Strain Green's Tensors, Reciprocity, and Their Applications to Seismic Source and Structure Studies

Green's function approach is widely used in modeling seismic wave- form. The representation theorem expresses the wave field as the inner product of the moment tensor and the spatial gradients of the Green's tensor. Standard practice in waveform calculations has been to compute the Green's tensors first and then obtain their gradients by numerical differentiation. The reciprocity of the Green's tensor enables us to express the wave field explicitly in terms of the strain Green's tensor, a third-order tensor composed of the spatial gradients of the Green's tensor elements. We propose here to use the strain Green's tensors rather than the Green's tensors themselves in computing the waveforms. By bypassing the need for Green's tensors and directly using the strain Green's tensors, we can improve the computa- tional efficiency in waveform modeling while eliminating the possible errors from numerical differentiation. The strain Green's tensor elements are also directly related to the partial derivatives of the waveforms with respect to moment tensor elements and structural parameters. Through the inversion of the focal mechanisms of 27 small events in the Los Angeles region, we demonstrate the effectiveness of the strain Green's tensor database approach in quickly recovering source parameters based on realistic 3D models. We show that the same database can also be used to improve the efficiency and accuracy in computing the Frechet kernels for tomography inver- sions.