• Alexander, G. D., , J. A. Weinman, , and J. L. Schols, 1998: The use of digital warping of microwave integrated water vapor imagery to improve forecasts of marine extratropical cyclones. Mon. Wea. Rev., 126 , 14691496.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129 , 28842903.

  • 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.

    • 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.

    • Search Google Scholar
    • Export Citation
  • Brewster, K. A., 2003a: Phase-correcting data assimilation and application to storm-scale numerical weather prediction. Part I: Method description and simulation testing. Mon. Wea. Rev., 131 , 480492.

    • Search Google Scholar
    • Export Citation
  • Brewster, K. A., 2003b: Phase-correcting data assimilation and application to storm-scale numerical weather prediction. Part II: Application to a severe storm outbreak. Mon. Wea. Rev., 131 , 493507.

    • Search Google Scholar
    • Export Citation
  • Burgers, G., , P. van Leeuwen, , and G. Evensen, 1998: Analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126 , 17191724.

  • Cohn, S. E., 1993: Dynamics of short-term univariate forecast error covariances. Mon. Wea. Rev., 121 , 31233149.

  • Cohn, S. E., 1997: An introduction to estimation theory. J. Meteor. Soc. Japan, 75 , 257288.

  • Dee, D. P., 1991: Simplification of the Kalman filter for meteorological data assimilation. Quart. J. Roy. Meteor. Soc., 117 , 365384.

    • Search Google Scholar
    • Export Citation
  • Dee, D. P., , and A. M. da Silva, 1998: Data assimilation in the presence of forecast bias. Quart. J. Roy. Meteor. Soc., 124 , 269295.

  • DeMaria, M., cited. 1997: Summary of the Tropical Prediction Center/National Hurricane Center tropical cyclone track and intensity guidance models. [Available online at http://www.nhc.noaa.gov/aboutmodels.html.].

  • Dickinson, S., , and R. A. Brown, 1996: A study of near-surface winds in marine cyclones using multiple satellite sensors. J. Appl. Meteor., 35 , 769781.

    • Search Google Scholar
    • Export Citation
  • Drazin, P. G., , and R. S. Johnson, 1989: Solitons: An Introduction. Cambridge University Press, 226 pp.

  • Epstein, E., 1969: Stochastic dynamic prediction. Tellus, 21 , 739759.

  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99 , 1014310162.

    • Search Google Scholar
    • Export Citation
  • Ghil, M., , and P. Malanotte-Rizzoli, 1991: Data assimilation in meteorology and oceanography. Advances in Geophysics, Vol. 33, Academic Press 141–266.

  • Gordon, N. J., , D. J. Salmond, , and A. F. M. Smith, 1993: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F: Radar Signal Process., 140 , 107113.

    • Search Google Scholar
    • Export Citation
  • Grassotti, C., , H. Iskenderian, , and R. N. Hoffman, 1999: Fusion of surface radar and satellite rainfall data using feature calibration and alignment. J. Appl. Meteor., 38 , 677695.

    • Search Google Scholar
    • Export Citation
  • Henry, A. J., 1916: Weather forecasting—Preliminary statement of the problem. Weather Forecasting in the United States, W. B. 583, A. J. Henry et al., Eds., U.S. Department of Agriculture, Weather Bureau, 69–76.

    • Search Google Scholar
    • Export Citation
  • Hoffman, R. N., , and C. Grassotti, 1996: A technique for assimilating SSM/I observations of marine atmospheric storms: Tests with ECMWF analyses. J. Appl. Meteor., 35 , 11771188.

    • Search Google Scholar
    • Export Citation
  • Hoffman, R. N., , Z. Liu, , J-F. Louis, , and C. Grassotti, 1995: Distortion representation of forecast errors. Mon. Wea. Rev., 123 , 27582770.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., , and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126 , 796811.

    • Search Google Scholar
    • Export Citation
  • Jazwinski, A. H., 1970: Stochastic Processes and Filtering Theory. Academic Press, 376 pp.

  • Lawson, W. G., , and J. A. Hansen, 2004: Implications of stochastic and deterministic filters as ensemble-based data assimilation methods in varying regimes of error growth. Mon. Wea. Rev., 132 , 19661981.

    • Search Google Scholar
    • Export Citation
  • Leslie, L. M., , and G. J. Holland, 1995: On the bogussing of tropical cyclones in numerical models: A comparison of vortex profiles. Meteor. Atmos. Phys., 56 , 101110.

    • Search Google Scholar
    • Export Citation
  • Li, Y., , and D. H. Sattinger, 1998: Matlab codes for nonlinear dispersive wave equations. University of Minnesota Tech. Rep., 18 pp. [Available online at http://www.math.usu.edu/dhs/codes.ps.].

  • Lorenc, A., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112 , 11771194.

  • Mardia, K. V., 1970: Measures of multivariate skewness and kurtosis with applications. Biometrika, 57 , 519530.

  • Mariano, A. J., 1990: Contour analysis: A new approach for melding geophysical fields. J. Atmos. Oceanic Technol., 7 , 285295.

  • Maybeck, P. S., 1979: Stochastic Models, Estimation, and Control. Vol. 141, Mathematics in Science and Engineering, Academic Press, 423 pp.

  • Miller, R. N., , M. Ghil, , and F. Gauthiez, 1994: Advanced data assimilation in strongly nonlinear dynamical systems. J. Atmos. Sci., 51 , 10371056.

    • Search Google Scholar
    • Export Citation
  • Miller, R. N., , E. F. Carter Jr., , and S. T. Blue, 1999: Data assimilation into nonlinear stochastic models. Tellus, 51A , 167194.

  • Muccino, J. C., , and A. F. Bennett, 2002: Generalized inversion of the Korteweg-de Vries equation. Dyn. Atmos. Oceans, 35 , 227263.

  • Nehrkorn, T., , R. N. Hoffman, , C. Grassotti, , and J-F. Louis, 2003: Feature calibration and alignment to represent model forecast errors: Empirical regularization. Quart. J. Roy. Meteor. Soc., 129 , 195218.

    • 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 , 17471763.

    • Search Google Scholar
    • Export Citation
  • Surgi, N., , H-L. Pan, , and S. J. Lord, 1998: Improvement of the NCEP global model over the Tropics: An evaluation of model performance during the 1995 hurricane season. Mon. Wea. Rev., 126 , 12871305.

    • Search Google Scholar
    • Export Citation
  • van Leeuwen, P. J., 2003: A variance-minimizing filter for large-scale applications. Mon. Wea. Rev., 131 , 20712084.

  • Whitaker, J. S., , and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130 , 19131924.

  • Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 40 40 8
PDF Downloads 25 25 7

Alignment Error Models and Ensemble-Based Data Assimilation

View More View Less
  • 1 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts
© Get Permissions
Restricted access

Abstract

The concept of alternative error models is suggested as a means to redefine estimation problems with non-Gaussian additive errors so that familiar and near-optimal Gaussian-based methods may still be applied successfully. The specific example of a mixed error model including both alignment errors and additive errors is examined. Using the specific form of a soliton, an analytical solution to the Korteweg–de Vries equation, the total (additive) errors of states following the mixed error model are demonstrably non-Gaussian for large enough alignment errors, and an ensemble of such states is handled poorly by a traditional ensemble Kalman filter, even if position observations are included. Consideration of the mixed error model itself naturally suggests a two-step approach to state estimation where the alignment errors are corrected first, followed by application of an estimation scheme to the remaining additive errors, the first step aimed at removing most of the non-Gaussianity so the second step can proceed successfully. Taking an ensemble approach for the soliton states in a perfect-model scenario, this two-step approach shows a great improvement over traditional methods in a wide range of observational densities, observing frequencies, and observational accuracies. In cases where the two-step approach is not successful, it is often attributable to the first step not having sufficiently removed the non-Gaussianity, indicating the problem strictly requires an estimation scheme that does not make Gaussian assumptions. However, in these cases a convenient approximation to the two-step approach is available, which trades obtaining a minimum variance estimate ensemble mean for more physically sound updates of the individual ensemble members.

Corresponding author address: W. Gregory Lawson, 77 Massachusetts Avenue, Room 54-1626, Cambridge, MA 02139. Email: wglawson@mit.edu

Abstract

The concept of alternative error models is suggested as a means to redefine estimation problems with non-Gaussian additive errors so that familiar and near-optimal Gaussian-based methods may still be applied successfully. The specific example of a mixed error model including both alignment errors and additive errors is examined. Using the specific form of a soliton, an analytical solution to the Korteweg–de Vries equation, the total (additive) errors of states following the mixed error model are demonstrably non-Gaussian for large enough alignment errors, and an ensemble of such states is handled poorly by a traditional ensemble Kalman filter, even if position observations are included. Consideration of the mixed error model itself naturally suggests a two-step approach to state estimation where the alignment errors are corrected first, followed by application of an estimation scheme to the remaining additive errors, the first step aimed at removing most of the non-Gaussianity so the second step can proceed successfully. Taking an ensemble approach for the soliton states in a perfect-model scenario, this two-step approach shows a great improvement over traditional methods in a wide range of observational densities, observing frequencies, and observational accuracies. In cases where the two-step approach is not successful, it is often attributable to the first step not having sufficiently removed the non-Gaussianity, indicating the problem strictly requires an estimation scheme that does not make Gaussian assumptions. However, in these cases a convenient approximation to the two-step approach is available, which trades obtaining a minimum variance estimate ensemble mean for more physically sound updates of the individual ensemble members.

Corresponding author address: W. Gregory Lawson, 77 Massachusetts Avenue, Room 54-1626, Cambridge, MA 02139. Email: wglawson@mit.edu

Save