2.5‐D time‐domain finite‐differencing of the quasistatic Maxwell's equations

Abstract

This paper describes a new electromagnetic forward modeling algorithm. The algorithm is a explicit finitedifference solution to reduced quasistatic Maxwell’s equations for electromagnetic fields excited by 3-D finite sources in a 2-D medium (i.e. a 2.5-D model). The reduced quasistatic Maxwell’s equations are derived from the standard Maxwell’s equations by taking the advantage of 2-D media. In these reduced equations, three components, h, , h,, and er , ae decoupled from the six components of h and e and can be solved directly. The main advantages of the new algorithm are: (1) the optimal efficiency in both computer time and memory because the time step are adapted to the diffusion physics of electromagnetic fields and only three components are need to be directly computed, (2) the applicability to complex resistivity models with possibly large resistivity contrasts (e.g. lOOO:l), (3) the flexibility of modeling varies sources, such as horizontal magnetic dipoles, vertical magnetic dipoles, ydirection electric dipoles and the superposition of these dipoles (e.g. long wire sources and horizontal ground loops), and (4) the simplicity of coding because the algorithm is explicit and the “time” step used in the algorithm is constant. The numerical results calculated by the finite-difference scheme match the analytical solutions very well. INTRODUCTION Mineral explorations are frequently performed with transient electromagnetic (TEM) systems. Physical and numerical modelings are important for the interpretations of real data and inversions of physical parameters. In some situations, the media are approximately 2-D (invariant in the y-direction), and often with sources localized to a small region or a point and receivers distributed in a plane. In such cases, a 2.5-D modeling method is adequate to calculate the electromagnetic response. A time domain 2.5-D TEM numerical modeling method has not been published to date. In this paper, I will derive reduced quasistatic Maxwell’s equations from the standard quasistatic Maxwell’s equations for a 2.5-D case, and propose a time domain explicit finite-difference scheme for implementing these reduced equations. This 2.5-D modeling method will be referred to ET, which stands for “Electromagnetic field modeling in the time domain”. Fist, the theory of ET method will be formulated, followed by the discussion on the numerical implementation of the ET, and then some numerical results calculated by the finite-difference scheme based on the ET. Numerical results for homogeneous media will be compared with analytical solutions. Here, e is the electric field intensity in V/m, b is the magnetic Induction in Wb/ m2, d is the dielectric displacement in C/ m , h is the magnetic field intensity A/m, j is the electric current density in A/ m2, sity in coulombs. and pe is the electric charge denDue to attenuations in the earth, geophysical EM surveys usually utilize only low frequen or late time data, for which the displacement currents (quasistatic approximation). Letting