Bayesian inverse problems and seismic inversion - Lim, S (2016)
The Bayesian formulation for inverse problems gives a way of making inferences about unknown quantities not directly observable. The application of Bayes' Theorem combines the prior information and the observation to give a posterior measure, which contains information about the quantity we are trying to estimate. In this thesis, we review a particular formulation, conventionally known as the strong constraint formulation of inverse problems. We describe methods to obtain summaries of information from the posterior measure. We also describe how prior measures are constructed using linear differential operators, to quantify as accurately as possible our knowledge of the parameters, independent of any observations. Then, we note that the strong constraint formulation of inverse problems makes it hard to obtain summaries of information of the posterior measure, typically obtained through an optimization of a misfit functional. Therefore, we introduce the weak constraint formulation in a Bayesian context for inverse problems, which eases the task of optimization. We use this formulation to perform sampling of the posterior measure. This method is tested on some simple test problems. We also compare the results between a strong and weak constraint formulation of inverse problems by studying a one-dimensional example. Finally, we apply the weak constraint formulation to the problem of full waveform inversion, which is a common problem in seismology. The forward problem we use here is the Laplace transform of the acoustic wave equation, and the inverse problem is solved in several frequencies. There are two approaches when observations at several frequencies are available. First is the sequential method, which processes the observation at different frequencies individually. The second method, which is the simultaneous method, processes the observations at all frequencies at once. We use the simultaneous method here, and used a non-trivial model problem, which yields promising results.