A recursive set of invariants of the magnetotelluric impedance tensor

Abstract

We present a recursive relation that produces pairs of rotation invariants of the magnetotelluric impedance tensor starting from the series and parallel impedances (S-P), which are readily computed from field data. Subsequent iterations combine the two impedances again as series and parallel impedances until, as the number of iterations tends to infinity, they converge to a single quantity. This single quantity is the well known impedance derived from the determinant of the magnetotelluric tensor. This particular way of splitting the determinant impedance into pairs of invariants, with progressive dilution of information, behaves as a natural regularizer when inverting the data. To tackle inversion in 2D we develop corresponding formulas for the partial derivatives with respect to the ground resistivities. We illustrate the performance of this set of invariant responses with synthetic data produced by a 3D model as well as with real soundings from the BC87 data set. In both cases we present 2D inversions of the S-P responses as they compare with the determinant and a couple of intermediate recursions. The models are necessarily related, with decreasing resolution as the iterations increase.