Electromagnetic subsurface imaging at VLF using distributed optimization [electronic resource]
Keywords: Electrical Engineering.

The purpose of radio remote sensing is to learn more about the environment by observing how the environment interacts with radio waves. Radio remote sensing can be seen as three generic steps: measurement, processing, and interpretation. In this thesis, we discuss methods to address the latter components of radio remote sensing as they pertain to electromagnetic subsurface imaging. We describe a signal processing technique, known as sparse separation, that allows us to decompose the observed data into components that are of scientific value and those that are considered interference. We formulate electromagnetic subsurface imaging as an optimization problem constrained by a set of partial differential equations (PDEs), specifically Maxwell's equations, which govern electromagnetic wave propagation. Algorithms for approximating the solution to these optimization problems are presented and tested numerically. We image conductivities underground with natural sources of electromagnetic radio waves. In the VLF (Very Low Frequency, 3-30kHz) band there are many types of naturally occurring electromagnetic waves. We are primarily interested in a class of electromagnetic waves known as radio atmospherics, or sferics for short. Sferics are short-duration, broadband radio bursts produced by lightning discharges worldwide. Sferics provide three basic advantages for electromagnetic subsurface imaging. First sferics are plentiful; lightning occurs at a rate of 45 discharges per second. Second, because of the global distribution of lightning discharges, we are able to observe sferics incident from multiple different directions of arrival. Each independent incident direction provides more information about the subsurface conductivities. Third, sferics are broadband in nature. Each sferic contains information at multiple frequencies, further augmenting the information we have collected about the subsurface area of interest. The techniques developed in this thesis are designed to take advantage of these three basic properties. To better understand the data, we introduce and develop the technique of sparse separation in an overcomplete dictionary. By solving for a sparse, higher dimensional representation of our data we are able to partition the data into components that are scientifically valuable and components that are considered interference. We develop methods based on soft thresholding, the proximal operator for the sparsity promoting l1 bound. These first order methods scale well, thus permitting the processing of large datasets. Electromagnetic subsurface imaging is formulated as a nonlinear, non-convex, PDE-constrained optimization problem. Using the finite difference frequency domain (FDFD) method for modeling Maxwell's equations on a discrete grid, we can predict the electromagnetic fields everywhere within a computational domain for any wave illumination and any set of scattering conductivities underground. The goal of electromagnetic subsurface imaging is to find an optimal set of conductivities within our model of Maxwell's equations that predict electromagnetic fields that closely match a set of electromagnetic field observations. Many techniques exist for approximating the solution to non-convex optimization problems including sequential linear approximation, alternating projections, and semidefinite relaxations and embeddings. Adaptations of the alternating directions method of multipliers (ADMM) to this non-convex problem have shown the best performance for obtaining the best accuracy and incorporating the multiple frequency and multiple direction of arrival information from sferics in a distributed and scalable manner. Development of these algorithms leads to computational experiments to discover the real-world performance bounds. Adding more information to the problem increases estimation accuracy of subsurface conductivities. We investigate performance as a function of background conductivity and sensor noise. Varying these parameters allows us to predict the influence of basic physics principles and test the algorithms in a variety of possible real-world scenarios. Furthermore, we explore the relationship between the number of sensor measurements necessary and the accuracy we can obtain for determining the subsurface conductivity parameters. Numerical testing leads to a subsampling principle which gives a relationship between accuracy and the number of radio observations.