Recent researches in data assimilation lead to the introduction of the parametric Kalman filter (PKF): an implementation of the Kalman filter, where the covariance matrices are approximated by a parameterized covariance model. In the PKF, the dynamics of the covariance during the forecast step relies on the prediction of the covariance parameters. Hence, the design of the parameter dynamics is crucial while it can be tedious to do this by hand. This contribution introduces a python package, SymPKF, able to compute PKF dynamics for univariate statistics and when the covariance model is parameterized from the variance and the local anisotropy of the correlations. The ability of SymPKF to produce the PKF dynamics is shown on a non-linear diffusive advection (Burgers equation) over a 1D domain and the linear advection over a 2D domain. The computation of the PKF dynamics is performed at a symbolic level, but an automatic code generator is also introduced to perform numerical simulations. A final multivariate example illustrates the potential of SymPKF to go beyond the univariate case.

中文翻译：

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SymPKF：用于单变量参数卡尔曼滤波器动力学设计的符号和计算工具箱

数据同化的最新研究导致引入参数卡尔曼滤波器（PKF）：卡尔曼滤波器的一种实现，其中协方差矩阵由参数化协方差模型近似。在PKF中，预测步骤期间协方差的动态取决于协方差参数的预测。因此，参数动力学的设计至关重要，而手工操作可能很繁琐。这一贡献引入了一个Python软件包SymPKF，该软件包能够计算单变量统计信息的PKF动态性，并且可以根据相关性的方差和局部各向异性对协方差模型进行参数化。SymPKF产生PKF动力学的能力在1D域上的非线性扩散对流（Burgers方程）和2D域上的线性对流中得到了证明。PKF动力学的计算是在符号级别执行的，但是还引入了自动代码生成器以执行数值模拟。最后一个多变量示例说明了SymPKF超越单变量情况的潜力。