The Effect of the Equivalent-Weights Particle Filter on Dynamical Balance in a Primitive Equation Model

Melanie Ades Department of Meteorology, University of Reading, Reading, United Kingdom

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Peter Jan van Leeuwen Department of Meteorology, University of Reading, Reading, United Kingdom

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Abstract

The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed.

The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed.

This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.

Denotes Open Access content.

Corresponding author address: Melanie Ades, Dept. of Meteorology, University of Reading, Earley Gate, P.O. Box 243, Reading RG6 6BB, United Kingdom. E-mail: m.ades@reading.ac.uk

Abstract

The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed.

The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed.

This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.

Denotes Open Access content.

Corresponding author address: Melanie Ades, Dept. of Meteorology, University of Reading, Earley Gate, P.O. Box 243, Reading RG6 6BB, United Kingdom. E-mail: m.ades@reading.ac.uk
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  • Adcroft, A. J., C. N. Hill, and J. C. Marshall, 1999: A new treatment of the Coriolis terms in C-grid models at both high and low resolutions. Mon. Wea. Rev., 127, 19281936, doi:10.1175/1520-0493(1999)127<1928:ANTOTC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ades, M., and P. J. Van Leeuwen, 2013: An exploration of the equivalent weights particle filter. Quart. J. Roy. Meteor. Soc., 139, 820840, doi:10.1002/qj.1995.

    • Search Google Scholar
    • Export Citation
  • Ades, M., and P. J. Van Leeuwen, 2014: The equivalent-weights particle filter in a high-dimensional system. Quart. J. Roy. Meteor. Soc., doi:10.1002/qj.2370, in press.

    • Search Google Scholar
    • Export Citation
  • 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
  • Bannister, R. N., 2008: A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances. Quart. J. Roy. Meteor. Soc., 134, 19511970, doi:10.1002/qj.339.

    • 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, doi:10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Brodeau, L., B. Barnier, A. M. Treguier, T. Penduff, and S. Gulev, 2010: An ERA40-based atmospheric forcing for global ocean circulation models. Ocean Modell., 31, 88104, doi:10.1016/j.ocemod.2009.10.005.

    • Search Google Scholar
    • Export Citation
  • Buehner, M., P. Houtekamer, C. Charette, H. L. Mitchell, and B. He, 2010: 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
  • 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
  • Courtier, P., and O. Talagrand, 1987: Variational assimilation of meteorological observations with the adjoint vorticity equation. II: Numerical results. Quart. J. Roy. Meteor. Soc., 113, 13291347, doi:10.1002/qj.49711347813.

    • Search Google Scholar
    • Export Citation
  • Daley, R., 1991: Atmospheric Data Analysis. Cambridge University Press, 457 pp.

  • Dance, S. L., 2004: Issues in high resolution limited area data assimilation for quantitative precipitation forecasting. Physica D, 196, 127, doi:10.1016/j.physd.2004.05.001.

    • Search Google Scholar
    • Export Citation
  • Doucet, A., N. de Freitas, and N. Gordon, Eds., 2001: Sequential Monte Carlo Methods in Practice. Springer, 581 pp.

  • 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
  • Evensen, G., 2003: The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dyn., 53, 343367, doi:10.1007/s10236-003-0036-9.

    • Search Google Scholar
    • Export Citation
  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • 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
  • Kitagawa, G., 1996: Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graphical Stat., 10, 253259. [Available online at www.jstor.org/stable/1390750.]

    • Search Google Scholar
    • Export Citation
  • Le Dimet, F.-X., and O. Talagrand, 1986: Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus, 38A, 97110, doi:10.1111/j.1600-0870.1986.tb00459.x.

    • Search Google Scholar
    • Export Citation
  • Liu, C., Q. Xiao, and B. Wang, 2008: An ensemble-based four-dimensional variational data assimilation scheme. Part I: Technical formulation and preliminary test. Mon. Wea. Rev., 136, 33633373, doi:10.1175/2008MWR2312.1.

    • Search Google Scholar
    • Export Citation
  • Lorenc, A., 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., 2003a: Modelling of error covariances by 4D-Var data assimilation. Quart. J. Roy. Meteor. Soc., 129, 31673182, doi:10.1256/qj.02.131.

    • Search Google Scholar
    • Export Citation
  • Lorenc, A., 2003b: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-Var. Quart. J. Roy. Meteor. Soc., 129, 31833203, doi:10.1256/qj.02.132.

    • Search Google Scholar
    • Export Citation
  • Lorenc, A., and Coauthors, 2000: The Met. Office global three-dimensional variational data assimilation scheme. Quart. J. Roy. Meteor. Soc., 126, 29913012, doi:10.1002/qj.49712657002.

    • Search Google Scholar
    • Export Citation
  • Morzfeld, M., X. Tu, E. Atkins, and A. J. Chorin, 2012: A random map implementation of implicit particle filters. J. Comput. Phys., 231, 20492066, doi:10.1016/j.jcp.2011.11.022.

    • Search Google Scholar
    • Export Citation
  • Papadakis, N., E. Mmin, A. Cuzol, and N. Gengembre, 2010: Data assimilation with the weighted ensemble Kalman filter. Tellus, 62A, 673697, doi:10.1111/j.1600-0870.2010.00461.x.

    • Search Google Scholar
    • Export Citation
  • Snyder, C., T. Bengtsson, P. Bickel, and J. Anderson, 2008: Obstacles to high-dimensional particle filtering. Mon. Wea. Rev., 136, 46294640, doi:10.1175/2008MWR2529.1.

    • Search Google Scholar
    • Export Citation
  • Sun, J., 2005: Convective-scale assimilation of radar data: Progresses and challenges. Quart. J. Roy. Meteor. Soc., 131, 34393463, doi:10.1256/qj.05.149.

    • Search Google Scholar
    • Export Citation
  • Talagrand, O., and P. Courtier, 1987: Variational assimilation of meteorological observations with the adjoint vorticity equation. I. Theory. Quart. J. Roy. Meteor. Soc., 113, 13111328, doi:10.1002/qj.49711347812.

    • Search Google Scholar
    • Export Citation
  • Van Leeuwen, P. J., 2009: Particle filtering in geophysical systems. Mon. Wea. Rev., 137, 40894114, doi:10.1175/2009MWR2835.1.

  • Van Leeuwen, P. J., 2010: Nonlinear data assimilation in geosciences: An extremely efficient particle filter. Quart. J. Roy. Meteor. Soc., 136, 19911999, doi:10.1002/qj.699.

    • Search Google Scholar
    • Export Citation
  • Weare, J., 2009: Particle filtering with path sampling and an application to a bimodal ocean current model. J. Comput. Phys., 228, 43124331, doi:10.1016/j.jcp.2009.02.033.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., and T. H. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131923, doi:10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
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