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：将集成卡尔曼滤波器应用于高维问题时，只能使用小规模集成。这需要对分析（观测更新）问题进行某种正则化处理。我们提出了一种基于球面上新型非平稳、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，是通用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、作为距离函数的正定性以及作为位置函数的平滑性。该模型允许对局部谱进行严格定义，要求局部谱是空间波数的平滑函数。我们提出并测试了一种集成滤波器，其中先验协方差被假定服从局部平稳卷积模型。该模型通过两阶段流程在线估计：首先，对集成扰动在多个波数带进行带通滤波，以提取聚合的局部空间谱；其次，神经网络从滤波场的样本方差中恢复局部谱。在模拟实验中，新滤波器能够优于多种现有技术。在小到中等规模集成下，改进效果显著。