Surface wave scattering theory is presented as a new method for analyzing teleseismic surface wave data. Using surface wave scattering integrals the effect of lateral heterogeneity both on the surface wave coda generation and on the direct surface wave is described. Since the employed scattering theory for the forward problem is linear, the inverse problem can conveniently be solved in the least squares sense using an iterative matrix solver. For waveform inversions of the direct surface wave, only near forward scattering contributes. For this case the isotropic approximation is introduced, which makes it possible to retrieve phase velocity information from scattering theory. It is shown that for practical waveform inversions the resulting system of linear equations is extremely large and how row action methods can be used conveniently for carrying out the inversion on moderate size computers. The performance of the inversions is illustrated with two numerical examples. In the first example the surface wave coda generated by one point scatter is inverted. It is shown that the reconstruction in this case is similar to Kirchhoff migration methods as used in exploration seismics. In the second example, ray geometrical effects (focusing and phase shifting) are obtained from the linear inversion with scattering theory. It follows from this example that linear waveform inversion can simultaneously fit the amplitude and the phase of surface wave data. ¿ American Geophysical Union 1988