Numerical solution of the inverse problem of magnetotelluric sounding

This study focuses on the coefficient inverse problem arising in magnetotelluric (MT) sounding, which plays a crucial role in geophysical exploration and subsurface characterization. The main objective is twofold: first, to construct a reliable forward numerical model based on the Helmholtz equation with a complex-valued conductivity coefficient, and second, to develop a stable inversion procedure for reconstructing the conductivity distribution from boundary measurements. The forward problem is discretized using a finite-difference approximation, ensuring numerical stability and accuracy for both the direct and adjoint formulations. To address the ill-posed nature of the inverse problem, a misfit functional is introduced, measuring the discrepancy between simulated and observed boundary data. This functional is minimized using the iterative Landweber method, which provides a simple yet robust tool for stabilizing reconstructions. Numerical experiments are carried out for a synthetic conductivity model consisting of a smooth background medium with an embedded localized anomaly. The obtained results demonstrate the ability of the proposed method to recover key structural features of the anomaly. The presented framework offers a promising foundation for the development of practical inversion algorithms applicable to real geophysical MT data.