Uncertainty Quantification for Source Localization and Passive Imaging in Random Media

This thesis presents work on inverse problems based on partial differential equations (PDEs) skewed toward the inverse source problem. The core of the thesis is divided into two major parts, passive measurements and stochasticity, although they need not be mutually exclusive. Furthermore, only two PDE models will be considered, namely the time-harmonic case based on the Helmholtz equation and the time-dependent one using the Cauchy problem of the wave equation.

Passive measurements are measurements obtained without the possibility of probing the system and are often used in fields where it is infeasible to excite the quantity of interest. The method has successfully been used in seismology and has seen a rise in interest in recent years. This work offers a more analytical approach to passive measurements using cross-correlation.

Additionally, it is well known that reality is too complicated to describe exactly, and approximation is the best we can do. An important tool in describing these approximations is stochastic models. Stochastic models can sufficiently describe the random effects seen in the real world while allowing new phenomena to be discovered. Addressing randomness can be done in various ways; we relied on the theory of stochastic partial differential equations, and when working with randomness, we handled this by assuming various quantities can be modelled by additive and multiplicative random fields.

In particular, we consider the spectral properties for the forward operator of the Helmholtz equation when the model allows for an inhomogeneous and potentially stochastic medium. In doing so, we rely on obtaining a sufficient transformation to guarantee a convergent Neumann series. We then approximate the forward map based on the transformed Neumann series of the Lippmann-Schwinger operator, guaranteeing convergence. The series expansion of the forward map is used to demonstrate a spectral leakage caused by the presence of the medium. Furthermore, we show a stability estimate for the inverse source problem for the Helmholtz equation when only access to the imaginary part of the wave field is obtainable, which amounts to a certain type of passive measurement.

The analysis in the time-dependent setup builds upon the cross-correlation of passive measurement; we obtain a source reconstruction method for the inverse source problem to the Cauchy wave equation in one dimension. We show that the cross-correlation can be viewed as a solution to a wave equation, and one can reconstruct the initial conditions of the cross-correlation from the passive boundary data. Having rebuild the initial conditions for the cross-correlation, solving the wave-equation yields cross-correlation in the entire domain and makes reconstructing the original source function possible. Finally, we provide some numerical results to validate the theoretical outcomes.