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

The problem of expressing a general 3-D magnetotelluric (MT) impedance tens or in the form of a 2-D tensor that has been distorted in some way is addre ssed 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 m atrices, all that is necessary to make a matrix with 2-D characteristics fr om one with 3-D characteristics is to allow the electric and magnetic obser ving axes to rotate independently. The process is then examined in terms of the operations of twist and pure s hear ("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 initia lly 1-D, with the parameters of split contributing 2-D characteristics to t he final matrix. Taken in reverse, these sequences offer two basic paths fo r the decomposition of a 3-D matrix, and are seen to be linked to the initi al theorem. The various operations on matrices are expressed diagrammatically using the Mohr circle construction, of which it is demonstrated two types are possib le. Mohr circles of an observed MT tensor display all the information held by the tensor, and the two types of circle construction respectively make c lear whether particular data are well suited to modeling by either split af ter twist, or twist after split. Generally, tensor decompositions may be di splayed by charting their progress in Mohr space. The Mohr construction also displays the invariants of a tensor and shows th at tensor decomposition can be viewed as a process of determining an approp riate set of invariants. An expectation that the origin of axes should be o utside every circle categorizes as irregular any tensors which, in either t he real or quadrature part, do not satisfy a Z(xy)Z(xy) < Z(xx)Z(yy) criter ion. The theory of the present paper applies equally to procedures for distortin g 1-D and 2-D model calculations for the purpose of matching observed 3-D d ata.