Discussion

Early tests with synthetic models demonstrate that the offset continuation equation is a useful and efficient regularization operator for interpolating seismic reflection data. I plan to perform more tests in order to evaluate the performance of this method on real data. An extension to 3-D data is simple in theory, but it will require several modifications in the computational framework.

In the range of possible interpolation methods Mazzucchelli et al. (1998), the offset continuation approach clearly stands on the more efficient side. The efficiency is achieved both by the small size of the finite-difference filter and by the method's ability to decompose and parallelize the method across different frequencies. Part of the efficiency gain could probably be sacrificed for achieving more accurate results. Here are some interesting ideas one could try:

• Instead of fixing the offset continuation filter in a data-independent way, one could estimate some of its coefficients from the data. In particular, the second term in equation (3) can be varied to better account for the effects of variable velocity and amplitude variation with offset. Theoretical extensions of offset continuation to the variable velocity case were studied by Hong et al. (1997) and Luo et al. (1999).
• Formulating the problem in the pre-NMO domain would allow us to consider several velocities by convolving several continuation filters. This could be an appropriate approach for interpolating both primary and multiple reflections.
• Missing data interpolation problems can be greatly accelerated by preconditioning Fomel et al. (1997); Fomel (1997). Finding an appropriate preconditioning for the offset continuation method is an open problem. The non-stationary nature of the continuation filter make this problem particularly challenging.

I plan to devote my remaining time at the Stanford Exploration Project to investigating these fascinating ideas.