Ensemble forecasting and data assimilation in coupled systems
Eugenia Kalnay
Department of Meteorology and the Chaos Group
University of Maryland
Ensemble forecasting, low dimensionality, and data assimilation. Examples with a QG model and the NCEP global model
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)
Implications for data assimilation

References and thanks:
Ott, Hunt, Szunyogh, Zimin, Kostelich, Corazza, Kalnay, Patil, Yorke, 2003: Local Ensemble Kalman Filtering, MWR, under review.
Patil, Hunt, Kalnay, Yorke and Ott, 2001: Local low-dimensionality of atmospheric dynamics, PRL.
Corazza, Kalnay, Patil, Yang, Hunt, Szunyogh, Yorke, 2003: Relationship between bred vectors and the errors of the day. NPG.
Hunt et al, 2003: 4DEnKF. Submitted to Tellus.
-----Coupled systems-----
Cai, Kalnay, and Toth, 2003: Bred Vectors of the Zebiak-Cane Model and their Application to ENSO Predictions.  J. Climate, 16, 40-56.
Yang et al 2003: Bred vectors in the NASA coupled ocean-atmosphere system. EGS.
Pena and Kalnay, 2003: Separating fast and slow modes in coupled chaotic systems. Submitted to Nonlinear Proc. in Physics
Kalnay, 2003: Atmospheric modeling, data assimilation and predictability, Cambridge University Press, 341 pp.

"An ensemble forecast starts from..."
An ensemble forecast starts from initial perturbations to the analysis…
In a good ensemble “truth” looks like a member of the ensemble
The initial perturbations should reflect the analysis “errors of the day”.
In a coupled system (e.g. ENSO) they should contain the slow errors.

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The errors of the day are instabilities of the background flow. At the same verification time, the forecast uncertainties have the same shape

Strong instabilities of the background tend to have simple shapes (perturbations lie in a low-dimensional subspace)

Errors of the day
They are instabilities of the background flow
They dominate the analysis and forecast errors
They are not taken into account in data assimilation except for 4D-Var and Kalman Filtering (very expensive methods)
Their shape can be estimated with breeding
Their shape is frequently simple (low dimensionality, Patil et al, 2001)
A coupled ENSO system contains “errors of the month”

One approach to create initial perturbations for ensemble forecasting with errors of the day: breeding
Breeding is simply running the nonlinear model a second time, from perturbed initial conditions

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Data assimilation: combine a forecast with observations. We make a temperature forecast Tb and then take an observation To.
A popular way to optimally estimate the truth (analysis) is to minimize the “3D-Var” cost function:

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The solution: Ensemble Kalman Filtering
1) Perturbed observations and ensembles of data assimilation
Evensen, 1994
Houtekamer and Mitchell, 1998
2) Square root filter, no need for perturbed observations:
Tippett, Anderson, Bishop, Hamill, Whitaker, 2003
Anderson, 2001
Whitaker and Hamill, 2002
Bishop, Etherton and Majumdar, 2001
3) Local Ensemble Kalman Filtering: done in local patches
Ott et al, 2003, MWR under review
Hunt et al, 2003, Tellus: 4DEnKF

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This advantage continues into the 3-day forecasts

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Results with Lorenz 40 variable model
Used by Anderson (2001), Whitaker and Hamill (2002) to validate their ensemble square root filter (EnSRF)
A very large global ensemble Kalman Filter converges to an “optimal” analysis rms error=0.20
This “optimal” rms error is achieved by the LEKF for a range of small ensemble members
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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Preliminary LEKF results with NCEP’s global model
T62, 28 levels (1.5 million d.o.f.)
The method is model independent: essentially the same code was used for the L40 model as for the NCEP global spectral model
Simulation with observations at every grid point (1.5 million obs)
Very parallel! Each grid point analysis done independently
Very fast! 20 minutes in a single 1GHz Intel processor with 10 ensemble members

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However, two remaining problems…
Model deficiencies
Coupled models with multiple time scales

"The atmosphere has coupled instabilities..."
The atmosphere has coupled instabilities that span many scales, from ENSO to brownian motion:
ENSO has a doubling time of about one month
Baroclinic weather waves – 2 days doubling time
Mesoscale phenomena – a few hours
Cumulus convection – 10 minutes
Brownian motion – …
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
This has major implications for ensemble forecasting and data assimilation…

To help understand the problem in coupled systems, test breeding in a coupled system. The results should be valid for other nonlinear approaches such as EnKF.
The local breeding growth rate is given by

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The two rules are very robust, with threat scores >90%

Breeding in a coupled system
Breeding: finite-amplitude, finite-time instabilities of the system (~Lyapunov vectors)
In a coupled system there are fast and slow modes, and a linear Lyapunov approach will only capture fast modes.
Can we do breeding of the slow modes?

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Results from Lorenz coupled models
Coupling a fast and a slow Lorenz model, we can do breeding of the slow modes
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
Can be applied to the ENSO coupled instabilities (Cai, Kalnay and Toth, 2002, for the Zebiak-Cane model)
We have also had promising results with the NASA NSIPP coupled ocean-atmosphere GCM (Yang, Cai and Kalnay, 2003)

Example of bred vectors (contours) associated with equatorial unstable waves (color) in the NASA coupled GCM. The bred vectors (contours) give the most unstable perturbations, a powerful tool for a dynamical analysis. (Yang et al, 2003)

Experiments with coupled systems
Zebiak-Cane model (Cai et al, 2002, J of Cl.):
We found the instabilities of the ENSO evolution, and their dependence on the annual cycle and the ENSO phase
Minimizing the projection on bred vectors on the initial conditions reduces a lot the “spring barrier”
NSIPP coupled GCM
We performed two independent breeding experiments
Results suggest we can isolate the ENSO instabilities
Breeding with the NSIPP operational system

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Breeding with the NSIPP
coupled GCM (10 year run)
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)
We performed two independent breeding cycles
Performed correlation matrix EOFs, results similar to regressing with own Nino-3 index
Results are extremely robust, and almost identical for BV1 and BV2, computed independently.

Regression maps with BV NINO3 index

Regression maps with BV NINO3 index

Background ENSO vs. ENSO “embryo”

Summary about breeding in a coupled system
Breeding is a simple, finite-time, finite-amplitude generalization of Lyapunov vectors: just run the model twice…
The only free parameters are the amplitude and frequency of renormalization (does not depend on the norm)
Breeding on the Lorenz (1963) model yields very robust prediction rules for regime change and duration
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.

Tentative conclusions about data assimilation in coupled systems with multiple time scales
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.
The results using breeding suggest that a coupled Ensemble Kalman Filter could be designed for data assimilation using long time steps (assimilation intervals)