Post-stack velocity analysis by separation and imaging of seismic diffractions |

How can one detect the spatially-variable velocity necessary for
focusing of different diffraction events? A good measure of
focusing is the *varimax norm* used by Wiggins (1978) for
minimum-entropy deconvolution and by Levy and Oldenburg (1987) for
zero-phase correction. The varimax norm is defined as

Rather than working with data windows, we turn focusing into a
continuously variable attribute using the technique of
*local attributes* (Fomel, 2007a). Noting that the correlation
coefficient of two sequences and is defined as

one can interpret the varimax measure in equation 1 as the inverse of the squared correlation coefficient between and a constant: . Well-focused signals have low correlation with a constant and correspondingly high varimax.

Going further toward a continuously variable focusing attribute, notice that the squared correlation coefficient can
be represented as the product of two quantities
,
where

and is the solution of the least-squares minimization

This allows us to define a continuously variable attribute by using continuously variable quantities and , which are defined as solutions of regularized optimization problems

where is a regularization operator designed to avoid trivial solutions by enforcing a desired behavior (such as smoothness). Shaping regularization (Fomel, 2007b) provides a particularly convenient method for enforcing smoothing in an iterative optimization scheme.

We apply the local focusing measure to obtain migration-velocity panels for every
point in the image. First, we follow the procedure outlined in the
previous section to replace a stacked or zero-offset section with a
section containing only separated diffractions. Next, we migrate the
data multiple times with different migration velocities. This is
accomplished by *velocity continuation*
(Fomel, 2003a), a method that performs time-migration velocity analysis
by continuing seismic images in velocity with the process also called
``image waves'' (Hubral et al., 1996). The velocity continuation theory
(Fomel, 2003b) shows that one can accomplish time migration with a set of
different velocities by making differential steps in velocity
similarly to the method of cascaded migrations
(Larner and Beasley, 1987) but described and implemented as a
continuous process. While comparable in theory to an ensemble of Stolt
migrations (Fowler, 1984; Mikulich and Hale, 1992), velocity
continuation has the advantage of working directly in the image
domain. It is implemented with an efficient and robust algorithm based
on the Fast Fourier Transform.

Finally, we compute for every sample point in each of the
migrated images. Thus, in equations 7 and 8
refers to the total number of sample points in an image. The output is
*focusing image gathers* (FIGs), exemplified in Figure 1.
A FIG is analogous to a conventional migration-velocity analysis panel
and suitable for picking migration velocities. The main difference is
that the velocity information is obtained from analysis of diffraction
focusing as opposed to semblance of flattened image gathers used in
prestack analysis.

Post-stack velocity analysis by separation and imaging of seismic diffractions |

2016-03-16