Learning from Manifold-Valued Data: An Application to Seismic Signal Processing

Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. Over the past several years, advances in sensor technology has lead to increases in the demand for computerized methods for analyzing seismological signals. Central to the understanding of the mechanisms generating seismic signals is the knowledge of the phases of seismic waves. Being able to specify the type of wave leads to better performing seismic early warning systems and can also aid in nuclear weapons testing ban treaty verification. In this thesis, we propose a new method for the classification of seismic waves measured from a three-channel seismograms. The seismograms are divided into overlapping time windows, where each time-window is mapped to a set of multi-scale three-dimensional unitary vectors that describe the orientation of the seismic wave present in the window at several physical scales. The problem of classifying seismic waves becomes one of classifying points on several two-dimensional unit spheres. We solve this problem by using kernel-based machine learning methods that are uniquely adapted to the geometry of the sphere. The classification of the seismic wave relies on our ability to learn the boundaries between sets of points on the spheres associated with the different types of seismic waves. At each signal scale, we define a notion of uncertainty attached to the classification that takes into account the geometry of the distribution of samples on the sphere. Finally, we combine the classification results obtained at each scale into a unique label. We validate our approach using a dataset of seismic events that occurred in Idaho, Montana, Wyoming, and Utah, between 2005 and 2006. Dedication To my wife. v Acknowledgements I would like to express my gratitude to my research advisor, Professor François G. Meyer, for his guidance, patience, and inspiration. Working with him has truly been a life changing experience. I hope to one day follow in his footsteps and become a great teacher and mentor. I would also like to thank Professor Shannon M. Hughes and Professor Juan G. Restrepo for being part of my thesis committee. Finally, I would like to thank my family members and friends for their encouragement and support.