Izvestiya, Atmospheric and Oceanic Physics

, Volume 46, Issue 6, pp 677–712 | Cite as

Problems of variational assimilation of observational data for ocean general circulation models and methods for their solution

  • V. I. Agoshkov
  • V. M. Ipatova
  • V. B. Zalesnyi
  • E. I. Parmuzin
  • V. P. Shutyaev


Problems of the variational assimilation of satellite observational data on the temperature and level of the ocean surface, as well as data on the temperature and salinity of the ocean from the ARGO system of buoys, are formulated with the use of the global three-dimensional model of ocean thermodynamics developed at the Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS). Algorithms for numerical solutions of the problems are developed and substantiated, and data assimilation blocks are developed and incorporated into the global three-dimensional model. Numerical experiments are performed with the use of the Indian Ocean or the entire World Ocean as examples. These numerical experiments support the theoretical conclusions and demonstrate that the use of a model with an assimilation block of operational observational data is expedient.


variational data assimilation satellite observational data ocean surface temperature salinity field ocean surface level problem of initialization of geophysical fields mathematical model of World Ocean circulation 


  1. 1.
    A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, London, 1970).Google Scholar
  2. 2.
    V. I. Agoshkov, Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics (IVM RAN, Moscow, 2003) [in Russian].Google Scholar
  3. 3.
    J.-L. Lions, Quelques methods de Resolution des Problemes aux Limites Nonlineares (Dunod, Paris, 1968).Google Scholar
  4. 4.
    G. I. Marchuk, Adjoint Equations and Analysis of Complex (Kluwer, Dordrecht, 1995).Google Scholar
  5. 5.
    A. F. Bennett, Inverse Modeling of the Ocean and Atmosphere (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
  6. 6.
    R. Daley, Atmospheric Data Analysis (Cambridge University Press, Cambridge, 1991).Google Scholar
  7. 7.
    M. Ghil and P. Malanotte-Rizzoli, “Data Assimilation in Meteorology and Oceanography,” Adv. Geophys. 33, 141–266 (1991).Google Scholar
  8. 8.
    E. Kalnay, Atmospheric Modeling. Data Assimilation and Predictibility (Cambridge University Press, Cambridge, 2003).Google Scholar
  9. 9.
    H. Panofsky, “Objective Weather-Map Analysis,” J. Appl. Meteorol. 6, 386–392 (1949).Google Scholar
  10. 10.
    B. Gilchrist and G. Cressman, “An Experiment in Objective Analysis,” Tellus 6, 309–318 (1954).CrossRefGoogle Scholar
  11. 11.
    P. Bergthorsson and B. Doos, “Numerical Weather Map Analysis,” Tellus 7, 329–340 (1955).CrossRefGoogle Scholar
  12. 12.
    G. Cressman, “An Operational Objective Analysis System,” Mon. Wea. Rev. (1959).Google Scholar
  13. 13.
    L. Gandin, Impartial Analysis of Hydrometeorological Fields (Gidrometizdat, Leningrad, 1963) [in Russian].Google Scholar
  14. 14.
    A. C. Lorenc, “A Global Three-Dimensional Multivariate Statistical Analysis Scheme,” Mon. Wea. Rev. 109, 701–721 (1981).CrossRefGoogle Scholar
  15. 15.
    A. C. Lorenc, “Analysis Methods for Numerical Weather Prediction,” Quart. J. R. Meteorol. Soc. 112, 1177–1194 (1986).CrossRefGoogle Scholar
  16. 16.
    J. C. Derber and A. Rosati, “A Global Ocean Data Assimilation System,” J. Phys. Oceanogr. 19, 1333–1347 (1989).CrossRefGoogle Scholar
  17. 17.
    R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Trans. ASME J. Basic Eng. 82D, 35–45 (1961).Google Scholar
  18. 18.
    R. E. Kalman and R. S. Bucy, “New Results in Linear Filtering and Prediction Theory,” Trans. ASME J. Basic Eng. 83D, 95–108 (1961).Google Scholar
  19. 19.
    G. Evensen, Data Assimilation: The Ensemble Kalman Filter (Springer, Berlin, 2007).Google Scholar
  20. 20.
    G. K. Korotaev and V. N. Eremeev, Introduction to Operative Oceanography of the Black Sea (NPTs EKOSI-Gidrofizika, Sevastopol, 2006) [in Russian].Google Scholar
  21. 21.
    B. A. Nelepo, V. V. Knysh, A. S. Sarkisyan, et al., “Study of Synoptic Variability of the Ocean Based on Dynamic-Stochastic Approach,” Dokl. Akad. Nauk 246(4), 974–978 (1979).Google Scholar
  22. 22.
    A. S. Sarkisyan, Ocean Dynamics Simulation (Gidrometeoizdat, St. Petersburg, 1991) [in Russian].Google Scholar
  23. 23.
    A. S. Sarkisyan, S. G. Demyshev, G. K. Korotaev, et al., “An Example of Quaternary Analysis of Observational Data Obtained with the Razrezy Program for the Newfoundland EAZO,” in Advances in Science and Technology: Atmosphere, Ocean, and Space—Razrezy Program (VINITI, Moscow, 1986), Vol. 6, pp. 88–89 [in Russian].Google Scholar
  24. 24.
    E. J. Fertig, J. Harlim, and B. R. Hunt, “A Comparative Study of 4D-var and a 4D Ensemble Kalman Filter: Perfect Model Simulations with Lorenz-96,” Tellus 59A, 96–100 (2007).Google Scholar
  25. 25.
    E. Kalnay, H. Li, T. Miyoshi, et al., “4D-var or Ensemble Kalman Filter?,” Tellus (A59), 758–773 (2007).Google Scholar
  26. 26.
    F. Q. Zhang, M. Zhang, and J. A. Hansen, “Coupling Ensemble Kalman Filter with Four Dimensional Variational Data Assimilation,” Adv. Atmos. Sci. 26(1), 1–8 (2009).CrossRefGoogle Scholar
  27. 27.
    Y. Sasaki, “Some Basic Formalisms in Numerical Variational Analysis,” Mon. Wea. Rev. 98, 875–883 (1970).CrossRefGoogle Scholar
  28. 28.
    C. Provost and R. Salmon, “A Variational Method for Inverting Hydrographic Data,” J. Mar. Res. 44, 1–34 (1986).CrossRefGoogle Scholar
  29. 29.
    V. V. Penenko and N. V. Obraztsov, “Variational Method for Fields of Meteorological Elements,” Meteorol. Gidrol., No. 11, 1–11 (1976).Google Scholar
  30. 30.
    F.-X. Le Dimet and O. Talagrand, “Variational Algorithms for Analysis and Assimilation of Meteorological Observations: Theoretical Aspects,” Tellus 38A, 97–110 (1986).CrossRefGoogle Scholar
  31. 31.
    G. I. Marchuk and V. V. Penenko, Application of Optimization Methods to the Problem of Mathematical Simulation of Atmospheric Processes and Environment, in Modelling and Optimization of Complex Systems: Proc. of the IFIP-TC7 Working Conf., Ed. by G. I. Marchuk (Springer, New York, 1978), pp. 240–252.Google Scholar
  32. 32.
    V. I. Agoshkov, E. I. Parmuzin, and V. P. Shutyaev, “Numerical Algorythm of Variational Assimilation of Observational Data on Oceanic Surface Temperature,” Zh. Vysch. Matem. Matem. Fiz. 48(8), pp. 1371–1391 (2008).Google Scholar
  33. 33.
    M. Ventsel’ and V. B. Zalesnyi, “Data Assimilation in One-Dimensional Model of Heat Convection-Diffusion in the Ocean,” Izv. Akad. Nauk, Fiz. Atm. Okeana 32(5), 613–629 (1996).Google Scholar
  34. 34.
    V. P. Shutyaev, Control Operators and Iterative Algorithms in Variational Data Assimilation Problems (Nauka, Moscow, 2001) [in Russian].Google Scholar
  35. 35.
    V. I. Agoshkov and G. I. Marchuk, “On Solvability and Numerical Solution of Data Assimilation Problems,” Russ. J. Numer. Anal. Math. Modelling 8, 1–16 (1993).CrossRefGoogle Scholar
  36. 36.
    P. Courtier, E. Andersson, W. Heckley, et al., “The ECMWF Implementation of Three-Dimensional Variational Assimilation (3D-var). I: Formulation,” Quart. J. R. Meteorol. Soc. 124, 1783–1807 (1998).Google Scholar
  37. 37.
    G. I. Marchuk and V. P. Shutyaev, “Iteration Methods for Solving a Data Assimilation Problem,” Russ. J. Numer. Anal. Math. Modelling 9(3), 265–279 (1994).CrossRefGoogle Scholar
  38. 38.
    G. I. Marchuk and V. B. Zalesny, “A Numerical Technique for Geophysical Data Assimilation Problem using PontryaginOS Principle and Splitting-Up Method,” Russ. J. Numer. Anal. Math. Modelling 8(4), 311–326 (1993).CrossRefGoogle Scholar
  39. 39.
    I. M. Navon, “A Review of Variational and Optimization Methods in Meteorology,” in Variational Methods in Geosciences, Ed. by Y. K. Sasaki, (Elsevier, New York, 1986), pp. 29–34.Google Scholar
  40. 40.
    D. F. Parrish and J. C. Derber, “The National Meteorological CenterOS Spectral Statistical Interpolation Analysis Scheme,” Mon. Wea. Rev. 120, 1747–1763 (1992).CrossRefGoogle Scholar
  41. 41.
    V. M. Ipatova, Data Assimilation Problems for the General Ocean Circulation Model in the Quasigeostrophic Approximation, (VINITI, Moscow, 1992) [in Russian].Google Scholar
  42. 42.
    V. I. Agoshkov and V. M. Ipatova, “Solvability of the Observational Data Assimilation Problem for a 3D Ocean Dynamics Model,” Differential Equations 43(8), 1064–1075 (2007).CrossRefGoogle Scholar
  43. 43.
    V. I. Agoshkov and V. M. Ipatova, “Existence Theorems for a 3D Ocean Dynamics Model and Data Assimilation Problems,” Dokl. Akad. Nauk 412(2), 151–153 (2007).Google Scholar
  44. 44.
    V. M. Ipatova, “Solvability of the Ocean Hydrothermodynamics Problem under a Nonlinear State Equation,” Russ. J. Numer. Anal. Math. Modelling 23(2), 185–196 (2008).CrossRefGoogle Scholar
  45. 45.
    E. I. Parmuzin, V. P. Shutyaev, and N. A. Diansky, “Numerical Solution of a Variational Data Assimilation Problem for a 3D Ocean Thermohydrodynamics Model with a Nonlinear Vertical Heat Exchange,” Russ. J. Numer. Anal. Math. Modelling 22(2), 177–198 (2007).CrossRefGoogle Scholar
  46. 46.
    Z. Sirkes and E. Tziperman, “Finite Difference of Adjoint or Adjoint of Finite Difference?,” Mon. Weather Rev. 125, 3373–3378 (1997).CrossRefGoogle Scholar
  47. 47.
    W. C. Chao and L.-P. Chang, “Development of a Four-Dimensional Variational Analysis System using the Adjoint Method at GLA. Pt I, Dynamics,” Mon. Weather Rev. 120, 1661–1672 (1992).CrossRefGoogle Scholar
  48. 48.
    I. Charpentier and M. Ghemires, Generation Automatique de Codes Adjoints: Strategies d’Utilisation pour le Logiciel Odysse, Application au Code Meteorologique Meso-NH, Rapport de Recherche INRIA RR-3251 (1997).Google Scholar
  49. 49.
    I. Gejadze, F.-X. Le Dimet, and V. Shutyaev, “On Analysis Error Covariances in Variational Data Assimilation,” SIAM J. Sci. Comput. 30(4), 1847–1874 (2008).CrossRefGoogle Scholar
  50. 50.
    F.-X. Le Dimet and V. P. Shutyaev, “On Deterministic Error Analysis in Variational Data Assimilation,” Nonlin. Proces. Geophys. 12, 481–490 (2005).CrossRefGoogle Scholar
  51. 51.
    V. P. Shutyaev and E. I. Parmuzin, “Some Algorithms for Studying Solution Sensitivity in the Problem of Variational Assimilation of Observation Data for a Model of Ocean Thermodynamics,” Russ. J. Numer. Anal. Math. Modelling 24(2), 145–160 (2009).CrossRefGoogle Scholar
  52. 52.
    A. Caya, J. Sun, and C. Snyder, “A Comparison between the 4D-var and the Ensemble Kalman Filter Techniques for Radar Data Assimilation,” Monthly Weather Rev. 133(11), 3081–3094 (2005).CrossRefGoogle Scholar
  53. 53.
    X. Tian, J. Xie, and A. Dai, “An Ensemble-Based Explicit 4D-Var Assimilation Method,” J. Geophys. Res. 113(D21124) (2008).Google Scholar
  54. 54.
    V. I. Agoshkov, F. P. Minyuk, A. S. Rusakov, et al., “Study and Solution of Identification Problems for Nonstationary 2D- and 3D-Convection-Diffusion Equation,” Russ. J. Numer. Anal. Math. Modelling 20(1), 19–43 (2005).CrossRefGoogle Scholar
  55. 55.
    V. I. Agoshkov, A. V. Gusev, N. A. Dianski, et al., “An Algorithm for the Solution of the Ocean Hydrothermodynamics Problem with Variational Assimilation of the Sea Level Function Data,” Russ. J. Numer. Anal. Math. Modelling 22(2), 133–161 (2007).CrossRefGoogle Scholar
  56. 56.
    “DYNAMO Group, Dynamics of North Atlantic Models and Assimilation with High-Resolution Models,” Ber. Inst. f. Meereskunde Kiel. 294 (1997).Google Scholar
  57. 57.
    I. N. Sinitsyn, Kalman and Pugachev Filters (Logos, Moscow, 2006) [in Russian].Google Scholar
  58. 58.
    V. V. Alekseev and V. B. Zalesnyi, “Numerical Model of Large-Scale Ocean Dynamics,” in Computational Processes and Systems (Nauka, Moscow, 1993), pp. 232–253 [in Russian].Google Scholar
  59. 59.
    N. A. Diansky, A. V. Bagno, and V. B. Zalesny, “Sigma Model of Global Ocean Circulation and Its Sensitivity to Variations in Wind Stress,” Izv. Akad. Nauk, Fiz. Atm. Okeana 38(4), 537–556 (2002) [Izv., Atmos. Ocean. Phys. 38 (4), 477–494 (2002).Google Scholar
  60. 60.
    N. A. Diansky, V. B. Zalesny, S. N. Moshonkin, et al., “High Resolution Modeling of the Monsoon Circulation in the Indian Ocean,” Okeanologiya 46(4), 11–12 (2006) [Oceanology 46 (5), 608–628 (2006)].Google Scholar
  61. 61.
    A. S. Sarkisyan, V. B. Zalesnyi, N. A. Dianskii, et al., “Mathematical Models of Ocean and Sea Circulation,” in Modern Problems in Computational Mathematics and Mathematical Simulation, (Nauka, Moscow, 2005), Vol. 2, pp. 175–278 [in Russian].Google Scholar
  62. 62.
    V. I. Agoshkov, S. A. Lebedev, and E. I. Parmuzin, “Numerical Solution to the Problem of Variational Assimilation of Operational Observational Data on the Ocean Surface Temperature,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 45(1), 76–107 (2009) [Izv., Atmos. Ocean. Phys. 45 (1), 69–101 (2009).Google Scholar
  63. 63.
    G. I. Marchuk, V. P. Dymnikov, and V. B. Zalesnyi, Mathematical Models in Geophysical Hydrodynamics and Numerical Methods of Their Realization (Gidrometeoizdat, Leningrad, 1987) [in Russian].Google Scholar
  64. 64.
    V. I. Agoshkov, E. I. Parmuzin, and V. P. Shutyaev, “A Numerical Algorithm of Variational Data Assimilation for Reconstruction of Salinity Fluxes on the Ocean Surface,” Russ. J. Numer. Anal. Math. Modelling 23(2), 135–161 (2008).CrossRefGoogle Scholar
  65. 65.
    G. I. Marchuk, Methods of Computational Mathematics (Nauka, Moscow, 1989) [in Russian].Google Scholar
  66. 66.
    A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977) [in Russian].Google Scholar
  67. 67.
    R. C. Pacanovsky and S. M. Griffies, “The MOM 3 Manual,” Geophys. Fluid Dynam. Labor. (NOAA, Princenton, 1999).Google Scholar
  68. 68.
    A. F. Blumberg and G. L. Mellor, A Description of a Three-Dimensional Coastal Ocean Circulation Model. Three-Dimensional Coastal Models, Ed. by N. S. Heaps (Amer. Geophys. Union, 1987).Google Scholar
  69. 69.
    V. I. Kuzin, Finite Element Method in Simulation of Oceanic Processes (VTs SO AN SSSR, Novosibirsk, 1985) [in Russian].Google Scholar
  70. 70.
    G. I. Marchuk, A. S. Rusakov, V. B. Zalesny, et al., “Splitting Numerical Technique with Application to the High Resolution Simulation of the Indian Ocean Circulation,” in Pure and Applied Geophysics (Birkhauser Verlag, Basel, 2005), No. 162, pp. 1407–1429.Google Scholar
  71. 71.
    V. I. Agoshkov, “Inverse Problems of the Mathematical Theory of Tides: Boundary-Function Problem,” Russ. J. Numer. Anal. Math. Modelling 20(1), 1–18 (2005).CrossRefGoogle Scholar
  72. 72.
    F. P. Vasil’ev, Numerical Methods of Solving Extremal Problems (Nauka, Moscow, 1988) [in Russian].Google Scholar
  73. 73.
    J.-C. Gilbert and C. Lemarechal, “Some Numerical Experiment with Variable Storage Quasi-Newton Algorithms,” Math. Program. 25, 408–435 (1989).Google Scholar
  74. 74.
    N. B. Zakharova, “Computational Algorithms of Interpolation and Extrapolation of Numerical Solution of Problems of Variational Assimilation of Geophysical Observational Data,” in Collected Abstracts of Best Diploma Works of 2009 (Izdat. Otdel Fak. VMK MGU, Moscow, 2009), pp. 26–27 [in Russian].Google Scholar
  75. 75.
    N. B. Zakharova and S. A. Lebedev, “Algorithms of Interpolation and Extrapolation of Operative Geophysical Observational Data,” Collect. Papers Young Sci. Fac. Comp. Mathem. Cybern., Mosc. State Univ., No. 6, 177–188 (2009).Google Scholar
  76. 76.
    V. B. Zalesny and A. V. Gusev, “Mathematical Model of the World Ocean Dynamics with Algorithms of Variational Assimilation of Temperature and Salinity Fields,” Russ. J. Numer. Anal. Math. Modelling 24(2), 171–190 (2009).CrossRefGoogle Scholar
  77. 77.
    G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems (CRC Press, New York, 1996).Google Scholar
  78. 78.
    G. Marchuk, V. Shutyaev, and V. Zalesny, Approaches to the Solution of Data Assimilation Problems, in Optimal Control and Partial Differential Equations, Ed. by J. Menaldi, E. Rofman, and A. Sulem (IOS Press, Amsterdam, 2001), pp. 489–497.Google Scholar
  79. 79.
    V. B. Zalesny and A. S. Rusakov, “Numerical Algorithm of Data Assimilation Based on Splitting and Adjoint Equation Methods,” Russ. J. Numer. Anal. Math. Modelling 22(2), 199–219 (2007).CrossRefGoogle Scholar
  80. 80.
    Yu. D. Resnyanskii and A. A. Zelen’ko, “Automated System for Predicting the Surface Ocean Layer Temperature for Five Days,” Meteorol. Gidrol., No. 8, 71–80 (1987).Google Scholar
  81. 81.
    E. V. Semenov, “Numerical Modeling of White Sea Dynamics and Monitoring Problem,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 40(1), 128–141 (2004) [Izv., Atmos. Ocean. Phys. 40 (1), 114–126 (2004)].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • V. I. Agoshkov
    • 1
  • V. M. Ipatova
    • 2
  • V. B. Zalesnyi
    • 1
  • E. I. Parmuzin
    • 1
  • V. P. Shutyaev
    • 1
  1. 1.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyi, Moscow oblastRussia

Personalised recommendations