The sample covariance matrix of a random vector is a good estimate of the true covariance matrix if the sample size is much larger than the length of the vector. In high-dimensional problems, this condition is never met. As a result, in high dimensions the EnKF ensemble does not contain enough information to specify the prior covariance matrix accurately. This necessitates the need for regularization of the analysis (observation update) problem. We propose a regularization technique based on a new spatial model on the sphere. The model is a constrained version of the general Gaussian process convolution model. The constraints on the location-dependent convolution kernel include local isotropy, positive definiteness as a function of distance, and smoothness as a function of location. The model allows for a rigorous definition of the local spectrum, which, in addition, is required to be a smooth function of spatial wavenumber. We regularize the ensemble Kalman filter by postulating that its prior covariances obey this model. The model is estimated online in a two-stage procedure. First, ensemble perturbations are bandpass filtered in several wavenumber bands to extract aggregated local spatial spectra. Second, a neural network recovers the local spectra from sample variances of the filtered fields. We show that with the growing ensemble size, the estimator is capable of extracting increasingly detailed spatially non-stationary structures. In simulation experiments, the new technique led to substantially better EnKF performance than several existing techniques.

翻译：随机向量的样本协方差矩阵在样本量远大于向量长度时，才能准确估计真实协方差矩阵。在高维问题中，这一条件通常无法满足，导致集合卡尔曼滤波（EnKF）集合无法提供足够信息来精确表征先验协方差矩阵，因此需要正则化分析（观测更新）问题。本文提出一种基于球面新型空间模型的正则化技术。该模型是一般高斯过程卷积模型的约束版本，其位置相关卷积核需满足局部各向同性、基于距离的正定性及位置平滑性等约束。模型允许对局部谱进行严格定义，且要求该谱成为空间波数的平滑函数。我们通过假设先验协方差服从该模型对集合卡尔曼滤波进行正则化。该模型采用两阶段在线估计：首先对集合扰动进行多波数带通滤波，提取聚合的局部空间谱；随后利用神经网络从滤波场的样本方差中重建局部谱。研究表明，随着集合规模增大，该估计器能够提取更精细的空间非平稳结构。在模拟实验中，该技术较现有多种方法显著提升了EnKF性能。