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Ensemble forecasting, low dimensionality, and
data assimilation. Examples with a QG model and the NCEP global model |
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Breeding, Lyapunov Vectors and Singular Vectors
in a coupled system with fast and slow time scales. Examples with a coupled
Lorenz model, Zebiak-Cane model and NASA coupled GCM (NSIPP) |
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Implications for data assimilation |
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Ott, Hunt, Szunyogh, Zimin, Kostelich, Corazza,
Kalnay, Patil, Yorke, 2003: Local Ensemble Kalman Filtering, MWR, under
review. |
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Patil, Hunt, Kalnay, Yorke and Ott, 2001: Local
low-dimensionality of atmospheric dynamics, PRL. |
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Corazza, Kalnay, Patil, Yang, Hunt, Szunyogh,
Yorke, 2003: Relationship between bred vectors and the errors of the day.
NPG. |
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Hunt et al, 2003: 4DEnKF. Submitted to Tellus. |
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-----Coupled systems----- |
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Cai, Kalnay, and Toth, 2003: Bred Vectors of the
Zebiak-Cane Model and their Application to ENSO Predictions. J. Climate, 16, 40-56. |
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Yang et al 2003: Bred vectors in the NASA
coupled ocean-atmosphere system. EGS. |
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Pena and Kalnay, 2003: Separating fast and slow
modes in coupled chaotic systems. Submitted to Nonlinear Proc. in Physics |
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Kalnay, 2003: Atmospheric modeling, data
assimilation and predictability, Cambridge University Press, 341 pp. |
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An ensemble forecast starts from initial
perturbations to the analysis… |
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In a good ensemble “truth” looks like a member
of the ensemble |
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The initial perturbations should reflect the
analysis “errors of the day”. |
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In a coupled system (e.g. ENSO) they should
contain the slow errors. |
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They are instabilities of the background flow |
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They dominate the analysis and forecast errors |
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They are not taken into account in data
assimilation except for 4D-Var and Kalman Filtering (very expensive
methods) |
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Their shape can be estimated with breeding |
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Their shape is frequently simple (low
dimensionality, Patil et al, 2001) |
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A coupled ENSO system contains “errors of the
month” |
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Breeding is simply running the nonlinear model a
second time, from perturbed initial conditions |
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1) Perturbed observations and ensembles of data
assimilation |
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Evensen, 1994 |
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Houtekamer and Mitchell, 1998 |
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2) Square root filter, no need for perturbed
observations: |
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Tippett, Anderson, Bishop, Hamill, Whitaker,
2003 |
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Anderson, 2001 |
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Whitaker and Hamill, 2002 |
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Bishop, Etherton and Majumdar, 2001 |
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3) Local Ensemble Kalman Filtering: done in
local patches |
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Ott et al, 2003, MWR under review |
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Hunt et al, 2003, Tellus: 4DEnKF |
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Used by Anderson (2001), Whitaker and Hamill
(2002) to validate their ensemble square root filter (EnSRF) |
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A very large global ensemble Kalman Filter
converges to an “optimal” analysis rms error=0.20 |
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This “optimal” rms error is achieved by the LEKF
for a range of small ensemble members |
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We performed experiments for different size
models: M=40 (original), M=80 and M=120, and compared a global KF with the
LEKF |
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T62, 28 levels (1.5 million d.o.f.) |
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The method is model independent: essentially the
same code was used for the L40 model as for the NCEP global spectral model |
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Simulation with observations at every grid point
(1.5 million obs) |
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Very parallel! Each grid point analysis done
independently |
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Very fast! 20 minutes in a single 1GHz Intel
processor with 10 ensemble members |
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Model deficiencies |
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Coupled models with multiple time scales |
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The atmosphere has coupled instabilities that
span many scales, from ENSO to brownian motion: |
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ENSO has a doubling time of about one month |
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Baroclinic weather waves – 2 days doubling time |
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Mesoscale phenomena – a few hours |
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Cumulus convection – 10 minutes |
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Brownian motion – … |
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Linear approaches, like Lyapunov and Singular
Vectors can only handle the fastest
growing instability present in the model, nonlinear integrations allow fast
instabilities to saturate |
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This has major implications for ensemble
forecasting and data assimilation… |
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The local breeding growth rate is given by |
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Breeding: finite-amplitude, finite-time
instabilities of the system (~Lyapunov vectors) |
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In a coupled system there are fast and slow
modes, and a linear Lyapunov approach will only capture fast modes. |
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Can we do breeding of the slow modes? |
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Coupling a fast and a slow Lorenz model, we can
do breeding of the slow modes |
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Valid for other nonlinear approaches (e.g.,
EnKF) but not of linear approaches (e.g., LVs and SVs) which are dominated
by the fastest component |
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Can be applied to the ENSO coupled instabilities
(Cai, Kalnay and Toth, 2002, for the Zebiak-Cane model) |
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We have also had promising results with the NASA
NSIPP coupled ocean-atmosphere GCM (Yang, Cai and Kalnay, 2003) |
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Zebiak-Cane model (Cai et al, 2002, J of Cl.): |
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We found the instabilities of the ENSO
evolution, and their dependence on the annual cycle and the ENSO phase |
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Minimizing the projection on bred vectors on the
initial conditions reduces a lot the “spring barrier” |
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NSIPP coupled GCM |
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We performed two independent breeding
experiments |
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Results suggest we can isolate the ENSO
instabilities |
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Breeding with the NSIPP operational system |
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Underway |
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As in the Lorenz coupled system, we rescale
using a slow variable (Nino 3 SST) and an interval long compared to the
“weather noise” (one month) |
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We performed two independent breeding cycles |
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Performed correlation matrix EOFs, results
similar to regressing with own Nino-3 index |
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Results are extremely robust, and almost
identical for BV1 and BV2, computed independently. |
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Breeding is a simple, finite-time,
finite-amplitude generalization of Lyapunov vectors: just run the model
twice… |
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The only free parameters are the amplitude and
frequency of renormalization (does not depend on the norm) |
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Breeding on the Lorenz (1963) model yields very
robust prediction rules for regime change and duration |
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In coupled models, it is possible to isolate the
fast and the slow modes by a physically based choice of the amplitude and
frequency of the normalization. |
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In a system with instabilities with multiple
time scales, methods that depend on linearization to get the “errors of the
day” such as 4D-Var and KF may not work. |
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The results using breeding suggest that a
coupled Ensemble Kalman Filter could be designed for data assimilation using
long time steps (assimilation intervals) |
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