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

Data processing path in 3D case

The processing path for the 2D $f$ -$x$ SPF is shown in Fig. 1b, which initializes the filter at the beginning of each line and updates the filter until the end of the line along the spatial direction. We defined a new processing path for the 3D case; the 3D $f$ -$x$ -$y$ SPF will process data from the beginning of a line to the end, and then from the end of the next line to the beginning in the spatial directions (Fig. 2b). This snaky processing path avoids the weakness that the filter should be initialized at the beginning of each line, which guarantees the effectiveness of initial filters.

For calculating the 3D $f$ -$x$ -$y$ SPF, the neighboring filters $\mathbf{A}_{m-1,n,l}$ , $\mathbf{A}_{m,n-1,l}$ , and $\mathbf{A}_{m,n,l-1}$ need to be stored, and these filters will be used when the stream reaches the adjacent point. For easy program implementation, we designed the cache space to store the neighboring filters. The cache space for $\mathbf{A}_{m-1,n,l}$ , $\mathbf{A}_{m,n,l-1}$ , and $\mathbf{A}_{m,n-1,l}$ are $N_{a}N_{x}N_{y}$ , $N_{a}N_{x}$ , and $N_{a}$ , respectively, where $N_a$ is the filter size, $N_x$ is the data size along the $x$ axis, and $N_y$ is the data size along the $y$ axis. Compared with the 3D $f$ -$x$ -$y$ RNA Liu and Chen (2013), the proposed method calculates the filter coefficients without iterations, which reduces the requirement of computational resources (Table 1). Repeatedly processing the data with the SPF can further suppress the noise, but part of the signal will also be weakened. Computational cost increases with the number of repeated calculations, and a balance between computational cost and noise suppression effect is needed.

The key steps of using the 3D $f$ -$x$ -$y$ SPF to attenuate the random noise are as follows:
1. Initializing filter coefficients $\mathbf{A}_{m-1,n,l}$ , $\mathbf{A}_{m,n-1,l}$ , and $\mathbf{A}_{m,n,l-1}$ with zeros.
2. Selecting reasonable parameters $\lambda_{f}$ , $\lambda_{x}$ , and $\lambda_{y}$ and computing $\lambda^{2} = \lambda_{f}^{2} + \lambda_{x}^{2} + \lambda_{y}^{2}$ .
3. Calculating $\mathbf{S^{T}}_{m,n,l}\mathbf{S^{*}}_{m,n,l}$ and $\mathbf{S^{T}}_{m,n,l}\mathbf{\hat{A}}_{m,n,l}$ in (14).
4. Computing residual $r_{m,n,l}$ with (15) and updating filter $\mathbf{A}_{m,n,l}$ with (14).
5. Estimating noise-free data $\tilde{X}_{m,n,l}$ with (16).
6. Looping steps 2-5 in snaky processing path until the entire process is completed.

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