    Seismic data interpolation using nonlinear shaping regularization  Next: Connection with iterative shrinkage Up: Theory Previous: Review of nonlinear shaping

## Interpolation using shaping regularization

The basic target of seismic interpolation is to solve the following equation: (3)

where is the observed data which is regularly or irregularly sampled, is the unknown data we want to reconstruct and is the sampling matrix (Liu and Sacchi, 2004) or so-called mask operator (Liu and Fomel, 2012b; Naghizadeh and Sacchi, 2010). The mask operator has a diagonal structure, which is composed by zero and identity matrix: (4)

Each in equation 4 corresponds to sampling a trace, and each corresponds to missing a trace. As equation 3 is under-determined, addition constraint is required in order to solve the equation. By applying a regularization term, we get a least-squares minimization solution for solving equation 3: (5)

where is a regularization operator and denotes the square of norm. Alternatively, we can use the shaping regularization framework 2, and substitute with , then, is obtained through the following iteration equation: (6)

In equation 6, can be selected as any linear or nonlinear operator for reasonable constraint, as long as the equation converges. Thus it offers us much freedom to control the model behavior. The shaping regularization framework is a very general iterative framework, which can be generalized into different commonly known interpolation techniques. Because the shaping operator can be chosen as a constraining operators such as soft thresholding in a sparsity promoting transformed domain, or as rank reduction in the Fourier transform domain (Oropeza and Sacchi, 2011).    Seismic data interpolation using nonlinear shaping regularization  Next: Connection with iterative shrinkage Up: Theory Previous: Review of nonlinear shaping

2015-11-24