The Classical Inverse Problem

The classical inverse problem (CIP) assumes no model structure error and, thus, only model parameters need to be identified. In this chapter, we consider the solution of CIP for single state variable models. Using the “fitting data” criterion of inverse problem formulation, we can obtain a quasi-solution by solving an optimization problem. We will show that when the inverse problem is extended well-posed and the conditions of quasi-identifiability are satisfied, thequasi-solution will approach the accurate inverse solution when the observation error reaches zero. The singular value decomposition (SVD) is introduced for linear model inversion, followed by a general procedure of linearization for mildly nonlinear model inversion. Optimization problems resulting from nonlinear inversion can be difficult to solve and the solutions are generally not unique. In this chapter, we shall restrict ourselves to cases in which the CIP is extended well-posed in a known region and the objective function for optimization is convex in the region. Consequently, the inverse solution can be found by using a local optimization algorithm. Common numerical methods for local optimization and norm selection problems will be briefly discussed.