Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 15 1 2008 10.5194/npg-15-109-2008 http://www.nonlin-processes-geophys.net/15/109/2008/ http://www.nonlin-processes-geophys.net/15/109/2008/npg-15-109-2008.html http://www.nonlin-processes-geophys.net/15/109/2008/npg-15-109-2008.pdf 109 114 2008-02-13 Spatiotemporal characterization of Ensemble Prediction Systems – the Mean-Variance of Logarithms (MVL) diagram J. M. Gutiérrez gutierjm@unican.es C. Primo M. A. Rodríguez J. Fernández University of Cantabria, Department of Applied Mathematics, Santander, Spain European Centre for Medium-Range Weather Forecasts, Reading, UK Instituto de Física de Cantabria, CSIC-UC, Santander, Spain We present a novel approach to characterize and graphically represent the spatiotemporal evolution of ensembles using a simple diagram. To this aim we analyze the fluctuations obtained as differences between each member of the ensemble and the control. The lognormal character of these fluctuations suggests a characterization in terms of the first two moments of the logarithmic transformed values. On one hand, the mean is associated with the exponential growth in time. On the other hand, the variance accounts for the spatial correlation and localization of fluctuations. In this paper we introduce the MVL (Mean-Variance of Logarithms) diagram to intuitively represent the interplay and evolution of these two quantities. We show that this diagram uncovers useful information about the spatiotemporal dynamics of the ensemble. Some universal features of the diagram are also described, associated either with the nonlinear system or with the ensemble method and illustrated using both toy models and numerical weather prediction systems. Barabási, A L. and Stanley, H E.: Fractal Concepts in Surface Growth, Cambridge University Press, 1995. Gneiting, T. and Raftery, A.: Weather Forecasting with Ensemble Methods, Science, 310, 248–249, 2005. Hagedorn, R., Doblas-Reyes, F J., and Palmer, T.: The rationale behind the success of multi–model ensembles in seasonal forecasting – I. Basic concept, Tellus, 57A, 219–233, 2005. López, J M., Primo, C., Rodríguez, M A., and Szendro, I.: Scaling Properties of Growing Noninfinitesimal Perturbations in Space-Time Chaos, Phys. Rev. E., 70, 056224, 2004. Lorenz, E \rm N.: Predictability-a problem partly solved. in Proceedings of ECMWF seminar Predictability., ECMWF Seminar Proceedings, ECMWF, Reading, UK., Seminar on Predictability, 1, 1–19, 1996. Lorenz, E \rm N. and Emanuel, K.: Optimal sites for supplementary weather observations: Simulation with a small model., J. Atmos. Sci., 55, 399–414, 1998. Molteni, F., Buizza, R., Palmer, T., and Petroliagis, T.: The ECMWF Ensemble Prediction System: Methodology and validation, Q. J. Roy. Meteorol. Soc., 122, 73–119, 1996. Palmer, T N.: The economic value of ensemble forecasts as a tool for risk assessment: From days to decades, Q. J. Roy. Meteorol. Soc., 128, 747–774, 2002. Palmer, T N., Alessandri, A., Andersen, U., Cantelaube, P., Daveyand, M., Délécluse, P., Déqué, M., D\'iez, E., Doblas-Reyes, F J., Feddersen, H., Graham, R., Gualdi, S., Guérémy, J F., Hagedorn, R., Hoshen, M., Keenlyside, N., Latif, M., Lazar, A., Maisonnave, E., Marletto, V., Orfila, A. P. M B., Rofel, P., Terres, J M., and Thomson, M C.: Development of a European multimodel ensemble system for seasonal–to–interannual prediction DEMETER, B. Am. Meteorol. Soc., 85, 853–872, 2004. Primo, C., Szendro, I G., Rodriguez, M A., and López, J M.: Dynamic scaling of bred vectors in spatially extended chaotic systems, Europhys. Lett., 76, 767–773, 2006. Primo, C., Szendro, I., Rodríguez, M A., and Gutiérrez, J M.: Error growth analysis in systems with spatial chaos: Coupled map lattices and global weather models, Phys. Rev. Lett., 98, 108501, 2007. Smith, L.: Disentangling uncertainty and error: On the predictability of nonlinear systems, Birkhauser, Boston, 2000. Toth, Z. and Kalnay, E.: Ensemble Forecasting at NMC: The Generation of Perturbations, B. Am. Meteorol. Soc., 74, 2317–2330, 1993. Vitart, F.: Monthly forecasting at ECMWF, Mon. Weather Rev., 132, 2761–2779, 2004. Wilks, D S.: Effects of stochastic parameterization in the Lorenz ’96 system, Q. J. Roy. Meteorol. Soc., 131, 389–407, 2005.