On volume-source representations based on the representation theorem

Abstract

We discuss different ways to characterize a moment tensor associated with an actual volume change of ΔV C , which has been represented in terms of either the stress glut or the corresponding stress-free volume change ΔV T . Eshelby’s virtual operation provides a conceptual model relating ΔV C to ΔV T and the stress glut, where non-elastic processes such as phase transitions allow ΔV T to be introduced and subsequent elastic deformation of −ΔV T is assumed to produce the stress glut. While it is true that ΔV T correctly represents the moment tensor of an actual volume source with volume change ΔV C , an explanation as to why such an operation relating ΔV C to ΔV T exists has not previously been given. This study presents a comprehensive explanation of the relationship between ΔV C and ΔV T based on the representation theorem. The displacement field is represented using Green’s function, which consists of two integrals over the source surface: one for displacement and the other for traction. Both integrals are necessary for representing volumetric sources, whereas the representation of seismic faults includes only the first term, as the second integral over the two adjacent fault surfaces, across which the traction balances, always vanishes. Therefore, in a seismological framework, the contribution from the second term should be included as an additional surface displacement. We show that the seismic moment tensor of a volume source is directly obtained from the actual state of the displacement and stress at the source without considering any virtual non-elastic operations. A purely mathematical procedure based on the representation theorem enables us to specify the additional imaginary displacement necessary for representing a volume source only by the displacement term, which links ΔV C to ΔV T . It also specifies the additional imaginary stress necessary for representing a moment tensor solely by the traction term, which gives the “stress glut.” The imaginary displacement-stress approach clarifies the mathematical background to the classical theory.