Computational methods for inverse problems subject to wave equations in heterogeneous media

异质介质中波动方程反问题的计算方法

基本信息

批准号：
EP/V050400/1
负责人：
Erik Burman
金额：
$ 68.06万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV050400%2F1
关键词：
Computational methods inverse problems subject

项目摘要

There are a very wide variety of applications where sound waves are used to provide information about physical processes, such approaches are known as Acoustic imaging. A well known example is ultrasound scans in medical science, where high-frequency sound waves captures live images from the inside of your body. Another important field of application is geoscience, where vibrations measured on the earths surface are used to extract information on structures or processes inside the earth, this is known as Seismic imaging. Important uses for such methods include warning systems for earthquakes or tsunamis and the identification of geological structures with the purpose of locating underground oil, gas, or other resources. All these imaging techniques rely on computational algorithms based on mathematics. To understand precisely how well an imaging method works in a certain situation one can apply a mathematical analysis. Different analyses can be applied on the one hand to the computational algorithm and on the other to the physical wave propagation itself, both with the purpose of seeing how accurately and efficiently an image is reconstructed from the acoustic data. To form a complete picture of the imaging process not only must these two aspects (computational and physical) be analysed separately, but the two analyses must be made to match so that the computational algorithm is optimised using the parameters set by the physical problems at hand. This is an ambitious programme that requires understanding both of the stability properties inherent to the physical and computational processes. The objective of the present project is to realise this goal in the context of seismic imaging. In particular we aim to understand how the accuracy of the imaging is influenced by the heterogeneous nature of the subsurface environment: the earth consists of different types of material intersected by fractures. The quantity that we wish to reconstruct is typically the source of the wave, that is what was the amplitude of and position of the initial vibration. This is a key data for the analysis of earthquakes. In that case, the source problem is further complicated by the fact that the seismic wave is initiated by a nonlinear process on the fault line. Often only the total energy of the source is computed. The promise of the proposed method is to recover refined information on the source by exploiting the fact that it is constrained by the friction law. It should be stressed, however, that the project does not aim to apply the planned method directly to practical geophysical imaging problems, rather the aim is to demonstrate the feasibility of the method, communicate the results to geophysicists, and get them to adopt the method. Throughout the project there will be a parallel development of mathematical analysis and computational methodology. The final aim is delivery of proof of concept computational software that returns, provably, the best imaging result possible from the point of view of accuracy.

在多种应用中，使用声波来提供有关物理过程的信息，这种方法被称为声学成像。一个众所周知的例子是医学上的超声扫描，高频声波从您的身体内部捕获了现场图像。应用的另一个重要领域是地球科学，在地球表面测量的振动用于提取有关地球内部结构或过程的信息，这被称为地震成像。此类方法的重要用途包括用于地震或海啸的警告系统以及识别地质结构，目的是找到地下油，天然气或其他资源。所有这些成像技术都取决于基于数学的计算算法。为了确切地了解成像方法在某种情况下的工作状况如何，可以应用数学分析。一方面可以将不同的分析应用于计算算法，另一方面可以应用于物理波传播本身，既是为了查看从声学数据中重建图像的准确和有效性。要形成成像过程的完整图片，不仅必须分别分析这两个方面（计算和物理），而且必须对两个分析进行匹配，以便使用手头物理问题设置的参数优化了计算算法。这是一个雄心勃勃的程序，需要了解物理和计算过程固有的稳定性属性。本项目的目的是在地震成像的背景下实现这一目标。特别是我们旨在了解成像的准确性如何受地下环境的异质性影响：地球由裂缝相交的不同类型的材料组成。我们希望重建的数量通常是波的来源，这就是初始振动的幅度和位置。这是分析地震的关键数据。在这种情况下，由于断层线上的非线性过程引发了地震波，因此源问题更加复杂。通常，仅计算源的总能量。提出的方法的承诺是通过利用它受摩擦法约束的事实来恢复有关来源的精致信息。但是，应该强调的是，该项目并不是要直接将计划的方法直接应用于实用的地球物理成像问题，而是目的是证明该方法的可行性，将结果传达给地球物理学家，并使他们采用该方法。在整个项目中，数学分析和计算方法论将有一个平行的发展。最终目的是提供概念计算软件的证明，从准确的角度来看，可以证明这是最佳成像结果。