The main result obtained in this paper is this: If it is possible to do forward modeling on a physical problem in a way that permits the output (i.e., the predicted values of some physical parameter that could be measured) and the first derivative of that same output with respect to model parameters (whatever they may be) to be calculated numerically, then (at least in principle) it is possible to solve the inverse problem using the method described. The main trick learned in this analysis comes from the realization that the steps in the model updates may have to be quite small in some cases for the implied guarantees of convergence to be realized.
We have concentrated on least-squares methods in the presentation in order to keep the analysis simple. However, it is clear that the general method presented could be applied as well to any norm of the data discrepancies that possesses the same necessary qualities, the most important of these being the existence of a first derivative with respect to the model parameters. One interesting example of such an alternative is Huber's hybrid of l1 and l2 [see Clearbout (1996)]. Other choices include special cases of the more general lp norm that also have a first derivative. These other methods will be discussed in future work.