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..."**

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 |

**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 | |

**This advantage continues
into the 3-day forecasts**

**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 |

**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 |

**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… |

The local breeding growth rate is given by |

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

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? |

**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) |

**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 |

**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) |