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.

Income inequality in the process of economic development : An empirical approach

This thesis studies the effect of income inequality on economic growth. This is done by analyzing panel data from several countries with both short and long time dimensions of the data. Two of the chapters study the direct effect of inequality on growth, and one chapter also looks at the possible indirect effect of inequality on growth by assessing the effect of inequality on savings. In Chapter two, the effect of inequality on growth is studied by using a panel of 70 countries and a new EHII2008 inequality measure. Chapter contributes on two problems that panel econometric studies on the economic effect of inequality have recently encountered: the comparability problem associated with the commonly used Deininger and Squire s Gini index, and the problem relating to the estimation of group-related elasticities in panel data. In this study, a simple way to 'bypass' vagueness related to the use of parametric methods to estimate group-related parameters is presented. The idea is to estimate the group-related elasticities implicitly using a set of group-related instrumental variables. The estimation results with new data and method indicate that the relationship between income inequality and growth is likely to be non-linear. Chapter three incorporates the EHII2.1 inequality measure and a panel with annual time series observations from 38 countries to test the existence of long-run equilibrium relation(s) between inequality and the level of GDP. Panel unit root tests indicate that both the logarithmic EHII2.1 inequality measure and the logarithmic GDP per capita series are I(1) nonstationary processes. They are also found to be cointegrated of order one, which implies that there is a long-run equilibrium relation between them. The long-run growth elasticity of inequality is found to be negative in the middle-income and rich economies, but the results for poor economies are inconclusive. In the fourth Chapter, macroeconomic data on nine developed economies spanning across four decades starting from the year 1960 is used to study the effect of the changes in the top income share to national and private savings. The income share of the top 1 % of population is used as proxy for the distribution of income. The effect of inequality on private savings is found to be positive in the Nordic and Central-European countries, but for the Anglo-Saxon countries the direction of the effect (positive vs. negative) remains somewhat ambiguous. Inequality is found to have an effect national savings only in the Nordic countries, where it is positive.