## Shaping regularization

Fomel (2009,2007) introduces shaping regularization in inversion problem, which regularizes the under-determined linear system by mapping the model to the space of acceptable models. Consider a linear system given as , where is the forward-modeling map, is the model vector, and is the data vector. Tikhonov regularization method amounts to minimize the least square problem bellow (Tikhonov, 1963):

 (28)

where is the regularization operator, and is a scalar parameter. The solution for equation is:

 (29)

Where is the least square approximated of , is the adjoint operator. If the forward operator is simply the identity operator, the solution of equation is the form below:

 (30)

which can be viewed as a smoothing process. If we let:

 (31)

or

 (32)

Substituting equation into equation yields a solution by shaping regularization:

 (33)

The forward operator may has physical units that require scaling. Introducing scaling into , equation 33 be written as:

 (34)

If with square and invertible . Equation can be written as:

 (35)

The conjugate gradient algorithm can be used for the solution of the equation.

2020-07-18