Noncausal $f$ -$x$ -$y$ regularized nonstationary prediction filtering for random noise attenuation on 3D seismic data

Introduction

Seismic noise attenuation is very important for seismic data processing and interpretation, especially for 3D seismic data. Among the methods of seismic noise attenuation, prediction filtering is one of the most effective and most commonly used methods, e.g., (Galbraith, 1984; Sacchi and Kuehl, 2001; Gulunay, 1986; Gulunay et al., 1993). Prediction filtering can be implemented in $f$ -$x$ domain or $t-x$ domain (Hornbostel, 1991; Abma and Claerbout, 1995). Abma and Claerbout (1995) compared $f$ -$x$ method and $t-x$ method and gave the advantages and disadvantages of both these methods. The proposed method in our paper belongs to the category of $f$ -$x$ domain methods. The $f$ -$x$ prediction technique was introduced for random noise attenuation on 2D poststack data by Canales (1984) and further developed by Gulunay (1986). Wang and West (1991) and Hornbostel (1991) used noncausal filters for random noise attenuation on stacked seismic data and obtain a good result. Linear prediction filtering states that the signal can be described by an autoregressive (AR) model, which means that a superposition of linear events transforms into a superposition of complex sinusoids in the $f$ -$x$ domain. Sacchi and Kuehl (2001) utilized the autoregressive-moving average (ARMA) structure of the signal to estimate a prediction error filter (PEF) and applied ARMA model to attenuate random noise. Liu et al. (2009) applied ARMA-based noncausal spatial prediction filtering to avoid the model inconsistency.

As already noted, these above mentioned $f$ -$x$ methods assume seismic section is composed of a finite number of linear events with constant dip in $t-x$ domain. To cope with the assumption continuous changes dips, short temporal and spatial analysis windows are usually used in $f$ -$x$ prediction filtering. Except using windowing strategy, several nonstationary prediction filters are proposed and used in seismic noise attenuation and interpolation. Naghizadeh and Sacchi (2009) proposed an adaptive $f$ -$x$ prediction filter, which was used to interpolate waveforms that have spatially variant dips. Fomel (2009) developed a general method of regularized nonstationary aoturegression (RNA) with shaping regularization (Fomel, 2007) for time domain inverse problems. Liu et al. (1991) propose a method for random noise attenuation in seismic data by applying noncausal regularized nonstationary autoregression (NRNA) in frequency domain, which is implemented for 2D seismic data. These nonstationary methods can control algorithm’s adaptability to changes in local dip so that they can process curved events.

If using $f$ -$x$ prediction filter to suppress random noise on 3D seismic data, one need to run the 2D algorithm slice by slice (along inline x or crossline y). To use more information to predict the effective signal in 3D data, several geophysicists extended $f$ -$x$ prediction filtering to 3D case. Chase (1992) designs and applies 2-D prediction filters in the plane defined by the inline and crossline directions for each temporal frequency slice of the 3-D data volume. Ozdemir et al. (1999) applied f-x-y projection filtering to attenuate random noise of seismic data with low poor signal to noise ratio (SNR), in which the crucial step of 2-D spectral factorization is achieved through the causal helical filter. Gulunay (2000) proposed using full-plane noncausal prediction filters to process each frequency slice of the 3-D data. Wang (2002) applied $f$ -$x$ -$y$ 3D prediction filter to implement seismic data interpolation and gave a good result. Hodgson et al. (2002) presented a novel method of noise attenuation for 3D seismic data, which applies a smoothing filter to each targeted frequency slice and allows targeted filtering of selected parts of the frequency spectrum.

In this paper, we extend $f$ -$x$ NRNA method (Liu et al., 1991) to $f$ -$x$ -$y$ case and use $f$ -$x$ -$y$ NRNA to attenuate random noise for 3D seismic data. The coefficients of 3D NRNA method are smooth along two space coordinates (x and y) in $f$ -$x$ -$y$ domain. This paper is organized as follows: First, we provide the theory for random noise on 3D seismic data, paying particular attention to establishment of $f$ -$x$ -$y$ NRNA equations with constraints and implementation of it with shaping regularization. Then we evaluate and compare the proposed method with $f$ -$x$ NRNA using synthetic and real data examples and discuss the parameter selection problem associated with our algorithm.

Noncausal $f$ -$x$ -$y$ regularized nonstationary prediction filtering for random noise attenuation on 3D seismic data