Developing learned regularization for geophysical inversions
Creators: name: Hewson, Chad
Publisher: name: University of British Columbia
Year: 2009

description: Geophysical inverse problems can be posed as the minimization of an objective function where one term (ϕ[sub d]) is a data misfit function and another (ϕ[sub m]) is a model regularization. In current practice, ϕ[sub m] is posed as a mathematical operator that potentially includes limited prior information on the model, m. This research focusses on the specification of learned forms of &lt;pm from information on the model contained in a training set, M[sub T]. This is accomplished via three routes: probabilistic, deterministic (model based) and the Haber- Tenorio (HT) algorithm. In order to adopt a pure probabilistic method for finding a learned ϕ[sub m], equivalence between Gibbs distributions and Markov random fields is established. As a result, the prior probability of any given model is reduced to the interactions of cells in a local neighbourhood. Gibbs potentials are defined to represent these interactions. The case of the multivariate Gaussian is used due to its expressible form of normalization. ϕ[sub m] is parameterized by a set of coefficients, θ, and the recovery of these parameters is obtained via an optimization method given M[sub T]. For non-Gaussian distributions θ is recovered via Markov chain Monte Carlo (MCMC) sampling techniques and a strategy to compare different forms of ϕ[sub m] is introduced and developed. The model based deterministic route revolves around independent identically distributed (i.i.d.) assumptions on some filter property of the model, z = f(m). Given samples of z, two methods of expressing its corresponding ϕ[sub m] are developed. The first requires the expression of a generic distribution to which all the samples of z are assumed to belong. Methodology to translate z into usable data and recover the corresponding ϕ[sub m] is developed. Although there are ramifications of the statistical assumptions, this method is shown to translate significant information on z into ϕ[sub m]. Specifically, the shape of the ϕ[sub m] functional is maintained and, as a result, the deterministic ϕ[sub m] performs well in geophysical inversions. This method is compared with the parametrization of the generalized Gaussian (p-norm) for z. Agreement between the generic ϕ[sub m] and generalized Gaussian helps validate the specific choice of norm in the probabilistic route. The HT algorithm is based around the notion that the geophysical forward operator should help determine the form of ϕ[sub m]. The strategy of Haber and Tenorio [16] is introduced and an algorithm for the recovery of 9 is developed. Two examples are used to show a case where the HT algorithm is advantageous and one where it does not differ significantly from the probabilistic route. Finally, a methodology to invert geophysical data with generic learned regularization is developed and a simple example is shown. For this example, the generic deterministic method is shown to transfer the most information from the training set to the recovered model. Difficulties with extremely non-linear objective functions due to learned regularization are discussed and research into more effective search algorithms is suggested.