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We propose a method to invert surface wave dispersion data directly for 3-D variations of shear wave speed, that is, without the intermediate step of phase or group velocity maps, using frequency-dependent ray tracing and a wavelet-based sparsity-constrained tomographic inversion. A fast marching method is used to compute, at each period, surface wave traveltimes and ray paths between sources and receivers. This avoids the assumption of great-circle propagation that is used in most surfacewave tomographic studies, butwhich is not appropriate in complex media. To simplify the problem we consider quasi-stratified media with smoothly varying seismic properties. We represent the 3-D shear wave speed model by means of 1-D profiles beneath grid points, which are determined from all dispersion data simultaneously using a wavelet-based sparsity-constrained tomographic method. The wavelet coefficients of the wave speed model are estimated with an iteratively reweighted least squares algorithm, and upon iteration the surface wave ray paths and the data sensitivity matrix are updated using the newly obtained wave speed model. 

To demonstrate its feasibility, we apply the method to determine the 3-D shallow crustal shear wave speed variations in the Taipei basin of Taiwan using short period interstation Rayleigh wave phase velocity dispersion measurements extracted from the ambient noise cross-correlation method. The results are consistent with previous studies and reveal strong shallow crustal heterogeneity that correlates with surface geology.

This method has been recently published in GJI  (Fang H., Yao, H. *, Zhang H, Huang, YC, van der Hilst R.D., 2015. Direct inversion of surface wave dispersion for 3-D shallow crustal structure based on ray tracing: methodology and application, Geophys. J. Int. , 2011, 1251-1263). 

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