Asymptotic pseudounitary stacking operators |

and its asymptotic pseudo-unitary adjoint as

According to equation (10),

According to equation (22),

Combining equations (26) and (27) uniquely determines both weighting functions, as follows:

Equations (28) and (29) complete the definition of asymptotic pseudo-unitary operator pair.

The notion of pseudo-unitary operators is directly applicable in the
situations where we can arbitrarily construct both forward and inverse
operators. One example of such a situation is the velocity transform
considered in the next section of this paper. In the more common
case, the forward operator is strictly defined by the physics of a
problem. In this case, we can include asymptotic inversion in the
iterative least-squares inversion by means of *preconditioning*
(Lambaré et al., 1992; Jin et al., 1992). The linear preconditioning operator should
transform the forward stacking-type operator to the form
(24) with the weighting function (28).
Theoretically, this form of preconditioning should lead to the fastest
convergence of the iterative least-squares inversion with respect to
the high-frequency parts of the model.

If the forward pseudo-unitary operator can be related to the forward modeling operator as , where and are weighting operators in the data and model domains correspondingly, then preconditioning simply amounts to replacing the least-squares equation

with the equation

where is the preconditioned model. The advantage of using equation (31) is in the the fact that the normal operator is closer (asymptotically) to identity and therefore should be easier to invert than the original operator in the least-squares solution (13).

Asymptotic pseudounitary stacking operators |

2013-03-03