NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-21-187-2014 Diagnostics on the cost-function in variational assimilations for meteorological models Michel Y. 1 Météo-France and CNRS, CNRM-GAME, UMR3589, Toulouse, France 05 02 2014 21 1 187 199 Copyright: © 2014 Y. Michel 2014 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://npg.copernicus.org/articles/21/187/2014/npg-21-187-2014.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/21/187/2014/npg-21-187-2014.pdf

Several consistency diagnostics have been proposed to evaluate variational assimilation schemes. The "Bennett-Talagrand" criterion in particular shows that the cost-function at the minimum should be close to half the number of assimilated observations when statistics are correctly specified. It has been further shown that sub-parts of the cost function also had statistical expectations that could be expressed as traces of large matrices, and that this could be exploited for variance tuning and hypothesis testing. <br><br> The aim of this work is to extend those results using standard theory of quadratic forms in random variables. The first step is to express the sub-parts of the cost function as quadratic forms in the innovation vector. Then, it is possible to derive expressions for the statistical expectations, variances and cross-covariances (whether the statistics are correctly specified or not). As a consequence it is proven in particular that, in a perfect system, the values of the background and observation parts of the cost function at the minimum are positively correlated. These results are illustrated in a simplified variational scheme in a one-dimensional context. <br><br> These expressions involve the computation of the trace of large matrices that are generally unavailable in variational formulations of the assimilation problem. It is shown that the randomization algorithm proposed in the literature can be extended to cover these computations, yet at the price of additional minimizations. This is shown to provide estimations of background and observation errors that improve forecasts of the operational ARPEGE model.

 References Aune, E., Simpson, D., and Eidsvik, J.: Parameter estimation in high dimensional Gaussian distributions, Stat. Comput., online first, 1–17, <a href="http://dx.doi.org/10.1007/s11222-012-9368-y">https://doi.org/10.1007/s11222-012-9368-y</a>, 2012. Avron, H. and Toledo, S.: Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix, Journal of the ACM, 58, 8:1–8:34, 2011. Bannister, R. N.: A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics, Q. J. Roy. Meteor. Soc., 134, 1971–1996, 2008. Bekas, C., Kokiopoulou, E., and Saad, Y.: An estimator for the diagonal of a matrix, Appl. Numer. Math., 57, 1214–1229, 2007. Belo-Pereira, M. and Berre, L.: The Use of an Ensemble Approach to Study the Background Error Covariances in a Global NWP Model, Mon. Weather Rev., 134, 2466–2489, 2006. Bennett, A., Leslie, L., Hagelberg, C., and Powers, P.: Tropical Cyclone Prediction Using a Barotropic Model Initialized by a Generalized Inverse Method, Mon. Weather Rev., 121, 1714–1729, 1993. Bennett, A. F., Chua, B. S., Harrison, D. E., and McPhaden, M. J.: Generalized Inversion of Tropical Atmosphere-Ocean (TAO) Data and a Coupled Model of the Tropical Pacific. Part II: The 1995 La Nina and 1997 El Nino, J. Climate, 13, 2770–2785, 2000. Chapnik, B., Desroziers, G., Rabier, F., and Talagrand, O.: Properties and first application of an error-statistics tuning method in variational assimilation, Q. J. Roy. Meteor. Soc., 130, 2253–2275, 2004. Chapnik, B., Desroziers, G., Rabier, F., and Talagrand, O.: Diagnosis and tuning of observational error in a quasi-operational data assimilation setting, Q. J. Roy. Meteor. Soc., 132, 543–565, 2006. 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, 1994. Courtier, P., Andersson, E., Heckley, W., Vasiljevic, D., Hamrud, M., Hollingsworth, A., Rabier, F., Fisher, M., and Pailleux, J.: The ECMWF implementation of three-dimensional variational assimilation (3D-Var). I: Formulation, Q. J. Roy. Meteor. Soc., 124, 1783–1807, 1998. Daley, R.: Estimating observation error statistics for atmospheric data assimilation, Annales Geophysicae, 11, 634–647, 1993. Dee, D. and da Silva, A.: Maximum-Likelihood Estimation of Forecast and Observation Error Covariance Parameters. Part I: Methodology, Mon. Weather Rev., 127, 1822–1834, 1999. Dee, D. P.: On-line Estimation of Error Covariance Parameters for Atmospheric Data Assimilation, Mon. Weather Rev., 123, 1128–1145, 1995. Derber, J. and Bouttier, F.: A reformulation of the background error covariance in the ECMWF global data assimilation system, Tellus A, 51, 195–221, 1999. Desroziers, G. and Ivanov, S.: Diagnosis and adaptive tuning of observation-error parameters in a variational assimilation, Q. J. Roy. Meteor. Soc., 127, 1433–1452, 2001. Desroziers, G., Berre, L., Chabot, V., and Chapnik, B.: A posteriori diagnostics in an ensemble of perturbed analyses, Mon. Weather Rev., 137, 3420–3436, 2009. Evensen, G.: The Ensemble Kalman Filter: theoretical formulation and practical implementation, Ocean Dynam., 53, 343–367, 2003. Fisher, M.: Background error covariance modelling, in: ECMWF Seminar on recent developments in data assimilation for atmosphere and ocean, ECMWF, 45–63, 2003. Gauthier, P. and Thépaut, J.-N.: Impact of the digital filter as a weak constraint in the preoperational 4Dvar assimilation system of Météo-France, Mon. Weather Rev., 129, 2089–2102, 2001. Gauthier, P., Charette, C., Fillion, L., Koclas, P., and Laroche, S.: Implementation of a 3D variational data assimilation system at the Canadian Meteorological Centre. Part I: The global analysis, Atmos. Ocean, 37, 103–156, 1999. Genton, M. G., He, L., and Liu, X.: Moments of skew-normal random vectors and their quadratic forms, Stat. Probabil. Lett., 51, 319–325, 2001. Girard, D.: Asymptotic optimality of the fast randomized versions of GCV and CL in ridge regression and regularization, Ann. Stat., 19, 1950–1963, 1991. Guidard, V. and Fischer, C.: Introducing the coupling information in a limited-area variational assimilation, Q. J. Roy. Meteor. Soc., 134, 723–735, 2008. Hollingsworth, A. and Lonnberg, P.: The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field, Tellus A, 38, 111–136, 1986. Houtekamer, P. and Mitchell, H.: A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation, Mon. Weather Rev., 129, 123–137, 2001. Hutchinson, M. F.: A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines, Commun. Stat.-Simul. C., 18, 1059–1076, 1989. Kucukkaraca, E. and Fisher, M.: Use of analysis ensembles in estimating flow-dependent background error variances, Tech. Rep. 492, ECMWF Technical Memorandum, 2006. Le Dimet, F. X. and Talagrand, O.: Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus A, 38, 97–110, 1986. Lorenc, A. C.: Analysis methods for numerical weather prediction, Q. J. Roy. Meteor. Soc., 112, 1177–1194, 1986. Lorenc, A. C.: Development of an operational variational assimilation scheme, J. Meteorol. Soc. Jpn, 75, 339–346, 1997. Mathai, A. and Provost, S. B.: Quadratic forms in random variables: theory and applications, Marcel Dekker, Inc. New York, statistics, a series of textbooks and monographs, 1992. Mathai, A., Provost, S., and Hayakawa, T.: Bilinear Forms and Zonal Polynomials, Springer-Verlag, New York, Lect. Notes Stat., 102, <a href="http://dx.doi.org/10.1007/978-1-4612-4242-0">https://doi.org/10.1007/978-1-4612-4242-0</a>, 1995. Ménard, R. and Chang, L.-P.: Assimilation of Stratospheric Chemical Tracer Observations Using a Kalman Filter. Part II: $\chi^2$-Validated Results and Analysis of Variance and Correlation Dynamics, Mon. Weather Rev., 128, 2672–2686, 2000. Ménard, R., Cohn, S. E., Chang, L.-P., and Lyster, P. M.: Assimilation of Stratospheric Chemical Tracer Observations Using a Kalman Filter. Part I: Formulation, Mon. Weather Rev., 128, 2654–2671, 2000. Michel, Y.: Estimating deformations of random processes for correlation modelling: methodology and the one-dimensional case, Q. J. Roy. Meteor. Soc., 139, 771–783, 2013. Muccino, J. C., Hubele, N. F., and Bennett, A. F.: Significance testing for variational assimilation, Q. J. Roy. Meteor. Soc., 130, 1815–1838, 2004. Pailleux, J., Geleyn, J.-F., and Legrand, E.: La prévision numérique du temps avec les modèles Arpege et Aladin: Bilan et perspectives = Numerical weather prediction with the models Arpege and Aladin; assessment and prospects, La Météorologie, 32–60, 2000 in French. Parrish, D. and Derber, J.: The National Meteorological Center's Spectral Statistical-Interpolation Analysis System, Mon. Weather Rev., 120, 1747–1763, 1992. Purser, R. J. and Parrish, D. F.: A Bayesian technique for estimating continuously varying statistical parameters of a variational assimilation, Meteorol. Atmos. Phys., 82, 209–226, 2003. Rabier, F.: Overview of global data assimilation developments in numerical weather-prediction centres, Q. J. Roy. Meteor. Soc., 131, 3215–3233, 2005. Rabier, F., Jarvinen, 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, 2000. Rasmussen, C. E. and Williams, C.: Gaussian Processes for Machine Learning, the MIT Press, 2006. Raynaud, L., Berre, L., and Desroziers, G.: Accounting for model error in the Météo-France ensemble data assimilation system, Q. J. Roy. Meteor. Soc., 138, 249–262, 2012. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd Edn., Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2003. Sadiki, W. and Fischer, C.: A posteriori validation applied to the 3D-VAR Arpege and Aladin data assimilation systems, Tellus A, 57, 21–34, 2005. Simmons, A. and Hollingsworth, A.: Some aspects of the improvement in skill of Numerical Weather Prediction, Q. J. Roy. Meteor. Soc., 128, 647–677, 2002. Stein, M. L., Chen, J., and Anitescu, M.: Stochastic Approximation of Score Functions for Gaussian Processes, Ann. Appl. Stat., 7, 1162–1191, 2013. Talagrand, O.: A posteriori verification of analysis and assimilation algorithms, in: Workshop on diagnosis of data assimilation systems, 2–4 November 1998, ECMWF, Reading, UK, 1999. Talagrand, O.: Data Assimilation: Making Sense of Observations, in: Evaluation of Assimilation Algorithms, Springer-Verlag Berlin Heidelberg, 217–240, 2010.