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To overcome the limitation of the scattering-angle-domain illumination for planar reflectors discussed in the preceding section,
we further decompose the illumination into dip-angle domain, resulting in dip-dependent scattering-angle-domain illumination.
From equations 9 and 10, it is easy to obtain the tangent of the dip angle using either
![$\displaystyle \tan \alpha = - \frac{p_{m_x}}{p_{m_z}} = -\frac{k_{m_x}}{k_{m_z}},$](img90.png) |
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(14) |
or
![$\displaystyle \tan \alpha = \frac{p_{h_z}}{p_{h_x}} = \frac{k_{h_z}}{k_{h_x}},$](img91.png) |
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(15) |
where
,
and
,
are the horizontal and vertical components
of the midpoint wavenumber vector
and the subsurface-offset wavenumber vector, respectively.
Dip decomposition using either equation 14 or 15 has
its own pros and cons.
Equation 15 is suitable for computing dip-angle gathers for sparsely isolated
image points, because it does not require any CMP information, i.e.,
and
,
and outputting gathers for sparsely isolated image points may mitigate the extra
computer time and storage spent in computing
both the horizontal and vertical subsurface offsets,
and
.
On the other hand, equation 14 is computationally less demanding, because
it does not require computing vertical subsurface offsets. However, it estimates dips using
the CMP information, hence a block of densely sampled image points in the CMP domain should be
output to avoid dip aliasing. In the following numerical examples, we use equation
14 for dip decomposition due to the fact that it is relatively inconvenient
to output vertical subsurface offsets by using one-way wave-equation-based extrapolators.
After transforming the subsurface-offset-domain sensitivity kernel into the dip-dependent scattering-angle
domain, we can proceed to compute the corresponding Hessian using
or the illumination using
![$\displaystyle H_{\gamma,\alpha}({\bf x},\gamma,\alpha) = \sum_{\omega}\sum_{{\b...
...ert L_{\gamma,\alpha}({\bf x},\gamma,\alpha,{\bf x}_r,{\bf x}_s,\omega)\vert^2,$](img101.png) |
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(17) |
where
is the dip-dependent scattering-angle-domain sensitivity kernel.
The complete procedure can
be summarized as follows:
- for fixed
,
and
, apply 3-D Fourier transform along axes
,
and
(if only horizontal subsurface offsets are computed)
or along
,
and
(if both horizontal and vertical subsurface offsets are computed)
- perform the mapping
according to relations 11 and 14, or
according to relations 11 and 15;
- apply inverse 2-D Fourier transform along axes
and
or inverse 1-D Fourier transform along axis
to obtain the dip-dependent scattering-angle-domain sensitivity kernel.
- compute either the dip-dependent scattering-angle-domain Hessian using equation 16 or
the illumination using equation 17.
For the same constant velocity example,
Figures 7 and 8 show the dip-dependent scattering-angle-domain
illumination for
and
dip angles, respectively.
The acquisition geometry is the same as that in Figure 5, i.e.,
shot and
receivers.
The illumination gathers
(Figures 7(b) and 8(b))
successfully predict the angle-dependent illumination for both the horizontal and dipping reflectors
(Figure 7(c) and Figure 8(c)).
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const-imag-hess-dip-adcig1
Figure 7. Dip-dependent scattering-angle-domain illumination.
Panel (a) is the illumination for a constant dip angle
and a constant scattering angle
;
(b) is the illumination angle gather for a constant dip angle
and at spatial location
m;
(c) is the reflectivity angle gather for the horizontal reflector extracted a
m, it is the same as Figure 6(a). [CR]
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const-imag-hess-dip-adcig2
Figure 8. Dip-dependent scattering-angle-domain illumination.
Panel (a) is the illumination for a constant dip angle
and a constant scattering angle
;
(b) is the illumination angle gather for a constant dip angle
and at spatial location
m;
(c) is the reflectivity angle gather for the dipping reflector extracted at
m, it is the same as Figure 6(b). [CR]
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Previous: Scattering-angle-domain illumination
2010-05-19