Multi-dimensional Fourier transforms in the helical coordinate system

Application to the multiple prediction

Multiple prediction is the first step in the class of adaptive multiple suppression methods (Verschuur et al., 1992). In a laterally homogeneous earth, Kelamis and Verschuur (2000) show that surface-related multiples can be predicted by taking the multi-dimensional auto-convolution of a common midpoint (CMP) gather. This auto-convolution reduces to a multiplication in the f-k domain, and so it can be performed rapidly with multi-dimensional FFT's.

Since multi-dimensional FFT's can be computed with a one-dimensional Fourier transform in helical coordinates, we can predict multiples by wrapping a CMP gather onto a helix, taking its 1-D FFT, squaring the result, and returning to the original domain.

We tested this algorithm on a single CMP from the synthetic BP multiple dataset (Clapp, 1999). Figure 6 displays the multiple prediction result using the helical coordinate system and only a single one-dimensional FFT. Theoretically, only first-order multiples should have correct relative amplitudes, and the source wavelet appears twice in the multiple prediction. However, the kinematics of all multiples are almost exact, even for higher-order multiples below 5 s two-way traveltime.

BP2
Figure 6. The left panel shows the multiple model obtained with the helix and a 1-D FFT. The right panel shows the input CMP gather with the offset axis reversed to facilitate the comparison. Some wrap-around effects appear at the top of the multiple model.

Multi-dimensional Fourier transforms in the helical coordinate system