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Ocean Dynamics

, Volume 66, Issue 12, pp 1651–1664 | Cite as

Local ensemble assimilation scheme with global constraints and conservation

  • Alexander BarthEmail author
  • Yajing Yan
  • Aida Alvera-Azcárate
  • Jean-Marie Beckers
Article
Part of the following topical collections:
  1. Topical Collection on the 47th International Liège Colloquium on Ocean Dynamics, Liège, Belgium, 4-8 May 2015

Abstract

Ensemble assimilation schemes applied in their original, global formulation respect linear conservation properties if the ensemble perturbations are set up accordingly. For realistic ocean systems, only a relatively small number of ensemble members can be calculated. A localization of the ensemble increment is therefore necessary to filter out spurious long-range correlations. The conservation of the global properties will be lost if the assimilation is performed locally, since the conservation requires a coupling between all model grid points which is removed by the localization. The distribution of ocean observations is often highly inhomogeneous. Systematic errors of the observed parts of the ocean state can lead to spurious adjustment of the non-observed parts via data assimilation and thus to a spurious increase or decrease in long-term simulations of global properties which should be conserved. In this paper, we propose a local assimilation scheme (with different variants and assumptions) which can satisfy global conservation properties. The proposed scheme can also be used for non-local observation operators. Different variants of the proposed scheme are tested in an idealized model and compared to the traditional covariance localization with an ad-hoc step enforcing conservation. It is shown that the inclusion of the conservation property reduces the total RMS error and that the presented stochastic and deterministic schemes avoiding error space rotation provide better results than the traditional covariance localization.

Keywords

Data assimilation Ensemble Kalman filter Localization Covariance modeling Conservation 

Notes

Acknowledgments

This work was funded by the SANGOMA EU project (grant FP7-671 SPACE-2011-1-CT-283580-SANGOMA), by the project PREDANTAR (SD/CA/04A) from the federal Belgian Science policy and the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS). Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11. We would like to thank the reviewers and the editor for their careful reading of the manuscript and their constructive criticism.

References

  1. Anderson JL (2007) Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D: Nonlinear Phenomena 230(1-2):99–111. doi: 10.1016/j.physd.2006.02.011 CrossRefGoogle Scholar
  2. Barth A, Alvera-Azcárate A, Beckers JM, Rixen M, Vandenbulcke L (2007) Multigrid state vector for data assimilation in a two-way nested model of the Ligurian Sea. J Mar Syst 65(1-4):41–59. doi: 10.1016/j.jmarsys.2005.07.006. http://hdl.handle.net/2268/4260 CrossRefGoogle Scholar
  3. Barth A, Beckers JM, Troupin C, Alvera-Azcárate A, Vandenbulcke L (2014) divand-1.0: n-dimensional variational data analysis for ocean observations. Geosci Model Dev 7(1):225–241. doi: 10.5194/gmd-7-225-2014. http://www.geosci-model-dev.net/7/225/2014/ CrossRefGoogle Scholar
  4. Bishop CH, Hodyss D (2007) Flow-adaptive moderation of spurious ensemble correlations and its use in ensemble-based data assimilation. Q J R Meteorol Soc 133(629):2029–2044. doi: 10.1002/qj.169 CrossRefGoogle Scholar
  5. Bishop CH, Hodyss D (2009a) Ensemble covariances adaptively localized with ECO-RAP. Part 1: tests on simple error models. Tellus A 61(1):84–96. doi: 10.1111/j.1600-0870.2008.00371.x
  6. Bishop CH, Hodyss D (2009b) Ensemble covariances adaptively localized with ECO-RAP. Part 2: a strategy for the atmosphere. Tellus A 61(1):97–111. doi: 10.1111/j.1600-0870.2008.00372.x
  7. Bishop CH, Hodyss D (2011) Adaptive ensemble covariance localization in Ensemble 4D-VAR State Estimation. Mon Weather Rev 139:1241–1255. doi: 10.1175/2010MWR3403.1 CrossRefGoogle Scholar
  8. Bishop CH, Etherton B, Majumdar SJ (2001) Adaptive sampling with the ensemble transform Kalman filter. Part I: theoretical aspects. Mon Weather Rev 129:420–436. doi: 10.1175/1520-0493(2001)129%3C0420:ASWTET%3E2.0.CO;2
  9. Brankart JM, Testut CE, Brasseur P, Verron J (2003) Implementation of a multivariate data assimilation scheme for isopycnic coordinate ocean models: application to a 1993-96 hindcast of the North Atlantic Ocean circulation. J Geophys Res 108(C3):3074. doi: 10.1029/2001JC001198 CrossRefGoogle Scholar
  10. Brankart JM, Ubelmann C, Testut CE, Cosme E, Brasseur P, Verron J (2009) Efficient parameterization of the observation error covariance matrix for square root or ensemble Kalman filters: Application to Ocean Altimetry. Mon Weather Rev 137:1908–1927. doi: 10.10.1175/2008MWR2693.1 CrossRefGoogle Scholar
  11. Burgers G, van Leeuwen PJ, Evensen G (1998) Analysis scheme in the ensemble Kalman filter. Mon Weather Rev 126:1719–1724. doi: 10.1175/1520-0493(1998)126%3C1719:ASITEK%3E2.0.CO;2 CrossRefGoogle Scholar
  12. Chen Y, Davis TA, Hager WW, Rajamanickam S (2008) Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate. ACM Trans Math Softw 35(3):22:1–22:14. doi: 10.1145/1391989.1391995 CrossRefGoogle Scholar
  13. Cox S, Matthews P (2002) Exponential time differencing for stiff systems. J Comput Phys 176(2):430–455. doi: 10.1006/jcph.2002.6995 CrossRefGoogle Scholar
  14. Danilov S (2013) Ocean modeling on unstructured meshes. Ocean Model 69:195–210. doi: 10.1016/j.ocemod.2013.05.005 CrossRefGoogle Scholar
  15. Davis TA, Hager WW (2009) Dynamic supernodes in sparse Cholesky update/downdate and triangular solves. ACM Trans Math Softw 35(4):27:1–27:23. doi: 10.1145/1462173.1462176 CrossRefGoogle Scholar
  16. Evensen G (2007) Data assimilation: the Ensemble Kalman Filter. Springer, p 279Google Scholar
  17. Gaspari G, Cohn SE (1999) Construction of correlation functions in two and three dimensions. Q J R Meteorol Soc 125(554):723–757. doi: 10.1002/qj.49712555417 CrossRefGoogle Scholar
  18. Greybush SJ, Kalnay E, Miyoshi T, Ide K, Hunt BR (2011) Balance and Ensemble Kalman Filter Localization Techniques. Mon Weather Rev 139:511–522. doi: 10.1175/2010MWR3328.1 CrossRefGoogle Scholar
  19. Hamill TM, Whitaker JS, Snyder C (2001) Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon Weather Rev 129:2776–2790. doi: 10.1175/1520-0493(2001)129%3C2776:DDFOBE%3E2.0.CO;2 CrossRefGoogle Scholar
  20. Hoteit I, Pham DT, Blum J (2002) A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific. J Mar Syst 36:101–127. doi: 10.1016/S0924-7963(02)00129-X CrossRefGoogle Scholar
  21. Houtekamer PL, Mitchell HL (1998) Data assimilation using ensemble Kalman filter technique. Mon Weather Rev 126:796–811. doi: 10.1175/1520-0493(1998)126%3C0796:DAUAEK%3E2.0.CO;2 CrossRefGoogle Scholar
  22. Houtekamer PL, Mitchell HL (2001) A sequential ensemble Kalman filter for atmospheric data assimilation. Mon Weather Rev 129:123–137. doi: 10.1175/1520-0493(2001)129%3C0123:ASEKFF%3E2.0.CO;2 CrossRefGoogle Scholar
  23. Hunt BR, Kostelich EJ, Szunyogh I (2007) Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Phys D 230:112–126. doi: 10.1016/j.physd.2006.11.008 CrossRefGoogle Scholar
  24. Ide K, Bennett P, Courtier M, Ghil M, Lorenc AC (1995) Unified notation for data assimilation: Operational, sequential and variational. J Meteorol Soc Jpn 75(1B):181–189Google Scholar
  25. Janjić T, Nerger L, Albertella A, Schröter J, Skachko S (2011) On domain localization in ensemble based Kalman filter algorithms. Mon Weather Rev 139(7):2046–2060. doi: 10.1175/2011MWR3552.1 CrossRefGoogle Scholar
  26. Janjić T, McLaughlin DB, Cohn SE (2012) Preservation of physical properties with ensemble based Kalman filter algorithms. In: Mathematical and Algorithmic Aspects of Atmosphere-Ocean Data Assimilation, Mathematisches Forschungsinstitut Oberwolfach. doi: 10.4171/OWR/2012/58, vol 58, pp 17–20
  27. Janjić T, McLaughlin D, Cohn SE, Verlaan M (2014) Conservation of mass and preservation of positivity with ensemble-type Kalman filter algorithms. Mon Weather Rev 142:755–773. doi: 10.1175/MWR-D-13-00056.1
  28. Kepert JD (2009) Covariance localisation and balance in an Ensemble Kalman Filter. Q J R Meteorol Soc 135(642):1157–1176. doi: 10.1002/qj.443 CrossRefGoogle Scholar
  29. Keppenne CL, Rienecker MM (2003) Assimilation of temperature into an isopycnal ocean general circulation model using a parallel ensemble Kalman filter. J Mar Syst 40-41:363–380. doi: 10.1016/S0924-7963(03)00025-3 CrossRefGoogle Scholar
  30. Khellat F, Vasegh N (2014) The Kuramoto-Sivashinsky equation revisited: Low-dimensional corresponding systems. Commun Nonlinear Sci Numer Simul 19(9):3011–3022. doi: 10.1016/j.cnsns.2014.01.015 CrossRefGoogle Scholar
  31. Lorenc AC (2003) The potential of the ensemble Kalman filter for NWP—a comparison with 4D-Var. Q J R Meteorol Soc 129(595):3183–3203. doi: 10.1256/qj.02.132 CrossRefGoogle Scholar
  32. Löhner R D, Ambrosiano J (1990) A vectorized particle tracer for unstructured grids. J Comput Phys 91 (1):22–31. doi: 10.1016/0021-9991(90)90002-I CrossRefGoogle Scholar
  33. Madec G (2014) NEMO ocean engine (Draft edition r6039). No. 27 in Note du Pôle de modélisation Institut Pierre-Simon Laplace. IPSL, FranceGoogle Scholar
  34. Moore AM, Arango HG, Broquet G, Powell BS, Weaver AT, Zavala-Garay J (2011) The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems: Part I - System overview and formulation. Prog Oceanogr 91(1):34 – 49. doi: 10.1016/j.pocean.2011.05.004 CrossRefGoogle Scholar
  35. Nerger L (2015) On serial observation processing in localized ensemble Kalman filters. Mon Weather Rev 143:1554–1567. doi: 10.1175/MWR-D-14-00182.1 CrossRefGoogle Scholar
  36. Nerger L, Hiller W (2013) Software for ensemble-based data assimilation systems-Implementation strategies and scalability. Comput Geosci 55:110 – 118. doi: 10.1016/j.cageo.2012.03.026 CrossRefGoogle Scholar
  37. Nerger L, Hiller W, Schröter J (2005) A Comparison of Error Subspace Kalman Filters. Tellus Ser A Dyn Meteorol Oceanogr 57A(5):715–735. doi: 10.1111/j.1600-0870.2005.00141.x CrossRefGoogle Scholar
  38. Nerger L, Janjić T, Schröter J, Hiller W (2012a) A regulated localization scheme for ensemble-based kalman filters. Q J R Meteorol Soc 138(664):802–812. doi: 10.1002/qj.945
  39. Nerger L, Janjić T, Schröter J, Hiller W (2012b) A unification of ensemble square root kalman filters. Mon Weather Rev 140. doi: 10.1175/MWR-D-11-00102.1
  40. Pan M, Wood EF (2006) Data assimilation for estimating the terrestrial water budget using a constrained Ensemble Kalman Filter. J Hydrometeor 7:534–547. doi: 10.1175/JHM495.1 CrossRefGoogle Scholar
  41. Shchepetkin A, McWilliams J (2005) The regional oceanic modeling system: a split-explicit, free-surface, topography-following-coordinate ocean model. Ocean Model 9:347–404. doi: 10.1016/j.ocemod.2004.08.002 CrossRefGoogle Scholar
  42. Troupin C, Barth A, Sirjacobs D, Ouberdous M, Brankart JM, Brasseur P, Rixen M, Alvera-Azcárate A, Belounis M, Capet A, Lenartz F, Toussaint ME, Beckers JM (2012) Generation of analysis and consistent error fields using the Data Interpolating Variational Analysis (DIVA). Ocean Model 52–53:90–101. doi: 10.1016/j.ocemod.2012.05.002. http://hdl.handle.net/2268/125731 CrossRefGoogle Scholar
  43. Wang Q, Zhou W, Wang D, Dong D (2013) Ocean model open boundary conditions with volume, heat and salinity conservation constraints. Adv Atmos Sci 31(1):188–196. doi: 10.1007/s00376-013-2269-y CrossRefGoogle Scholar
  44. Weaver AT, Courtier P (2001) Correlation modelling on the sphere using a generalized diffusion equation. Q J R Meteorol Soc 127:1815–1842. doi: 10.1002/qj.49712757518 CrossRefGoogle Scholar
  45. 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 :1360–1378. doi: 10.1175/1520-0493(2003)131%3C1360:TAFVAW%3E2.0.CO;2
  46. Whitaker JS, Hamill TM (2002) Ensemble data assimilation without perturbed obsevations. Mon Weather Rev 130:1913–1924. doi: 10.1175/1520-0493(2002)130%3C1913:EDAWPO%3E2.0.CO;2 CrossRefGoogle Scholar
  47. White L, Legat V, Deleersnijder E (2008) Tracer Conservation for Three-Dimensional, Finite-Element, Free-Surface, Ocean Modeling on Moving Prismatic Meshes. Mon Weather Rev 136:420–442. doi: 10.1175/2007MWR2137.1 CrossRefGoogle Scholar
  48. Zhu J, Zheng F, Li X (2011) A new localization implementation scheme for ensemble data assimilation of non-local observations. Tellus A 63:2. doi: 10.1111/j.1600-0870.2010.00486.x CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.GeoHydrodynamic and Environmental Research (GHER)University of LiègeLiègeBelgium
  2. 2.LISTIC, Polytech Annecy-ChambéryUniversité Savoie Mont-BlancAnnecy-le-VieuxFrance

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