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:
is a regularization operator and
denotes the square of
Alternatively, we can use the shaping regularization framework 2, and substitute
is obtained through the following iteration equation:
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