The Maximum Likelihood Ensemble Filter with State Space Localization

New method for ensemble data assimilation that incorporates state space covariance localization, global numerical optimization, and implied Bayesian inference, is presented. The method is referred to as the MLEF with State Space Localization (MLEF-SSL) due to its similarity with the Maximum Likelihood Ensemble Filter (MLEF). One of the novelties introduced in MLEF-SSL is the calculation of a reduced-rank localized forecast error covariance using random projection. The Hessian preconditioning is accomplished via Cholesky decomposition of the Hessian matrix, accompanied with solving triangular system of equations instead of directly inverting matrices. For ensemble update the MLEF-SSL system employs resampling of posterior perturbations. The MLEF-SSL was applied to Lorenz model II and compared to Ensemble Kalman Filter with state space localization and to MLEF with observation space localization. The observations include linear and nonlinear observation operators, each applied to integrated and point observations. Results indicate improved performance of MLEF-SSL, particularly in assimilation of integrated nonlinear observations. Resampling of posterior perturbations for ensemble update also indicates a satisfactory performance. Additional experiments were conducted to examine the sensitivity of the method to the rank of random matrix and to compare it to truncated eigenvectors of the localization matrix. The two methods are comparable in application to low-dimensional Lorenz model, except that the new method outperforms the truncated eigenvector method in case of severe rank reduction. The random basis method is simple to implement and may be more promising for realistic high-dimensional applications.

Assimilation of Nonlinear Observations with the Maximum Likelihood Ensemble Filter

<p>Satellite observations have been a growing source for data assimilation in the operational numerical weather prediction. Remotely sensed observations require a nonlinear observation operator.  Most ensemble-based data assimilation methods are formulated for tangent linear observation operators, which are often substituted by nonlinear observation operators. By contrast, the Maximum Likelihood Ensemble Filter (MLEF), which has features of both variational and ensemble approaches, is formulated for linear and nonlinear operators in an identical form and can use non-differentiable observation operators.<span> </span></p><p>In this study, we investigate the performance of MLEF and Ensemble Transform Kalman Filter (ETKF) with the tangent linear and nonlinear observation operators in assimilation experiments of nonlinear observations with a one-dimensional Burgers model.</p><p>The ETKF analysis with the nonlinear operator diverges when the observation error is small due to unrealistically large increments associated with the high order observation terms. The filter divergence can be avoided by localization of the extent of observation influence, but the analysis error is still larger than that of MLEF. In contrast, MLEF is found to be more stable and accurate without localization owing to the minimization of the cost function. Notably, MLEF can make an accurate analysis solution even without covariance inflation, eliminating the labor of parameter adjustment. In addition, the smaller observation error is, or the stronger observation nonlinearity is, MLEF with the nonlinear operators can assimilate observations more effectively than MLEF with the tangent linear operators. This result indicates that MLEF can incorporate nonlinear effects and evaluate the observation term in the cost function appropriately. These encouraging results imply that MLEF is suitable for assimilation of satellite observations with high nonlinearity.</p>

A Maximum Likelihood Ensemble Filter via a Modified Cholesky Decomposition for Non-Gaussian Data Assimilation

This paper proposes an efficient and practical implementation of the Maximum Likelihood Ensemble Filter via a Modified Cholesky decomposition (MLEF-MC). The method works as follows: via an ensemble of model realizations, a well-conditioned and full-rank square-root approximation of the background error covariance matrix is obtained. This square-root approximation serves as a control space onto which analysis increments can be computed. These are calculated via Line-Search (LS) optimization. We theoretically prove the convergence of the MLEF-MC. Experimental simulations were performed using an Atmospheric General Circulation Model (AT-GCM) and a highly nonlinear observation operator. The results reveal that the proposed method can obtain posterior error estimates within reasonable accuracies in terms of ℓ − 2 error norms. Furthermore, our analysis estimates are similar to those of the MLEF with large ensemble sizes and full observational networks.

Development of a hybrid variational-ensemble data assimilation technique for observed lightning tested in a mesoscale model

Lightning measurements from the Geostationary Lightning Mapper (GLM) that will be aboard the Geostationary Operational Environmental Satellite – R Series will bring new information that can have the potential for improving the initialization of numerical weather prediction models by assisting in the detection of clouds and convection through data assimilation. In this study we focus on investigating the utility of lightning observations in mesoscale and regional applications suitable for current operational environments, in which convection cannot be explicitly resolved. Therefore, we examine the impact of lightning observations on storm environment. Preliminary steps in developing a lightning data assimilation capability suitable for mesoscale modeling are presented in this paper. World Wide Lightning Location Network (WWLLN) data was utilized as a proxy for GLM measurements and was assimilated with the Maximum Likelihood Ensemble Filter, interfaced with the Nonhydrostatic Mesoscale Model core of the Weather Research and Forecasting system (WRF-NMM). In order to test this methodology, regional data assimilation experiments were conducted. Results indicate that lightning data assimilation had a positive impact on the following: information content, influencing several dynamical variables in the model (e.g., moisture, temperature, and winds), and improving initial conditions during several data assimilation cycles. However, the 6 h forecast after the assimilation did not show a clear improvement in terms of root mean square (RMS) errors.