Convolutional neural network-based adaptive localization for an ensemble Kalman filter

Flow-dependent background error covariances estimated from short-term ensemble forecasts suffer from sampling errors due to limited ensemble sizes. Covariance localization is often used to mitigate the sampling errors, especially for high dimensional geophysical applications. Most applied localization methods, empirical or adaptive ones, multiply the Kalman gain or background error covariances by a distance-dependent parameter, which is a simple linear filtering model. Here two localization methods based on convolutional neural networks (CNNs) learning from paired data sets are proposed. The CNN-based localization function (CLF) aims to minimize the sampling error of the estimated Kalman gain, and the CNN-based empirical localization function (CELF) aims to minimize the posterior error of state variables. These two CNN-based localization methods can provide localization functions that are nonlinear, spatially and temporally adaptive, and non-symmetric with respect to displacement, without requiring any prior assumptions for the localization functions. Results using the Lorenz05 model show that CLF and CELF can better capture the structures of the Kalman gain than the best Gaspari and Cohn (GC) localization function and the adaptive reference localization method. For both perfect- and imperfect-model experiments, CLF produces smaller errors of the Kalman gain, prior and posterior than the best GC and reference localization, especially for spatially averaged observations. Without model error, CELF has smaller prior and posterior errors than the best GC and reference localization for spatially averaged observations, while with model error, CELF has smaller prior and posterior errors than the best GC and reference localization for single-point observations.