# Presentation: 182094

## Implementation of the LEKF on the NCEP GFS: “Perfect Model” Experiments 

**Chaos-Weather Team**
- University of Maryland College Park

- Camp Springs, MD, February 27, 2004

## UMCP Chaos-Weather Team

**Founders**: E. Kalnay and J. Yorke
**Theory**: B. Hunt and E. Ott et al.
**4D Extension**: B. Hunt, T. Sauer, and J. Yorke et al.
**GFS Implementation**: I. Szunyogh, E. Kostelich and G. Gyarmati et al.

- An **interdisciplinary team** of experts in dynamical systems
- theory,  meteorology, mathematics, and scientific computing

## Outline

- The LEKF Concept
- 4-D Extension
- Implementation on the NCEP GFS
- Conclusions
- Future Plans

## Data Assimilation

**t****1**

**t****3**

**Analysis pdf**
**For the Grid**

**Forecast pdf **
**For the Grid **

**Initial Condition:**
**Analysis Mean**

**Observation pdf**

**Kalman Filter: ****The analysis uncertainty is evolved using the model dynamics to obtain the forecast uncertainty**

**t****2**

## Ensemble Kalman Filtering

- t1

- t2

- Ensemble of
- Initial conditions

- Background Ensemble

-  **Ensemble Kalman Filters (EnKF):** (i) Multiple analyses are prepared, and then (ii) the most likely state is determined by the mean of the analysis ensemble
-  **Ensemble Square-root Filters (EnSF):** (ii) The most likely state is determined first, and then (ii) the mean state is perturbed to obtain the analysis ensemble

## Examples

**EnKF:** Evensen (1994), Houtekamer and Mitchell (1998, 2001), Hamill and Snyder (2000), Anderson and Anderson (1999), Keppene and Rienecker (2002)
**EnSF**: Anderson (2001), Bishop et al. (2001), Whitaker and Hamill (2002), Tippett et al. (2002), **LEKF** (Ott et al., 2003a, b)
- What is the difference between our scheme and other square-root filters? **The LEKF is not a sequential data assimilation scheme.**

## Sequential Schemes

- The **observations** are **assimilated sequentially**
- A local region around the grid point is defined by the Gaussian filter of Gaspari and Cohn (1998)
- The **state estimate** is **updated at all grid-points** within the local region

## Local Ensemble Kalman Filter

- The **state estimate** is **updated concurrently** at the different grid-points
-  A local region is defined (the shape is arbitrary, no forced distance dependent reduction of the correlation)
**All observations** within the local region are **assimilated**

## Potential Advantages of the LEKF

- Allows for significant reduction of the dimension, based on estimating the complexity of the dynamics (efficient filtering of redundant information from the observations)
- Efficient for observations with correlated errors
- The computation can be performed concurrently for each grid point
**We believe that this formulation is advantageous, when many observations, especially those with correlated errors, are assimilated **

## Potential Disadvantages of the LEKF

- When the observations are sparse (when the local regions must be large) the Gaussian filter may be a better tool to localize the covariance information (more efficient filtering of spurious long distance correlations)
- When the number of observations is significantly lower than the number of grid points, the LEKF is probably more expensive than a sequential scheme
**We expect that sequential schemes are more suitable than the LEKF, when relatively few observations are assimilated (e.g.; Whitaker et al.  2003: Reanalysis without radio-sondes ...)**

## Schematic of the LEKF

**Analysis ensemble at t-1**

**Background ensemble at t**

**Local background vectors**

**Global analysis ensemble at t**

**...**

- Evolve model from t-1 to t

- Form local
- vectors

- Do local analysis

- Obtain
- Global
- Analysis Ensemble

**Local analysis and Analysis Error covariance matrix**

**Local Analysis Ensembles**

- Obtain
- Local
- Analysis Ensembles

## Illustration by the Lorenz-96 model

- xM=x1

- x2

- x3

- xM-1

- “Eastward group velocity”

- Unstable non-linear waves
- with a typical wavelength of
- 5-6 grid points (the wavelength
- is independent of the M)

- For M=40 the Lyapunov dimension is 27.1

- Prototype of a spatio-temporally chaotic system
- with a finite correlation length

## LEKF vs. Global EKF

- The global scheme requires an increasing number of ensemble members as the size of the system increases
- The number of ensemble members needed in the LEKF is much lower than in the global scheme and independent of the system size

- Global

- Local

- M=120

- M=80

- M=40

- ~Lyapunov
- dimension

## 4D Ensemble Kalman Filter

- Ensemble Kalman Filters allow for assimilating observations
- at the correct time:

- y=h(**x**)

- observation

- observational operator

- model state

**x****b**=**E****b****e**

- background mean

- background ensemble

- vector of linear weights

**y**-**Hx**o=**y**-**H****E**o**e**=**y**-**H**’**x**b

- innovation at obs. time

- ensemble at obs. time

- modified **H**

- Hunt et. al 2004a,b:

**H**’ does not exist for a
- single background

## 4D Ensemble Kalman Filter

- 4D Ens. KF

- Approach used in
- 3D-Var Schemes

- Only data at analysis
- Times are used

- Illustration by the 40-variable Lorenz model

## NCEP GFS

- A time series of  “true states” is generated by a long integration of the model started from the operational NCEP analysis at January 1, 2000.
- The ”observations” are created by adding Gaussian random noise to the true state (the errors: 1 K for temperature, 1.1 m/s for wind vector components, and 1 hPa for surface pressure)
- The lower boundary condition of an analysis ensemble member is copied from the associated background ensemble member
- The simulated observations are assimilated and the result is compared to the `true state’
- This experimental design allows for** ****testing our fundamental hypothesis**, that the local dimensionality of the model is low (whether we can stay close to the true state by estimating the state in local low-dimensional spaces)
- This is a crucial step, which is needed for us to be able to **distinguish between problems in our formulation** (and implementation) and the effects of model errors

## Experiments

**Red (“Base Line”):** 80-member, the surface pressure, temperature and wind are observed at each grid point, ozone and humidity copied from “truth”
**Green:** Same as Red, except 40-members and 90% of the observations are removed
**Dark Blue:** Same as Red, except 40-members are removed, and ozone and humidity are copied from background
**Light Blue (“Realistic”):** Same as Red, except 40 members and 90% of the observations are removed, ozone and humidity are copied from background

## Observational error

- All implementations are
- Close to the “Base line”

- Similarly good convergence was
- Also found for the other variables

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## Conclusions

- While some modest improvement is expected from further tuning parameters in the “Perfect Model” environment, the LEKF is ready for testing with real observations
- Based on the “Perfect Model” experiments it is expected that the LEKF analyses will have the highest quality in the extra-tropic (where a modest size ensemble, and a modest number of observations are sufficient), while  analyses in the Tropics will improve with increased observational density. Some modest improvement can be expected when the ensemble size is increased

## “Near Future” Work

- Implementation of the 4D extension (in progress)
- Assimilation of radiosonde observations (in progress)
- Implementation on the NCEP RSM (in progress)
- Surface Analysis (your help is wanted!)
- Implementation on the NASA model (starts soon)
- Handling of model errors (in the theoretical phase)
- Direct minimization of the cost function (in the theoretical phase)