Adaptive prediction filtering in $t$ -$x$ -$y$ domain for random noise attenuation using regularized nonstationary autoregression

Introduction

Attenuation of noise is a persistent problem in seismic exploration. Random noise can come from various sources. Although stacking can at least partly suppress random noise in prestack data, residual random noise after stacking will decrease the accuracy of the final data interpretation. In recent years, several authors have developed effective methods of eliminating random noise. For example, Ristau and Moon (2001) compared several adaptive filters, which they applied in an attempt to reduce random noise in geophysical data. Karsli et al. (2006) applied complex-trace analysis to seismic data for random-noise suppression, recommending it for low-fold seismic data. Some transform methods were also used to deal with seismic random noise, e.g., the discrete cosine transform (Lu and Liu, 2007), the curvelet transform (Neelamani et al., 2008), and the seislet transform (Fomel and Liu, 2010).

Seismic reflections are recorded according to special geometry and always appear lateral continuity, which is used to distinguish events from the background noise. If events of interest are linear (lines in 2D data and planes in 3D data), one can predict linear events by using prediction techniques in the frequency-space domain or the time-space domain (Abma and Claerbout, 1995). The $f$ -$x$ prediction technique was introduced by Canales (1984) and further developed by Gulunay (1986), and is a standard industry method known as ``FXDECON''. Sacchi and Kuehl (2001) utilized the autoregressive-moving average (ARMA) structure of the signal to estimate $f$ -$x$ prediction-error filter (PEF) and the noise sequence. Liu et al. (2012) developed $f$ -$x$ regularized nonstationary autoregression (RNA) to attenuate random noise. Liu and Chen (2013) further extended 2D $f$ -$x$ RNA to 3D noncausal regularized nonstationary autoregression (NRNA) for random noise elimination.

The prediction process can be also achieved in $t$ -$x$ domain (Claerbout, 1992). Abma and Claerbout (1995) discussed $f$ -$x$ and $t$ -$x$ approaches to predict linear events from random noise. $t$ -$x$ prediction always passes less random noise than $f$ -$x$ prediction method because the $f$ -$x$ prediction technique, while dividing the prediction problem into separate problems for each frequency, produces a filter as long as the data series in time. However, seismic data are naturally nonstationary, and a standard $t$ -$x$ prediction filter can only be used to process stationary data (Claerbout, 1992). Patching is a common method to handle nonstationarity (Claerbout, 2010), although it occasionally fails in the assumption of piecewise constant dips. Crawley et al. (1999) proposed smoothly nonstationary prediction-error filters (PEFs) with ``micropatches'' and radial smoothing, which typically produces better results than the rectangular patching approach. Fomel (2002) developed a plane-wave destruction (PWD) filter (Claerbout, 1992) as an alternative to $t$ -$x$ PEF and applied a PWD operator to represent nonstationary. However, PWD method depends on the assumption of a small number of smoothly variable seismic dips. When background random noise is strong, the dip estimation is a difficult problem. Sacchi and Naghizadeh (2009) proposed an algorithm to compute time and space variant prediction filters for noise attenuation, which is implemented by a recursive scheme where the filter is continuously adapted to predict the signal.

In this paper, we propose a $t$ -$x$ adaptive prediction filter (APF) to preserve nonstationary signal and attenuate random noise. The key idea is the use of shaping regularization (Fomel, 2007) to constrain the time and space smoothness of filter coefficients in the corresponding ill-posed autoregression problem. The structure of space-causal and space-noncausal APF is also discussed. We test nonstationary characteristics of APF by using a 2D synthetic curved model, a 2D synthetic poststack model, and a 3D prestack French model. Results of applying the proposed method to the field example demonstrate that regularized APF can be effective in eliminating random noise.

Adaptive prediction filtering in $t$ -$x$ -$y$ domain for random noise attenuation using regularized nonstationary autoregression