Model Parameterization and Model Selection in Geophysical Inverse Problems. Designing Inverse Problems that Respect a priori Geophysical Knowledge

The vast majority of the Earth system is inaccessible to direct observation. Consequently, the structure and dynamics of the Earth can only be determined indirectly, via geophysical sensing. These methods have the mathematical form of an inverse problem, in which the data and the unknowns are linked by a physical process, such as seismic wave propagation. From the possibly noisy data, we have indirect access to the unknowns. The vast majority of geophysical inverse problems are ill-posed, and require the provision of a priori knowledge to stabilize the solution. This thesis investigates methods for designing inverse problems to better take advantage of geophysical or geological constraints, to allow better resolution or more interpretability of the solutions. Four major themes are investigated: In Chapter 2, we study the collection of a novel dataset of Rayleigh wave horizontal-to-vertical ratios to provide stronger constraints on upper-crustal structure in Southern California. In Chapters 3 and 4, we develop a method for wavefield-reconstruction of sparse seismic data, including heterogeneous networks consisting of both displacement and strain instruments. This method amounts to an inversion in data-space, and promises to unlock the potential of wavefield based methods for complex datasets. In Chapters 5 and 6, we investigate a new structural parameterization based on a combination of Gaussian processes and the level-set method, that better models discontinuous geological features such as sedimentary basins. We test our method on a variety of synthetic and real datasets, culminating in a detailed study of the northeastern Los Angeles basin, which we found to be significantly deeper and steeper than in previous models. Finally, we develop a method of model selection for noisy historical datasets, which we investigate using the case study of correcting Oldham's data misinterpretation in the 1906 paper that "discovered" Earth's core.