Bayesian wavefield separation by transform-domain sparsity promotion

Introduction

Successful separation of coherent signal components constitutes an important step in the seismic processing flow. Coherent noise separation, such as primary-multiple separation as part of Surface-Related Multiple Elimination (SRME) (Verschuur et al., 1992; Weglein et al., 1997; Fokkema and van den Berg, 1993; Berkhout and Verschuur, 1997) or interferometric ground-roll removal (Dong and Schuster, 2006), typically involves two stages: a noise-prediction stage and a primary-reflection noise separation stage. During the second stage, measures need to be taken to compensate for moderate kinematic and dynamic errors in the predictions of the noise components (see e.g.  Verschuur et al., 1992; Ikelle et al., 1997). For instance, 3-D complexity in the Earth produces these types of errors during SRME (Verschuur, 2006; Verschuur et al., 1992; Ikelle et al., 1997; Ross et al., 1999; Dragoset and Jericevic, 1998), while multiple (van Borselen et al., 2004; Moore and Dragoset, 2004; van Dedem and Berkhout, 2005; Lin et al., 2004) and interferometric ground-roll predictions (Dong and Schuster, 2006) become unreliable for incomplete data acquisitions. As a result, separation methods based on conventional (windowed) convolutional matched filtering may not compensate adequately for these errors, rendering noise removal ineffective.

Our main concern in this letter is the construction of an iterative separation algorithm that is stable with respect to moderate errors in the noise predictions. The traditional ``separation by subtraction after matching'' approach yields predictions of signal components with residual amplitude, timing, dip, phase, and wavelet errors. The separation obtained via such methods can be improved via an additional curvelet-domain nonlinear matching procedure that can handle significant amplitude errors provided they vary smoothly as a function of the location and dips of the noise predictions (Herrmann et al., 2007b). Assuming that the resulting predictions (possibly after this additional matched filtering stage) have moderate residual errors, the algorithm proposed in this paper complements such $\ell_2$ -matched-filtering techniques where noise in the total data and errors in the noise predictions lead to residual noise energy, high-frequency clutter, and deterioration of primary energy (Herrmann et al., 2007a; Abma et al., 1005; Chen et al., 2004). In particular, our algorithm takes the predictions obtained using the traditional methods as input, and produces improved estimates of the primaries and multiples. (For simplicity, we will refer to such predictions as the ``SRME-predictions'' although our algorithm is generic and the predictions could be obtained using any method appropriate for the type of wavefield-separation problem.) This is achieved by exploiting the wavefront detection and hence separation capability of curvelets (Herrmann et al., 2007a,2008) while offering additional control over the energy mismatch between the SRME-predicted noise components (the input) and the estimated signal components (the output). Previous separation algorithms based on curvelet-domain sparsity lack this type of control (Herrmann et al., 2007a,2008). This letter builds explicitly on more general results on transform-based signal separation that will be reported elsewhere.

Bayesian wavefield separation by transform-domain sparsity promotion