A Machine Learning Approach to Adaptive Covariance Localization
Virginia Polytechnic Institute and State University（弗吉尼亚理工学院暨州立大学）；Argonne National Laboratory（阿贡国家实验室）；Statistical and Applied Mathematical Science Institute（统计与应用数学科学研究所）
Data assimilation plays a key role in large-scale atmospheric weather forecasting, where the state of the physical system is estimated from model outputs and observations, and is then used as initial condition to produce accurate future forecasts. The Ensemble Kalman Filter (EnKF) provides a practical implementation of the statistical solution of the data assimilation problem and has gained wide popularity as. This success can be attributed to its simple formulation and ease of implementation. EnKF is a Monte-Carlo algorithm that solves the data assimilation problem by sampling the probability distributions involved in Bayes theorem. Because of this, all flavors of EnKF are fundamentally prone to sampling errors when the ensemble size is small. In typical weather forecasting applications, the model state space has dimension $10^{9}-10^{12}$, while the ensemble size typically ranges between $30-100$ members. Sampling errors manifest themselves as long-range spurious correlations and have been shown to cause filter divergence. To alleviate this effect covariance localization dampens spurious correlations between state variables located at a large distance in the physical space, via an empirical distance-dependent function. The quality of the resulting analysis and forecast is greatly influenced by the choice of the localization function parameters, e.g., the radius of influence. The localization radius is generally tuned empirically to yield desirable results.This work, proposes two adaptive algorithms for covariance localization in the EnKF framework, both based on a machine learning approach. The first algorithm adapts the localization radius in time, while the second algorithm tunes the localization radius in both time and space. Numerical experiments carried out with the Lorenz-96 model, and a quasi-geostrophic model, reveal the potential of the proposed machine learning approaches.
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We introduce a new sparse estimator of the covariance matrix for high-dimensional models in which the variables have a known ordering. Our estimator, which is the solution to a convex optimization problem, is equivalently expressed as an estimator which tapers the sample covariance matrix by a Toeplitz, sparsely-banded, data-adaptive matrix. As a result of this adaptivity, the convex banding estimator enjoys theoretical optimality properties not attained by previous banding or tapered estimators. In particular, our convex banding estimator is minimax rate adaptive in Frobenius and operator nor…
