The propagation of seismic waves may be described in the space-frequency domain by the Rayleigh-Sommerfeld convolution integral. The kernel of this integral is called a spatial wavelet and it embodies the physics and geometry of the propagation problem. The concepts of spatial convolution and spatial wavelet are simple and are similar to other topics studied by geophysicists. With a view to understanding these concepts, some aspects of spatial wavelets and their application to two-dimensional, zero-offset, acoustic seismic modelling were investigated. In studying the spatial wavelet, two topics in particular were examined: spatial aliasing and wavelet truncation. Spatial aliasing arises from the need to compute a discrete wavelet for implementation on a computer. This problem was solved by using an analytic expression for the spatial wavelet in the Fourier (wavenumber) domain. In the wavenumber domain the wavelet was windowed by a fourth order Butterworth operator, which removed aliasing. This technique is simple and flexible in its use. The second problem of wavelet truncation is due to the necessity of having a wavelet of finite length. A length limiting scheme based upon on the energy content of a wavelet was developed. It was argued that if that if a large portion of the wavelet energy was contained in a finite number of samples, then truncation at that sample would incur a minimal loss of information. Numerical experiments showed this to be true. The smallest length wavelet was found to depend on temporal frequency, medium velocity and extrapolation increment. The combined effects of these two solutions to the practical problem of computing a spatial wavelet resulted in two drawbacks. First, the wavelets provide modelling capabilities up to structural dips of 30 degrees. Second, there is a potential for instability due to recursive application of the wavelet. However, neither of these difficulties hampered the modelling of fairly complex structures. The spatial wavelet concept was applied to seismic modelling for media of varying complexity. Homogeneous velocity models were used to demonstrate diffraction evolution, dip limitations and imaging of curved structures. The quality of modelling was evaluated by migrating the modelled data to recover the time-image model of the reflection structure. Migrations of dipping and synform structures indicated that the modelled results were of a high calibre. Horizontally stratified velocity models were also examined for dipping and synform structures. Modelling these reflection structures showed that the introduction of a depth variable velocity profile has a tremendous influence on the synthetic seismic section. Again, migration proved that the quality of the data was excellent. Finally, the spatial wavelet algorithm was extended to the case of laterally varying velocity structures. The effects of space variant spatial convolution in the presence of a smoothed velocity field were examined. Smoothed velocity fields were computed by a simple weighted averaging procedure. The weighting function used was a decaying exponential whose decay rate determined the amount of smoothing. Seis-mograms computed for this case showed that the algorithm gave smoother and more continuous reflection signatures when the velocity field has been smoothed so that the largest lateral velocity gradient corresponded to the lower end of the temporal frequency band of the spatial wavelets. In this respect, the results are similar to those of geometric ray theory. Also, the travel times of these models compared favourably with those of ray tracings. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

This thesis is concerned with the problem of estimating broadband acoustic impedance from normal incidence reflection seismograms. This topic is covered by following the linear inverse formalisms described by Parker (1977) and Oldenburg (1980). The measured seismogram is modelled as a convolution of subsurface reflectivity with a source wavelet. Then an appraisal of the seismogram is performed to obtain unique bandlimited reflectivity information. This bandlimited reflecitivity information is then utilized in two different construction algorithms which provide a broadband estimate of reflectivity; from which a broadband impedance function may be computed. The first construction method is a maximum entropy method which uses an autoregressive representation of a small portion of the reflectivity spectrum to predict spectral values outside that small portion. The second and most versatile construction method is the linear programming approach of Levy and Fullagar (1981) which utilizes the unique bandlimited spectral information obtained from an appraisal and provides a broadband reflectivity function which has a minimum 1( norm. Both methods have been tested on synthetic and real seismic data and have shown good success at recovering interpretable broadband impedance models. Errors in the data and the uniqueness of constructed reflectivity models play important roles in estimating the impedance function and in assessing its uniqueness. The Karhunen-Loeve transformation is discussed and applied on real data to stabilize the construction results in the presence of noise. The generally accepted idea that low frequency impedance information must be supplied from well log or velocity analyses because of the bandlimited nature of seismic data has been challenged. When accurate, bandlimited reflectivity information can be recovered from the seismic trace, then an interpretable, broadband impedance model may be recovered using the two construction algorithms presented in this thesis. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

Thesis advisor: Vincent Cho / Leading for equity is a challenging endeavor. One leadership practice that fosters equitable learning environments is engaging in dialogue and reflection. When district leaders participate in dialogue and reflection, their discourse helps them derive meaning, and in turn, shapes their understanding of the critical and complex issues related to fostering equity. As part of a group qualitative case study about district leadership practices that foster equity in one diverse Massachusetts school district, the purpose of this individual study was to better understand how district leaders used framing during dialogue and reflection. More specifically it addressed how they used framing processes (Bedford and Snow, 2000) when engaging in equity talk. Utilizing inductive reasoning for data gathered by semi-structured interviews, observations, and document review, this study identified equity talk manifesting as one of three themes: diversity as an asset, decision-making processes, and use of data and feedback. Understanding how and when specific framing processes are used can empower district leaders to be more strategic in impacting stakeholder thinking and language and maintaining an equity focus. / Thesis (EdD) — Boston College, 2020. / Submitted to: Boston College. Lynch School of Education. / Discipline: Educational Leadership and Higher Education.

Paper folding is an underappreciated method with a number of interesting applications in real life. From Japanese traditional origami we moved to today's use in space research or medicine, all thanks to the application of mathematical principles on it. The work explains the basic rules of paper geometry and its difference from Euclidean geometry. The problems of doubling the cube or trisection of the angle, which cannot be solved in Euclidean geometry, can be solved quite easily by translating the paper. The thesis contains ideas for mathematics teachers from elementary school to high school on how to use paper folding directly in mathematics teaching. Among the ideas we can find a simple verification of the Pythagorean theorem and the construction of conic sections. Physical handling of paper helps to consolidate the topic, develops spatial imagination and dexterity.

In order to study the detailed spectral intensity distribution of light reflected from a solid-gas interface, the extinction theorem in optics is extended to include an absorbing medium and the thermal motion of the gas molecules near the interface. The theoretical spectral intensity distribution in the region of anomalous dispersion is found to be strongly modified compared to that predicted by existing theory. An important consequence of this theory in the line shapes of the reflected light is the possibility of using recently developed saturation spectroscopic techniques to study atoms and molecules near surfaces. In order to investigate the feasibility of these new techniques for obtaining solid-gas molecule interaction potentials, models of solid-gas interfaces were studied with and without interactions of the type 1/zP (p = 2,3,4), where z is the distance between gas molecules and solid surface. A marked difference in the line shapes of the reflected light among the possible interactions suggests that the forms of interaction at the interface can be measured using known techniques. Furthermore, the possibility of measuring the flow of gas near walls where currently available laser-Doppler anemometers can not spatially reach is investigated. It is shown that the shift and width of the numerically calculated line shape of the reflected light in our model flow is directly related to the mean and the fluctuating velocity fields respectively.

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<p>The central problems investigated in this thesis concern the theory and practice of critical ethical reflection from the standpoint of hermeneutical philosophy. The overriding questions addressed in the thesis are, does hermeneutical philosophy leave us with an attenuated conception of, or diminished capacity for, ethical critique, as certain of its critics maintain? How is critical reflection possible in lieu of foundations and formal decision procedures, and what philosophical resources are at its disposal? More fundamentally, what is involved in the practice of critical reflection? In arguing that such reflection is best viewed as a mode of heIp hermeneutic discourse, questions arise concerning the role served by moral imagination in the practice of critique. It also raises questions concerning the role, if any, which ethical theory serves in informing a critique of human practices. Does critical reflection require the assistance of an ethical theory? If, as I contend, it does, then what method of theorizing i.s consistent with the principles of hermeneutics? Finally, how does hermeneutical philosophy view the relation between theory and practice in moral philosophy?</p> / Doctor of Philosophy (PhD)

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