Nonlinear effects in 4D-Var

Nonlinear Processes in Geophysics - Tập 25 Số 3 - Trang 713-729
Massimo Bonavita1, Peter Lean1, Elías Hólm1
1European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading RG2 9AX, UK

Tóm tắt

Abstract. The ability of a data assimilation system to deal effectively with nonlinearities arising from the prognostic model or the relationship between the control variables and the available observations has received a lot of attention in theoretical studies based on very simplified test models. Less work has been done to quantify the importance of nonlinearities in operational, state-of-the-art global data assimilation systems. In this paper we analyse the nonlinear effects present in ECMWF 4D-Var and evaluate the ability of the incremental formulation to solve the nonlinear assimilation problem in a realistic NWP environment. We find that nonlinearities have increased over the years due to a combination of increased model resolution and the ever-growing importance of observations that are nonlinearly related to the state. Incremental 4D-Var is well suited for dealing with these nonlinear effects, but at the cost of increasing the number of outer loop relinearisations. We then discuss strategies for accommodating the increasing number of sequential outer loops in the tight schedules of operational global NWP.

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Tài liệu tham khảo

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.

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Nonlinear Processes in Geophysics

Tập 25 Số 3

713-729

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