Abstract

In geophysics, the direct application of covariance matrix dynamics described by the Kalman filter (KF) is limited by the high dimension of such problems. The parametric Kalman filter (PKF) is a recent alternative to the ensemble Kalman filter, where the covariance matrices are approximated by a covariance model featured by a set of parameters. The covariance dynamics is then described by the time evolution of these parameters during the analysis and forecast cycles. This study focuses on covariance model parametrized by the variance and the local anisotropic tensor fields (VLATcov). The analysis step of the PKF for VLATcov in a 2D/3D domain is first introduced. Then, using 2D univariate numerical investigations, the PKF is shown to be able to provide a low numerical cost approximation of the Kalman filter analysis step, even for anisotropic error correlation functions. Moreover the PKF has been shown able to reproduce the KF over several assimilation cycles in a transport dynamics. An extension toward the multivariate situation is theoretically studied in a 1D domain.

摘要

在地球物理学中，由卡尔曼滤波器 (KF) 描述的协方差矩阵动力学的直接应用受到此类问题的高维数的限制。参数卡尔曼滤波器 (PKF) 是集成卡尔曼滤波器的最新替代方案，其中协方差矩阵由以一组参数为特征的协方差模型近似。然后通过这些参数在分析和预测周期中的时间演变来描述协方差动态。本研究侧重于由方差和局部各向异性张量场 (VLATcov) 参数化的协方差模型。首先介绍了 2D/3D 域中 VLATcov 的 PKF 分析步骤。然后，使用 2D 单变量数值研究，显示 PKF 能够提供卡尔曼滤波器分析步骤的低数值成本近似值，即使对于各向异性误差相关函数。此外，PKF 已被证明能够在运输动力学中的几个同化循环中重现 KF。在一维域中理论上研究了对多元情况的扩展。