In tomographic seismology, traveltimes and amplitudes of acoustic/elastic waves can be obtained by either ray-tracing or solving eikonal and transport equations.
We consider second-order finite difference schemes to solve these equations accurately and efficiently as well. Since the transport equation includes the gradient and the Laplacian of the computed traveltime, special superconvergent difference formulas must be employed to guarantee accurate amplitudes. Upwind differences are requisite to sharply resolve discontinuities in the traveltime derivatives, while centered differences improve accuracy and enjoy extra-order accuracy for the averaged gradient of the traveltime field. A second-order upwind ENO (Essentially Non-Oscillatory) scheme satisfies these requirements; the scheme is implemented with a dynamic DNO (Down 'N' Out) marching and an effective PS (Post Sweeping); the resulting algorithm, ENO-DNO-PS, turns out to be unconditionally ray-stable. It is numerically verified that the computed amplitude is first-order accurate, while the first-arrival traveltime shows second-order accuracy. The algorithm has been successfully tested for real velocity models having large contrasts, and extended for anisotropic media (vertical/tilted-axis transverse isotropy). We will discuss in detail important/interesting numerical and technical issues, including boundary treatment, velocity averaging, and initialization schemes for the DNO marching.
I will NOT forget to discuss open/challenging problems for which engineers in oil industry are boiling their brains.