Introduction

Signals recorded on seismic records are not stationary. The frequency content of the seismic waves change as they propagate through the subsurface because of the anelastic attenuation and scattering. At the same time recorded data is nonstationary in space because different events have different slopes that change with time as well. That is why it is natural to try to find filters that adapt to the local characteristics of the signals.

Prediction error filters arise from the theory of linear prediction, which deals with the problem of estimating the future samples of the signal using a linear combination of its previous samples weighted by sought-for filter coefficients. The error of this prediction turns out to be white (not correlated), while the filter has the inverse spectrum of the signal (that allows estimation of the original signal). Consequently, one of the most common applications of these filters is deconvolution - the process that tries to widen the spectrum by removing the convolutional effects of the bandlimited signal. Having the information of the original signals, these filters can also be used for multidimensional interpolation.

There are different ways of finding a PEF, all based on minimizing the norm of the prediction error. Probably the very first successful way of solving this problem was based on an efficient way of solving normal equations (finding the inverse of a Toeplitz matrix) using the Levinson-Durbin algorithm (Claerbout, 1985). This problem can also be approached using optimization theory tools such as methods of steepest descent, conjugate gradient, etc (Claerbout, 2014). While being very efficient and stable, the gradient-based methods, however, require significant effort to be implemented in a nonstationary environment. On the other hand, it turns out that the methods based on the original Levinson-Durbin recursion are very fast, efficient and easily adjusted to account for nonstationarity.

One of these filters considered herein is called the lattice predictor filter and arises from the Levinson-Durbin algorithm. It is well studied in the electrical engineering society, it can be easily extended to be adaptive and it is known to be the most efficient structure to give orthogonal (not correlated) forward and backward prediction errors.

Another popular adaptive algorithm is called recursive least squares (RLS) that essentially is a way of adaptively finding the inverse correlation matrix (coming from the normal equations) as the data streams through the filter.


2018-06-10