Representation of complex seismic sources by orthogonal moment-tensor fields

Seismic radiation from indigenous sources can be represented by the excess of model stress over actual stress, a second-order tensor field that Backus & Mulcahy named the stress glut. We prove a new representation theorem that exactly and uniquely decomposes any stress-glut (or strain-glut) density into a set of orthogonal tensor fields of increasing degree alpha, up to six in number, ordered by their first non-zero polynomial moments. The zeroth-degree field (alpha = 0) is the projection of the stress-glut density onto its zeroth polynomial moment, which defines the seismic moment tensor, Aki seismic moment M-0, and centroid-moment tensor (CMT) point source. The higher-degree fields describe mechanism complexity-source variability that arises from the space-time variations in the orientation of the stress glut, which is the main subject of this theoretical study. The representation theorem generalizes the point-source approximation to a sum of multipoles that features the CMT monopole as its leading term. The first-degree field contributes a dipole tensor with an mechanism orthogonal to the CMT, the second-degree field contributes a quadrupole tensor, and so on, up to six orthogonal fields in all. We define the total scalar moment M-T to be the integral of the scalar moment density, and we use the representation theorem to partition this total moment into a sum of fractional moments for each degree alpha. If the faulting is simple enough, M-T approximate to M-0, and the higher-degree terms will be small. When the faulting is more complex, however, M-T > M-0; the higher-degree fields will contribute more to the radiation, and this contribution will increase with frequency. Application to simple planar faulting shows that out-of-plane variations in slip-vector orientation reduce M-0/M-T more than in-plane variations of similar magnitude, consistent with Kagan's early study of mechanism complexity. We decompose stress-glut realizations from the Graves & Pitarka rupture simulator; typical values of M-0/M-T are 0.82-0.92, consistent with analytical results. The higher-degree fields of the Graves-Pitarka sources typically radiate up to alpha = 4; only the isotropic term is zero. We compute synthetic seismograms to illustrate the radiation patterns of the higher-degree fields and their frequency dependence. Decomposition of the Duputel & Rivera source model for the 2016 Kaikoura earthquake (M-w 7.8) indicates that the radiation from the higher-degree fields was large enough (M-0/M-T = 0.82) that it may be possible to invert global datasets for low-degree multipoles. Other applications of the stress-glut representation theorem are discussed.