The canonical decomposition and its relationship to other forms of magnetotelluric impedance tensor analysis

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E. Yee
K.V. Paulson


A technique for magnetotelluric (MT) data analysis, known as the canonical decomposition, is developed from first principles. This analysis is based on the canonical decomposition of the impedance tensor Z and explicitly parametrizes Z in terms of eight physically relevant structural parameters which specify the transfer characteristics of the Earth system (i.e. the maximum and minimum principal apparent resistivities and the associated principal phases) as well as the principal or intrinsic coordinate system for Z (i.e. the two principal orthogonal electric and magnetic field polarization states). It is shown that the formulation of canonical decomposition in which the polarization descriptors are specified in terms of elliptic parameters results in the MT impedance tensor analysis presented by LaTorraca et al. The relationships between canonical decomposition and several other forms of magnetotelluric data analysis are explored. Specifically, we compare the canonical decomposition with the "conventional" analysis, the maximum coherency analysis, the associate and conjugate directions analysis developed by Counil et al., Eggers' eigenstate analysis and Spitz's rotation analysis. It is shown that canonical decomposition is a natural generalization of the conventional analysis in that both the rotation and ellipticity properties of Z are utilized in the definition of a principal coordinate system. A generalization of the maximum coherency analysis is shown to yield the same parameters as those extracted in canonical decomposition. By imposing a specific restriction on the generalized maximum coherency analysis, we next show how to extract the parameters (i.e. the directions of maximum and minimum current and induction and the corresponding electric and magnetic sheet impedances) that were obtained by Counil et al. in their associate and conjugate directions analysis. The relationship between canonical decomposition and Eggers' eigenstate analysis is developed and it is shown that the primary deficiency in the eigenstate formulation resides in the incorporation of an artificial field constraint. Spitz's rotation analysis extracts two analytical rotation angles from the matrix factors in the Cayley factorization of Z. It is shown that the Cayley factorization of Z is nothing more than the repackaging of the information in canonical decomposition and, as a consequence, Spitz's rotation analysis is not required to extract a principal or intrinsic coordinate system of Z.

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Yee, E., and K.V. Paulson. 1987. “The Canonical Decomposition and Its Relationship to Other Forms of Magnetotelluric Impedance Tensor Analysis”. Journal of Geophysics 61 (1), 173-89.
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