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The generic geophysical inverse problem
Claerbout (1998a); Tarantola (1987) can be summarized as follows - given a linear
forward modeling operator , and some recorded data ,estimate a model such that .
If the system is over-determined, the model that minimizes the
expected (*L*2) error in predicted data is given by the solution to
the normal equations ():
| |
(92) |

Typically the matrices involved in industrial-scale geophysical
inverse problems are too large to invert directly, and we depend on
iterative gradient-based linear solvers to estimate solutions.
However, operators such as prestack depth migration are so expensive
to apply that we can only afford to iterate a handful of times, at
best.
To attempt to speed convergence, we can always change model-space
variables from to through a linear operator
, and solve the following new system for ,

| |
(93) |

When we find a solution, we can then recover the model estimate,
.
If we choose the operator such that , then even simply applying the adjoint ()will yield a good model estimate; furthermore, gradient-based solvers
should converge to a solution of the new system rapidly in only a few
iterations.

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** Up:** Model versus data normalization
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Stanford Exploration Project

5/27/2001