Stanford Earth imaging Project
Thesis

Selected topics in seismic imaging and algebra SEP-192 (2024)

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Rahul Sarkar Thesis (34.71 MB)

Table of contents

  • Chapter 1: Introduction
  • Part I: Lippmann-Schwinger equation & its applications in seismic imaging
    • Chapter 2: Lippmann-Schwinger equation with horizontal homogeneity
    • Chapter 3: Waveform inversion with the Lippmann-Schwinger equation constraint
    • Chapter 4: Application to a real data set
  • Part II: Time-dependent Born scattering
    • Chapter 5: Born scattering from a time-dependent perturbation
    • Chapter 6: Microlocal analysis of singularities in time-dependent Born scattering
  • Part III: Some problems in algebra and analysis
    • Chapter 7: Index of invariance
    • Chapter 8: Density theorems with applications in quantum signal processing
    • Chapter 9: Maximal sets of commuting and anticommuting Pauli operators
    • Chapter 10: Some structural properties of the qudit Pauli group

Abstract

This thesis primarily comprises of two distinct themes and broad application areas in mathematics: the first being the topic of wave propagation problems and their applications in seismic imaging, and the second being algebra and its applications in the development of iterative linear equation solvers and quantum information theory. Part I and Part II of the thesis deal with the first topic, while Part III covers the latter category.

The key focus of Part I, starting in Chapter 2, is a generalization of a numerical scheme for solving the Lippmann-Schwinger equation, known as the truncated kernel method, to accommodate the case of background sound speeds which are either a linearly increasing function or a more general function of a single coordinate. This setting is much more natural for real world problems, arising from applications in seismic imaging and inversion. The methods discussed and developed in this part of the thesis pertain to frequency domain techniques, and they are particularly relevant today in the context of recent advances in acquisition systems, such as marine resonators and vibrators capable of producing sound waves, either at a few discrete frequencies or within a narrow frequency range. Several strategies to efficiently implement the integral kernel of the Lippmann-Schwinger equation, and how to use symmetries to reduce the pre-computation and storage costs of the necessary Green’s functions are discussed, along with some numerical experiments. In Chapter 3, we develop an alternating minimization algorithm for solving the waveform inversion problem, posed as a joint optimization problem over perturbations in slowness squared and the wavefields, with the Lippmann-Schwinger equation acting as the constraint between these two sets of physical quantities. The inversion algorithm is demonstrated on a synthetic 2D example for the inversion of guided waves. Finally, in Chapter 4, we perform 3D inversion of direct arrivals of a distributed acoustic sensing real data set, using the same constrained inversion algorithm.

In Part II of the thesis, in Chapter 5, we develop a new class of image extensions, called the time-dependent Born scattering extension, which is closely related to the time-lag extension of Biondi and Almonin, but have important differences. Based on this extension, we introduce the concept of time-dependent Born scattering common-image gathers as a tool for detecting sound speed errors, which have a potential for future use in extended inversion algorithms. To support this, we demonstrate that this extended model is reasonably effective at modeling residuals in various settings, a property widely considered to be a key requirement for the success of extended waveform inversion algorithms. In the last part of this chapter, and continuing into Chapter 6, we undertake the study of the normal operator, first demonstrating numerically that it is different in nature as compared to the normal operator corresponding to the time-independent Born scattering problem, in the sense that it does not preserve singularities, leading to the formation of artifacts during the imaging process. This effect is studied in Chapter 6 using Hörmander’s machinery of wavefront sets of distributions and other known facts from microlocal analysis.

Part III of the thesis is dedicated to problems arising in linear algebra and quantum information theory. In Chapter 7, we generalize a result of Hallman and Gu, in connection to an iterative Krylov subspace based method called LSMB, where they showed that the solutions to the LSMB objective function lie on a straight line. In order to generalize this result, we introduce a quantity called the index of invariance, and then proceed to show that there is a specific optimization problem, which when posed over arbitrary subspaces, has the property that the solutions lie in a low-dimensional affine subspace, whose dimension is upper bounded by the index of invariance. From this, one recovers the result of Hallman and Gu as a special case. We also study in considerable detail the question of how tight is this upper bound, and this leads to some surprising conclusions. In Chapter 8, we prove a generalization of Weierstrass approximation theorem in the presence of constraints on polynomials outside the domain of approximation, in connection to constructibility questions arising in quantum signal processing. Chapter 9 is dedicated to the study of maximal sets of commuting and anticommuting (qubit) Pauli operators, and provides necessary and sufficient conditions for sets to have this maximality property. We also furnish an efficient randomized algorithm that takes any non-maximal anticommuting set, and extends it to its maximum possible size. Finally, in Chapter 10, we study some structural properties of the qudit Pauli group, motivated by a question of minimizing qudit resource requirements for implementing quantum Hamiltonian simulation algorithms and some classes of quantum error-correction protocols. Our results generalize known results in the literature to the case when the qudit dimensions are composite. We prove non-trivial results about maximum sizes of non-commuting pairs of Paulis, maximum sizes of non-commuting sets, and also elucidate some group theoretic properties of the qudit Pauli group, with potential applications in the development of qudit stabilizer codes.

Reproducibility and source codes

This thesis has been tested for reproducibility. The source code and reproducibility steps are available at these GitHub repositories for the following chapters:

 

Author(s)
Rahul Sarkar
Publication Date
September, 2024

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