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
, we seek the velocity function *v* and the data perturbation
of root mean square relative size at most 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 . The final section summarizes our conclusions and formulates a few directions for further research.

4/20/1999