Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 15 1 2008 10.5194/npg-15-127-2008 http://www.nonlin-processes-geophys.net/15/127/2008/ http://www.nonlin-processes-geophys.net/15/127/2008/npg-15-127-2008.html http://www.nonlin-processes-geophys.net/15/127/2008/npg-15-127-2008.pdf 127 143 2008-02-18 Inverse modelling of atmospheric tracers: non-Gaussian methods and second-order sensitivity analysis M. Bocquet Université Paris-Est, CEREA, Research and Teaching Centre in Atmospheric Environment, Joint laboratory École Nationale des Ponts et Chaussées / EDF R&D, avenue Blaise Pascal, 77455 Champs sur Marne, France INRIA, Paris-Rocquencourt research centre, France For a start, recent techniques devoted to the reconstruction of sources of an atmospheric tracer at continental scale are introduced. A first method is based on the principle of maximum entropy on the mean and is briefly reviewed here. A second approach, which has not been applied in this field yet, is based on an exact Bayesian approach, through a maximum a posteriori estimator. The methods share common grounds, and both perform equally well in practice. When specific prior hypotheses on the sources are taken into account such as positivity, or boundedness, both methods lead to purposefully devised cost-functions. These cost-functions are not necessarily quadratic because the underlying assumptions are not Gaussian. As a consequence, several mathematical tools developed in data assimilation on the basis of quadratic cost-functions in order to establish a posteriori analysis, need to be extended to this non-Gaussian framework. Concomitantly, the second-order sensitivity analysis needs to be adapted, as well as the computations of the averaging kernels of the source and the errors obtained in the reconstruction. All of these developments are applied to a real case of tracer dispersion: the European Tracer Experiment [ETEX]. Comparisons are made between a least squares cost function (similar to the so-called 4D-Var) approach and a cost-function which is not based on Gaussian hypotheses. Besides, the information content of the observations which is used in the reconstruction is computed and studied on the application case. A connection with the degrees of freedom for signal is also established. As a by-product of these methodological developments, conclusions are drawn on the information content of the ETEX dataset as seen from the inverse modelling point of view. Bocquet, M.: Grid resolution dependence in the reconstruction of an atmospheric tracer source, Nonlin. Processes Geophys., 12, 219–234, 2005a. Bocquet, M.: Reconstruction of an atmospheric tracer source using the principle of maximum entropy, I: Theory, Q. J. Roy. Meteor. Soc., 131, 2191–2208, 2005b. Bocquet, M.: Reconstruction of an atmospheric tracer source using the principle of maximum entropy, II: Applications, Q. J. Roy. Meteor. Soc., 131, 2209–2223, 2005c. Bocquet, M.: High resolution reconstruction of a tracer dispersion event, Q. J. Roy. Meteor. Soc., 133, 1013–1026, 2007. Borwein, J. M. and Lewis, A. S.: Convex analysis and nonlinear optimization: theory and examples, Springer, 2000. 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., 543–565, 2006. Courtier, P.: Dual formulation of four-dimensional variational assimilation, Q. J. Roy. Meteor. Soc., 2449–2461, 1997. Cover, T M. and Thomas, J A.: Elements of information theory, Wiley series in telecommunications, Wiley-Interscience, 1991. Davoine, X. and Bocquet, M.: Inverse modelling-based reconstruction of the Chernobyl source term available for long-range transport, Atmos. Chem. Phys., 7, 1549–1564, 2007. Ellis, R S.: Entropy, Large Deviations, and Statistical Mechanics, Springer, 1985. Fisher, M.: Estimation of Entropy Reduction and Degrees of Freedom for Signal for Large Variational Analysis Systems, Tech. Rep. 397, ECMWF, 2003. Hourdin, F. and Talagrand, O.: Eulerian backtracking of atmospheric tracers. I: Adjoint derivation and parametrization of subgid-scale transport., Q. J. Roy. Meteor. Soc., 132, 567–583, 2006. Issartel, J.-P. and Baverel, J.: Inverse transport for the verification of the Comprehensive Nuclear Test Ban Treaty, Atmos. Chem. Phys., 3, 475–486, 2003. Krysta, M. and Bocquet, M.: Source reconstruction of an accidental radionuclide release at European scale, Q. J. Roy. Meteor. Soc., 133, 529–544, 2007. Krysta, M., Bocquet, M., and Brandt, J.: Probing ETEX-II data set with inverse modelling, Atmos. Chem. Phys. Discuss., 2795–2819, 2008. Lagarde, T.: Nouvelle approche des méthodes d'assimilation de données: les algorithmes de point selle, Ph.D. thesis, Université Paul Sabatier, Toulouse III, France, 2000. Le~Besnerais, G., Bercher, J.-F., and Demoment, G.: A new look at entropy for solving linear inverse problems, Information Theory, IEEE T. Inform. Theory, 45, 1565–1578, 1999. Le~Dimet, F., Ngodock, H.-E., Luong, B., and Verron, J.: Sensitivity analysis in variational data assimilation, J. Meteorol. Soc. Jpn., 75, 245–255, 1997. Mohammad-Djafari, A.: A comparison of two approaches: Maximum enropy on the mean (MEM) and Bayesian (BAYES) for inverse problems, Kluwer Academic, berg-en-Dal, South Africa, 1996. Ngodock, H.-E.: Assimilation de données et Analyse de sensibilité: Une application à la circulation océanique, Ph.D. thesis, Université Joseph-Fourier – Grenoble I, France, 1996. Nocedal, J. and Wright, S J.: Numerical Optimization, Springer Series in Operations Research, Springer, 2006. Nodop, K., Connolly, R., and Girardi, F.: The field campaigns of The European tracer experiment (ETEX): overview and results, Atmos. Environ., 32, 4095–4108, 1998. Pudykiewicz, J A.: Application of adjoint transport tracer equations for evaluating source parameters, Atmos. Environ., 32, 3039–3050, 1998. Quélo, D., Krysta, M., Bocquet, M., Isnard, O., Minier, Y., and Sportisse, B.: Validation of the Polyphemus platform on the ETEX, Chernobyl and Algeciras cases, Atmos. Environ., 41, 5300–5315, 2007. Rodgers, C D.: Information content and optimization of high spectral resolution measurements, Proc. SPIE, 2830, 136–147, 1996. Rodgers, C D.: Inverse Methods for atmospheric sounding, World Scientific, Series on Atmopsheric, Oceanic and Planetary Physics, 2000. Talagrand, O.: Assimilation of Observations, an Introduction, J. Meteorol. Soc. Jpn., 75, 191–209, 1997. Tarantola, A. and Valette, B.: Generalized nonlinear inverse problems solved using the least square criterion, Rev. Geophys. Space Ge., 20, 219–232, 1982. Uliasz, M.: Application of the Perturbation Theory to the Sensitivity Analysis of an Air Pollution Model., Z. Meteorol., 33, 355–362, 1983.