Semi-analytical solutions of seismo-electromagnetic signals arising from the motional induction in 3-D multi-layered media: part I-theoretical formulations

Taking into account the motional induction effect, in which the Earth crust that has finite electrical conductivity vibrates in the ambient geomagnetic field resulting in motionally induced electric current, we derive semi-analytical solutions of seismo-electromagnetic signals generated by an earthquake source in 3-D multi-layered media, which consists of an air half-space and multiple solid layers. First, both the elastic and electromagnetic (EM) wave-fields involved in the governing equations, which have the form of Maxwell's equations coupled with elastodynamic equations, are expanded by a set of vector basis functions in cylindrical coordinate system. Then, we reorganize the transformed governing equations expressed by expansion coefficients and obtain corresponding first-order ordinary differential equations for the wave-fields in air and solid media. The expansion of the motionally induced electric current and the reorganization of Maxwell's equations are the most important part, and also the most complicated and tedious part of this work. Thereafter, we solve the first-order ordinary differential equations through the Luco-Apsel-Chen generalized reflection and transmission method gaining solutions of the expansion coefficients. Finally, we obtain the frequency-space-domain semi-analytical solutions written as integrations of corresponding expansion coefficients over wavenumber domain, which can be numerically calculated by the discrete wavenumber method. The time-domain solutions can be achieved by further applying the discrete inverse Fourier transform. To have a numerical stability at any high frequency, we adopt the analytical regularization approach in the derivation process by introducing two artificial interfaces with infinitely small distance from the source. On the basis of the semi-analytical solutions, we can tell that only EM fields of TM mode (in which magnetic fields are transversely polarized) will be induced by SH waves, whereas EM fields of both TE mode (in which magnetic fields are transversely polarized) and TM mode will be induced by P and SV waves. The derived semi-analytical solutions can be used to calculate seismo-electromagnetic signals either below or above the free surface.