On the Foundation and Different Interpretations of Ensemble Sensitivity

Le Duc aInstitute of Engineering Innovation, The University of Tokyo, Tokyo, Japan
bMeteorological Research Institute, Tsukuba, Japan

Search for other papers by Le Duc in
Current site
Google Scholar
PubMed
Close
https://orcid.org/0000-0002-0529-076X
,
Takuya Kawabata bMeteorological Research Institute, Tsukuba, Japan

Search for other papers by Takuya Kawabata in
Current site
Google Scholar
PubMed
Close
, and
Daisuke Hotta bMeteorological Research Institute, Tsukuba, Japan

Search for other papers by Daisuke Hotta in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

In sensitivity analysis, ensemble sensitivity is defined as the regression coefficients resulting from a simple linear regression of changes of a response function on initial perturbations. One of the interpretations for ensemble sensitivity considers this a simplified version of regression-based adjoint sensitivity called univariate ensemble sensitivity whose derivation involves the so-called diagonal approximation. This approximation, which replaces the analysis error covariance matrix by a diagonal matrix with the same diagonal, helps to avoid inversion of the analysis error covariance, but, at the same time causes confusion in understanding and practical application of ensemble sensitivity. However, some authors have challenged such a controversial interpretation by showing that univariate ensemble sensitivity is multivariate in nature, which raises the necessity for the foundation of ensemble sensitivity. In this study, we have tried to resolve the confusion by establishing a robust foundation for ensemble sensitivity without relying on the controversial diagonality assumption. As employed in some studies, we adopt an impact-based definition for ensemble sensitivity by taking into account probability distributions of analysis perturbations. The mathematical results show that standardized ensemble sensitivity carries in itself three important quantities at the same time: 1) standardized changes of the forecast response with one standard deviation changes of individual state variables, 2) correlations between the forecast response and individual state variables, and 3) the most sensitive analysis perturbation. The theory guarantees validity of ensemble sensitivity, demonstrates its multivariate nature, and explains why ensemble sensitivity is effective in practice.

© 2023 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: Le Duc, leduc@sogo.t.u-tokyo.ac.jp

Abstract

In sensitivity analysis, ensemble sensitivity is defined as the regression coefficients resulting from a simple linear regression of changes of a response function on initial perturbations. One of the interpretations for ensemble sensitivity considers this a simplified version of regression-based adjoint sensitivity called univariate ensemble sensitivity whose derivation involves the so-called diagonal approximation. This approximation, which replaces the analysis error covariance matrix by a diagonal matrix with the same diagonal, helps to avoid inversion of the analysis error covariance, but, at the same time causes confusion in understanding and practical application of ensemble sensitivity. However, some authors have challenged such a controversial interpretation by showing that univariate ensemble sensitivity is multivariate in nature, which raises the necessity for the foundation of ensemble sensitivity. In this study, we have tried to resolve the confusion by establishing a robust foundation for ensemble sensitivity without relying on the controversial diagonality assumption. As employed in some studies, we adopt an impact-based definition for ensemble sensitivity by taking into account probability distributions of analysis perturbations. The mathematical results show that standardized ensemble sensitivity carries in itself three important quantities at the same time: 1) standardized changes of the forecast response with one standard deviation changes of individual state variables, 2) correlations between the forecast response and individual state variables, and 3) the most sensitive analysis perturbation. The theory guarantees validity of ensemble sensitivity, demonstrates its multivariate nature, and explains why ensemble sensitivity is effective in practice.

© 2023 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: Le Duc, leduc@sogo.t.u-tokyo.ac.jp
Save
  • Ancell, B. C., and G. J. Hakim, 2007: Comparing adjoint- and ensemble-sensitivity analysis with applications to observation targeting. Mon. Wea. Rev., 135, 41174134, https://doi.org/10.1175/2007MWR1904.1.

    • Search Google Scholar
    • Export Citation
  • Ancell, B. C., and A. A. Coleman, 2022: New perspectives on ensemble sensitivity analysis with applications to a climatology of severe convection. Bull. Amer. Meteor. Soc., 103, E511E530, https://doi.org/10.1175/BAMS-D-20-0321.1.

    • Search Google Scholar
    • Export Citation
  • Bednarczyk, C. N., and B. C. Ancell, 2015: Ensemble sensitivity analysis applied to a southern plains convective event. Mon. Wea. Rev., 143, 230249, https://doi.org/10.1175/MWR-D-13-00321.1.

    • Search Google Scholar
    • Export Citation
  • Bishop, C. M., 2006: Pattern Recognition and Machine Learning. Springer, 738 pp.

  • Brown, B. R., and G. J. Hakim, 2015: Sensitivity of intensifying Atlantic hurricanes to vortex structure. Quart. J. Roy. Meteor. Soc., 141, 25382551, https://doi.org/10.1002/qj.2540.

    • Search Google Scholar
    • Export Citation
  • Chang, E. K. M., M. Zheng, and K. Raeder, 2013: Medium-range ensemble sensitivity analysis of two extreme Pacific extratropical cyclones. Mon. Wea. Rev., 141, 211231, https://doi.org/10.1175/MWR-D-11-00304.1.

    • Search Google Scholar
    • Export Citation
  • Errico, R. M., and T. Vukicevic, 1992: Sensitivity analysis using an adjoint of the PSU–NCAR mesoscale model. Mon. Wea. Rev., 120, 16441660, https://doi.org/10.1175/1520-0493(1992)120<1644:SAUAAO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Garcies, L., and V. Homar, 2009: Ensemble sensitivities of the real atmosphere: Application to Mediterranean intense cyclones. Tellus, 61A, 394406, https://doi.org/10.1111/j.1600-0870.2009.00392.x.

    • Search Google Scholar
    • Export Citation
  • Hacker, J. P., and L. Lei, 2015: Multivariate ensemble sensitivity with localization. Mon. Wea. Rev., 143, 20132027, https://doi.org/10.1175/MWR-D-14-00309.1.

    • Search Google Scholar
    • Export Citation
  • Hakim, G. J., and R. D. Torn, 2008: Ensemble synoptic analysis. Synoptic-Dynamic Meteorology and Weather Analysis and Forecasting: A Tribute to Fred Sanders, Meteor. Monogr., No. 33, Amer. Meteor. Soc., 147–162, https://doi.org/10.1175/0065-9401-33.55.147.

  • Hanley, K. E., D. J. Kirshbaum, N. M. Roberts, and G. Leoncini, 2013: Sensitivities of a squall line over central Europe in a convective-scale ensemble. Mon. Wea. Rev., 141, 112133, https://doi.org/10.1175/MWR-D-12-00013.1.

    • Search Google Scholar
    • Export Citation
  • Hill, A. J., C. C. Weiss, and B. C. Ancell, 2016: Ensemble sensitivity analysis for mesoscale forecasts of dryline convection initiation. Mon. Wea. Rev., 144, 41614182, https://doi.org/10.1175/MWR-D-15-0338.1.

    • Search Google Scholar
    • Export Citation
  • Ito, K., and C.-C. Wu, 2013: Typhoon-position-oriented sensitivity analysis. Part I: Theory and verification. J. Atmos. Sci., 70, 25252546, https://doi.org/10.1175/JAS-D-12-0301.1.

    • Search Google Scholar
    • Export Citation
  • Langland, R. H., R. L. Elsberry, and R. M. Errico, 1995: Evaluation of physical processes in an idealized extratropical cyclone using adjoint sensitivity. Quart. J. Roy. Meteor. Soc., 121, 13491386, https://doi.org/10.1002/qj.49712152608.

    • Search Google Scholar
    • Export Citation
  • Limpert, G. L., and A. L. Houston, 2018: Ensemble sensitivity analysis for targeted observations of supercell thunderstorms. Mon. Wea. Rev., 146, 17051721, https://doi.org/10.1175/MWR-D-17-0029.1.

    • Search Google Scholar
    • Export Citation
  • Ren, S., L. Lei, Z.-M. Tan, and Y. Zhang, 2019: Multivariate ensemble sensitivity analysis for Super Typhoon Haiyan (2013). Mon. Wea. Rev., 147, 34673480, https://doi.org/10.1175/MWR-D-19-0074.1.

    • Search Google Scholar
    • Export Citation
  • Saltelli, A., M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola, 2008: Global Sensitivity Analysis: The Primer. John Wiley and Sons, 304 pp.

  • Torn, R. D., 2010: Ensemble-based sensitivity analysis applied to African easterly waves. Wea. Forecasting, 25, 6178, https://doi.org/10.1175/2009WAF2222255.1.

    • Search Google Scholar
    • Export Citation
  • Torn, R. D., and G. J. Hakim, 2008: Ensemble-based sensitivity analysis. Mon. Wea. Rev., 136, 663677, https://doi.org/10.1175/2007MWR2132.1.

    • Search Google Scholar
    • Export Citation
  • Torn, R. D., and G. J. Hakim, 2009: Initial-condition sensitivity of western Pacific extratropical transitions determined using ensemble-based sensitivity analysis. Mon. Wea. Rev., 137, 33883406, https://doi.org/10.1175/2009MWR2879.1.

    • Search Google Scholar
    • Export Citation
  • Yokota, S., H. Seko, M. Kunii, H. Yamauchi, and H. Niino, 2016: The tornadic supercell on the Kanto Plain on 6 May 2012: Polarimetric radar and surface data assimilation with EnKF and ensemble-based sensitivity analysis. Mon. Wea. Rev., 144, 31333157, https://doi.org/10.1175/MWR-D-15-0365.1.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 367 366 11
Full Text Views 205 205 3
PDF Downloads 237 237 4