We followed the approach described by Aoi (2002). The observation equation is first linearized and then solved iteratively by singular value decomposition. Synthetic waveforms as well as sensitivity functions are needed to obtain the solution. We used the elastic finite-difference code E3D (Larsen and Schultz, 1995) to calculate the waveforms and obtain the sensitivity functions numerically, by taking the difference of waveforms from perturbed and unperturbed models.

The results of the inversion using the waveforms for the $M_{w}$=6.4 Near Coast of Central Chile event are shown in Figure 2.21. The top two rows show the first 20 s of the waveforms as only this time window was used in the inversion. The bottom two rows show a 60 s time window. In addition to the P-wave, the pP-wave can be seen arriving just after the first 20 s. The results show that although the waveforms for only the first 20 s were used in the inversion, the final model also better describes the P-wave coda following the pP arrival.

Figure 2.22 compares the velocities in the SCV basins for the USGS v2 model (below station 238), initial, and final model after 3 iterations. Density and density gradient were not free parameters but were determined from P-wave velocity using the Nafe-Drake equation (Brocher, 2005).

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