Numerical Solution of Next Generation Inverse Problems for Geophysical Applications

我的研究的广泛目标是推进地球物理反演技术，从而改善地球地下的非侵入性调查。地球物理反演生成地下的模型（三维空间表示），这些模型可以从地球物理调查中产生测量数据，例如，在机载系统、地面或钻孔上进行的地震、重力、磁、电和电磁测量。反演是一个计算密集的过程，依赖于描述所涉及的物理现象的微分方程的精确数值解，以及针对手头特定逆问题量身定制的数值优化例程的开发。它是一个跨学科的领域，结合了数学、计算机科学、物理和地球科学的各个方面，但前两者对我的研究具有主要意义。地球物理反演问题通常是非唯一的：由于数据的不确定性以及反演问题的物理现象或数学所固有的其他方面，有无数的模型可以将测量到的地球物理数据拟合到所需的程度。因此，仅凭测量数据不足以找到有意义、可解释和可靠的唯一解决方案。为了减少非唯一性，必须加入额外的信息，提供数学上表现良好的问题，最重要的是，从应用的角度来看，生成与任何现有地下信息一致的解决方案。地球物理反演的中心研究问题之一，也是指导我的研究的问题之一，是如何最好地减少反问题的非唯一性，从而减少所得到的解的模糊性。地球物理反演提供了一种非侵入性的方法，可以提供地下模型，在进行更有侵入性的检查之前，可以用来通知决策者。事实证明，这些技术有助于解决各种应用领域的各种实际问题，包括资源勘探、减轻自然灾害、考古学、农业和许多其他环境和土木工程调查，所有这些在加拿大都很重要。地球物理反演是提高对我们所生活的世界和我们的文明史的认识的基本工具。我的研究项目将开发新的地球物理反演方法，并研究将这些方法应用于各种重要的全球问题的最佳实践。

The broad objective of my research is to advance geophysical inversion technology and thereby improve non-invasive investigation of the Earth's subsurface. Geophysical inversion generates models (three-dimensional spatial representations) of the subsurface that could have given rise to measured data from a geophysical survey, for example, seismic, gravity, magnetic, electrical and electromagnetic measurements taken on airborne systems, at ground level, or down drillholes. Inversion is a computationally intensive procedure, relying on accurate numerical solution of the differential equations that describe the physical phenomena involved, and development of numerical optimization routines tailored to the specific inverse problem at hand. It is very much an interdisciplinary field, combining aspects of mathematics, computer science, physics and Earth sciences, but the former two are of primary significance to my research. Geophysical inverse problems are generally nonunique: because of data uncertainty and other aspects inherent to the physical phenomena or mathematics of the inverse problem, there are an infinite number of models that can fit the measured geophysical data to the desired degree. Hence, the measured data alone are not enough to find a unique solution that is meaningful, interpretable and reliable. Additional information must be incorporated to reduce the nonuniqueness, provide mathematically well-behaved problems and, most importantly from an applied perspective, generate solutions that are consistent with any existing information regarding the subsurface. One of the central research questions in geophysical inversion, and one which guides my research, is how to best reduce the nonuniqueness of the inverse problem and thereby reduce ambiguity in the solutions obtained. Geophysical inversion provides a non-invasive method for providing models of the subsurface that can be used to inform decision makers before more invasive inspection is undertaken. These techniques have proven helpful for addressing a wide variety of practical problems in various fields of application, including resource exploration, natural hazard mitigation, archaeology, agriculture and many other environmental and civil engineering investigations, all of which are important across Canada. Geophysical inversion is a fundamental tool for advancing knowledge of the world we live in and the history of our civilization. My research program will develop new geophysical inversion methods and research best practices for applying those methods to various important worldwide problems.