I will introduce, justify and illustrate a fast method for calculating finite-frequency waveforms for the first arriving signals due to a point source in a three-dimensional, heterogeneous, acoustic media. In analogy to the use of Feynman path-integrals in quantum physics, a solution to the scalar wave equation is obtained by Monte Carlo summation over the contributions of elementary signals propagating along a representative sample of all possible paths between the source and receiver. The elementary signal, related to the source-time function convolved with the Green's function for the homogeneous problem, is summed into a waveform at the time equal to the travel-time (integral of slowness) along each path. The constructive and destructive interference of the contributions from the paths converges to an estimate of the complete response for the portion of the medium and range of travel times covered by the sampled paths.

This path-integral technique is applicable to strongly varying, complicated, three-dimensional media; the resulting waveform includes refracted, reflected and diffracted waves with appropriate geometrical spreading, focusing, defocusing and phase changes. In contrast to geometrical ray methods, this technique is valid at finite frequency and does not exhibit singular behaviour. Relative to numerical methods the path-integral method requires insignificant computer memory and one to two orders of magnitude less CPU time. Because the portion of the medium that contributes to the signal at any point on the waveform is known, this method is applicable with iterative inversion techniques. In addition, the speed and accuracy of the path-integral method may be sufficient to allow three-dimensional, full-waveform, tomographic imaging using stochastic, non-linear, global search methods.