Theory

When doing downward continuation in the offset domain, we begin by organizing our data cube as a function of midpoint x, offset h, and frequency f. We then apply the double square root (DSR) equation to move the wavefield down one depth step $\Delta z$ Claerbout (1995). We apply an imaging condition, and then repeat the procedure. This methodology can be quite expensive even in 2-D because the cost C is approximately

$$\begin{eqnarray} C &\approx&nz *f * ( FFT(nx,nh) + nx*nh* CEXP) \\ C &\approx&nz * f * ( nh*nx log(nh) + nx*nh log(nx) + CEXP(nx,nh)) \nonumber ,\end{eqnarray}$$ (1)

where FFT(nx,nh) is the expense of doing a 2-D FFT on a nx by nh dataset and CEXP is the cost of multiplying by a complex exponential. In 3-D the cost is even more substantial.

Equation (1) indicates that the number of depths can greatly affect the cost of the migration. As a result, the choice of depth sampling is a major decision. Too fine a depth sampling will make the cost exorbitant; too coarse will cause resolution and aliasing problems.

The required depth sampling is also depth varying. We need finer depth sampling near the surface, while coarser depth sampling is appropriate in the deeper section. So the first obvious way to speed up our migration is to vary the sampling as a function of depth.