Next: Numerical comparisons Up: Data-space weighting functions Previous: Data-space weighting functions

## Combining model-space and data-space weighting functions

With two possible preconditioning operators, and , the question remains, what is the best strategy for combining them?

Chemingui (1999) calculated both and from operator fold and data fold respectively, but observed that because they both contain the inverse units to the operator, , he should not apply them both at once. Instead he combined the two preconditioning operators, and solved the system
 (102) (103)
where is an adjustable parameter. Chemingui (1999) provided no advice on the choice of n, but for the problem he was solving, he observed that applying both and with n=1/2 converged to a solution more rapidly than either end member ().

The first alternative strategy that I propose is to calculate a model-space weighting function, , and use it to create a new preconditioned system with the form of

Now probe the composite operator, , for a data-space weighting function for the new system,
 (104)
The new data-space weighting function is dimensionless, and can be applied in consort with the model-space operator. This leads to a new system of equations,
 (105) (106)
with appropriate model-space and data-space preconditioning operators. The adjoint solution to this system is given by
 (107)

A second alternative strategy is the corollary of this: create a new system that is preconditioned by an appropriate data-space weighting function, and then calculate a model-space weighing function based on the new system. The first step is to calculate with equation (), and set up a new system of equations,
 (108)
The second step is to calculate a model-space weighting function based on this new operator,
 (109)
The preconditioned composite system of equations is now
 (110) (111)

Next: Numerical comparisons Up: Data-space weighting functions Previous: Data-space weighting functions
Stanford Exploration Project
5/27/2001