Fourier methods of seismic data regularisation

QDome Data Example

To use a more realistic seismic example the QDome synthetic data from Claerbout's Image Estimation By Example (Claerbout, 1995) was adapted to contain an irregular x-axis, details of the data computation can be found in Claerbout (1995). This dataset features flat beds at the surface, a Gaussian centre with a fault and dipping beds at the bottom - this model is useful since it incorporates events that are flat, dipping, curving, aliased and discontinuous, and so is an appropriate model to test the robustness of the algorithm.

An irregular axis was created as in the previous examples and the QDome data was calculated for this irregular axis. A 3D cube of data was created and presented herein are two slices from the cube orthogonal to the (regular) y axis - one slice containing strongly aliased events and one containing a discontinous fault

aliasg
Figure 7. QDome slice containing aliased data (a), DFT regularisation result (b) and ALFT regularisation result (c).

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Figure 8. QDome slice containing a fault (a), DFT regularisation result (b) and ALFT regularisation result (c).

The ALFT shows a much improved result in both of these cases. In this example the DFT particularly struggles with the amplitudes of the traces and some appear much diminished - these are the traces that were the most irregular. The fault appears more continuous and defined in the ALFT result than in the irregular input image, as do the steep events in Figure 8. However the algorithm still struggles somewhat in the aliased sections of the data, and the spectrum (as seen in Figure 7), is noisier in aliased regions. Schonewille (2009) discusses how this problem can be addressed, and this can be done by extrapolating the known spectrum to higher $\omega$ and $k$ values, estimating weights for these higher frequencies and computing the ALFT for these.

Whilst the results of using the ALFT show a marked improvement over DFT regularisation and a robustness in the presence of aliased and discontinuous events it is fundamentally limited by its reliance on multiple domain transformations. For large datasets the number of operations required by this technique is unreasonable and so a different method, or maybe one that augments the ALFT to require fewer transforms, is desired. It is possible that the answer to this problem lies in tapering the input data in such a way that Fourier components can be made more orthogonal. This is discussed in the subsequent section.

Fourier methods of seismic data regularisation