Collaborative Research: Computational techniques for nonlinear joint inversion

为了管理地下水等自然资源和监测来自工业垃圾填埋场的污染物，需要准确地表示地球的地下部分。地球物理勘探技术是对地下进行成像的非侵入性策略。在这些方法中，电场被诱导到亚表面，并测量随后的衰减响应。在反演方法中，通过将这些测量与物理模型相结合，这些测量被转换为关于地下的信息。通常情况下，这些问题在数学上是不适定的，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，将复电阻率测量和探地雷达测量相结合，在102-109赫兹的大范围内综合材料特性。这一范围的信息将被组合在一个联合反演中，提供比传统用于成像地下的更多的观测信息。我们将通过用实验统计适当地加权测量和模型来容纳不一致的信息。在该项目下开发的算法在计算上是高效的，可以用于大型数据集或复杂的数学模型，因为它们基于现代数值线性代数技术。不适定线性反问题的正则化解已被广泛研究，主要涉及应用正则化子的选择和相关权重的影响。然而，在解不适定的非线性逆问题的背景下，将雅可比逆稳定在牛顿修正内，从而有效地正则化解的影响似乎没有那么好地被认识到。此外，可以以某种特别的方式选择拉格朗日参数，该拉格朗日参数连接用于联合或多个反演的一个或多个模型和数据，并控制反演过程各组成部分之间的关系。在线性框架中生成收敛解序列的计算成本限制了对非线性框架中大多数线性方法的认真考虑。这个项目通过应用适当地包括基于物理的建模约束的技术，以及基于数据中潜在的噪声统计来选择正则化参数，来转换相关非线性问题的解决方案。这种方法为将不确定性纳入非线性问题的解决方案开辟了有效的途径，强调了允许分析内在测量和数值误差在求解过程中传播的求解技术。因此，基本的计算算法有可能产生重大影响，超出本项目的具体要求。

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.