Introduction

Sparse transforms aim to represent the most important information of an image with few coefficients in the transform domain while obtaining a good quality approximation of the original image. Over the past several decades, different types of wavelet-like transforms have been proposed and successfully applied in image compression and denoising (Li and Gao, 2013; Hennenfent and Herrmann, 2006), including curvelets (Starck et al., 2002; Ma and Plonka, 2010), contourlets (Do and Vetterli, 2005), shearlets (Labate et al., 2005), directionlets (Velisavljevic et al., 2006), bandelets (Le Pennec and Mallat, 2005). The strong anisotropic selectivity of these wavelet-like transforms helps achieve excellent data compression and accurate reconstruction for seismic images.

Fomel and Liu (2010) introduced the seislet transform, which is a digital wavelet-like transform designed specifically for seismic data. Based on the lifting scheme used in digital wavelet transform (DWT) construction (Sweldens, 1995), the seislet transform follows dominant local slopes obtained by plane-wave destruction (PWD) (Fomel, 2002; Claerbout, 2000) to predict seismic events. Instead of using PWD, Liu and Fomel (2010) used offset continuation (OC) to construct updating and prediction operators. The OC-seislet transform has better performance than the PWD-seislet transform in characterizing and compressing structurally complex prestack reflection data. To reduce sensitivity to strong noise interference, Liu et al. (2015) proposed a velocity-dependent (VD) seislet transform. In this method, the normal moveout equation was introduced to serve as a connection between local slope and scanned velocities. Chen and Fomel (2018) utilized empirical mode decomposition (EMD) to obtain smoothly non-stationary data, and then applied the 1D non-stationary seislet transform to the data. This new approach was called the EMD-seislet transform and has shown excellent performance in attenuating random noise. Recently, several studies show the superiority of the seislet transform on sparse representation of seismic data over the Fourier transform, the wavelet transform and the curvelet transform (Chen and Fomel, 2018; Fomel and Liu, 2010; Gan et al., 2015).

With the ability to compress and reconstruct seismic images, the seislet transform has been successfully applied in seismic data processing such as noise attenuation (Chen et al., 2016; Chen, 2016), deblending (Gan et al., 2016; Chen et al., 2014) and data interpolation (Liu et al., 2013; Gan et al., 2016). However, the original implementation of PWD-seislet transform requires recursively computation when predicting distant traces, which increases its computation cost, especially when the dataset is very large. Additionally, the smooth slopes from PWD in the original seislet transform can fail to correctly follow reflections across discontinuities including faults and unconformities. Therefore, geologically meaningful discontinuities may not be optimally compressed.

The relative time (RT) volume, $\tau(x,y,t)$, has different meanings in different domains. In seismic image domain, the RT is the same as the relative geologic time (Stark, 2004,2003) and each RT contour corresponds to a geologic horizon. However, RT volumes of seismic gathers do not necessarily have geologic meaning, since in this case, constant RT contours align with seismic events. There are several ways to generate an RT volume. One can always pick out as many horizons or events as possible to obtain it, which is simple but laborious. The RT volume can also be generated by unwrapping seismic instantaneous phase (Wu and Zhong, 2012; Stark, 2003). Fomel (2010) used local slopes of seismic events estimated by PWD (Fomel, 2002) to generate the RT volume, which has superior computational performance. The RT volume already has been successfully applied in missing well-log data prediction (Bader et al., 2018) and seismic horizons construction (Wu and Hale, 2015).

In this paper, we propose a new implementation of the seislet transform, called RT-seislet transform, by using the RT obtained from the predictive painting (Fomel, 2010) to construct prediction operators for the seislet transform. In this way, prediction of one trace from distant traces is computed directly and accurately, which saves a lot of computation and has better performance with faults and unconformities. By applying the new formulation of the seislet transform to synthetic and real data, we demonstrate that the proposed method is efficient and able to preserve discontinuities, while achieving excellent sparse representation of seismic data.

This paper is organized as follows: we first review the seislet transform and the estimation of an RT volume. Then, we incorporate RT attribute to construct new formulation of prediction and update operators for the seislet transform. Finally, we test the proposed RT-seislet transform on both synthetic and real datasets and compare its performance with the PWD-seislet transform.