Phenomenological representation and kinematics of general seismic sources including the seismic vector modes

Abstract

SUMMARY A unified mathematical description and kinematics are presented for the general indigenous seismic sources including the single force and torque, as well as the moment tensor. A new class of earthquakes associated with a release of gravitational energy and/or a reduction of pressure gradient in the Earth's interior are shown to be observed through the detection of the ‘seismic vector modes’. The Earth is divided into two subsystems: a source region spanning a finite volume and containing a finite mass, and the remainder of the Earth surrounding this volume. The phenomenological source of the seismic waves radiated outside the source volume is given by the traction on the boundary through which the momentum exchange takes place between the two subsystems. The traction is expressed in terms of the orthogonal GSH (Generalized Spherical Harmonics) components, each of which is identified with the independent radiation pattern characterized by the harmonic order l and modal type k (toroidal, spheroidal 1 or spheroidal 2). By investigating each traction mode in terms of both the excitation mechanism in the source and the efficiency of wave radiation outside of the source region, we have recognized the practical importance of the three ‘vector-type’ components with harmonic order l= 1, in addition to the the conventional ‘tensor-type’ force components (moment tensor). The three are (1)single force, produced by the motion of the gravitational centre of the source region relative to the surrounding medium; (2)torque, produced by the change of the angular momentum and the gravitational loading in the source region; (3)pressure dipole, produced by the change of the pressure gradient in the source region. We have shown the possible occurrence of a new class of earthquakes, which dominantly radiate the vector type wave modes in the Earth's interior, especially in a partially molten region beneath volcanic areas. We have proposed to detect the vector-type components in the anomalous seismic events not well characterized by the moment tensor model. Conversion formula are derived between the force system obtained as a set of present traction modes in the 2-D (surface) description, and that obtained by Backus & Mulcahy (1976) in terms of the polynomial moments representing the 3-D description of the equivalent volume force. The traction modes provide a necessary and sufficient set of source parameters or ‘seismic observables’, which are determinable only by studying the wavefield observed far outside a source volume by the angular spectral decomposition of the radiation pattern.