The distortion tensor of magnetotellurics: a tutorial on some properties

Abstract

Abstract.A 2 Â 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 Â 2 matrix has a negative determinant by then having a value greater than unity.

Monitoring hydraulic stimulation using telluric sounding

“…local site M. Therefore, a change in E at rotated site B produces a change in E along the corresponding but differently rotated axis at site M, with the change generally amplified or attenuated (Lilley 2015). This rotation of the local and regional axes reduces the telluric distortion tensor to an ideal 2D form.…”

“…The presence of galvanic distortion will be manifest as an amplitude shift, and a twist and shear operation (Lilley 2015). For 2D structures, the distortion matrix can be rotated to reflect the changes in E along and across strike.…”

“…This local distortion of E can be described by a real-valued second-rank tensor D that relates the electric field measured at a local (E M ) and regional (E B ) site according to (Chave and Jones 2012) where In this case, D is relative to axes X and Y, typically north and east. Introducing a rotation matrix R(θ), and the transpose R T (θ), the value of D relative to axes X ′ and Y ′ (which are rotated θ ′ clockwise from north and east) is given by the matrix D ′ (Lilley 2015) (1)…”

“…Following from Lilley (2015), EA of the telluric distortion matrix involves finding a direction of E M for which the change in E B is in the same direction, with the eigenvalue of the direction giving the gain of the process. When real eigenvectors exist, the characteristic equation for D, can be solved for the two eigenvalues ζ 1 and ζ 2 with solutions Eigenvectors can then be found corresponding to these eigenvalues.…”

“…The injection of conductive fluids into the subsurface will alter the telluric distortion matrix which can be mapped to show the lateral constraints of fluid migration. Lilley (2015) proposed the application of both eigenvalue analysis (EA) and singular value decomposition (SVD) on the telluric distortion matrix, with Mohr diagrams introduced as a versatile way of visualising properties of the matrix. These diagrams can be used to determine the extent to which the matrix is diagnostic of 1D, 2D or 3D geological structure as well as determining a strike direction (with 90 • ambiguity) and a relative amplitude change.…”