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## Initial ensemble perturbations - basic concepts

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**Initial ensemble perturbations-basic concepts**Linus Magnusson Acknowledgements: ErlandKällén, Jonas Nycander, Martin Leutbecher, …, …**Introduction**Optimal analysis Perturbed forecast = + Perturbation Perturbations added to a subset or all model variables**Desirable properties of initial perturbations**• Sampling analysis uncertainty • Mean amplitude • Geographical distribution • Spatial scale • ”Error of the day” • Sustainable growth • High quality of probabilistic forecast**Ensemble mean RMSE and ensemble spread**Saturation level dE/dt Initial uncertainty cf. Bengtsson, Magnusson and Källén, MWR (2008) Optimally: Ensemble mean RMSE = ensemble spread**Random Perturbations (why does it not work?)**Analysis error variance estimate x global tuning factor Random grid-point number (-1 to 1) x**Random Field (RF) perturbation (for benchmarking)**Analysis Random date 1 Analysis Random date 2 - Random Field Perturbation x normalizing factor = Not flow-dependent, but linear balances maintained cf. Magnusson, Nycander and Källén, Tellus A (2009)**Geostrophic balance and perturbations**Initially +48h Blue – random pert., Red – random field, climatology - grey**Singular vectors**Optimize perturbation growth for a time interval Atmospheric state Norm dependent! M tangent linear operator. x(t)=M(t,x0)x(0) Singular vector perturbation (Will be further explained by Simon Lang)**Breeding perturbations**Perturbed Forecast +06h Unperturbed Forecast +06h - Breeding vector x normalizing factor =**Ensemble transform perturbations (further development of BV)**(Wei et al., Tellus A, 2008) (x=pf-em), compare with BV Make the perturbations orthogonal C and Γ from eigenvalue problem: Error norm • Simplex transformation • Regional re-scaling • ETKF perturbations similar idea (Wang and Bishop, JAS, 2003)**Perturbation methods Lorenz-63**Initial point After 1 time unit • Random pert. • ✚ 1st SV • ✖ 2nd SV • BV**Exponential perturbation growth**Lorenz-63 NWP-model (ECMWF) SV – red, BV – blue, Random Field Pert. – Green, Random Pert. - black**Evolution of ensemble spread(one case, total pert. energy**700 hPa) Initially Maximum –red**Evolution of ensemble spread(one case, total pert. energy**700 hPa) +48h Maximum –red, Scale 48: twice the scale for +00h**Connections between perturbations and baroclinic zones**Fastest growth rate of normal modes**Correlation Eady index – Ens. Stdev z500**SV - Red ET - Blue RF – Green RP - Black (20N-70N)**Mean initial perturbation distribution**Figure 2. Magnusson, Leutbecher and Källén. 2008 (MWR)**Mean perturbation distribution after 48 hours**Figure 3. Magnusson, Leutbecher and Källén. 2008 (MWR)**Ranked Probability skill score - t850**Different initial perturbations (N.Hem) SV - Red ET - Blue RF –Green Different Centres (from Park et al.(2008), Courtesy R. Buizza )**Other things to consider - Perturbation symmetry**• +/- symmetry -> rank N/2 • Simplex transformation -> rank N-1 • No clear advantage for simplex transformation in our metrics**Other things to consider - Importance of initial amplitude**scaling Perturbation growth is highly model dependent