Model error covariances play a central role in the performance of data assimilation methods applied to nonlinear state-space models. However, these covariances are largely unknown in most of the applications. A misspecification of the model error covariance has a strong impact on the computation of the posterior probability density function, leading to unreliable estimations and even to a total failure of the assimilation procedure. In this work, we propose the combination of the expectation maximization (EM) algorithm with an efficient particle filter to estimate the model error covariance using a batch of observations. Based on the EM algorithm principles, the proposed method encompasses two stages: the expectation stage, in which a particle filter is used with the present updated value of the model error covariance as given to find the probability density function that maximizes the likelihood, followed by a maximization stage, in which the expectation under the probability density function found in the expectation step is maximized as a function of the elements of the model error covariance. This novel algorithm here presented combines the EM algorithm with a fixed point algorithm and does not require a particle smoother to approximate the posterior densities. We demonstrate that the new method accurately and efficiently solves the linear model problem. Furthermore, for the chaotic nonlinear Lorenz-96 model the method is stable even for observation error covariance 10 times larger than the estimated model error covariance matrix and also is successful in moderately large dimensional situations where the dimension of the estimated matrix is 40 x 40.


  1. particle filters
  2. state-space models
  3. model error covariance
  4. EM algorithm

MSC codes

  1. 62M05
  2. 62M20
  3. 60G25
  4. 93E10
  5. 93E11

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Information & Authors


Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 681 - 707
ISSN (online): 2166-2525


Submitted: 4 November 2019
Accepted: 16 February 2021
Published online: 20 May 2021


  1. particle filters
  2. state-space models
  3. model error covariance
  4. EM algorithm

MSC codes

  1. 62M05
  2. 62M20
  3. 60G25
  4. 93E10
  5. 93E11



Funding Information

H2020 European Research Council https://doi.org/10.13039/100010663 : 694509

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