Conditioning the gradient

Often people do calculations by the method of steepest descent without realizing it. Often a result is improved in a single step, or with a small number of steps, many fewer than the number needed to achieve convergence. This is especially true with images where the dimensionality is huge and where a simple improvement to the adjoint operator is sought. Three-dimensional migration is an example. In these cases it may be worthwhile to make some ad hoc improvements to the gradient that acknowledge the gradient will be a perturbation to the image $\bold x$ and so should probably have an amplitude and spectrum like that of $\bold x$.A more formal mathematical discussion of preconditioning is on page .