Lecture Notes on Inverse Theory

Inverse theory is the art of inferring properties of a physical system from observations. It is used to transform recordings of wavefields into estimates of medium properties in seismic tomography, medical imaging and nondestructive testing. It serves to constrain the distribution of density inside the Earth from gravity measurements, and to obtain probabilities of life on exoplanets based on telescope observations of electromagnetic spectra. The list of examples is endless. The primary goal of these lecture notes is to equip the reader with a solid theoretical background and with an extensive toolbox that can be used to solve real-world inverse problems. A central theme is the recognition that the best theory or the best method do not exist. A good solution strategy is application- and data-specific, and so these notes are intended to help with making a reasonable choice.