Kirchhoff migration using eikonal equation traveltimes

Abstract

The use of ray shooting followed by interpolation of traveltimes onto a regular grid is a popular and robust method for computing diffraction curves for Kirchhoff migration. An alternative to this method is to compute the traveltimes by directly solving the eikonal equation on a regular grid, without computing raypaths. Solving the eikonal equation on such a grid simplifies the problem of interpolating times onto the migration grid, but this method is not well defined at points where two different branches of the traveltime field meet. Also, computational and data storage issues that are relatively unimportant for performance in two dimensions limit the applicability of both schemes in three dimensions. A new implementation of a gridded eikonal equation solver has been designed to address these problems. A 2-D version of this algorithm is tested by using it to generate traveltimes to migrate the Marmousi synthetic data set using the exact velocity model. The results are compared with three other images: an F-X migration (a standard for comparison), a Kirchhoff migration using ray tracing, and a Kirchhoff migration using traveltimes generated by a commonly used eikonal equation solver. The F-X‐migrated image shows the imaging objective more clearly than any of the Kirchhoff migrations, and we advance a heuristic reason to explain this fact. Of the Kirchhoff migrations, the one using ray tracing produces the best image, and the other two are of comparable quality.

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

“…In the control framework, a semi-Lagrangian scheme is obtained for H-J equations by discretizing in time the dynamic programming principle [19,20]. Another approach to obtaining a "time" dependent H-J equation from the static H-J equation is using the so called paraxial formulation in which a preferred spatial direction is assumed in the characteristic propagation [21,17,29,36,37]. High order numerical schemes are well developed for the time dependent H-J equation on structured and unstructured meshes [34,25,51,24,33,7,26,31,35,1,3,4,6,8]; see a recent review on high order numerical methods for time dependent H-J equations by Shu [46].…”

A level set based Eulerian method for paraxial multivalued traveltimes

“…In some applications, for example, wave propagation in reflection seismics [8], the traveltimes of interest are carried by the so-called sub-horizontal rays [17,34,40], where sub-horizontal means ''oriented in the positive z-direction''. A convenient characterization for sub-horizontal rays is that dz dt P c cos h max > 0; ð10Þ for some 0 < h max < p=2.…”

“…One may apply the high frequency asymptotics to a parabolic wave equation to obtain a ''time''-dependent eikonal equation as well. However, as pointed out in [12], the paraxial approximation in geometrical optics signifies another simplification which can be made when there is one preferred wave propagation direction so that a stationary eikonal equation can be rewritten as an evolution equation in one of the spatial variables [17,35]; as long as a version of the so-called sub-horizontal condition holds, there is no approximation involved. The paraxial eikonal equation we will use first appeared in [17] and was rigorously justified in [42].…”

“…We notice that the paraxial formulation for eikonal equations [17] is different from the paraxial (or parabolic) formulation for wave equations [8]. The paraxial formulation for wave equations first appeared in reflection seismology [8], and the resulting equation is usually called the parabolic wave equation; later the idea was used in underwater acoustics and many other applications.…”

“…Although the paraxial assumption in the method does not allow overturning rays in the physical space, it is sufficient for many geophysical applications [17,33,34]. We notice that the paraxial formulation for eikonal equations [17] is different from the paraxial (or parabolic) formulation for wave equations [8].…”