Advertisement

Background Error Correlation Modeling with Diffusion Operators

  • Max YaremchukEmail author
  • Matthew Carrier
  • Scott Smith
  • Gregg Jacobs
Chapter

Abstract

Many background error correlation (BEC) models in data assimilation are formulated in terms of a positive-definite smoothing operator B that is employed to simulate the action of correlation matrix on a vector in state space. In this chapter, a general procedure for constructing a BEC model as a rational function of the diffusion operator D is presented and analytic expressions for the respective correlation functions in the homogeneous case are obtained. It is shown that this class of BEC models can describe multi-scale stochastic fields whose characteristic scales can be expressed in terms of the polynomial coefficients of the model. In particular, the connection between the inverse binomial model and the well-known Gaussian model \(\mathbf{\mathsf{B}}_{g} =\exp \mathbf{\mathsf{D}}\) is established and the relationships between the respective decorrelation scales are derived.By its definition, the BEC operator has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of B) is a computationally expensive procedure, therefore an efficient numerical approximation is needed. Under the assumption of local homogeneity of D, a heuristic method for computing the diagonal elements of B is proposed. It is shown that the method is sufficiently accurate for realistic applications, and requires 102 times less computational resources than other methods of diagonal estimation that do not take into account prior information on the structure of B.

Keywords

Diagonal Element Diffusion Tensor Polynomial Coefficient Hadamard Matrix Diffusion Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This study was supported by the Office of Naval Research (Program element 0602435N) as part of the project “Exploring error covariances in variational data assimilation”.

References

  1. Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs and mathematical tables. Dover Publications, New YorkGoogle Scholar
  2. Avramidi IG (1999) Covariant techniques for computation of the heat kernel. Rev Math Phys 11:947–980CrossRefGoogle Scholar
  3. Bekas CF, Kokiopoulou E, Saad Y (2007) An estimator for the diagonal of a matrix. Appl Numer Math 57:1214–1229CrossRefGoogle Scholar
  4. Belo-Pereira M, Berre L (2006) The use of ensemble approach to study the background error covariances in a global NWP model. Mon Weather Rev 134:2466CrossRefGoogle Scholar
  5. Berre L, Desroziers G (2010) Filtering of background error variances and correlations by local spatial averraging: a review. Mon Weather Rev 138:3693CrossRefGoogle Scholar
  6. Bishop C, Hodyss D (2007) Flow adaptive moderation of spurious ensemble correlations and its use in ensemble data assimilation. Q J R Meteorol Soc 133:2029CrossRefGoogle Scholar
  7. Bishop C, Hodyss D (2011) Adaptive ensemble covariance localization in ensemble 4D-VAR estimation. Mon Weather Rev 139:1241CrossRefGoogle Scholar
  8. Carrier M, Ngodock H (2010) Background error correlation model based on the implicit solution of a diffusion equation. Ocean Model 35:45–53CrossRefGoogle Scholar
  9. Derber J, Rosati A (1989) A global oceanic data assimilation system. J Phys Oceanogr 19:1333CrossRefGoogle Scholar
  10. Di Lorenzo E, Moore AM, Arango HG, Cornuelle BD, Miller AJ, Powell BS, Chua BS, Bennett AF (2007) Weak and strong constraint data assimilation in the Inverse Ocean Modelling System (ROMS): development and application for a baroclinic coastal upwelling system. Ocean Model 16:160CrossRefGoogle Scholar
  11. Dong S-J, Liu K-F (1994) Stochastic estimation with Z 2 noise. Phys Lett B 328:130–136CrossRefGoogle Scholar
  12. Egbert GD, Bennett AF, Foreman MGG (1994) Topex/Poseidon tides estimated using a global inverse model. J Geophys Res 99:24821CrossRefGoogle Scholar
  13. Gaspari G, Cohn SE, Guo J, Pawson S (2006) Construction and application of covariance functions with variable length-fields. Q J R Meteorol Soc 132:1815CrossRefGoogle Scholar
  14. Girard DF (1987) Un algorithme simple et rapide pour la validation croissee generalisee sur des problemes de grande taillee. RR 669-M, Grenoble, France: Informatique et Mathématiques Appliquées de GrenobleGoogle Scholar
  15. Gregori P, Porcu E, Mateu J, Sasvari Z (2008) On potentially negative space time covariances obtained as sum of products of marginal ones. Ann Inst Stat Math 60:865CrossRefGoogle Scholar
  16. Gusynin VP, Kushnir VA (1991) On-diagonal heat kernel expansion in covariant derivatives in curved space. Class Quantum Gravity 8:279–285CrossRefGoogle Scholar
  17. Hristopulos DT, Elogne SN (2007) Analytic properties and covariance functions of a new class of generalized Gibbs random fields. IEEE Trans Inf Theory 53:4467–4679CrossRefGoogle Scholar
  18. Hristopulos DT, Elogne SN (2009) Computationally efficient spatial interpolators based on Spartan spatial random fields. IEEE Trans Signal Process 57:3475–3487CrossRefGoogle Scholar
  19. Hutchison MF (1989) A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. J Commun Stat Simul 18:1059–1076CrossRefGoogle Scholar
  20. Martin P, Barron C, Smedstad L, Wallcraft A, Rhodes R, Campbell T, Rowley C (2008) Software design description for the navy coastal ocean model (NCOM) Ver. 4.0. Naval Research Laboratory Report NRL/MR/7320-08-9149, p 151Google Scholar
  21. Mirouze I, Weaver A (2010) Representation of the correlation functions in variational data assimilation using an implicit diffusion operator. Q J R Meteorol Soc 136:1421CrossRefGoogle Scholar
  22. Ngodock HE, Chua BS, Bennett (2000) Generalized inversion of a reduced gravity primitive equation ocean model and tropical atmosphere ocean data. Mon Weather Rev 128:1757Google Scholar
  23. 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 134:1425CrossRefGoogle Scholar
  24. Pannekoucke O, Berre L, Desroziers G (2008) Background-error correlation length-scale estimates and their sampling statistics. Q J R Meteorol Soc 134:497CrossRefGoogle Scholar
  25. Purser RJ (2008a) Normalization of the diffusive filters that represent the inhomogeneous covariance operators of variational assimilation, using asymptotic expansions and the techniques of non-euclidean geometry: part I: analytic solutions for symmetrical configurations and the validation of practical algorithms. NOAA/NCEP Office Note 456, p 48Google Scholar
  26. Purser RJ (2008b) Normalization of the diffusive filters that represent the inhomogeneous covariance operators of variational assimilation, using asymptotic expansions and the techniques of non-euclidean geometry: part II: Riemannian geometry and the generic parametrix expansion method. NOAA/NCEP Office Note 457, p 55Google Scholar
  27. Purser RJ, Wu W, Parrish DF, Roberts NM (2003) Numerical aspects of the application of recursive filters to variational statistical analysis. Part II: spatially inhomogeneous and anisotropic general covariances. Mon Weather Rev 131:1536–1548CrossRefGoogle Scholar
  28. Stein ML (1999) Interpolation of spatial data. Some theory for kriging. Springer, New YorkCrossRefGoogle Scholar
  29. Weaver A, Courtier P (2001) Correlation modelling on the sphere using a generalized diffusion equation. Q J R Meteorol Soc 127:1815CrossRefGoogle Scholar
  30. Weaver AT, Vialard J, Anderson DLT (2003) Three and four-dimensional variational assimilation with a general circulation model of the Tropical Pacific Ocean. Part I: formulation, internal diagnostics and consistency checks. Mon Weather Rev 131:1360Google Scholar
  31. Xu Q (2005) Representations of inverse covariances by differential operators. Adv Atmos Sci 22(2):181CrossRefGoogle Scholar
  32. Yaremchuk M, Carrier M (2012) On the renormalizaiton of the covariance operators. Mon Weather Rev 140:639–647CrossRefGoogle Scholar
  33. Yaremchuk M, Nechaev D (2013) Covariance localization with diffusion-based correlation models. Mon Weather Rev 141:848–860CrossRefGoogle Scholar
  34. Yaremchuk M, Sentchev A (2012) Multi-scale correlation functions associated with the polynomials of the diffusion operator. Q J R Meteorol Soc 138:1948–1953CrossRefGoogle Scholar
  35. Yaremchuk M, Smith S (2011) On the correlation functions associated with polynomials of the diffusion operator. Q J R Meteorol Soc 137:1927–1932CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Max Yaremchuk
    • 1
    Email author
  • Matthew Carrier
    • 1
  • Scott Smith
    • 1
  • Gregg Jacobs
    • 1
  1. 1.Naval Research Laboratory, Stennis Space CenterHancockUSA

Personalised recommendations