Discussion

The results exhibited in the last section are characteristic of dual regularization: it robustly identifies the dominant moveout trend, so long as the linear solves in the Moré-Hebden algorithm are sufficiently precise. It is also a great deal more expensive than ``raw'' differential semblance (J₀) minimization, often by a factor of 10-15, due to the need to solve linear systems.

Note that dual regularization does not eliminate the need for preprocessing to reduce coherent noise: the desired primary reflection phase must be energetically dominant, whereas very strong multiple reflection phases are common in some areas.

Speed improvements should be possible through better heuristics and algorithmic tuning, and perhaps through more effective constrained optimization. Since differential semblance is also effective in estimating laterally heterogeneous models Chauris et al. (1998); Symes and Versteeg (1993), dual regularization can also be applied in that context; of course, algorithmic efficiency will then become even more of an issue.

Dual regularization also implies a strategy for determination of $\sigma$: it should assume the smallest value for which the minimum of $J_{\sigma}$ is (essentially) zero. Application of this noise level determination algorithm requires a method for estimating a tolerance for this minimum, a matter currently under study.