Improving the performance of f-x prediction filtering at low signal-to-noise ratios

The conventional method of f-x filtering for random noise reduction suffers from three drawbacks. Firstly, the wavenumber response of the filter does not peak exactly at the wavenumbers of the signal components. Secondly, the amplitude of the filter response is less than one at the signal component wavenumbers, causing attenuation of the signal. Finally, sidelobes in the filter response cause noise at wavenumbers well separated from the signal components to leak into the filtered output. Singular value decomposition (SVD) of the data matrix shows that the problems may be reduced by using a transient-free formulation of the data matrix; that is, minimizing the squared errors over a finite data length rather than minimizing the expected value of the squared errors under the assumption of an infinite length of available data. Using the transient-free formulation, noise-free signal can be predicted perfectly, unlike the conventional method. The SVD analysis of the transient-free case shows that the noise-reduction performance may be improved further at all signal-to-noise ratios (SNRs). This is achieved because in the noise-free case, the correlation matrix is rank-deficient. For noisy data, an estimate of the correlation matrix is made by selecting appropriate eigenvectors to construct the filter. The use of selected eigenvectors ensures stability, thus permitting the use of much longer filters than the usual methods, with a consequent improvement in SNR gain. Conventional techniques for estimating the effective rank of the correlation matrix focus on variations in the size of the eigenvalues, selecting only the largest. It was found that these methods severely overestimate the effective rank. Furthermore, in synthetic tests it was found that some eigenvectors corresponding to noise may have eigenvalues larger than some of the signal eigenvectors. The eigenvector selection is therefore based on the observation that the phase of a noise eigenvector has a random walk appearance, whereas the phase of the signal eigenvectors varies in a smooth manner. Statistical criteria permit the selection of the signal eigenvectors in a robust way. Tests on synthetic data show that the SNR gain may typically be 10 dB for the selected eigenvector method, as opposed to 5 dB to 0 dB at different input signal-to-noise ratios for the other methods. The optimal filter lengths were about twice the optimal lengths found for the conventional method. Tests on real stacked data also show considerable improvement in performance. Care must be taken in areas of complicated structure, particularly when strongly curved events are presents, to select sufficient eigenvectors, but this may be achieved at the price of a slight loss in noise reduction.