In this paper, we examine an efficient, practical method to calculate approximate, finite-frequency waveforms for the early signals from a point source in three-dimensional, acoustic media with smoothly varying velocity and constant density. In analogy to the use of Feynman path-integrals in quantum physics, we obtain an approximate waveform solution for the scalar wave equation by a Monte Carlo summation of elementary signals over a representative sample of all possible paths between a source and observation point. The elementary signal is formed from the convolution of the source-time function with a time-derivative of the Green's function for the homogeneous problem. For each path, this elementary signal is summed into a time-series at a travel-time obtained from an integral of slowness along the path. The constructive and destructive interference of these signals produces the approximate, waveform response for the range of travel-times covered by the sampled paths. We justify the path-summation technique for a smooth medium using a heuristic construction involving the Helmholtz-Kirchhoff integral theorem. The technique can be applied to smooth, but strongly varying and complicated velocity structures. The approximate waveform includes geometrical spreading, focusing, defocusing and phase changes, but does not fully account for multiple scattering. We compare path-summation waveforms to the exact solution for a three-dimensional geometry involving a low- velocity spherical inclusion, and to finite-difference waveforms for a two-dimensional structure with realistic, complicated velocity variations.
In contrast to geometrical ray methods, the path-summation approach reproduces finite-frequency wave phenomena such as diffraction and does not exhibit singular behaviour. Relative to the finite-differences numerical method, the path-summation approach requires insignificant computer memory and, depending on the number of waveforms required, up to one to two orders of magnitude less computing time. The sampled paths and associated travel-times produced by the path-summation give a relation between the medium and the signal on the waveform that is not available with finite-differences and finite elements methods. Furthermore, the speed and accuracy of the path-summation method may be sufficient to allow three-dimensional, waveform inversion using stochastic, non-linear, global search methods.
Key words: wave equation, wave propagation, synthetic waveforms, inversion, lateral heterogeneity