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Geometric Inverse Problems Arising from Seismology

地震学引起的几何反问题

基本信息

批准号：
2204997
负责人：
Teemu Saksala
金额：
$ 17万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204997&HistoricalAwards=false
关键词：
Geometric Inverse Problems Arising Seismology

项目摘要

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.

本计画旨在探讨地震学中新的几何反问题。通过求解逆问题，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，它使用地震波探测地球的地下，并在石油和天然气勘探以及关键的矿物勘探和地下CO2封存中得到大量利用。这种成像技术的基础是测量地震波的到达时间，并求解穿过行星的波速。地震波的速度取决于材料的性质，因此恢复它提供了有关地球结构的信息。物理学中的费马原理指出，波在两个位置之间的路径可以在最短的时间内传播。因此，波的传播时间定义了距离的数学模型，其中两个位置之间的距离是使用时钟而不是尺子测量的。这种物理驱动的数学框架通常在微分几何领域中进行研究。这项研究进一步发展了地震学的数学理论，提出和解决越来越现实和复杂的逆问题建模行星内部。作为应用地震学的理论基础，这项研究中的技术和问题将有助于更好地描述地球，它的地震事件的传输，以及它的深部内部结构。此外，该项目将为学生和早期职业研究人员提供培训机会。该项目的重点是推进三个类别的几何逆问题，实现增加物理现实主义的共同目标。这是通过从根本上改变几何设置为非黎曼来完成的。在间接测量的数学理论中，地球通常被建模为具有边界的黎曼流形。在此假设下，相应的反问题是从边界距离函数恢复黎曼度量，即波在一对边界点之间的旅行时间。然而，黎曼几何作为一个数学框架往往是一个不充分的地球物理现实的代表。为了追求物理精度，Finsler度量是各向异性弹性介质中最快qP偏振波的一个很好的几何模型。然而，类的所有芬斯勒指标是如此之大，旅行时间测量不能单独确定这些指标唯一。由于这个原因，这个项目将调查Finsler度量的某些子类产生的线性弹性，特别是专注于Berwald度量，有许多黎曼性质。最重要的是，他们有一个典型的列维奇维塔连接。在这项研究中解决的三个类别的问题将包括：1）完整数据集的非黎曼旅行时间问题，2）旅行时间问题，考虑部分数据集的源和接收器，和3）各向异性介质中的积分几何问题。这几类问题都是相互关联和相互支持的。最终，这项工作将指导反问题领域朝着越来越现实的地震学场景发展。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。