An alternative Born normal mode method to compute synthetic seismograms in the 3D Earth

We present a new technique to compute the long-period synthetic waveforms in the 3D Earth with the formulism of the first order normal mode perturbation theory. Unlike the commonly used normal mode summation based approaches, in which the coupling effects among modes are calculated only along the great-circle path (e.g., along the same branch, path-average approximation; or across branches, non-linear asymptotic theory), this technique takes into account the complete global 3D effects by numerical integrations. To compromise the time-consuming numerical integrations, spherical harmonics are used as model coefficients, and the procedure will be implemented in two steps. First, the coupling effects among modes from each source moment tensor and spherical harmonics of a particular geometry are computed and stored as a table file. Second, for any given 3D Earth model, the model coefficients are rotated to the desired geometry in which the tabled file can be directly used as "green's functionA", and the computation of synthetics is then straightforward. While the numerical cost in the first step is expensive for modes with large angular order, the resulting table file can be applied to any combination of seismic sources, receivers and any 3D Earth model. To validate the method, we compare the resulting synthetics to the Spectral Element Method solution. We also present a preliminary model for upper mantle based on real data with this technique.