Buehner, M., Houtekamer, P., Charette, C., Mitchell, H. L., and He, B.: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part I: Description and single-observation experiments, Mon. Weather Rev., 138, 1550–1566,
https://doi.org/10.1175/2009MWR3157.1, 2010.
a
Buehner, M., McTaggart-Cowan, R., Beaulne, A., Charette, C., Garand, L., Heilliette, S., Lapalme, E., Laroche, S., Macpherson, S. R., Morneau, J., and Zadra, A.: Implementation of deterministic weather forecasting systems based on ensemble–variational data assimilation at Environment Canada. Part I: The global system, Mon. Weather Rev., 143, 2532–2559,
https://doi.org/10.1175/MWR-D-14-00354.1, 2015.
a
Buehner, M., Caron, J.-F., Lapalme, E., Caya, A., Du, P., Rochon, Y., Skachko, S., Bani Shahabadi, M., Heilliette, S., Deshaies-Jacques, M., Chang, W., and Sitwell, M.: The Modular and Integrated Data Assimilation System at Environment and Climate Change Canada (MIDAS v3.9.1), Geosci. Model Dev., 18, 1–18,
https://doi.org/10.5194/gmd-18-1-2025, 2025.
a
Cao, H., Henze, D. K., Shephard, M., Dammers, E., Cady-Pereira, K., Alvarado, M., Lonsdale, C., Luo, G., Yu, F., Zhu, L., Danielson, C., and Edgerton, E.: Inverse modeling of NH3 sources using CrIS remote sensing measurements, Environ. Res. Lett., 15, 104082,
https://doi.org/10.1088/1748-9326/abb5cc, 2020.
a
Chen, K., Bai, L., Ling, F., Ye, P., Chen, T., Luo, J.-J., Chen, H., Xiao, Y., Chen, K., Han, T., and Ouyang, W.: Towards an end-to-end artificial intelligence driven global weather forecasting system, arXiv [preprint],
https://doi.org/10.48550/arXiv.2312.12462, 2023a.
a
Chen, K., Han, T., Ling, F., Gong, J., Bai, L., Wang, X., Luo, J.-J., Fei, B., Zhang, W., Chen, X., Ma, L., Zhang, T., Su, R., Ci, Y., Li, B., Yang, X., and Ouyang, W.: The operational medium-range deterministic weather forecasting can be extended beyond a 10-day lead time, Commun. Earth Environ., 6, 518,
https://doi.org/10.1038/s43247-025-02502-y, 2025.
a
Chen, L., Zhong, X., Zhang, F., Cheng, Y., Xu, Y., Qi, Y., and Li, H.: FuXi: A cascade machine learning forecasting system for 15-day global weather forecast, npj Clim. Atmos. Sci., 6, 190,
https://doi.org/10.1038/s41612-023-00512-1, 2023b.
a
Côté, J., Desmarais, J.-G., Gravel, S., Méthot, A., Patoine, A., Roch, M., and Staniforth, A.: The operational CMC–MRB global environmental multiscale (GEM) model. Part II: Results, Mon. Weather. Rev., 126, 1397–1418,
https://doi.org/10.1175/1520-0493(1998)126<1397:TOCMGE>2.0.CO;2, 1998a.
a
Côté, J., Gravel, S., Méthot, A., Patoine, A., Roch, M., and Staniforth, A.: The operational CMC–MRB global environmental multiscale (GEM) model. Part I: Design considerations and formulation, Mon. Weather. Rev., 126, 1373–1395,
https://doi.org/10.1175/1520-0493(1998)126<1373:TOCMGE>2.0.CO;2, 1998b.
a
Daley, R.: Atmospheric data analysis, 2, Cambridge university press, 1993. a
Desroziers, G., Camino, J.-T., and Berre, L.: 4DEnVar: link with 4D state formulation of variational assimilation and different possible implementations, Q. J. Roy. Meteor. Soc., 140, 2097–2110,
https://doi.org/10.1002/qj.2325, 2014.
a
Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res.-Oceans, 99, 10143–10162,
https://doi.org/10.1029/94JC00572, 1994.
a
Fairbairn, D., Pring, S. R., Lorenc, A. C., and Roulstone, I.: A comparison of 4DVar with ensemble data assimilation methods, Q. J. Roy. Meteor. Soc., 140, 281–294,
https://doi.org/10.1002/qj.2135, 2014.
a
Garrigues, S., Remy, S., Chimot, J., Ades, M., Inness, A., Flemming, J., Kipling, Z., Laszlo, I., Benedetti, A., Ribas, R., Jafariserajehlou, S., Fougnie, B., Kondragunta, S., Engelen, R., Peuch, V.-H., Parrington, M., Bousserez, N., Vazquez Navarro, M., and Agusti-Panareda, A.: Monitoring multiple satellite aerosol optical depth (AOD) products within the Copernicus Atmosphere Monitoring Service (CAMS) data assimilation system, Atmos. Chem. Phys., 22, 14657–14692,
https://doi.org/10.5194/acp-22-14657-2022, 2022.
a
Girard, C., Plante, A., Desgagné, M., McTaggart-Cowan, R., Côté, J., Charron, M., Gravel, S., Lee, V., Patoine, A., Qaddouri, A., Roch, M., Spacek, L., Tanguay, M., Vaillancourt, P., and Zadra, A.: Staggered vertical discretization of the Canadian Environmental Multiscale (GEM) model using a coordinate of the log-hydrostatic-pressure type, Mon. Weather. Rev., 142, 1183–1196,
https://doi.org/10.1175/MWR-D-13-00255.1, 2014.
a
Gong, W., Makar, P., Zhang, J., Milbrandt, J., Gravel, S., Hayden, K., Macdonald, A., and Leaitch, W.: Modelling aerosol–cloud–meteorology interaction: A case study with a fully coupled air quality model (GEM-MACH), Atmos. Environ., 115, 695–715,
https://doi.org/10.1016/j.atmosenv.2015.05.062, 2015.
a
He, K., Zhang, X., Ren, S., and Sun, J.: Deep residual learning for image recognition, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 770–778,
https://doi.org/10.1109/CVPR.2016.90, 2016.
a
Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Horányi, A., Muñoz-Sabater, J., Nicolas, J., Peubey, C., Radu, R., Schepers, D., Simmons, A., Soci, C., Abdalla, S., Abellan, X., Balsamo, G., Bechtold, P., Biavati, G., Bidlot, J., Bonavita, M., De Chiara, G., Dahlgren, P., Dee, D., Diamantakis, M., Dragani, R., Flemming, J., Forbes, R., Fuentes, M., Geer, A., Haimberger, L., Healy, S., Hogan, R. J., Hólm, E., Janisková, M., Keeley, S., Laloyaux, P., Lopez, Philippe abd Lupu, C., Radnoti, G., de Rosnay, P., Rozum, I., Vamborg, F., Villaume, S., and Thépaut, J.-N.: The ERA5 global reanalysis, Q. J. Roy. Meteor. Soc., 146, 1999–2049,
https://doi.org/10.1002/qj.3803, 2020.
a
Ho, J., Jain, A., and Abbeel, P.: Denoising diffusion probabilistic models, Adv. Neur. In., 33, 6840–6851,
https://proceedings.neurips.cc/paper_files/paper/2020/file/4c5bcfec8584af0d967f1ab10179ca4b-Paper.pdf (last access: 26 October 2026), 2020.
a,
b
Huijnen, V., Miyazaki, K., Flemming, J., Inness, A., Sekiya, T., and Schultz, M. G.: An intercomparison of tropospheric ozone reanalysis products from CAMS, CAMS interim, TCR-1, and TCR-2, Geosci. Model Dev., 13, 1513–1544,
https://doi.org/10.5194/gmd-13-1513-2020, 2020.
a
Inness, A., Baier, F., Benedetti, A., Bouarar, I., Chabrillat, S., Clark, H., Clerbaux, C., Coheur, P., Engelen, R. J., Errera, Q., Flemming, J., George, M., Granier, C., Hadji-Lazaro, J., Huijnen, V., Hurtmans, D., Jones, L., Kaiser, J. W., Kapsomenakis, J., Lefever, K., Leitão, J., Razinger, M., Richter, A., Schultz, M. G., Simmons, A. J., Suttie, M., Stein, O., Thépaut, J.-N., Thouret, V., Vrekoussis, M., Zerefos, C., and the MACC team: The MACC reanalysis: an 8 yr data set of atmospheric composition, Atmos. Chem. Phys., 13, 4073–4109,
https://doi.org/10.5194/acp-13-4073-2013, 2013.
a
Inness, A., Ades, M., Agustí-Panareda, A., Barré, J., Benedictow, A., Blechschmidt, A.-M., Dominguez, J. J., Engelen, R., Eskes, H., Flemming, J., Huijnen, V., Jones, L., Kipling, Z., Massart, S., Parrington, M., Peuch, V.-H., Razinger, M., Remy, S., Schulz, M., and Suttie, M.: The CAMS reanalysis of atmospheric composition, Atmos. Chem. Phys., 19, 3515–3556,
https://doi.org/10.5194/acp-19-3515-2019, 2019.
a
Lahoz, W. A., Errera, Q., Swinbank, R., and Fonteyn, D.: Data assimilation of stratospheric constituents: a review, Atmos. Chem. Phys., 7, 5745–5773,
https://doi.org/10.5194/acp-7-5745-2007, 2007.
a
Lam, R., Sanchez-Gonzalez, A., Willson, M., Wirnsberger, P., Fortunato, M., Alet, F., Ravuri, S., Ewalds, T., Eaton-Rosen, Z., Hu, W., Merose, A., Hoyer, S., Holland, G., Vinyals, O., Stott, J., Pritzel, A., Mohamed, S., and Battaglia, P.: Learning skillful medium-range global weather forecasting, Science, 382, 1416–1421,
https://doi.org/10.1126/science.adi2336, 2023.
a
Li, L., Carver, R., Lopez-Gomez, I., Sha, F., and Anderson, J.: Generative emulation of weather forecast ensembles with diffusion models, Sci. Adv., 10, eadk4489,
https://doi.org/10.1126/sciadv.adk4489, 2024a.
a,
b
Li, Y., Han, W., Li, H., Duan, W., Chen, L., Zhong, X., Wang, J., Liu, Y., and Sun, X.: Fuxi-en4dvar: An assimilation system based on machine learning weather forecasting model ensuring physical constraints, Geophys. Res. Lett., 51, e2024GL111136,
https://doi.org/10.1029/2024GL111136, 2024b.
a
Lorenc, A. C.: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-Var, Q. J. Roy. Meteor. Soc., 129, 3183–3203,
https://doi.org/10.1256/qj.02.132, 2003.
a
Ménard, R. and Deshaies-Jacques, M.: Evaluation of analysis by cross-validation, Part II: Diagnostic and optimization of analysis error covariance, Atmosphere, 9, 70,
https://doi.org/10.3390/atmos9020070, 2018b.
a
Ménard, R. and Deshaies-Jacques, M.: Error covariance estimation methods based on analysis residuals and its application to air quality surface observation networks, in: Air Pollution Modeling and its Application XXV 35, 249–253, Springer,
https://doi.org/10.1007/978-3-319-57645-9_40, 2018c.
a,
b
Ménard, R., Cossette, J.-F., and Deshaies-Jacques, M.: On the complementary role of data assimilation and machine learning: An example derived from air quality analysis, in: Computational Science–ICCS 2020: 20th International Conference, Amsterdam, the Netherlands, 3–5 June 2020, Proceedings, Part VI 20, 212–224, Springer,
https://doi.org/10.1007/978-3-030-50433-5_17, 2020.
a
Miyazaki, K., Maki, T., Patra, P., and Nakazawa, T.: Assessing the impact of satellite, aircraft, and surface observations on CO2 flux estimation using an ensemble-based 4-D data assimilation system, J. Geophys. Res.-Atmos., 116,
https://doi.org/10.1029/2010JD015366, 2011.
a
Miyazaki, K., Eskes, H. J., and Sudo, K.: Global NOx emission estimates derived from an assimilation of OMI tropospheric NO2 columns, Atmos. Chem. Phys., 12, 2263–2288,
https://doi.org/10.5194/acp-12-2263-2012, 2012a.
a
Miyazaki, K., Eskes, H. J., Sudo, K., Takigawa, M., van Weele, M., and Boersma, K. F.: Simultaneous assimilation of satellite NO
2, O
3, CO, and HNO
3 data for the analysis of tropospheric chemical composition and emissions, Atmos. Chem. Phys., 12, 9545–9579,
https://doi.org/10.5194/acp-12-9545-2012, 2012.
a,
b
Moran, M., Ménard, S., Talbot, D., Huang, P., Makar, P., Gong, W., Landry, H., Gravel, S., Gong, S., Crevier, L.-P., Kallaur, A., and Sassi, M.: Particulate-matter forecasting with GEM-MACH15, a new Canadian air-quality forecast model, Air Pollution Modelling and Its Application XX, Springer, 289–292,
https://doi.org/10.1007/978-90-481-3812-8, 2010.
a
Park, S.-Y., Dash, U. K., Yu, J., Yumimoto, K., Uno, I., and Song, C. H.: Implementation of an ensemble Kalman filter in the Community Multiscale Air Quality model (CMAQ model v5.1) for data assimilation of ground-level PM2.5, Geosci. Model Dev., 15, 2773–2790,
https://doi.org/10.5194/gmd-15-2773-2022, 2022.
a
Pathak, J., Subramanian, S., Harrington, P., Raja, S., Chattopadhyay, A., Mardani, M., Kurth, T., Hall, D., Li, Z., Azizzadenesheli, K., Hassanzadeh, P., Kashinath, K., and Anandkumar, A.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators, arXiv [preprint],
https://doi.org/10.48550/arXiv.2202.11214, 2022.
a
Pavlovic, R., Chen, J., Anderson, K., Moran, M. D., Beaulieu, P.-A., Davignon, D., and Cousineau, S.: The FireWork air quality forecast system with near-real-time biomass burning emissions: Recent developments and evaluation of performance for the 2015 North American wildfire season, J. Air. Waste. Manage., 66, 819–841,
https://doi.org/10.1080/10962247.2016.1158214, 2016.
a
Pendergrass, D. C., Jacob, D. J., Nesser, H., Varon, D. J., Sulprizio, M., Miyazaki, K., and Bowman, K. W.: CHEEREIO 1.0: a versatile and user-friendly ensemble-based chemical data assimilation and emissions inversion platform for the GEOS-Chem chemical transport model, Geosci. Model Dev., 16, 4793–4810,
https://doi.org/10.5194/gmd-16-4793-2023, 2023.
a
Peters, W., Krol, M., Van Der Werf, G., Houweling, S., Jones, C., Hughes, J., Schaefer, K., Masarie, K., Jacobson, A., Miller, J., Cho, C., Ramonet, M., Schmidt, M., Ciattaglia, L., Apadula, F., Heltai, D., Meinhardt, F., Di Sarra, A., Piacentino, S., Sferlazzo, D., Aalto, T., Hatakka, J., Ström, J., Haszpra, L., Meijer, H., Van Der Laan, S., Neubert, R., Jordan, A., Rodó, X., Morguí, J., Vermeulen, A., Popa, E., Rozanski, K., Zimnoch, M., Manning, A., Leuenberger, M., Uglietti, C., Dolman, A., Ciais, P., Heimann, M., and Tans, P.: Seven years of recent European net terrestrial carbon dioxide exchange constrained by atmospheric observations, Glob. Change Biol., 16, 1317–1337,
https://doi.org/10.1111/j.1365-2486.2009.02078.x, 2010.
a
Price, I., Sanchez-Gonzalez, A., Alet, F., Andersson, T. R., El-Kadi, A., Masters, D., Ewalds, T., Stott, J., Mohamed, S., Battaglia, P., Lam, R., and Willson, M.: GenCast: Diffusion-based ensemble forecasting for medium-range weather, arXiv [preprint],
https://doi.org/10.48550/arXiv.2312.15796, 2023.
a,
b
Rabier, F., Järvinen, H., Klinker, E., Mahfouf, J.-F., and Simmons, A.: The ECMWF operational implementation of four-dimensional variational assimilation. I: Experimental results with simplified physics, Q. J. Roy. Meteor. Soc., 126, 1143–1170,
https://doi.org/10.1002/qj.49712656415, 2000.
a
Robichaud, A.: Surface data assimilation of chemical compounds over North America and its impact on air quality and Air Quality Health Index (AQHI) forecasts, Air Qual. Atmos. Hlth., 10, 955–970,
https://doi.org/10.1007/s11869-017-0485-9, 2017.
a
Rochon, Y. J., Sitwell, M., and Cho, Y.-M.: A study on harmonizing total ozone assimilation with multiple sensors, Atmos. Chem. Phys., 19, 9431–9451,
https://doi.org/10.5194/acp-19-9431-2019, 2019.
a,
b
Ronneberger, O., Fischer, P., and Brox, T.: U-net: Convolutional networks for biomedical image segmentation, in: Medical image computing and computer-assisted intervention–MICCAI 2015: 18th international conference, Munich, Germany, 5–9 October 2015, Proceedings, part III 18, 234–241, Springer,
https://doi.org/10.1007/978-3-319-24574-4_28, 2015.
a,
b
Shephard, M. W. and Cady-Pereira, K. E.: Cross-track Infrared Sounder (CrIS) satellite observations of tropospheric ammonia, Atmos. Meas. Tech., 8, 1323–1336,
https://doi.org/10.5194/amt-8-1323-2015, 2015.
a,
b
Shephard, M. W., Cady-Pereira, K. E., Luo, M., Henze, D. K., Pinder, R. W., Walker, J. T., Rinsland, C. P., Bash, J. O., Zhu, L., Payne, V. H., and Clarisse, L.: TES ammonia retrieval strategy and global observations of the spatial and seasonal variability of ammonia, Atmos. Chem. Phys., 11, 10743–10763,
https://doi.org/10.5194/acp-11-10743-2011, 2011.
a
Shephard, M. W., Dammers, E., Cady-Pereira, K. E., Kharol, S. K., Thompson, J., Gainariu-Matz, Y., Zhang, J., McLinden, C. A., Kovachik, A., Moran, M., Bittman, S., Sioris, C. E., Griffin, D., Alvarado, M. J., Lonsdale, C., Savic-Jovcic, V., and Zheng, Q.: Ammonia measurements from space with the Cross-track Infrared Sounder: characteristics and applications, Atmos. Chem. Phys., 20, 2277–2302,
https://doi.org/10.5194/acp-20-2277-2020, 2020.
a
Sitwell, M., Shephard, M. W., Rochon, Y., Cady-Pereira, K., and Dammers, E.: An ensemble-variational inversion system for the estimation of ammonia emissions using CrIS satellite ammonia retrievals, Atmos. Chem. Phys., 22, 6595–6624,
https://doi.org/10.5194/acp-22-6595-2022, 2022.
a,
b,
c,
d
Sitwell, M., Shephard, M. W., and Kharol, S. K.: Ammonia bidirectional flux model tailored for satellite retrieval parameter inversions, Geosci. Model Dev., 19, 4797–4815,
https://doi.org/10.5194/gmd-19-4797-2026, 2026.
a,
b,
c,
d,
e
Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B.: Score-based generative modeling through stochastic differential equations, arXiv [preprint],
https://doi.org/10.48550/arXiv.2011.13456, 2020.
a
Thepaut, J.-N. and Courtier, P.: Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model, Q. J. Roy. Meteor. Soc., 117, 1225–1254,
https://doi.org/10.1002/qj.49711750206, 1991.
a
Van Damme, M., Clarisse, L., Franco, B., Sutton, M. A., Erisman, J. W., Kruit, R. W., van Zanten, M., Whitburn, S., Hadji-Lazaro, J., Hurtmans, D., Clerbaux, C., and Coheur, P.-F.: Global, regional and national trends of atmospheric ammonia derived from a decadal (2008–2018) satellite record, Environ. Res. Lett., 16, 055017,
https://doi.org/10.1088/1748-9326/abd5e0, 2021.
a
Vaughan, A., Markou, S., Tebbutt, W., Requeima, J., Bruinsma, W. P., Andersson, T. R., Herzog, M., Lane, N. D., Chantry, M., Hosking, J. S., and Turner, R. E.: Aardvark weather: end-to-end data-driven weather forecasting, arXiv [preprint],
https://doi.org/10.48550/arXiv.2404.00411, 2024.
a
White, E., Shephard, M. W., Cady-Pereira, K. E., Kharol, S. K., Ford, S., Dammers, E., Chow, E., Thiessen, N., Tobin, D., Quinn, G., O'Brien, J., and Bash, J.: Accounting for non-detects: Application to satellite ammonia observations, Remote Sens., 15, 2610,
https://doi.org/10.3390/rs15102610, 2023.
a
Xiao, Y., Bai, L., Xue, W., Chen, K., Han, T., and Ouyang, W.: Fengwu-4dvar: Coupling the data-driven weather forecasting model with 4d variational assimilation, arXiv [preprint],
https://doi.org/10.48550/arXiv.2312.12455, 2023.
a
Zhu, L., Henze, D., Cady-Pereira, K., Shephard, M., Luo, M., Pinder, R., Bash, J., and Jeong, G.-R.: Constraining US ammonia emissions using TES remote sensing observations and the GEOS-Chem adjoint model, J. Geophys. Res.-Atmos., 118, 3355–3368,
https://doi.org/10.1002/jgrd.50166, 2013.
a
