Deterministic Predictability of the Most Probable State and Reformulation of Variational Data Assimilation

Abstract

If the divergence in phase space of the evolution equation of a deterministic nonlinear system does not depend on the state variables (hereafter referred to as the divergence condition), the deterministic prediction starting from the mode of a probability density function (PDF) of the state variables remains the mode of the PDF at forecast time. For a system that does not satisfy the divergence condition, a condition for the forecast state to remain sufficiently close to the mode of the PDF is derived under assumption of a small forecast error. Calculation of the divergence in phase space for finite-dimensional analogs of several Eulerian equations of hydrodynamics shows that the divergence condition holds for the quasigeostrophic equations with lateral boundaries and the shallow water equations on a sphere.On the basis of the above results, a new formulation of four-dimensional variational data assimilation (4DVar) is presented. A Gaussian prior PDF at the beginning of an assimilation window is evolved up to the end of that window according to the Liouville equation. It is found that if the divergence condition holds, the cost function with the prior PDF thus evolved is equivalent to the conventional cost function of 4DVar. This result reveals that a non-Gaussian prior PDF which evolves according to the Liouville equation is implicitly used in 4DVar. Data assimilation experiments with toy models are conducted to demonstrate this advantage of 4DVar. The background error covariance at the beginning of the assimilation window is obtained from ensemble Kalman filter (EnKF). To alleviate the difficulty of multiple minima, when the convergence value of the cost function exceeds a certain threshold, the 4DVar analysis is replaced by the corresponding EnKF analysis. Results demonstrate that 4DVar cycles with the abovementioned modifications outperform EnKF cycles in terms of the accuracy of analysis in strong nonlinearity as well as in weak nonlinearity.

Ensemble Kalman Filtering Based on Potential Vorticity for Atmospheric Multi-scale Data Assimilation

“…This is one of the benefits of 4D-Var over EnKF. For instance, Tsuyuki (2014) shows that a non-Gaussian prior probability density function (PDF) that evolves according to the Liouville equation (e.g., Ehrendorfer 1994) is implicitly used in 4D-Var under certain conditions even if a prior PDF at the beginning of the assimilation window is Gaussian. On the other hand, the approximations introduced by using PV inversion in EnKF are directly reflected in the analysis through the Kalman gain, which implies that more care may be needed for the application of PV to EnKF than that to 4D-Var.…”

Comparison between Four-Dimensional LETKF and Ensemble-Based Variational Data Assimilation with Observation Localization

“…If the non-Gaussianity of the PDF is weak, then exp(−J ) is a good approximation of the PDF (Tsuyuki 2014). Therefore, the result from EnVAR, which minimizes globally defined J without assuming linearity of H k (x), should be closer to the true value than that of LETKF when H k (x) is non-linear or when observations globally affect physically distant analysis points.…”

“…However, the analytical method in EnKF is simplified to explicitly solve the analysis from the first guess, assuming linearity of the observation operator and Gaussianity of the probability density function (PDF). These two assumptions in EnKF cause problems in data assimilation with a multi-scale model (Tsuyuki 2014). One way to avoid such assumptions is to minimize the cost function implicitly with a four-dimensional ensemble-based variational method (4D-EnVAR, Zupanski 2005;Zupanski et al 2008;Liu et al 2008Liu et al , 2009.…”

Nonlinear Data Assimilation by Deep Learning Embedded in an Ensemble Kalman Filter

“…This may be plausible since non-Gaussian data assimilation needs some information on higher-order moments of probability density functions (PDFs). As for the 4-dimensional variational method (4D-Var), Tsuyuki (2014) showed that the 4D-Var with a conventional cost function implicitly used a non-Gaussian prior PDF that evolved according to the Liouville equation (Ehrendorfer, 1994) if a certain condition was satisfied, and that the difficulty caused by multiple minima could be alleviated by combining with the EnKF. The iterative ensemble Kalman filter/smoother (IEnKF/IEnKS) have been shown to be the missing link between the PF and the EnKF and 4D-Var, and can work very well with mild nonlinearity and generate a much better analysis than the above data assimilation methods (Sakov et al, 2012;Bocquet andSakov, 2013, 2014;Bocquet, 2016).…”