Kinematics of Residual NMO

The residual NMO differential equation is the second term in (1):  

$${{\partial \tau} \over {\partial v}} = {{h^2} \over {v^3\,\tau}}\;.$$ (22)

Equation (22) is independent from the midpoint x. This fact indicates the one-dimensional nature of normal moveout. The general solution of equation (22) is obtained by simple integration. It takes the form  

$$\tau^2(v) = C - {h^2 \over v^2} = \tau_1^2 + h^2\,\left({1 \over v_1^2} - {1 \over v^2}\right)\;,$$ (23)

where C is an arbitrary velocity-independent constant, and I have chosen the constants $\tau_1$ and v₁ so that $\tau(v_1) = \tau_1$.

For the case of a point diffractor, solution (23) easily combines with the zero-offset solution (16). The result is a simplified version of the prestack residual migration summation path:  

$$\tau(x)=\sqrt{\tau_d^2 + {{(x - x_d)^2} \over {v_d^2 - v^2}} + h^2\,\left({1 \over v_d^2} - {1 \over v^2}\right)}\;.$$ (24)

Summation paths of the form (24) for a set of diffractors with different depths are plotted in Figures 3 and 4. The parameters chosen in these plots allow a direct comparison with Etgen's Figures 2.4 and 2.5 Etgen (1990), based on the exact solution and reproduced in Figures 9 and 10. The comparison shows that the approximate solution (24) captures the main features of the prestack residual migration operator, except for the residual DMO cusps appearing in the exact solution when the diffractor depth is smaller than the offset.

 

vlcve1 Figure 3 Summation paths of the simplified prestack residual migration for a series of depth diffractors. Residual slowness v/vd is 1.2; offset h is 1 km. This figure is to be compared with Etgen's Figure 2.4.

 

vlcve2 Figure 4 Summation paths of the simplified prestack residual migration for a series of depth diffractors. Residual slowness v/vd is 0.8; offset h is 1 km. This figure is to be compared with Etgen's Figure 2.5.

Neglecting the residual DMO term in residual migration is approximately equivalent in accuracy to neglecting the DMO step in conventional processing. Indeed, as follows from the geometric analogue of equation (1) derived in Appendix A, dropping the residual DMO term corresponds to the condition  

$$\tan^2{\alpha}\,\tan^2{\gamma} \ll 1\;,$$ (25)

where $\alpha$ is the dip angle, and $\gamma$ is the reflection angle. As shown by Yilmaz and Claerbout 1980, the conventional processing sequence without the DMO step corresponds to the separable approximation of the double-square-root equation (67):  

$$\sqrt{1 - v^2\,\left({{\partial t} \over {\partial s}}\right... ...- v^2\,\left({{\partial t} \over {\partial h}}\right)^2} - 2\;.$$ (26)

In geometric terms, approximation (26) transforms to  

$$\cos{\alpha}\,\cos{\gamma} \approx \sqrt{1 - \sin^2{\alpha}\... ...s^2{\gamma}} + \sqrt{1 - \sin^2{\gamma}\,\cos^2{\alpha}} - 1\;.$$ (27)

Estimating the accuracy of the separable approximation by the first term of the Taylor series for small $\alpha$ and $\gamma$ yields the estimate of ${3 \over 4}\,\tan^2{\alpha}\,\tan^2{\gamma}$Yilmaz and Claerbout (1980), which agrees qualitatively with (25). Though approximation (24) fails in situations where the dip moveout correction is necessary, it is significantly more accurate than the 15-degree approximation of the double-square-root equation, implied in the migration velocity analysis method of Yilmaz and Chambers 1984 and MacKay and Abma 1992. The 15-degree approximation  

$$\sqrt{1 - v^2\,\left({{\partial t} \over {\partial s}}\right... ...t)^2 + \left({{\partial t} \over {\partial s}}\right)^2\right)$$ (28)

corresponds geometrically to the equation  

$$2\,\cos{\alpha}\,\cos{\gamma} \approx {{3 + \cos{2\alpha}\,\cos{2\gamma}} \over 2}\;.$$ (29)

Its estimated accuracy is ${1 \over 8}\,\tan^2{\alpha} + {1 \over 8}\,\tan^2{\gamma}$. Unlike the separable approximation, which is accurate separately for zero offset and zero dip, the 15-degree approximation fails at zero offset in the case of a steep dip and at zero dip in the case of a large offset.