Regularization is a method of imposing additional conditions for
solving inverse problems with optimization methods. When model
parameters are not fully constrained by the problem (the inverse
problem is mathematically ill-posed), regularization limits the
variability of the model and guides the iterative optimization to the
desired solution by adding assumptions about the model power,
smoothness, predictability, etc. In other words, it constrains the
model null space to an *a priori* chosen pattern. A thorough
mathematical theory of regularization has been introduced by works of
Tikhonov's school Tikhonov and Arsenin (1977).

In this paper, I discuss two alternative formulations of
regularized least-square inversion problems. The first formulation,
which I call *model-space* , extends the data space and
constructs a composite column operator. The second,
*data-space* , formulation extends the model space and constructs a
composite row operator. This second formulation is intrinsically
related to the concept of model preconditioning. I illustrate the
general theory with examples from *Three-Dimensional Filtering*
Claerbout (1994).

Two excellent references cover almost all of the theoretical material in this note. One is the paper by Ryzhikov and Troyan (1991). The other one is a short note by Harlan, available by courtesy of the author on the World Wide Web Harlan (1995). I have attempted to translate some of the ideas in these two references to the linear operator language, familiar to the readers of Claerbout (1992, 1994).

11/11/1997