Combining the traditional normal mode theory for vertical structure with the newly developed Gaussian beam approach for laterally heterogeneous structure, a new method has been developed for synthesizing waveforms of surface waves for the laterally heterogeneous seismic velocity structure of the earth. In the frequency domain the wave field is constructed from single-mode surface waves which propagate along ray paths on the surface of the earth following the phase velocity mapping for that frequency. Then the wave field at the observation point is constructed using the Gaussian beam method, a high-frequency asymptotic solution. We use the ray-centered coordinate system previously developed for body wave synthesis to derive parabolic equations. The method for surface wave synthesis differs from body wave synthesis by the following: (1) the speed of the wave packet along the ray is the local group velocity even though the ray path itself is determined by the phase velocity distribution in an isotropic or transversely isotropic medium, (2) the energy flow density of a ray is determined by the product of geometrical spreading, group velocity, and energy integral of the eigenfunction of the normal mode; spreading may be an important factor in most cases, and (3) surface waves traveling on a spherical earth may be mapped into Cartesian coordinates (two-dimensional) by using the Mercator transformation including the effect of ellipticity. We found that the weighting factors of each Gaussian beam for a moment tensor representation of an earthquake are equivalent to those of a far-field radiation pattern in laterally homogeneous model. Synthetic seismograms of narrow bandwidth with several different center frequencies are compared to real band-pass-filtered data to delineate the anomalies of three-dimensional structures. Although there are some ambiguities in the selection of parameters used to synthesize seismograms by the Gaussian beam method, physically appropriate values may be estimated, and the choice of these parameters is not critical to the results.

The amplitudes of surface waves are determined mainly from the spatial second derivative of the phase velocity distribution, and several forward tests on regionalized models with periods 20--40 s show that this waveform synthesis is sensitive to slight variations of laterally heterogeneous structure which conventional methods using only phase information cannot resolve. Results of tests for heterogeneous structure in the Pacific Ocean imply that this method may help to resolve weak and small-scale velocity anomalies such as the Hawaiian hot spots or details of lateral changes in seismic velocities near spreading ridges.