Spectral decomposition with f − x − y preconditioning

Spectral decomposition, or local time‐frequency analysis, tries to enhance the amount of information one can obtain from a seismic volume by finding the frequency content of the seismic data at each time sample. However, if a small amount of noise is present within the seismic amplitude volume, it has the potential to become more prominent in the spectrally decomposed data especially if high‐resolution or sparsity promoting methods are utilized. To combat this problem post‐processing noise removal has commonly been employed, but these techniques can potentially degrade the resolution of small‐scale geological structures in their attempt to remove this noise. Rather than de‐noising the spectrally decomposed data after they are generated, we propose to incorporate the ideas of f − x − y deconvolution within the spectral decomposition process to create an algorithm that has the ability to de‐noise the time‐frequency representation of the data as they are being generated. By incorporating the spatial prediction error filters that are utilized for f − x − y deconvolution with the spectral decomposition problem, a spatially smooth time‐frequency representation that maintains its sparsity, or high‐resolution characteristics, can be obtained. This spatially smooth high‐resolution time‐frequency representation is less likely to exhibit the random noise that was present in the more conventionally obtained time‐frequency representation. Tests on a real data set demonstrate that by de‐noising while the time‐frequency representation is being constructed, small‐scale geological structures are more likely to maintain their resolution since the de‐noised time‐frequency representation is specifically built to reconstruct the data.