Introduction

Velocity analysis quantifies and parametrizes moveout in terms of velocity functions. In common with most other parts of conventional processing, velocity analysis rests on the linearized model of reflections, which treats short scale components of Earth mechanical structure function as perturbations of the large scale components. In fact, most processing derives from the acoustic version of this model, which predicts only one family of moveout curves, or phases, those of so-called compressional wave primary reflections.

Some seismic data exhibit the characteristics predicted by this acoustic primaries-only model to good approximation. Other data exhibit several phases, however; while one of these usually appears to be a compressional wave primary phase, others may represent multiple reflections, mode conversions, 3D reflection phenomena in data treated as 2D, and so forth. These other phases may carry considerable energy. Multiple reflection energy is suppressed to some extent by various multiple removal techniques, but none is universally effective in removing all phases but the primary. Residual non-primary phases pose an obstacle to velocity estimation, in that a single velocity function cannot predict several moveout families simultaneously (within the linearized acoustic approximation). The conventional approach to moveout ambiguity is visual and interpretive: the processor is expected to reject coherent noise by interactively updating velocity functions to recognize and flatten selectively the primary events in image gathers, recognize and fit only primary reflection peaks, and so on. It seems odd that the most robust information in seismic data must in the end be teased out by hand, especially as the size of 3D datasets precludes visual inspection of all but a small fraction of the prestack data.

This paper presents an alternative approach to coherent noise rejection, based on a formulation of velocity analysis as an inverse problem, when primary reflection energy dominates the (preprocessed) data. The idea is quite simple. Flatness of image gathers diagnoses the success of a velocity analysis. Image gathers created from data containing multiple phases are impossible to flatten. Therefore creation of flat image gathers requires data perturbation. If the primary phase is dominant, then the smallest data perturbation permitting flat image gathers will be that which removes the non-primary phases.

The mathematical embodiment of this idea is the dual regularization theory of velocity inversion, introduced in Gockenbach et al. (1995); Gockenbach and Symes (1997). Given a relative noise level $\sigma$, we seek the velocity function v and the data perturbation of root mean square relative size at most $\sigma$ which together yield the flattest image gather. To measure flatness, we use the differential semblance criterion, introduced in Symes (1986) and developed in a series of papers [for example Chauris et al. (1998); Symes (1998b)]. In fact, differential semblance is the only semblance measure providing the good theoretical properties needed to ensure the reliability of coherent noise rejection Kim and Symes (1998); Symes (1998a).

Thie next section describes a simple algorithm for solution of this constrained optimization problem. An example using a marine CMP and layered acoustic modeling demonstrates the coherent noise rejection permitted by reasonable estimates of noise level $\sigma$. The final section summarizes our conclusions and formulates a few directions for further research.