Random noise attenuation using $f$-$x$ regularized nonstationary autoregression

Introduction

Random noise attenuation in seismic data can be implemented in the frequency-space ($f$-$x$) and time-space ($t$-$x$) domain using prediction filters (Abma and Claerbout, 1995). Linear prediction filtering assumes that the signal can be described by an autoregressive (AR) model. When the data are contaminated by random noise, the signal is considered to be predicted by the AR filter and the noise is the residual (Bekara and van der Baan, 2009). A number of approaches in $f$-$x$ domain have been proposed and been used for attenuating random noise. The $f$-$x$ prediction technique was introduced by Canales (1984) and further developed by Gulunay (1986). The $f$-$x$ domain prediction technique is also referred as $f$-$x$ deconvolution by Gulunay (1986). Sacchi and Kuehl (2001) utilized the autoregressive-moving average (ARMA) structure of the signal to estimate a prediction error filter (PEF) and the noise sequence is estimated by self-deconvolving the PEF from the filtered data. Hodgson et al. (2002) presented a novel method of noise attenuation for 3D seismic data, which applies a smoothing filter (e.g. 2D median filter) to each targeted frequency slice and allows targeted filtering of selected parts of the frequency spectrum. The conventional $f$-$x$ domain prediction uses windowing strategies to avoid that the seismic events are not linear. The data are assumed to be piecewise linear and stationary in an analysis temporal and spatial window. To overcome the potentially low performance of $f$-$x$ deconvolution that arises with processing structural complex data, Bekara and van der Baan (2009) proposed a new filtering technique for random and coherent noise attenuation in seismic data by applying empirical mode decomposition (EMD) (Huang et al., 1998) on constant-frequency slices in the $f$-$x$ domain and removing the first intrinsic mode function. In addition, in the research field of seismic data interpolation, Naghizadeh and Sacchi (2009) proposed an adaptive $f$-$x$ prediction filter, which was used to interpolate waveforms that have spatially variant dips. The $f$-$x$ domain prediction technique can be implemented in the frequency slice and also in pyramid domain (Sun and Ronen, 1996). The implemented in pyramid domain makes the operators more efficient because one only needs to estimate one prediction filter from many different frequencies (Sun and Ronen, 1996; Guitton and Claerbout, 2010; Hung et al., 2004).

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 and concluded that $f$-$x$ prediction is equivalent to $t$-$x$ prediction with a long time length. Crawley et al. (1999) proposed smooth nonstationary PEFs with micropatches and radial smoothing in the application of seismic interpolation, which typically produces better results than the rectangular patching approach. Izquierdo et al. (2006) proposed a technique for structural noise reduction in ultrasonic nondestructive examination using time-varying prediction filter. 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.

Fomel (2009) developed a general method of nonstationary regression with shaping regularization (Fomel, 2007). Shaping regularization has an advantage of a fast iterative convergence. Regularized nonstationary regression (RNA) has been used in multiple subtraction Fomel (2009), time-frequency analysis (Liu et al., 2011b), and nonstationary polynomial fitting (Liu et al., 2011a). Liu and Fomel (2010) introduced an adaptive PEFs using RNA in $t$-$x$ domain which has been used for trace interpolation.

In this paper, we investigate the $f$-$x$ domain prediction technique and propose $f$-$x$ domain RNA to attenuate random noise in seismic data. Firstly, we review the theory of $f$-$x$ stationary autoregression. Then, we describe the $f$-$x$ RNA and extend to complex number domain. Next we provide the methodology of random noise attenuation using $f$-$x$ RNA. Finally, we use synthetic and real data examples to evaluate and compare the proposed method with other noise attenuation techniques, such as $f$-$x$ domain and $t$-$x$ domain prediction techniques.

Random noise attenuation using $f$-$x$ regularized nonstationary autoregression