NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-19-127-2012 On closure parameter estimation in chaotic systems Hakkarainen J. 1 2 4 Ilin A. 3 4 Solonen A. 1 2 4 Laine M. 1 Haario H. 2 Tamminen J. 1 Oja E. 3 Järvinen H. 1 Finnish Meteorological Institute, Helsinki, Finland Lappeenranta University of Technology, Lappeenranta, Finland Aalto University School of Science, Espoo, Finland These authors contributed equally to this work. 15 02 2012 19 1 127 143 Copyright: © 2012 J. Hakkarainen et al. 2012 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/19/127/2012/npg-19-127-2012.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/19/127/2012/npg-19-127-2012.pdf

Many dynamical models, such as numerical weather prediction and climate models, contain so called closure parameters. These parameters usually appear in physical parameterizations of sub-grid scale processes, and they act as "tuning handles" of the models. Currently, the values of these parameters are specified mostly manually, but the increasing complexity of the models calls for more algorithmic ways to perform the tuning. Traditionally, parameters of dynamical systems are estimated by directly comparing the model simulations to observed data using, for instance, a least squares approach. However, if the models are chaotic, the classical approach can be ineffective, since small errors in the initial conditions can lead to large, unpredictable deviations from the observations. In this paper, we study numerical methods available for estimating closure parameters in chaotic models. We discuss three techniques: off-line likelihood calculations using filtering methods, the state augmentation method, and the approach that utilizes summary statistics from long model simulations. The properties of the methods are studied using a modified version of the Lorenz 95 system, where the effect of fast variables are described using a simple parameterization.

 References Annan, J. and Hargreaves, J.: Efficient estimation and ensemble generation in climate modelling, Phil. Trans. R. Soc. A, 365, 2077–2088, <a href="http://dx.doi.org/10.1098/rsta.2007.2067">https://doi.org/10.1098/rsta.2007.2067</a>, 2007. Annan, J. D., Lunt, D. J., Hargreaves, J. C., and Valdes, P. J.: Parameter estimation in an atmospheric GCM using the Ensemble Kalman Filter, Nonlin. Processes Geophys., 12, 363–371, <a href="http://dx.doi.org/10.5194/npg-12-363-2005">https://doi.org/10.5194/npg-12-363-2005</a>, 2005. Auvinen, H., Bardsley, J.&nbsp;M., Haario, H., and Kauranne, T.: Large-Scale Kalman Filtering Using the Limited Memory BFGS Method, Electron. T. Numer. Ana., 35, 217–233, 2009. Auvinen, H., Bardsley, J., Haario, H., and Kauranne, T.: The variational Kalman filter and an efficient implementation using limited memory BFGS, Int. J. Numer. Meth. Fl., 64, 314–335, 2010. Bishop, C.&nbsp;M.: Pattern Recognition and Machine Learning, Information Science and Statistics, Springer, New York, 2nd Edn., 2006. Bonavita, M., Torrisi, L., and Marcucci, F.: The ensemble Kalman filter in an operational regional NWP system: preliminary results with real observations, Q. J. Roy. Meteor. Soc., 134, 1733–1744, <a href="http://dx.doi.org/10.1002/qj.313">https://doi.org/10.1002/qj.313</a>, 2008. Cappe, O., Godsill, S., and Moulines, E.: An overview of existing methods and recent advances in sequential Monte Carlo, Proceedings of IEEE, 95, 899–924, <a href="http://dx.doi.org/10.1109/JPROC.2007.893250">https://doi.org/10.1109/JPROC.2007.893250</a>, 2007. Cornuet, J.-M., Santos, F., Beaumont, M.&nbsp;A., Robert, C.&nbsp;P., Marin, J.-M., Balding, D.&nbsp;J., Guillemaud, T., and Estoup, A.: Inferring population history with {DIY ABC}: a user-friendly approach to approximate {B}ayesian computation, Bioinformatics, 24, 2713–2719, 2008. DelSole, T. and Yang, X.: State and Parameter Estimation in Stochastic Dynamical Models, Physica D, 239, 1781–1788, 2010. Dowd, M.: Estimating parameters for a stochastic dynamic marine ecological system, Environmetrics, 22, 501–515, <a href="http://dx.doi.org/10.1002/env.1083">https://doi.org/10.1002/env.1083</a>, 2011. Evensen, G.: Data assimilation: The ensemble Kalman filter, Springer, 2007. 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, 2007. Gelman, A., Carlin, J., Stern, H., and Rubin, D.: Bayesian Data Analysis, Chapman & Hall, 2nd Edn., 2003. Haario, H., Saksman, E., and Tamminen, J.: An adaptive Metropolis algorithm, Bernoulli, 7, 223–242, 2001. Haario, H., Laine, M., Mira, A., and Saksman, E.: DRAM: Efficient adaptive MCMC, Stat. Comput., 16, 339–354, <a href="http://dx.doi.org/10.1007/s11222-006-9438-0">https://doi.org/10.1007/s11222-006-9438-0</a>, 2006. Houtekamer, P., Herschel, L., Mitchell, G., Buehner, M., Charron, M., Spacek, L., and Hansen, B.: Atmospheric Data Assimilation with an Ensemble Kalman Filter: Results with Real Observations, Mon. Weather Rev., 133, 604–620, 2005. Hunt, B.&nbsp;R., Kostelich, E.&nbsp;J., and Szunyogh, I.: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter, Physica D, 230, 112–126, 2007. Ionides, E., Breto, C., and King, A.: Inference for nonlinear dynamical systems, Proc. Nat. Aca. Sci., 103, 18438–18443, 2006. Jackson, C.&nbsp;S., Sen, M.&nbsp;K., Huerta, G., Deng, Y., and Bowman, K.&nbsp;P.: Error Reduction and Convergence in Climate Prediction, J. Climate, 21, 6698–6709, <a href="http://dx.doi.org/10.1175/2008JCLI2112.1">https://doi.org/10.1175/2008JCLI2112.1</a>, 2008. Järvinen, H., Räisänen, P., Laine, M., Tamminen, J., Ilin, A., Oja, E., Solonen, A., and Haario, H.: Estimation of ECHAM5 climate model closure parameters with adaptive MCMC, Atmos. Chem. Phys., 10, 9993–10002, <a href="http://dx.doi.org/10.5194/acp-10-9993-2010">https://doi.org/10.5194/acp-10-9993-2010</a>, 2010. Järvinen, H., Laine, M., Solonen, A., and Haario, H.: Ensemble prediction and parameter estimation system: the concept, Q. J. Roy. Meteor. Soc., <a href="http://dx.doi.org/10.1002/qj.923">https://doi.org/10.1002/qj.923</a>, published online, 2011. Kalman, R.: A New Approach to Linear Filtering and Prediction Problems, Transactions of the ASME – Journal of Basic Engineering, Series D, 82, 35–42, 1960. Kitagawa, G.: A self-organizing state space model, J. Am. Stat. Assoc., 93, 1203–1215, 1998. Kivman, G. A.: Sequential parameter estimation for stochastic systems, Nonlin. Processes Geophys., 10, 253–259, <a href="http://dx.doi.org/10.5194/npg-10-253-2003">https://doi.org/10.5194/npg-10-253-2003</a>, 2003. Laine, M., Solonen, A., Haario, H., and Järvinen, H.: Ensemble prediction and parameter estimation system: the method, Q. J. Roy. Meteor. Soc., published online, <a href="http://dx.doi.org/10.1002/qj.922">https://doi.org/10.1002/qj.922</a>, 2011. Leutbecher, M.: Predictability and Ensemble Forecasting with L}orenz-95 systems, Lecture notes, {ECMWF meteorological training course on Predictability, diagnostics and seasonal forecasting, available at: <a href="http://www.ecmwf.int/newsevents/training/meteorological_presentations/pdf/PR/Practice_L95.pdf">http://www.ecmwf.int/newsevents/training/meteorological_presentations/pdf/PR/Practice_L95.pdf</a>, 2010. Lorenz, E.: Predictability: A problem partly solved, Proceedings of the Seminar on Predicability, European Center on Medium Range Weather Forecasting, 1, 1–18, 1995. Marjoram, P. and Tavaré, S.: Modern computational approaches for analysing molecular genetic variation data, Nat. Rev. Genet., 7, 759–770, 2006. Martin, G.&nbsp;M., Johnson, D.&nbsp;W., and Spice, A.: The measurement and parameterization of effective radius of droplets in warm stratocumulus, J. Atmos. Sci., 51, 1823–1842, 1994. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E.: Equations of State Calculations by Fast Computing Machines, J. Chem. Phys., 21, 1087–1092, 1953. Molteni, F., Buizza, R., Palmer, T., and Petroliagis, T.: The ECMWF Ensemble Prediction System: Methodology and validation, Q. J. Roy. Meteor. Soc., 122, 73–119, 1996. Nocedal, G. and Wright, S.: Numerical Optimization, Springer, Berlin, 1999. Ott, E., Hunt, B., Szunyogh, I., Zimin, A., Kostelich, E., Corazza, M., Kalnay, E., Pati, D., and Yorke, J.: A local ensemble Kalman filter for atmospheric data assimilation, Tellus A, 56, 415–428, 2004. Palmer, T., Shutts, G., Hagedorn, R., Doblas-Reyes, F., Jung, T., and Leutbecher, M.: Representing Model Uncertainty in Weather and Climate Prediction, Ann. Rev. Earth Planet. Sci., 33, 163–193, 2005. Rabier, F., Järvinen, H., Klinker, E., Mahfouf, J.-F., and Simmons, A.: The ECMWF implementation of four dimensional variational assimilation, Part I: Experimental results with simplified physics, Q. J. Roy. Meteor. Soc., 126, 1143–1170, 2000. Rougier, J.&nbsp;C.: Efficient Emulators for Multivariate Deterministic Functions, J. Comput. Graphic. Stat., 17, 827–843, 2008. Sexton, D., Murphy, J., Collins, M., and Webb, M.: Multivariate Prediction Using Imperfect Climate Models Part {I}: Outline of Methodology, Clim. Dynam., 1–30, <a href="http://dx.doi.org/10.1007/s00382-011-1208-9">https://doi.org/10.1007/s00382-011-1208-9</a>, 2011. Shapiro, A., Dentcheva, D., and Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory, MOS-SIAM Series on Optimization 9, Soc. Ind. Appl. Math., 2009. Singer, H.: Parameter estimation of nonlinear stochastic differential equations: Simulated maximum likelihood versus extended Kalman filter and Ito-Taylor expansion, J. Comput. Graph. Stat., 11, 972–995, 2002. Tippett, M., Anderson, J., Bishop, G., Hamill, T., and Whitaker, J.: Ensemble Square Root Filters*, Mon. Weather Rev., 131, 1485–1490, <a href="http://dx.doi.org/10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2">https://doi.org/10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2</a>, 2003. Villagran, A., Huerta, G., Jackson, C.&nbsp;S., and Sen, M.&nbsp;K.: Computational Methods for Parameter Estimation in Climate Models, Bayesian Analysis, 3, 823–850, <a href="http://dx.doi.org/10.1214/08-BA331">https://doi.org/10.1214/08-BA331</a>, 2008. Whitaker, J. and Hamill, T.: Ensemble data assimilation without perturbed observations, Mon. Weather Rev., 130, 1913–1924, 2002. Wilks, D.: Effects of stochastic parametrizations in the {L}orenz '96 system, Q. J. Roy. Meteor. Soc., 131, 389–407, <a href="http://dx.doi.org/10.1256/qj.04.03">https://doi.org/10.1256/qj.04.03</a>, 2005. Zupanski, M.: Maximum Likelihood Ensemble Filter: Theoretical Aspects, Mon. Weather Rev., 133, 1710–1726, 2005.