A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances

Quarterly Journal of the Royal Meteorological Society

Volumn 134, Issue 637, 2008, Pages 1951-1970

  Bannister, R N ^a  

^a UNIVERSITY OF READING   (United Kingdom)

Author keywords

Background error statistics; Balance; Calibration; Multivariate; Sperability

Indexed keywords

CALIBRATION; COVARIANCE MATRIX; FORECASTING; NUMERICAL METHODS; STATISTICS; WEATHER FORECASTING;

BACKGROUND ERROR STATISTICS; CALIBRATION METHODS; DATA ASSIMILATIONS; ENSEMBLE METHODS; FORECAST ERRORS; METEOROLOGICAL DATUMS; MULTIVARIATE; SPECIAL TREATMENTS; SPERABILITY; UNIVARIATE; VARIATIONAL DATA ASSIMILATIONS;

ERROR STATISTICS;

CALIBRATION; COVARIANCE ANALYSIS; DATA ASSIMILATION; ENSEMBLE FORECASTING; METEOROLOGY; NUMERICAL METHOD; WEATHER FORECASTING;

EID: 58049201686     PISSN: 00359009     EISSN: None     Source Type: Journal    
DOI: 10.1002/qj.339     Document Type: Review

References (80)

1. Bannister RN. 2006. Collection of results on background error covariance models. DARC Internal Report, 7 (available from Data Assimilation Research Centre, Department of Meteorology, University of Reading, Reading, RG6 6BB, UK; http://darc.nerc.ac.uk/v2/internalreports.html).
2. Bannister RN. 2008. A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics. Q. J. R. Meteorol. Soc. 134. DOI: 10.1002/qj.340.
3. Barker DM, Huang W, Guo YR, Bourgeois AJ, Xiao QN. 2004. A three-dimensional variational data assimilation system for MM5: Implementation and initial results. Mon. Weather Rev. 132: 897-914.
4. Barlow R. 1989. Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences. John Wiley: New York.
5. Bartello P, Mitchell HL. 1992. A continuous three-dimensional model of short-range forecast error covariances. Tellus 44A: 217-235.
6. Belo Pereira MB, Berre L. 2006. The use of an ensemble approach to study the background error covariances in a global NWP model. Mon. Weather Rev. 134: 2466-2489.
7. Berre L. 2000. Estimation of synoptic and mesoscale forecast error covariances in a limited area model. Mon. Weather Rev. 128: 644-667.
8. Berre L, Ştefánescu SE, Pereira MB. 2006. The representation of the analysis effect in three error simulation techniques. Tellus 58A: 196-209.
9. Bishop CH, Etherton BJ, Majumdar SJ. 2001. Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Weather Rev. 129: 420-436.
10. Bouttier F. 1996. 'Application of Kalman filtering to numerical weather prediction'. Pp. 61-90 in Proceedings of the 1996 ECMWF seminar on data assimilation, 2-6 September 1996. ECMWF: Reading, UK.
11. Buehner M. 2005. Ensemble derived stationary and flow dependent background error covariances: Evaluation in a quasi-operational NWP setting. Q. J. R. Meteorol. Soc. 131: 1013-1043.
12. Buehner M, Gauthier P, Liu Z. 2005. Evaluation of new estimates of background and observation error covariances for variational assimilation. Q. J. R. Meteorol. Soc. 131: 3373-3384.
13. Charney JG, Drazin PG. 1961. Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res. 66: 83-109.
14. Courtier P, Thépaut J-N, Hollingsworth A. 1994. A strategy for operational implementation of 4d-Var, using an incremental approach. Q. J. R. Meteorol. Soc. 120: 1367-1387.
15. Courtier P, Andersson E, Heckley W, Pailleux J, Vasiljevic D, Hollingsworth A, Fisher M, Rabier F. 1998. The ECMWF implementation of three-dimensional variational assimilation (3D-Var). I: Formulation. Q. J. R. Meteorol. Soc. 124: 1783-1807.
16. Daley R. 1991. Atmospheric Data Analysis. Cambridge University Press: UK.
17. Dee DP. 2004. 'Variational bias correction of radiance data in the ECMWF system'. Pp. 97-112 in ECMWF Workshop on Assimilation of high spectral resolution sounders in NWP, 28 June-1 July 2004. ECMWF: Reading, UK.
18. Dethof A, Hólm EV. 2004. Ozone assimilation in the ERA-40 reanalysis project. Q. J. R. Meteorol. Soc. 130: 2851-2872.
19. Derber J, Bouttier F. 1999. A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus 51A: 195-221.
20. Ehrendorfer M. 2007. A review of issues in ensemble-based Kalman filtering Meteorol. Z. 16: 795-818.
21. Evensen G. 2003. The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dynamics 53: 343-367.
22. Eyring V, Butchart N, Waugh DW, Akiyoshi H, Austin J, Bekki S, Bodeker GE, Boville BA, Bruhl C, Chipperfield MP, Cordero E, Dameris M, Deushi M, Fioletov VE, Frith SM, Garcia RR, Gettelman A, Giorgetta MA, Grewe V, Jourdain L, Kinnison DE, Mancini E, Manzini E, Marchand M, Marsh DR, Nagashima T, Newman PA, Nielsen JE, Pawson S, Pitari G, Plummer DA, Rozanov E, Schraner M, Shepherd TG, Shibata K, Stolarski RS, Struthers H, Tian W, Yoshiki M. 2006. Assessment of temperature, trace species, and ozone in chemistry-climate model simulations of the recent past. J. Geophys. Res. 111(:): D22308 (doi:10.1029/2006JD007327).
23. Fisher M. 2001. 'Assimilation techniques (4): 4dVar'. ECMWF Meteorological Training Course Lecture Series (available from ECMWF, Shinfield Park, Reading, Berkshire, RG2 9AX, UK).
24. Fisher M. 2003. 'Background error covariance modelling'. Pp. 45-64 in ECMWF Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, 8-12 September 2003. ECMWF: Reading UK.
25. Fisher M, Andersson E. 2001. Developments in 4d-Var and Kalman filtering. ECMWF Tech. Memo., 347 (available from ECMWF, Shinfield Park, Reading, Berkshire, RG2 9AX, UK).
26. Fisher M, Leutbecher M, Kelly GA. 2005. On the equivalence between Kalman smoothing and weak constraint four-dimensional variational data assimilation. Q. J. R. Meteorol. Soc. 131: 3235-3246.
27. Fletcher SJ, Zupanski M. 2006. A data assimilation method for log-normally distributed observational errors. Q. J. R. Meteorol. Soc. 132: 2505-2519.
28. Gaspari G, Cohn SE. 1999. Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc. 125: 723-757.
29. Gustafsson N, Berre L, Hornquist S, Huang XY, Lindskog M, Navascues B, Mogensen KS, Thornsteinsson S. 2001. Three-dimensional variational data assimilation for a limited area model. Tellus 53A: 425-446.
30. Haltiner GJ, Williams RT. 1980. Numerical Prediction and Dynamic Meteorology, 2nd edition. John Wiley: London.
31. Hamill TM, Mullen SL, Snyder C, Toth Z, Baumhefner DP. 2000. Ensemble forecasting in the short to medium range: Report from a workshop. Bull. Am. Meteorol. Soc. 81: 2653-2664.
32. Hamill TM, Whitaker JS, Snyder C. 2001. Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filer. Mon. Weather Rev. 129: 2776-2790.
33. Hollingsworth A, Lönnberg P. 1986. The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field. Tellus 38A: 111-136.
34. Hólm E, Andersson E, Beljaars A, Lopez P, Mahfouf JF, Simmons A, Thépaut J-N. 2002. Assimilation and modelling of the hydrological cycle: ECMWF's status and plans. ECMWF Tech. Memo., 383 (available from ECMWF, Shinfield Park, Reading, Berkshire, RG2 9AX, UK).
35. Holton JR. 1992. An Introduction to Dynamic Meteorology. Academic Press: New York.
36. Houtekamer PL, Mitchell HL. 2001. A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 129: 123-137.
37. Houtekamer PL, Lefaivre L, Derome J, Ritchie H, Mitchell HL. 1996. A system simulation approach to ensemble prediction. Mon. Weather Rev. 124: 1225-1242.
38. Ide K, Coutier P, Ghil M, Lorenc AC. 1997. Unified notation for data assimilation: operational, sequential and variational. J. Meteorol. Soc. Jpn 75: 181-189.
39. Ingleby NB. 2001. The statistical structure of forecast errors and its representation in the Met Office global three-dimensional variational data assimilation system. Q. J. R. Meteorol. Soc. 127: 209-231.
40. Isaksen L, Fisher M, Berner J. 2007. 'Use of analysis ensembles in estimating flow-dependent background error variance'. Pp. 65-86 in ECWMF Workshop Proceedings on Flow-dependent Aspects of Data Assimilation, 11-13 June 2007. ECMWF: Reading, UK.
41. Järvinen H. 2001. Temporal evolution of innovation and residual statistics in the ECMWF variational data assimilation systems. Tellus 53A: 333-347.
42. Johnson C, Hoskins BJ, Nichols NK. 2005. A singular vector perspective of 4d-Var: Filtering and interpolation. Q. J. R. Meteorol. Soc. 131: 1-19.
43. Kalman RE. 1960. A new approach to linear filter and prediction problems. J. Basic. Eng. 82: 35-45.
44. Kalnay E. 2003. Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press: Cambridge, UK.
45. Katz D. 2007. The application of PV-based control variable transformations in variational data assimilation. PhD Thesis, Reading University, Department of Mathematics.
46. Knopf B, Held H, Schellnhuber HJ. 2005. Forced versus coupled dynamics in Earth system modelling and prediction. Nonlinear Processes in Geophys. 12: 311-320.
47. Lahoz WA, Geer AJ, Bekki S, Bormann N, Ceccherini S, Elbern H, Errera Q, Eskes HJ, Fonteyn D, Jackson DR, Khattatov B, Massart S, Peuch V-H, Rharmill S, Ridolfi M, Segers A, Talagrand O, Thornton HE, Vik AF, von Clarmann T. 2006. The assimilation of Envisat Data (ASSET) project. Atmos. Chem. Phys. Discuss. 6: 12769-12824.
48. Lahoz WA, Fonteyn D, Swinbank R. 2007. Data assimilation of atmospheric constituents: A review. Atmos. Chem. Phys. Discuss. 7: 9561-9633.
49. Lanczos C. 1988. Applied Analysis. Dover Publications: New York.
50. Lee PM. 1997. Bayesian Statistics: An Introduction. Arnold Publishers: London.
51. Lewis JM, Lakshmivarahan S, Dhall SK. 2006. Dynamic Data Assimilation: A Least-squares Approach. Cambridge University Press: Cambridge, UK.
52. Li ZJ, Navon IM. 2001. Optimality of variational data assimilation and its relationship with the Kalman filter and smoother. Q. J. R. Meteorol. Soc. 127: 661-683.
53. Lönnberg P, Hollingsworth A. 1986. The statistical structure of short-range forecast errors as determined from radiosonde data. Part II: The covariance of height and wind errors. Tellus 38A: 137-161.
54. Lorenc AC. 1981. A global three-dimensional multivariate statistical interpolation scheme. Mon. Weather Rev. 109: 701-721.
55. Lorenc AC. 1986. Analysis methods for numerical weather prediction. Q. J. R. Meteorol. Soc. 112: 1177-1194.
56. Lorenc AC. 2003a. Modelling of error covariances by 4d-Var assimilation. Q. J. R. Meteorol. Soc. 129: 3167-3182.
57. Lorenc AC. 2003b. The potential of the ensemble Kalman filter for NWP - a comparison with 4d-Var. Q. J. R. Meteorol. Soc. 129: 3183-3203.
58. Lorenc AC, Rawlins F. 2005. Why does 4D-Var beat 3D-Var? Q. J. R. Meteorol. Soc. 131: 3247-3257.
59. Lui Z-Q, Rabier F. 2002. The interaction between model resolution, observation resolution and observation density in data assimilation: A one-dimensional study. Q. J. R. Meteorol. Soc. 128: 1367-1386.
60. Lynch P, Huang X-Y. 1992. Initialization of the HIRLAM model using a digital filter. Mon. Weather Rev. 120: 1019-1034.
61. Ménard R, Daley R. 1996. The application of Kalman smoother theory to the estimation of 4DVAR error statistics. Tellus 48A: 221-237.
62. Pailleux J. 1997. Large-scale data assimilation systems: atmospheric applications to numerical weather prediction. J. Meterol. Soc. Jpn 75: 347-358.
63. Parrish DF, Derber JC. 1992. The National Meteorological Center's spectral statistical interpolation analysis system. Mon. Weather Rev. 120: 1747-1763.
64. Phillips NA. 1986. The spatial statistics of random geostrophic modes and first-guess errors. Tellus 38A: 314-332.
65. Polavarapu S, Ren S, Rochon Y, Sankey D, Ek N, Koshyk J, Tarasick D. 2005. Data assimilation with the Canadian middle atmosphere model. Atmos.-Ocean 43: 77-100.
66. Rabier F. 2005. Overview of global data assimilation developments in numerical weather prediction centres. Q. J. R. Meteorol. Soc. 131: 3215-3233.
67. Rabier F, Mahfouf J-F, Fisher M, Järvinen H, Simmons AJ, Andersson E, Bouttier F, Courtier P, Hamrud M, Haseler J, Hollingsworth A, Isaksen L, Klinker E, Saarinen S, Temperton C, Thépaut J-N, Undén P, Vasiljevic D. 1997. Recent experimentation on 4D-Var and first results from a Simplified Kalman Filter. ECMWF Tech. Memo., 240 (available from ECMWF, Shinfield Park, Reading, Berkshire, RG2 9AX, UK).
68. Rabier F, McNally A, Andersson E, Courtier P, Undén P, Eyre J, Hollingsworth A, Bouttier F. 1998. The ECMWF implementation of three-dimensional variational assimilation (3D-Var). II: Structure functions. Q. J. R. Meteorol. Soc. 124: 1809-1829.
69. Rutherford ID. 1972. Data assimilation by statistical interpolation of forecast error fields. J. Atmos. Sci. 29: 809-815.
70. Sadiki W, Fischer C. 2005. A posteriori validation applied to the 3D-VAR Arpège and Aladin data assimilation systems Tellus 57A: 21-34.
71. Saha S, Nadiga S, Thiaw C, Wang J, Wang W, Zhang Q, Van den Dool HM, Pan HL, Moorthi S, Behringer D, Stokes D, Pena M, Lord S, White G, Ebisuzaki W, Peng P, Xie P. 2006. The NCEP Climate Forecast System. J. Clim. 19: 3483-3517.
72. Schlatter TW. 2000. Variational assimilation of meteorological observations in the lower atmosphere: a tutorial on how it works. J. Atmos. Solar-Terr. Phys. 62: 1057-1070.
73. Seaman NL. 2003. Future directions of meteorology related to air quality research. Environ. Int. 29: 245-252.
74. Shutts G. 2005. A kinetic energy backscatter algorithm for use in ensemble prediction systems. Q. J. R. Meteorol. Soc. 131: 3079-3102.
75. Široká M, Fischer C, Cässé V, Brožkova R, Geleyn J-F. 2003. The definition of mesoscale selective forecast error covariances for a limited area variational analysis. Meteorol. Atmos. Phys. 82: 227-244.
76. Thépaut J-N, Courtier P, Belaud G, Lemaitre G. 1996. Dynamical structure functions in a four-dimensional variational assimilation: A case study. Q. J. R. Meteorol. Soc. 122: 535-561.
77. Weaver AT, Deltel C, Machu E, Ricci S, Daget N. 2005. A multivariate balance operator for variational ocean data assimilation. Q. J. R. Meteorol. Soc. 131: 3605-3626.
78. Žagar N, Andersson E, Fisher M. 2005. Balanced tropical data assimilation based on a study of equatorial waves in ECMWF short-range forecast errors. Q. J. R. Meteorol. Soc. 131: 987-1011.
79. Zhang F. 2005. Dynamics and structure of mesoscale error covariance of a winter cyclone estimated through short-range ensemble forecasts. Mon. Weather Rev. 133: 2876-2893.
80. Zupanski D. 1997. A general weak constraint applicable to operational 4d-Var data assimilation systems. Mon. Weather Rev. 125: 2274-2291.