Adaptive prediction filtering in - - domain for random noise attenuation using regularized nonstationary autoregression

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## 3D space-noncausal adaptive prediction filtering

Equation 2 and 4 show space-forward prediction equations, which are similar as the causal prediction filtering equations in - domain (Gulunay, 2000). Furthermore, - prediction filter also includes time-noncausal coefficients. In the case of both space-forward and space-backward prediction equations (space-noncausal prediction filter), equation 4 can be written (Naghizadeh and Sacchi, 2009; Liu et al., 2012; Liu and Chen, 2013; Spitz, 1991) with the results averaged: 2pt
 (5)

Equation 5 shows that one sample in - domain can be predicted by the samples in adjacent traces with weight coefficients , which is time- and space-varying. The equation assumes that the seismic data only consist of plane waves and random noise that corresponds to a least-squares error.

Figure 1a shows a 2D space-causal APF structure, which is time-noncausal filter. White grids stand for prediction samples and the dark-grey grid is the output (or target) position, while light-grey grids are unused samples. The filter size of the space-causal APF is . Meanwhile, space-noncausal APF (Figure 1b) has a symmetric structure along time and space axes. The filter size of the space-noncausal APF is . The 3D - - APF also has space-causal or space-noncausal structure, Figure 2 shows the noncausal one. In a 3D seismic datacube, the plane events can be predicted along two different spatial directions. A 2D - APF will have difficulty preserving accurate plane waves because it only uses the information in or direction, however, a 3D - - APF provides a more natural structure. - - adaptive prediction filtering for random noise attenuation follows two steps:

1. Estimating 3D space-noncausal APF coefficients by solving the regularized least-squares problem (equation 4 or 5 in 2D): 2pt

 (6)

2. Calculating noise-free signal according to 2pt

 (7)

causal2d,noncausal2d
Figure 1.
Schematic illustration of a 2D - APF. A space-causal filter (a) and a space-noncausal filter (b).

noncausal3d
Figure 2.
Schematic illustration of a 3D - - space-noncausal APF.

 Adaptive prediction filtering in - - domain for random noise attenuation using regularized nonstationary autoregression

Next: Synthetic data test Up: Theory Previous: - adaptive prediction filtering

2014-12-07