Supposing
is a model vector and
is the data after applying a forward operator
. Nonlinear shaping regularization is used for solving the following equation:
(1)
using an iterative framework:
(2)
where
means the forward operator
is not limited to linear case.
is the shaping operator which shapes the model to an admissible model iteratively and
is the backward operator which provides an approximate mapping from data space to model space (Fomel, 2008). Specially, when
is taken as the adjoint of the
(in the linear case) or the adjoint of the Frechet derivative of
(in the nonlinear case), and take
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