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

有非常广泛的应用，其中声波被用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中的超声波扫描，其中高频声波从您的身体内部捕获实时图像。另一个重要的应用领域是地球科学，在地球表面测量的振动用于提取地球内部结构或过程的信息，这被称为地震成像。这些方法的重要用途包括地震或海啸预警系统，以及确定地质结构以确定地下石油、天然气或其他资源的位置。所有这些成像技术都依赖于基于数学的计算算法。为了准确地理解成像方法在特定情况下的工作情况，可以应用数学分析。不同的分析可以一方面应用于计算算法，另一方面应用于物理波传播本身，两者的目的都是为了了解如何准确和有效地从声学数据重建图像。为了形成成像过程的完整图像，不仅必须分别分析这两个方面（计算和物理），而且必须使这两个分析相匹配，以便使用手头的物理问题所设置的参数来优化计算算法。这是一个雄心勃勃的计划，需要了解物理和计算过程所固有的稳定性。本项目的目标是在地震成像的背景下实现这一目标。特别是，我们的目标是了解成像的准确性是如何受到地下环境的异质性的影响：地球由不同类型的材料组成，这些材料被裂缝包裹。我们希望重建的量通常是波的来源，也就是初始振动的振幅和位置。这是分析地震的关键数据。在这种情况下，震源问题由于地震波是由断层线上的非线性过程引发的这一事实而进一步复杂化。通常只计算源的总能量。所提出的方法的承诺是通过利用它受到摩擦定律约束的事实来恢复源上的精细信息。然而，应该强调的是，该项目的目的并不是将计划的方法直接应用于实际的地球物理成像问题，而是为了证明该方法的可行性，将结果传达给地球物理学家，并让他们采用该方法。在整个项目中，将有一个数学分析和计算方法的平行发展。最终目标是交付概念验证计算软件，从准确性的角度来看，该软件可以证明返回最佳成像结果。

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.