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Notice that the physical units of an operator
(such as the meters or feet implied by )are the
same
as the physical units of the adjoint operator.
The units of an inverse operator, however, are
inverse
to the units of the original operator.
Thus it is hard to imagine that an adjoint operator
could ever be a satisfactory approximation to the inverse.
We know, however, that adjoints often are a satisfactory approximation
to an inverse,
which means then that either
(1) such operators do not have physical units, or
(2) a scaling factor in the final result is irrelevant.
With the tomographic operator,
the adjoint is quite far from the inverse
so practicioners typically work from the adjoint toward the inverse.
Some operators are arrays with different physical units
for different array elements.
For these operators the adjoint is unlikely to be a satisfactory
approximation to the inverse
since changing the units changes the adjoint.
A way to bring all components to the same units
is to redefine each member of data space and model space
to be itself divided by its variance.
Alternately, again we can abandon the idea of finding immediate utility
in the adjoint of an operator and
and we could progress from the adjoint toward the inverse.
EXERCISES:

Show how to adapt tomography for ``fat'' rays
of thickness N_{z} points along the zaxis.
Next: STOLT MIGRATION
Up: INTRODUCTION TO TOMOGRAPHY
Previous: INTRODUCTION TO TOMOGRAPHY
Stanford Exploration Project
10/21/1998