Collaborative Research: Computational techniques for nonlinear joint inversion

为了管理地下水等自然资源和监测工业垃圾填埋场等污染物，需要准确地表示地球地下的情况。地球物理勘探技术是非侵入性的地下成像技术。在这些方法中，电场被诱导到地下，随后的衰减响应被测量。通过将这些测量结果与反演方法中的物理模型相结合，将其转换为有关地下的信息。通常情况下，这些问题在数学上是病态的，因为测量和数学模型提供了不一致或不完整的信息。该项目将提供一种新的电磁地球物理表征方法，该方法结合了复杂电阻率和探地雷达测量，整合了102 - 109 Hz大范围频段的材料特性。这些信息将结合在一起进行联合反演，提供比传统的地下成像更多的观测信息。我们将通过适当加权测量和实验统计模型来容纳不一致的信息。本项目开发的算法计算效率高，可用于大型数据集或复杂的数学模型，因为它们以现代数值线性代数技术为基础。不适定线性逆问题的正则解由于正则化器的选择和相关权重的影响而得到了广泛的研究。然而，在求解病态非线性逆问题的背景下，在牛顿更新内稳定雅可比反演的影响，它有效地使解正则化，似乎不太受重视。此外，连接一个或多个模型和数据进行联合或多次反演的拉格朗日参数，以及控制反演过程中各分量之间关系的拉格朗日参数，可能会以某种特殊的方式选择。在线性框架中生成一个收敛的解序列的计算代价限制了在非线性框架中对大多数线性方法的认真考虑。该项目通过应用适当的技术，包括基于物理的建模约束，以及基于数据中的底层噪声统计选择正则化参数，改变了相关非线性问题的解决方案。该方法通过强调在求解过程中允许分析内在测量和数值误差传播的求解技术，为将不确定性纳入非线性问题的解决方案开辟了有效途径。因此，潜在的计算算法有可能对这个项目的具体细节产生重大影响。

An accurate representation of the Earth's subsurface is needed to manage natural resources such as groundwater and to monitor pollutants such as those from industrial landfills. Geophysical exploration techniques are non-invasive strategies for imaging the subsurface. In these approaches, electric fields are induced into the subsurface and the subsequent decay response is measured. These measurements are converted into information about the subsurface by combining them with a physical model in an inversion methodology. It is often the case that these problems are mathematically ill-posed because the measurements and mathematical model provide inconsistent or incomplete information. This project will provide a new method of electromagnetic geophysical characterization that combines complex resistivity and ground-penetrating radar measurements, integrating material properties across a vast range of frequency bands: 102 - 109 Hz. This range of information will be combined in a joint inversion that offers more observational information than is traditionally used to image the subsurface. We will accommodate inconsistent information by appropriately weighting measurements and models with experimental statistics. The algorithms developed under this project are computationally efficient and can be used with large data sets or complex mathematical models because they are grounded in modern numerical linear algebra techniques. Regularizing solutions for ill-posed linear inverse problems have been widely studied with respect to the impact of the choice and relevant weighting of applied regularizers. Yet, in the context of the solution of ill-posed nonlinear inverse problems the impact of stabilizing a Jacobian inversion within a Newton update, which effectively regularizes the solution, appears to be less well-appreciated. In addition, Lagrange parameters that connect one or more models and data for joint or multiple inversion, and control the relationship between components of an inversion process, may be chosen in a somewhat ad-hoc manner. The computational cost of generating a convergent sequence of solutions in the linear framework limits serious consideration of most linear approaches in the nonlinear framework. This project transforms the solution of relevant nonlinear problems by applying techniques that appropriately include physically based modeling constraints, and choosing regularization parameters based on underlying noise statistics in data. This methodology opens efficient avenues for incorporating uncertainty in solutions of nonlinear problems by emphasizing solution techniques that permit analysis of the propagation of intrinsic measurement and numerical error through the solution process. Thus the underlying computational algorithms have the potential for significant impact beyond the specifics of this project.