Fourier methods of seismic data regularisation

Results using Slepian tapers

The use of Slepian tapers for seismic data regularisation requires some initial ingenuity, since previously their purpose has been in estimating the spectra of complex processes and not in reconstructing data. Figure 10 shows a windowed irregular spectrum containg two main frequencies and its equivalent spectrum after applying four $\pi$ Slepian tapers and weighting appropriately. The irregular spectum indicates leakage has occured, both around the peak and at higher frequencies, whilst the spectrum after tapering shows less contamination around the frequencies of interest and an overly cleaner spectrum. In this case the tapering has reduced the spectral leakage with little computation time. It should be noted that the ALFT can often not remove the leaked frequencies surrounding the main peaks, whilst the DPSS tapers have reduced this.

tape
Figure 10. The result of applying the four tapers in the $\pi$ DPSS sequence to a spectrum exhibiting leakage.

The choice of number of tapers and the time bandwidth product is very situation dependent. In a lot of cases a poor choice of the time bandwidth product merely serves the smear spectral peaks and enhance leakage, even giving zero frequency energies.

The taper weights are designed to reduce the bias of the power spectrum, and not the amplitude. Using these weights on the amplitude gives peak splitting, which theory tells us should happen, and if the root of the power is used then phase information of the data is lost. Ongoing efforts involve calculating amplitude and/or phase weights to reduce bias, rather than within the power. Currently in data reconstruction taper imprints are still visible, which again was expected by theory. Increasing the number of tapers alters this problem, and a method of amplitude preservation is yet to be finalised. Currently the method outlined by Park (1992) where the reconstruction is posed as an inverse problem is being attempted.

Fourier methods of seismic data regularisation