by several published methods for decomposing or parameterizing the measured MT impedance tensor. The stability of these parameters in the presence of noise is also examined. The parameterizations studied include the conventional 2-D parameterization (Swift, 1967), Eggers's (1982) and Spitz's (1985) eigenstate formulations, LaTorraca et al.'s (1986) SVD decomposition, and the Groom and Bailey (1989) method designed specifically for 3-D galvanic electric scattering. The relationships between the impedance or eigenvalue estimates of each method and the true regional impedances are examined, as are the azimuthal (e.g., regional 2-D strike, eigenvector orientation and local strike) and ellipticity parameters.The 3-D structure causes the conventional 2-D estimates of impedances to be site-dependent mixtures of the regional impedance responses, with the strike estimate being strongly determined by the orientation of the local current. For strong 3-D electric scattering, the local current polarization azimuth is mainly determined by the local 3-D scattering rather than the regional currents. There are strong similarities among the 2-D rotation estimates of impedance and the eigenvalue estimates of impedance both by Eggers's and Spitz's first parameterization as well as the characteristic values of LaTorraca et al. There are striking similarities among the conventional estimate of strike, the orientations given by the Eggers's, Spitz's (Q), and LaTorraca et al.'s decompositions, as well as the estimate of local current polarization azimuth given by Groom and Bailey. It was found that one of the ellipticities of Eggers, LaTorraca et al., and Spitz is identically zero for all sites and all periods, indicating that one eigenvalue or characteristic value is linearly polarized. There is strong evidence that this eigenvalue is related to the local current. For these three methods, the other ellipticity differs from zero only when there are significant differences in the phases of the regional 2-D impedances (i.e., strong 2-D inductive effects), implying the second ellipticity indicates a multidimensional inductive response.Spitz's second parameterization (U), and the Groom and Bailey decomposition, were able to recover information regarding the actual regional 2-D strike and the separate character of the 2-D regional impedances. Unconstrained, both methods can suffer from noise in their ability to resolve structural information especially when the current distortion causes the impedance tensor to be approximately singular. The method of Groom and Bailey, designed specifically for quantifying the fit of the measured tensors to the physics of the parameterization, constraining a model, and resolving parameters, can recover much of the information in the two regional impedances and some information about the local structure.