Seismic data interpolation using nonlinear shaping regularization

Review of nonlinear shaping regularization

Supposing $\mathbf{m}$ is a model vector and $\mathbf{d}$ is the data after applying a forward operator $\mathbf{F}$ . Nonlinear shaping regularization is used for solving the following equation:

$$\mathbf{F}[\mathbf{m}]=\mathbf{d},$$ (1)

using an iterative framework:

$$\mathbf{m}_{n+1} = \mathbf{S}[\mathbf{m}_n+\mathbf{B}[\mathbf{d}-\mathbf{F}[\mathbf{m}_n]]],$$ (2)

where $[\cdot]$ means the forward operator $\mathbf{F}$ is not limited to linear case. $\mathbf{S}$ is the shaping operator which shapes the model to an admissible model iteratively and $\mathbf{B}$ is the backward operator which provides an approximate mapping from data space to model space (Fomel, 2008). Specially, when $\mathbf{B}$ is taken as the adjoint of the $\mathbf{F}$ (in the linear case) or the adjoint of the Frechet derivative of $\mathbf{F}$ (in the nonlinear case), and take $\mathbf{S}$ as an identity operator, iteration 2 becomes a famous Landweber iteration (Landweber, 1951). Iteration 2 can get converged if the spectral radius of the operator on the right hand side is less than one (Collatz, 1966).

Seismic data interpolation using nonlinear shaping regularization