Seismic data interpolation without iteration using $t$ -$x$ -$y$ streaming prediction filter with varying smoothness

Step 2: Data interpolation with $t$ -$x$ -$y$ SPF

The SPF error $r(t,x,y)$ can be expressed as follows:

$$r(t,x,y)=d(t,x,y)-\mathbf{d}(t,x,y)^{T}\mathbf{a}(t,x,y)=\xi^2\fr... ...t,x,y)^T\mathbf{\bar{a}}(t,x,y)} { \xi^2+\mathbf{d}(t,x,y)^T\mathbf{d}(t,x,y)},$$ (10)

equation 10 shares the same form as the second term in the right hand side of equation 9. Substituting equation 10 into equation 9, we obtain equation 11:

$$\mathbf{a}(t,x,y)=\mathbf{\bar{a}}(t,x,y)+\frac{r(t,x,y)}{\xi^2}\mathbf{d}(t,x,y).$$ (11)

When a missing data is encountered, $r(t,x,y)$ can be assigned as zero, and equation 9 can be reduced to:

$$\mathbf{a}(t,x,y)= \mathbf{\bar{a}}(t,x,y).$$ (12)

Therefore, the data interpolation is also implemented in a streaming manner, where the missing data are reconstructed right after the unknown filter gets updated, and the interpolated data is shown as:

$$\widehat{d}(t,x,y)=\mathbf{d}(t,x,y)^{T}\mathbf{a}(t,x,y)=\mathbf{d}(t,x,y)^{T} \mathbf{\bar{a}}(t,x,y).$$ (13)

The field data always includes noise, hence the interpolated traces with noise is more realistic, where the prediction error $r(t,x,y)$ is set to a small random noise.

To use the available data for SPF estimation, we designed a 3D $t$ -$x$ -$y$ non-causal in space SPF shown in Figure 1a, the light-gray grids represent prediction samples and the dark-gray ones exhibit target positions, whereas white grids represent unused samples. Non-causal in space SPF utilizes more adjoining traces around the target traces to predict signals, therefore, it can provide more accurate interpolated results than the causal in space version. The interpolation steps in the 2D case are illustrated schematically in Figure 1b. The black and white circles represent the known data and the missing data, respectively. Meanwhile, the dotted part is the prior non-causal SPF position, and dark-grey triangle is the target position. When the target position is known, the SPF coefficients $\mathbf{a}(t,x,y)$ can be obtained from equation 9. In streaming computation, we can use the time or space axis as the interpolation direction. The light-gray area in Figure 1b is the position where the SPF moves next, and the target trace becomes missing data. Further, spatial gaps are reconstructed according to equation 12 and 13. Note that the prior filter coefficients are required in this calculation. If the first target position is missing trace, e.g., marine data with near-offset missing, one may use the mirror data to initialize the coefficients of SPF in the space directions.

We also interpolated the results in the forward and backward spatial directions; adding the two results $\widehat{d}_{sum}=(\widehat{d}_{forw}+\widehat{d}_{back})/2$ can reduce the interpolated error caused by the directional properties of the streaming computation, where $d_{forw}$ is the forward interpolated result and $d_{back}$ is the backward one. The proposed method uses local varying smoothness of SPF to characterize time-space variation of nonstationary data, the analytical calculation of the inverse matrix in equation 6 avoids iteration, which results in superior computational speed. Table 1 compares the computational cost between 3D Fourier POCS (Abma and Kabir, 2006) and the 3D $t$ -$x$ -$y$ SPF. The proposed method occupies less computational resources by reducing the cost to a single convolution.

Method	Cost	Filter storage
3D Fourier POCS	$O(N_t N_x N_y log(N_t N_x N_y)N_{iter})$	$O(N_f N_{k_{x}}N_{k_{y}})$
$t$ -$x$ -$y$ SPF	$O(N_a N_t N_x N_y)$	$O(N_a N_x N_y)$

Table 1. Rough cost comparison between 3D Fourier POCS and $t$ -$x$ -$y$ SPF. $N_a$ is the filter size, $N_{iter}$ is the number of iterations length, $N_t$ is the data length in the time direction, $N_x$ and $N_y$ are the data length in the space directions, $N_f$ is the data size along frequency axis, and $N_{k_{x}}$ and $N_{k_{y}}$ are the data size along wavenumber $k_{x}$ and $k_{y}$ axis, respectively.

filter3d,filter2d
Figure 1. (a) Schematic illustration of a 3D non-causal in space SPF and (b) interpolation process.

Seismic data interpolation without iteration using $t$ -$x$ -$y$ streaming prediction filter with varying smoothness