Recovering the initial data of the wave equation from boundary measurements including directivity

Waves and their characteristics are not only a main subject in physics, they are also being studied extensively in mathematics in the context of partial differential equations. One of these equations, the scalar wave equation, is used as a model for describing the behavior and propagation of waves phenomena in mathematical terms. Based on so-called initial conditions, describing the solution of the wave equation or approximating its solution on a computer with numerical methods are main objectives. However, in practice, the values of the solution of the wave equation are often measured and then used to determine its initial conditions. The resulting inverse problem of the wave equation arises, for example, in photoacoustic tomography (PAT), which utilizes acoustic pressure signals of materials such as biological tissues in order to investigate their internal structure by creating visual representations. Physically speaking, in PAT, the initial condition corresponds to the local initial pressure rise in the sample, induced by short pulses of electromagnetic radiation. In standard PAT, it is assumed that acoustic signals, which are measured by sensors outside of the body, equal the acoustic pressure itself. Nevertheless, in general, sensors such as piezoelectric sensors exhibit a directivity effect which is not included in this assumption. In this thesis, a measurement model including these directivity effects and consisting of a linear combination of the acoustic pressure and its normal derivative is being presented. Based on this model, the thesis provides theoretical results such as exact inversion formulas for recovering the initial data of the wave equation. Numerical simulations as well as algorithms for the numerical implementation of the derived formulas are being shown. The final chapter provides theoretical considerations that are related to the discretization of the initial pressure in PAT. (author)