Separability of 3-d migration (Jakubowicz)

In an operations environment, 3-D is much harder to cope with than 2-D. Therefore, it may be expedient to suppose that 3-D migration can be achieved merely by application of 2-D migration twice, once in the x-direction and once in the y-direction. The previous section would lead you to believe that such an expedient process would result in a significant degradation of accuracy. In fact, the situation is much better than might be supposed. It has been shown by Jakubowicz and Levin [1983] that, wonder of wonders, for a constant-velocity medium, the expedient process is exact.

The explanation is this: migration consists of more than downward continuation. It also involves imaging, that is, the selection of data at t=0. In principle, downward continuation is first completed, for both the x and the y directions. After that, the imaging condition is applied. In the expedient process there are four steps: downward continuation in x, imaging, downward continuation in y, and finally a second imaging. Why it is that the expedient procedure gives the correct result seems something of a puzzle, but the validity of the result is easy to demonstrate.

First note that substitution of (7) into (8) gives (9) where

$$\begin{eqnarray} t_1^2 &=& t_0^2 \ +\ (x - x_0 )^2 / v^2 \\ t^2 &=& t_1^2 \ +\ (... ...2 \\ t^2 &=& t_0^2 \ +\ (x - x_0 )^2 / v^2 \ +\ (y - y_0 )^2 / v^2\end{eqnarray}$$ (7) (8) (9)

Equation (9) represents travel time to an arbitrary point scatterer. For a 2-D survey recorded along the y-axis, i.e., at constant x, equation (8) is the travel-time curve. In-line hyperbolas cannot be distinguished from sideswipe hyperbolas. 2-D migration with equation (8) brings the energy up to t₁. Subsequently migrating the other direction with equation (7) brings the energy up the rest of the way to t₀. This is the same result as the one given by the more costly 3-D procedure migrating with (9).

The Jakubowicz justification is somewhat more mathematical, but may be paraphrased as follows. First note that substitution of (10) into (11) gives (12) where

$$\begin{eqnarray} k_{\tau}^2 &=& \omega^2 \ -\ v^2 \, k_x^2 \\ k_z^2 &=& {k_{\tau... ...\ -\ k_y^2 \\ k_z^2 &=& {\omega^2 \over v^2} \ -\ k_x^2 \ -\ k_y^2\end{eqnarray}$$ (10) (11) (12)

Two-dimensional Stolt migration over x may be regarded as a transformation from travel-time depth t to a pseudodepth $\tau$ by use of equation (10). The second two-dimensional migration over y may be regarded as a transformation from pseudodepth $\tau$ to true depth z by use of equation (11). The composite is the same as equation (12), which depicts 3-D migration.

The validity of the Jakubowicz result goes somewhat beyond its proof. Our two-dimensional geophysicist may be migrating other offsets besides zero offset. (In chapter nonzero-offset data is migrated). If a good job is done, all the reflected energy moves up to the apex of the zero-offset hyperbola. Then the cross-plane migration can handle it if it can handle zero offset. So offset is not a problem. But can a good job be done of bringing all the energy up to the apex of the zero-offset hyperbola?

Difficulty arises when the velocity of the earth is depth-dependent, as it usually is. Then the Jakubowicz proof fails, and so does the expedient 3-D method. With a 2-D survey you have the problem that the sideswipe planes require a different migration velocity than the vertical plane. Rays propagating to the side take longer to reach the high-velocity media deep in the earth. So sideswipes usually require a lower migration velocity. If you really want to do three-dimensional migration with v(z), you should forget about separation and do it the hard way. Since we know how to transpose the hard way really isn't much harder.