Diagonal weighting preconditioning

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 $\bold L$ 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 $\bold L^T$ followed by a scaling of the model by the diagonal operator. Consequently, we refer to this approach as model-space preconditioning.