This thesis proposes a new proof of the acoustic transparency theorem for material with a bounded variation. The theorem states that if the material properties (density, bulk modulus) is of bounded variation, the net power transmitted through the point z = 0 over a time interval [−T,T] is greater than some constant times the energy at the time zero over a spatial interval [0,Z], provided that T equals the time of travel of a wave from 0 to Z. This means the reflected energy of an input into the earth will be received. Otherwise, the reflections may not arrive at the surface. A proof gives a lower bound for material properties (density, bulk modulus) with bounded variation using sideways energy estimate. A different lower bound that works only for piecewise constant coefficients is also given. It gives a lower bound by analyzing reflections and transmissions of the waves at the jumps of the material properties. This thesis also gives an example to illustrate that the bounded variation assumption may not be necessary for the medium to be transparent. This thesis also discusses relations between the transparency property and the data of an inverse problem.

I propose two Galerkin methods based on the transfer-of-approximation property for static and dynamic acoustic boundary value problems in seismic applications. For problems with heterogeneous coefficients, the polynomial finite element spaces are no longer optimal unless special meshing techniques are employed. The transfer-of-approximation property provides a general framework to construct the optimal approximation subspace on regular grids. The transfer-of-approximation finite element method is theoretically attractive for that it works for both scalar and vectorial elliptic problems. However the numerical cost is prohibitive. To compute each transfer-of-approximation finite element basis, a problem as hard as the original one has to be solved. Furthermore due to the difficulty of basis localization, the resulting stiffness and mass matrices are dense. The 2D harmonic coordinate finite element method (HCFEM) achieves optimal second-order convergence for static and dynamic acoustic boundary value problems with variable coefficients at the cost of solving two auxiliary elliptic boundary value problems. Unlike the conventional FEM, no special domain partitions, adapted to discontinuity surfaces in coe cients, are required in HCFEM to obtain the optimal convergence rate. The resulting sti ness and mass matrices are constructed in a systematic procedure, and have the same sparsity pattern as those in the standard finite element method. Mass-lumping in HCFEM maintains the optimal order of convergence, due to the smoothness property of acoustic solutions in harmonic coordinates, and overcomes the numerical obstacle of inverting the mass matrix every time update, results in an efficient, explicit time step.

The elastic wave equation describes the propagation of elastic disturbances produced by seismic events in the Earth or vibrations in plates and beams. The acoustic wave equation governs the propagation of sound. The description of the wave fields resulting from an initial configuration or time dependent forces is a valuable tool when gaining insight into the effects of the layering of the Earth, the propagation of earthquakes or the behavior of underwater sound. In the most general case exact solutions to both the elastic wave equation and the acoustic wave equation are impossible to construct. Numerical methods that produce approximative solutions to the underlaying equations now become valuable tools. In this thesis we construct numerical solvers for the elastic and acoustic wave equations with focus on stability, high order of accuracy, boundary conditions and geometric flexibility. The numerical solvers are used to study wave boundary interactions and effects of curved geometries. We also compare the methods that we have constructed to other methods for the simulation of elastic and acoustic wave motion.

Low frequency acoustic room behaviour can be approximated using numerical methods. Traditionally, music studio control rooms are built with complex geometries, making their eigenmodes difficult to predict mathematically. Hence, a summation-by-parts method with simultaneous-approximation-terms is derived to approximate the time dependent acoustic wave equation in three dimensions. The derived model is limited to rectangular prismatic rooms but planned to be expanded to handle complex geometries in the future. Semi-reflecting boundary conditions are used, corresponding to tabulated reflection and absorption properties of real. walls. Two speakers are modeled as omnidirectional point sources placed on a boundary, to mimic common studio setups. Through tests and examination of eigenvalues of the matrices in the method, conditions for stability and reflection coefficients are derived. Simulations of sound pressure distribution produced by the model correlate well to room mode theory, suggesting the model to be accurate in the application of predicting low frequency acoustic room behaviour. However, the convergence rate of the model turns out to be lower than expected when point sources are introduced. Future steps towards applying the model to real music studio control rooms include modeling the walls as changes in density and wave speed rather than boundaries of the domain. This would potentially allow more complex geometries to be modeled within a larger, rectangular domain.

This report presents a comprehensive investigation into the accuracy and stability of Finite Difference Methods (FDM) when applied to the acoustic wave equation. The analysis focuses on comparing the classical 2nd order FDM with highly-accurate computational stencil of order 2p = 2,4 and 6 with Summation-by-Parts (SBP) and Simultaneous Approximation Term (SAT) technique of the Finite Difference Method. The objective of the study is to investigate complex numerical techniques that contributes to highly-accurate and stable solutions to many hyperbolic PDEs.  The report starts by introducing the governing problem and studies its well-posedness to ensure stable and unique solutions of the governing equations. It continues with basic introduction to the classic spatial discretization of the FDM and introduces the SBP-SAT implementation of the method. The governing equations are rewritten as a semi-discrete problem, such that it can be written as a system of ordinary differential equations (ODEs) only dependent on the temporal evolution. This system can be solved with classic Runge-Kutta methods to ensure robust and accurate time-stepping schemes.  The results show that the implementation of the higher-order SBP-SAT Finite Difference Method provides highly accurate solutions of the acoustic wave equation compared to the classic FDM. The results also show that the method provides stable solutions with no visible oscillations (dispersion), which can be a challenge for higher order methods. Overall, this paper contributes with valuable insights into the analysis of accuracy and stability in finite difference methods for acoustic wave equation.

Mes travaux de thèse portent sur la construction de conditions aux limites absorbantes (CLAs) pour des problèmes de propagation d'ondes posés dans des milieux limités par des surfaces régulières. Ces conditions sont nouvelles car elles prennent en compte non seulement les ondes proagatives (comme la plupart des CLAs existantes) mais aussi les ondes évanescentes et rampantes. Elles sont donc plus performantes que les conditions existantes. De plus, elles sont facilement implémentables dans un schéma d'éléments finis de type Galerkine Discontinu (DG) et ne modifie pas la condition de stabilité de Courant-Friedrichs-Lewy (CFL). Ces CLAs ont été implémentées dans un code simulant la propagation des ondes acoustiques ainsi que dans un code simulant la propagation des ondes en régime harmonique. Les comparaisons réalisées entre les nouvelles conditions et celles qui sont les plus utilisées dans la littérature montrent que prendre en compte les ondes évanescentes et les ondes rampantes permet de diminuer les réflexions issues de la frontière artificielle et donc de rapprocher la frontière artificielle du bord de l'obstacle. On limite ainsi les coûts de calcul, ce qui est un des avantages de mes travaux. De plus, compte tenu du fait que les nouvelles CLAs sont écrites pour des frontières quelconques, elles permettent de mieux adapter le domaine de calcul à la forme de l'obstacle et permettent ainsi de diminuer encore plus les coûts de calcul numérique. / In my PhD, I have worked on the construction of absorbing boundary conditions (ABCs) designed for wave propagation problems set in domains bounded by regular surfaces. These conditions are new since they take into account not only propagating waves (as most of the existing ABCs) but also evanescent and creeping waves. Therefore, they outperform the existing ABCs. Moreover, they can be easily implemented in a discontinuous Galerkin finite element scheme and they do not change the Courant-Friedrichs-Lewy stability condition. These ABCs have been implemented in two codes that respectively simulate the propagation of acoustic waves and harmonic waves. The comparisons performed between these ABCs and the ABCs mostly used in the litterature show that when we take into account evanescent and creeping waves, we reduce the reflections coming from the artificial boundary. Therefore, thanks to these new ABCs, the artificial boundary can get closer to the obstacle. Consequently, we reduce the computational costs which is one of the advantages of my work. Moreover, since these new ABCs are written for any kind of boundary, we can adapt the shape of the computational domain and thus we can reduce again the computational costs.

Les méthodes d'éléments finis de type Galerkine discontinu (DG FEM) ont démontré précision et efficacité pour résoudre des problèmes d'ondes dans des milieux complexes. Cependant, elles nécessitent un très grand nombre de degrés de liberté, ce qui augmente leur coût de calcul en comparaison du coût des méthodes d'éléments finis continus.Parmi les différentes approches variationnelles pour résoudre les problèmes aux limites, se distingue une famille particulière, basée sur l'utilisation de fonctions tests qui sont des solutions locales exactes des équations à résoudre. L'idée vient de E.Trefftz en 1926 et a depuis été largement développée et généralisée. Les méthodes variationnelles de type Trefftz-DG appliquées aux problèmes d'ondes se réduisent à des intégrales de surface, ce qui devrait contribuer à réduire les coûts de calcul.Les approches de type Trefftz ont été largement développées pour les problèmes harmoniques, mais leur utilisation pour des simulations en domaine transitoire est encore limitée. Quand elles sont appliquées dans le domaine temporel, les méthodes de Trefftz utilisent des maillages qui recouvrent le domaine espace-temps. C'est une des paraticularités de ces méthodes. En effet, les méthodes DG standards conduisent à la construction d'un système semi-discret d'équations différentielles ordinaires en temps qu'on intègre avec un schéma en temps explicite. Mais les méthodes de Trefftz-DG appliquées aux problèmes d'ondes conduisent à résoudre une matrice globale, contenant la discrétisation en espace et en temps, qui est de grande taille et creuse. Cette particularité gêne considérablement le déploiement de cette technologie pour résoudre des problèmes industriels.Dans ce travail, nous développons un environnement Tre#tz-DG pour résoudre des problèmes d'ondes mécaniques, y compris les équations couplées de l'élasto-acoustique. Nous prouvons que les formulations obtenues sont bien posées et nous considérons la difficulté d'inverser la matrice globale en construisant un inverse approché obtenu à partir de la décomposition de la matrice globale en une matrice diagonale par blocs. Cette idée permet de réduire les coûts de calcul mais sa précision est limitée à de petits domaines de calcul. Etant données les limitations de la méthode, nous nous sommes intéressés au potentiel du "Tent Pitcher", en suivant les travaux récents de Gopalakrishnan et al. Il s'agit de construire un maillage espace-temps composé de macro-éléments qui peuvent être traités indépendamment en faisant une hypothèse de causalité. Nous avons obtenu des résultats préliminaires très encourageants qui illustrent bien l'intérêt du Tent Pitcher, en particulier quand il est couplé à une méthode de Trefftz-DG formulée à partir d'intégrales de surface seulement. Dans ce cas, le maillage espace-temps est composé d'éléments qui sont au plus de dimension 3. Il est aussi important de noter que ce cadre se prête à l'utilisation de pas de temps locaux ce qui est un plus pour gagner en précision avec des coûts de calcul réduits. / Discontinuous Finite Element Methods (DG FEM) have proven flexibility and accuracy for solving wave problems in complex media. However, they require a large number of degrees of freedom, which increases the corresponding computational cost compared with that of continuous finite element methods. Among the different variational approaches to solve boundary value problems, there exists a particular family of methods, based on the use of trial functions in the form of exact local solutions of the governing equations. The idea was first proposed by Trefftz in 1926, and since then it has been further developed and generalized. A Trefftz-DG variational formulation applied to wave problems reduces to surface integrals that should contribute to decreasing the computational costs.Trefftz-type approaches have been widely used for time-harmonic problems, while their implementation for time-dependent simulations is still limited. The feature of Trefftz-DG methods applied to time-dependent problems is in the use of space-time meshes. Indeed, standard DG methods lead to the construction of a semi-discrete system of ordinary differential equations in time which are integrated by using an appropriate scheme. But Trefftz-DG methods applied to wave problems lead to a global matrix including time and space discretizations which is huge and sparse. This significantly hampers the deployment of this technology for solving industrial problems.In this work, we develop a Trefftz-DG framework for solving mechanical wave problems including elasto-acoustic equations. We prove that the corresponding formulations are well-posed and we address the issue of solving the global matrix by constructing an approximate inverse obtained from the decomposition of the global matrix into a block-diagonal one. The inversion is then justified under a CFL-type condition. This idea allows for reducing the computational costs but its accuracy is limited to small computational domains. According to the limitations of the method, we have investigated the potential of Tent Pitcher algorithms following the recent works of Gopalakrishnan et al. It consists in constructing a space-time mesh made of patches that can be solved independently under a causality constraint. We have obtained very promising numerical results illustrating the potential of Tent Pitcher in particular when coupled with a Trefftz-DG method involving only surface terms. In this way, the space-time mesh is composed of elements which are 3D objects at most. It is also worth noting that this framework naturally allows for local time-stepping which is a plus to increase the accuracy while decreasing the computational burden.

Dans cette thèse, nous nous intéressons à la modélisation mathématique des hétérogénéités de longueurs caractéristiques beaucoup plus petites que la longueur d'ondes. La thèse consiste en deux parties. La partie théorique est dédiée à l'obtention d'un développement asymptotique raccordé: la solution est décrite à l'aide d'un développement de champ proche au voisinage de l'obstacle et par un développement de champ lointain hors de ce voisinage. Le développement de champ lointain met en jeu des solutions singulières de l'équation des ondes tandis que le champ proche lui est régi par un modèle quasi-statique. Ces deux développements sont alors raccordés dans une zone intermédiaire dite de raccord. Nous obtenons alors des estimations d'erreurs permettant de rendre rigoureux ce développement asymptotique formel. La deuxième partie est numérique. Elle décrit à la fois la méthode de Galerkine discontinue, une méthode de raffinement de maillage espace-temps et propose une discrétisation des modèles asymptotiques obtenues précédemment. Elle est illustrée par un certain nombre de tests numériques. / In this thesis, we focus our attention on the modeling of heterogeneities which are smaller than the wavelength. The document is decomposed into two parts : a theoretical one and a numerical one. In the first part, we derive a matched asymptotic expansion composed of a far-field expansion and a near-field expansion. The terms of the far-field expansion are singular solutions of the wave equation whereas the terms of the near-field expansion satisfy quasistatic problems. These expansions are matched in an intermediate region. We justify mathematically this theory by proving error estimates. In the second part, we describe the Discontinuous Galerkin method, a local time stepping method and the implementation of the matched asymptotic method. Numerical simulations illustrate these results.

The primary focus of this thesis is to examine the detailed seismic structure of the northern Cascadia margin, including the Cascadia basin, the deformation front and the continental shelf. The results of this study are contributing towards understanding sediment deformation and tectonics on this margin. They also have important implications for exploration of hydrocarbons (oil and gas) and natural hazards (submarine landslides, earthquakes, tsunamis, and climate change). The first part of this thesis focuses on the role of gas hydrate in slope failure observed from multibeam bathymetry data on a frontal ridge near the deformation front off Vancouver Island margin using active-source ocean bottom seismometer (OBS) data collected in 2010. Volume estimates (∼ 0.33 km^3) of the slides observed on this margin indicate that these are capable of generating large (∼ 1 − 2 m) tsunamis. Velocity models from travel time inversion of wide angle reflections and refractions recorded on OBSs and vertical incidence single channel seismic (SCS) data were used to estimate gas hydrate concentrations using effective medium modeling. Results indicate a shallow high velocity hydrate layer with a velocity of 2.0 − 2.1 km/s that corresponds to a hydrate concentration of 40% at a depth of 100 m, and a bottom simulating reflector (BSR) at a depth of 265 − 275 m beneath the seafloor (mbsf). These are comparable to drilling results on an adjacent frontal ridge. Margin perpendicular normal faults that extend down to BSR depth were also observed on SCS and bathymetric data, two of which coincide with the sidewalls of the slump indicating that the lateral extent of the slump is controlled by these faults. Analysis of bathymetric data indicates, for the first time, that the glide plane occurs at the same depth as the shallow high velocity layer (100±10 mbsf). In contrast, the glide plane coincides with the depth of the BSR on an adjacent frontal ridge. In either case, our results suggest that the contrast in sediments strengthened by hydrates and overlying or underlying sediments where there is no hydrate is what causing the slope failure on this margin. The second part of this dissertation focuses on obtaining the detailed structure of the Cascadia basin and frontal ridge region using mirror imaging of few widely spaced OBS data. Using only a small airgun source (120 cu. in.), our results indicate structures that were previously not observed on the northern Cascadia margin. Specifically, OBS migration results show dual-vergence structure, which could be related to horizontal compression associated with subduction and low basal shear stress resulting from over-pressure. Understanding the physical and mechanical properties of the basal layer has important implications for understanding earthquakes on this margin. The OBS migrated image also clearly shows the continuity of reflectors which enabled the identification of thrust faults, and also shows the top of the igneous oceanic crust at 5−6 km beneath the seafloor, which were not possible to identify in single-channel and low-fold multi-channel seismic (MCS) data. The last part of this thesis focuses on obtaining detailed seismic structure of the Vancouver Island continental shelf from MCS data using frequency domain viscoacoustic full waveform inversion, which is first of its kind on this margin. Anelastic velocity and attenuation models, derived in this study to subseafloor depths of ∼ 2 km, are useful in understanding the deformation within the Tofino basin sediments, the nature of basement structures and their relationship with underlying accreted terranes such as the Crescent and the Pacific Rim terranes. Specifically, our results indicate a low-velocity zone (LVZ) with a contrast of 200 m/s within the Tofino basin sediment section at a depth 600 − 1000 mbsf over a lateral distance of 10 km. This LVZ is associated with high attenuation values (0.015 − 0.02) and could be a result of over pressured sediments or lithology changes associated with a high porosity layer in this potential hydrocarbon environment. Shallow high velocities of 4 − 5 km/s are observed in the mid-shelf region at depths > 1.5 km, which is interpreted as the shallowest occurrence of the Eocene volcanic Crescent terrane. The sediment velocities sharply increase about 10 km west of Vancouver Island, which probably corresponds to the underlying transition to the Mesozoic marine sedimentary Pacific Rim terrane. High attenuation values of 0.03 − 0.06 are observed at depths > 1 km, which probably corresponds to increased clay content and the presence of mineralized fluids. / Graduate / 0373 / 0372 / 0605 / subbarao@uvic.ca