Solution of the Time-Domain Inverse Resistivity Problem in the Model Reduction Framework Part I. One-Dimensional Problem with SISO Data

Many time-domain problems in engineering applications can be described by means of a parameter dependent time-invariant dynamic systems. We are interested in parameter estimation, by fitting available transient measurements using the nonlinear least square method. As the main application, we consider the control source electromagnetic method (CSEM) of geophysical exploration governed by the diffusion Maxwell system, where the unknown parameters describe the spatial distribution of electrical resistivity. We propose a novel model reduction approach for constructing an efficient approximation of the Jacobian. The reduction is based on projection of the state variable onto a Rational Krylov subspace (RKS), and it allows us to split the time and space dependence of the derivative. We examine several popular RKSs and single out the H2-optimal subspace that not only minimizes the approximation error but completely annuls its influence on the inversion result. Preliminary numerical experiments with a simplified one-dimensional, singleinput/single-output CSEM setting are reported to validate our strategy.