Modeling and inversion of seismic data using multiple scattering, renormalization and homotopy methods

Seismic scattering theory plays an important role in seismic forward modeling and is the theoretical foundation for various seismic imaging methods. Full waveform inversion is a powerful technique for obtaining a high-resolution model of the subsurface. One objective of this thesis is to develop convergent scattering series solutions of the Lippmann-Schwinger equation in strongly scattering media using renormalization and homotopy methods. Other objectives of this thesis are to develop efficient full waveform inversion methods of time-lapse seismic data and, to investigate uncertainty quantification in full waveform inversion for anisotropic elastic media based on integral equation approaches and the iterated extended Kalman filter. The conventional Born scattering series is obtained by expanding the Lippmann-Schwinger equation in terms of an iterative solution based on perturbation theory. Such an expansion assumes weak scattering and may have the problems of convergence in strongly scattering media. This thesis presents two scattering series, referred to as convergent Born series (CBS) and homotopy analysis method (HAM) scattering series for frequency-domain seismic wave modeling. For the convergent Born series, a physical interpretation from the renormalization prospective is given. The homotopy scattering series is derived by using homotopy analysis method, which is based on a convergence control parameter $h$ and a convergence control operator $H$ that one can use to ensure convergence for strongly scattering media. The homotopy scattering scattering series solutions of the Lippmann-Schwinger equation, which is convergent in strongly scattering media. The homotopy scattering series is a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. The Fast Fourier Transform (FFT) is employed for efficient implementation of matrix-vector multiplication for the convergent Born series and the homotopy scattering series. This thesis presents homotopy methods for ray based seismic modeling in strongly anisotropic media. To overcome several limitations of small perturbations and weak anisotropy in obtaining the traveltime approximations in anisotropic media by expanding the anisotropic eikonal equation in terms of the anisotropic parameters and the elliptically anisotropic eikonal equation based on perturbation theory, this study applies the homotopy analysis method to the eikonal equation. Then this thesis presents a retrieved zero-order deformation equation that creates a map from the anisotropic eikonal equation to a linearized partial differential equation system. The new traveltime approximations are derived by using the linear and nonlinear operators in the retrieved zero-order deformation equation. Flexibility on variable anisotropy parameters is naturally incorporated into the linear differential equations, allowing a medium of arbitrarily anisotropy. This thesis investigates efficient target-oriented inversion strategies for improving full waveform inversion of time-lapse seismic data based on extending the distorted Born iterative T-matrix inverse scattering to a local inversion of a small region of interest (e. g. reservoir under production). The target-oriented approach is more efficient for inverting the monitor data. The target-oriented inversion strategy requires properly specifying the wavefield extrapolation operators in the integral equation formulation. By employing the T-matrix and the Gaussian beam based Green’s function, the wavefield extrapolation for the time-lapse inversion is performed in the baseline model from the survey surface to the target region. I demonstrate the method by presenting numerical examples illustrating the sequential and double difference strategies. To quantify the uncertainty and multiparameter trade-off in the full waveform inversion for anisotropic elastic media, this study applies the iterated extended Kalman filter to anisotropic elastic full waveform inversion based on the integral equation method. The sensitivity matrix is an explicit representation with Green’s functions based on the nonlinear inverse scattering theory. Taking the similarity of sequential strategy between the multi-scale frequency domain full waveform inversion and data assimilation with an iterated extended Kalman filter, this study applies the explicit representation of sensitivity matrix to the the framework of Bayesian inference and then estimate the uncertainties in the full waveform inversion. This thesis gives results of numerical tests with examples for anisotropic elastic media. They show that the proposed Bayesian inversion method can provide reasonable reconstruction results for the elastic coefficients of the stiffness tensor and the framework is suitable for accessing the uncertainties and analysis of parameter trade-offs

Exciton gas transport through nano-constrictions

An indirect exciton is a bound state of an electron and a hole in spatially separated layers. Two-dimensional indirect excitons can be created optically in heterostructures containing double quantum wells or atomically thin semiconductors. We study theoretically transmission of such bosonic quasiparticles through nano-constrictions. We show that quantum transport phenomena, e.g., conductance quantization, single-slit diffraction, two-slit interference, and the Talbot effect, are experimentally realizable in systems of indirect excitons. We discuss similarities and differences between these phenomena and their counterparts in electronic devices.Comment: (v2) Updated title, text, and references; 12 pages, 9 figure

Momentum distributions from bichromatic ionization of atoms and molecules

When a sufficiently strong laser field acts on an atom or a molecule, ionization can occur. Electrons released in this process are accelerated by the laser field and the distribution of their final momenta can be measured. As opposed to using linear or circular polarization to drive the ionization process, tailored fields provide additional degrees of freedom to create field shapes with special properties. The present thesis investigates the interaction of atoms and molecules with such fields through numerical calculation of photoelectron momentum distributions, and their application towards a time-resolved picture of strong-field ionization. For a two-color scheme where a weak orthogonal second harmonic is used to probe the ionization process in a strong linearly polarized laser field by observing the modulation of the signal as a function of the delay between the two colors, we solve the time-dependent Schrödinger equation in three dimensions and find the time of ionization resolved by final photoelectron momentum. We demonstrate that the delay scan is sensitive to Coulomb focusing and reveals signatures of photoelectron holography. While two-color schemes can be used to measure ionization times in linear polarization, the attoclock is used in circular polarization. There, the ionization time is inferred from the detection angle of the photoelectron. Because of Coulomb effects, a theoretical model is always required to determine the precise mapping. Contrary to models that are typically used, we obtain this mapping without relying on the notion of an electron trajectory. This is achieved by considering the stationary points of the Dyson integral representation of the time-dependent Schrödinger equation. We find these stationary points using numerical wave function propagation in complex time and confirm that the maximum of the momentum distributions corresponds well to the time of peak field strength. Using a counter-rotating bicircular laser field, the concept of the attoclock can be transferred to other types of polarization. For suitable field strength ratio, the electric field is approximately linearly polarized around the time of peak field strength while the shape of the vector potential is similar to the attoclock. First, we apply the trajectory-free theory to this field to find the most probable time of ionization. Second, we combine the bicircular field with the two-color scheme. This allows us to compare the ionization time measured in the two-color scheme with the one measured in the attoclock. We find that the orthogonal two-color scheme measures ionization time as if the Coulomb potential were not present. However, switching to parallel polarization, we obtain meaningful ionization times in accordance with the attoclock principle that ionization takes place most likely at the peak of the pulse. Applying the bicircular field to a polar molecule, we find that the momentum distribution shows a dependence on the orientation, but this does not imply an orientation dependence of the ionization time

Ion-Atom Collisions: A Time-Dependent Density-Functional-Theory Perspective

Time-dependent density functional theory (TDDFT) is an alternate formulation of time-dependent N-body quantum mechanics which allows one to describe a system via the single-particle density, n, rather than the full N-body wave function. While this reformulation is in theory exact in practice it necessitates at least two approximations. First, the exchange-correlation potential which encodes the two-particle interactions present in the time-dependent Schrdinger equation into the language of the single-particle description is not precisely known. Even if one had perfect knowledge of this potential a further approximation would be required when attempting to extract the values of observables as the exact relation between the one-particle density and most observables of interest is also unknown. This dissertation investigates these issues using ion-atom collision systems as a testbed. First, the observable problem is explored in antiproton-helium, proton-helium, and He^{2+}-He collision systems. Total cross sections for all charge transfer processes in these systems, the observables of choice in the present situation, are determined using a two-centred extension of a correlation-integral model that was originally applied to single-centred situations. Following this theoretical total cross section results for all ionization/capture processes in the He^{+}-He collision system are presented in the approximate impact energy range 10-1000 keV/amu. Calculations were performed within the framework of a spin-dependent extension of TDDFT. These cross sections are used as a benchmark to test an accurate exchange-correlation potential generated via the Krieger-Li-Iafrate approximation applied within the exchange-only limit in which correlation is ignored. The results of two models, one where electron translation factors in the orbitals used to calculate the potential are ignored and another where partial electron translation factors are included, are compared with available experimental data as well as a selection of previous theoretical calculations