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In the case of common-offset migration in a general variable-velocity
medium, the weighting function (46) cannot be simplified to a
different form, and all its components need to be calculated
explicitly by dynamic ray tracing (Cerveny and de Castro, 1993). In the
constant-velocity case, we can differentiate the explicit expression
for the summation path
![$\displaystyle \widehat{\theta}(y;z,x) = z + {{\rho_s(x,y) + \rho_r(x,y)} \over v}\;,$](img129.png) |
(50) |
where
and
are the lengths of the incident and reflected rays:
For simplicity, the vertical component of the midpoint
is set here to zero. Evaluating the second derivative term in formula
(46) for the common-offset geometry leads, after some heavy
algebra, to the expression
![$\displaystyle \left\vert{{\partial^2 T\left(s(y),x\right)} \over {\partial x \...
...{{\rho_s + \rho_r} \over {v \rho_s \rho_r}}\right)^{m-1} \cos{\alpha(x)}\;.$](img137.png) |
(53) |
Substituting (53) into the general formula (46) yields
the weighting function for the common-offset true-amplitude
constant-velocity migration:
![$\displaystyle \widehat{w}_{CO}(y;z,x) = {1\over{\left(2 \pi\right)^{m/2}}} ...
...+ \rho_r)^{m-1} (\rho_s^2 + \rho_r^2)} \over {v (\rho_s \rho_r)^{m/2+1}}}\;.$](img138.png) |
(54) |
Equation (54) is similar to the result obtained by
Sullivan and Cohen (1987). In the case of zero offset
,
it reduces to equation (49). Note that the value
of
in (54) corresponds to the two-dimensional (cylindric)
waves recorded on the seismic line. A special case is the 2.5-D inversion,
when the waves are assumed to be spherical, while the recording is on a line,
and the medium has cylindric symmetry. In this case, the modeling weighting
function (42) transforms to
(Deregowski and Brown, 1983; Bleistein, 1986)
![$\displaystyle w(x;t,y) = {1\over{\left(2 \pi\right)^{1/2}}} {\sqrt{v} {C\left(s(y),x,r(y)\right)} \over {\sqrt{\rho_s \rho_r (\rho_s + \rho_r)}}}\;,$](img141.png) |
(55) |
and the time filter is
. Combining this result with formula (53)
for
, we obtain the weighting function for the 2.5-D
common-offset migration in a constant velocity medium
(Sullivan and Cohen, 1987):
![$\displaystyle \widehat{w}_{CO;2.5D}(y;z,x) = {1\over{\left(2 \pi\right)^{1/2}}...
... + \rho_r} (\rho_s^2 + \rho_r^2)} \over {\sqrt{v} (\rho_s \rho_r)^{3/2}}}\;.$](img143.png) |
(56) |
The corresponding time filter for 2.5-D migration is
.
In the
common-offset case, the pseudo-unitary weighting is defined from
(47) and (53) as follows:
![$\displaystyle w^{(-)}_{CO}(y;z,x) = {1\over{\left(2 \pi v\right)^{m/2}}} {...
...r 2} \sqrt{\rho_s^2 + \rho_r^2}} \over {(\rho_s \rho_r)^{{m+1} \over 2}}}\;,$](img145.png) |
(57) |
where
![$\displaystyle \cos{\alpha} = \left({ {(x - y)^2 + \rho_s \rho_r - h^2} \over {2 \rho_s \rho_r}} \right)^{1/2}\;.$](img146.png) |
(58) |
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Next: Post-Stack Time Migration
Up: Migration
Previous: 2. Zero-offset migration
2013-03-03