Design of spatial prediction filters
Spatially random noise is uncorrelated from trace to trace independent of temporal frequency. Therefore, spatial prediction filters can be conveniently designed and applied in the frequency-space domain. As such, spatial prediction filtering has come to be known in practice by the term f − x deconvolution. While the theoretical review of the filter design and application is provided in Section F.5, below is a step-by-step procedure for f − x deconvolution.
- Start with the CMP-stacked section P(x, t) and apply Fourier transform in the time direction to obtain the complex matrix P(x, ω).
- Transpose the complex matrix P(x, ω), so that, for each frequency component, P(x, ω) is represented by the n–length complex vector P : (P0, P1, P2, …, Pn−1), where n is the number of traces in the stacked section.
- For each frequency ω within a specified bandwidth, design a complex prediction filter F(x) with unit prediction lag (Section F.6) represented by the m–length complex vector F : (F0, F1, F2, …, Fm−1), where m is the number of coefficients in the filter.
- Apply the complex filter F(x) to the input data component P(x, ω).
- Repeat steps (c) and (d) for all frequency components within the specified bandwidth, combine the results, and transpose to trace format.
- Inverse Fourier transform to obtain the filtered stacked section.
In practice, the prediction filter is designed using a group of stacked traces rather than the entire stacked section itself. The filter length m usually is chosen between 7-21 points, and the typical length n of the input complex vector P is set to ten times the filter length. Starting at one end of the stacked section, say from the left-hand side, the filter is applied to output one predicted sample at (m + 1)st trace for each frequency component. The design gate is then moved one trace to the right by dropping the first trace within the design window on the left and picking up the next trace on the right. The design gate is slid from one end of the section to the other one trace at a time. The prediction filter may be designed and applied in two opposite directions, and the outputs from the two applications may be averaged.