Noniterative f-x-y streaming prediction filtering for random noise attenuation on seismic data

Introduction

In seismic exploration, random noise is unavoidable because it is composed of environmental noise, the interference of wind motion, and the noise from geophones Yilmaz (2001). Meanwhile, complex subsurface media typically cause the energy loss experienced by seismic signals, which show low amplitude in deep exploration conditions. These factors result in low-quality data and make signal-to-noise ratio (SNR) gradually low. In obtaining high-quality seismic images and improving the SNR, one key problem is the nonstationary characteristics of the signal. Seismic data are time-varying in nature, and the nonstationary properties of seismic data display that energy, track, time-frequency spectra of seismic events, and statistical characteristics of random noise change with time and space. The denoising methods that consider such nonstationary features can better preserve the valid signals. The other problem with denoising is the increasing computational costs, although broadband, wide-azimuth, and high-density data acquisition can lead to high-resolution and high-fidelity images. Many authors have proposed methods for random noise attenuation based on different theories. The mean filter Bonar and Sachhi (2012) and median filter Liu et al. (2009); Wu et al. (2018) are effective denoising methods for images, but they may somewhat smear seismic signals when complicated structures and low SNR are encountered. Mathematical transforms such as wavelet transforms Langston and Mousavi (2019); Berkner and Wells (1998); Yu et al. (2007) and seislet transforms Fomel and Liu (2010); Liu et al. (2015); Liu and Fomel (2010) can characterize the nonstationary properties of seismic signals and provide reasonable signal and noise separation based on their compression ability. Recently, deep learning or machine learning techniques Kimiaefar et al. (2016); Yu et al. (2019); Zhu et al. (2019); Djarfour et al. (2014) have also been proposed to suppress random noise; however, the initialization of neural networks requires a large number of samples and high computational cost. For supervised and semi-supervised learning, the preparation of the training set may need the help of traditional methods to generate denoised results as a reference.

Prediction filters (PFs) have proved effective for random noise attenuation, and they can be implemented in the time-space or frequency-space domain. When seismic events have varying slope, the configuration of the filter size influences the filtering results, especially the filter size of $t$ -$x$ PFs along the time axis. There are few impacts on $f$ -$x$ PFs because they only estimate data along the spatial directions. Besides, seismic events with different dominant frequencies are overlapped in the $t$ -$x$ domain, and they can be naturally separated in the $f$ -$x$ domain. Abma and Claerbout Abma and Claerbout (1995) discussed the differences in PFs in the $f$ -$x$ and $t$ -$x$ domains. The $f$ -$x$ prediction filter for denoising was first introduced by Canales Canales (1984), and further developed by Gülünay Gülünay (1986) to a standard industry method known as “FXDECON”, which is equivalent to a $t$ -$x$ domain prediction filter, selecting the entire trace along the time direction. Liu et al. Liu et al. (2012) developed the $f$ -$x$ adaptive prediction filters to suppress random noise by using regularized nonstationary autoregression (RNA); the regularization term was used to limit the global smoothness of the filter coefficients. The method was further extended from a two-dimensional (2D) to three-dimensional (3D) case for random noise attenuation Liu and Chen (2013). The $f$ -$x$ -$y$ RNA provides preferable adaptive features because it uses an iterative algorithm to calculate the frequency-space-varying filter coefficients, which leads to a large storage and high computational time, especially in large-scale data processing.

Local similarity constraints have been proposed to directly calculate the adaptive prediction filter without iterations, which can save the computational resources. Starting with the prediction equation for a certain data point, Sacchi and Naghizadeh Sacchi and Naghizadeh (2009) transformed the ill-posed problem of adaptive prediction filter into a local smoothing problem, and introduced a quadratic regularization term to stabilize the solution of the local prediction filter. Fomel and Claerbout Fomel and Claerbout (2016) proposed the concept of streaming prediction error filter (SPEF) to update the filter as each new data value arrives in the time-space domain. This method combines the prediction equation with locally similar constraints to solve the overdetermined linear system. Arising from different starting points, these two methods share the same least-squares solution and significantly reduce the computational cost by avoiding the iterative algorithm. Liu and Li Liu and Li (2018) further proposed a streaming orthogonal prediction filter (SOPF) in the $t$ -$x$ domain, which applies signal-and-noise orthogonalization based on the streaming prediction theory and provides a fast solution for the adaptive prediction filter to suppress random noise. Guo et al. Guo et al. (2020) attempted to eliminate seismic random noise by using the $f$ -$x$ SPF only with 1D spatial constraint.

In this study, we derived the theory of the new $f$ -$x$ -$y$ streaming prediction filter based on the local smoothness constraints in high dimensions. The multi-dimensional constraints make the filter involve the property of local similarity not only along the spatial directions (space $x$ and space $y$ ), but also the frequency direction. It is not a simple 3D extension with more spatial axis from 2D SPF Guo et al. (2020), we took advantage of 3D data with space $y$ constraint and suppressed oscillation with frequency constraint, meanwhile, a special filter-updating path was developed to help the SPF solve the random noise attenuation problem in higher dimensions. We compared the feasibility of the 3D $f$ -$x$ -$y$ SPF in attenuating random noise with the 2D $f$ -$x$ SPF and the 3D $f$ -$x$ -$y$ RNA on two synthetic models. The field data example confirms that the 3D $f$ -$x$ -$y$ SPF with the matching processing path has a reasonable denoising ability and a low computational cost in practice.

Noniterative f-x-y streaming prediction filtering for random noise attenuation on seismic data