Approximation of Rupture Directivity in Regional Phases Using Upgoing and Downgoing Wave Fields

Recent broadband modeling of regional events suggests that vertical directivity is particularly important at high frequency. Conventionally, such directivity is obtained by summing a grid of point sources. This relatively time-consuming procedure can be greatly reduced by introducing directivity time histories appropriate for the various crustal phases in terms of upgoing and downgoing paths that are calculated at only one depth. To achieve this, we formulated frequency-wavenumber solutions for a simultaneous computation of surface displacement for three wave fields, upgoing, downgoing, and the total from a seismic source buried in a layered medium (Appendix A). The concept of upgoing and downgoing wave field is introduced in the source layer matrix explicitly before allowing the source coefficients to interact with the propagation of the stress-displacement vector. Using this new algorithm, we generated a set of upgoing and downgoing wave fields at a fixed depth for different crustal models. We also simulated the effects of rupture propagation using distributed point-source summations and predicted the same effect by summing the upgoing and downgoing wave fields calculated at a single depth, each convolved with a separate analytical boxcar function representing the far-field rupture. A library of these new Green's functions should prove much more effective in modeling recorded motions than using point-source Green's functions alone.