Data assimilation for multi-scale models is an important contemporary research topic. Especially the role of unresolved scales and model error in data assimilation needs to be systematically addressed. Here we examine these issues using the Ensemble Kalman filter (EnKF) with the two-level Lorenz-96 model as a conceptual prototype model of the multi-scale climate system. We use stochastic parameterization schemes to mitigate the model errors from the unresolved scales. Our results indicate that a third-order autoregressive process performs better than a first-order autoregressive process in the stochastic parameterization schemes, especially for the system with a large time-scale separation.Model errors can also arise from imprecise model parameters. We find that the accuracy of the analysis (an optimal estimate of a model state) is linearly correlated to the forcing error in the Lorenz-96 model. Furthermore, we propose novel observation strategies to deal with the fact that the dimension of the observations is much smaller than the model states. We also propose a new analog method to increase the size of the ensemble when its size is too small.
[1] J. L. Anderson, An Ensemble Adjustment Kalman Filter for Data Assimilation, Monthly Weather Review, 129 (2001), pp. 2884-2903.
[2] , An adaptive covariance inflation error correction algorithms for ensemble filters, Tellus, 59 (2007), pp. 210-224.
[3] J. L. Anderson and S. Anderson, A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts , Mon. Weather Rev., 127 (1999), pp. 2741-2758.
[4] J. Berner,U.Achatz, L. Batte, L. Bengtsson, A. d. l. Camara, H. M. Christensen, M. Colangeli,D. R. Coleman,D. Crom- melin, S. I. Dolaptchiev, et al., Stochastic parameterization: Toward a new view of weather and climate models, Bulletin of the American Meteorological Society, 98 (2017), pp. 565-588.
[5] T. Berry and J. Harlim, Linear theory for filtering nonlinear multiscale systems with model error, Proc. Roy. Soc. London, 470A (2014).
[6] M. Bonavita, L. Isaksen, and E. Holm, On the use of EDA background error variances in the ECMWF 4D-Var, Quarterly Journal of the Royal Meteorological Society, 138 (2012), pp. 1540-1559.
[7] M. Bonavita, L. Raynaud, and L. Isaksen, Estimating background-error variances in the ECMWF Ensemble of Data Assimilations system: some effects of ensemble size and day-to-day variability, Quarterly Journal of the Royal Meteorological Society, 137 (2011), pp. 423-434.
[8] H. Christensen, I. Moroz, and T. Palmer, Simulating weather regimes: impact of stochastic and perturbed parameter schemes in a simple atmospheric model, Clim Dyn (2015), 44 (2015), pp. 2195-2214.
[9] A. Clayton, A. Lorenc, and D. Barker, Operational implementation of a hybrid ensemble/4D-Var global data assimilation system at the Met Office, Quarterly Journal of the Royal Meteorological Society, 139 (2013), pp. 1445-1461.
[10] P. Courtier, J.-N. Thepaut, and A. Hollingsworth, A strategy for operational implementation of 4D-VAR, using an incremental approach, Quarterly Journal of the Royal Meteorological Society, 120 (1994), pp. 1367-1387.
[11] D. Crommelin and E. Vanden-Eijnden, Subgrid-scale parameterization with conditional Markov chains, J. Atmos. Sci., 65 (2008), pp. 2661-2675.
[12] A. Dalcher and E. Kalnay, Error growth and predictability in operational ECMWF forecasts, Tellus, 39A (1987), pp. 474-491.
[13] R. Daley, Atmospheric data assimilation, Journal of the Meteorological Society of Japan, 75 (1997), pp. 319-329.
[14] V. Echevin, P. Mey, and G. Evensen, Horizontal and vertical structure of the representer functions for sea surface measurements in a coastal circulation model, J. Phys. Oceanogr., 30 (2000), pp. 2627-2635.
[15] M. Ehrendorfer and J. Tribbia, Optimal Prediction of Forecast Error Covariances through Singular Vectors, J. Atmos. Sci., 54 (1997), pp. 286-313.
[16] G. Evensen, Inverse Methods and data assimilation in nonlinear ocean models, Physica (D), 77 (1994a), pp. 108-129.
[17] , Sequential data assimilation with a nonlinear quasi-geostrophic model usingMonte Carlo methods to forecast error statistics, Journal of Geophysical Research, 99 (1994b), pp. 10143-10162.
[18] , Advanced data assimilation for strongly nonlinear dynamics, Monthly Weather Review, 125 (1997), pp. 1342-1354.
[19] , The Ensemble Kalman Filter: theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), pp. 343-367.
[20] I. Fatkullin and E. Vanden-Eijnden, A computational strategy for multiscale systems with applications to Lorenz 96 model, Journal of Computational Physics, 200 (2004), pp. 605-638.
[21] M. Fisher, M. Leutbecher, and G. Kelly, On the equivalence between Kalman smoothing and weak-constraint fourdimensional variational data assimilation, Quarterly Journal of the Royal Meteorological Society, 131 (2005), pp. 3235-3246.
[22] C. L. E. Franzke, A. J.Majda, and E. Vanden-Eijnden, Low-order stochastic mode reduction for a realistic barotropic model climate, J. Atmos. Sci., 62 (2005), pp. 1722-1745.
[23] C. L. E. Franzke, T. O’Kane, J. Berner, P. Williams, and V. Lucarini, Stochastic climate theory and modelling, WIREs Climate Change, 6 (2015), pp. 63-78. [24] G. Gaspari and S. E. Cohn, Construction of correlation functions in two and three dimensions, Quart. J. Roy. Meteor. Soc., 125 (1999), pp. 723-757.
[25] M. Ghil and P.Malanotte-Rizzoli, Data assimilation inmeteorology and oceanography, Adv. Geophys., 33 (1991), pp. 141-266.
[26] G. Gottwald, D. Crommelin, and C. L. E. Franzke, Stochastic climate theory, in Nonlinear and Stochastic Climate Dynamics, C. L. E. Franzke and T. O’Kane, eds., Cambridge University Press, Cambridge, 2017.
[27] I. Grooms, Y. Lee, and A. J. Majda, Ensemble Filtering and Low-Resolution Model Error: Covariance Inflation, Stochastic Parameterization, and Model Numerics, Mon. Weather Rev., 143 (2015), pp. 3912-3924.
[28] T. M. Hamil, J. S. Whitaker, and C. Snyder, Distance-dependent filtering of back-ground error covariance estimates in an ensemble Kalman filter, Mon. Weather Rev., 129 (2001).
[29] J. Harlim,Model Error in Data Assimilation, in Nonlinear and Stochastic Climate Dynamics, C. L. E. Franzke and T. J. O’Kane, eds., Cambridge University Press, Cambridge, 2017, ch. 10, pp. 276-317.
[30] V. Haugen and G. Evensen, Assimilation of SLA and SST data into an OGCM for the Indian Ocean, Ocean Dynamics, 52 (2002), pp. 133-151.
[31] P. Houtekamer and H. Mitchell, Data Assimilation Using an Ensemble Kalman Filter Technique, Monthly Weather Review, 126 (1998), pp. 796-811.
[32] B. Hunt, E. Kostelich, and I. Szunyogh, Efficient data assimilation for spatiotemporal chaos: a local ensemble transform Kalman filter, Physica, 230 (2007), pp. 112-126.
[33] H. Jarvinen, E. Andersson, and F. Bouttier, Variational assimilation of time sequences of surface observation with serially correlated errors, Tellus, 51A (1999), pp. 469-488.
[34] R. Kalman, A new approach to linear filtering and prediction problem, Trans. AMSE J. Basic Eng., 82D (1960), pp. 35-45.
[35] R. Kalman and R. Bucy, New results in linear filtering and prediction theory, Trans. AMSE J. Basic Eng., 83D (1961), pp. 95-108.
[36] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge Univ. Press, 2002.
[37] E. Kalnay, H. Li, T. Miyoshi, S.-C. Yang, and J. Ballabrera-poy, 4-D-Var or ensemble Kalman filter?, Tellus, 59A (2007), pp. 758-773.
[38] C. Keppenne and M. Rienecker, Assimilation of temperature into an isopycnal ocean general circulation model using a parallel Ensemble Kalman Filter, J. Mar. Sys., 40-41 (2003), pp. 363-380.
[39] A. Lorenc, Analysis methods for numerical weather prediction, Quarterly Journal of the Royal Meteorological Society, 112 (1986), pp. 1177-1194.
[40] , The potential of the ensemble Kalman filter for NWP-a comparison with 4D-Var, Quarterly Journal of the Royal Meteorological Society, 129 (2003), pp. 3183-3203.
[41] E. Lorenz, Atmospheric predictability experiments with a large numerical model, Tellus, 34 (1982), pp. 505-513.
[42] , Predictability - a problem partly solved, in In: Proceedings of seminar on predictability, vol. 1, Shinfield Park,Reading, United Kingdom, 4-8 September 1995, In: Proceedings of seminar on predictability, ECMWF, pp. 1-18.
[43] , Designing chaotic models, J. Atmos. Sci., 62 (2005), pp. 1574-1587.
[44] , Regimes in simple systems, J. Atmos. Sci., 63 (2006), pp. 2056-2073.
[45] E. Lorenz and K. Emanuel, Optimal sites for supplementary weather observations: simulation with a small model, J. Atmos. Sci., 55 (1998), pp. 399-414.
[46] H. Madsen and R. Canizares, Comparison of Extended and Ensemble Kalman filters for data assimilation in coastal area modelling, Int. J. Numer. Meth. Fluids, 31 (1999), pp. 961-981.
[47] A. Majda, C. L. E. Franzke, and D. Crommelin, Normal forms for reduced stochastic climate models, Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 3649-3653.
[48] A. J. Majda, C. L. E. Franzke, and B. Khouider, An applied mathematics perspective on stochastic modelling for climate, Phil. Trans. R. Soc. A, 366 (2008), pp. 2429-2455.
[49] A. J. Majda, I. Timofeyev, and E. V. Eijnden, Models for stochastic climate prediction, Proc. Nat. Acad. Sci. USA, 96 (1999), pp. 14687-14691.
[50] Z. Meng and F. Zhang, Tests of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation. Part II: Imperfect Model experiments., Mon. Wea. Rev., 135 (2007), pp. 1403-1423.
[51] R. N. Miller, M. Ghil, and F. Gauthiez, Advanced Data Assimilation in Strongly Nonlinear Dynamical Systems, J. Atmos. Sci., 51 (1994), pp. 1037-1056.
[52] H. Mitchell, P. Houtekamer, and G. Pellerin, Ensemble size, and model-error representation in an Ensemble Kalman Filter, Monthly Weather Review, 130 (2002), pp. 2791-2808.
[53] L. Mitchell and A. Carrassi, Accounting for model error due to unresolved scales within ensemble Kalman filtering, Quart. J. Roy. Meteor. Soc., 141 (2015), pp. 1417-1428.
[54] T. N. Palmer, A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parameterization in weather and climate prediction models, Quart. J. Roy. Meteor. Soc., 127 (2001), pp. 279-304.
[55] D. T. Pham, Stochastic Methods for Sequential Data Assimilation in Strongly Nonlinear Systems, Mon. Weather Rev., 129 (2001), pp. 1194-1207. [56] Y. Sasaki, Numerical variational analysis with weak constraint and application to surface analysis of severe storm gust, Monthly Weather Review, 98 (1970), pp. 899-910.
[57] A. J. Simmons, R. Mureau, and T. Petroliagis, Error growth estimates of predictability from the ECMWF forecasting system, Quarterly Journal of the Royal Meteorological Society, 121 (1995), pp. 1739-1771.
[58] R. Tardif, G. Hakim, and C. Snyder, Coupled atmosphere-ocean data assimilation experiments with a low-order model and CMIP5 model data, Clim. Dyn., 45 (2015), pp. 1415-1427.
[59] Z. Toth and E. Kalnay, Ensemble forecasting at NMC: the generation of perturbations, Bull. Amer. Meteor. Soc., 74 (1993), pp. 2317-2330.
[60] M. van Loon, P. Builtjes, and A. Segers, Data assimilation of ozone in the atmospheric transport chemistry model LOTUS, Environ. Modelling Software, 15 (2000), pp. 603-609.
[61] G. Walker, On Periodicity in Series of Related Terms, Proceedings of the Royal Society of London, 131 (1931), pp. 518-532.
[62] X.Wang,D. Parrish,D. Kleist, and J.Whitaker, GSI 3DVar-Based Ensemble-Variational Hybrid Data Assimilation for NCEP Global Forecast System: Single Resolution Experiments, Monthly Weather Review, 141 (2013), pp. 4098-4117.
[63] J. Whitaker, G. Compo, and J.-N. Thepaut, A Comparison of Variational and Ensemble-Based Data Assimilation Systems for Reanalysis of Sparse Observations, Monthly Weather Review, 137 (2009), pp. 1991-1999.
[64] J. Whitaker and T. Hamill, Ensemble Data Assimilation without Perturbed Observations, Monthly Weather Review, 130 (2002), pp. 1913-1924.
[65] D. Wilks, Effiects of stochastic parameterizations in the Lorenz ’96 model, Quart. J. Roy. Meteor. Soc., 131 (2005), pp. 389-407.
[66] G. U. Yule, On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer’s Sunspot Numbers, Philosophical Transactions of the Royal Society of London, 226 (1927), pp. 267-298