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，需要准确表示地球的地下情况。 地球物理勘探技术是用于对地下成像的非侵入性策略。 在这些方法中，电场被感应到地下，并测量随后的衰减响应。 通过将这些测量值与反演方法中的物理模型相结合，将其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，该方法将复电阻率和探地雷达测量相结合，整合了102 - 109 Hz宽频带范围内的材料特性。 这一范围的信息将结合在一个联合反演，提供更多的观测信息比传统上用于成像的地下。 我们将通过适当加权测量和模型与实验统计来适应不一致的信息。 该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，因为它们基于现代数值线性代数技术。 关于应用正则化子的选择和相关权重的影响，不适定线性逆问题的正则化解已得到广泛研究。然而，在不适定的非线性反问题的解决方案的背景下，稳定的雅可比反演牛顿更新，有效地正则化的解决方案，似乎是不太好的赞赏。 此外，可以以某种特别的方式选择连接用于联合或多重反演的一个或多个模型和数据并且控制反演过程的分量之间的关系的拉格朗日参数。 在线性框架中生成一个收敛的解序列的计算成本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包括基于物理的建模约束的技术，并根据数据中的潜在噪声统计选择正则化参数，来转换相关非线性问题的解决方案。 这种方法开辟了有效的途径，将不确定性的非线性问题的解决方案，强调解决方案的技术，允许通过解决方案的过程中的固有测量和数值误差的传播分析。 因此，底层的计算算法有可能产生超出本项目具体范围的重大影响。

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.