Algorithms for Assessing and Improving Joint Inversion

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.