Skip to main content
SpringerLink
Log in

An efficient matrix-free algorithm for the ensemble Kalman filter

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

In this work, we present an efficient matrix-free ensemble Kalman filter (EnKF) algorithm for the assimilation of large data sets. The EnKF has increasingly become an essential tool for data assimilation of numerical models. It is an attractive assimilation method because it can evolve the model covariance matrix for a non-linear model, through the use of an ensemble of model states, and it is easy to implement for any numerical model. Nevertheless, the computational cost of the EnKF can increase significantly for cases involving the assimilation of large data sets. As more data become available for assimilation, a potential bottleneck in most EnKF algorithms involves the operation of the Kalman gain matrix. To reduce the complexity and cost of assimilating large data sets, a matrix-free EnKF algorithm is proposed. The algorithm uses an efficient matrix-free linear solver, based on the Sherman–Morrison formulas, to solve the implicit linear system within the Kalman gain matrix and compute the analysis. Numerical experiments with a two-dimensional shallow water model on the sphere are presented, where results show the matrix-free implementation outperforming an singular value decomposition-based implementation in computational time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bishop, C.H., Etherton, B.J., Majumdar, S.J.: Adaptive sampling with the ensemble transform Kalman filter. Part I: theoretical aspects. Mon. Weather. Rev. 129(3), 420–436 (2001)

    Article  Google Scholar 

  2. Dongarra, J.J., Moler, C.B., Bunch, J.R., Stewart, G.W.: LINPACK User’ Guide. SIAM, Philadelphia (1979)

    Book  Google Scholar 

  3. Egidi, N., Maponi, P.: A Sherman–Morrison approach to the solution of linear systems. J. Comput. Appl. Math. 189, 703–718 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99(C5), 10143–10162 (1994)

    Article  Google Scholar 

  5. Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003)

    Article  Google Scholar 

  6. Evensen, G., van Leeuwen, P.J.: Assimilation of Geosat altimeter data for the Agulhas current using the ensemble Kalman filter with a quasigeostrophic model. Mon. Weather. Rev. 124, 85–96 (1996)

    Article  Google Scholar 

  7. Hager, W.W.: Updating the inverse of a matrix. SIAM Rev. 31, 221–239 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Houtekamer, P.L., Mitchell, H.L., Pellerin, G., Buehner, M., Charron, M., Spacek, L., Hansen, B.: Atmospheric data assimilation with an ensemble Kalman filter: results with real observations. Mon. Weather. Rev. 133, 604–620 (2005)

    Article  Google Scholar 

  9. Houtekamer, P.L., Mitchell, H.L.: Data assimilation using an ensemble Kalman filter technique. Mon. Weather. Rev. 126, 796–811 (1998)

    Article  Google Scholar 

  10. Hunt, B.R., Kostelich, E.J., Szunyogh, I.: Efficient data assimilation for spatiotemporal chaos: a local ensemble transform Kalman filter. Arxiv preprint:physics/0511236 (2005)

  11. Keppenne, C.L., Rienecker, M.M.: Initial testing of a massively parallel ensemble Kalman filter with the Poseidon isopycnal ocean general circulation model. Mon. Weather. Rev. 130(12), 2951–2965 (2002)

    Article  Google Scholar 

  12. Lin, S.J., Chao, W.C., Sud, Y.C., Walker, G.K.: A class of the van Leer-type transport schemes and its application to the moisture transport in a general circulation model. Mon. Weather. Rev. 122, 1575–1593 (1994)

    Article  Google Scholar 

  13. Lin, S.J., Rood, R.B.: An explicit flux-form semi-Lagrangian shallow-water model on the sphere. Q. J. R. Meteorol. Soc. 123, 2477–2498 (1997)

    Article  Google Scholar 

  14. Mandel, J.: Efficient implementation of the ensemble Kalman filter. Tech. Report 231, University of Colorado at Denver and Health Sciences Center (2006)

  15. Maponi, P.: The solution of linear systems by using the Sherman–Morrison formula. Linear Alg. Appl. 420, 276–294 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. McDonald, A., Bates, J.R.: Semi-Lagrangian integration of a gridpoint shallow water model on the sphere. Mon. Weather Rev. 117(1), 130–137 (1989)

    Article  Google Scholar 

  17. Neta, B., Giraldo, F.X., Navon, I.M.: Analysis of the Turkel–Zwas scheme for the two-dimensional shallow water equations in spherical coordinates. J. Comp. Phys. 133, 102–112 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Reichle, R.H., McLaughlin, D.B., Entekhabi, D.: Hydrologic data assimilation with the ensemble Kalman filter. Mon. Weather. Rev. 130, 103–114 (2002)

    Article  Google Scholar 

  19. Sherman, J., Morrison, W.J.: Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Stat. 21, 124–127 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  20. Whitaker, J.S., Hamill, T.M.: Ensemble data assimilation without perturbed observations. Mon. Weather. Rev. 130, 1913–1924 (2002)

    Article  Google Scholar 

  21. Xu, L., Daley, R.: Data Assimilation with a barotropically unstable shallow water system using representer algorithms. Tellus 54A, 125–137 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics and Plasma Physics, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Humberto C. Godinez & J. David Moulton

Authors
  1. Humberto C. Godinez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. David Moulton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Humberto C. Godinez.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Godinez, H.C., Moulton, J.D. An efficient matrix-free algorithm for the ensemble Kalman filter. Comput Geosci 16, 565–575 (2012). https://doi.org/10.1007/s10596-011-9268-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-011-9268-9

Keywords

Search

Navigation