Representation of correlation functions in variational assimilation using an implicit diffusion operator

Quarterly Journal of the Royal Meteorological Society

Volumn 136, Issue 651, 2010, Pages 1421-1443

  Mirouze, I ^a   Weaver, A T ^a  

^a CNRS   (France)

Author keywords

3D var; 4D var; Background error; Covariance modelling; Diffusion equation; Recursive filter

Indexed keywords

3D-VAR; 4D-VAR; BACKGROUND ERRORS; COVARIANCE MODELLING; DIFFUSION EQUATION; RECURSIVE FILTERS;

CORRELATION METHODS; DATA PROCESSING; DIFFUSION; MATHEMATICAL OPERATORS; PARTIAL DIFFERENTIAL EQUATIONS; THREE DIMENSIONAL; VALUE ENGINEERING; VIDEO SIGNAL PROCESSING;

MATHEMATICAL MODELS;

COVARIANCE ANALYSIS; DATA ASSIMILATION; FORECASTING METHOD; MODELING; NUMERICAL MODEL;

EID: 77957115234     PISSN: 00359009     EISSN: 1477870X     Source Type: Journal    
DOI: 10.1002/qj.643     Document Type: Article

References (73)

1. Bannister RN. 2007. Can wavelets improve the representation of forecast error covariances in variational data assimilation? Mon. Weather Rev. 135: 387-408.
2. Bannister RN. 2008a. A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances. Q. J. R. Meteorol. Soc. 134: 1951-1970.
3. Bannister RN. 2008b. 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: 1971-1996.
4. Bennett AF. 2002. Inverse Modeling of the Ocean and Atmosphere. Cambridge University Press: Cambridge.
5. Bennett AF, Chua BS, Leslie LM. 1996. Generalized inversion of a global numerical weather prediction model. Meteorol. Atmos. Phys. 60: 165-178.
6. Bennett AF, Chua BS, Leslie LM. 1997. Generalized inversion of a global numerical weather prediction model. II: analysis and implementation. Meteorol. Atmos. Phys. 62: 129-140.
7. Bennett AF, Chua BS, Harrison DE, McPhaden MJ. 1998. Generalized inversion of tropical atmosphere-ocean data and a coupled model of the Tropical Pacific. J. Climate 11: 1768-1792.
8. Berre L, Stefǎnescu SE, BelPereira M. 2006. The representation of the analysis effect in three error simulation techniques. Tellus 58A: 196-209.
9. Briggs WL, Henson VE, McCormick SF. 2000. A Multigrid Tutorial, 2nd edn. Siam: Philadelphia.
10. Buehner M. 2005. Ensemble-derived stationary and flow-dependent background-error covariances: evaluation ina quasi-operational NWP setting. Q. J. R. Meteorol. Soc. 131: 1013-1043.
11. Buell CE. 1972. Correlation functions for wind and geopotential on isobaric surfaces. J. Appl. Meteorol. 11: 51-59.
12. Chua BS, Bennett AF. 2001. An inverse ocean modeling system. Ocean Modelling 3: 137-165.
13. Cohn S. 1997. An introduction to estimation theory. J. Meteorol. Soc. Jpn 75: 257-288.
14. Courtier P, Thépaut JN, Hollingsworth A. 1994.Astrategy for operational implementation of 4D-Var, using an incremental approach. Q. J. R. Meteorol. Soc. 120: 1367-1388.
15. Cummings JA. 2005. Operational multivariate ocean data assimilation. Q. J. R. Meteorol. Soc. 131: 3583-3604.
16. Daget N, Weaver AT, Balmaseda MA. 2009. Ensemble estimation of background-error variances in a three-dimensional variational data assimilation system for the global ocean. Q. J. R. Meteorol. Soc. 135: 1071-1094.
17. Daley R. 1991. Atmospheric Data Analysis. Cambridge University Press: Cambridge.
18. Daley R, Barker E. 2001. 'NRL atmospheric variational data assimilation system', NAVDAS Source Book NRL/PU/7530-01-441. Naval Research Laboratory: Monterey, CA.
19. Deckmyn A, Berre L. 2005. A wavelet approach to representing background error covariances in a limited-area model. Mon. Weather Rev. 133: 1279-1294.
20. Derber J, Bouttier F. 1999. A reformulation of the background error covariance in the ECMWFglobal data assimilation system. Tellus 51A: 195-221.
21. Derber JC, Rosati A. 1989. A global oceanic data assimilation system. J. Phys. Oceanogr. 19: 1333-1347.
22. Dobricic S, Pinardi N. 2008. An oceanographic three-dimensional variational data assimilation scheme. Ocean Modelling 22: 89-105.
23. Egbert G, Bennett A, Foreman M. 1994. Topex/Poseidon tides estimated using a global inverse model. J. Geophys. Res. 99: 24821-24852.
24. Gaspari G, Cohn S. 1999. Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc. 125: 723-757.
25. Gaspari G, Cohn S., Guo J., Pawson S. 2006. Construction and application of correlation functions with variable length-fields. Q. J. R. Meteorol. Soc. 132: 1815-1838.
26. Gelb A. 1974. Applied Optimal Estimation. MIT Press: Cambridge.
27. Gneiting T. 1999. Correlation functions for atmospheric data analysis. Q. J. R. Meteorol. Soc. 125: 2449-2464.
28. Gneiting T. 2002. Compactly supported correlation functions. J. Multivariate Anal. 83: 493-508.
29. Gneiting T, Kleiber W, Schlather M. 2009. Mattern cross-covariance functions for multivariate random fields. Technical Report No. 549, University of Washington, Department of Statistics. Available at http://www.stat.washington.edu/research/reports/2009/tr549.pdf.
30. Golub GH, van Loan CF. 1996. Matrix Computations, 3rd edn. Johns Hopkins University Press: Baltimore.
31. Gradshteyn IS, Ryzhik IM. 1980. Table of Integrals, Series, and Products, Corrected and Enlarged edn. Academic Press: San Diego.
32. Gregori P, Porcu E, Mateu J, Sasvári Z. 2008. On potentially negative space time covariances obtained as sum of products of marginal ones. Ann. Inst. Stat. Math. 60: 865-882.
33. Guttorp P, Gneiting T. 2006. Miscellanea studies in the history of probability and statistics XLIX: On the Matérn correlation family. Biometrika. 93: 989-995.
34. Haberman R. 1987. Elementary Applied Partial Differential Equations, 2nd edn. Prentice-Hall: Englewood Cliffs.
35. Hayden CM, Purser RJ. 1995. Recursive filter objective analysis of meteorological fields: applications to NESDIS operational processing. J. Appl. Meteorol. 34: 3-15.
36. Hristopulos DT. 2003. Spartan Gibbs random field models for geostatistical applications. SIAM J. Sci. Comput. 24: 2125-2162.
37. Hristopulos DT, Elogne SN. 2007. Analytic properties and covariance functions of a new class of generalized Gibbs random fields. IEEE Trans. Inform. Theory 53: 4467-4679.
38. Hristopulos DT, Elogne SN. 2009. Computationally efficient spatial interpolators based on Spartan spatial random fields. IEEE Trans. Signal Processing 57: 3475-3487.
39. Jeffrey A. 1995. Handbook of Mathematical Formulas and Integrals. Academic Press: San Diego.
40. Jones DS. 1982. The Theory of Generalised Functions, 2nd edn. Cambridge University Press: Cambridge.
41. Julian PR, Thiébaux HJ. 1975. On some properties of correlation functions used in optimum interpolation schemes. Mon. Weather Rev. 103: 605-616.
42. Kelbert M, Leonenko N, Ruiz-Medina MD. 2005. Fractional random fields associated with stochastic fractional heat equations. Adv. Appl. Prob. 37: 108-133.
43. Liu H, Xue M, Purser RJ, Parrish DF. 2007. Retrieval of moisture from simulated GPS slant-path water vapor observations using 3DVAR with anisotropic recursive filter. Mon. Weather Rev. 135: 1506-1521.
44. Liu Y, Zhu J, She J, Zhuang S, Fu W, Gao J. 2009. Assimilating temperature and salinity profile observations using an anisotropic recursive filter in a coastal ocean model. Ocean Modelling 30: 75-87.
45. Lorenc AC. 1992. Iterative analysis using covariance functions and filters. Q. J. R. Meteorol. Soc. 118: 569-591.
46. Ma C. 2005. Linear combinations of space-time covariance functions and variograms. IEEE Trans. Signal Process. 53: 857-864.
47. Madec G. 2008. 'NEMO ocean engine'. Note du Pole de mod ́elisation No 27 ISSN No 1288-1619, Institut Pierre-Simon Laplace: Paris.
48. Martin MJ, Hines A, Bell MJ. 2007. Data assimilation in the FOAM operational short-range ocean forecasting system: a description of the scheme and its impact. Q. J. R. Meteorol. Soc. 133: 981-995.
49. Mogensen KS, Balmaseda MA, Weaver AT, Martin M, Vidard A. 2009. 'NEMOVAR: a variational data assimilation system for the NEMO ocean model'. In ECMWF Newsletter 120 - Summer 2009, ECMWF: Reading, UK. pp 17-21.
50. Ngodock HE. 2005. Efficient implementation of covariance multiplication for data assimilation with the representer method. Ocean Modelling 8: 237-251.
51. Oliver DS. 1998. Calculation of the inverse of the covariance. Math. Geology 30: 911-933.
52. Paciorek CJ, Schervish MJ. 2006. Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17: 483-506.
53. Pannekoucke O, Massart S. 2008. Estimation of the local diffusion tensor and normalization for heterogeneous correlation modelling using a diffusion equation. Q. J. R. Meteorol. Soc. 134: 1425-1438.
54. Pannekoucke O, Berre L, Desroziers G. 2007. Filtering properties of wavelets for local background-error correlations. Q. J. R. Meteorol. Soc. 133: 363-379.
55. Purser RJ. 2005. 'A geometrical approach to the synthesis of smooth anisotropic covariance operators for data assimilation', Office Note 447. National Centers for Environmental Prediction: Camp Springs.
56. Purser RJ, Wu WS, Parrish DF, Roberts NM. 2003a. Numerical aspects of the application of recursive filters to variational statistical analysis. Part I: spatially homogeneous and isotropic Gaussian covariances. Mon. Weather Rev. 131: 1524-1535.
57. Purser RJ, Wu WS, Parrish DF, Roberts NM. 2003b. Numerical aspects of the application of recursive filters to variational statistical analysis. Part II: Spatialy inhomogeneous and anisotropic general. covariances. Mon. Weather Rev. 131: 1536-1548.
58. Rabier F. 2005. Overview of global data assimilation developments in numerical weather-prediction centres. Q. J. R. Meteorol. Soc. 131: 3215-3233.
59. Raymond WH, Garder A. 1991. A review of recursive and implicit filters. Mon. Weather Rev. 119: 477-495.
60. Stein ML. 1999. Interpolation of Spatial Data. Some Theory for Kriging. Springer: New York.
61. Stein ML. 2005. 'Nonstationary spatial covariance functions', Technical Report No. 21. University of Chicago, Center for Integrating Statistical and Environmental Science: Chicago. Available at http://www.stat.uchicago.edu/cises/research.
62. Strang G. 1986. Introduction to Applied Mathematics. Wellesley- Cambridge Press: Wellesley.
63. Tarantola A. 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM: Philadelphia.
64. Trémolet Y. 2006. Accounting for an imperfect model in 4D-Var Q. J. R. Meteorol. Soc. 132: 2483-2504.
65. Trémolet Y. 2007. Model-error estimation in 4D-Var. Q. J. R. Meteorol. Soc. 133: 1267-1280.
66. von Storch H, Zwiers FW. 1999. Statistical Analysis in Climate Research. Cambridge University Press: Cambridge.
67. Weaver AT, Courtier P. 2001. Correlation modelling on the sphere using a generalized diffusion equation. Q. J. R. Meteorol. Soc. 127: 1815-1846.
68. Weaver AT, Ricci S. 2004. 'Constructing a background-error correlation model using generalized diffusion operators'. In ECMWF Proceedings of the Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean. ECMWF: Reading, UK; pp 327-339.
69. 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: 3605-3625.
70. Whittle P. 1963. Stochastic processes in several dimensions. Bull. Inst. Internat. Statist. 40: 974-994.
71. Wu WS, Purser RJ, Parrish DF. 2002. Three-dimensional variational analysis with spatially inhomogeneous covariances. Mon. Weather Rev. 130: 2905-2916.
72. Xu Q. 2005. Representations of inverse covariances by differential operators. Adv. Atmos. Sci. 22: 181-198.
73. Yaglom AM. 1987. Correlation Theory of Stationary and Related Random Functions. I: Basic Results. Springer-Verlag: New York, NY.