• Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903, https://doi.org/10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., 2007: Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D, 230, 99111, https://doi.org/10.1016/j.physd.2006.02.011.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., and L. Lei, 2013: Empirical localization of observation impact in ensemble Kalman filters. Mon. Wea. Rev., 141, 41404153, https://doi.org/10.1175/MWR-D-12-00330.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., and D. Hodyss, 2007: Flow-adaptive moderation of spurious ensemble correlations and its use in ensemble-based data assimilation. Quart. J. Roy. Meteor. Soc., 133, 20292044, https://doi.org/10.1002/qj.169.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., B. J. Etherton, and S. J. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420436, https://doi.org/10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., B. Huang, and X. Wang, 2015: A nonvariational consistent hybrid ensemble filter. Mon. Wea. Rev., 143, 50735090, https://doi.org/10.1175/MWR-D-14-00391.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., J. S. Whitaker, and L. Lei, 2017: Gain form of the ensemble transform Kalman filter and its relevance to satellite data assimilation with model space ensemble covariance localization. Mon. Wea. Rev., 145, 45754592, https://doi.org/10.1175/MWR-D-17-0102.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buehner, M., and M. Charron, 2007: Spectral and spatial localization of background-error correlations for data assimilation. Quart. J. Roy. Meteor. Soc., 133, 615630, https://doi.org/10.1002/qj.50.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 14310 162, https://doi.org/10.1029/94JC00572.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fairbairn, D., S. R. Pring, A. C. Lorenc, and I. Roulstone, 2014: A comparison of 4DVar with ensemble data assimilation methods. Quart. J. Roy. Meteor. Soc., 140, 281294, https://doi.org/10.1002/qj.2135.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gasperoni, N. A., and X. Wang, 2015: Adaptive localization for the ensemble-based observation impact estimate using regression confidence factors. Mon. Wea. Rev., 143, 19812000, https://doi.org/10.1175/MWR-D-14-00272.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Greybush, S. J., E. Kalnay, T. Miyoshi, K. Ide, and B. R. Hunt, 2011: Balance and ensemble Kalman filter localization techniques. Mon. Wea. Rev., 139, 511522, https://doi.org/10.1175/2010MWR3328.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., 2006: Ensemble-based atmospheric data assimilation. Predictability of Weather and Climate, T. Palmer and R. Hagedorn, Eds., Cambridge University Press, 124–156.

    • Crossref
    • Export Citation
  • Holland, B., and X. Wang, 2013: Effects of sequential or simultaneous assimilation of observations and localization methods on the performance of the ensemble Kalman filter. Quart. J. Roy. Meteor. Soc., 139, 758770, https://doi.org/10.1002/qj.2006.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796811, https://doi.org/10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 2001: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129, 123137, https://doi.org/10.1175/1520-0493(2001)129<0123:ASEKFF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 2005: Ensemble Kalman filtering. Quart. J. Roy. Meteor. Soc., 131, 32693289, https://doi.org/10.1256/qj.05.135.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and F. Zhang, 2016: Review of the ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 144, 44894532, https://doi.org/10.1175/MWR-D-15-0440.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huang, B., and X. Wang, 2018: On the use of cost-effective valid-time-shifting (VTS) method to increase ensemble size in the GFS hybrid 4DEnVar system. Mon. Wea. Rev., 146, 29732998, https://doi.org/10.1175/MWR-D-18-0009.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112126, https://doi.org/10.1016/j.physd.2006.11.008.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ide, K., P. Courtier, M. Ghil, and A. C. Lorenc, 1997: Unified notation for data assimilation: Operational, sequential and variational. J. Meteor. Soc. Japan, 75, 181189, https://doi.org/10.2151/jmsj1965.75.1B_181.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Janjić, T., L. Nerger, A. Albertella, J. Schröter, and S. Skachko, 2011: On domain localization in ensemble-based Kalman filter algorithms. Mon. Wea. Rev., 139, 20462060, https://doi.org/10.1175/2011MWR3552.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kalman, R. E., and R. S. Bucy, 1961: New results in linear filtering and prediction theory. J. Basic Eng., 83, 95–108, https://doi.org/10.1115/1.3658902.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kretschmer, M., B. R. Hunt, and E. Ott, 2015: Data assimilation using a climatologically augmented local ensemble transform Kalman filter. Tellus, 67A, 26617, https://doi.org/10.3402/tellusa.v67.26617.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kuhl, D., and Coauthors, 2007: Assessing predictability with a local ensemble Kalman filter. J. Atmos. Sci., 64, 11161140, https://doi.org/10.1175/JAS3885.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lei, L., and J. S. Whitaker, 2017: Evaluating the trade-offs between ensemble size and ensemble resolution in an ensemble-variational data assimilation system. J. Adv. Model. Earth Syst., 9, 781789, https://doi.org/10.1002/2016MS000864.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., 2017: Improving ensemble covariances in hybrid variational data assimilation without increasing ensemble size. Quart. J. Roy. Meteor. Soc., 143, 10621072, https://doi.org/10.1002/qj.2990.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1996: Predictability: A problem partly solved. Proc. Seminar on Predictability, Shinfield Park, Reading, United Kingdom, European Centre for Medium-Range Weather Forecasts, 1–18.

  • Lorenz, E. N., 2005: Designing chaotic models. J. Atmos. Sci., 62, 15741587, https://doi.org/10.1175/JAS3430.1.

  • Miyoshi, T., and S. Yamane, 2007: Local ensemble transform Kalman filtering with an AGCM at a T159/L48 resolution. Mon. Wea. Rev., 135, 38413861, https://doi.org/10.1175/2007MWR1873.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Miyoshi, T., K. Kondo, and T. Imamura, 2014: The 10,240-member ensemble Kalman filtering with an intermediate AGCM. Geophys. Res. Lett., 41, 52645271, https://doi.org/10.1002/2014GL060863.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nerger, L., T. Janjić, J. Schröter, and W. Hiller, 2012: A regulated localization scheme for ensemble-based Kalman filters. Quart. J. Roy. Meteor. Soc., 138, 802812, https://doi.org/10.1002/qj.945.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Oczkowski, M., I. Szunyogh, and D. J. Patil, 2005: Mechanisms for the development of locally low-dimensional atmospheric dynamics. J. Atmos. Sci., 62, 11351156, https://doi.org/10.1175/JAS3403.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Patil, D. J., B. R. Hunt, E. Kalnay, J. A. Yorke, and E. Ott, 2001: Local low dimensionality of atmospheric dynamics. Phys. Rev. Lett., 86, 58785881, https://doi.org/10.1103/PhysRevLett.86.5878.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rainwater, S., and B. Hunt, 2013: Mixed-resolution ensemble data assimilation. Mon. Wea. Rev., 141, 30073021, https://doi.org/10.1175/MWR-D-12-00234.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sakov, P., and L. Bertino, 2011: Relation between two common localisation methods for the EnKF. Comput. Geosci., 15, 225237, https://doi.org/10.1007/s10596-010-9202-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., and C. H. Bishop, 2003: A comparison of breeding and ensemble transform Kalman filter ensemble forecast schemes. J. Atmos. Sci., 60, 11401158, https://doi.org/10.1175/1520-0469(2003)060<1140:ACOBAE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., C. H. Bishop, and S. J. Julier, 2004: Which is better, an ensemble of positive–negative pairs or a centered spherical simplex ensemble? Mon. Wea. Rev., 132, 15901605, https://doi.org/10.1175/1520-0493(2004)132<1590:WIBAEO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., T. M. Hamill, J. S. Whitaker, and C. H. Bishop, 2007: A Comparison of Hybrid Ensemble Transform Kalman Filter–Optimum Interpolation and Ensemble Square Root Filter Analysis Schemes. Mon. Wea. Rev., 135, 10551076, https://doi.org/10.1175/MWR3307.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924, https://doi.org/10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Xu, Q., L. Wei, H. Lu, C. Qiu, and Q. Zhao, 2008: Time-expanded sampling for ensemble-based filters: Assimilation experiments with a shallow-water equation model. J. Geophys. Res., 113, 112, https://doi.org/10.1029/2007JG000450.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 16 16 16
PDF Downloads 14 14 14

The High-Rank Ensemble Transform Kalman Filter

View More View Less
  • 1 School of Meteorology, University of Oklahoma, Norman, Oklahoma
  • | 2 School of Earth Sciences, University of Melbourne, Melbourne, Australia
Restricted access

Abstract

The ensemble Kalman filter is typically implemented either by applying the localization on the background error covariance matrix (B-localization) or by inflating the observation error variances (R-localization). A mathematical demonstration suggests that for the same effective localization function, the background error covariance matrix from the B-localization method shows a higher rank than the R-localization method. The B-localization method is realized in the ensemble transform Kalman filter (ETKF) by extending the background ensemble perturbations through modulation (MP-localization). Specifically, the modulation functions are constructed from the leading eigenvalues and eigenvectors of the original B-localization matrix. Because of its higher rank than the classic R-localized ETKF, the B-/MP-localized ETKF is termed as the high-rank ETKF (HETKF). The performances of the HETKF and R-localized ETKF were compared through cycled data assimilation experiments using the Lorenz model II. The results show that the HETKF outperforms the R-localized ETKF especially for a small ensemble. The improved analysis in the HETKF is likely associated with the higher rank from the B-/MP-localization method, since its higher rank is expected to contribute more positively to alleviating the rank deficiency issue and thus improve the analysis for a small ensemble. The HETKF is less sensitive to the localization length scales and inflation factors. Furthermore, the experiments suggest that the above conclusion comparing the HETKF and R-localized ETKF does not depend on how the analyzed ensemble perturbations are subselected in the HETKF.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xuguang Wang, xuguang.wang@ou.edu

Abstract

The ensemble Kalman filter is typically implemented either by applying the localization on the background error covariance matrix (B-localization) or by inflating the observation error variances (R-localization). A mathematical demonstration suggests that for the same effective localization function, the background error covariance matrix from the B-localization method shows a higher rank than the R-localization method. The B-localization method is realized in the ensemble transform Kalman filter (ETKF) by extending the background ensemble perturbations through modulation (MP-localization). Specifically, the modulation functions are constructed from the leading eigenvalues and eigenvectors of the original B-localization matrix. Because of its higher rank than the classic R-localized ETKF, the B-/MP-localized ETKF is termed as the high-rank ETKF (HETKF). The performances of the HETKF and R-localized ETKF were compared through cycled data assimilation experiments using the Lorenz model II. The results show that the HETKF outperforms the R-localized ETKF especially for a small ensemble. The improved analysis in the HETKF is likely associated with the higher rank from the B-/MP-localization method, since its higher rank is expected to contribute more positively to alleviating the rank deficiency issue and thus improve the analysis for a small ensemble. The HETKF is less sensitive to the localization length scales and inflation factors. Furthermore, the experiments suggest that the above conclusion comparing the HETKF and R-localized ETKF does not depend on how the analyzed ensemble perturbations are subselected in the HETKF.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xuguang Wang, xuguang.wang@ou.edu
Save