A Four-Dimensional Incremental Analysis Update for the Ensemble Kalman Filter

Lili Lei Key Laboratory of Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China, and Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, Colorado

Search for other papers by Lili Lei in
Current site
Google Scholar
PubMed
Close
and
Jeffrey S. Whitaker NOAA/Earth System Research Laboratory/Physical Sciences Division, Boulder, Colorado

Search for other papers by Jeffrey S. Whitaker in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced during a model integration is often computed once and assumed to be constant for an assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus, the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems.

To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) used with the EnKF is presented. It constructs time-varying analysis increments by applying all observations in an assimilation window to state variables at different times during the assimilation window. It then gradually applies these time-varying analysis increments through the assimilation window. Results from a dry two-layer primitive equation model and the NCEP GFS show that EnKF with 4DIAU (EnKF-4DIAU) and 3DIAU (EnKF-3DIAU) reduce imbalances in the analysis compared to EnKF without initialization (EnKF-RAW). EnKF-4DIAU retains the time-varying information in the analysis increments better than EnKF-3DIAU, and produces better analysis and forecast than either EnKF-RAW or EnKF-3DIAU.

Corresponding author address: Lili Lei, School of Atmospheric Sciences, Nanjing University, 163 Xianlin Ave., Nanjing 210023, China. E-mail: lililei@nju.edu.cn

Abstract

The analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced during a model integration is often computed once and assumed to be constant for an assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus, the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems.

To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) used with the EnKF is presented. It constructs time-varying analysis increments by applying all observations in an assimilation window to state variables at different times during the assimilation window. It then gradually applies these time-varying analysis increments through the assimilation window. Results from a dry two-layer primitive equation model and the NCEP GFS show that EnKF with 4DIAU (EnKF-4DIAU) and 3DIAU (EnKF-3DIAU) reduce imbalances in the analysis compared to EnKF without initialization (EnKF-RAW). EnKF-4DIAU retains the time-varying information in the analysis increments better than EnKF-3DIAU, and produces better analysis and forecast than either EnKF-RAW or EnKF-3DIAU.

Corresponding author address: Lili Lei, School of Atmospheric Sciences, Nanjing University, 163 Xianlin Ave., Nanjing 210023, China. E-mail: lililei@nju.edu.cn
Save
  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903, doi:10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., 2009: Spatially and temporally varying adaptive covariance inflation for ensemble filters. Tellus, 61A, 7283, doi:10.1111/j.1600-0870.2008.00361.x.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 27412758, doi:10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Baer, F., and J. Tribbia, 1977: On complete filtering of gravity modes through non-linear initialization. Mon. Wea. Rev., 105, 15361539, doi:10.1175/1520-0493(1977)105<1536:OCFOGM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Bergemann, K., and S. Reich, 2010: A mollified ensemble Kalman filter. Quart. J. Roy. Meteor. Soc., 136, 16361643, doi:10.1002/qj.672.

    • Search Google Scholar
    • Export Citation
  • Bloom, S. C., L. L. Takacs, A. M. Da Silva, and D. Ledvina, 1996: Data assimilation using incremental analysis updates. Mon. Wea. Rev., 124, 12561271, doi:10.1175/1520-0493(1996)124<1256:DAUIAU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Buehner, M., P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, 2010a: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part I: Description and single-observation experiments. Mon. Wea. Rev., 138, 15501566, doi:10.1175/2009MWR3157.1.

    • Search Google Scholar
    • Export Citation
  • Buehner, M., P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, 2010b: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part II: One-month experiments with real observations. Mon. Wea. Rev., 138, 15671586, doi:10.1175/2009MWR3158.1.

    • Search Google Scholar
    • Export Citation
  • Buehner, M., and Coauthors, 2015: Implementation of deterministic weather forecasting systems based on ensemble–variational data assimilation at Environment Canada. Part I: The global system. Mon. Wea. Rev., 143, 25322559, doi:10.1175/MWR-D-14-00354.1.

    • Search Google Scholar
    • Export Citation
  • Burgers, G., P. J. van Leeuwen, and G. Evensen, 1998: Analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126, 17191724, doi:10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Carton, J. A., G. Chepurin, X. Cao, and B. Giese, 2000: A Simple Ocean Data Assimilation analysis of the global upper ocean 1950–95. Part I: Methodology. J. Phys. Oceanogr., 30, 294309, doi:10.1175/1520-0485(2000)030<0294:ASODAA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Courtier, P., and Coauthors, 1998: The ECMWF implementation of three-dimensional variational assimilation (3D-Var). I: Formulation. Quart. J. Roy. Meteor. Soc., 124, 17831808, doi:10.1002/qj.49712455002.

    • 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, doi:10.1029/94JC00572.

    • Search Google Scholar
    • Export Citation
  • Fujita, T., D. J. Stensrud, and D. C. Dowell, 2007: Surface data assimilation using an ensemble Kalman filter approach with initial condition and model physics uncertainties. Mon. Wea. Rev., 135, 18461868, doi:10.1175/MWR3391.1.

    • Search Google Scholar
    • Export Citation
  • Gaspari, G., and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723757, doi:10.1002/qj.49712555417.

    • Search Google Scholar
    • Export Citation
  • Gottwald, G. A., 2014: Controlling balance in an ensemble Kalman filter. Nonlinear Processes Geophys., 21, 417426, doi:10.5194/npg-21-417-2014.

    • 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, doi:10.1175/2010MWR3328.1.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., and J. S. Whitaker, 2005: Accounting for the error due to unresolved scales in ensemble data assimilation: A comparison of different approaches. Mon. Wea. Rev., 133, 31323147, doi:10.1175/MWR3020.1.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., J. S. Whitaker, and C. Snyder, 2001: Distance-dependent filtering of background-error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 27762790, doi:10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2.

    • 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, doi:10.1175/1520-0493(2001)129<0123:ASEKFF>2.0.CO;2.

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

    • Search Google Scholar
    • Export Citation
  • Huang, B., J. L. Kinter III, and P. S. Schopf, 2002: Ocean data assimilation using intermittent analyses and continuous model error correction. Adv. Atmos. Sci., 19, 965993, doi:10.1007/s00376-002-0059-z.

    • Search Google Scholar
    • Export Citation
  • Huang, X.-Y., and P. Lynch, 1993: Diabatic digital-filtering initialization: Application to the HIRLAM model. Mon. Wea. Rev., 121, 589603, doi:10.1175/1520-0493(1993)121<0589:DDFIAT>2.0.CO;2.

    • 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, doi:10.1016/j.physd.2006.11.008.

    • Search Google Scholar
    • Export Citation
  • Juckes, M., and B. Lawrence, 2009: Inferred variables in data assimilation: Quantifying sensitivity to inaccurate error statistics. Tellus, 61A, 129143, doi:10.1111/j.1600-0870.2008.00376.x.

    • Search Google Scholar
    • Export Citation
  • Kalman, R. E., 1960: A new approach to linear filtering and prediction problems. J. Basic Eng., 82, 3545, doi:10.1115/1.3662552.

  • Kalnay, E., 2002: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 368 pp.

  • Kepert, J., 2009: Covariance localisation and balance in an ensemble Kalman filter. Quart. J. Roy. Meteor. Soc., 135, 11571176, doi:10.1002/qj.443.

    • Search Google Scholar
    • Export Citation
  • Kleist, D. T., and K. Ide, 2015: An OSSE-based evaluation of hybrid variational–ensemble data assimilation for the NCEP GFS. Part II: 4DEnVar and hybrid variants. Mon. Wea. Rev., 143, 452470, doi:10.1175/MWR-D-13-00350.1.

    • Search Google Scholar
    • Export Citation
  • Kleist, D. T., D. F. Parrish, J. C. Derber, R. Treadon, W. Wu, and S. Lord, 2009: Introduction of the GSI into NCEP Global Data Assimilation System. Wea. Forecasting, 24, 16911705, doi:10.1175/2009WAF2222201.1.

    • Search Google Scholar
    • Export Citation
  • Lei, L., D. R. Stauffer, S. E. Haupt, and G. S. Young, 2012a: A hybrid nudging- ensemble Kalman filter approach to data assimilation. Part I: Application in the Lorenz system. Tellus, 64A, 18484, doi:10.3402/tellusa.v64i0.18484.

    • Search Google Scholar
    • Export Citation
  • Lei, L., D. R. Stauffer, and A. Deng, 2012b: A hybrid nudging-ensemble Kalman filter approach to data assimilation. Part II: Application in a shallow- water model. Tellus, 64A, 18485, doi:10.3402/tellusa.v64i0.18485.

    • Search Google Scholar
    • Export Citation
  • Lei, L., D. R. Stauffer, and A. Deng, 2012c: A hybrid nudging-ensemble Kalman filter approach to data assimilation in WRF/DART. Quart. J. Roy. Meteor. Soc., 138, 20662078, doi:10.1002/qj.1939.

    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112, 11771194, doi:10.1002/qj.49711247414.

    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., N. Bowler, A. Clayton, S. Pring, and D. Fairbairn, 2015: Comparison of hybrid-4DEnVar and hybrid-4DVar data assimilation methods for global NWP. Mon. Wea. Rev., 143, 212229, doi:10.1175/MWR-D-14-00195.1.

    • Search Google Scholar
    • Export Citation
  • Lynch, P., and X.-Y. Huang, 1992: Initialization of the HIRLAM model using a digital filter. Mon. Wea. Rev., 120, 10191034, doi:10.1175/1520-0493(1992)120<1019:IOTHMU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Machenhauer, B., 1977: On the dynamics of gravity oscillations in a shallow water model with applications to normal mode initialization. Contrib. Atmos. Phys., 50, 253271.

    • Search Google Scholar
    • Export Citation
  • Mitchell, H. L., P. L. Houtekamer, and G. Pellerin, 2002: Ensemble size, balance, and model-error representation in an ensemble Kalman filter. Mon. Wea. Rev., 130, 27912808, doi:10.1175/1520-0493(2002)130<2791:ESBAME>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • NCAR Developmental Testbed Center, 2015: NOAA ensemble Kalman filter (beta release v1.0 compatible with GSI community release v3.3) user’s guide. NCAR, 55 pp. [Available online at http://www.dtcenter.org/com-GSI/users/docs/enkf_users_guide/EnKF_UserGuide_v1.0Beta.pdf.]

  • Ourmières, Y., J.-M. Brankart, L. Berline, P. Brasseur, and J. Verron, 2006: Incremental analysis update implementation into a sequential ocean data assimilation system. J. Atmos. Oceanic Technol., 23, 17291744, doi:10.1175/JTECH1947.1.

    • Search Google Scholar
    • Export Citation
  • Palmer, T., R. Buizza, F. Doblas-Reyes, T. Jung, M. Leutbecher, G. Shutts, M. Steinheimer, and A. Weisheimer, 2009: Stochastic parameterization and model uncertainty. ECMWF Tech. Memo. 598, 42 pp.

  • Polavarapu, S., S. Ren, A. M. Clayton, D. Sankey, and Y. Rochon, 2004: On the relationship between incremental analysis updating and incremental digital filtering. Mon. Wea. Rev., 132, 24952502, doi:10.1175/1520-0493(2004)132<2495:OTRBIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rienecker, M., and Coauthors, 2007: The GEOS-5 data assimilation system—Documentation of versions 5.0.1 and 5.1.0. NASA GSFC Tech. Rep. Series on Global Modeling and Data Assimilation, NASA/TM-2007-104606, Vol. 27, 92 pp.

  • Thepaut, J.-N., R. Hoffman, and P. Courtier, 1993: Interactions of dynamics and observations in a four-dimensional variational data assimilation. Mon. Wea. Rev., 121, 33933414, doi:10.1175/1520-0493(1993)121<3393:IODAOI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Wang, X., and T. Lei, 2014: GSI-based four-dimensional ensemble–variational (4DEnsVar) data assimilation: Formulation and single-resolution experiments with real data for NCEP Global Forecast System. Mon. Wea. Rev., 142, 33033325, doi:10.1175/MWR-D-13-00303.1.

    • Search Google Scholar
    • Export Citation
  • Wang, X., D. Parrish, D. Kleist, and J. S. Whitaker, 2013: GSI 3DVar-based ensemble-variational hybrid data assimilation for NCEP Global Forecast System: Single-resolution experiments. Mon. Wea. Rev., 141, 40984117, doi:10.1175/MWR-D-12-00141.1.

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

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., and T. M. Hamill, 2012: Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev., 140, 30783089, doi:10.1175/MWR-D-11-00276.1.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., T. M. Hamill, X. Wei, Y. Song, and Z. Toth, 2008: Ensemble data assimilation with the NCEP Global Forecast System. Mon. Wea. Rev., 136, 463482, doi:10.1175/2007MWR2018.1.

    • Search Google Scholar
    • Export Citation
  • Wu, W. S., R. J. Purser, and D. F. Parrish, 2002: Three-dimensional variational analysis with spatially inhomogeneous covariances. Mon. Wea. Rev., 130, 29052916, doi:10.1175/1520-0493(2002)130<2905:TDVAWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Yan, Y., A. Barth, and J. M. Beckers, 2014: Comparison of different assimilation schemes in a sequential Kalman filter system. Ocean Modell., 73, 123137, doi:10.1016/j.ocemod.2013.11.002.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., C. Snyder, and J. Sun, 2004: Impacts of initial estimate and observation availability on convective-scale data assimilation with ensemble Kalman filter. Mon. Wea. Rev., 132, 12381253, doi:10.1175/1520-0493(2004)132<1238:IOIEAO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Zhu, Y., R. Todling, J. Guo, S. E. Cohn, I. M. Navon, and Y. Yang, 2003: The GEOS-3 Retrospective Data Assimilation System: The 6-hour lag case. Mon. Wea. Rev., 131, 21292150, doi:10.1175/1520-0493(2003)131<2129:TGRDAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Zou, X., A. Barcilon, I. M. Navon, J. Whitaker, and D. G. Cacuci, 1993: An adjoint sensitivity study of blocking in a two-layer isentropic model. Mon. Wea. Rev., 121, 28332857, doi:10.1175/1520-0493(1993)121<2833:AASSOB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 1366 491 30
PDF Downloads 1065 401 35