Inverse problem for the wave equation : partial data and novel boundary sources

An inverse problem for the wave equation is a mathematical formulation of the problem to convert measurements of sound waves to information about the wave speed governing the propagation of the waves. This doctoral thesis extends the theory on the inverse problems for the wave equation in cases with partial measurement data and also considers detection of discontinuous interfaces in the wave speed. A possible application of the theory is obstetric sonography in which ultrasound measurements are transformed into an image of the fetus in its mother's uterus. The wave speed inside the body can not be directly observed but sound waves can be produced outside the body and their echoes from the body can be recorded. The present work contains five research articles. In the first and the fifth articles we show that it is possible to determine the wave speed uniquely by using far apart sound sources and receivers. This extends a previously known result which requires the sound waves to be produced and recorded in the same place. Our result is motivated by a possible application to reflection seismology which seeks to create an image of the Earth s crust from recording of echoes stimulated for example by explosions. For this purpose, the receivers can not typically lie near the powerful sound sources. In the second article we present a sound source that allows us to recover many essential features of the wave speed from the echo produced by the source. Moreover, these features are known to determine the wave speed under certain geometric assumptions. Previously known results permitted the same features to be recovered only by sequential measurement of echoes produced by multiple different sources. The reduced number of measurements could increase the number possible applications of acoustic probing. In the third and fourth articles we develop an acoustic probing method to locate discontinuous interfaces in the wave speed. These interfaces typically correspond to interfaces between different materials and their locations are of interest in many applications. There are many previous approaches to this problem but none of them exploits sound sources varying freely in time. Our use of more variable sources could allow more robust implementation of the probing.

Kemian ylioppilaskokeen tehtävät summatiivisen arvioinnin välineenä

The main purpose of the research was to illustrate chemistry matriculation examination questions as a summative assessment tool, and represent how the questions have evolved over the years. Summative assessment and its various test item classifications, Finnish goal-oriented curriculum model, and Bloom’s Revised Taxonomy of Cognitive Objectives formed the theoretical framework for the research. The research data consisted of 257 chemistry questions from 28 matriculation examinations between 1996 and 2009. The analysed test questions were formulated according to the national upper secondary school chemistry curricula 1994, and 2003. Qualitative approach and theory-driven content analysis method were employed in the research. Peer review was used to guarantee the reliability of the results. The research was guided by the following questions: (a) What kinds of test item formats are used in chemistry matriculation examinations? (b) How the fundamentals of chemistry are included in the chemistry matriculation examination questions? (c) What kinds of cognitive knowledge and skills do the chemistry matriculation examination questions require? The research indicates that summative assessment was used diversely in chemistry matriculation examinations. The tests included various test item formats, and their combinations. The majority of the test questions were constructed-response items that were either verbal, quantitative, or experimental questions, symbol questions, or combinations of the aforementioned. The studied chemistry matriculation examinations seldom included selected-response items that can be either multiple-choice, alternate choice, or matching items. The relative emphasis of the test item formats differed slightly depending on whether the test was a part of an extensive general studies battery of tests in sciences and humanities, or a subject-specific test. The classification framework developed in the research can be applied in chemistry and science education, and also in educational research. Chemistry matriculation examinations are based on the goal-oriented curriculum model, and cover relatively well the fundamentals of chemistry included in the national curriculum. Most of the test questions related to the symbolism of chemical equation, inorganic and organic reaction types and applications, the bonding and spatial structure in organic compounds, and stoichiometry problems. Only a few questions related to electrolysis, polymers, or buffer solutions. None of the test questions related to composites. There were not any significant differences in the emphasis between the tests formulated according to the national curriculum 1994 or 2003. Chemistry matriculation examinations are cognitively demanding. The research shows that the majority of the test questions require higher-order cognitive skills. Most of the questions required analysis of procedural knowledge. The questions that only required remembering or processing metacognitive knowledge, were not included in the research data. The required knowledge and skill level varied slightly between the test questions in the extensive general studies battery of tests in sciences and humanities, and subject-specific tests administered since 2006. The proportion of the Finnish chemistry matriculation examination questions requiring higher-order cognitive knowledge and skills is very large compared to what is discussed in the research literature.