Shaping regularization in geophysical estimation problems |

Taking equation 5 and using it as the
definition of the regularization operator , we can write

Substituting equation 10 into 1 yields a formal solution of the estimation problem regularized by shaping:

The meaning of equation 11 is easy to interpret in some special cases:

- If (no shaping applied), we obtain the solution of unregularized problem.
- If ( is a unitary operator), the solution is simply and does not require any inversion.
- If (shaping by scaling), the solution approaches as goes to zero.

The operator may have physical units that require
scaling. Introducing scaling of by in
equation 11, we can rewrite it as

Iterative inversion with the conjugate-gradient algorithm requires
symmetric positive definite operators (Hestenes and Steifel, 1952). The inverse
operator in equation 12 can be symmetrized when the
shaping operator is symmetric and representable in the form
with a square and invertible . The
symmetric form of equation 12 is

Shaping regularization in geophysical estimation problems |

2013-07-26