Imagine yourself doing a big modeling job by these methods.
At each iteration you will be pushing the residual into
the adjoint operator to see which direction you will move next,
If each iteration is taking you a day,
you will be looking at all your intermediate results each day.
You will be looking at your current model and your
current residual .
But your residual can be carried into model space
with the operator.
We have been calling this the gradient and the model change
and it is, but it is also the current residual
as it appears in model space.
Viewing it, you'll see where in model space
your theoretical data mismatches your observed data.
In some applications it might be more informative than itself.