Conventional processing--separable approximation

The DSR operator is now defined as the parenthesized operator in equation (1b):  

$$\hbox{DSR} (Y,H) \eq \sqrt { 1 \ -\ (Y - H)^2 } \ +\ \sqrt { 1 \ -\ (Y + H )^2 }$$ (58)

In Fourier space, downward continuation is done with the operator $\exp ( i \omega v^{-1} \$DSR$\, z)$.

There is a serious problem with this operator: it is not separable into a sum of an offset operator and a midpoint operator. Nonseparable means that a Taylor series for (58) contains terms like $Y^2 \, H^2$.Such terms cannot be expressed as a function of Y plus a function of H. Nonseparability is a data-processing disaster. It implies that migration and stacking must be done simultaneously, not sequentially. The only way to recover pure separability would be to return to the space of S and G. (That is a drastic alternative, far from conventional processing. We will return to it later).

Let us review the general issue of separability. The obvious way to get a separable approximation of the operator $\sqrt{1 \,-\, X^2 \,-\, Y^2}$ is to form a Taylor series expansion, and then drop all the cross terms. A more clever approximation is $\sqrt{1\,-\,X^2}\ +$$\sqrt{1\,-\, Y^2 }\ -\ 1$,which fits all Y exactly when $X\,=\,0$ and all X exactly when $Y\,=\,0$.Applying this idea (though not the same equation) to the DSR operator gives

$$\begin{eqnarray} \hbox{SEP}(Y,H) &=& 2 \ +\ [\hbox{DSR}(Y,0) \ -\ 2] \ +\ [\hbo... ...rt { 1 \ -\ Y^2 } \ -\ 1 ) \ +\ ( \sqrt { 1 \ -\ H^2 } \ -\ 1 ) ]\end{eqnarray}$$ (59) (60)

Notice that at H = 0 (59) and (60) become equal to the DSR operator. At Y = 0 (59) and (60) also become equal to the DSR operator. Only when both H and Y are nonzero does SEP depart from DSR.

The splitting of (59) and (60) into a sum of three operators offers an advantage like the one offered by the 2-D Fourier kernel $\exp (ik_y y \ +\ ik_h h )$,which has a phase that is the sum of two parts. It means that Fourier integrals may have either y or h nested on the inside. So downward continuation with SEP could be done in (k_h , k_y)-space as implied by (55), or we could choose to Fourier transform to (h, k_y ), (k_h , y ), or (y,h) by appropriate nesting operations.

It is convenient to give familiar names to the three terms in (60). The first is associated with time-to-depth conversion, the second with migration, and the third with normal moveout.  

$$\hbox{SEP}(Y,H) \eq \hbox{TD} \ +\ \hbox{MIG}(Y) \ +\ \hbox{NMO}(H)$$ (61)

The approximation (59) and (60) can be interpreted as ``standard processing.'' The first stage in standard processing is NMO correction. In (59) and (60) the NMO operator downward continues all offsets at the earth's surface, to all offsets at depth. Selecting zero offset is no more than abandoning all other offsets. Like stacking over offset, selecting zero offset reduces the amount of data under consideration.

Ordinarily the abandoned offsets are not migrated. (Alternately, a clever procedure for changing stacking velocities after migration involves migrating several offsets near zero offset).

Since all terms in the SEP operator are interchangeable, it would seem wasteful to use it to migrate all offsets before stack. The result of doing so should be identical to after-stack migration.