The distortion tensor of magnetotellurics: a tutorial on some properties
Frederick E. M. LilleyResearch School of Earth Sciences, Australian National University, Canberra, ACT 0200, Australia. Email: ted.lilley@anu.edu.au
Exploration Geophysics 47(2) 85-99 https://doi.org/10.1071/EG14093
Submitted: 29 September 2014 Accepted: 12 March 2015 Published: 1 May 2015
Journal Compilation © ASEG 2016
Abstract
A 2 × 2 matrix is introduced which relates the electric field at an observing site where geological distortion applies to the regional electric field, which is unaffected by the distortion. For the student of linear algebra this matrix provides a practical example with which to demonstrate the basic and important procedures of eigenvalue analysis and singular value decomposition.
The significance of the results can be visualised because the eigenvectors of such a telluric distortion matrix have a clear practical meaning, as do their eigenvalues. A Mohr diagram for the distortion matrix displays when real eigenvectors exist, and tells their magnitudes and directions.
The results of singular value decomposition (SVD) also have a clear practical meaning. These results too can be displayed on a Mohr diagram. Whereas real eigenvectors may or may not exist, SVD is always possible. The ratio of the two singular values of the matrix gives a condition number, useful to quantify distortion. Strong distortion causes the matrix to approach the condition known as ‘singularity’. A closely-related anisotropy number may also be useful, as it tells when a 2 × 2 matrix has a negative determinant by then having a value greater than unity.
Key words: distortion, eigenanalysis, magnetotelluric, Mohr, SVD, telluric.
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