Andersson, E., Fisher, M., Holm, E., Isaksen, L., Radnòti, G., and
Trémolet, Y.: Will the 4D-Var approach be defeated by nonlinearity? ECMWF
Tech. Memo. 479, available at:
https://www.ecmwf.int/sites/default/files/elibrary/2005/7768-will-4d-var-approach-be-defeated-nonlinearity (last access: 1 September 2018),
2005.

Bauer, P., Geer, A. J., Lopez, P., and Salmond, D.: Direct 4D-Var
assimilation of all-sky radiances. Part I: Implementation, Q. J. Roy. Meteor.
Soc., 136, 1868–1885. https://doi.org/10.1002/qj.659, 2010.

Björck, A.: Numerical methods for least squares problems, SIAM,
Philadelphia, ISBN 0-89871-360-9, 1996.

Bonavita, M., Trémolet, Y., Holm, E., Lang, S. T. K., Chrust, M.,
Janisková, M., Lopez, P., Laloyaux, P., De Rosnay, P., Fisher, M.,
Hamrud, M., and English, S.: A Strategy for Data Assimilation, ECMWF
Technical Memorandum n. 800, available at:
https://www.ecmwf.int/en/elibrary/17179-strategy-data-assimilation (last access: 1 September 2018),
2017a.

Bonavita, M., Dahoui, M., Lopez, P., Prates, F., Hólm, E., De Chiara, G.,
Geer, A., Isaksen, L., and Ingleby, B.: On the initialization of Tropical
Cyclones. ECMWF Technical Memorandum n. 810, available at
https://www.ecmwf.int/en/elibrary/17677-initialization-tropical-cyclones (last access: 1 September 2018),
2017b.

Carrassi, A., Ghil, M., Trevisan, A., and Uboldi, F.: Data assimilation as a
nonlinear dynamical system problem: Stability and convergence of the
prediction-assimilation system, Chaos, 18, 023112, https://doi.org/10.1063/1.2909862,
2008.

Courtier, P., Thépaut, J.-N., and Hollingsworth, A.: A strategy for
operational implementation of 4D-Var, using an incremental approach, Q. J.
Roy. Meteor. Soc., 120, 1367–1387, https://doi.org/10.1002/qj.49712051912, 1994.

Fisher, M.: Minimization Algorithms for Variational Data Assimilation.
Proceedings of the ECMWF Seminar on Recent Developments in Numerical Methods
for Atmospheric Modelling, available at:
https://www.ecmwf.int/en/elibrary/9400-minimization-algorithms-variational-data-assimilation (last access: 1 September 2018),
1998.

Fisher, M.: Estimation of entropy reduction and degrees of freedom for
signal for large variational analysis systems, ECMWF Technical Memorandum n.
397, available at:
https://www.ecmwf.int/en/elibrary/9402-estimation-entropy-reduction-and-degrees-freedom-signal-large-variational-analysis (last access: 1 September 2018), 2003.

Gauthier, P.: Chaos and quadri-dimensional data assimilation: a study based
on the Lorenz model, Tellus A, 44, 2–17,
https://doi.org/10.1034/j.1600-0870.1992.00002.x, 1992.

Gauthier, P., Tanguay, M., Laroche, S., Pellerin, S.,
and Morneau, J.: Extension of 3DVAR to 4DVAR: implementation of 4DVAR at the meteorological service of
Canada, Mon. Weather Rev., 135, 2339–2354, https://doi.org/10.1175/MWR3394.1, 2007.

Geer, A. J. and Bauer, P.: Observation errors in all-sky data
assimilation, Q. J. R. Meteor. Soc., 137, 2024–2037, https://doi.org/10.1002/qj.830,
2011.

Geer, A. J., Baordo, F., Bormann, N., Chambon, P., English, S. J., Kazumori,
M., Lawrence, H., Lean, P., Lonitz, K., and Lupu, C.: The growing impact of
satellite observations sensitive to humidity, cloud and precipitation, Q. J.
Roy. Meteor. Soc., 143, 3189–3206, https://doi.org/10.1002/qj.3172, 2017.

Gratton, S., Lawless, A., and Nichols, N. K.: Approximate Gauss–Newton
methods for nonlinear least squares problems, SIAM J. Optimiz., 18, 106–132,
https://doi.org/10.1137/050624935, 2007.

Hólm, E. V., Andersson, E., Beljaars, A. C. M., Lopez, P., Mahfouf, J.-F.,
Simmons, A., and Thépaut, J.-J.: Assimilation and Modelling of the
Hydrological Cycle: ECMWF's Status and Plans. ECMWF Tech. Memo. 383,
available at:
https://www.ecmwf.int/sites/default/files/elibrary/2002/9996-assimilation-and-modelling-hydrological-cycle-ecmwfs-status-and-plans.pdf (last access: 1 September 2018),
2002.

Hoteit, I.: A reduced-order simulated annealing approach for
four-dimensional variational data assimilation in meteorology and
oceanography, Int. J. Numer. Meth. Fl., 58, 1181–1199, https://doi.org/10.1002/fld.1794,
2008.

Isaksen, L., Bonavita, M., Buizza, R., Fisher, M., Haseler, J., Leutbecher,
M., and Raynaud, L.: Ensemble of data assimilations at ECMWF. ECMWF Tech.
Memo. 636, available at:
https://www.ecmwf.int/en/elibrary/10125-ensemble-data-assimilations-ecmwf (last access: 1 September 2018),
2010.

Janisková, M. and Lopez, P.: Linearized physics for data assimilation
at ECMWF, in: Data assimilation for Atmospheric, Oceanic and Hydrological
Applications (Vol. II), edited by: Park, S. K. and Xu, L., Springer-Verlag
Berlin Heidelberg, 251–286, https://doi.org/10.1007/978-3-642-35088-7, 2013.

Jarvinen, H., Thépaut, J. N., and Courtier, P.: Quasi-continuous
variational data assimilation, Q. J. Roy. Meteor. Soc., 122, 515–534, 1996.

Kadowaki, T.: A 4-Dimensional Variational Assimilation System for the JMA
Global Spectrum Model, CAS/JAC WGNE Research Activities in Atmospheric and
Oceanic Modelling, 34, 1–17, 2005.

Laroche, S. and Gauthier, P.: A validation of the incremental formulation
of 4D variational data assimilation in a nonlinear barotropic flow, Tellus A,
50, 557–572, https://doi.org/10.3402/tellusa.v50i5.14558, 1998.

Lawless, A. S., Gratton, S., and Nichols, N. K.: Approximate iterative methods
for variational data assimilation, Int. J. Numer. Meth. Fl., 47, 1129–1135,
https://doi.org/10.1002/fld.851, 2005.

Lorenc, A. C. and Payne, T.: 4D-Var and the butterfly effect: Statistical
four-dimensional data assimilation for a wide range of scales, Q. J. Roy.
Meteor. Soc., 133, 607–614, https://doi.org/10.1002/qj.36, 2007.

Maddox, R. A.: Mesoscale Convective Complexes, B. Am. Meteorol. Soc., 61,
1374–1387, https://doi.org/10.1175/1520-0477(1980)061<1374:MCC>2.0.CO;2, 1980.

Malardel, S., Wedi, N., Deconinck, W., Diamantakis, M., Kühnlein, C.,
Mozdzynsky, G., Hamrud, M., and Smolarkiewicz, P.: A new grid for the IFS.
ECMWF Newsletter No. 146, Winter 2015/16, available at:
https://www.ecmwf.int/sites/default/files/elibrary/2016/17262-new-grid-ifs.pdf (last access: 1 September 2018),
2016.

Miller, R. N., Ghil, M., and Gauthiez, F.: Advanced Data Assimilation in
Strongly Nonlinear Dynamical Systems, J. Atmos. Sci., 51, 1037–1056,
https://doi.org/10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2, 1994.

Pires, C., Vautard, R., and Talagrand, O.: On extending the limits of
variational assimilation in nonlinear chaotic systems, Tellus A, 48, 96–121,
1996.

Rabier, F. and Courtier, P.: Four-Dimensional Assimilation in the Presence
of Baroclinic Instability, Q. J. Roy. Meteor. Soc., 118, 649–672,
https://doi.org/10.1002/qj.49711850604, 1992.

Rabier, F., Järvinen, H., Klinker, E., Mahfouf, J.-F., and Simmons, A.:
The ECMWF operational implementation of four-dimensional variational
assimilation. Part I: Experimental results with simplified physics, Q. J.
Roy. Meteor. Soc., 126, 1143–1170, https://doi.org/10.1002/qj.49712656415, 2000.

Radnòti, G., Trémolet, Y., Andersson, E., Isaksen, L., Hólm, E. V.,
and Janiskova, M.: Diagnostics of linear and incremental approximations in
4D-Var revisited for higher resolution analysis, ECMWF Tech Memo 479,
available at:
https://www.ecmwf.int/en/elibrary/11816-diagnostics-linear-and-incremental-approximations-4d-var-revisited-higher (last access: 1 September 2018),
2005.

Rawlins, F., Ballard, S. P., Bovis, K. J., Clayton, A. M., Li, D.,
Inverarity, G. W., Lorenc, A. C., and Payne, T. J.: The Met Office global
four-dimensional variational data assimilation scheme, Q. J. Roy. Meteor.
Soc., 133, 347–362, https://doi.org/10.1002/qj.32, 2007.

Rodwell, M. J., Magnusson, L., Bauer, P., Bechtold, P., Bonavita, M., Cardinali, C., and Diamantakis, M.:
Characteristics of occasional poor
medium-range weather forecasts for Europe, B. Am. Meteorol. Soc., 94,
1393–1405, https://doi.org/10.1175/BAMS-D-12-00099.1, 2013.

Rosmond, T. and Xu, L.: Development of NAVDAS-AR: nonlinear formulation and
outer loop tests, Tellus A., 58, 45–58,
https://doi.org/10.1111/j.1600-0870.2006.00148.x, 2006.

Tanguay, M., Bartello, P., and Gauthier, P.: Four-dimensional data assimilation
with a wide range of scales, Tellus A, 47, 974–997,
https://doi.org/10.1034/j.1600-0870.1995.00204.x, 1995.

Tavolato, C. and Isaksen, L.: On the use of a Huber norm for observation
quality control in the ECMWF 4D-Var, Q. J. Roy. Meteor. Soc., 141,
1514–1527, https://doi.org/10.1002/qj.2440, 2015.

Trémolet, Y.: Diagnostics of linear and incremental approximations in
4D-Var, Q. J. Roy. Meteor. Soc., 130, 2233–2251, https://doi.org/10.1256/qj.03.33, 2004.

Trémolet, Y.: Incremental 4D-Var convergence study, Tellus A,
59, 706–718, https://doi.org/10.1111/j.1600-0870.2007.00271.x, 2007.

Trevisan, A. and Uboldi, F.: Assimilation of standard and targeted observations
within the unstable subspace of the observation – analysis –forecast cycle
system, J. Atmos. Sci., 61, 103–113, 2004.

Trevisan, A., D'Isidoro, M., and Talagrand, O.: Four-dimensional variational
assimilation in the unstable subspace and the optimal subspace dimension, Q.
J. Roy. Meteor. Soc., 136, 487–496, 2010.

Veerse, F. and Thépaut, J.-N.: Multiple-truncation incremental approach
for four-dimensional data assimilation, Q. J. Roy. Meteor. Soc., 124,
1889–1908, https://doi.org/10.1002/qj.49712455006, 1998.