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A hybrid data assimilation algorithm is developed for complex dynamicalsystems with partial observations. The method starts with applying a spectraldecomposition to the entire spatiotemporal fields, followed by creating amachine learning model that builds a nonlinear map between the coefficients ofobserved and unobserved state variables for each spectral mode. A cheaplow-order nonlinear stochastic parameterized extended Kalman filter (SPEKF)model is employed as the forecast model in the ensemble Kalman filter to dealwith each mode associated with the observed variables. The resulting ensemblemembers are then fed into the machine learning model to create an ensemble ofthe corresponding unobserved variables. In addition to the ensemble spread, thetraining residual in the machine learning-induced nonlinear map is furtherincorporated into the state estimation that advances the quantification of theposterior uncertainty. The hybrid data assimilation algorithm is applied to aprecipitating quasi-geostrophic (PQG) model, which includes the effects ofwater vapor, clouds, and rainfall beyond the classical two-level QG model. Thecomplicated nonlinearities in the PQG equations prevent traditional methodsfrom building simple and accurate reduced-order forecast models. In contrast,the SPEKF model is skillful in recovering the intermittent observed states, andthe machine learning model effectively estimates the chaotic unobservedsignals. Utilizing the calibrated SPEKF and machine learning models under amoderate cloud fraction, the resulting hybrid data assimilation remainsreasonably accurate when applied to other geophysical scenarios with nearlyclear skies or relatively heavy rainfall, implying the robustness of thealgorithm for extrapolation.

Combining Stochastic Parameterized Reduced-Order Models with Machine  Learning for Data Assimilation and Uncertainty Quantification with Partial  Observations