The distortion tensor of magnetotellurics: a tutorial on some properties

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.