Seismic data interpolation using streaming prediction filter in the frequency domain

Two-step interpolation strategy

The classical method for restoring missing data is to ensure that the restored data, after specified filtering, have minimum energy (Claerbout, 1992). Data interpolation by prediction filtering can be divided into two problems (Liu and Fomel, 2011): filter estimation and data reconstruction with filters. The first step is to calculate the filter by minimizing the following autoregression problem:

$$\min_{\mathbf{F}} \Vert \mathbf{g} - \mathbf{G}^{\mathsf{T}} \mathbf{F} \Vert _{2}^{2},$$ (1)

where $\{\bullet\}^{\mathsf{T}}$ denotes the transpose operator, $\mathbf{F}$ is adaptive PF, $\mathbf{g}$ is the data matrix, and $\mathbf{G}$ is the translation of $\mathbf{g}$ . In the second step, the missing data are restored by minimizing the least-squares problem:

$$\min_{\hat{\mathbf{g}}} \Vert \hat{\mathbf{g}} - \mathbf{G}^{\mathsf{T}}\mathbf{F} \Vert _{2}^{2},$$ (2)

where $\hat{\mathbf{g}}$ is the reconstructed data.

In the time-space or frequency-space domain, Eq. (1) and (2), established by the prediction filtering theory, are designed to solve different variables, and they are solved individually. The number of unknown filter coefficients in adaptive PF $\mathbf{F}$ is often greater than the number of known data in $\mathbf{g}$ , in other words, the number of unknowns is greater than the number of equations. Eq. (1) usually presents an ill-posed problem, and different regularization terms have been introduced to stabilize the filter solution $\mathbf{F}$ . Further discussion will be provided in the next section. Meanwhile, we extended the framework of the streaming computation in the frequency domain and proposed a local and multidimensional smoothness to constrain Eq. (1) in the frequency domain.

Seismic data interpolation using streaming prediction filter in the frequency domain