Practical Implementation

The algorithm described by equations 4 and 5 is theoretically attractive because it shows in one equation that downward extrapolation can be done entirely in the $\omega$-$\mathbf{k}$ domain, even for arbitrary spatial velocity variations. From the practical point of view, however, the algorithm is proportional, at each depth step and each frequency, to the cube of the spatial dimensions. Clearly, the cost is associated with the unreasonable demand that we extrapolated, at each depth step, as many wavefields as spatial positions. This is not really necessary, as we will see below.
 

• Computation of the vertical wavenumber
• Computation of the horizontal wavenumbers
• Subsampling in $\omega$-$\mathbf{k}$ Space
• Other issues