In this thesis a parameter recovery method for use in ultrasound elasticity imaging is developed. Elasticity imaging is a method for using a series of ultrasound images (and the displacement field between them) to estimate the spatial variation of the stiffness of the tissue being imaged. Currently iterative methods are used to do this: a model of tissue mechanics is assumed and a large number of simulations using varying parameters are compared to the actual displacement field. The aim of this work is to develop a solution method that works back from the known displacement field to determine the tissue properties, reducing the number of simulations that must be performed to one. The parameter recovery method is based on the formulation and direct solution of the 2-d linear elasticity inverse problem using finite element methods. The inverse problem is analyzed mathematically and the existence and uniqueness of solutions is described for varying numbers of displacement fields and appropriate boundary conditions. It is shown to be hyperbolic (and so difficult to solve numerically) and then reformulated as a minimization problem with hyperbolic Euler-Lagrange equations. A finite element solution of the minimization problem is developed and implemented. The results of the finite element implementation are shown to work well in recovering the parameters used in numerical simulations of the linear elasticity forward problem so long as these are continuous. The method is shown to be robust in dealing with small errors in displacement estimation and larger errors in the boundary values of the parameters. The method is also tested on displacement fields calculated from series of real ultrasound images. The validity of modelling the ultrasound elasticity imaging process as a 2-d problem is discussed. The assumption of plane strain is shown not to be valid and methods for extending the parameter recovery method to 3 dimensions once 3-d ultrasound becomes more widely used are described (but not implemented).

This thesis presents a Lagrangian formulation of the Euler equations for subsonic flows. A special coordinate transformation is used to define the Lagrangian coordinates, namely the stream function and the Lagrangian distance, in function of the Cartesian coordinates. This Lagrangian formulation introduces two new geometry state variables, and a Lagrangian behavior parameter defining a pseudo-Lagrangian time used during the iteration procedure to obtain the solution for subsonic flows. / The eigenstructure and characteristics analysis for the new system of equations is based on a linear Jacobian matrix-mapping procedure, which starts from the well-known eigenstructure and characteristics in the Eulerian plane and uses the coordinate transformation to find their counterparts in the Lagrangian plane. This analysis studies the basic properties of the Euler equations in the Lagrangian formulation, such as hyperbolicity, homogeneity and rotational invariance. The Riemann problem in the Lagrangian plane is also studied. Those elements are used to construct the numerical scheme for solving the Euler equations in the Lagrangian formulation. / The numerical scheme is constructed using first and second-order dimensional-splitting with hybrid flux operators, based on flux vector splitting and Godunov methods, which include a 2-D Riemann solver in the Lagrangian plane. The numerical method is validated by comparing the present solutions with the results obtained with an Eulerian formulation for several internal flows. / This numerical method based on a Lagrangian formulation has also been extended for the solution of unsteady subsonic flows by using a dual time approach. The method validation in this case has been done by comparison with the Eulerian formulation solutions for several internal subsonic flows with oscillating boundaries.

This thesis concerns two topics: the kinematics of Pacific cross-equatorial flow ??? the location, timing and magnitude of the flow; and their dynamics???what are the driving forces controlling the flow? Despite extensive observations in the central and eastern Pacific, observations of these flows remain contradictory. We use output from an Ocean General Circulation Model (OGCM) viewed from a Lagrangian framework on density layers. This addresses the problem of high variability due to features such as Tropical Instability Waves. The annual mean flow is found to be southward nearly everywhere, east of 140??W. Flow becomes stronger in the second half of the year due to a bolus transport of very light surface water, introduced by Tropical Instability Waves. A Tropical Cell pattern occurs along the equator that does not require diapycnal downwelling. From 160??E to 160??W the annual mean flow is northward, occurring mostly in the mixed layer, appearing to originate partly from the Equatorial Undercurrent surfacing in the east. The northward flow is strongest in March and becomes southward in September. The wind stress and nonlinear terms are shown to be the key driving features, with a prescribed biharmonic Smagorinsky horizontal friction scheme having negligible impact. From 160??E to 160??W, the flow is partly accounted for by an Ekman forcing, with the curl of the nonlinear term providing a crucial additional torque, more than doubling the magnitude in some instances. From 160??W to 120??W the wind stress curl provides a weak southward flow of about 1 Sv, which increases by the nonlinear addition to around 5 Sv. The curl of the steady component of the nonlinear term, derived from annual mean currents, is similar in structure to the total nonlinear term, but higher in magnitude. The structure of the variable term, which was mostly of opposite sign to the steady term, suggests damping occurs in place of friction. While our study is limited to an examination of the model's characteristics, our results provide important clues to the observed flow patterns not resolved by present-day measurements. This study also highlights the importance of time-space variability and both horizontal and vertical density structure in controlling the flow and its feedback on the system.

[Truncated abstract] Estimating internal biogeochemical fluxes is essential to the understanding of the dynamic of aquatic ecosystems. Different ecological approaches have been used to gain insight into the internal cycling, but success has been limited. A critical point is the identification of the characteristic scales of patterns and the underlying processes affecting the behaviour of biological and chemical species. Failing to capture these scales leads to misinterpretation of field and numerical data. In this study, key aspects in the design of ecological surveys are identified to ensure that the internal biogeochemical processes are well represented. In the first part of this thesis, a 1D reaction-diffusion-advection equation is used to investigate the formation of patterns and relevant time and spatial scales. This is used to define an approach for the determination of a critical domain size that allows differentiation of the role of local and internal cycling from advective fluxes across the open boundaries in a shallow coastal ecosystem. By using a 3D numerical model, in conjunction with an extensive field data set, it is shown that domain sizes must be larger than this critical value in order to capture the patterns generated within the system. For smaller domains, transport processes control the evolution of the system across the boundaries misleading the interpretation of the internal ecological dynamics. The study of the influence of boundary fluxes on ecological patchiness was motivated by the need to define the size of the domain necessary for the assessment of the impact of a sewage outflow on a coastal regime. The quantification of biogeochemical processes has proven to be difficult to achieve especially under conditions of high spatial and temporal hydrodynamic and biogeochemical variability. In the second part of this thesis, a Lagrangian experimental design is employed to estimate biogeochemical rate coefficients in situ. A set of four drogues and a cross-transect sampling design is used to capture the patchy distribution of phytoplankton and nutrient species, and high transport and mixing rates. ... Total chlorophyll from both models shows similar behavior when the variability in the 3D model, expressed as Chlamax/Chlamin, is low. When Chlamax/Chlamin is high, the difference between the biomass predicted by the two models reaches 30% due to the generation of localised patches. Comparison of the 1D and 3D results highlights the need of using models that are able to resolve the spatial complexity to some extent, as the use of averaged properties may produce misleading results. This is especially important in the presence of patches with differential physiological and biogeochemical characteristics, and nonlinear processes, in which case biomass average is not necessarily linearly related to the averaged environment.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--Oregon Graduate Institute of Science and Technology, 1997.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

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