Interpreting the seismic glut moments of total degree two or less

Abstract

The stress glut Γ(x, t) (Backus & Mulcahy 1976a) is a symmetric second-order tensor field which completely describes the mechanical properties of an indigenous seismic source. The equivalent force γ(x,t)=-ΔΓ(x,t) is all that can be determined by observing the motion. The glut moment tensor Λ(m,n)(ζ, r) is the polynomial moment of Λ=ΔtΓ relative to position ζ and time r with spatial degree m and temporal degree n. It affects the seismic motion only through the force moment tensor γ(m+1,n)(ζ,τ), which is obtained by symmetrizing Λ(m,n)f(ζ,τ) on its last m+1 indices. Here we show how to obtain the force moment tensors of low total degree m+ n+ 1 from the long-period motion produced by the source. We show that in faulting with rupture propagation, all the Λ(m,n)(ζ, τ) with m + n = p will usually produce motion of comparable intensity, and that they are all parts of a single tensor Λ(ρ, r) of order p+ 2 over R4, four-dimensional space time. We extend the definitions of centroid and variance tensor to tensor fields and show that the centroid of Λ in R4 gives the position and time of the source centroid. The variance tensor estimates the time duration of anelastic behaviour at the source, the rupture velocity, and the size, shape and orientation of the source region. The hypocentral coordinates, the time duration, and the rupture velocity are always calculable from the glut moments with m+n < 2, m < 1, and these can be measured from the long-period motion. The geometry of the source region requires knowledge of Γ(2,0), only part of which can be calculated from γ(3,0) for a general indigenous seismic source. We illustrate the general theory with four different types of seismic sources, and we find that when all six tensors γ(m+1,n)(ζ, τ) with m + n<2 have been measured, there are three ways to resolve the fault-plane ambiguity, one using long-period information, and the other two using measurements of the final static deformation produced by the source.