Higher moment excitation of normal modes and surface waves

Following the formalism introduced initially by Backus and Malcuhy, and developed for body waves by Stump, we investigate the excitation of normal modes and surface waves in the Earth by higher moments, characteristic of seismic sources with extended lateral dimensions. We derive coefficients of excitation in the most general case for second order moments (third-order tensors), and show that they can be readily computed from the double-couple excitation coefficients for all geometries, and for both normal modes and surface waves. We apply our results to a number of simple fault models, including ruptures involving vertical propagation, which cannot be treated by the classic directivity method. While it is in principle possible to correct for the vertical extent of the source through the use of the centroidal double-couple in the case of simple ruptures, our formalism can be applied to any geometry. Our results show that vertical rupture, even over short distances (10 km), can substantially modify the excitation of normal modes in the case of dip-slip sources. This may have important consequences for moment tensor inversion, and in the search for deep lateral heterogeneities. Finally, this formalism may be transposed to the theory of tsunami generation.