Maximum-Likelihood Estimation of Forecast and Observation Error Covariance Parameters. Part I: Methodology

Dick P. Dee General Sciences Corporation, Laurel, Maryland, and Data Assimilation Office, NASA/Goddard Space Flight Center, Greenbelt, Maryland

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Arlindo M. da Silva NASA/Goddard Space Flight Center, Greenbelt, Maryland

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Abstract

The maximum-likelihood method for estimating observation and forecast error covariance parameters is described. The method is presented in general terms but with particular emphasis on practical aspects of implementation. Issues such as bias estimation and correction, parameter identifiability, estimation accuracy, and robustness of the method, are discussed in detail. The relationship between the maximum-likelihood method and generalized cross-validation is briefly addressed.

The method can be regarded as a generalization of the traditional procedure for estimating covariance parameters from station data. It does not involve any restrictions on the covariance models and can be used with data from moving observers, provided the parameters to be estimated are identifiable. Any available a priori information about the observation and forecast error distributions can be incorporated into the estimation procedure. Estimates of parameter accuracy due to sampling error are obtained as a by-product.

Corresponding author address: Dr. Dick P. Dee, NASA/GSFC Data Assimilation Office, Mail Code 910.3, Greenbelt, MD 20771.

Email: ddee@dao.gsfc.nasa.gov

Abstract

The maximum-likelihood method for estimating observation and forecast error covariance parameters is described. The method is presented in general terms but with particular emphasis on practical aspects of implementation. Issues such as bias estimation and correction, parameter identifiability, estimation accuracy, and robustness of the method, are discussed in detail. The relationship between the maximum-likelihood method and generalized cross-validation is briefly addressed.

The method can be regarded as a generalization of the traditional procedure for estimating covariance parameters from station data. It does not involve any restrictions on the covariance models and can be used with data from moving observers, provided the parameters to be estimated are identifiable. Any available a priori information about the observation and forecast error distributions can be incorporated into the estimation procedure. Estimates of parameter accuracy due to sampling error are obtained as a by-product.

Corresponding author address: Dr. Dick P. Dee, NASA/GSFC Data Assimilation Office, Mail Code 910.3, Greenbelt, MD 20771.

Email: ddee@dao.gsfc.nasa.gov

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  • Bartello, P., and H. L. Mitchell, 1992: A continuous three-dimensional model of short-range forecast error covariances. Tellus,44A, 217–235.

  • Burg, J. P., D. G. Luenberger, and D. L. Wenger, 1982: Estimation of structured covariance matrices. Proc. IEEE,70, 963–974.

  • Chavent, G., 1979: Identification of distributed parameter systems: About the output least square method, its implementation, and identifiability. Identification and System Parameter Estimation: Proceedings of the Fifth IFAC Symposium, R. Iserman, Ed., Vol. 1, Pergamon Press, 85–97.

  • Cohn, S. E., 1997: Introduction to estimation theory. J. Meteor. Soc. Japan,75, 257–288.

  • Cramér, H., 1946: Mathematical Methods of Statistics. Princeton University Press, 575 pp.

  • Daley, R., 1985: The analysis of synoptic scale divergence by a statistical interpolation scheme. Mon. Wea. Rev.,113, 1066–1079.

  • ——, 1991: Atmospheric Data Analysis. Cambridge University Press, 457 pp.

  • ——, 1993: Estimating observation error statistics for atmospheric data assimilation. Ann. Geophys.,11, 634–647.

  • DAO, 1996: Algorithm theoretical basis document version 1.01. Data Assimilation Office, NASA/Goddard Space Flight Center, Greenbelt, MD. [Available online at http://dao.gsfc.nasa.gov/subpages/atbd.html.].

  • Dee, D. P., 1995: On-line estimation of error covariance parameters for atmospheric data assimilation. Mon. Wea. Rev.,123, 1128–1145.

  • ——, and A. M. da Silva, 1998: Data assimilation in the presence of forecast bias. Quart. J. Roy. Meteor. Soc.,124, 269–295.

  • ——, G. Gaspari, C. Redder, L. Rukhovets, and A. M. da Silva, 1999:Maximum-likelihood estimation of forecast and observation error covariance parameters. Part II: Applications. Mon. Wea. Rev., 1835–1849.

  • Devenyi, D., and T. W. Schlatter, 1994: Statistical properties of 3-hour prediction errors derived from the mesoscale analysis and prediction system. Mon. Wea. Rev.,122, 1263–1280.

  • Fisher, R. A., 1922: On the mathematical foundations of theoretical statistics. Philos. Trans. Roy. Soc. London,222A, 309–368.

  • Gandin, L. S., 1963: Objective Analysis of Meteorological Fields (in Russian). Israel Program for Scientific Translation, 242 pp.

  • Gaspari, G., and S. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc.,125, 723–757.

  • Hollingsworth, A., and P. Lönnberg, 1986: The statistical structure of short-range forecast errors as determined from rawinsonde data. Part I: The wind field. Tellus,38A, 111–136.

  • Lönnberg, P., and A. Hollingsworth, 1986: The statistical structure of short-range forecast errors as determined from rawinsonde data. Part II: The covariance of height and wind errors. Tellus,38A, 137–161.

  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc.,112, 1177–1194.

  • Lupton, R., 1993: Statistics in Theory and Practice. Princeton University Press, 188 pp.

  • Muirhead, R. J., 1982: Aspects of Multivariate Statistical Theory. Wiley, 673 pp.

  • Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992: Numerical Recipes in FORTRAN: The Art of Scientific Computing. 2d ed. Cambridge University Press, 963 pp.

  • Riishøjgaard, L.-P., 1998: A direct way of specifying flow-dependent background error correlations for meteorological analysis systems. Tellus,50A, 42–57.

  • Rutherford, I., 1972: Data assimilation by statistical interpolation of forecast error fields. J. Atmos. Sci.,29, 809–815.

  • Sorenson, H. W., 1980: Parameter Estimation: Principles and Problems. Marcel Dekker, 382 pp.

  • Strang, G., 1988: Linear Algebra and its Applications. 3d ed. Academic Press, 505 pp.

  • Thiébaux, H. J., H. L. Mitchell, and D. W. Shantz, 1986: Horizontal structure of hemispheric forecast error correlations for geopotential and temperature. Mon. Wea. Rev.,114, 1048–1066.

  • ——, L. L. Morone, and R. L. Wobus, 1990: Global forecast error correlation. Part 1: Isobaric wind and geopotential. Mon. Wea. Rev.,118, 2117–2137.

  • Wahba, G., and J. Wendelberger, 1980: Some new mathematical methods for variational objective analysis using splines and cross-validation. Mon. Wea. Rev.,108, 1122–1145.

  • ——, D. R. Johnson, F. Gao, and J. Gong, 1995: Adaptive tuning of numerical weather prediction models: Randomized GCV in three- and four-dimensional data assimilation. Mon. Wea. Rev.,123, 3358–3369.

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