Geometric Inverse Problems Arising from Seismology

这个项目旨在研究地震学产生的新的几何反问题。通过求解反问题，可以通过间接的、不完整的、经常有噪声的测量来获得关于未知介质的信息。一个常见的例子是地震成像，它使用地震波来探测地球的地下，并被广泛用于石油和天然气勘探以及关键的矿产勘探和地下二氧化碳封存。这种成像技术是基于测量地震波的到达时间和求解通过地球的波速。地震波的速度取决于材料的性质，因此恢复它可以提供有关地球结构的信息。物理学中的费马原理指出，波在两个地点之间走一条路径，可以在最短的时间内传播。因此，波的传播时间定义了一个距离的数学模型，其中两个位置之间的距离是使用时钟而不是尺子来测量的。这种基于物理动机的数学框架在微分几何领域中被普遍研究。这项研究通过提出和解决模拟行星内部的日益现实和复杂的反问题，进一步发展了地震学的数学理论。作为应用地震学的理论基础，本研究中的技术和问题将有助于更好地描述地球、地震事件的传播及其深部结构。此外，该项目将为学生和早期职业研究人员提供培训机会。这个项目集中于推进三类几何反问题，共同的目标是实现更高的物理真实感。这是通过从根本上将几何设置更改为非黎曼设置来实现的。在间接测量的数学理论中，地球通常由带有边界的黎曼流形来建模。在这种假设下，相应的反问题是从边界距离函数恢复黎曼度量，即波在两个边界点之间的传播时间。然而，作为数学框架的黎曼几何往往不足以代表地球物理现实。为了追求物理精度，Finsler度规是各向异性弹性介质中最快的QP极化波的一个很好的几何模型。然而，所有Finsler指标的类别是如此之大，以至于仅靠旅行时间测量不能唯一地确定这些指标。出于这个原因，本项目将研究由线性弹性产生的某些Finsler度量的子类，特别是具有许多黎曼性质的Berwald度量。最重要的是，他们有一种典型的利维-西维塔联系。本研究将解决三类问题：1)具有完整数据集的非黎曼走时问题；2)同时考虑震源和接收者部分数据集的走时问题；3)各向异性介质中的积分几何问题。这些类别的问题都是相互关联和相互支持的。最终，这项工作将引导反问题领域走向越来越现实的地震学情景。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.