In the inversion relation equ1, the number of equations is
the number of traces in the input data
and the number of unknowns is the number of output traces or bins.
Since is unbalanced, we can improve its condition
by diagonal weighting Ronen (1994). We apply the
row and column normalization operators described
earlier as preconditioners.
Similar approaches based on diagonal scaling are
discussed in the mathematical literature using different norms for
the columns. Often they are referred to
as left and right preconditioners; we prefer to call them
*data-space* and *model-space* preconditioners. The rationale in
the terminology is based on the fact that the scaled adjoint is the
first step of the inversion. With left preconditioning the adjoint operator
is applied after the data have been normalized by the diagonal operator.
We therefore refer to this solution as *data-space* preconditioning.
Right preconditioning is equivalent to applying the adjoint operator
followed by a scaling of the model by the diagonal
operator. Consequently, we refer to this approach as
*model-space* preconditioning.

7/5/1998