Adaptive curvelet-domain primary-multiple separation

Introduction

Surface-Related Multiple Elimination (SRME) (Verschuur et al., 1992; Weglein et al., 1997; Fokkema and van den Berg, 1993; Berkhout and Verschuur, 1997) involves two stages, namely multiple prediction and primary-multiple separation. During the second stage, measures are taken to compensate for imperfections in the multiple predictions. For SRME, predicted multiples often include source signatures and directivity patterns that differ from those present in the data (see e.g.  Verschuur et al., 1992; Ikelle et al., 1997). Moreover, 2D SRME produces errors in the predicted multiples due to 3D complexity of the Earth (Verschuur, 2006; Ross et al., 1999; Dragoset and Jericevic, 1998), while recently-developed full 3D-SRME algorithms may suffer from imperfections related to incomplete acquisitions (see e.g. van Borselen et al., 2004; Moore and Dragoset, 2004; van Dedem and Verschuur, 2005; Lin et al., 2004), including erroneous reconstructions of missing near offsets (Dragoset and Jericevic, 1998). For field data, these factors preclude iterative SRME, resulting in amplitude errors that vary for different multiple orders (see e.g.  Pfaffenholz et al., 2002; Verschuur and Berkhout, 1997).

In practice, the second separation stage appears particularly challenging since adaptive $\ell_2$ -matched-filtering techniques are known to lead to residual multiple energy, high-frequency clutter and deterioration of the primaries (Herrmann et al., 2007b; Abma et al., 2005; Chen et al., 2004). By employing the curvelet transform's ability (Candes et al., 2006; Hennenfent and Herrmann, 2006) to detect wavefronts with conflicting dips (e.g. caustics), Herrmann et al. (2007b,d) were able to derive a non-adaptive (independent of the total data) separation scheme that uses the original data and SRME-predicted multiples as input and produces an estimate for the primaries. This threshold-based method proved robust with respect to moderate errors (sign, phase and timing) in the predicted multiples, and derived its success from the sparsifying property of curvelets for data with wavefronts. Despite recent advances in thresholding, by a Bayesian formulation (Saab et al., 2007; Wang et al., 2007), and mitigation of the effects of missing data (Herrmann et al., 2007a; Hennenfent and Herrmann, 2007), curvelet-domain separation deteriorates when the predicted multiples have significant amplitude errors. Thresholding in these cases may give rise to inadvertent removal of primary energy or to remnant multiple energy.

Adaptive curvelet-domain primary-multiple separation