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Second-Order Methods in Variational Data Assimilation

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Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (Vol. IV)

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Abstract

The chapter presents the second-order methods in variational data assimilation. The algorithms to compute the Hessian of the cost function are discussed, the second-order adjoint method among them. General sensitivity analysis for the  optimality system is presented. Using the Hessian, the sensitivity of the optimal solution and its functionals is studied with respect to observations and uncertainties in model parameters. Numerical examples for joint state and parameter estimation for a sea thermodynamics model are presented.

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References

  • Agoshkov VI, Sheloput TO (2017) The study and numerical solution of some inverse problems in simulation of hydrophysical fields in water areas with ‘liquid’ boundaries. Russ J Numer Anal Math Modelling 32:147–164

    Google Scholar 

  • Agoshkov VI, Parmuzin EI, Shutyaev VP (2008) Numerical algorithm of variational assimilation of the ocean surface temperature data. Comp Math Math Phys 48:1371–1391

    Article  Google Scholar 

  • Agoshkov VI, Parmuzin EI, Zalesny VB, Shutyaev VP et al (2015) Variational assimilation of observation data in the mathematical model of the Baltic Sea dynamics. Russ J Numer Anal Math Modelling 30:203–212

    Google Scholar 

  • Alifanov OM, Artyukhin EA, Rumyantsev SV (1996) Extreme methods for solving ill-posed problems with applications to inverse heat transfer problems. Begell House Publishers, Danbury

    Google Scholar 

  • Asch M, Bocquet M, Nodet M (2016) Data assimilation: methods, algorithms, and applications. SIAM, Philadelphia

    Book  Google Scholar 

  • Bocquet M (2012) Parameter-field estimation for atmospheric dispersion: application to the Chernobyl accident using 4D-Var. Q J R Meteor Soc 138:664–681

    Article  Google Scholar 

  • Cacuci DG (1981) Sensitivity theory for nonlinear systems: II. Extensions to additional classes of responses. J Math Phys 22:2803–2812

    Article  Google Scholar 

  • Carrassi A, Bocquet M, Bertino L, Evensen G (2018) Data assimilation in the geosciences: an overview of methods, issues, and perspectives. WIREs Clim Change 9:1–80

    Article  Google Scholar 

  • Chavent G (1983) Local stability of the output least square parameter estimation technique. Math Appl Comp 2:3–22

    Google Scholar 

  • Cioaca A, Sandu A, de Sturler E (2013) Efficient methods for computing observation impact in 4D-Var data assimilation. Comput Geosci 17:975–990

    Article  Google Scholar 

  • Daescu DN (2008) On the sensitivity equations of four-dimensional variational (4D-Var) data assimilation. Mon Weather Rev 136:3050–3065

    Article  Google Scholar 

  • Daescu DN, Langland RH (2013) Error covariance sensitivity and impact estimation with adjoint 4D-Var: Theoretical aspects and first applications to NAVDAS-AR. Q J R Meteorol Soc 139:226–241

    Article  Google Scholar 

  • Dontchev AL (1983) Perturbations, approximations and sensitivity analysis of optimal control systems. Lecture notes in control and information sciences, vol 52. Springer, Berlin

    Google Scholar 

  • Fletcher SJ (2017) Data assimilation for the geosciences: from theory to application. Elsevier, Amsterdam

    Book  Google Scholar 

  • Gejadze I, Le Dimet FX, Shutyaev V (2008) On analysis error covariances in variational data assimilation. SIAM J Sci Comp 30:1847–1874

    Article  Google Scholar 

  • Gejadze I, Le Dimet FX, Shutyaev V (2010) On optimal solution error covariances in variational data assimilation problems. J Comput Phys 229:2159–2178

    Article  CAS  Google Scholar 

  • Gejadze IYu, Copeland GJM, Le Dimet FX, Shutyaev VP (2011) Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics. J Comput Phys 230:7923–7943

    Article  Google Scholar 

  • Gejadze IYu, Shutyaev VP, Le Dimet FX (2013) Analysis error covariance versus posterior covariance in variational data assimilation. Q J R Meteor Soc 139:1826–1841

    Article  Google Scholar 

  • Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, London

    Google Scholar 

  • Godinez HC, Daescu DN, (2009) A second order adjoint method to targeted observations. In: Allen G, Nabrzyski J, Seidel E, van Albada GD, Dongarra J, Sloot PMA (eds) Computational Science - ICCS 2009. ICCS, (2009) Lecture notes in computer science, vol 5545. Springer, Berlin, Heidelberg, pp 332–341

    Google Scholar 

  • Griesse R, Vexler B (2007) Numerical sensitivity analysis for the quantity of interest in pde-constrained optimization. SIAM J Sci Comp 29:22–48

    Article  Google Scholar 

  • Kalnay E, Ota Y, Miyoshi T et al (2012) A simpler formulation of forecast sensitivity to observations: Application to ensemble Kalman filters. Tellus 64A:18462

    Article  Google Scholar 

  • Karagali I, Hoyer J, Hasager CB (2012) SST diurnal variability in the North Sea and the Baltic Sea. Remote Sens Environ 121:159–170

    Article  Google Scholar 

  • Kolmogorov AN (1946) On the proof of the method of least squares. Uspekhi Mat Nauk 1:57–70

    Google Scholar 

  • Langland RH, Baker NL (2004) Estimation of observation impact using the NRL atmospheric variational data assimilation adjoint system. Tellus 56A:189–201

    Article  Google Scholar 

  • Le Dimet FX (1982) A general formalism of variational analysis. Report OK-73091-22-1, CIMMS, Norman

    Google Scholar 

  • Le Dimet FX, Talagrand O (1986) Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A 38:97–110

    Article  Google Scholar 

  • Le Dimet FX, Ngodock HE, Luong B, Verron J (1997) Sensitivity analysis in variational data assimilation. J Meteorol Soc Japan 75:245–255

    Article  Google Scholar 

  • Le Dimet FX, Navon IM, Daescu DN (2002) Second-order information in data assimilation. Mon Weather Rev 130:629–648

    Article  Google Scholar 

  • Lions JL (1968) Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris

    Google Scholar 

  • Marchuk GI, Penenko VV (1978) Application of optimization methods to the problem of mathematical simulation of atmospheric processes and environment. In: Modelling and optimization of complex systems. Proceedings of the IFIP-TC7 conference. Springer, Heidelberg, pp 240–252

    Google Scholar 

  • Marchuk GI (1995) Adjoint equations and analysis of complex systems. Kluwer, Dordrecht

    Book  Google Scholar 

  • Marchuk GI, Dymnikov VP, Zalesny VB (1987) Mathematical models in geophysical hydrodynamics and numerical methods for their realization. Gidrometeoizdat, Leningrad

    Google Scholar 

  • Marchuk GI, Agoshkov VI, Shutyaev VP (1996) Adjoint equations and perturbation algorithms in nonlinear problems. CRC Press, New York

    Google Scholar 

  • Markov AA (1900) Ischislenie veroyatnostej. Imperial Academy of Sciences, St Petersburg

    Google Scholar 

  • Navon IM (1998) Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography. Dyn Atmos Oceans 27:55–79

    Google Scholar 

  • Penenko V, Obraztsov NN (1976) A variational initialization method for the fields of the meteorological elements. Soviet Meteorol Hydrol (English transl) 11:1–11

    Google Scholar 

  • Polak E (1997) Optimization: algorithms and consistent approximations. Appl Math Sci 124. Springer, New York

    Google Scholar 

  • Pontryagin LS, Boltyansky VG, Gamkrelidze RV, Mishchenko EF (1964) The mathematical theory of optimal processes. International Series of Monographs in Pure and Applied Mathematics, vol 55. Pergamon Press, Oxford, New York

    Google Scholar 

  • Sasaki Y (1958) An objective analysis based on the variational method. J Meteorol Soc Japan 36:77–88

    Article  Google Scholar 

  • Schirber S, Klocke D, Pincus R et al (2013) Parameter estimation using data assimilation in an atmospheric general circulation model: From a perfect toward the real world. J Adv Model Earth Syst 5:58–70

    Article  Google Scholar 

  • Shutyaev VP, Le Dimet FX, Gejadze IYu et al (2012) Optimal solution error covariance in highly nonlinear problems of variational data assimilation. Nonlin Processes Geophys 19:177–184

    Article  Google Scholar 

  • Shutyaev V, Le Dimet FX, Shubina E (2017) Sensitivity with respect to observations in variational data assimilation. Russ J Numer Anal Math Modelling 32:61–71

    Article  Google Scholar 

  • Shutyaev VP, Le Dimet FX, Parmuzin EI (2018) Sensitivity analysis with respect to observations in variational data assimilation for parameter estimation. Nonlin Processes Geophys 25:429–439

    Article  Google Scholar 

  • Smith PJ, Thornhill GD, Dance SL et al (2013) Data assimilation for state and parameter estimation: application to morphodynamic modelling. Q J R Meteor Soc 139:314–327

    Article  Google Scholar 

  • Storch RB, Pimentel LCG, Orlande HRB (2007) Identification of atmospheric boundary layer parameters by inverse problem. Atmos Environ 41:141–1425

    Google Scholar 

  • Sun NZ (1994) Inverse problems in groundwater modeling. Kluwer, Dordrecht

    Google Scholar 

  • Talagrand O, Courtier P (1987) Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. Q J R Meteor Soc 113:1311–1328

    Article  Google Scholar 

  • Thacker WC (1989) The role of the Hessian matrix in fitting models to measurements. J Geophys Res 94:6177–6196

    Article  Google Scholar 

  • White LW, Vieux B, Armand D, Le Dimet FX (2003) Estimation of optimal parameters for a surface hydrology model. Adv Water Resour 26:337–348

    Article  Google Scholar 

  • Yuepeng W, Yue C, Navon IM et al (2018) Parameter identification techniques applied to an environmental pollution model. J Ind Manage Optim 14:817–831

    Article  Google Scholar 

  • Zakharova NB, Agoshkov VI, Parmuzin EI (2013) The new method of ARGO buoys system observation data interpolation. Russ J Numer Anal Math Modelling 28:67–84

    Article  Google Scholar 

  • Zalesny V, Agoshkov V, Aps R, Shutyaev V et al (2017) Numerical modeling of marine circulation, pollution assessment and optimal ship routes. J Mar Sci Eng 5:1–20

    Article  Google Scholar 

  • Zhu Y, Navon IM (1999) Impact of parameter estimation on the performance of the FSU global spectral model using its full-physics adjoint. Mon Weather Rev 127:1497–1517

    Article  Google Scholar 

Download references

Acknowledgements

The research (Sections 5–7) was supported by the Russian Science Foundation (project No.20-11-20057).

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Authors and Affiliations

  1. LJK, Université Grenoble Alpes, 700 Avenue Centrale, 38401 Domaine Universitaire de Saint-Martin-d’Hères, Grenoble, France

    François-Xavier Le Dimet

  2. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, 119333 Gubkina 8, Moscow, Russia

    Victor Shutyaev

Authors
  1. François-Xavier Le Dimet
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  2. Victor Shutyaev
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Correspondence to Victor Shutyaev .

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Editors and Affiliations

  1. Climate and Energy Systems Engineering, Ewha Womans University, Seoul, Korea (Republic of)

    Seon Ki Park

  2. Marine Meteorology Division, Naval Research Laboratory, Monterey, CA, USA

    Liang Xu

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Le Dimet, FX., Shutyaev, V. (2022). Second-Order Methods in Variational Data Assimilation. In: Park, S.K., Xu, L. (eds) Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (Vol. IV). Springer, Cham. https://doi.org/10.1007/978-3-030-77722-7_7

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