Seismic data interpolation using streaming prediction filter in the frequency domain

$f$ -$x$ -$y$ streaming prediction filter

The extension to the 3D $f$ -$x$ -$y$ domain is straightforward as the $f$ -$x$ -$y$ SPF can also efficiently perform data interpolation in high dimensions. In the $f$ -$x$ -$y$ domain, the prediction relationship for seismic data in a certain frequency slice is expressed as

$$\begin{aligned}g_{m,n,l} & \approx \sum_{p} \sum_{q} g_{m,n-p,l... ...\mathbf{G}_{m,n,l}^{\mathsf{T}} \mathbf{F}_{m,n,l}, \end{aligned}$$

where $m$ , $n$ , and $l$ denote the indices of data sample $g_{m,n,l}$ along the $f$ , $x$ , and $y$ axes, respectively. Vector $\mathbf{F}_{m,n,l}$ is a group of filter coefficients in the $f$ -$x$ -$y$ SPF, vector $\mathbf{G}_{m,n,l}$ contains the data points with spatial shifts $p$ ($x$ axis) and $q$ ($y$ axis) near $g_{m,n,l}$ . Similar to the case of the $f$ -$x$ SPF, the range of spatial shifts $p$ and $q$ depends on the number of seismic events in the local window of space $x$ and space $y$ . As shown in Fig. 2, when $p \in \{ -1, 1\}$ and $q \in \{ 1, 3\}$ , vector $\mathbf{G}_{m,n,l}$ can be flattened as $\mathbf{G}_{m,n,l} = \{ g_{m,n+1,l-1}, g_{m,n+1,l-2}, \cdots ,g_{m,n-1,l-3}\}$ , and $\mathbf{F}_{m,n,l} = \{ f_{m,n,l,-1, 1}, f_{m,n,l,-1, 2}, \cdots ,f_{m,n,l, 1, 3}\}$ . For the space-noncausal case, $p \in \{ -1, 1\}$ , $q \in \{-3, 3\}$ , and $\left \vert p \right \vert + \left \vert q \right \vert \neq 0$ , $\mathbf{G}_{m,n,l}$ can be expressed as follows: $\mathbf{G}_{m,n,l} = \{ g_{m,n+1,l+3},\ g_{m,n+1,l+2}, \cdots, g_{m,n,l+1}, g_{m,n,l-1}, \cdots , g_{m,n-1,l-3} \}$ , and $\mathbf{F}_{m,n,l} = \{ f_{m,n,l, -1, -3}, f_{m,n,l, -1, -2},\\ \cdots, f_{m,n,l, 0, -1}, f_{m,n,l, 0, 1}, \cdots, f_{m,n,l, 1, 3} \}$ .

The cost function is defined as

$$J_{m,n,l} = \Vert g_{m,n,l} - \mathbf{G}_{m,n,l}^{\mathsf{T}} \mathbf{F}_{m,n,l} \Vert _{2}^{2}.$$ (13)

Assuming that the adaptive PF at position $(m,n,l)$ is similar to another near positions $(m-1,n,l)$ , $(m,n-1,l)$ , and $(m,n,l-1)$ in the $f$ -$x$ -$y$ domain, we can minimize the new cost function Eq. (14) to solve the overdetermined problem:

$$\begin{aligned}\hat{J}_{m,n,l} & = J_{m,n,l} + \lambda_{f}^{2} ... ...f{F}_{m,n,l} - \mathbf{F}_{m,n,l-1} \Vert _{2}^{2}, \end{aligned}$$

where $\lambda_{f}$ , $\lambda_{x}$ , and $\lambda_{y}$ are the constant weights of the regularization terms, which describe the local smoothness constraints along different directions. The simplified block matrix can be written as

$$\begin{bmatrix}\mathbf{G}_{m,n,l}^{\mathsf{T}} \ \lambda_{f} \ma... ...bda_{x} \mathbf{F}_{m,n-1,l} \ \lambda_{y} \mathbf{F}_{m,n,l-1} \end{bmatrix}.$$ (15)

Similar to the 2D case, the analytic solution of the $f$ -$x$ -$y$ SPF is given by

$$\mathbf{F}_{m,n,l} = \mathbf{\tilde{F}}_{m,n,l} + \frac{ g_{m,n,l... ...mathbf{G}_{m,n,l}^{\mathsf{T}} \mathbf{G}_{m,n,l}^{*} } \mathbf{G}_{m,n,l}^{*},$$ (16)

where

$$\begin{cases}\lambda^{2} = \lambda_{f}^{2}+\lambda_{x}^{2}+\l... ...}_{m,n-1,l} + \lambda_{y}^{2} \mathbf{F}_{m,n,l-1} \end{cases}.$$ (17)

Furthermore, the unknown data sample in the 3D data cube can be calculated as

$$\hat{g}_{m,n,l} = \mathbf{G}_{m,n,l}^{\mathsf{T}} \mathbf{F}_{m,n,l}.$$ (18)

Because the $f$ -$x$ -$y$ SPF predicts data along two spatial directions, we defined a new processing path in the space plane with zigzag shape (Fig. 2a) to prevent unnecessary filter initialization. Meanwhile, the $f$ -$x$ -$y$ SPF was assigned to the proposed filter form (Fig. 2a), which can better reduce the influence of unknown data samples than the noncausal filter in spatial directions (Fig. 2b). Following the processing path of Algorithm 2 for data processing, $\mathbf{F}_{m-1,n,l}$ , $\mathbf{F}_{m,n-1,l}$ , and $\mathbf{F}_{m,n,l-1}$ can be seen as known, requiring only calculating Eq. (16) and (18) to obtain the results.

causal3d,noncausal3d
Figure 2. Proposed filter form (a) and space-noncausal filter form (b) in the $f$ -$x$ -$y$ domain. Solid circle denotes known trace, and hollow circle denotes unknown trace.

Seismic data interpolation using streaming prediction filter in the frequency domain