Three-dimensional first arrival traveltimes and amplitudes via eikonal and transport finite difference solvers

First arrival traveltimes and associated amplitudes are essential components in Kirchhoff migration and modeling. Traditionally they have been determined by ray tracing. However ray tracing does not give traveltimes on a regular grid, and is not guaranteed to produce the minimum traveltime. Seismic traveltimes in three dimensions can be computed efficiently and accurately on a regular grid using an essentially nonoscillatory ("ENO") Hamilton-Jacobi (HJ) second order scheme. The scheme can be implemented in fully vectorizable form. Several examples illustrate the effectiveness of this approach to traveltime computation. A similar accurate scheme is required to solve the transport equation for the amplitudes associated with the first arrival traveltimes. A second-order Runge-Kutta upwind finite difference scheme is constructed for this purpose. The transport equation involves the traveltime gradient and Laplacian which must be evaluated using the output of the eikonal scheme $\tau.$ The error in $\tau$ is second order accurate, hence the approximation to the traveltime Laplacian is zeroth order accurate, and there is no reason to expect the traveltime Laplacian, hence the amplitude, to converge. One remedy to ensure that the traveltime Laplacian is sufficiently accurate to guarantee convergence, is to use a higher order scheme, say third order ENO upwind scheme to solve the eikonal equation. Preliminary numerical results are presented to demonstrate the third-order accuracy of the HJ-ENO numerical flux in spatial directions (x and y) and of TVD (total variation diminishing) Runge-Kutta method in z-direction