Lognormal and Mixed Gaussian–Lognormal Kalman Filters

Steven J. Fletcher aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Steven J. Fletcher in
Current site
Google Scholar
PubMed
Close
https://orcid.org/0000-0002-1662-7460
,
Milija Zupanski aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Milija Zupanski in
Current site
Google Scholar
PubMed
Close
,
Michael R. Goodliff aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Michael R. Goodliff in
Current site
Google Scholar
PubMed
Close
,
Anton J. Kliewer aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Anton J. Kliewer in
Current site
Google Scholar
PubMed
Close
,
Andrew S. Jones aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Andrew S. Jones in
Current site
Google Scholar
PubMed
Close
,
John M. Forsythe aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by John M. Forsythe in
Current site
Google Scholar
PubMed
Close
,
Ting-Chi Wu aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Ting-Chi Wu in
Current site
Google Scholar
PubMed
Close
,
Md. Jakir Hossen aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Md. Jakir Hossen in
Current site
Google Scholar
PubMed
Close
, and
Senne Van Loon aCooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Search for other papers by Senne Van Loon in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

In this paper we present the derivation of two new forms of the Kalman filter equations; the first is for a pure lognormally distributed random variable, while the second set of Kalman filter equations will be for a combination of Gaussian and lognormally distributed random variables. We show that the appearance is similar to that of the Gaussian-based equations, but that the analysis state is a multivariate median and not the mean. We also show results of the mixed distribution Kalman filter with the Lorenz 1963 model with lognormal errors for the background and observations of the z component, and compare them to analysis results from a traditional Gaussian-based extended Kalman filter and show that under certain circumstances the new approach produces more accurate results.

Michael R. Goodliff’s current affiliation: Cooperative Institute for Research in the Environmental Studies, University of Colorado Boulder and NOAA/Physical Sciences Laboratory, Boulder, Colorado.

Anton J. Kliewer’s current affiliation: Cooperative Institute for Research in the Atmosphere, NOAA/OAR/ESRL/Global Systems Laboratory, Boulder, Colorado.

Ting-Chi Wu’s current affiliation: Ministry of Science and Technology, Taiwan.

Md. Jakir Hossen’s current affiliation: I.M. System’s Group, Inc., College Park, Maryland.

© 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: Steven J. Fletcher, steven.fletcher@colostate.edu

Abstract

In this paper we present the derivation of two new forms of the Kalman filter equations; the first is for a pure lognormally distributed random variable, while the second set of Kalman filter equations will be for a combination of Gaussian and lognormally distributed random variables. We show that the appearance is similar to that of the Gaussian-based equations, but that the analysis state is a multivariate median and not the mean. We also show results of the mixed distribution Kalman filter with the Lorenz 1963 model with lognormal errors for the background and observations of the z component, and compare them to analysis results from a traditional Gaussian-based extended Kalman filter and show that under certain circumstances the new approach produces more accurate results.

Michael R. Goodliff’s current affiliation: Cooperative Institute for Research in the Environmental Studies, University of Colorado Boulder and NOAA/Physical Sciences Laboratory, Boulder, Colorado.

Anton J. Kliewer’s current affiliation: Cooperative Institute for Research in the Atmosphere, NOAA/OAR/ESRL/Global Systems Laboratory, Boulder, Colorado.

Ting-Chi Wu’s current affiliation: Ministry of Science and Technology, Taiwan.

Md. Jakir Hossen’s current affiliation: I.M. System’s Group, Inc., College Park, Maryland.

© 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: Steven J. Fletcher, steven.fletcher@colostate.edu
Save
  • Aseev, N. A., and Y. Y. Shprits, 2019: Reanalysis of ring current electron phase space densities using Van Allen probe observations, convection model, and log-normal Kalman filter. Space Wea., 17, 619638, https://doi.org/10.1029/2018SW002110.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. E., 1997: An introduction to estimation error theory. J. Meteor. Soc. Japan, 75, 257288, https://doi.org/10.2151/jmsj1965.75.1B_257.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., and N. Fabio, 1997: Solving for the generalized inverse of the Lorenz model. J. Meteor. Soc. Japan, 75, 229243, https://doi.org/10.2151/jmsj1965.75.1B_229.

    • Search Google Scholar
    • Export Citation
  • Fletcher, S. J., 2010: Mixed lognormal-Gaussian four-dimensional data assimilation. Tellus, 62A, 266287, https://doi.org/10.1111/j.1600-0870.2010.00439.x.

    • Search Google Scholar
    • Export Citation
  • Fletcher, S. J., 2017: Data Assimilation for the Geosciences: From Theory to Applications. Elsevier, 976 pp.

  • Fletcher, S. J., and M. Zupanski, 2006a: A data assimilation method for log-normally distributed observational errors. Quart. J. Roy. Meteor. Soc., 132, 25052519, https://doi.org/10.1256/qj.05.222.

    • Search Google Scholar
    • Export Citation
  • Fletcher, S. J., and M. Zupanski, 2006b: A hybrid multivariate normal and lognormal distribution for data assimilation. Atmos. Sci. Lett., 7, 4346, https://doi.org/10.1002/asl.128.

    • Search Google Scholar
    • Export Citation
  • Fletcher, S. J., and M. Zupanski, 2007: Implications and impacts of transforming lognormal variables into normal variables in VAR. Meteor. Z., 16, 755765, https://doi.org/10.1127/0941-2948/2007/0243.

    • Search Google Scholar
    • Export Citation
  • Fletcher, S. J., and A. S. Jones, 2014: Multiplicative and additive incremental variational data assimilation for mixed lognormal-Gaussian errors. Mon. Wea. Rev., 142, 25212544, https://doi.org/10.1175/MWR-D-13-00136.1.

    • Search Google Scholar
    • Export Citation
  • Goodliff, M., S. Fletcher, A. Kliewer, J. Forsythe, and A. Jones, 2020: Detection of non-Gaussian behavior using machine learning techniques: A case study on the Lorenz 63 model. J. Geophys. Res. Atmos., 125, e2019JD031551, https://doi.org/10.1029/2019JD031551.

    • Search Google Scholar
    • Export Citation
  • Kalman, R. E., 1960: A new approach to linear filtering and prediction problems. J. Basic Eng., 82, 3545, https://doi.org/10.1115/1.3662552.

    • Search Google Scholar
    • Export Citation
  • Kalman, R. E., and R. S. Bucy, 1961: New results in linear filtering and prediction theory. J. Basic Eng., 83, 95108, https://doi.org/10.1115/1.3658902.

    • Search Google Scholar
    • Export Citation
  • Kliewer, A. J., S. J. Fletcher, A. S. Jones, and J. M. Forsthye, 2016: Comparison of Gaussian, logarithmic transform and mixed Gaussian-log-normal distribution based 1DVAR microwave temperature-water vapour mixing ratio retrievals. Quart. J. Roy. Meteor. Soc., 142, 274286, https://doi.org/10.1002/qj.2651.

    • Search Google Scholar
    • Export Citation
  • Kondrashov, D., M. Ghil, and Y. Shprits, 2011: Lognormal Kalman filter for assimilating phase space density data in the radiation belts. Space Wea., 9, S11006, https://doi.org/10.1029/2011SW000726.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130141, https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Zupanski, M., 2005: Maximum likelihood ensemble filter. Part I: Theoretical aspects. Mon. Wea. Rev., 133, 17101726, https://doi.org/10.1175/MWR2946.1.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 584 275 15
Full Text Views 229 109 6
PDF Downloads 290 143 19