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 T_b and then take an observation T_o.
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

Slide 32

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

Underway

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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

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)


	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


	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 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.


	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”


	Breeding is simply running the nonlinear model a second time, from perturbed initial conditions


	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


	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


	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


	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…


	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?


	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)


	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
		Underway


	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.


	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.


	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)