To estimate displacements of physical fields, a general framework is proposed. Considering that for each state variable, a tensor field can be associated, ways these displacements act on different state variables will differ according to the tensor field definitions. This perspective provides a differential-geometry-based reformulation of the generalized optical flow (OF) algorithm. Using the proposed framework, optimisation procedures can explicitly ensure the conservation of certain physical quantities (total mass, total vorticity, total kinetic energy, etc.). Existence and uniqueness of the solutions to the local optimisation problem are demonstrated, leading to a new nudging strategy using all-available observations to infer displacements of both observed and unobserved state variables. Using the proposed nudging method before EnKF, numerical results show that ensemble data assimilation better preserves the intrinsic structure of underline physical processes if the ensemble members are aligned with the observations.