Magnetotelluric tensor decomposition: Part I, Theory for a basic procedure

Abstract

The problem of expressing a general 3-D magnetotelluric (MT) impedance tensor in the form of a 2-D tensor that has been distorted in some way is addressed first in terms of a general theorem. This theorem shows that when the real and quadrature parts of a tensor are analyzed separately as distinct matrices, all that is necessary to make a matrix with 2-D characteristics from one with 3-D characteristics is to allow the electric and magnetic observing axes to rotate independently. The process is then examined in terms of the operations of twist and pure shear (“split”) on such matrices. Both of two basic sequences of split after twist, and twist after split, produce a typical 3-D matrix from one initially 1-D, with the parameters of split contributing 2-D characteristics to the final matrix. Taken in reverse, these sequences offer two basic paths for the decomposition of a 3-D matrix, and are seen to be linked to the initial theorem. The various operations on matrices are expressed diagrammatically using the Mohr circle construction, of which it is demonstrated two types are possible. Mohr circles of an observed MT tensor display all the information held by the tensor, and the two types of circle construction respectively make clear whether particular data are well suited to modeling by either split after twist, or twist after split. Generally, tensor decompositions may be displayed by charting their progress in Mohr space. The Mohr construction also displays the invariants of a tensor and shows that tensor decomposition can be viewed as a process of determining an appropriate set of invariants. An expectation that the origin of axes should be outside every circle categorizes as irregular any tensors which, in either the real or quadrature part, do not satisfy a [Formula: see text] criterion. The theory of the present paper applies equally to procedures for distorting 1-D and 2-D model calculations for the purpose of matching observed 3-D data.

Magnetotelluric resistivity modeling for 3D characterization of geothermal reservoirs in the Western side of Mt. Aso, SW Japan

“…In addition, an analysis on Mohr circle proposed by Lilley (1993Lilley ( , 1998a is carried out to evaluate the anisotropic strength of resistivity structures from the impedance tensor. The center point (X,Y) and the radius R of Mohr circle are defined by the components of the impedance tensor as The real and imaginary parts of the impedance tensor derive two difference Mohr circles.…”

A magnetotelluric study of the sensitivity of an area to seismoelectric signals

“…The application of Mohr circles in MT interpretation was introduced by Lilley (1976). Nowadays, this simple technique is used to display the dimensionality of a structure and to infer the regional strike angle (Lilley, 1993(Lilley, , 1998aMakris et al, 1999;Ingham et al, 2001). By simply rotating and then plotting specific rotated components of the measured impedance tensor (e.g.…”

Magnetotelluric Tensor Decomposition: Insights from Linear Algebra and Mohr Diagrams

“…Different points on the diagram can be checked to confirm that Equations 6 to 15 for the rotation of axes are obeyed. The diagram is "Type 1" of Lilley (1998Lilley ( , p.1889, and shows how axes for Z ′ yx and Z ′ yy can be included. The diagram in Fig.…”

“…Equations for this form of the SVD of an MT tensor were derived in an earlier paper (Lilley, 1998). Taking phase relative to the H signal, so that H q is zero, Equation 1 is written in its in-phase and quadrature parts as…”

“…The analysis of in-phase and quadrature parts separately is seen to give further insight into the question of whether, as suspected by Lilley (1998), the determinants of the parts taken separately should always be positive.…”