Algorithms for Assessing and Improving Joint Inversion

了解地球次表层的结构对现代社会至关重要。例如，这种知识有助于建造安全的结构，使我们能够定位矿物、碳氢化合物、地下水和污染物；绘制隧道、管道和矿山的地图。地球次表层的结构可以通过岩石、土壤和水等物质特性来了解。但是，这些特性不能在地下直接测量。因此，非侵入性的地球物理测量被用来观察它们。这包括测量电磁波或电流注入地下后地球表面的能量响应。从这些能量观测中推断地下图像的数学方法被称为反演。恢复地下图像的主要挑战是没有一个单一的图像是由给定的能量响应产生的。最近，科学家们通过使用多种类型的能量来恢复图像来解决这个问题，这被称为联合反转。在这个项目中，PI使用联合反演来结合来自大范围频率范围的能量转移的观测。PI假设，利用来自大频率范围的信息，可以创建准确地表示地下表面的唯一图像。这种方法可以扩展到其他领域，如无线通信和视频处理。因此，PI还将开发一个软件包，用于确定特定数据类型是否互补。同时联合反演涉及利用来自多种类型数据的信息来优化单个目标函数。PI将结合地球地下的复电阻率(ER)和探地雷达(GPR)测量，并确定使用这两种数据的有效性。该方法确定描述不同类型数据收集技术的物理学是否以及如何有助于消除对方的零空间。更广泛地说，PI将决定不同类型的数据如何才能最有效地相互规范。这将通过量化描述ER和GPR数据基本物理的积分解的奇异值的衰减率来实现。这需要开发和实施一个框架，通过该框架可以在连续设置下计算联合反演中的奇异值。衰减率和有效联合反演之间的关系将用博伊西水文研究基地的ER和GPR现场数据进行检验。该项目通过在每个领域的新实现和分析来帮助弥合正反向建模之间的差距。

Understanding the structure of the earth's subsurface is essential to modern society. For example, this knowledge aids in building safe structures, allows us to locate minerals, hydrocarbons, groundwater and contaminants; map tunnels, pipes and mines. The structure of the earth's subsurface can be understood by its material properties such as rock, soil and water. However, these properties cannot be measured directly in the subsurface. Therefore, non-invasive geophysical measurements are used to observe them. This involves measuring an energy response at the earth's surface, after electromagnetic waves or electric currents are injected into the subsurface. The mathematical approach to inferring an image of the subsurface from these energy observations is called inversion. The major challenge in recovering a subsurface image is that no one single image results from a given energy response. Recently, scientists have addressed this issue by using multiple types of energy to recover an image, which is referred to as joint inversion. In this project, the PIs use joint inversion to combine observations of energy transfer from a large range of frequencies. The PIs hypothesize that with information from a large frequency range, a unique image can be created that is an accurate representation of the subsurface. This approach can be extended to other fields such as wireless communication and video processing. Therefore, the PIs will also develop a software package that can be used to decide if particular data types are complimentary.Simultaneous joint inversion involves optimizing a single objective function with information from multiple types of data. The PIs will combine complex electrical resistivity (ER) and ground penetrating radar (GPR) measurements in the Earth's subsurface, and determine the effectiveness of using both types of data. The approach determines if and how the physics describing different types of data collection techniques contribute to abolishing the other's null space. More generally, the PIs will determine how different types of data can most effectively regularize each other. This will be done by quantifying decay rates of singular values of integral solutions describing the fundamental physics for ER and GPR data. This requires the development and implementation of a framework through which the singular values in a joint inversion can be computed in a continuous setting. The relationship between decay rates and an effective joint inversion will be tested with ER and GPR field data from the Boise Hydrological Research Site. This project helps bridge the gap between forward and inverse modeling by novel implementations and analyses in each field.