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## Representation of Complex Seismic Sources by Orthogonal Moment-Tensor Fields

Thomas H. Jordan, & Alan Juarez

Published November 21, 2018, SCEC Contribution #8887

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 (1976) 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 , up to six in number, ordered by their first nonzero polynomial moments. The zeroth-degree field () is the projection of the stress-glut density onto its zeroth polynomial moment, which defines the seismic moment tensor, Aki seismic moment , and centroid-moment tensor (CMT) point source. Our 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 to the point source, the second-degree field () contributes a quadrupole, and so on. We define the total scalar moment 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 . If the faulting is simple enough, and the higher-degree terms will be small; however, when the faulting is more complex, and radiation from the higher-degree fields becomes significant, especially at higher frequencies. We decompose various fault-rupture models to illustrate how the higher-degree terms characterize the source complexities, and we compute synthetic seismograms to assess the radiation amplitudes. Application to simple planar faulting shows that out-of-plane variations in slip-vector orientation reduce more than in-plane variations of similar magnitude. We decompose stress-glut realizations from the Graves & Pitarka (2016) rupture simulator; typical values of are 0.82-0.92, consistent with analytical results. The higher-degree fields of the Graves-Pitarka sources typically radiate up to ; only the isotropic term is zero. We describe new inverse problems posed by the representation theorem, and we speculate on methods for their solution. Source models for the 2016 Kaikoura earthquake (Mw 7.8) indicate that the radiation from the higher-degree fields was large enough ( = 0.82) that it may possible to invert global datasets for at least some of the higher-degree multipoles.

Key Words
Source theory, earthquakes, seismic moment, stress glut, source mechanism

Citation
Jordan, T. H., & Juarez, A. (2018). Representation of Complex Seismic Sources by Orthogonal Moment-Tensor Fields. Geophysical Journal International,. doi: 10.1093/gji/ggy492.

Related Projects & Working Groups
CISM