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Next: Conclusions Up: EXAMPLES Previous: Velocity Transform

Offset Continuation and DMO

Offset continuation is the operator that transforms seismic reflection data from one offset to another (Bolondi et al., 1982; Salvador and Savelli, 1982). If the data are continued from half-offset $ h_1$ to a larger offset $ h_2$ , the summation path of the post-NMO integral offset continuation has the following form (Stovas and Fomel, 1996; Fomel, 2001b; Biondi and Chemingui, 1994):

$\displaystyle \theta(x;t,y) = {t \over h_2} \sqrt{{U+V} \over 2}\;,$ (73)

where $ U = h_1^2 + h_2^2 - (x - y)^2$ , $ V = \sqrt{U^2 -
4 h_1^2 h_2^2}$ , and $ x$ and $ y$ are the midpoint coordinates before and after the continuation. The summation path of the reverse continuation is found from inverting (73) to be

$\displaystyle \widehat{\theta}(y;z,x) = {z  h_2} \sqrt{2 \over {U+V}} = {z \over h_1} \sqrt{{U-V} \over 2}\;.$ (74)

The Jacobian of the time coordinate transformation in this case is simply

$\displaystyle \left\vert\partial \widehat{\theta} \over \partial z\right\vert = {t \over z}\;.$ (75)

Differentiating summation paths (73) and (74), we can define the product of the weighting functions according to formula (10), as follows:

$\displaystyle w \widehat{w}={1\over{2 \pi}}   {\sqrt{\left\vert F \widehat{...
...rt}} = {t\over{2 \pi}}  {{\left(h_2^2-h_1^2\right)^2 - (x-y)^4} \over V^3}\;.$ (76)

The weighting functions of the amplitude-preserving offset continuation have the form (Fomel, 2001b)
$\displaystyle w(x;t,y)$ $\displaystyle =$ $\displaystyle \sqrt{z \over {2 \pi}}\;
{{h_2^2-h_1^2-(x-y)^2} \over {V^{3/2}}}\;,$ (77)
$\displaystyle \widehat{w}(y;z,x)$ $\displaystyle =$ $\displaystyle {{t/\sqrt{z}} \over \sqrt{2 \pi}}\;
{{h_2^2-h_1^2 + (x-y)^2} \over {V^{3/2}}}\;.$ (78)

It easy to verify that they satisfy relationship (76); therefore, they appear to be asymptotically inverse to each other.

The weighting functions of the asymptotic pseudo-unitary offset continuation are defined from formulas (28) and (29), as follows:

$\displaystyle w^{(+)}$ $\displaystyle =$ $\displaystyle {1\over{\left(2 \pi\right)^{1/2}}} \;
\left\vert F \widehat{F}...
...{{\left(\left(h_2^2-h_1^2\right)^2 - (x-y)^4\right)^{1/2}}
\over {V^{3/2}}}\;,$ (79)
$\displaystyle w^{(-)}$ $\displaystyle =$ $\displaystyle {1\over{\left(2 \pi\right)^{1/2}}} \;
\left\vert F \widehat{F}...
...{{\left(\left(h_2^2-h_1^2\right)^2 - (x-y)^4\right)^{1/2}}
\over {V^{3/2}}}\;.$ (80)

The most important case of offset continuation is the continuation to zero offset. This type of continuation is known as dip moveout (DMO). Setting the initial offset $ h_1$ equal to zero in the general offset continuation formulas, we deduce that the inverse and forward DMO operators have the summation paths

$\displaystyle \theta(x;t,y)$ $\displaystyle =$ $\displaystyle {t \over h_2} \sqrt{h_2^2-(x-y)^2}\;,$ (81)
$\displaystyle \widehat{\theta}(y;z,x)$ $\displaystyle =$ $\displaystyle {{z  h_2} \over \sqrt{h_2^2-(x-y)^2}}\;.$ (82)

The weighting functions of the amplitude-preserving inverse and forward DMO are
$\displaystyle w(x;t,y)$ $\displaystyle =$ $\displaystyle \sqrt{z \over {2 \pi}}\;{1 \over h_2}\;,$ (83)
$\displaystyle \widehat{w}(y;z,x)$ $\displaystyle =$ $\displaystyle {{t/\sqrt{z}} \over \sqrt{2 \pi}}\;
{{h_2 \left(h_2^2 + (x-y)^2\right)} \over
{\left(h_2^2-(x-y)^2\right)^2}}\;,$ (84)

and the weighting functions of the asymptotic pseudo-unitary DMO are
$\displaystyle w^{(+)}$ $\displaystyle =$ $\displaystyle \sqrt{z \over {2 \pi}}\;
{\sqrt{h_2^2 + (x-y)^2} \over {h_2^2-(x-y)^2}}\;,$ (85)
$\displaystyle w^{(-)}$ $\displaystyle =$ $\displaystyle {{t/\sqrt{z}} \over \sqrt{2 \pi}}\;
{\sqrt{h_2^2 + (x-y)^2} \over {h_2^2-(x-y)^2}}\;.$ (86)

Equations similar to (83) and (84) have been published by Stovas and Fomel (1996). Equation (84) differs from the similar result of Black et al. (1993) by a simple time multiplication factor. This difference corresponds to the difference in definition of the amplitude preservation criterion. Equation (84) agrees asymptotically with the frequency-domain Born DMO operators (Bleistein and Cohen, 1995; Bleistein, 1990; Liner, 1991). Likewise, the stacking operator with the weighting function (83) corresponds to Ronen's inverse DMO (Ronen, 1987), as discussed by Fomel (2001b). Its adjoint, which has the weighting function

$\displaystyle \widetilde{w}(x;t,y) = {{t/\sqrt{z}} \over {2 \pi}}\;{1 \over h_2}\;,$ (87)

corresponds to Hale's DMO (Hale, 1984).


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Next: Conclusions Up: EXAMPLES Previous: Velocity Transform

2013-03-03