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Algorithms of variational data assimilation for problems of ocean dynamics

  • Valery I. Agoshkov , Vladimir B. Zalesny , Victor P. Shutyaev EMAIL logo , Eugene I. Parmuzin , Natalia B. Zakharova und Tatiana O. Sheloput
Veröffentlicht/Copyright: 12. Juni 2025

Abstract

The current state of research in the field of variational assimilation of observational data in models of ocean dynamics developed at the INM RAS is presented. The developed technology of four-dimensional variational data assimilation (4D-Var) is based on the method of multicomponent splitting of the mathematical model of ocean dynamics and minimization of the cost functional associated with observational data by solving an optimality system that includes adjoint equations and covariance matrices of observation errors and errors of initial approximation. Effective algorithms for solving variational problems of data assimilation based on iterative processes using direct and adjoint equations with a special choice of iterative parameters are proposed, as well as algorithms for studying the sensitivity of model characteristics to errors in observational data. The methodology is illustrated for a model of the Black Sea hydrothermodynamics with variational data assimilation to restore heat fluxes on the sea surface.

MSC 2010: 65K10

Funding statement: The work was supported by the Russian Science Foundation (project 19–71–20035, studies in Section 4) and by the Moscow Center of Fundamental and Applied Mathematics at INM RAS (Agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2025-347).

References

[1] V. I. Agoshkov, Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics. INM RAS, Moscow, 2003 (in Russian).Suche in Google Scholar

[2] V. I. Agoshkov, A. V. Gusev, N. A. Diansky, and R. V. Oleinikov, 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 (2007), 133–161.10.1515/RJNAMM.2007.007Suche in Google Scholar

[3] V. I. Agoshkov and G. I. Marchuk, On solvability and numerical solution of data assimilation problems. Russ. J. Numer. Anal. Math. Modelling. 8 (1993), 1–16.10.1515/rnam.1993.8.1.1Suche in Google Scholar

[4] V. I. Agoshkov, E. I. Parmuzin, and V. P. Shutyaev, Numerical algorithm for variational assimilation of sea surface temperature data. Comp. Math. Math. Physics 48 (2008), No. 8, 1293–131210.1134/S0965542508080046Suche in Google Scholar

[5] V. I. Agoshkov, V. P. Shutyaev, E. I. Parmuzin, N. B. Zakharova, T. O. Sheloput, and N. R. Lezina, Variation data assimilation in the mathematical model of the Black Sea dynamics. Phys. Oceanogr. 26 (2019), No. 6, 387–396.Suche in Google Scholar

[6] M. Asch, M. Bocquet, and M. Nodet, Data Assimilation: Methods, Algorithms, and Applications. SIAM, Philadelphia, USA, 2016.10.1137/1.9781611974546Suche in Google Scholar

[7] D. G. Cacuci, Sensitivity theory for nonlinear systems: II. Extensions to additional classes of responses. J. Math. Phys. 22 (1981), 2803–2812.10.1063/1.524870Suche in Google Scholar

[8] A. Carrassi, M. Bocquet, L. Bertino, and G. Evensen, Data assimilation in the geosciences: an overview of methods, issues, and perspectives. WIREs Clim. Change 9 (2018), 1–80.10.1002/wcc.535Suche in Google Scholar

[9] F. Chen, G. Shapiro, and R. Thain, Sensitivity of sea surface temperature simulation by an ocean model to the resolution of the meteorological forcing. Int. Sch. Res. Not. Oceanography 2013 (2013), 215715.10.5402/2013/215715Suche in Google Scholar

[10] A. Cioaca, A. Sandu, and E. de Sturler, Efficient methods for computing observation impact in 4D-Var data assimilation. Comput. Geosci. 17 (2013), 975–990.10.1007/s10596-013-9370-2Suche in Google Scholar

[11] D. N. Daescu, On the sensitivity equations of four-dimensional variational (4D-Var) data assimilation. Mon. Weather Rev. 136 (2008), 3050–3065.10.1175/2007MWR2382.1Suche in Google Scholar

[12] 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., Atmos. Ocean. Phys. 38 (2002), No. 4, 477–494.Suche in Google Scholar

[13] V. P. Dymnikov and V. B. Zalesny, Fundamentals of Computational Geophysical Fluid Dynamics. GEOS, Moscow, 2019.Suche in Google Scholar

[14] V. P. Dymnikov, D. V. Kulyamin, P. A. Ostanin, and V. P. Shutyaev, Data assimilation for the two-dimensional ambipolar diffusion equation in Earth’s ionosphere model. Comput. Math. Math. Phys. 63 (2023), No. 5, 845–867.10.1134/S0965542523050093Suche in Google Scholar

[15] S. J. Fletcher, Data Assimilation for the Geosciences: From Theory to Application. Elsevier, Amsterdam, the Netherlands, 2017.Suche in Google Scholar

[16] I. Gejadze, F.-X. Le Dimet, and V. P. Shutyaev, On analysis error covariances in variational data assimilation. SIAM J. Sci. Comput. 30 (2008), No. 4, 1847–1874.10.1137/07068744XSuche in Google Scholar

[17] I. Gejadze, F.-X. Le Dimet, and V. P. Shutyaev, On optimal solution error covariances in variational data assimilation problems. J. Comp. Phys. 229 (2010), 2159–2178.10.1016/j.jcp.2009.11.028Suche in Google Scholar

[18] I. Gejadze, V. P. Shutyaev, and F.-X. Le Dimet, Analysis error covariance versus posterior covariance in variational data assimilation. Q. J. R. Meteorol. Soc. 139 (2013), 1826–1841.10.1002/qj.2070Suche in Google Scholar

[19] G. Gualtieri, Analyzing the uncertainties of reanalysis data used for wind resource assessment: A critical review. Renew. Sustain. Energy Rev. 167 (2022), 112741.10.1016/j.rser.2022.112741Suche in Google Scholar

[20] H. Hersbach et al., The ERA5 global reanalysis. Q. J. R. Meteorol. Soc. 146 (2020), 1999–2049.10.1002/qj.3803Suche in Google Scholar

[21] A. N. Kolmogorov, On the proof of the method of least squares. Uspekhi Matem. Nauk 1 (1946), 57–70 (in Russian).Suche in Google Scholar

[22] F. X. Le Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus 38A (1986), 97–110.10.3402/tellusa.v38i2.11706Suche in Google Scholar

[23] F.-X. Le Dimet, I. M. Navon, and D. N. Daescu, Second-order information in data assimilation. Month. Wea. Rev. 130 (2002), No. 3, 629–648.10.1175/1520-0493(2002)130<0629:SOIIDA>2.0.CO;2Suche in Google Scholar

[24] F.-X. Le Dimet, H. E. Ngodock, B. Luong, and J. Verron, Sensitivity analysis in variational data assimilation. J. Meteorol. Soc. Japan 75 (1997), No. 1B, 245–255.10.2151/jmsj1965.75.1B_245Suche in Google Scholar

[25] F.-X. Le Dimet and V. Shutyaev, On deterministic error analysis in variational data assimilation. Nonlinear Processes in Geophysics 12 (2005), 481–490.10.5194/npg-12-481-2005Suche in Google Scholar

[26] F.-X. Le Dimet, V. Shutyaev, and E. Parmuzin, Sensitivity of functionals with respect to observations in variational data assimilation. Russ. J. Numer. Anal. Math. Modelling 31 (2016), No. 2, 81–91.10.1515/rnam-2016-0009Suche in Google Scholar

[27] J. L. Lions, Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles. Dunod, Paris, 1968.Suche in Google Scholar

[28] A. C. Lorenc, Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc. 112 (1986), 1177–1194.10.1002/qj.49711247414Suche in Google Scholar

[29] E. A. Lupyan, A. A. Matveev, I. A. Uvarov, T. Yu. Bocharova, O. Yu. Lavrova, M. I. Mityagina, See the Sea: A satellite service for studying ocean surface processes and phenomena, Sovrem. Probl. Distantsionnogo Zondirovaniya Zemli Kosmosa 9 (2012), No. 2, 251–261 (in Russian).Suche in Google Scholar

[30] G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems. Kluwer, Dordrecht, 1995.10.1007/978-94-017-0621-6Suche in Google Scholar

[31] G. I. Marchuk, Splitting and Alternating Direction Methods. Handbook of Numerical Analysis; Vol. 1 (Eds. P. G. Ciarlet and J. L. Lions). North-Holland, Amsterdam, the Netherlands, 1990, pp. 197–462.10.1016/S1570-8659(05)80035-3Suche in Google Scholar

[32] G. I. Marchuk, Adjoint equations and sensitivity of functionals. Earth Research from Space (1997), No. 4, 100–125.Suche in Google Scholar

[33] G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems of Mathematical Physics. CRC Press, New York, 1996.Suche in Google Scholar

[34] G. I. Marchuk and V. V. Penenko, Study of sensitivity of discrete models of atmosphere and ocean dynamics. Izvestiya AN SSSR. Physics of the Atmosphere and Ocean 15 (1979), No. 11, 1123–1131.Suche in Google Scholar

[35] G. I. Marchuk and V. P. Shutyaev, Iteration methods for solving a data assimilation problem. Russ. J. Numer. Anal. Math. Modelling 9 (1994), No. 3, 265–279.10.1515/rnam.1994.9.3.265Suche in Google Scholar

[36] G. I. Marchuk and V. B. Zalesny, A numerical technique for geophysical data assimilation problem using Pontryagin’s principle and splitting-up method. Russ. J. Numer. Anal. Math. Modelling 8 (1993), No. 4, 311–326.10.1515/rnam.1993.8.4.311Suche in Google Scholar

[37] A. A. Markov, Ischislenie Veroyatnostej. Imperial Academy of Sciences, St. Petersburg, Russia, 1900.Suche in Google Scholar

[38] K. Mogensen, M. A. Balmaseda, A. T. Weaver, M. Martin, and A. Vidard, NEMOVAR: a variational data assimilation system for the NEMO ocean model. ECMWF Technical Memorandum (2009), No. 120.Suche in Google Scholar

[39] A. Penenko, Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements. Inverse Problems & Imaging 14 (2020), No. 5, 757–782.10.3934/ipi.2020035Suche in Google Scholar

[40] V. V. Penenko and N. V. Obraztsov, A variational initialization method for the fields of the meteorological elements. Meteorol. Gidrol. 11 (1976), 1–11.Suche in Google Scholar

[41] Y. K. Sasaki, An objective analysis based on the variational method. J. Meteor. Soc. Japan 36 (1958), 77–88.10.2151/jmsj1923.36.3_77Suche in Google Scholar

[42] G. I. Shapiro and M. Salim, How efficient is model-to-model data assimilation at mitigating atmospheric forcing errors in a regional ocean model? J. Mar. Sci. Eng. 11 2023, No. 5, 935.10.3390/jmse11050935Suche in Google Scholar

[43] V. P. Shutyaev, Control Operators and Iterative Algorithms in Problems in Variational Data Assimilation Problems. Nauka, Moscow, 2001.10.1515/jiip.2001.9.2.177Suche in Google Scholar

[44] V. P. Shutyaev, Methods for observation data assimilation in problems of physics of atmosphere and ocean. Izv. Atmos. Ocean. Phys. 55 (2019), 17–31.10.1134/S0001433819010080Suche in Google Scholar

[45] V. P. Shutyaev and F.-X. Le Dimet, Sensitivity of functionals of variational data assimilation problems. Dokl. Math. 99 (2019), 295–298.10.1134/S1064562419030153Suche in Google Scholar

[46] V. Shutyaev, V. Zalesny, V. Agoshkov, E. Parmuzin, and N. Zakharova, 4D-Var data assimilation and sensitivity of ocean model state variables to observation errors. J. Mar. Sci. Eng. 11 (2023), 1253.10.3390/jmse11061253Suche in Google Scholar

[47] V. P. Shutyaev, V. I. Agoshkov, E. I. Parmuzin, V. B. Zalesny, and N. B. Zakharova, 4D technology of variational data assimilation for sea dynamics problems. Supercomputing Frontiers and Innovations 9 (2022), No. 1, 4–16.Suche in Google Scholar

[48] N. B. Zakharova, Verification of observational SST data, Sovrem. Probl. Distantsionnogo Zondirovaniya Zemli Kosmosa 13 (2016), No. 3, 106–113 (in Russian).Suche in Google Scholar

[49] V. B. Zalesny, V. I. Agoshkov, V. P. Shutyaev, F.-X. Le Dimet, and B. O. Ivchenko, Numerical modeling of ocean hydrodynamics with variational assimilation of observational data. Izv. Atmos. Ocean. Phys. 52 (2016), 431–442.10.1134/S0001433816040137Suche in Google Scholar

[50] V. Zalesny, V. Agoshkov, V. Shutyaev, E. Parmuzin, N. Zakharova, Numerical modeling of marine circulation with 4D variational data assimilation. J. Mar. Sci. Eng. 8 (2020), No. 503, 1–19.10.3390/jmse8070503Suche in Google Scholar

[51] V. B. Zalesny, N. A. Diansky, V. V. Fomin, S. N. Moshonkin, and S. G. Demyshev, Numerical model of the circulation of the Black Sea and the Sea of Azov. Russ. J. Numer. Anal. Math. Modelling 27 (2012), No. 1, 95–112.10.1515/rnam-2012-0006Suche in Google Scholar

[52] V. B. Zalesny, G. I. Marchuk, V. I. Agoshkov, A. V. Bagno, A. V. Gusev, N. A. Diansky, S. N. Moshonkin, R. Tamsalu, and E. M. Volodin, Numerical simulation of large-scale ocean circulation based on the multicomponent splitting method. Russ. J. Numer. Anal. Math. Modelling 25 (2010), 581–609.10.1515/rjnamm.2010.036Suche in Google Scholar

Received: 2025-04-28
Accepted: 2025-04-30
Published Online: 2025-06-12
Published in Print: 2025-06-26

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 15.1.2026 von https://www.degruyterbrill.com/document/doi/10.1515/rnam-2025-0014/pdf
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