195-616B
Topics in Geophysical Fluid Dynamics (Winter Term 2002)
Department of Atmospheric and Oceanic
Sciences,
McGill University
Introduction
to Data Assimilation and applications to atmospheric data analysis
Professor: Pierre Gauthier
Email: pierre.gauthier@uqam.ca
Description
Lecture
No.1 Introduction: data assimilation and the inverse
problem
- Data assimilation and its
relationship
to
the initial value problem of numerical weather prediction.
- Inverse problem: how to
reconstruct
the instantaneous
state of the atmosphere when the number of observations is less than
the
number of degrees of freedom. Regularization based on an a priori estimate,
the background state.
- Random variable, probability distribution. Variance, covariance
and
correlation.
Univariate and multivariate linear regression.
Lecture
No.3 univariate statistical interpolation
- Minimum variance estimate.
- Statistical interpolation
algorithm:
univariate
case.
- Forecast error variance and
covariance.
- Impact of the forecast error
statistics on
the analysis.
- The variational form of
statistical
interpolation:
the 3D-Var.
Lecture
No.4 a Bayesian approach to statistical data assimilation
- Bayes' theorem and generalization of statistical estimation to
non-Gaussian
error statistics
- Application to the quality control of observations: variational
form
- Use of innovations for observation monitoring and estimation of
error
statistics
- Diagnosing the optimality of an assimilation system: the
Chi-square
test.
Lecture
No.5 multivariate statistical interpolation.
- Embedding of dynamical constraints within the background-error
statistics.
- Forward models of observation operators. Observation quality
control.
Lecture
No.6 Extension of statistical interpolation to the 4D case:
the
Kalman filter
- Sequential estimation algorithm for 4D data assimilation: the
Kalman
filter.
- Derivation of the basic equations of the Kalman filter.
- Extension to the nonlinear case: the extended Kalman filter, the
reduced
rank Kalman filter and the ensemble Kalman filter.
Lecture
No.7 Quadri-dimensional variational data assimilation: the
4D-Var.
- Continuous variational assimilation of observations distributed
in time.
- The tangent-linear and adjoint models.
- Implicit dynamical structures of background-error covariances of
4D-Var.
- Singular vectors and their relationship to precursors to
dynamical
instability.
References
General
Burden, R.L., J.D. Faires et A.C. Reynolds, 1978: Numerical
analysis.
Ed Prindle, Weber and Schmidt, 579 pages.
Golub, G.H. et C.F. van Loan, 1983: Matrix computations.
John
Hopkins University Press, 476 pages.
Press, W.H., S.A. Teukolsky, W.T. Vetterling et B.P. Flannery, 1992:
Numerical
recipes: the art of scientific computing (Second Edition).
Cambridge
University Press, 963 pages
Applications to geophysical flows
Bennett, A.F., 1992: Inverse methods in physical oceanography.
Cambridge University Press, 346 pages.
Daley, R., 1991: Atmospheric data analysis. Cambridge
University
Press, Atmospheric and Space Science Series, 457 pages.
Rodgers, R.D., 2000: Inverse Methods for Atmospheric Sounding:
theory
and practice. World Scientific Series On Atmospheric and Planetary
Physics, vol.2, 238 pages.
Tarantola, A., 1987: Inverse problem theory: methods for data
fitting
and model parameter estimation. Elsevier, Amsterdam, 613 pages.
Wunsch, C., 1996: The ocean circulation inverse problem.
Cambridge
University Press, 442 pages.
Statistics
Jazwinski, A., 1970: Stochastic processes and filtering theory.
Academic Press, 376 pages.
Maybeck, P.S., 1979: Stochastic models, estimation and control:
vol.
1 et 2. Academic Press, Mathematics in science and engineering, 712
pages.
Training course:
Fisher, M., 2001: Assimilation techniques: 3D-Var
Fisher, M., 2001: Assimilation techniques: 4D-Var
Fisher, M., 2001: Assimilation techniques: Approximate Kalman
Filters
and Singular Vectors
Bouttier, F. and P. Courtier, 2000: Data assimilation concepts and
methods
Järvinen, H., 1998: Observations and diagnostic tools for
data
assimilation: