Seismic data interpolation using streaming prediction filter in the frequency domain

Introduction

With the limitation of field geometries, uniform data acquisition is rarely achieved in practice; instead, it always displays irregular or undersampled data distribution in spatial directions. However, many subsequent processing steps, such as multiple elimination and migration, are based on the prerequisite of regular data distribution. Seismic data interpolation has become a key technique in seismic data processing workflows. A failed interpolation method may create artifacts, which affect the accuracy of seismic imaging. Recently, high-density and wide-azimuth seismic acquisition can achieve large-scale field data; however, increasing computational costs and nonstationary signal recovery have posed challenges to data interpolation. Many interpolation methods have been proposed to reconstruct missing data, which are based on signal processing or seismic kinematic/dynamic principles. Low-rank methods (Gao et al., 2017,2013; Trickett et al., 2010; Chen et al., 2016) assume that seismic data are low rank in the mapping domain, and rank-reduction operators are used to recover the missing data. Plane wave decomposition (Hellman and Boyer, 2016; Fomel, 2002) reconstructs missing data by using local slope information. Compressive sensing framework with different sparse domains, e.g., Fourier transform (Wang et al., 2010a; Abma and Kabir, 2006) and seislet transform (Fomel and Liu, 2010; Gao et al., 2015; Liu and Fomel, 2010), iteratively recover the missing traces. Machine learning has become a popular research direction, and is also used for data interpolation (Jia and Ma, 2017; Kaur et al., 2019; Oliveira et al., 2018; Zhang et al., 2020; Mandelli et al., 2019). However, these methods always encounter high computational cost.

Prediction filter (PF) provide an important approach for seismic data interpolation. Spitz (1991) proved that high-frequency components can be estimated with the PF calculated from low-frequency components, and an $f$ -$x$ interpolation method beyond aliasing was proposed. Sacchi and Ulrych (1997) used an autoregression moving average (ARMA) model to calculate the PF and interpolated near-offset missing gaps in the $f$ -$x$ domain. Porsani (1999) proposed a half-step PF to efficiently interpolate missing data. Wang (2002) extended different kinds of PFs in high dimensions to implement the $f$ -$x$ interpolation algorithm. Abma and Kabir (2005) made comparison of several PF interpolation methods. Naghizadeh and Sacchi (2008) used an exponentially weighted recursive least squares (EWRLS) for the adaptive PF to interpolate data in the $f$ -$x$ domain. Wang et al. (2010b) created the virtual traces from marine seismic data and utilized the matching filter or nonstationary prediction error filter (PEF) to fill gaps. Liu and Fomel (2011) proposed an approach to interpolate aliased data based on adaptive PEF and regularized nonstationary autoregression (RNA) in the $t$ -$x$ -$y$ domain. Li et al. (2017) proposed multidimensional adaptive PEF to reconstruct seismic data in the frequency domain. Liu and Chen (2018) introduced an efficient method based on the $f$ -$x$ RNA for regular and irregular missing data interpolation in the $f$ -$x$ domain. Liu et al. (2019) designed the multiscale and multidirectional PEF to improve the accuracy of filter coefficients while reconstructing seismic data.

It is difficult to balance the computational cost and the interpolation accuracy while dealing with large-scale data in most iterative methods. However, the framework of the streaming computation (Fomel and Claerbout, 2016; Sacchi and Naghizadeh, 2009) can directly solve the nonstationary autoregression problem. Liu and Li (2018) proposed an orthogonal $t$ -$x$ streaming PF (SOPF) for random noise attenuation, and Guo et al. (2020) provided an initial idea for the $f$ -$x$ SPF with a 1D spatial constraint and tried to efficiently eliminate seismic random noise. Therefore, we further improved the noniterative framework of the streaming computation in the frequency domain and used the SPF with new frequency-domain constraints for seismic data interpolation. The new adaptive PF can reduce the computational cost in the interpolation problem while capturing details of nonstationary seismic data. In this study, we first discussed the two-step interpolation strategy for PF. With the two-step strategy, we derived the theory of the new $f$ -$x$ SPF and $f$ -$x$ -$y$ SPF based on the streaming framework. The relevant interpolation algorithm and filter design were developed to help the SPF solve the problem of the nonstationary data reconstruction with a feasible computational cost. Numerical examples demonstrate the effectiveness and efficiency of the proposed methods.

Seismic data interpolation using streaming prediction filter in the frequency domain