Seismic interpolation

The basic target of seismic interpolation is to solve the following equation:

$$\mathbf{d}_{obs}=\mathbf{Md},$$ (1)

where $\mathbf{d}_{obs}$ is the observed data which is regularly or irregularly sampled, $\mathbf{d}$ is the unknown data we want to reconstruct and $\mathbf{M}$ is the sampling matrix. The sampling operator has a diagonal structure, which is composed by zero and identity matrix:

$$\mathbf{M} = \left[\begin{array}{cccccccc} \mathbf{I} & & & & & \... ...I}& & \\ & & & &\mathbf{\ddots} & \\ & & & & & \mathbf{I} \end{array}\right].$$ (2)

Each $\mathbf{I}$ in equation 2 corresponds to sampling a trace, and each $\mathbf{O}$ corresponds to missing a trace.

As equation 1 is under-determined, additional constraint is required in order to solve the equation. By applying a regularization term, we get a least-squares minimization solution for solving equation 1:

$$\hat{\mathbf{d}}=\arg\min_{\mathbf{d}}\Arrowvert \mathbf{d}_{obs}-\mathbf{Md}\Arrowvert_2^2+\mathbf{R}(\mathbf{d}),$$ (3)

where $\mathbf{R}$ is a regularization operator and $\Arrowvert\cdot\Arrowvert_2^2$ denotes the square of $L_2$ norm.