Multidimensional filters are used to characterize and separate seismic signal and noise. This separation may be achieved with either simple filtering or by an inversion process that involves solving a system of regressions. The system of regressions describes the expected properties of the noise and signal by using filters and weights.
Signal and noise separation by filtering may be done by either f-x prediction or t-x prediction. Both these techniques are prediction-error filters that define the noise as the prediction error. The f-x prediction is shown to be equivalent to a t-x prediction with a very long filter length in time. While filtering is simple, it can produce spurious events and attenuate the amplitude of the signal. A technique that separates signal and noise with an inversion can eliminate these weaknesses of simple filtering.
An important issue in signal and noise separation is the removal of high-amplitude noise before filtering or inversion, since high-amplitude noise corrupts the estimation of prediction-error filters and impairs least-squares inversion techniques. To detect automatically these high-amplitude noises, trace-to-trace predictions are examined for large residuals that correspond to bad samples. After the bad samples are eliminated, the inversions are arranged to predict the missing data simultaneously with the signal and noise separation. The missing data may be data that has been removed because of the high-amplitude noise removal, or it may be data missing because it was not recorded.
Two general forms of inversion are used. One form requires only a filter that describes the signal. This form of inversion is useful in removing noise that is unpredictable from trace to trace. The other form requires filters that describe both the signal and the noise. This second form allows high-amplitude and coherent noise to be well separated from the signal, but it often requires a more complicated weighting function to properly distribute the data between noise and signal.
These techniques are applied to synthetic and real seismic data to demonstrate the weaknesses and strengths of the various approaches.