Reuniting optimization theory and wave theory

Let's take a quick peek beyond this book into the future. A seismic image is typically a 1000$\times$1000 plane, derived from a volume of about 1000³ interrelated data points. There are unknowns present everywhere, not only in the earth model, but also in the data, as noise, as gaps, and as insufficient spatial density and extent of data recording. To assemble an interpretation we must combine principles from physics with principles from statistics. Presumably this could be done in some monster optimization formulation. A look at the theory of optimization shows that solution techniques converge in a number of iterations that is greater than the number of unknowns. Thus the solution to the problem, once we learn how to pose the problem properly, seems to require about a million times as much computing power as is available. What a problem!

But the more you look at the problem, the more interesting it becomes. First we have an optimization problem. Since we are constrained to make only a few iterations, say, three, we must go as far as we can in those three steps. Now, not only do we have the original optimization problem, but we also have the new problem of solving it in an optimum way. First we have correlated randomness in the raw data. Then, during optimization, the earth model changes in a correlated random way from one iteration to the next. Not only is the second optimization problem the practical one--it is deeper at the theoretical level.