Start Submission Become a Reviewer

Reading: Second-order approximation to the 3DVAR cost function: application to analysis/forecast

Download

A- A+
Alt. Display

Original Research Papers

Second-order approximation to the 3DVAR cost function: application to analysis/forecast

Authors:

S. Lakshmivarahan ,

School of Computer Science, University of Oklahoma, 200 Felgar str., Norman, OK 73019, US
X close

Yuki Honda,

School of Meteorology, University of Oklahoma, US
About Yuki
On leave from the Japan Meteorological Agency.
X close

J. M. Lewis

National Severe Storms Laboratory, US
About J. M.
On assignment to the Desert Research Institute.
X close

Abstract

The analysis/forecast cycling scheme used in operational weather prediction centers involves repeated application of a 3DVAR assimilation phase followed by a forecast phase. In this paper we first derive a closed-form expression for the complete Hessian of the 3DVAR objective function. This facilitates the use of Newton-like algorithms in the 3DVAR assimilation phase. It is also shown that this algorithm includes the standard first-order method as a special case. Notwithstanding the method used in this optimization phase, if the iterative optimization is terminated prematurely before convergence (generally for the sake of economy), the resulting analysis incurs an error that will corrupt the initial condition of the model and thereby affect the forecast. Using Lorenz’s maximum simplification form of the barotropic vorticity equation, we examine the assimilation error and its dependency on the following factors: observational and background error covariances and the optimization strategy.

How to Cite: Lakshmivarahan, S., Honda, Y. and Lewis, J.M., 2003. Second-order approximation to the 3DVAR cost function: application to analysis/forecast. Tellus A: Dynamic Meteorology and Oceanography, 55(5), pp.371–384. DOI: http://doi.org/10.3402/tellusa.v55i5.12106
  Published on 01 Jan 2003
 Accepted on 15 Apr 2003            Submitted on 18 Mar 2002

References

  1. Andersson , E. and Järvinen , H. 1999 . Variational quality con-trol . Q. J. R. Meteorol. Soc . 125 , 697 – 722 .  

  2. Bergthorsson , P. and Diiiis , B. 1955 . Numerical weather map analysis . Tellus 7 , 229 – 240 .  

  3. Daley , R. 1991 . Atmospheric data analysis . Cambridge University Press , Cambridge , 457 pp .  

  4. Daley , R. and Barker , E. 2001. NAVDAS: Formulation and Diagnostics. Mon. Wea. Rev . 129 , 129 – 883 .  

  5. Dennis , J. E. , Gay , D. M. and Welsch , R.E. 1981 . An adaptive nonlinear least-squares algorithm . ACM Trans. Math. Software , 7 , 369 – 383 .  

  6. Dennis , J. E. Jr. and Schnabel , R. B. 1996 . Numerical methods for unconstrained optimization and nonlinear equa-tions . SIAM Classics in Applied Mathematics, vol . 16 . 378 pp .  

  7. Epstein , E. 1969 . Stochastic dynamic prediction . Tellus 21 , 739 – 759 .  

  8. Huang , X.-Yu. 2000. Variational analysis using spatial filters. Mon. Wea. Rev . 128 , 128 – 2600 .  

  9. Lewis , J. M. and Derber J. C. 1985 . The use of adjoint equations to solve a variational adjustment problem with advective constraints . Tellus 37A , 309 – 322 .  

  10. Le Dimet , F. X. and Talagrand , O. 1986 . Variational algorithms for analysis and assimilation of meteorological ob-servations: theoretical aspects . Tellus 38A , 97 – 110 .  

  11. Le Dimet , E X. , Navon , I. M. and Descau , D. N. 2002 Second-order information in data assimilation . Mon. Wea. Rev . 130 , 629 – 648 .  

  12. Lorenc , A. C. 1986 . Analysis methods for numerical weather prediction . Quart. J. R. MeteoroL Soc . 112 , 1177 – 1194 .  

  13. Lorenz , E. N. 1960 . Maximum simplification of the dynamic equations . Tellus 12 , 243 – 254 .  

  14. Nash , G. and Sofer , A. 1996 . Linear and nonlinear program-ming . McGraw Hill , New York , 692 pp .  

  15. Press , W. , Teukolsky , S. , Vetterling , W. and Flannery , B. 1992 . Numerical recipes in FORTRAN (art of scientific comput-ing) . Cambridge University Press , Cambridge , 963 pp .  

  16. Sasaki , Y. 1958 . An objective analysis based on variational methods . J. MeteoroL Soc. Jpn . 36 , 77 – 88 .  

  17. Talagrand , O . and Courtier , P. 1987 . Variational assimilation of meteorological observations with adjoint voracity equation. I: Theory . Q. J. R. MeteoroL Soc . 113 , 113 – 1328 .  

  18. Tarantola , A. 1987 . Inverse problem theory: Methods for data fitting and model parameter estimation . Elsevier , Amster-dam , 613 pp .  

  19. Thi´ebaux , H. J. and Pedder , M. A. 1987 . Spatial objective analysis . Academic Press , New York , 299 pp.  

  20. Thompson , P. D . 1957 . Uncertainty of initial state as a factor in the predictability of large scale atmosphere pattern . Tellus 9 , 275 – 295.  

  21. Wang , Z. , Navon , I. , Le Dimet , F. and Zhou , X. 1992 . The second-order adjoint analysis: theory and application. Me-teoroL Atmos. Phys . 50 , 50 – 20 .  

  22. Wang , Z. , Navon , I. , Zhou , X. and LeDimet , F. 1995 . A truncated Newton optimization algorithm in meteorological application with analytic Hessian/vector products . Corn-put. Optim. Appl . 4 , 4 – 262 .  

  23. Wang , Z. , Droegemeier , K. , White , L. and Navon , I. 1997 . Application of a new adjoint Newton algorithm to the 3D ARPS storm scale model using simulated data . Mon. Wea. Rev . 125 , 125 – 2478 .  

  24. Wiin-Nielsen , A. 1991 . The birth of numerical weather pre-diction . Tellus 43A , 36 – 52 .  

comments powered by Disqus