** Next:** Approximation of by application
** Up:** Normalized partial stacking and
** Previous:** Normalized partial stacking and

We now analyze the properties of the approximate inverse
that is defined by the substitution of
with a diagonal matrix ,when the diagonal elements of are
equal to the sum of the corresponding rows of .Notice that is symmetric, and thus
summing over the columns would be equivalent
to summing over the rows.
It can be noticed that the diagonal elements of
are equal
to the sum of the columns of the original operator as defined in equation (1);
i.e.,
they are equal to the fold computed for each corresponding model trace.

The weights *a*_{lj} are interpolator weights,
and they fulfill the constraint
expressed in
equation (1).
It immediately follows that

| |
(10) |

which is equivalent to equation (6).
In summary, we just derived,
for the model-space inverse,
an approximation
that is easy to compute, and it is equal
to the fold normalization that
was defined by heuristic considerations.
However, the definition of the approximate inverse
can be applied when more complex imaging operators are
involved,
and for which
we do not have a simple heuristic to define the weights
that improve the results obtained by application of the adjoint operator.

** Next:** Approximation of by application
** Up:** Normalized partial stacking and
** Previous:** Normalized partial stacking and
Stanford Exploration Project

9/18/2001