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.
The distance to M31 in the era of precision cosmology
