In seismic exploration, predictions are used along both the time and space axis of seismic sections, but the applications of these predictions are different. Along the time axis, the desired output is the unpredictable part of the trace. The predicatable part contains the source wavelet and multiple reflections that reduce the apparent resolution. Along the spatial axis the desired output is the predictable part of the traces, and the unpredictable part is often noise that doesn't correlate between traces. While this lateral prediction is understandable in a simple layered earth, even a subsurface made up of random scatterers produces lateraly continuous diffraction events in unmigrated sections.
FX-decon is a useful noise-attenuation technique that has become a standard seismic processing tool. As introduced by Canales1984, it predicts linear events with a complex least-squares prediction technique in the frequency-space domain.
The two-dimensional deconvolution discussed here uses the routines developed by Claerbout and shown in the references Claerbout (1992a) and inClaerbout (1992b). These routines calculate filters using a conjugate-gradient method directly from the data in the time-space domain rather than Fourier transforming the data before doing the predictions.
Where FX-decon does a spatial prediction for each frequency, the two-dimensional deconvolution calculates a single filter. Although the number of coefficients in the filter is typically greater for two-dimensional deconvolution than for FX-decon, FX-decon creates a different filter for each frequency. Thus, two-dimensional deconvolution is more constrained than FX-decon in making its predictions.
In this paper, the two-dimensional deconvolution differs from Claerbout's work by extending the filter in the x-direction and limiting it in the time-direction. Where Claerbout was trying to predict single plane waves, I am attempting to predict anything that is laterally predictable with a reasonably small two-dimensional filter. We hope that the two-dimensional deconvolution allows more control over the filter action by modifying the filter shape.
I show that two-dimensional deconvolution and FX-decon produce similar results for most cases. It appears that two-dimensional deconvolution's advantage is that it does not line up noise as much as FX-decon when no predictable signal is present. Also, two-dimensional deconvolution does not produce the artificial lineups along strong signals to the extent FX-decon does.