T. Auligné, A. P. Mcnally, and D. P. Dee, Adaptive bias correction for satellite data in a numerical weather prediction system, Quarterly Journal of the Royal Meteorological Society, vol.133, pp.631-642, 2007.

R. N. Bannister, A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances, Quarterly Journal of the Royal Meteorological Society, vol.134, pp.1951-1970, 2008.

R. N. Bannister, A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics, Quarterly Journal of the Royal Meteorological Society, vol.134, pp.1971-1996, 2008.

R. N. Bannister, A review of operational methods of variational and ensemble-variational data assimilation, Quarterly Journal of the Royal Meteorological Society, vol.143, pp.607-633, 2017.

A. Barth, J. M. Beckers, C. Troupin, A. Alvera-azeárate, and L. Vandenbulcke, Divand-1.0: -dimensional variational data analysis for ocean observations, Geoscientific Model Development, vol.7, pp.225-241, 2014.

M. Belo-pereira and L. Berre, The use of an ensemble approach to study the background error covariances in a global NWP model, Monthly Weather Review, vol.134, pp.2466-2489, 2006.

D. Bolin and F. Lindgren, A comparison between Markov approximations and other methods for large spatial data sets, Computational Statistics and Data Analysis, vol.61, pp.7-32, 2013.
DOI : 10.1016/j.csda.2012.11.011
URL : https://purehost.bath.ac.uk/ws/files/86995149/bolin_lindgren_csda_v3.pdf

N. Bormann and P. Bauer, Estimates of spatial and interchannel observation-error characteristics for current sounder radiances for numerical weather prediction. I: Methods and application to ATOVS data, Quarterly Journal of the Royal Meteorological Society, vol.136, pp.1036-1050, 2010.

N. Bormann, A. Collard, and P. Bauer, Estimates of spatial and interchannel observation-error characteristics for current sounder radiances for numerical weather prediction. II: Application to AIRS and IASI data, Quarterly Journal of the Royal Meteorological Society, vol.136, pp.1051-1063, 2010.

J. M. Brankart, C. Ubelmann, C. E. Testut, E. Cosme, P. Brasseur et al., Efficient parameterization of the observation error covariance matrix for square root or Ensemble Kalman Filters: application to ocean altimetry, Monthly Weather Review, vol.137, pp.1908-1927, 2009.

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, 2013.

T. Bui-thanh, O. Ghattas, J. Martin, and G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion, SIAM Journal on Scientific Computing, vol.35, pp.2494-2523, 2013.

W. F. Campbell, E. Satterfield, B. Ruston, and N. Baker, Accounting for correlated observation error in a dual formulation 4D variational data assimilation system, Monthly Weather Review, vol.145, pp.1019-1032, 2017.
DOI : 10.1175/mwr-d-16-0240.1

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Dynamics, 1987.

M. J. Carrier and H. Ngodock, Background-error correlation model based on the implicit solution of a diffusion equation, Ocean Modelling, vol.35, pp.45-53, 2010.

V. Chabot, M. Nodet, N. Papadakis, and A. Vidard, Accounting for observation errors in image data assimilation, Tellus A, vol.67, issue.1, 2015.
DOI : 10.3402/tellusa.v67.23629
URL : https://hal.archives-ouvertes.fr/hal-01119039

P. G. Ciarlet, The finite element method for elliptic problems, 2002.

R. Daley, Atmospheric Data Analysis, 1991.

M. L. Dando, A. J. Thorpe, and J. R. Eyre, The optimal density of atmospheric sounder observations in the Met Office NWP system, Quarterly Journal of the Royal Meteorological Society, vol.133, pp.1933-1943, 2007.

T. S. Davis, Direct Methods for Sparse Linear Systems, 2006.
DOI : 10.1137/1.9780898718881

G. Desroziers, L. Berre, B. Chapnik, and P. Poli, Diagnosis of observation-, background-and analysis-error statistics in observation space, Quarterly Journal of the Royal Meteorological Society, vol.131, pp.3385-3396, 2005.

I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices, 1989.

H. Edelsbrunner, T. Tan, and R. Waupotitsch, An ( 2 log ) time algorithm for the minmax angle triangulation, SIAM Journal on Scientific Computing, vol.13, pp.994-1008, 1992.
DOI : 10.1145/98524.98535

A. Ern and J. Guermond, Theory and practice of finite elements, 2004.

M. Fisher, The sensitivity of analysis errors to the specification of background-error covariances, Proceedings of Workshop on Flow-dependent Aspects of Data Assimilation, pp.27-36, 2007.

G. Gaspari and S. E. Cohn, Construction of correlation functions in two and three dimensions, Quarterly Journal of the Royal Meteorological Society, vol.125, pp.723-757, 1999.

S. Gratton, P. Toint, and J. Tshimanga, A comparison between conjugate gradients and multigrid solvers for covariance modelling in data assimilation, Quarterly Journal of the Royal Meteorological Society, vol.139, pp.1481-1487, 2011.

P. Guttorp and T. Gneiting, Studies in the history of probability and statistics XLIX: on the Matérn correlation family, Biometrika, vol.93, pp.989-995, 2006.

T. Janji?, N. Bormann, M. Bocquet, J. A. Carton, S. E. Cohn et al., On the representation error in data assimilation, Quarterly Journal of the Royal Meteorological Society, vol.144, pp.1257-1278, 2018.

H. Järvinen, E. Andersson, and F. Bouttier, Variational assimilation of time sequences of surface observations with serially correlated errors, Tellus A, vol.51, pp.469-488, 1999.

G. H. Jones, The Theory of Generalised Functions, 1982.
DOI : 10.1017/cbo9780511569210

U. Khristenko, L. Scarabosio, P. Swierczynski, E. Ullmann, and B. Wohlmuth, Analysis of boundary effects on PDE-based sampling of Whittle-Matérn random fields, 2018.

F. Lindgren, H. Rue, and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, Journal of the Royal Statistical Society. Series B, vol.73, pp.423-498, 2011.
DOI : 10.1111/j.1467-9868.2011.00777.x
URL : https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2011.00777.x

Z. Q. Liu and F. Rabier, The interaction between model resolution, observation resolution and observation density in data assimilation: a one-dimensional study, Quarterly Journal of the Royal Meteorological Society, vol.128, pp.1367-1386, 2002.

Z. Q. Liu and F. Rabier, The potential of high-density observations for numerical weather prediction: a study with simulated observations, Quarterly Journal of the Royal Meteorological Society, vol.129, pp.3013-3035, 2003.

A. C. Lorenc, Development of an operational variational assimilation scheme, Journal of the Meteorological Society of Japan, vol.75, pp.339-346, 1997.
DOI : 10.2151/jmsj1965.75.1b_339
URL : https://www.jstage.jst.go.jp/article/jmsj1965/75/1B/75_1B_339/_pdf

Y. Michel, Revisiting Fisher's approach to the handling of horizontal spatial correlations of the observation errors in a variational framework, Quarterly Journal of the Royal Meteorological Society, vol.144, 2011.

I. Mirouze and A. Storto, Handling boundaries with the one-dimensional first-order recursive filter, Quarterly Journal of the Royal Meteorological Society, vol.142, pp.2478-2487, 2016.
DOI : 10.1002/qj.2840

I. Mirouze and A. T. Weaver, Representation of correlation functions in variational assimilation using an implicit diffusion operator, Quarterly Journal of the Royal Meteorological Society, vol.136, pp.1421-1443, 2010.

T. Montmerle, F. Rabier, and C. Fischer, Relative impact of polar-orbiting and geostationary satellite radiances in the ALADIN/France numerical weather prediction system, Quarterly Journal of the Royal Meteorological Society, vol.133, pp.655-671, 2007.

R. J. Purser, W. S. Wu, D. F. Parrish, and N. M. Roberts, Numerical aspects of the application of recursive filters to variational statistical analysis. Part II: spatially inhomogeneous and anisotropic general covariances, Monthly Weather Review, vol.131, pp.1536-1548, 2003.

F. Rabier, Importance of data: a meteorological perspective, Ocean Weather Forecasting: An Integrated View of Oceanography, pp.343-360, 2006.

G. A. Ruggiero, E. Cosme, J. M. Brankart, L. Sommer, and J. , An efficient way to account for observation error correlations in the assimilation of date from the future SWOT high-resolution altimeter mission, Journal of Atmospheric and Oceanic Technology, vol.33, pp.2755-2768, 2016.

Y. Saad, Iterative Methods for Sparse Linear Systems, 2003.
DOI : 10.1137/1.9780898718003

J. Schmetz, P. Pili, S. Tjemkes, D. Just, J. Kerkmann et al., An introduction to Meteosat Second Generation (MSG), Bulletin of the American Meteorological Society, vol.83, pp.977-992, 2002.
DOI : 10.1175/1520-0477(2002)083<0977:aitmsg>2.3.co;2

Y. Seity, P. Brousseau, S. Malardel, G. Hello, P. Bénard et al., The AROME-France convective-scale operational model, Monthly Weather Review, vol.139, pp.976-991, 2011.
DOI : 10.1175/2010mwr3425.1

D. Simpson, F. Lindgren, and H. Rue, In order to make spatial statistics computationally feasible, we need to forget about the covariance function, Environmetrics, vol.23, pp.65-74, 2012.
DOI : 10.1002/env.1137

M. L. Stein, Interpolation of spatial data; Some Theory for Kriging, 1999.

L. M. Stewart, S. L. Dance, and N. K. Nichols, Data assimilation with correlated observation errors: experiments with a 1-D shallow water model, Tellus A, vol.65, issue.1, 2013.
DOI : 10.3402/tellusa.v65i0.19546
URL : https://doi.org/10.3402/tellusa.v65i0.19546

L. M. Stewart, S. L. Dance, and N. K. Nichols, Correlated observation errors in data assimilation, International Journal for Numerical Methods in Fluids, vol.56, pp.1521-1527, 2008.
DOI : 10.1002/fld.1636
URL : http://www.reading.ac.uk/web/FILES/maths/03-07.pdf

L. M. Stewart, S. L. Dance, N. K. Nichols, J. R. Eyre, and J. Cameron, Estimating interchannel observation-error correlations for IASI radiance data in the Met Office system, Quarterly Journal of the Royal Meteorological Society, vol.140, pp.1236-1244, 2014.

R. Stuhlmann, A. Rodriguez, S. Tjemkes, J. Grandell, A. Arriaga et al., Plans for EUMETSAT's Third Generation Meteosat geostationary satellite programme, Advances in Space Research, vol.36, issue.5, pp.975-981, 2005.
DOI : 10.1016/j.asr.2005.03.091

M. Szyndel, G. Kelly, and J. Thépaut, Evaluation of potential benefit of assimilation of SEVIRI water vapour radiance data from Meteosat-8 into global numerical weather prediction analyses, Atmospheric Science Letters, vol.6, pp.105-111, 2005.

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

C. Ubelmann, L. Gaultier, and L. Fu, SWOT Simulator for Ocean Science, 2016.

J. A. Waller, S. Ballard, S. L. Dance, G. Kelly, N. K. Nichols et al., Diagnosing horizontal and inter-channel observation error correlations for SEVIRI observations using observation-minus-background and observation-minus-analysis statistics. Remote sensing, vol.8, p.581, 2016.
DOI : 10.3390/rs8070581
URL : https://www.mdpi.com/2072-4292/8/7/581/pdf

J. A. Waller, S. L. Dance, and N. K. Nichols, Theoretical insight into diagnosing observation-error correlations using observation-minus-background and observation-minus-analysis residuals, Quarterly Journal of the Royal Meteorological Society, vol.142, pp.418-431, 2016.
DOI : 10.1002/qj.2661
URL : https://rmets.onlinelibrary.wiley.com/doi/pdf/10.1002/qj.2661

J. A. Waller, D. Simonin, S. L. Dance, N. K. Nichols, and S. Ballard, Diagnosing observation-error correlations for Doppler radar radial winds in the Met Office UKV model using observation-minus-background and observation-minus-analysis statistics, Monthly Weather Review, vol.144, pp.3533-3551, 2016.

A. T. Weaver and P. Courtier, Correlation modelling on the sphere using a generalized diffusion equation, Quarterly Journal of the Royal Meteorological Society, vol.127, pp.1815-1846, 2001.

A. T. Weaver, S. Gürol, J. Tshimanga, M. Chrust, and A. Piacentini, Time"-parallel diffusion-based correlation operators, Quarterly Journal of the Royal Meteorological Society, vol.144, pp.2067-2088, 2018.
DOI : 10.1002/qj.3302

A. T. Weaver and I. Mirouze, On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation, Quarterly Journal of the Royal Meteorological Society, vol.139, pp.242-260, 2013.

A. T. Weaver, J. Tshimanga, and A. Piacentini, Correlation operators based on an implicitly formulated diffusion equation solved with the Chebyshev iteration, Quarterly Journal of the Royal Meteorological Society, vol.142, pp.455-471, 2016.
DOI : 10.1002/qj.2664

P. P. Weston, W. Bell, and J. R. Eyre, Accounting for correlated error in the assimilation of high-resolution sounder data, Quarterly Journal of the Royal Meteorological Society, vol.140, pp.2420-2429, 2014.

P. Whittle, Stochastic processes in several dimensions, Bulletin of the International Statistical, vol.40, pp.974-994, 1963.

M. Yaremchuk and M. Carrier, On the renormalization of the covariance operators, Monthly Weather Review, vol.140, pp.637-649, 2012.

M. Yaremchuk, J. M. D&apos;addezio, G. Panteleev, and G. Jacobs, On the approximation of the inverse error covariances of high-resolution satellite altimetry data, Quarterly Journal of the Royal Meteorological Society, vol.144, pp.1927-1932, 2018.

M. Yaremchuk and D. Nechaev, Covariance localization with the diffusion-based correlations models, Monthly Weather Review, vol.141, pp.848-860, 2013.
DOI : 10.1175/mwr-d-12-00089.1

M. Yaremchuk and S. Smith, On the correlation functions associated with polynomials of the diffusion operator, Quarterly Journal of the Royal Meteorological Society, vol.137, pp.1927-1932, 2011.