Heterogeneous correlation modeling based on the wavelet diagonal assumption and on the diffusion operator

Monthly Weather Review

Volumn 137, Issue 9, 2009, Pages 2995-3012

  Pannekoucke, Olivier ^a  

^a CENTRE NATIONAL DE RECHERCHES MÉTÉOROLOGIQUES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

BACKGROUND ERRORS; CORRELATION FUNCTION; CORRELATION MODELING; DIFFUSION OPERATORS; GEOGRAPHICAL VARIATIONS; HYBRID FORMULATIONS; LEGENDRE; LENGTH SCALE; LOCAL CORRELATIONS; NONSEPARABLE; SECOND GENERATION WAVELET; SPHERICAL WAVELETS;

BOUNDARY LAYER FLOW; CORRELATION METHODS; DIFFUSION; FEATURE EXTRACTION; MATHEMATICAL TRANSFORMATIONS; SPHERES;

WAVELET TRANSFORMS;

CORRELATION; ENSEMBLE FORECASTING; GEOGRAPHICAL VARIATION; MODELING; RANDOM WALK METHOD; TRANSFORM; WAVELET ANALYSIS;

EID: 70350778485     PISSN: 00270644     EISSN: None     Source Type: Journal    
DOI: 10.1175/2009MWR2783.1     Document Type: Article

References (62)

1. Antoine, J., L. Demanet, L. Jacques, and P. Vandergheynst, 2002: Wavelets on the sphere: Implementation and approximation. Appl. Comput. Harmon. Anal., 13, 177-200.
2. Asselin, R., 1972: Frequency filter for time integrations. Mon. Wea. Rev., 100, 487-490.
3. Auger, L., and A. Tangborn, 2004: A wavelet-based reduced rank Kalman filter for assimilation of stratospheric chemical tracer observations. Mon. Wea. Rev., 132, 1220-1237.
4. Belo Pereira, M., and L. Berre, 2006: The use of an ensemble approach to study the background error covariances in a global NWP model. Mon. Wea. Rev., 134, 2466-2489.
5. Berre, L., O. Pannekoucke, G. Desroziers, S. Stefanescu, B. Chapnik, and L. Raynaud, 2007: A variational assimilation ensemble and the spatial filtering of its error covariances: Increase of sample size by local spatial averaging. Proc.Workshop on Flow-Dependent Aspects of Data Assimilation, Reading, United Kingdom, ECMWF, 151-168.
6. Boer, G., 1983: Homogeneous and isotropic turbulence on the sphere. J. Atmos. Sci., 40, 154-163.
7. Boer, G., and T. Shepherd, 1983: Large-scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci., 40, 164-184.
8. Bouttier, F., 1993: The dynamics of error covariances in a barotropic model. Tellus, 45A, 408-423.
9. Bouttier, F., 1994: A dynamical estimation of error covariances in an assimilation system. Mon. Wea. Rev., 122, 2376-2390.
10. Boyd, J. P., 2001: Chebyshev and Fourier Spectral Methods. 2nd ed. Dover Publications, 688 pp.
11. Buehner, M., and M. Charron, 2007: Spectral and spatial localization of background error correlations for data assimilation. Quart. J. Roy. Meteor. Soc., 133, 615-630.
12. Coifman, R., and W. Wickerhauser, 1992: Entropy-based algorithms for best-basis selection. IEEE Trans. Info. Theory, 38, 713-718.
13. Courtier, P., and J. Geleyn, 1988: A global numerical weather prediction model with variable resolution: Application to the shallow-water equations. Quart. J. Roy. Meteor. Soc., 114, 1321-1346.
14. Courtier, P., and Coauthors, 1998: The ECMWF implementation of three-dimensional variational assimilation (3D-VAR). I: Formulation. Quart. J. Roy. Meteor. Soc., 124, 1783-1807.
15. Daley, R., 1991: Atmospheric Data Analysis. Cambridge University Press, 472 pp.
16. Daubechies, I., 1992: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, 377 pp.
17. Deckmyn, A., and L. Berre, 2005: A wavelet approach to representing background error covariances in a LAM. Mon. Wea. Rev., 133, 1279-1294.
18. Derber, J., and A. Rosati, 1989: A global oceanic data assimilation system. J. Phys. Oceanogr., 19, 1333-1347.
19. Derber, J., and F. Bouttier, 1999: A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus, 51A, 195-221.
20. Desroziers, G., 1997: A coordinate change for data assimilation in spherical geometry of frontal structures. Mon. Wea. Rev., 125, 3030-3038.
21. Dyn, N., D. Levin, and J. Andgregory, 1990: Butterfly subdivision scheme for surface interpolation with tension control. Trans. Graph., 2, 160-169.
22. Egbert, G. D., M. G. Foreman, and A. F. Bennett, 1994: TOPEX/Poseidon tides estimated using a global inverse model. J. Geophys. Res., 99, 24 821-24 852.
23. Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error. J. Geophys. Res., 99, 10 143-10 162.
24. Fisher, M., 2003: Background error covariance modelling. Proc. ECMWF Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, Reading, United Kingdom, ECMWF, 45-63.
25. Fisher, M., 2004: Generalized frames on the sphere, with application to background error covariance modelling. Proc. ECMWF Seminar on Recent Developments in Numerical Methods for Atmospheric and Ocean Modelling, Reading, United Kingdom, ECMWF, 87-102.
26. Fisher, M., and P. Courtier, 1995: Estimating the covariance matrices of analysis and forecast error in variational data assimilation. Tech. Rep. 220, ECMWF Tech. Memo., ECMWF, 28 pp.
27. Fisher, M., and E. Andersson, 2001: Developments in 4D-VAR and Kalman filtering. Tech. Rep. 347, ECMWF Tech. Memo., ECMWF, 36 pp.
28. Freeden, W., and U. Windheuser, 1996: Spherical wavelet transform and its discretization. Adv. Comput. Math., 5, 51-94.
29. Freeden, W., and M. Schreiner, 1998: Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere. Constr. Approx., 14, 493-515.
30. Gaspari, G., and S. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723-757.
31. Holshneider, M., 1990: Wavelet analysis on the circle. J. Math. Phys., 31, 39-44.
32. Holshneider, M., 1996: Continuous wavelet transforms on the sphere. J. Math. Phys., 37, 4156-4165.
33. Houtekamer, P., and H. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796-811.
34. Houtekamer, P., L. Lefaivre, J. Derome, H. Ritchie, and H. Mitchell, 1996: A system simulation approach to ensemble prediction. Mon. Wea. Rev., 124, 1225-1242.
35. Ingleby, B., 2001: The statistical structure of forecast errors and its representation in the Met Office global 3-D variational data assimilation scheme. Quart. J. Roy. Meteor. Soc., 127, 209-231.
36. Kalnay, E., 2002: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 364 pp.
37. Kevlahan, N., and M. Farge, 1997: Vorticity filaments in two-dimensional turbulence: Creation, stability and effect. J. Fluid Mech., 346, 49-76.
38. Lönnberg, P., 1988: Developments in the ECMWF analysis scheme. Proc. ECMWF Seminar on Data Assimilation and the Use of Satellite Data, Reading, United Kingdom, ECMWF, 75-120.
39. Lorenc, A., 2003: The potential of ensemble Kalman filter for NWP - A comparison with 4D-VAR. Quart. J. Roy. Meteor. Soc., 129, 3183-3203.
40. Mallat, S., 1999: A Wavelet Tour of Signal Processing. 2nd ed. Academic Press, 637 pp.
41. Mallat, S., G. Papanicolaou, and Z. Zhang, 1998: Adaptive covariance estimation of locally stationary processes. Ann. Stat., 26, 1-47.
42. McWilliams, J., 1984: The emergence of isolated coherent vorticies in turbulent flow. J. Fluid Mech., 146, 21-43.
43. Mehra, M., and N. Kevlahan, 2008: An adaptive wavelet collocation method for the solution of partial differential equations on the sphere. J. Comput. Phys., 227, 5610-5632.
44. Müller, C., 1966: Spherical Harmonics. Springer-Verlag, 45 pp.
45. Pannekoucke, O., 2008: Modelisation des structures locales de covariance des erreurs de prévision l'aide des ondelettes. Ph.D. thesis, Université de Toulouse, France, 208 pp.
46. Pannekoucke, O., and S. Massart, 2008: Estimation of the local diffusion tensor and normalization for heterogeneous correlation modelling using a diffusion equation. Quart. J. Roy. Meteor. Soc., 134, 1425-1438.
47. Pannekoucke, O., L. Berre, and G. Desroziers, 2007: Filtering properties of wavelets for local background-error correlations. Quart. J. Roy. Meteor. Soc., 133, 363-379.
48. Pannekoucke, O., L. Berre, and G. Desroziers, 2008: Background error correlation lengthscale estimates and their sampling statistics. Quart. J. Roy. Meteor. Soc., 134, 497-508.
49. Purser, R., W.-S. Wu,D. Parrish, and N. Roberts, 2003a: Numerical aspects of the application of recursive filters to variational statistical analysis. Part I: Spatially homogeneous and isotropic Gaussian covariances. Mon. Wea. Rev., 131, 1524-1535.
50. Purser, R., W.-S. Wu,D. Parrish, and N. Roberts 2003b: Numerical aspects of the application of recursive filters to variational statistical analysis. Part II: Spatially inhomogeneous and anisotropic general covariances. Mon. Wea. Rev., 131, 1536-1548.
51. Rabier, F., A. McNally, E. Andersson, P. Courtier, P. Undén, J. Eyre, A. Hollingsworth, and F. Bouttier, 1998: TheECMWF implementation of three-dimensional variational assimilation (3D-VAR). II: Structure functions. Quart. J. Roy. Meteor. Soc., 124, 1809-1829.
52. Rosca, D., 2005: Locally supported rational spline wavelets on the sphere. Math. Comput., 74, 1803-1829.
53. Sadourny, R., A. Arakawa, and Y. Mintz, 1968: Integration of the nondivergent barotropic vorticity equationwith an icosahedral-hexagonal grid for the sphere. Mon. Wea. Rev., 96, 351-356.
54. Schröder, P., and W. Sweldens, 1995a: Spherical wavelets: Efficiently representing functions on the sphere. Proc. 22nd Conf. on Computer Graphics and Interactive Techniques, Los Angeles, CA, SIGGRAPH, 161-172.
55. Schröder, P., and W. Sweldens, 1995b: Spherical wavelets: Texture processing. Tech. Rep. 4, Industrial Mathematics Initiative, Department of Mathematics, University of South California, 13 pp.
56. Sweldens, W., 1995: The lifting scheme: A new philosophy in biorthogonal wavelet constructions. Wavelet Appl. Signal Image Process., III, 68-79.
57. Sweldens, W., 1998: The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal., 29, 511-546.
58. Tangborn, A., 2004: Wavelet approximation of error covariance propagation in data assimilation. Tellus, 56A, 16-28.
59. Weaver, A., and P. Courtier, 2001: Correlation modelling on the sphere using a generalized diffusion equation. Quart. J. Roy. Meteor. Soc., 127, 1815-1846.
60. Weaver, A., and S. Ricci, 2003: Constructing a background-error correlation model using generalized diffusion operators. Proc. ECMWF Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, Reading, United Kingdom, ECMWF, 327-340.
61. Williamson, D., 1968: Integration of the barotropic vorticity equation on a spherical geodesic grid. Tellus, 20, 642-653.
62. Yoden, S., and M. Yamada, 1993: A numerical experiment on two-dimensional decaying turbulence on a rotating sphere. J. Atmos. Sci., 50, 631-643.