A singular vector perspective of 4D‐Var: Filtering and interpolation

Four-dimensional variational data assimilation (4D-Var) combines the information from a time sequence of observations with the model dynamics and a background state to produce an analysis. In this paper, a new mathematical insight into the behaviour of 4D-Var is gained from an extension of concepts that are used to assess the qualitative information content of observations in satellite retrievals. It is shown that the 4D-Var analysis increments can be written as a linear combination of the singular vectors of a matrix which is a function of both the observational and the forecast model systems. This formulation is used to consider the filtering and interpolating aspects of 4D-Var using idealized case-studies based on a simple model of baroclinic instability. The results of the 4D-Var case-studies exhibit the reconstruction of the state in unobserved regions as a consequence of the interpolation of observations through time. The results also exhibit the filtering of components with small spatial scales that correspond to noise, and the filtering of structures in unobserved regions. The singular vector perspective gives a very clear view of this filtering and interpolating by the 4D-Var algorithm and shows that the appropriate specification of the a priori statistics is vital to extract the largest possible amount of useful information from the observations. Copyright © 2005 Royal Meteorological Society

[1]  Monique Tanguay,et al.  Four‐dimensional data assimilation with a wide range of scales , 1995 .

[2]  Philippe Courtier,et al.  Unified Notation for Data Assimilation : Operational, Sequential and Variational , 1997 .

[3]  Jean-Noël Thépaut,et al.  Variational inversion of simulated TOVS radiances using the adjoint technique , 1990 .

[4]  P. D. Thompson,et al.  A Dynamical Method of Analyzing Meteorological Data , 1961 .

[5]  P. Courtier,et al.  Extended assimilation and forecast experiments with a four‐dimensional variational assimilation system , 1998 .

[6]  Ionel M. Navon,et al.  Incomplete observations and control of gravity waves in variational data assimilation , 1992 .

[7]  ProblemsPer Christian HansenDepartment.  The L-curve and its use in the numerical treatment of inverse problems , 2000 .

[8]  C. Wunsch,et al.  Determining the General Circulation of the Oceans: A Preliminary Discussion , 1977, Science.

[9]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. Ii: Numerical Results , 2007 .

[10]  Y. Sasaki.  SOME BASIC FORMALISMS IN NUMERICAL VARIATIONAL ANALYSIS , 1970 .

[11]  P. Courtier,et al.  The ECMWF implementation of three‐dimensional variational assimilation (3D‐Var). I: Formulation , 1998 .

[12]  A. Hollingsworth,et al.  Objective Analysis for Numerical Weather Prediction , 1986 .

[13]  G. Wahba,et al.  Adaptive Tuning of Numerical Weather Prediction Models: Simultaneous Estimation of Weighting, Smoothing, and Physical Parameters , 1998 .

[14]  C. Wunsch.  The Ocean Circulation Inverse Problem , 1996 .

[15]  Andrew C. Lorenc,et al.  Analysis methods for numerical weather prediction , 1986 .

[16]  B. Hoskins,et al.  Simple Initial Value Problems and Mechanisms for Baroclinic Growth , 2001 .

[17]  Jean-Noël Thépaut,et al.  The information content of clear sky IASI radiances and their potential for numerical weather prediction , 1998 .

[18]  Joab R. Winkler,et al.  Tikhonov regularisation in standard form for polynomial basis conversion , 1997 .

[19]  Jens Schröter,et al.  Variational Assimilation of Geosat Data into an Eddy-resolving Model of the Gulf Stream Extension Area , 1993 .

[20]  F. L. Dimet,et al.  Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects , 1986 .

[21]  David F. Shanno,et al.  Algorithm 500: Minimization of Unconstrained Multivariate Functions [E4] , 1976, TOMS.

[22]  Philippe Courtier,et al.  Four‐Dimensional Assimilation In the Presence of Baroclinic Instability , 1992 .

[23]  Clive D Rodgers,et al.  Inverse Methods for Atmospheric Sounding: Theory and Practice , 2000 .

[24]  Nancy Nichols,et al.  Very large inverse problems in atmosphere and ocean modelling , 2005 .

[25]  Carlton L. Mateer,et al.  On the Information Content of Umkehr Observations. , 1965 .

[26]  Florence Rabier,et al.  Channel selection methods for Infrared Atmospheric Sounding Interferometer radiances , 2002 .

[27]  Gene H. Golub,et al.  Matrix computations , 1983 .

[28]  A. Simmons,et al.  The ECMWF operational implementation of four‐dimensional variational assimilation. I: Experimental results with simplified physics , 2007 .

[29]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory , 2007 .

[30]  Anthony Hollingsworth,et al.  The statistical structure of short-range forecast errors as determined from radiosonde data , 1986 .

[31]  Christine Johnson,et al.  Information Content of Observations in Variational Data Assimilation , 2003 .

[32]  Philippe Courtier,et al.  Interactions of Dynamics and Observations in a Four-Dimensional Variational Assimilation , 1993 .

[33]  P. Gauthier,et al.  A validation of the incremental formulation of 4D variational data assimilation in a nonlinear barotropic flow , 1998 .

[34]  Gérald Desroziers,et al.  Diagnosis and adaptive tuning of observation‐error parameters in a variational assimilation , 2001 .

[35]  D. P. Eppel,et al.  On the sensitivity of a least-squares fit of discretized linear hyperbolic equations to data , 1995 .

[36]  Vincent Toumazou,et al.  Using a Lanczos Eigensolver in the Computation of Empirical Orthogonal Functions , 2001 .

[37]  Jean-Noël Thépaut,et al.  Four‐dimensional variational analyses of FASTEX situations using special observations , 1999 .

[38]  D. Dee.  On-line Estimation of Error Covariance Parameters for Atmospheric Data Assimilation , 1995 .