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.

Anaerobic ammonium oxidation (anammox) in sediments of the Gulf of Finland

Anaerobic ammonium oxidation (anammox) and denitrification were measured in the open sea and coastal accumulation basins of the Gulf of Finland. The different methods used gave conflicting results on the importance of the anammox process in the sediments. Anammox generally contributed less than 20 % to the total N-2 production, and no anammox was found in a shallow inner estuary basin. However, the discovery of the anammox process in the open sea sediments challenges the denitrification measurements made in the area, as the coexistence of anammox and denitrification compromises the central assumptions behind the method used in denitrification measurements and causes overestimates of the N-2 production. The high (NO3-)-N-15 incubation concentration used in Baltic Sea denitrification measurements exacerbates this overestimation, which is likely to have been substantial.