Score matching filters
Authors: Turčičová Marie , Mandel Jan , Eben Kryštof 
Published: March 2021

&lt;p&gt;A widely popular group of data assimilation methods in meteorological and geophysical sciences is formed by filters based on Monte-Carlo approximation of the traditional Kalman filter, e.g. &lt;span&gt;E&lt;/span&gt;&lt;span&gt;nsemble Kalman filter &lt;/span&gt;&lt;span&gt;(EnKF)&lt;/span&gt;&lt;span&gt;, &lt;/span&gt;&lt;span&gt;E&lt;/span&gt;&lt;span&gt;nsemble &lt;/span&gt;&lt;span&gt;s&lt;/span&gt;&lt;span&gt;quare-root filter and others. Due to the computational cost, ensemble &lt;/span&gt;&lt;span&gt;size &lt;/span&gt;&lt;span&gt;is &lt;/span&gt;&lt;span&gt;usually &lt;/span&gt;&lt;span&gt;small &lt;/span&gt;&lt;span&gt;compar&lt;/span&gt;&lt;span&gt;ed&lt;/span&gt;&lt;span&gt; to the dimension of the &lt;/span&gt;&lt;span&gt;s&lt;/span&gt;&lt;span&gt;tate &lt;/span&gt;&lt;span&gt;vector. &lt;/span&gt;&lt;span&gt;Traditional &lt;/span&gt; &lt;span&gt;EnKF implicitly uses the sample covariance which is&lt;/span&gt;&lt;span&gt; a poor estimate of the &lt;/span&gt;&lt;span&gt;background covariance matrix - singular and &lt;/span&gt;&lt;span&gt;contaminated by &lt;/span&gt;&lt;span&gt;spurious correlations. &lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;W&lt;/span&gt;&lt;span&gt;e focus on modelling the &lt;/span&gt;&lt;span&gt;background &lt;/span&gt;&lt;span&gt;covariance matrix by means of &lt;/span&gt;&lt;span&gt;a linear model for its inverse. This is &lt;/span&gt;&lt;span&gt;particularly &lt;/span&gt;&lt;span&gt;useful&lt;/span&gt; &lt;span&gt;in&lt;/span&gt;&lt;span&gt; Gauss-Markov random fields (GMRF), &lt;/span&gt;&lt;span&gt;where&lt;/span&gt; &lt;span&gt;the inverse covariance matrix has &lt;/span&gt;&lt;span&gt;a banded &lt;/span&gt;&lt;span&gt;structure&lt;/span&gt;&lt;span&gt;. &lt;/span&gt;&lt;span&gt;The parameters of the model are estimated by the&lt;/span&gt;&lt;span&gt; score matching &lt;/span&gt;&lt;span&gt;method which &lt;/span&gt;&lt;span&gt;provides&lt;/span&gt;&lt;span&gt; estimators in a closed form&lt;/span&gt;&lt;span&gt;, cheap to compute&lt;/span&gt;&lt;span&gt;. The resulting estimate&lt;/span&gt;&lt;span&gt; is a key component of the &lt;/span&gt;&lt;span&gt;proposed &lt;/span&gt;&lt;span&gt;ensemble filtering algorithms. &lt;/span&gt;&lt;span&gt;Under the assumption that the state vector is a GMRF in every time-step, t&lt;/span&gt;&lt;span&gt;he Score matching filter with Gaussian resamplin&lt;/span&gt;&lt;span&gt;g (SMF-GR) &lt;/span&gt;&lt;span&gt;gives&lt;/span&gt;&lt;span&gt; in every time-step a consistent (in the large ensemble limit) estimator of mean and covariance matrix &lt;/span&gt;&lt;span&gt;of the forecast and analysis distribution&lt;/span&gt;&lt;span&gt;. Further, we propose a filtering method called Score matching ensemble filter (SMEF), based on regularization of the EnK&lt;/span&gt;&lt;span&gt;F&lt;/span&gt;&lt;span&gt;. Th&lt;/span&gt;&lt;span&gt;is&lt;/span&gt;&lt;span&gt; filter performs well even for non-Gaussian systems with non-linear dynamic&lt;/span&gt;&lt;span&gt;s&lt;/span&gt;&lt;span&gt;. &lt;/span&gt;&lt;span&gt;The performance of both filters is illustrated on a simple linear convection model and Lorenz-96.&lt;/span&gt;&lt;/p&gt;