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With two possible preconditioning operators, and
, the question remains, what is the best strategy for
combining them?
The first 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,
| |
(12) |

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,
| |
(13) |

| |

with appropriate model-space and data-space preconditioning
operators.
The adjoint solution to this system is given by
| |
(14) |

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.

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Stanford Exploration Project

4/29/2001