A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations |
Let us consider the problem of solving for and instead of
and by recasting
equations 5, 6 and 7 in the time coordinates:
From equation A-3
On the first glance, equations A-6 and A-7 seem suitable for numerically extrapolating
and in direction using the boundary conditions A-4.
After such an extrapolation, one would be able to reconstruct from equation A-2 and
thus solve the original problem. However, by further decoupling the system using the equivalence of the
second-order mixed derivatives, we discover that the underlying PDEs are elliptic. For instance, applying
to both sides of equation A-7 results in
Solving elliptic equations A-10 and A-11 with the Cauchy-type boundary conditions A-4 is an ill-posed problem (Evans, 2010). A different formulation, leading to a non-linear elliptic PDE, was previously discussed by Cameron et al. (2009).
A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations |