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![]() | Measuring image focusing for velocity analysis | ![]() |
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In Biondi (2008b), I introduced a new semblance functional, that I dubbed Image-focusing semblance, aimed at quantitatively measuring image focusing simultaneously along the spatial directions and the reflection angle (or offset) axes. The underlying idea is to extend the conventional semblance evaluation by measuring image coherency also along the the structural-dip axes. However, the estimates provided by the image-focusing semblance presented in that report can be biased by reflectors' curvature. In this section, I modify the definition of the image-focusing semblance by explicitly exposing its dependency from the image local curvature. This enables a consistent evaluation of the image focusing across both the reflection-angle axis and the structural-dip axis and improves the interpretability of the results.
The starting point of my method is an ensemble of prestack images,
;
these images are function of a spatial coordinate vector
(with
depth and
the horizontal location),
the aperture angle
, and a velocity parameter
.
In the numerical examples that follow,
the ensemble of prestack images
is obtained by residual prestack migration in the angle domain
as I presented in Biondi (2008a).
The parameter
is the ratio between the new migration
velocity and the migration velocity used for the initial migration.
The proposed method could be easily adapted to the case when
residual prestack Stolt migration
(Sava, 2003),
or any other method that can efficiently generate ensembles
of prestack images dependent on a velocity parameter,
is used to compute
.
Although,
when using other methods to produce
the ensemble
,
the corrections equivalent
to equations 5, 8
and 9 might be different.
To measure coherency along the structural dip ,
I first decompose the prestack image and create
the dip-decomposed prestack image
.
When using either choices of residual prestack migration
discussed above,
the decomposition can be efficiently performed
in the Fourier domain
during the residual prestack migration.
If other methods are used to produce the ensemble
of prestack images
,
the dip decomposition could as efficiently performed in the space domain
by applying recursive filters
(Hale, 2007; Fomel, 2002).
Notice, that
the dip-decomposed images I use as input
have different kinematic characteristics
than the ones described in
Reshef and Rüger (2008), Landa et al. (2008), and Reshef (2008).
They obtain dip-decomposed images by not performing the
implicit summation over dips that is part of angle-domain Kirchoff migration
(Audebert et al., 2002),
whereas I decompose the migrated images.
In equation 5 in Biondi (2008b) I defined the 2D Image-focusing semblance as:
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![]() | Measuring image focusing for velocity analysis | ![]() |
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