An Empirical Green’s Function Approach for Isolating Directivity Effects in Earthquake Ground-Motion Amplitudes

ABSTRACT In this study, we apply an empirical Green’s function (eGf) method within a ground-motion modeling framework to mitigate trade-offs between source, path, and site effects. Many physical processes contribute to spatial variations in observed ground motions, including earthquake radiation pattern, directivity, variable path attenuation, and site effects. Current nonergodic ground-motion models use spatially varying coefficients for path and site effects, but they do not address trade-offs with complex earthquake source effects. To quantify the influence of directivity on ground-motion amplitudes, we use records from multiple smaller earthquakes with epicenters near that of a larger event. We use these small magnitude events as eGfs and estimate repeatable path and site effects at individual stations, assuming that the average adjustments are not controlled by directivity. We adjust residuals from the larger earthquake using the eGf terms, isolating effects related to the rupture. This method clearly enhances the observed broadband directivity observed in the 2022 M 5.1 and 2007 M 5.4 Alum Rock earthquake ground motions, reinforcing the conclusion that their ruptures were unilateral. For the 2004 M 6.0 Parkfield earthquake, we find a bilateral rupture model better fits the data because variations in rupture velocity, slip rate, and slip distribution seem to have a stronger effect on the ground motions than rupture direction alone. Applying eGf adjustments reduces the standard deviation of the rupture models over the three earthquakes by 32% on average and by up to 57% for the 2022 Alum Rock earthquake, confirming we have effectively removed repeatable effects related to the wave propagation path and site response. We propose a novel measure of the frequency-dependent directivity amplification strength as the reduction in ground-motion residual variability gained by fitting a directivity model; for the three earthquakes considered, this parameter varies between 25% and 75%, indicating that directivity can strongly influence ground motions and should be considered in ground-motion modeling.