Seislet-based morphological component analysis using scale-dependent exponential shrinkage

Introduction

A wide range of applications have been carried out by solving a series of linear inverse problems and using the fact that numerous varieties of signals can be sparsely represented with an appropriate dictionary, namely a certain kind of transform bases. Under the dictionary, the signals have fewer non-zeros of the representation coefficients. However, many complex signals are usually linear superposition of several elementary signals, and cannot be efficiently represented using only one dictionary. The concept of morphological diversity was therefore proposed by Starck et al. (2004,2005) to combine several dictionaries for sparse representations of signals and images. Then, the signal is considered as a superposition of several morphological components. One has to choose a dictionary whose atoms match the shape of the geometrical structures to sparsify, while leading to a non-sparse (or at least not as sparse) representation of the other signal content. That is the essence of so-called morphological component analysis (MCA) (Starck et al., 2004,2007; Woiselle et al., 2011).

Seislet transform and seislet frame are useful tools for seismic data compression and sparse representation (Fomel and Liu, 2010). Seislets are constructed by applying the wavelet lifting scheme (Sweldens, 1998) along the spatial direction, taking advantage of the prediction and update steps to characterize local structure of the seismic events. In the seislet transform, the locally dominant event slopes are found by plane-wave destruction (PWD), which is implemented using finite difference stencils to characterize seismic images by a superposition of local plane waves (Claerbout, 1992). By increasing the accuracy and dip bandwidth of PWD, Fomel (2002) demonstrated its competitive performance compared with prediction error filter (PEF) in the applications to fault detection, data interpolation, and noise attenuation. PWD keeps the number of adjustable parameters to a minimum, endows the estimated quantity with a clear physical meaning of the local plane-wave slope, and gets rid of the requirement of local windows in PEF. Recently, Chen et al. (2013a,b) accelerated the computation of PWD using an analytical estimator and improved its accuracy.

In this paper, we propose seislet-based MCA for seismic data processing. We reformulate MCA algorithm in the shaping-regularization framework (Fomel, 2007,2008). Successful seislet-based MCA depends on reliable slope estimation of seismic events, which can be done by plane-wave destruction (PWD) filtering. Due to the special importance of an effective shrinkage or thresholding function in sparsity-promoting shaping optimization, we propose a scale-dependent exponential shrinkage operator, which can flexibly approximate many well-known existing thresholding functions. Synthetic and field data examples demonstrate the potential of seislet-based MCA in the application to trace interpolation and multiple removal.

Seislet-based morphological component analysis using scale-dependent exponential shrinkage