Abstract
Abstract. Recent research in data assimilation has led to the introduction of the parametric Kalman filter (PKF): an implementation of the Kalman filter, whereby the covariance matrices are approximated by a parameterized covariance model. In the PKF, the dynamics of the covariance during the forecast step rely 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 nonlinear diffusive advection (the 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.
Highlights
The Kalman filter (KF) (Kalman, 1960) is one of the backbones of data assimilation
We focus on the parametric Kalman filter (PKF) applied to a particular family of covariance models, whose parameters are defined in grid points by the variance and the anisotropy fields: P = (V, g), where g will denotes the local anisotropy tensor of the local correlation function
We have covariance models parameterized from the variance and the local anisotropy, which are both related to the error field: knowing the dynamics of the error leads to the dynamics of the VLATcov parameters
Summary
The Kalman filter (KF) (Kalman, 1960) is one of the backbones of data assimilation. This filter represents the dynamics of a Gaussian distribution all along the analysis and forecast cycles and takes the form of two equations representing the evolution of the mean and of the covariance of the Gaussian distribution. While the equations of the KF are simple linear algebra, the large dimension of linear space encountered in the realm of data assimilation makes the KF impossible to handle, and this is true for the forecast step. This limitation has motivated some approximation of covariance matrix to make the KF possible. SymPKF comes with an automatic code generator to provide an end-to-end exploration of the PKF approach from the computation of the PKF dynamics to their numerical integration.
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