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 h₁ to a larger offset h₂, the summation path of the post-NMO integral offset continuation has the following form Biondi and Chemingui (1994); Fomel (1995b); Stovas and Fomel (1996):  

$$\theta(x;t,y) = {t \over h_2}\,\sqrt{{U+V} \over 2}\;,$$ (85)

where U = h₁² + h₂² - (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 (85) to be  

$$\widehat{\theta}(y;z,x) = {z \,h_2}\,\sqrt{2 \over {U+V}} = {z \over h_1}\,\sqrt{{U-V} \over 2}\;.$$ (86)

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

$$\left\vert\partial \widehat{\theta} \over \partial z\right\vert = {t \over z}\;.$$ (87)

Differentiating summation paths (85) and (86), we can define the product of the weighting functions according to formula (9), as follows:  

$$w\,\widehat{w}={1\over{2\,\pi}} \, {\sqrt{\left\vert F\,\wi... ...,\pi}} \,{{\left(h_2^2-h_1^2\right)^2 - (x-y)^4} \over V^3}\;.$$ (88)

The weighting functions of the amplitude-preserving offset continuation have the form

$$\begin{eqnarray} w(x;t,y) & = & \sqrt{z \over {2\,\pi}}\; {{h_2^2-h_1^2-(x-y)^2}... ...over \sqrt{2\,\pi}}\; {{h_2^2-h_1^2 + (x-y)^2} \over {V^{3/2}}}\;.\end{eqnarray}$$ (89) (90)

It easy to verify that they satisfy relationship (88); therefore, they appear to be asymptotically inverse to each other.

The weighting functions of the asymptotic pseudo-unitary offset continuation are defined from formulas (34) and (35), as follows:

$$\begin{eqnarray} w^{(+)} & = & {1\over{\left(2\,\pi\right)^{1/2}}} \; \left\ver... ...(h_2^2-h_1^2\right)^2 - (x-y)^4\right)^{1/2}} \over {V^{3/2}}}\;.\end{eqnarray}$$ (91) (92)

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 h₁ equal to zero in the general offset continuation formulas, we deduce that the inverse and forward DMO operators have the summation paths

$$\begin{eqnarray} \theta(x;t,y) & = & {t \over h_2}\,\sqrt{h_2^2-(x-y)^2}\;, \ \... ...hat{\theta}(y;z,x) & = & {{z \,h_2} \over \sqrt{h_2^2-(x-y)^2}}\;.\end{eqnarray}$$ (93) (94)

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

$$\begin{eqnarray} w(x;t,y) & = & \sqrt{z \over {2\,\pi}}\;{1 \over h_2}\;, \ \wi... ...t(h_2^2 + (x-y)^2\right)} \over {\left(h_2^2-(x-y)^2\right)^2}}\;,\end{eqnarray}$$ (95) (96)

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

$$\begin{eqnarray} w^{(+)} & = & \sqrt{z \over {2\,\pi}}\; {\sqrt{h_2^2 + (x-y)^2... ...\sqrt{2\,\pi}}\; {\sqrt{h_2^2 + (x-y)^2} \over {h_2^2-(x-y)^2}}\;.\end{eqnarray}$$ (97) (98)

Formulas similar to (95) and (96) have been published by Fomel 1995b and Stovas and Fomel 1996. Formula (96) 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. Formula (96) agrees asymptotically with the frequency-domain Born DMO operators Bleistein and Cohen (1995); Bleistein (1990). Likewise, the stacking operator with the weighting function (95) corresponds to Ronen's inverse DMO Ronen (1987), as I discussed in an earlier report Fomel (1995b). Its adjoint, which has the weighting function  

$$\widetilde{w}(x;t,y) = {{t/\sqrt{z}} \over {2\,\pi}}\;{1 \over h_2}\;,$$ (99)

corresponds to Hale's DMO Hale (1984).