Seismic data interpolation using nonlinear shaping regularization

INTRODUCTION

Seismic data interpolation plays a fundamental role in seismic data processing, which provides the regularly sampled seismic data for the following jobs like high-resolution processing, wave-equation migration, multiple suppression, amplitude-versus-offset (AVO) or amplitude-versus-azimuth (AVAZ) analysis, and time-lapse studies. There has been a number of effective methods to recover missing seismic traces, these methods can be generally divided into three types. The first type of approach is based on prediction, which utilizes a convolutional prediction filter computed from the low-frequency parts to predict the high-frequency components (Spitz, 1991; Porsani, 1999; Wang, 2002). However, the predictive filtering method can only be applied to regularly sampled seismic data. The second type is a transformed domain method (Abma and Kabir, 2006; Chen et al., 2014b; Candès et al., 2006a), which is based on compressive sensing theory (Donoho, 2006; Candès et al., 2006b) to achieve a successful recovery using highly incomplete available data (Wang, 2003; Chen et al., 2014a; Sacchi et al., 1998). Naghizadeh and Sacchi (2007) proposed a multistep autoregressive strategy which combines the first two types of methods to reconstruct irregular seismic data. The third type is based on the wave equation. This type of method utilize the inherent constraint of the seismic data from wave equation to interpolate seismic data, thus it depends on the known velocity model, which also becomes its limitation (Fomel, 2003; Canning and Gardner, 1996). Recently, more and more researchers have developed algorithms connecting the interpolation and deblending (Chen et al., 2014c; Chen, 2014) problems for the irregular sampled simultaneous-source data (Li et al., 2013), which provide new recipes for conventional seismic interpolation problem.

Linear shaping regularization was proposed by Fomel (2007b) for regularizing under-determined geophysical inverse problems. Compared with the commonly used Tikhonov regularization (Tikhonov, 1963), shaping regularization enjoys a number of advantages, including easier control on properties of the estimated model or in some cases significantly faster convergence. In the past five years, the linear form of shaping regularization has been utilized broadly. Fomel (2007a) used it to define the local correlation, which was then utilized by Liu et al. (2011a,2009) for optimal stacking and by Liu and Fomel (2012a); Liu et al. (2011b) for local time-frequency analysis. Du et al. (2010) proposed to use shaping regularized inversion to estimate the Q factor of seismic attenuation by treating a spectral division as a non-stationary inversion problem. Liu et al. (2012) applied shaping regularization to random noise attenuation, and achieved a better result than $f-x$ predictive filter in the case of non-stationary seismic signal. Chen et al. (2012) found an application of smooth shaping in Gabor deconvolution. All these methods use a linear operator to constrain the model when iteratively solving the inverse problem.

Fomel (2007b) extended linear shaping regularization to its nonlinear form. In the nonlinear form, the shaping operator is not limited to be linear, and thus produce more convenience in implementing the iterative framework. However, the properties and applications of nonlinear form of shaping regularization have been barely mentioned since that. In this abstract, we build a bridge among the well known iterative shrinkage thresholding (IST), projection onto convex sets (POCS) algorithms and shaping regularization. We derive the two well-known inversion formulations (IST and POCS) in the basis of shaping regularization. We also propose a faster version of shaping regularization, where a linear combination operator is used. The faster version utilizes the information of both the current and previous shaping regularized model to form a new model, which demonstrates an obviously faster convergence.

Seismic data interpolation using nonlinear shaping regularization