Stochastic Parametrisation

Please find below a range of resources and references to help you learn more about stochastic parametrisations, their development, and their validation.


Review articles

Berner, J., Achatz, U., Batté, L., Bengtsson, L., De La Cámara, A., Christensen, H. M., … Yano, J.-I. (2017). Stochastic parameterization: toward a new view of weather and climate models. Bulletin of the American Meteorological Society98(3).

Palmer, T.N. (2019) Stochastic weather and climate models. Nat Rev Phys 1, 463–471.



Ewald, B., and C. Penland, (2009). Numerical generation of stochastic differential equations in climate models, Handbook of numerical analysis: Computational Methods for the Atmosphere and the Oceans, 14, 279-306.

Gardiner, C.W. (1985). Handbook of Stochastic Methods, Springer, Berlin. (Chapter 6 gives a readable version of Papanicolaou and Kohler 1974).

Hasselmann, (1975). Stochastic climate models part I. Theory.Tellus,

Papanicolaou, G. C, and Kohler, W., (1974). "Asymptotic Theory of Mixing Stochastic Ordinary Differential Equations," Communications on Pure and Applied Mathematics, 27, pp. 641-668.

Penland, C., (2003). Noise out of chaos and why it won't go away. Bulletin of the American Meteorological Society, 84, 921-925.

Penland, C., (2003). A Stochastic approach to Nonlinear Dynamics: A Review (Electonic supplement to 'Noise out of chaos and why it won't go away'). Bulletin of the American Meteorological Society, 84, ES43-ES51.

Rüemelin, W., (1982), Numerical treatment of stochastic differential equations, SIAM J. Numer. Math., 19 (1982), no. 3, 604–613.


Data-driven approaches to developing stochastic parametrisations

Arnold, H. M., Moroz, I. M., & Palmer, T. N., (2013). Stochastic parametrizations and model uncertainty in the Lorenz ’96 system. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences371(1991), 20110479–20110479.

Christensen, H. M., (2020). Constraining stochastic parametrisation schemes using high-resolution simulations. Quarterly Journal of the Royal Meteorological Society146(727).

Shutts, G. J., & Palmer, T. N., (2007). Convective forcing fluctuations in a cloud-resolving model: Relevance to the stochastic parameterization problem. Journal of Climate20(2), 187–202.

Shutts, G. J. & Pallares, A. C., (2014), Assessing parametrization uncertainty associated with horizontal resolution in numerical weather prediction models. Phil. Trans. R. Soc. 372 (2018),


Physically motivated stochastic parametrisations

Bengtsson, L., Körnich, H., Källén, E., Svensson, G., 2011: Large-Scale Dynamical Response to Subgrid-Scale Organization Provided by Cellular Automata. Journal of the Atmospheric Sciences Volume 68, Issue 12 pp. 3132-3144.

Bengtsson, L., M. Steinheimer, P. Bechtold, and J.-F. Geleyn, 2013: A stochastic parameterization for deep convection using cellular automata, Quarterly Journal of the Royal Meteorological Society, 139 (675) .

Bengtsson, L., Körnich, H., 2016:  Impact of a stochastic parameterization of cumulus convection, using cellular automata, in a meso-scale ensemble prediction system. Quarterly Journal of the Royal Meteorological Society. Q.J.R. Meteorol. Soc., 142: 1150–1159. doi: 10.1002/qj.2720

Bengtsson, L., Dias, J., Tulich, S., Gehne, M., & Bao, J.‐W. (2021). A stochastic parameterization of organized tropical convection using cellular automata for global forecasts in NOAA's Unified Forecast System. Journal of Advances in Modeling Earth Systems, 13, e2020MS002260.

Sakradzija, M., A. Seifert, and A. Dipankar, 2016: A stochastic scale-aware parameterization of shallow cumulus convection across the convective gray zone, Journal of Advances in Modeling Earth Systems, 8, doi:10.1002/2016MS000634

Sakradzija, M., and D. Klocke, 2018: Physically constrained stochastic shallow convection in realistic kilometer-scale simulations, Journal of Advances in Modeling Earth Systems, 10, doi: 10.1029/2018MS001358

Sakradzija M., F. Senf, L. Scheck, M. Ahlgrimm, and D. Klocke, 2020: Local impact of stochastic shallow convection on clouds and precipitation in the tropical Atlantic, Mon. Wea. Rev., 148, 12, 5041–5062,


Operational approaches

Bengtsson, L., J. Bao, P. Pegion, C. Penland, S. Michelson, and J. Whitaker, 2019: A Model Framework for Stochastic Representation of Uncertainties Associated with Physical Processes in NOAA’s Next Generation Global Prediction System (NGGPS). Mon. Wea. Rev., 147, 893–911,

Lang, S. T. K., Lock, S.‐J., Leutbecher, M, Bechtold, P, Forbes, R. M., (2021), Revision of the Stochastically Perturbed Parametrisations model uncertainty scheme in the Integrated Forecasting System.  Q J R Meteorol Soc., 147: 1364– 1381.

Leutbecher, M., Lock, S.‐J., Ollinaho, P., Lang, S. T. K., Balsamo, G., Bechtold, P., ... and Weisheimer, A. (2017), Stochastic representations of model uncertainties at ECMWF: state of the art and future vision. Q.J.R. Meteorol. Soc, 143: 2315-2339.

Lock, S‐J, Lang, STK, Leutbecher, M, Hogan, RJ, Vitart, F, (2019). Treatment of model uncertainty from radiation by the Stochastically Perturbed Parametrization Tendencies (SPPT) scheme and associated revisions in the ECMWF ensembles. Q J R Meteorol Soc., 145 ( Suppl. 1): 75‐ 89.

Palmer, T. N., Buizza, R., Doblas-Reyes, F., Jung, T., Leutbecher, M., Shutts, G.J., Steinheimer, M. and Weisheimer, A., 2009. Stochastic Parametrization and Model Uncertainty. ECMWF Tech. Memo 598,


Verification of ensemble forecasts

Ferro, C. A. T., Richardson, D. S. and Weigel, A. P., (2008), On the effect of ensemble size on the discrete and continuous ranked probability scores. Met. Apps, 15: 19-24.

Ferro, C.A.T., (2014), Fair scores for ensemble forecasts. Q.J.R. Meteorol. Soc., 140: 1917-1923.

Leutbecher, M., (2019) Ensemble size: How suboptimal is less than infinity? Q J R Meteorol Soc. 145 ( Suppl. 1): 107– 128.