Seismic data interpolation using plane-wave shaping regularization

Plane-wave Shaping

The formulation of linear shaping regularization is (Fomel, 2007)

$$\mathbf{\hat{m}}=\left[\mathbf{I}+\mathbf{S}\left(\mathbf{L}... ...-\mathbf{I}\right)\right]^{-1}\mathbf{S}\mathbf{L}^T\mathbf{d}$$ (1)

where $\mathbf{\hat{m}}$ is a vector of model parameters; $\mathbf{S}$ is the shaping operator; $\mathbf{d}$ is the data; and $\mathbf{L}$ and $\mathbf{L}^T$ are the forward and adjoint operators respectively. In interpolation problems, $\mathbf{L}$ is forward interpolation (in the case of irregular sampling) or simple masking (in the case of missing-data interpolation on a regular grid). In 1-D, shaping in Z-transform notation can be triangle smoothing (Claerbout, 1992)

$$T_n=\frac{1}{n^2}\left(\frac{1-Z^n}{1-Z}\right)\left(\frac{1-Z^{-n}}{1-Z^{-1}}\right)$$ (2)

for a given smoothing radius $n$. One can visualize this as a convolution of two box filters producing a weighting triangle for a triangle / neighborhood radius of $n$. Increasing $n$ produces a smoother model. In 2-D the shift operator $Z$ translates into shifts along local slope. $Z$ corresponds to PWD - which can be thought of as a differentiation - while its inverse operator $\frac{1}{1-Z}=1+Z+Z^2+...+Z^n$ corresponds to PWC - similar to integration.

Seismic data interpolation using plane-wave shaping regularization