Adaptive multiple subtraction using regularized nonstationary regression

Introduction

Many natural phenomena, including geological events and geophysical data, are fundamentally nonstationary. They may exhibit stationarity on the short scale but eventually change their behavior in space and time. Nonstationary adaptive filtering is a well-developed field in signal processing (Haykin, 2001). In seismic signal processing, nonstationary filters were analyzed by Margrave (1998) and applied to many important problems, including multiple suppression (Rickett et al., 2001), data interpolation (Curry, 2003; Crawley et al., 1999), migration deconvolution (Valenciano et al., 2006; Guitton, 2004).

In this paper, I present a general approach to designing nonstationary operators, including the case of nonstationary matching and prediction-error filters. The key idea is the application of shaping regularization (Fomel, 2007b) for constraining the continuity and smoothness of the filter coefficients. Regularization makes the estimation problem well-posed and leads to fast numerical algorithms. Advantages of shaping regularization in comparison with the classic Tikhonov regularization include an easier control of the regularization parameters and a faster iterative convergence resulting from the better conditioning of the inverted matrix.

Adaptive subtraction is a method for matching and removing coherent noise, such as multiple reflections, after their prediction by a data-driven technique (Verschuur et al., 1992). Adaptive subtraction involves a matching filter to compensate for the amplitude, phase, and frequency distortions in the predicted noise model. Different techniques for matching filtering and adaptive subtraction have been developed and discussed by a number of authors (Verschuur et al., 1992; Lu and Mao, 2005; Monk, 1993; Abma et al., 2005; Denisov et al., 2006; Guitton and Verschuur, 2004; van Borselen et al., 2003; Spitz, 1999; Wang, 2003). The regularized non-stationary regression technique, proposed in this paper, allows the matching filter to become smoothly nonstationary without the need to break the input data into local windows.

The paper is organized as follows. I start with an overview of stationary and nonstationary regression theory and introduce a method of regularized nonstationary regression. Next, I demonstrate this method using toy examples of line fitting and nonstationary deconvolution. Finally, I apply it to the adaptive multiple suppression problem and test its performance using a number of benchmark synthetic data examples.

Adaptive multiple subtraction using regularized nonstationary regression