The problem of estimating a spatially distributed process described by a partial differential equation (PDE), whose observations are contaminated by a zero mean Gaussian noise, is considered in this work. The basic premise of this work is that a set of mobile sensors achieve better estimation performance than a set of immobile sensors. To enhance the performance of the state estimator, a network of sensors that are capable of moving within the spatial domain is utilized. Specifically, such an estimation process is achieved by using a set of spatially distributed mobile sensors. The objective is to provide mobile sensor control policies that aim to improve the state estimate. The metric for such an estimate improvement is taken to be the expected state estimation error. Using different spatial norms, two guidance policies are proposed. The current approach capitalizes on the efficient filter gain design in order to avoid intense computational requirements resulting from the solution to filter Riccati equations. Simulation studies implementing and comparing the two proposed control policies are provided.