**Graduate and Postdoctoral Studies**

**Computational and Applied Mathematics**

2:00 PM to 4:00 PM

1003 Duncan Hall

Accurate representation and estimation of seismic sources is essential to the seismic inversion problem. General sources can be approximated by a truncated series of multipoles, depending on the finiteness and anisotropy of the source. Most research in joint determination of source and medium parameters assumes isotropic point-sources resulting in an inability to fit the anisotropy observed in data, ultimately impacting the recovery of medium parameters. In this thesis I lay the groundwork for joint source-medium parameter inversion with generally anisotropic seismic sources via full waveform inversion through three key contributions: a mathematical and computational framework for the modeling and inversion of general sources via multipoles, construction and analysis of discretizations of multipole sources on regular grids, and preconditioners based on fractional time derivative/integral operators for the ill-conditioned multipole source estimation subproblem. My framework is based on the natural vector space structure, the space of multipoles, that comes from expressing sources as linear combinations of base multipoles parametrized by time-dependent coefficients and is generally applicable to scalar-, vector-, and tensor-valued source terms that appear in the acoustic and elasticity equations in first order form. I develop a flexible object-oriented implementation that encodes these multipole spaces and furthermore follows closely the underlying mathematics. In this thesis I address issues related to discretizing multipoles over uniform grids for finite difference implementations. In particular I extend the method of moment matching conditions for approximating derivatives of the delta function in higher dimensions by sequences of regular distributions that converge in the weak-* topology. Numerical results show that optimal convergence rates of finite difference solutions, when using a proper source approximation, can be achieved away from the source location. The multipole source inversion subproblem poses challenges stemming from ill-conditioning of the source-to-data map. I develop an approach that seeks to better condition the inversion problem from a more fundamental angle: redefine the domain space of the source-to-data map to yield a better bounded operator, thus improving the condition number associated with solving the normal equations that result from a least squares formulation. My preconditioners consist of fractional derivative/integral operators whose order is chosen semi-heuristically based on the analytical solutions of the acoustic wave equation in unbounded media with multipole source term. Numerical experiments conducted demonstrate dramatic accelerations of conjugate gradient iterates and accuracy of estimated sources with preconditioning. As an application of my multipole framework I also study the efficacy of the multipole model on the airgun-array source, the most common type of active source in marine seismic surveying. Synthetic data used for the numerical experiments was generated using open source airgun modeling software that takes into account the physics of airguns and airgun-arrays, in particular the nonlinear source-to-source interactions that occur in array setups. Inversion results recovered a dominating isotropic component of the multipole source model that accounted for 84% of the observed radiation pattern. An extra 10% of the observed output pressure field can be explained when incorporating dipole terms in the source representation, thus motivating the use of multipoles to capture source anisotropy.