A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations |

where .

Similarly to the constant velocity gradient medium, it is convenient to write down the ray-tracing system in
the form (Cervený, 2001)

For a particular image ray

Solving equation D-5 for as a function of and

Combining equations D-3 through D-6, we find

According to equation 4, D-7 can give rise to the geometrical spreading:

where we must resolve . This is done by revisiting equation D-8. For given and , is the root of a depressed cubic function of the following form:

After some algebraic manipulations, we find

Finally, inserting equations D-10 and D-11 into equation 3 results
in the Dix velocity:

A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations |

2015-03-25