Low-frequency electromagnetic scattering by a penetrable spherical underground cavity with magnetic dipole stimulation

Abstract The present study deals with the computation of the electromagnetic vector fields, being scattered by a penetrable spherical cavity that is situated beneath the Earth’s surface and needs to be identified in real-life applications. Aiming to solve the important direct problem, the excitation source is modeled as a time-harmonic magnetic dipole, randomly oriented in three-dimensional space and positioned on the Earth’s surface, operating in the low-frequency regime. In the context of low-frequency analysis, the incident, external and internal fields are expanded in terms of the low angular frequency, which is the same for each medium, namely, the conductive Earth and the lossless air cavity. The static zeroth-order Rayleigh term, along with the first three dynamic contributions, yield an accurate approximation of the solution. Higher-order terms are considered as insignificant within the low-frequency regime and are consequently omitted. Accordingly, Maxwell’s equations are reformulated into a finite set of interdependent elliptic partial differential equations, constrained by transmission conditions on the cavity interface and by asymptotic decay conditions as the spatial variable tends toward infinity, that is, to regions well beyond the domain of observation. From this point onward, the 3D scattering boundary value problems are solved incrementally, with the determination of the unknown constant coefficients, leading either to explicit expressions or to infinite systems of linear algebraic equations. These can be solved by employing standard cut-off techniques. Subsequently, we proceed to the visualization of the resulting fields, providing their graphical representation, which reveals the expected physical behavior and provides an efficient analytical tool for an inversion scheme.