Spatio-temporal dynamical processes in the physical and environmental sciences are often described by partial differential equations (PDEs). The inherent complexity of such processes due to high-dimensionality and multiple scales of spatial and temporal variability is often intensified by characteristics such as sparsity of data, complicated boundaries and irregular geometrical spatial domains, among others. In addition, uncertainties in the appropriateness of any given PDE for a real-world process, as well as uncertainties in the parameters associated with the PDEs are typically present. These issues necessitate the incorporation of efficient parameterizations of spatio-temporal models that are capable of addressing such characteristics. In this work, a hierarchical Bayesian model characterized by the PDE-based dynamics for spatio-temporal processes based on their Galerkin finite element method (FEM) representations is developed and discussed. As an example, spatio-temporal models based on advection-diffusion processes are considered. Finally, an application of the hierarchical Bayesian modeling approach is presented which considers the analysis of tracking data obtained from DST (data storage devices) sensors to mimic the pre-spawning upstream migration process of the declining shovelnose sturgeon.