On the Construction of Virtual Interior Point Source Travel Time Distances from the Hyperbolic Neumann-to-Dirichlet Map

We introduce a new algorithm to construct travel time distances between a point in the interior of a Riemannian manifold and points on the boundary of the manifold, and describe a numerical implementation of the algorithm. It is known that the travel time distances for all interior points determine the Riemannian manifold in a stable manner. We do not assume that there are sources or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map, or its restriction, as our data. Our algorithm is a variant of the Boundary Control method, and to our knowledge, this is the first numerical implementation of the method in a geometric setting

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain, and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semi-geodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.Comment: 24 pages, 6 figure

MACHINAL: SILENCE, STAGE DIRECTIONS AND SOPHIE TREADWELL

This thesis explores the legacy of silence that surrounds Sophie Treadwell and her work. It does so by investigating the abundant and poetic stage directions she provides within her play "Machinal"

Healthy Work Environment and Outcome Improvements for Nursing

The purpose of this study was to educate registered nurses about healthy work environments and the importance the relationship of the environment has on nursing outcomes. The project was conducted online using Qualtrics survey software. Data was collected from November 11th, 2021, to December 4th, 2021. Registered nurses currently working in a southeast Kansas hospital participated by completing a two-part survey. The survey consisted of a pretest, an educational resource, and finished with a post-test identical to the pretest except for the demographic questions collected in the pre-test. The data collected from completed surveys were used to measure participant knowledge and perceptions. The study demonstrated an increase in the participants knowledge and perception about healthy work environments through the overall mean scores, but the paired samples t-test only found statistical significance with two of the survey questions. Overall, the projects findings over healthy work environments would indicate that the replication of this study with a larger sample size would be beneficial to further support the data gathered

Techniques for Reconstructing a Riemannian Metric Via the Boundary Control Method

In this dissertation, we consider some new techniques related to the solution of the inverse boundary value problem for the wave equation with partial boundary data. Most results are formulated in a geometric setting, where waves propagate in the interior of a smooth manifold with smooth boundary M, and the wave speed is modelled by an unknown Riemannian metric g. For data, we focus mostly on using the Neumann-to-Dirichlet (N-to-D) map with sources and receivers restricted to a measurement set Γ ⊂ ∂M. The goal of the inverse problem, in this setting, is to use these wave boundary measurements to recover the geometry of (M, g) near the measurement set. We note that this geometric perspective accomodates, as special cases, both the scalar acoustic wave equation and elliptically anisotropic wave speeds. We consider three problems. In the first problem, we provide a technique to use the N-to-D map to construct the travel times between interior points with known semi-geodesic coordinates and boundary points belonging to Γ. Such travel times can be used to reconstruct the metric in semi-geodesic coordinates using one of several existing techniques, so this procedure can be viewed as providing a data processing step for a metric reconstruction procedure. In the second problem, we consider a redatuming procedure, where we use data on the boundary and known near-boundary geometry to synthesize wave measurements in this known near-boundary region. This allows us to construct a map which plays a similar role to the N-to-D map, but for interior sources and interior measurements. Our motivation for this procedure is that it can serve as a data propagation step for a layer stripping reconstruction method, in which one first reconstructs the metric near the boundary and then propagates data into this region to serve as data for an interior reconstruction step. In the third problem, we restrict attention to the case where M is a domain in Rn, and consider two related procedures to use the N-to-D map or Dirichlet-to-Neumann (D-to-N) map to directly reconstruct the metric. In the anisotropic case, we construct the metric in semi-geodesic coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case, we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. In addition to providing constructive procedures, we analyze the stability of some steps from these procedures. In particular we consider the stability of the redatuming procedure and the stability of the metric reconstruction procedure from internal data (for the third problem). Moreover, we provide computational experiments to demonstrate our three main procedures