A Nonlinear Data Assimilation Algorithm with Closed-Form Approximations for Multilayer Flow Fields

Abstract State estimation in multilayer turbulent flow fields with only a single layer of partial observation remains a challenging yet practically important task. Applications include inferring the state of the deep ocean by exploiting surface observations. Directly implementing an ensemble Kalman filter based on the full forecast model is usually expensive. One widely used method in practice projects the information of the observed layer to other layers via linear regression. However, large errors appear when nonlinearity in the highly turbulent flow field becomes dominant. In this paper, we develop a multistep nonlinear data assimilation method that involves the sequential application of nonlinear assimilation steps across layers. Unlike traditional linear regression approaches, a conditional Gaussian nonlinear system is adopted as the approximate forecast model to characterize the nonlinear dependence between adjacent layers. At each step, samples drawn from the posterior of the current layer are treated as pseudo-observations for the next layer. Each sample is assimilated using analytic formulas for the posterior mean and covariance. The resulting Gaussian posteriors are then aggregated into a Gaussian mixture. Therefore, the method can capture strongly turbulent features, particularly intermittency and extreme events, and more accurately quantify the inherent uncertainty. Applications to the two-layer quasigeostrophic system with Lagrangian data assimilation demonstrate that the multistep method outperforms the one-step method, particularly as the tracer number and ensemble size increase. Results also show that the multistep conditional Gaussian data assimilation (CGDA) is particularly effective for assimilating frequent, high-accuracy observations, which are scenarios where traditional EnKF methods may suffer from catastrophic filter divergence. Significance Statement This work addresses the challenges in state estimation of complex turbulent flow fields using partial observations, with applications such as inferring the state of the deep ocean. A new multistep nonlinear data assimilation method is developed, utilizing conditional Gaussian nonlinear systems as effective approximate forecast models. The new method outperforms traditional linear approaches. It has the advantages of better characterizing non-Gaussian features and improving the quantification of uncertainties, especially for intermittency and extreme events. Notably, the data assimilation solution is given by a Gaussian mixture distribution, with each Gaussian component solved via closed-form analytic formulas that are numerically robust and accurate. Future research will extend this method to more complex geophysical systems, simultaneously estimating the states and inferring ocean topography.