Inverse problem

Inverse problem

Starting from observations (measurements) we, geophysicists, often face the problem of trying to constraint the parameters describing the physical systems we study. This is called an Inverse Problem. Adjusting a straight line across a cloud of points or imaging the tri-dimensional distribution of seismic velocities inside the Earth by using arrival times of seismological signals are typical Inverse Problems.

Within this course we treat on one hand discrete linear or weakly non-linear problems and on the other hand strongly non-linear problems. We first study the Generalized Inverse method, based in the singular value decomposition. In this context we also introduce the concepts of Resolution matrix, Information matrix, Covariance matrix, etc.  We then study the stochastic approach developed by Tarantola and Valette and focus on the linear-gaussian case. Concerning the non-linear case we first present the local methods. These are typically iterative methods based on first and possibly higher derivatives of the model-posteriori density function. (Keywords: Gradient, steepest-descent, Newton, etc.). Finally, we present an overview of the Montecarlo methods used in the strongly non-linear case to sample the model space.  (Keywords : Bayesian, Metropolis-Hasting, Simulated Annealing, etc).