Time-to-depth conversion and seismic velocity estimation using time-migration velocity

The objective was to build an efficient algorithm 1 to esti- mate seismic velocity from time-migration velocity, and 2 to convert time-migrated images to depth. We established theoreti- cal relations between the time-migration velocity and seismic ve- locity in two and three dimensions using paraxial ray-tracing the- ory. The relation in two dimensions implies that the conventional Dix velocity is the ratio of the interval seismic velocity and the geometric spreading of image rays. We formulated an inverse problem of finding seismic velocity from the Dix velocity and de- veloped a numerical procedure for solving it. The procedure con- sists of two steps: 1 computation of the geometric spreading of image rays and the true seismic velocity in time-domain coordi- nates from the Dix velocity;2 conversion of the true seismic ve- locity from the time domain to the depth domain and computa- tion of the transition matrices from time-domain coordinates to depth. For step 1, we derived a partial differential equationPDE in two and three dimensions relating the Dix velocity and the geometric spreading of image rays to be found. This is a nonlin- ear elliptic PDE. The physical setting allows us to pose a Cauchy problem for it. This problem is ill posed, but we can solve it nu- merically in two ways on the required interval of time, if it is suf- ficiently short. One way is a finite-difference scheme inspired by the Lax-Friedrichs method. The second way is a spectral Cheby- shev method. For step 2, we developed an efficient Dijkstra-like solver motivated by Sethian's fast marching method. We tested numerical procedures on a synthetic data example and applied them to a field data example. We demonstrated that the algo- rithms produce a significantly more accurate estimate of seismic velocity than the conventional Dix inversion. This velocity esti- mate can be used as a reasonable first guess in building velocity models for depth imaging.