Path-summation waveforms

Abstract

In this paper, we examine an efficient, practical method to calculate approximate, finite‐frequency waveforms for the early signals from a point source in 3‐D 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 traveltime 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 traveltimes 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 with the exact solution for a 3‐D geometry involving a low‐velocity spherical inclusion, and with finite‐difference waveforms for a 2‐D 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‐difference 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 traveltimes produced by the path summation give a relation between the medium and the signal on the waveform that is not available with finite‐difference and finite‐element methods. Furthermore, the speed and accuracy of the path‐summation method may be sufficient to allow 3‐D waveform inversion using stochastic, non‐linear, global search methods.

Theory and Observations: Body Waves, Ray Methods, and Finite-Frequency Effects

“…Before going into these linearized approximations, we must emphasize the path integral approach where any trajectory connecting the source and the receiver may contribute to seismogram estimation (Lomax, 1999;Schlottmann, 1999;Thomson, 2001). Although we have not yet found a practical way of assessing convergence toward the true solution of these integrals, these methods exhibit nonlocal behaviors, which make them completely different from approaches we gave consider up to now.…”

A Feynman Path Integral Representation for Elastic Wave Scattering by Anisotropic Weakly Perturbations

“…[4][5][6], see also Appendix C for a toy-model of ours to understand the principles of path-integral tomography mechanism [7].…”

The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena

“…However, the Rytov approximation differs from the Born approximation in that the phase relation of the incident and scattered wavefield is linear, rather than the amplitude. Although computationally impractical, the path-integral approach [e.g., 22; 23] is conceptually attractive because it provides a link between many of the ray-based methods and the full-wave equation methods. Variations on the finite-difference approach are the one-way [e.g., 24; 25; 26] and the phasescreen [e.g., 27; 28; 29; 30] methods.…”