Numerical Modelling in Geophysics I: A Two-Dimensional Solution of Electromagnetic Induction Equations Using the Finite Difference Method

Abstract

Electromagnetic induction provides a fundamental physical basis for imaging the electrical conductivity structure of the Earth’s crust and mantle. In this study, we derive the governing equations for two-dimensional electromagnetic field modelling from Maxwell’s equations and formulate them within the finite difference method. Under the quasi-static approximation, appropriate for geophysical frequencies, the displacement current is neglected, simplifying the coupled electromagnetic system in the frequency domain. Assuming a 2-D Earth, the problem is decomposed into transverse electric and transverse magnetic modes, each governed by a scalar Helmholtz-type equation. These equations are subsequently discretized on a structured grid using central finite difference schemes, resulting in a system of linear algebraic equations suitable for numerical implementation. The presented formulation establishes a rigorous theoretical foundation for forward modelling of electromagnetic responses, particularly in magnetotelluric (MT) applications. This work serves as a basis for subsequent developments in numerical implementation, mesh design, and inversion frameworks for complex geo-electrical structures.