Application to 3-d downward continuation

The operator for migration of zero-offset reflection seismic data in three dimensions is expandable to second order by Taylor series expansion to the so-called 15$^\circ$ approximation  

$$\sqrt{ {(-i \omega ) {}^2 \over v^2}\ \,-\, { \ \partial^2 \... ...-\, {v \over - 2 i \omega} \ {\partial^2 \ \over \partial y^2 }$$ (5)

The most common case is when v is slowly variable or independent of x and y. Then the conditions of full separation do apply. This is good news because it means that we can use ordinary 2-D wave-extrapolation programs for 3-D, doing the in-line data and the out-of-line data in either order. The bad news comes when we try for more accuracy. Keeping more terms in the Taylor series expansion soon brings in the cross term $\partial^4 / {\partial x}^2 {\partial y}^2$.Such a term allows neither full separation nor splitting. Fortunately, present-day marine data-acquisition techniques are sufficiently crude in the out-of-line direction that there is little justification for out-of-line processing beyond the 15$^\circ$ equation. Francis Muir had the good idea of representing the square root as  

$$\sqrt{ {(-i \omega {)}^2 \over v^2}\ \ -\ \ {\partial^2 \ov... ...r - 2 i \omega} \ \ {\partial^2 \ \over \partial y^2 } \ \ \ \\ end{displaymath}$$ (6)

There may be justification for better approximations with land data. Fourier transformation of at least one of the two space axes will solve the computational problem. This should be a good approach when the medium velocity does not vary laterally so rapidly as to invalidate application of Fourier transformation.