EMD-seislet transform

Introduction

Sparse approximation aims at extracting the most important information of the given data by a linear combination of pre-specified atom signals with sparse linear coefficients. Sparse approximation theory has been a rapidly evolving field in digital image analysis, since many state-of-the-art signal and image processing tasks have been successfully handled with the concept of sparsity, including image inpainting and restoration (Elad et al., 2005; Cai et al., 2013; Mairal et al., 2009,2008), image denoising (Cai et al., 2013; Protter and Elad, 2009), data compression (Bryt, 2008), blind source separation (Zibulevsky and Pearlmutter, 2001), etc.

Over the past several decades, different types of sparse transforms have been explored for seismic data processing applications, and promising results have been reported. LePennec and Mallat (1992) applied a wavepacket transform to seismic data compression. Zhang and Ulrych (2003) developed a type of wavelet frame that takes into account the characteristics of seismic data both in time and space for denoising applications. Ioup and Ioup (1998) applied wavelet transform to both random noise removal and data compression with soft thresholding in the wavelet domain. Du and Lines (2000) applied multi-resolution property of wavelet transform to attenuate tube waves. Jafarpour et al. (2009) used a discrete cosine transform (DCT) to obtain sparse representations of fields with distinct geologic features and improve the solutions to traditional geophysical estimation problems. A number of researchers have also reported successful applications of the curvelet transform (Candès et al., 2006) in different seismic data processing tasks thanks to the multi-scale and sparse property of the curvelet domain (Hennenfent and Herrmann, 2006; Wang et al., 2008).

Fomel and Liu (2010) introduced a data-adaptive sparsity promoting transform, called the seislet transform. Following the lifting scheme used in the construction of second-generation digital wavelets (Sweldens, 1995), the seislet transform utilizes the spatial predictability property of seismic data to construct the predictive operator. Fomel and Liu (2010) used plane-wave construction (PWD) to aid the prediction process. Instead of using PWD, Liu and Fomel (2010) used differential offset continuation (OC) to construct seislet transform for pre-stack reflection data. OC-seislet transform can obtain better sparsity for conflicting-dip pre-stack seismic data in that OC seislet uses offset continuation (Fomel, 2003) instead of local slopes to connect different common offset gathers. In order to relieve the dependence of seislet transform on local slope estimation, Liu and Liu (2013) and Liu et al. (2015) proposed a velocity-dependent (VD) seislet transform based on conventional velocity analysis. The seislet transform has also found a recent application in simultaneous-source separation (Chen et al., 2014a). Instead of using a single slope or frequency map to sparsify the seismic data, Fomel and Liu (2010) proposed to apply several seislet transforms with smoothly variable slopes or frequencies, which are referred to as seislet frames.

Empirical mode decomposition (EMD) is a recently popular signal processing method (Huang et al., 1998), which was proposed to prepare a stable input for the Hilbert transform. The essence of EMD is to stabilize a highly non-stationary signal by decomposing it into smoothly variable frequency components, which are called intrinsic mode functions (IMF). EMD has found many successful applications in seismic data processing, such as time-frequency analysis (Han and van der Baan, 2013) and noise attenuation (Bekara and van der Baan, 2009; Chen et al., 2015; Chen and Ma, 2014; Chen et al., 2014b).

In this paper, we propose to use EMD to create sparse transforms for representing seismic data. We first transform the input seismic data from $t-x$ domain to $f-x$ domain, then apply EMD to each frequency slice to decompose input data into smoothly variable frequency components. Next, 1D non-stationary seislet transform is applied independently to each component. We evaluate the sparsity of the new representation (EMD-seislet transform) and apply it to noise attenuation of a seismic image.

EMD-seislet transform