Initial-Value Approach to the Electromagnetic Induction in a~Heterogeneous Sphere

We present an initial-value (IV) approach to the forward problem of large-scale electromagnetic induction in a heterogeneous sphere with 3-D conductivity structure, excited by an external source current. The formulation in the time domain is suitable to study the Earth's electromagnetic response to arbitrary variations of the external source currents, in particular to transient signals, which are difficultly treated by the traditional frequency-domain approach. The parabolic partial differential equation of electromagnetic induction is reformulated in the weak sense. The magnetic field is parameterized by vector spherical harmonic functions in lateral coordinates and by piecewise linear finite elements in radial coordinate. A semi-implicit Euler scheme is applied for the time integration. The terms containing the radial conductivity profile are treated implicitly, while the effect of lateral variations is taken explicitly from the previous time step. Conductivity models consisting of multiple homogeneous off-axis nested spheres are used to test the IV method and computer code against the known semi-analytic (SA) solutions for periodic external source signals. The differences between the results of IV and SA solutions can be attributed solely to the numerical errors and diminish as the spatial and temporal resolution is increased. The ability to treat arbitrary external source signals, which is the most important advantage of the IV approach over the frequency-domain methods comes at the cost of two drawbacks. Firstly, an initial magnetic field has to be specified in the whole sphere.Secondly, the common concept of frequency-dependent skin-depth can not be directly employed,because frequency is not introduced at all.