Applications of the ensemble Kalman filter to high-dimensional problems are feasible only with small ensembles. This necessitates a kind of regularization of the analysis (observation update) problem. We propose a regularization technique based on a new non-stationary, non-parametric spatial model on the sphere. The model termed the Locally Stationary Convolution 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 is required to be a smooth function of spatial wavenumber. We propose and test an ensemble filter in which prior covariances are postulated to obey the Locally Stationary Convolution 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. In simulation experiments, the new filter was capable of outperforming several existing techniques. With small to moderate ensemble sizes, the improvement was substantial.

翻译：集合卡尔曼滤波在高维问题中的应用仅在小集合样本下可行，这需要对分析（观测更新）问题实施正则化处理。本文提出一种基于球面新型非平稳非参数空间模型的正则化技术。该模型被称为局部平稳卷积模型，是高斯过程卷积模型的一种约束形式。对位置相关卷积核的约束包括：局部各向同性、距离函数的正定性以及位置函数的平滑性。该模型能够严格定义所需满足空间波数平滑性的局部谱。我们提出并测试了一种先验协方差服从局部平稳卷积模型的集合滤波器，该模型通过两阶段方法在线估计：首先，对集合扰动进行多波数带的带通滤波以提取聚合的局部空间谱；其次，利用神经网络从滤波场的样本方差中恢复局部谱。仿真实验表明，新型滤波器性能优于现有多种技术。在中小规模集合条件下，其改进效果尤为显著。