Abstract: The goal in many applications of inverse problems is to reconstruct static physical properties given a time series of measurements. Clearly, using data from different time points is very appealing as it helps reduce ill-posedness and thus can be expected to lead more reliable reconstructions. However, in many cases a straight-forward inversion is unfeasible without addressing dynamic behaviour of the overall system during the acquisition time, for example, motion between measurements. Mathematically, joint reconstruction of both static physical properties and dynamic phenomena can be modelled as an inverse problem with hyperbolic PDE constraints. Clearly the estimation of static and dynamic properties are tightly coupled: For instance, having precise knowledge of the underlying motion, static estimates of superior quality to those based on a single time point can be expected. Vice versa, accuracy of motion estimation in principle improves if high quality reconstructions at each time frame are available. This talk presents current work on numerical schemes for jointly estimating static and dynamic properties. It focusses on two inverse problems from geophysical and medical imaging in which the dynamic behaviour of the system can be related to a hyperbolic PDE. The talk addresses numerical challenges arising from discretization of the hyperbolic PDE constraints and present efficient solution techniques.