Advanced numerical techniques for inverse problems in geophysics

• Autores: Olga Ortega Gelabert
• Directores de la Tesis: Sergio Zlotnik (dir. tes.) , Pedro Díez Mejía (codir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2020
• Idioma: español
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• Resumen

  • This thesis presents an efficient methodology to couple Model Order Reduction techniques within the framework of geophysical probabilistic inversion problems. Accurate models of the interaction between Earth inner processes and surface features are essential to make reliable predictions of the observables which are a fundamental part of Bayesian inference. Markov Chain Monte Carlo (MCMC) methods have become standard in dealing with probabilistic inversions and they rely on sampling strategies that require solving forward problems many times. Computationally expensive large-scale forward problems are the principal bottleneck that can limit the capabilities and potential of multi-observable geophysical probabilistic inversions. In particular, dynamical effects arising from the sub-lithospheric mantle flow are not usually taken into account in the estimation of surface elevation due to the high computational cost of the associated 3D Stokes flow problem.

    The main idea of this thesis is to use the Reduced Basis (RB) method as a surrogate of the true forward problem (3D Stokes flow) to provide fast and accurate approximations. The surrogate is then used to generate samples of the posterior distribution at a much lower computational cost. RB strategies are based on expressing the solution of a problem in a low dimensional space, i.e. a reduced basis. Taking advantage of the convergence nature of the MCMC, we propose a greedy strategy that builds the reduced basis on the fly and as required by the inverse problem. In doing so, the basis is specifically tailored to the posterior features of the problem. In addition, to guarantee an accurate surrogate we define a goal-oriented error estimator which focuses on a particular Quantity of Interest of the problem and, therefore, it guides the basis to achieve the required accuracy in such particular features. All this translates into a problem-shaped basis that is more compact and smaller than if it had to be accurate everywhere in the domain. Moreover, to deal with the costly assembly of matrices, we use the specific parametrization of the problem and sampling strategy to define an assembly procedure that efficiently updates the matrices only with the contribution of the elements that changed between successive inversion steps.

    The benefits and limitations of the method are illustrated through several numerical examples. Finally, to demonstrate the applicability of the method two more realistic inverse problems are presented. The first one uses dynamic topography to infer the Lithosphere-Asthenosphere Boundary depth of a spherical domain representing a portion of Earth and the second one is applied to a larger problem in which the African lithospheric structure is discretized in 1225 inversion parameters.