T. Amemiya, Non-linear regression models, Handbook of Econometrics, pp.333-389, 1983.

H. Auvinen, J. M. Bardsley, H. Haario, and T. Kauranne, The variational Kalman filter and an efficient implementation using limited memory BFGS, International Journal for Numerical Methods in Fluids, vol.63, issue.1-3, pp.314-350, 2010.
DOI : 10.1002/fld.2153

R. Bellman, Introduction to Matrix Analysis, 1960.

P. Courtier, J. N. Thépaut, and A. Hollingsworth, A strategy for operational implementation of 4D-Var, using an incremental approach, Quarterly Journal of the Royal Meteorological Society, vol.45, issue.519, pp.1367-1388, 1994.
DOI : 10.1002/qj.49712051912

S. Dobricic, A Sequential Variational Algorithm for Data Assimilation in Oceanography and Meteorology, Monthly Weather Review, vol.137, issue.1, pp.269-287, 2009.
DOI : 10.1175/2008MWR2500.1

N. R. Draper and H. Smith, Applied Regression Analysis, 1966.
DOI : 10.1002/9781118625590

G. Evensen, Sequential data assimilation with a nonlinear quasigeostrophic model using Monte-Carlo methods to forecast error statistics, J. Geophys. Res, pp.9910143-10162, 1994.

G. Evensen, The Ensemble Kalman Filter: theoretical formulation and practical implementation, Ocean Dynamics, vol.53, issue.4, pp.343-367, 2003.
DOI : 10.1007/s10236-003-0036-9

M. Fisher and P. Courtier, Estimating the covariance matrices of analysis and forecast error in variational data assimilation, 1995.

S. Fu?ik and A. Kufner, Nonlinear Differential Equations. - Amsterdam, 1980.

I. Gejadze, L. Dimet, F. Shutyaev, and V. , On Analysis Error Covariances in Variational Data Assimilation, SIAM Journal on Scientific Computing, vol.30, issue.4, pp.1847-1874, 2008.
DOI : 10.1137/07068744X

URL : https://hal.archives-ouvertes.fr/inria-00391893

I. Gejadze, L. Dimet, F. Shutyaev, and V. , On optimal solution error covariances in variational data assimilation problems, Journal of Computational Physics, vol.229, issue.6, pp.2292159-2178, 2010.
DOI : 10.1016/j.jcp.2009.11.028

URL : https://hal.archives-ouvertes.fr/hal-00787292

I. Gejadze, L. Dimet, F. Shutyaev, and V. , Computation of the optimal solution error covariance in variational data assimilation problems with nonlinear dynamics, Journal of Computational Physics, pp.23079-7943, 2011.

L. Hascoët and V. Pascual, TAPENADE 2.1 user's guide, 2004.

H. O. Hartley and A. Booker, Nonlinear Least Squares Estimation, The Annals of Mathematical Statistics, vol.36, issue.2, pp.638-650, 1965.
DOI : 10.1214/aoms/1177700171

C. Heyde and I. Johnstone, On Asymptotic Posterior Normality for Stochastic Processes, J. Roy. Stat. Soc. B41, pp.184-189, 1979.
DOI : 10.1007/978-1-4419-5823-5_45

R. I. Jennrich, Asymptotic properties of nonlinear least square estimation, Annals of Mathematical Statistics, issue.40, pp.633-643, 1969.

J. Kim, Bayesian Asymptotic Theory in a Time Series Model with a Possible Nonstationary Process, Econometric Theory, vol.47, issue.3-4, pp.764-773, 1994.
DOI : 10.2307/2938280

A. S. Lawless, S. Gratton, and N. K. Nichols, Approximate iterative methods for variational data assimilation, International Journal for Numerical Methods in Fluids, vol.108, issue.10-11, pp.1-6, 2005.
DOI : 10.1002/fld.851

L. Dimet, F. Navon, I. M. Daescu, and D. N. , Second-Order Information in Data Assimilation*, Monthly Weather Review, vol.130, issue.3, pp.629-648, 2002.
DOI : 10.1175/1520-0493(2002)130<0629:SOIIDA>2.0.CO;2

URL : https://hal.archives-ouvertes.fr/hal-00787272

L. Dimet, F. Shutyaev, and V. , On deterministic error analysis in variational data assimilation. Nonlinear Processes in Geophysics, pp.481-490, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00331085

L. Dimet, F. X. Talagrand, and O. , Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus A, vol.109, issue.2, pp.97-110, 1986.
DOI : 10.1111/j.1600-0870.1986.tb00459.x

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Copyright c, Royal Meteorological Society Q. J. R. Meteorol. Soc, vol.00, pp.2-24, 1988.
DOI : 10.1137/1.9780898719628

J. L. Lions, Contrôle optimal des systèmes gouvernés par des ´ equations aux dérivées partielles, 1968.

D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale minimization, pp.503-528, 1989.

A. C. Lorenc and . Roy, Analysis methods for numerical weather prediction, Quarterly Journal of the Royal Meteorological Society, vol.108, issue.474, pp.1177-1194, 1986.
DOI : 10.1002/qj.49711247414

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems, 1996.

M. Moakher, A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices, SIAM Journal on Matrix Analysis and Applications, vol.26, issue.3, 2005.
DOI : 10.1137/S0895479803436937

W. K. Newey and D. Mcfadden, Large sample estimation and hypothesis testing, pp.2111-2245, 1994.

S. V. Patankar, Numerical Heat Transfer and Fluid Flow, 1980.

V. V. Penenko and N. N. Obraztsov, A variational initialization method for the fields of the meteorological elements, Soviet Meteorology and Hydrology (English translation), pp.1-11, 1976.

B. S. Powell and A. M. Moore, Estimating the 4DVAR analysis error of GODAE products. Ocean Dynamics, pp.121-138, 2009.

F. Rabier and P. Courtier, Four-Dimensional Assimilation In the Presence of Baroclinic Instability, Quarterly Journal of the Royal Meteorological Society, vol.119, issue.506, pp.649-672, 1992.
DOI : 10.1002/qj.49711850604

D. A. Ratkowsky, Nonlinear Regression Modelling: a Unified Practical Approach, 1983.

Y. Sasaki, A fundamental study of the numerical prediction based on the variational principles, J. Meteor. Soc. Japan, issue.33, pp.262-275, 1955.

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, vol.19, pp.451-559, 2010.
DOI : 10.1017/S0962492910000061

A. Tarantola, Inverse Problems Theory: Methods for Data Fitting and Model Parameter Estimation, 1987.

A. Tarantola, Inverse Problems Theory and Methods for Model Parameter Estimation, 2005.
DOI : 10.1137/1.9780898717921

W. C. Thacker, The role of the Hessian matrix in fitting models to measurements, Journal of Geophysical Research, vol.93, issue.C5, pp.6177-6196, 1989.
DOI : 10.1029/JC094iC05p06177

J. N. Thepaut and P. Courtier, Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model, Quarterly Journal of the Royal Meteorological Society, vol.110, issue.502, pp.1225-1254, 1991.
DOI : 10.1002/qj.49711750206

A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method. English translation of, Dokl. Akad. Nauk SSSR, issue.151, pp.501-504, 1963.

J. Tshimanga, S. Gratton, A. T. Weaver, and A. Sartenaer, Limitedmemory preconditioners, with application to incremental fourdimensional variational assimilation. Q, J. R. Meteorol. Soc, issue.134, pp.751-769, 2008.

C. K. Wikle and M. L. Berliner, A Bayesian tutorial for data assimilation, Physica D: Nonlinear Phenomena, vol.230, issue.1-2, pp.1-16, 2007.
DOI : 10.1016/j.physd.2006.09.017

H. White and I. Domowitz, Nonlinear Regression with Dependent Observations, Econometrica, vol.52, issue.1, pp.143-162, 1984.
DOI : 10.2307/1911465

Y. Yang, I. M. Navon, and P. Courtier, A New Hessian Preconditioning Method Applied to Variational Data Assimilation Experiments Using NASA General Circulation Models, Monthly Weather Review, vol.124, issue.5, pp.1000-1017, 1996.
DOI : 10.1175/1520-0493(1996)124<1000:ANHPMA>2.0.CO;2

K. Yuan and R. I. Jennrich, Asymptotics of Estimating Equations under Natural Conditions, Journal of Multivariate Analysis, vol.65, issue.2, pp.65245-260, 1998.
DOI : 10.1006/jmva.1997.1731

M. Zupanski, I. M. Navon, and D. Zupanski, The Maximum Likelihood Ensemble Filter as a non???differentiable minimization algorithm, Quarterly Journal of the Royal Meteorological Society, vol.133, issue.C5, 2008.
DOI : 10.1002/qj.251