Concerning dispersion relations for the magnetotelluric impedance tensor

Abstract

SUMMARY Adopting a purely systems-theoretic viewpoint, it is shown how a small set of well-established general physical principles such as causality, stability, and passivity, applied to the earth system, determines the analytical properties of the associated impedance tensor in the complex frequency plane. These physical principles and their consequences are utilized to obtain an integral representation for the impedance tensor based upon the Cauer representation for positive functions. This representation is developed primarily by the imposition of the passivity and symmetry requirement on the impedance tensor. As an application, the representation is employed to develop a set of inequality constraints (necessary conditions) that must be verified by the impedance tensor. Since linearity and passivity imply causality, the Cauer integral representation for the impedance tensor is then utilized to derive a subtracted dispersion relation that connects the real and imaginary Hermitian parts of the tensor on the real frequency axis. Furthermore, it is shown how subtracted dispersion relations for the impedance tensor may be developed directly from the causality principle without recourse to the Cauer representation. In this fashion, the dispersion relations for the scalar response function, developed by Weidelt and Fischer & Schnegg in the context of 1-D and 2-D earth systems, respectively, have been extended to the tensor response function corresponding to general 3-D earth systems. Although the focus in this paper is limited to the explicit development of dispersion relations for the impedance tensor, the applicability and scope of the methods utilized are of a more general nature and can be employed to develop integral relationships for other Earth response functions (e.g. tipper responses). For the general 3-D earth, the dispersion relations have direct applications in the formulation of consistency tests, the construction of consistent estimates of the impedance tensor, and the identification of non-linearities.