Non-Gaussian Variational Data Assimilation Embedded Reduced-Order Modeling Methods Through Statistical Error Transformations

Predictions made by current simulation methods exhibit deficiencies in predicting high-speed reactive flows featuring nonlinear physics, such as turbulence and shocks, due to uncertainties in flow rates (e.g., boundary and initial conditions) or model assumptions. Data assimilation (DA) offers a potential solution to address such deficiencies by combining the numerical model of a system with real observations to find the most likely state. Most DA methods predominantly assume Gaussian distributions for system errors, whereas high-speed reactive flow features physics exhibiting non-Gaussian errors. Direct application of Gaussian-based DA methods are likely to produce biased or even non-physical predictions. This work presents a non-Gaussian variational DA method utilizing a transfer function to transform model predictions and observations into a Gaussian space, along with explicit representations of arbitrary error distributions which can be used for variational DA to maximize the posteriori. We evaluate and compare this new DA method with the standard ensemble Kalman filter (EnKF), four-dimensional variational (4DVar), and a normal-score EnKF and 4DVar approaches for reduced-order model (ROM) development. We use a suite of 1D test problems with prescribed strongly non-Gaussian system error distributions to perform the evaluations. The results show that the new non-gaussian Gaussian method produces more accurate state predictions than the standard Gaussian DA methods.