Geometric Inverse Problems Arising from Seismology

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This project aims to investigate new geometric inverse problems arising from seismology. By solving inverse problems, information can be gained about an unknown medium via indirect, incomplete, and often noisy measurements. A common example of this is seismic imaging, which uses seismic waves to probe the subsurface of the Earth, and is heavily utilized in oil and gas exploration as well as critical mineral exploration and subsurface CO2 sequestration. This imaging technique is based on measuring the arrival times of seismic waves and solving the wave speed through the planet. The seismic wave speed depends on material properties, and thus recovering it provides information about the structure of the Earth. Fermat’s principle in physics states that a wave takes a path between two locations that can be traveled in the shortest time. Thus, the travel time of a wave defines a mathematical model for a distance, in which the distance between two locations is measured using a clock instead of a ruler. This type of physically-motivated mathematical framework is commonly studied in the field of differential geometry. This research further develops the mathematical theory of seismology by posing and solving increasingly realistic and complex inverse problems modeling planetary interiors. Serving as the theoretical foundation underlying applied seismology, the techniques and problems in this study will contribute towards better characterizing the Earth, it’s transmission of seismic events, and its deep interior structure. Additionally, this project will provide training opportunities for students and early career researchers. This project focuses on advancing three categories of geometric inverse problems with the common objective of achieving increased physical realism. This is done by fundamentally changing the geometric setting to be non-Riemannian. In the mathematical theory of indirect measurements, the Earth is commonly modeled by a Riemannian manifold with a boundary. Under this assumption, the corresponding inverse problem is to recover a Riemannian metric from the boundary distance function, that is the travel time of a wave between a pair of boundary points. However, Riemannian geometry as a mathematical framework is often an insufficient representation of the geophysical reality. To pursue physical accuracy, Finsler metrics are a good geometric model for the fastest qP-polarized waves in anisotropic elastic media. However, the class of all Finsler metrics is so large that travel time measurements alone cannot determine these metrics uniquely. For this reason, this project will investigate certain sub-classes of Finsler metrics arising from linear elasticity, specifically focusing on the Berwald metrics that have many Riemannian properties. Most importantly, they have a canonical Levi-Civita connection. The three categories of problems solved in this research will include: 1) non-Riemannian travel time problems with complete datasets, 2) travel time problems that consider partial datasets for both the sources and receivers, and 3) integral geometric problems in anisotropic media. These categories of problems are all interconnected and mutually supportive. Ultimately this work will guide the field of inverse problems towards increasingly realistic seismological scenarios.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.