Early electromagnetic waves from earthquake rupturing: I. theoretical formulations

Earthquake taking place in a fluid-saturated porous medium can generate electromagnetic (EM) waves because of the electrokinetic effect. These generated EM waves arrive at a distant observatory much earlier than the seismic waves because their velocities are much faster than those of the seismic waves. They may explain the early EM signals which have been detected before the detection of the seismic waves after the occurrences of earthquakes. In this study, we attempt to analyse such a kind of early EM signals induced by an earthquake because of the electrokinetic effect. The earthquake is assumed to be a fault slip and is modelled by a moment tensor point source. With Pride's equations quantifying the coupling between seismic and EM waves, we first present a real-axis integration (RAI) algorithm to calculate the seismoelectric wavefields in a layered porous formation. Although full waveforms can be calculated by such a RAI technique, individual waves cannot be easily separated from the full waveforms. The need to compute the individual waves is eminent for the purpose of investigating the early EM waves, because these EM waves are usually several orders weaker than and are masked by the EM signals accompanying the seismic waves in the full waveforms. Therefore, we further develop a branch-cut integration (BCI) algorithm, by transforming the original wavenumber integral along the real axis in the complex wavenumber plane for the RAI technique to a sum of integrals along the vertical branch cuts and the residues of the poles. For performing the integrations along the vertical branch cuts, determination of the Riemann sheets are explained and displayed. Finally, the seismoelectric wavefields are represented in forms allowing calculating individual waves.