A Matrix-Free Posterior Ensemble Kalman Filter Implementation Based on a Modified Cholesky Decomposition

On the mathematical modelling and data assimilation for air pollution assessment in the Tropical Andes

Air pollution assessment in the Tropical Andes requires a multidisciplinary approach. This can be supported from the understanding of the underlying biological dynamics and atmospheric behavior, to the mathematical approach for the proper use of all available information. This review paper touches on several aspects in which mathematical models can help to solve challenging problems regarding air pollution in reviewing the state-of-the-art at the global level and assessing the corresponding state of development as applied to the Tropical Andes. We address the complexities and challenges that modelling atmospheric dynamics in a mega-diverse region with abrupt topography entails. Understanding the relevance of monitoring and facing the problems of data scarcity, we call attention to the usefulness of data assimilation for uncertainty reduction, and how these techniques could help tackle the scarcity of regional monitoring networks to accelerate the implementation and development of modelling systems for air quality in the Tropical Andes. Finally, we suggest a cyberphysical framework for decision-making processes based on the data assimilation of chemical transport models, the forecast of scenarios, and their use in regulation and policy making.

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.

A Four Dimensional Variational Data Assimilation Framework for Wind Energy Potential Estimation

In this paper, we propose a Four-Dimensional Variational (4D-Var) data assimilation framework for wind energy potential estimation. The framework is defined as follows: we choose a numerical model which can provide forecasts of wind speeds then, an ensemble of model realizations is employed to build control spaces at observation steps via a modified Cholesky decomposition. These control spaces are utilized to estimate initial analysis increments and to avoid the intrinsic use of adjoint models in the 4D-Var context. The initial analysis increments are mapped back onto the model domain from which we obtain an estimate of the initial analysis ensemble. This ensemble is propagated in time to approximate the optimal analysis trajectory. Wind components are post-processed to get wind speeds and to estimate wind energy capacities. A matrix-free analysis step is derived from avoiding the direct inversion of covariance matrices during assimilation cycles. Numerical simulations are employed to illustrate how our proposed framework can be employed in operational scenarios. A catalogue of twelve Wind Turbine Generators (WTGs) is utilized during the experiments. The results reveal that our proposed framework can properly estimate wind energy potential capacities for all wind turbines within reasonable accuracies (in terms of Root-Mean-Square-Error) and even more, these estimations are better than those of traditional 4D-Var ensemble-based methods. Moreover, large variability (variance of standard deviations) of errors are evidenced in forecasts of wind turbines with the largest rate-capacity while homogeneous variability can be seen in wind turbines with the lowest rate-capacity.

A Random Line-Search Optimization Method via Modified Cholesky Decomposition for Non-linear Data Assimilation

This paper proposes a line-search optimization method for non-linear data assimilation via random descent directions. The iterative method works as follows: at each iteration, quadratic approximations of the Three-Dimensional-Variational (3D-Var) cost function are built about current solutions. These approximations are employed to build sub-spaces onto which analysis increments can be estimated. We sample search-directions from those sub-spaces, and for each direction, a line-search optimization method is employed to estimate its optimal step length. Current solutions are updated based on directions along which the 3D-Var cost function decreases faster. We theoretically prove the global convergence of our proposed iterative method. Experimental tests are performed by using the Lorenz-96 model, and for reference, we employ a Maximum-Likelihood-Ensemble-Filter (MLEF) whose ensemble size doubles that of our implementation. The results reveal that, as the degree of observational operators increases, the use of additional directions can improve the accuracy of results in terms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document}-norm of errors, and even more, our numerical results outperform those of the employed MLEF implementation.