Inverse problem of electroseismic conversion. I: Inversion of Maxwell's equations with internal data

TL;DR

This paper develops a mathematical method to uniquely and stably reconstruct electrical parameters in electroseismic conversion models using internal data and complex geometrical optics solutions, extending previous inverse problem studies.

Contribution

It introduces a new approach for reconstructing multiple parameters in Maxwell's equations with internal data, allowing for variable magnetic permeability.

Findings

Unique determination of conductivity, permittivity, and electrokinetic mobility.

Lipschitz stability estimate for the reconstruction.

Extension to variable magnetic permeability.

Abstract

Pride (1994, Phys. Rev. B 50 15678-96) derived the governing model of electroseismic conversion, in which Maxwell's equations are coupled with Biot's equations through an electrokinetic mobility parameter. The inverse problem of electroseismic conversion was first studied by Chen and Yang (2013, Inverse Problem 29 115006). By following the construction of Complex Geometrical Optics (CGO) solutions to a matrix \Schrodinger equation introduced by Ola and Somersalo (1996, SIAM J. Appl. Math. 56 No. 4 1129-1145), we analyze the reconstruction of conductivity, permittivity and the electrokinetic mobility parameter in Maxwell's equations with internal measurements, while allowing the magnetic permeability $\mu$ to be a variable function. We show that knowledge of two internal data sets associated with well-chosen boundary electric sources uniquely determines these parameters. Moreover, a Lipschitz-type stability is obtained based on the same set.