Sequential Ensemble Kalman Filtering of Compressible Flows With Shocks: Enforcing Positivity

The current work considers sequential ensemble Kalman filtering (EnKF) methods for the data assimilation (DA) of compressible flows exhibiting sharp gradients and shocks. Analysis of such configurations can generate non-positive thermodynamic values – which poses a significant challenge to the overall procedure, as the forecast model/equations cannot operate on non-realizable states. To side step such issues, a DA procedure that operates on logarithmic- transformed versions of the solution variables is employed, wherein positivity is automatically enforced via an inverse transform prior to passing the update to a forecast model. To demonstrate the utility of such an approach, the EnKF method is used to anchor unsteady ensemble simulations to a truth solution via synthetically generated sparse measurement targets that are in deviance from the initial ensemble predictions. Preliminary one-dimensional assessments on a non-reacting shock tube and a reacting detonation tube suggest that leveraging such variable transforms greatly improves the robustness of the overall DA procedure and forms a crucial building block for the assimilation of discontinuous flows.