While optimal experimental design for static linear inverse problems has been well studied, there is little in the way of experimental design methods for dynamical systems. In particular, dynamic problems where historic data is included in the optimal experimental design method, such that the experiment tracks the motion of the model. Thus, we propose an adaptive design method which minimizes the adapted mean square error (amse) of the regularized model estimate, defined by introducing a monitor function which scales the mean squared error according to historic model estimates. We present the adaptive design method for two model estimation formulations based on the classification of the error in the dynamic process. First, we consider the case where there is no error in the dynamical system and formulate the inverse problem as a PDE constrained optimization problem to recover the initial condition. And secondly, we include the error in the dynamics and formulate the inverse problem as a Kalman smoothing system to recover models for all times. We demonstrate our approach with A-optimal design criteria for a borehole seismic tomography experiment, where the motion of the model is governed by advection in a constant velocity field, and show that we are able to recover optimal designs which track the motion of the model.