Asymptotic pseudounitary stacking operators

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):

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

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

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

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

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

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

The weighting functions of the amplitude-preserving offset continuation have the form (Fomel, 2001b)

$$w(x;t,y) = \sqrt{z \over {2 \pi}}\; {{h_2^2-h_1^2-(x-y)^2} \over {V^{3/2}}}\;,$$ (77)

$$\widehat{w}(y;z,x) = {{t/\sqrt{z}} \over \sqrt{2 \pi}}\; {{h_2^2-h_1^2 + (x-y)^2} \over {V^{3/2}}}\;.$$ (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:

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

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

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

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

The weighting functions of the amplitude-preserving inverse and forward DMO are

$$w(x;t,y) = \sqrt{z \over {2 \pi}}\;{1 \over h_2}\;,$$ (83)

$$\widehat{w}(y;z,x) = {{t/\sqrt{z}} \over \sqrt{2 \pi}}\; {{h_2 \left(h_2^2 + (x-y)^2\right)} \over {\left(h_2^2-(x-y)^2\right)^2}}\;,$$ (84)

and the weighting functions of the asymptotic pseudo-unitary DMO are

$$w^{(+)} = \sqrt{z \over {2 \pi}}\; {\sqrt{h_2^2 + (x-y)^2} \over {h_2^2-(x-y)^2}}\;,$$ (85)

$$w^{(-)} = {{t/\sqrt{z}} \over \sqrt{2 \pi}}\; {\sqrt{h_2^2 + (x-y)^2} \over {h_2^2-(x-y)^2}}\;.$$ (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

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

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

Asymptotic pseudounitary stacking operators