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Introduction

In seismic exploration, high-quality seismic data acquisition is a key ingredient in creating accurate subsurface interpretations, due to cost and access limitations, it is often impossible to achieve ideal surface sampling of sources and receivers where all spatial directions are well sampled. Therefore, there is often a need to regularize and interpolate the recorded seismic data at an early stage of the seismic processing workflow.

In past decades, several methods for seismic data interpolation have been proposed, and these include two major categories, which are based on the theories of wave dynamics and image analysis. According to the physical characteristics of seismic wave propagation, different types of integral continuous operators have proved their effectiveness for seismic data interpolation, such as shot continuation operators (Mazzucchelli et al., 1999) and offset continuation operators (Fomel, 2003). Interferometry is also used for interpolation of missing seismic data (Wang et al., 2009). Recently, new theories in signal processing such as compressive sensing (CS) and machine learning (ML) have shown great potential in reconstructing seismic traces. CS-based methods for data interpolation assume that seismic data obey sparsity when transformed to an appropriate domain, such as curvelet transform (Shahidi et al., 2013; Yang and Gao, 2012; Naghizadeh and Sacchi, 2010; Herrmann and Hennenfent, 2008), Radon transform (Jager et al., 2002; Shao et al., 2017), dreamlet transform (Wang et al., 2015), and seislet transform (Liu et al., 2015; Liu and Fomel, 2010). Moreover, application of machine learning in seismic exploration is a hot topic and it is expected to aid in interpolating missing seismic traces. Jia et al. (2018) used a Monte Carlo method for intelligent interpolation to reduce the cost of training sets. Wang et al. (2019) designed an eight-layer residual learning networks (ResNets) for regularly missing data reconstruction. Wang et al. (2020) explored a convolutional auto-encoder (CAE) method for interpolating irregularly sampled shot gathers by introducing transfer learning strategy. Machine learning methods depend on the characteristics of the training data, which can overcome the assumptions of linear events, sparsity, or low rank (Jia and Ma, 2017); however, the accuracy of the interpolated results is limited by the similarity of the characteristics between the training data and the processed data.

Prediction-based interpolation methods are important approaches for seismic data interpolation, and it involves both characteristics of seismic phase-shift operator and signal convolution operator. Prediction filters (PFs) or prediction-error filters (PEFs) can be implemented in the time-space or frequency-space domain. Spitz (1991) initially proposed - PFs for the interpolation of missing seismic data. Porsani (1999) improved Spitz's approach by introducing a half-step PF. Wang (2002) further extended prediction interpolation from the - domain to the - - domain. Wang et al. (2007) also designed a localized - - PF to interpolate 3D seismic data. Naghizadeh and Sacchi (2009) used exponentially weighted recursive least squares to calculate adaptive PFs in the - domain. Curry and Shan (2010) used multiples and frequency domain PEFs to interpolate missing data near offsets. Li et al. (2017) proposed a multidimensional adaptive PEF to reconstruct seismic data in the frequency domain. Liu and Chen (2018) developed a prediction interpolation by using - regularized nonstationary autoregression (RNA), which can deal with the events that have space-varying dips. Zheng et al. (2019) developed a SPF in the - domain to interpolate missing traces, which reduces high computational cost by directly solving an inverse problem in the complex domain. Meanwhile, time-space PEFs were successfully applied to reconstruct datasets where the missing data might be regularly or irregularly represented. Claerbout (1992) first proposed missing-data restoration using PEFs in the - domain. Crawley et al. (1999) described a method for data interpolation with smoothly varying PEFs, which used ``steering filters'' to control the smoothness of the filters. Curry (2003) developed a nonstationary, multi-scale PEFs to interpolate irregularly-sampled data. Liu and Fomel (2011) restored decimated and randomly missing traces based on RNA in the time domain, which uses shaping regularization to control the smoothness of adaptive PEFs. Liu et al. (2018) proposed a 3D - - multiscale multidirectional adaptive PEF to simultaneously reconstruct randomly and regularly missing data. Compared with - PF, a - PF could avoid the generation of false events in the presence of strong parallel events (Abma and Claerbout, 1995). This is because of the ability of - prediction, to control the length of the PFs in time. To reduce computational time and storage, Fomel and Claerbout (2016) proposed noniterative streaming PEFs to recover holes in 2D images.

In this paper, we proposed an SPF with varying smoothness in the time-space domain to reconstruct irregular and regular missing seismic traces; in this method, SPFs are extended from one to two spatial dimensions. The proposed method involves a two-step strategy (Claerbout, 1992; Crawley et al., 1999). In comparison with streaming PEFs (Fomel and Claerbout, 2016), we presented a similarity matrix to restrict the underdetermined least-squares problem of SPF, which enables regularization term to change with seismic data. We also designed a non-causal in space SPF to further improve the accuracy of interpolation and we compare its results with those from a causal in space filter. The proposed method shows the superiority of synchronous data reconstruction with irregular and regular missing seismic traces. Synthetic and field data tests demonstrate the effectiveness and efficiency of the proposed SPF method in reconstructing missing seismic data.

Next: Theory Up: Liu et al.: Interpolation Previous: Liu et al.: Interpolation

2022-04-12