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

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.