On the Choice of an Optimal Localization Radius in Ensemble Kalman Filter Methods

Paul Kirchgessner Alfred Wegener Institute, Helmholtz Center for Polar and Marine Research, Bremerhaven, Germany

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Lars Nerger Alfred Wegener Institute, Helmholtz Center for Polar and Marine Research, Bremerhaven, Germany

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Angelika Bunse-Gerstner University of Bremen, Bremen, Germany

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Abstract

In data assimilation applications using ensemble Kalman filter methods, localization is necessary to make the method work with high-dimensional geophysical models. For ensemble square root Kalman filters, domain localization (DL) and observation localization (OL) are commonly used. Depending on the localization method, appropriate values have to be chosen for the localization parameters, such as the localization length and the weight function. Although frequently used, the properties of the localization techniques are not fully investigated. Thus, up to now an optimal choice for these parameters is a priori unknown and they are generally found by expensive numerical experiments. In this study, the relationship between the localization length and the ensemble size in DL and OL is studied using twin experiments with the Lorenz-96 model and a two-dimensional shallow-water model. For both models, it is found that the optimal localization length for DL and OL depends linearly on an effective local observation dimension that is given by the sum of the observation weights. In the experiments no influence of the model dynamics on the optimal localization length was observed. The effective observation dimension defines the degrees of freedom that are required for assimilating observations, while the ensemble size defines the available degrees of freedom. Setting the localization radius such that the effective local observation dimension equals the ensemble size yields an adaptive localization radius. Its performance is tested using a global ocean model. The experiments show that the analysis quality using the adaptive localization is similar to the analysis quality of an optimally tuned constant localization radius.

Corresponding author address: Paul Kirchgessner, Alfred Wegener Institute, Helmholtz Center for Polar and Marine Research, Am Handelshafen 12, D-27570 Bremerhaven, Germany. E-mail: paul.kirchgessner@awi.de

Abstract

In data assimilation applications using ensemble Kalman filter methods, localization is necessary to make the method work with high-dimensional geophysical models. For ensemble square root Kalman filters, domain localization (DL) and observation localization (OL) are commonly used. Depending on the localization method, appropriate values have to be chosen for the localization parameters, such as the localization length and the weight function. Although frequently used, the properties of the localization techniques are not fully investigated. Thus, up to now an optimal choice for these parameters is a priori unknown and they are generally found by expensive numerical experiments. In this study, the relationship between the localization length and the ensemble size in DL and OL is studied using twin experiments with the Lorenz-96 model and a two-dimensional shallow-water model. For both models, it is found that the optimal localization length for DL and OL depends linearly on an effective local observation dimension that is given by the sum of the observation weights. In the experiments no influence of the model dynamics on the optimal localization length was observed. The effective observation dimension defines the degrees of freedom that are required for assimilating observations, while the ensemble size defines the available degrees of freedom. Setting the localization radius such that the effective local observation dimension equals the ensemble size yields an adaptive localization radius. Its performance is tested using a global ocean model. The experiments show that the analysis quality using the adaptive localization is similar to the analysis quality of an optimally tuned constant localization radius.

Corresponding author address: Paul Kirchgessner, Alfred Wegener Institute, Helmholtz Center for Polar and Marine Research, Am Handelshafen 12, D-27570 Bremerhaven, Germany. E-mail: paul.kirchgessner@awi.de
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  • Anderson, J. L., 2007: Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D, 230, 99–111, doi:10.1016/j.physd.2006.02.011.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., 2012: Localization and sampling error correction in ensemble Kalman filter data assimilation. Mon. Wea. Rev., 140, 2359–2371, doi:10.1175/MWR-D-11-00013.1.

    • 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, 2029–2044, doi:10.1002/qj.169.

    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., and D. Hodyss, 2009: Ensemble covariances adaptively localized with ECO-RAP. Part 1: Tests on simple error models. Tellus, 61A, 84–96, doi:10.1111/j.1600-0870.2008.00371.x.

    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., B. Etherton, and S. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420–436, doi:10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Danilov, S., G. Kivman, and J. Schröter, 2004: A finite-element ocean model: Principles and evaluation. Ocean Modell., 6, 125–150, doi:10.1016/S1463-5003(02)00063-X.

    • 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, 10 143–10 162, doi:10.1029/94JC00572.

    • 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, 723–757, doi:10.1002/qj.49712555417.

    • 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, 511–522, doi:10.1175/2010MWR3328.1.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P., and H. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796–811, doi:10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P., and H. Mitchell, 2001: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129, 123–137, doi:10.1175/1520-0493(2001)129<0123:ASEKFF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hunt, B., E. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112–126, doi:10.1016/j.physd.2006.11.008.

    • 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, 2046–2060, doi:10.1175/2011MWR3552.1.

    • Search Google Scholar
    • Export Citation
  • Janjić, T., J. Schröter, R. Savcenko, W. Bosch, A. Albertella, R. Rummel, and O. Klatt, 2012: Impact of combining GRACE and GOCE gravity data on ocean circulation estimates. Ocean Sci., 8, 65–79, doi:10.5194/os-8-65-2012.

    • Search Google Scholar
    • Export Citation
  • Kang, J.-S., E. Kalnay, T. Miyoshi, J. Liu, and I. Fung, 2012: Estimation of surface carbon fluxes with an advanced data assimilation methodology. J. Geophys. Res.,117, D24101, doi:10.1029/2012JD018259.

  • Krysta, M., E. Cosme, and J. Verron, 2011: A consistent hybrid variational-smoothing data assimilation method: Application to a simple shallow-water model of the turbulent midlatitude ocean. Mon. Wea. Rev., 139, 3333–3347, doi:10.1175/2011MWR3150.1.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E., 1995: Predictability: A problem partly solved. Proc. Seminar on Predictability, Reading, United Kingdom, ECMWF, 1–18.

  • Losa, S. N., S. Danilov, J. Schröter, L. Nerger, S. Maßmann, and F. Janssen, 2012: Assimilating NOAA SST data into the BSH operational circulation model for the North and Baltic Seas: Inference about the data. J. Mar. Syst., 105–108, 152–162, doi:10.1016/j.jmarsys.2012.07.008.

    • Search Google Scholar
    • Export Citation
  • Migliorini, S., 2013: Information-based data selection for ensemble data assimilation. Quart. J. Roy. Meteor. Soc., 139, 2033–2054, doi:10.1002/qj.2104.

    • Search Google Scholar
    • Export Citation
  • Miyoshi, T., and S. Yamane, 2007: Local ensemble transform Kalman filtering with an AGCM at a T159/L48 resolution. Mon. Wea. Rev., 135, 3841–3861, doi:10.1175/2007MWR1873.1.

    • Search Google Scholar
    • Export Citation
  • Nerger, L., and W. Hiller, 2013: Software for ensemble-based data assimilation systems implementation strategies and scalability. Comput. Geosci., 55, 110–118, doi:10.1016/j.cageo.2012.03.026.

    • Search Google Scholar
    • Export Citation
  • Nerger, L., W. Hiller, and J. Schröter, 2005: PDAF—The parallel data assimilation framework: Experiences with Kalman filtering. Use of High Performance Computing in Meteorology—Proceedings of the 11th ECMWF Workshop, Reading, United Kingdom, ECMWF, 63–66.

  • Nerger, L., S. Danilov, W. Hiller, and J. Schröter, 2006: Using sea-level data to constrain a finite-element primitive-equation ocean model with a local SEIK filter. Ocean Dyn., 56, 634–649, doi:10.1007/s10236-006-0083-0.

    • Search Google Scholar
    • Export Citation
  • Nerger, L., T. Janjic, J. Schroeter, and W. Hiller, 2012: A regulated localization scheme for ensemble-based Kalman filters. Quart. J. Roy. Meteor. Soc., 138, 802–812, doi:10.1002/qj.945.

    • Search Google Scholar
    • Export Citation
  • Otkin, J. A., 2012: Assessing the impact of the covariance localization radius when assimilating infrared brightness temperature observations using an ensemble Kalman filter. Mon. Wea. Rev., 140, 543–561, doi:10.1175/MWR-D-11-00084.1.

    • Search Google Scholar
    • Export Citation
  • Patil, D., B. Hunt, E. Kalnay, J. Yorke, and E. Ott, 2001: Local low dimensionality of atmospheric dynamics. Phys. Rev. Lett., 86, 5878–5881, doi:10.1103/PhysRevLett.86.5878.

    • Search Google Scholar
    • Export Citation
  • Pham, D. T., 2001: Stochastic methods for sequential data assimilation in strongly nonlinear systems. Mon. Wea. Rev., 129, 1194–1207, doi:10.1175/1520-0493(2001)129<1194:SMFSDA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Robert, A. J., 1966: The integration of a low order spectral form of the primitive meteorological equations. J. Meteor. Soc. Japan, 44, 237–245.

    • Search Google Scholar
    • Export Citation
  • Sadourny, R., 1975: The dynamics of finite-difference models of the shallow-water equations. J. Atmos. Sci., 32, 680–689, doi:10.1175/1520-0469(1975)032<0680:TDOFDM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Sakov, P., and L. Bertino, 2011: Relation between two common localization methods for the EnKF. Comput. Geosci., 15, 225–237, doi:10.1007/s10596-010-9202-6.

    • Search Google Scholar
    • Export Citation
  • Sakov, P., F. Counillon, L. Bertino, K. A. Lisæter, P. R. Oke, and A. Korablev, 2012: TOPAZ4: An ocean-sea ice data assimilation system for the North Atlantic and Arctic. Ocean Sci., 8, 633–656, doi:10.5194/os-8-633-2012.

    • Search Google Scholar
    • Export Citation
  • Timmermann, R., S. Danilov, J. Schröter, C. Böning, D. Sidorenko, and K. Rollenhagen, 2009: Ocean circulation and sea ice distribution in a finite element global sea ice ocean model. Ocean Modell., 27, 114–129, doi:10.1016/j.ocemod.2008.10.009.

    • Search Google Scholar
    • Export Citation
  • Wang, Q., S. Danilov, and J. Schröter, 2008: Finite element ocean circulation model based on triangular prismatic elements, with application in studying the effect of topography representation. J. Geophys. Res.,113, C05015, doi:10.1029/2007JC004482.

  • Whitaker, J., and T. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 1913–1924, doi:10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Yoon, Y.-N., E. Ott, and I. Szunyogh, 2010: On the propagation of information and the use of localization in ensemble Kalman filtering. J. Atmos. Sci., 67, 3823–3834, doi:10.1175/2010JAS3452.1.

    • Search Google Scholar
    • Export Citation
  • Zupanski, D., A. S. Denning, M. Uliasz, M. Zupanski, A. E. Schuh, P. J. Rayner, W. Peters, and K. D. Corbin, 2007: Carbon flux bias estimation employing Maximum Likelihood Ensemble Filter (MLEF). J. Geophys. Res.,112, D17107, doi:10.1029/2006JD008371.

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