Editorial Type:
Article
Article Type:
Research Article

An Overlooked Issue of Variational Data Assimilation

Benjamin Ménétrier National Center for Atmospheric Research, Mesoscale and Microscale Meteorology Laboratory, Boulder, Colorado

Search for other papers by Benjamin Ménétrier in
Current site
Google Scholar
PubMed Close
and
Thomas Auligné National Center for Atmospheric Research, Mesoscale and Microscale Meteorology Laboratory, Boulder, Colorado

Search for other papers by Thomas Auligné in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Oct 2015
DOI:
https://doi.org/10.1175/MWR-D-14-00404.1
Page(s):
3925–3930
Restricted access

Abstract

The control variable transform (CVT) is a keystone of variational data assimilation. In publications using such a technique, the background term of the transformed cost function is defined as a canonical inner product of the transformed control variable with itself. However, it is shown in this paper that this practical definition of the cost function is not correct if the CVT uses a square root of the background error covariance matrix that is not square. Fortunately, it is then shown that there is a manifold of the control space for which this flaw has no impact, and that most minimizers used in practice precisely work in this manifold. It is also shown that both correct and practical transformed cost functions have the same minimum. This explains more rigorously why the CVT is working in practice. The case of a singular is finally detailed, showing that the practical cost function still reaches the best linear unbiased estimate (BLUE).

Corresponding author address: Benjamin Ménétrier, NCAR/MMM, 3450 Mitchell Lane, Boulder, CO 80301. E-mail: menetrie@ucar.edu

Abstract

The control variable transform (CVT) is a keystone of variational data assimilation. In publications using such a technique, the background term of the transformed cost function is defined as a canonical inner product of the transformed control variable with itself. However, it is shown in this paper that this practical definition of the cost function is not correct if the CVT uses a square root of the background error covariance matrix that is not square. Fortunately, it is then shown that there is a manifold of the control space for which this flaw has no impact, and that most minimizers used in practice precisely work in this manifold. It is also shown that both correct and practical transformed cost functions have the same minimum. This explains more rigorously why the CVT is working in practice. The case of a singular is finally detailed, showing that the practical cost function still reaches the best linear unbiased estimate (BLUE).

Corresponding author address: Benjamin Ménétrier, NCAR/MMM, 3450 Mitchell Lane, Boulder, CO 80301. E-mail: menetrie@ucar.edu
Save
Cover Monthly Weather Review
Monthly Weather Review

  • Bannister, R. N., 2008: A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics. Quart. J. Roy. Meteor. Soc., 134, 1971–1996, doi:10.1002/qj.340.

    • Search Google Scholar
    • Export Citation

  • Buehner, M., 2005: Ensemble-derived stationary and flow-dependent background-error covariances: Evaluation in a quasi-operational NWP setting. Quart. J. Roy. Meteor. Soc., 131, 1013–1043, doi:10.1256/qj.04.15.

    • Search Google Scholar
    • Export Citation

  • Courtier, P., J.-N. Thépaut, and A. Hollingsworth, 1994: A strategy for operational implementation of 4D-Var, using an incremental approach. Quart. J. Roy. Meteor. Soc., 120, 1367–1387, doi:10.1002/qj.49712051912.

    • Search Google Scholar
    • Export Citation

  • Derber, J., and A. Rosati, 1989: A global oceanic data assimilation system. J. Phys. Oceanogr., 19, 1333–1347, doi:10.1175/1520-0485(1989)019<1333:AGODAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • El Akkraoui, A., Y. Trémolet, and R. Todling, 2013: Preconditioning of variational data assimilation and the use of a bi-conjugate gradient method. Quart. J. Roy. Meteor. Soc., 139, 731–741, doi:10.1002/qj.1997.

    • Search Google Scholar
    • Export Citation

  • Fisher, M., 2003: Background error covariance modelling. Proc. ECMWF Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, Reading, United Kingdom, ECMWF, 45–63.

  • Gürol, S., A. T. Weaver, A. M. Moore, A. Piacentini, H. G. Arango, and S. Gratton, 2014: B-preconditioned minimization algorithms for variational data assimilation with the dual formulation. Quart. J. Roy. Meteor. Soc., 140, 539–556, doi:10.1002/qj.2150.

    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., 2003a: Modelling of error covariances by 4D-Var data assimilation. Quart. J. Roy. Meteor. Soc., 129, 3167–3182, doi:10.1256/qj.02.131.

    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., 2003b: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-Var. Quart. J. Roy. Meteor. Soc., 129, 3183–3203, doi:10.1256/qj.02.132.

    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., and Coauthors, 2000: The Met. Office global three-dimensional variational data assimilation scheme. Quart. J. Roy. Meteor. Soc., 126, 2991–3012, doi:10.1002/qj.49712657002.

    • Search Google Scholar
    • Export Citation

  • Parrish, D. F., and J. C. Derber, 1992: The National Meteorological Center’s spectral statistical-interpolation analysis system. Mon. Wea. Rev., 120, 1747–1763, doi:10.1175/1520-0493(1992)120<1747:TNMCSS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Woodbury, M. A., 1950: Inverting modified matrices. Memo. Rep. 42, Vol. 316, Statistical Research Group, Princeton University, Princeton, NJ, 106 pp.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 186 101 6
PDF Downloads 116 53 2