Reducing non-uniqueness in seismic inverse problems

The scientific investigation of the solid Earth's complex processes, including their interactions with the oceans and the atmosphere, is an interdisciplinary field in which seismology has one key role. Major contributions of modern seismology are (1) the development of high-resolution tomographic images of the Earth's structure and (2) the investigation of earthquake source processes. In both disciplines the challenge lies in solving a seismic inverse problem, i.e. in obtaining information about physical parameters that are not directly observable. Seismic inverse studies usually aim to find realistic models through the minimization of the misfit between observed and theoretically computed (synthetic) ground motions. In general, this approach depends on the numerical simulation of seismic waves propagating in a specified Earth model (forward problem) and the acquisition of illuminating data. While the former is routinely solved using spectral-element methods, many seismic inverse problems still suffer from the lack of information typically leading to ill-posed inverse problems with multiple solutions and trade-offs between the model parameters. Non-linearity in forward modeling and the non-convexity of misfit functions aggravate the inversion for structure and source. This situation requires an efficient exploitation of the available data. However, a careful analysis of whether individual models can be considered a reasonable approximation of the true solution (deterministic approach) or if single models should be replaced with statistical distributions of model parameters (probabilistic or Bayesian approach) is inevitable. Deterministic inversion attempts to find the model that provides the best explanation of the data, typically using iterative optimization techniques. To prevent the inversion process from being trapped in a meaningless local minimum an accurate initial low frequency model is indispensable. Regularization, e.g. in terms of smoothing or damping, is necessary to avoid artifacts from the mapping of high frequency information. However, regularization increases parameter trade-offs and is subjective to some degree, which means that resolution estimates tend to be biased. Probabilistic (or Bayesian) inversions overcome the drawbacks of the deterministic approach by using a global model search that provides unbiased measures of resolution and trade-offs. Critical aspects are computational costs, the appropriate incorporation of prior knowledge and the difficulties in interpreting and processing the results. This work studies both the deterministic and the probabilistic approach. Recent observations of rotational ground motions, that complement translational ground motion measurements from conventional seismometers, motivated the research. It is investigated if alternative seismic observables, including rotations and dynamic strain, have the potential to reduce non-uniqueness and parameter trade-offs in seismic inverse problems. In the framework of deterministic full waveform inversion a novel approach to seismic tomography is applied for the first time to (synthetic) collocated measurements of translations, rotations and strain. The concept is based on the definition of new observables combining translation and rotation, and translation and strain measurements, respectively. Studying the corresponding sensitivity kernels assesses the capability of the new observables to constrain various aspects of a three-dimensional Earth structure. These observables are generally sensitive only to small-scale near-receiver structures. It follows, for example, that knowledge of deeper Earth structure are not required in tomographic inversions for local structure based on the new observables. Also in the context of deterministic full waveform inversion a new method for the design of seismic observables with focused sensitivity to a target model parameter class, e.g. density structure, is developed. This is achieved through the optimal linear combination of fundamental observables that can be any scalar measurement extracted from seismic recordings. A series of examples illustrate that the resulting optimal observables are able to minimize inter-parameter trade-offs that result from regularization in ill-posed multi-parameter inverse problems. The inclusion of alternative and the design of optimal observables in seismic tomography also affect more general objectives in geoscience. The investigation of the history and the dynamics of tectonic plate motion benefits, for example, from the detailed knowledge of small-scale heterogeneities in the crust and the upper mantle. Optimal observables focusing on density help to independently constrain the Earth's temperature and composition and provide information on convective flow. Moreover, the presented work analyzes for the first time if the inclusion of rotational ground motion measurements enables a more detailed description of earthquake source processes. The complexities of earthquake rupture suggest a probabilistic (or Bayesian) inversion approach. The results of the synthetic study indicate that the incorporation of rotational ground motion recordings can significantly reduce the non-uniqueness in finite source inversions, provided that measurement uncertainties are similar to or below the uncertainties of translational velocity recordings. If this condition is met, the joint processing of rotational and translational ground motion provides more detailed information about earthquake dynamics, including rheological fault properties and friction law parameters. Both are critical e.g. for the reliable assessment of seismic hazards