Assessing Predictability with a Local Ensemble Kalman Filter

Istvan Szunyogh

“Chaos-Weather Team”

University of Maryland College Park

SAMSI DA Workshop, October 5, 2005

Components• Data assimilation scheme: Local Ensemble (Transform)

Kalman Filter (Ott et al. 2002, 2004; Hunt 2005) implemented by Eric Kostelich (ASU) and I. Sz.

• Model: Operational Global Forecast System (GFS) of the National Centers for Environmental Prediction/National Weather Service (Perfect model scenario)

• Model Resolution: T62 (~150 km) in the horizontal directions and 28 vertical level dimension of the state vector:1,137,024; dimension of the grid space (analysis space): 2,544,768]

• Observations: Uniformly distributed vertical soundings of wind, temperature and surface pressure (distribution of analysis/forecast errors is not affected by distribution of data coverage)

Local VectorsIllustration on a two dimensional grid

• The state estimate is updated at the center grid point

• The background state is considered only from a local region (yellow dots)

• All observations are considered from the local region (purple diamonds)

• The components of the local vectors are the grid-point variables at the yellow locations

E-dimensionA local measure of complexity

Illustration in 2D model grid space

1 Number of EnsembleMembers-1

E-dimension

Complexity:

Based on the eigenvalues

A spatio-temporallychanging scalar valueis assigned to each grid point

Introduced byPatil, Hunt et al. (2001)Studied in details byOczkowski et al (2005)

€

σiof the ensemble based estimate of the local covariancematrix:

€

€

2

i1/2σ

i∑ ⎛ ⎝ ⎜ ⎞

⎠ ⎟ iσ

i

∑

The more unevenly distributed the variance inthe ensemble space, the lower the E-dimension

Explained (Background) Error Variance

Illustration for a rank-2 covariance matrix (3-member ensemble)

Background mean

True state

Eigenvector 1

Eigenvector 2

b

be

Explained Variance: be2/ b2

Projection onto the plane of theeigenvectors

A perfect explained variance of 1 implies that the space of uncertainties iscorrectly captured by the ensemble, but it does not guarantee that the distribution of the variance within that space is correctly represented by the ensemble

Experiment

• Number of ensemble members: 40 • Local regions: 7x7xV grid point cubes;

V=1, 3, 5, 7 • Variance Inflation: Multiplicative,

uniform 4% (needed to compensate for the loss of variance due to nonlinearities and sampling errors)

• Observations: 2000 vertical soundings

Depth of Local Cubes

Mid-troposphere

Lower troposphere

Upper troposphere

Lower stratosphere

Dimension of Local

State Vector ~1,700

Time evolution of errors

analysis cycle (time)

Rmsanalysiserror

surface pressure

Observational error

The error settles at a similarly rapid speed for all variables15-days (60 cycles) is a safe upper bound estimate for the transient

Evolution of the Forecast Errors

45-day mean

As forecast timeincreases the extratropicalstorm track regions becomethe regions oflargest error

D. Kuhl et al.

Evolution of the E-dimension

The E-dimension rapidlyDecreases in the stormTrack regions

The error growth andthe decrease of theE-dimension is closelyrelated

D. Kuhl et al.

Evolution of the Explained Variance

The explained variance isthe largest in the storm trackregions and it increases withtime

Large error growth, low E-dimension, and large explained varianceare closely related

There seems to exist a ‘localanalogue’ to the unstable subspace

D. Kuhl et al.

E-d

imen

sion

Explained Variance

The scatter plots confirmthe increasingly closecorrespondence betweenlow E-dimensionality andhigh explained variance(improving ensemble performance)

D. Kuhl et al.

Time Mean Evolution of the Forecast Errors

0

0.5

1

1.5

2

2.5

3

3.5

4

0 12 24 36 48 60 72 84 96 108 120 132

Forecast (Hour)

Average Forecast Error

Extra Trop. NH

Extra Trop. SH

Tropics (linear growth)

(exponential growth)

Curves fitted forFirst 72 hours

The error doubling time in the extratropics is about35-37 hours

D. Kuhl et al.

Conclusions • For the LEKF data assimilation scheme the analysis errors

are the smallest where the growth of the forecast errors is the fastest

• This can be explained by the (i) strong anti-correlation between local dimensionality and the background error variance explained by the ensemble and by that (ii) the regions of local low dimensionality are the regions of most rapid error growth

• These results were obtained for a perfect model and homogeneous data coverage; model errors and the uneven distribution of observations can distort this behavior in practice

References• Kuhl, D., I. Szunyogh, E. J. Kostelich, G. Gyarmati, D.J. Patil, M. Oczkowski, B. Hunt,

E. Kalnay, E. Ott, J. A. Yorke, 2005: Assessing predictability with a Local Ensemble Kalman Filter (submitted)

• Szunyogh, I, E. J. Kostelich, G. Gyarmati, D. J. Patil, B. R. Hunt, E. Kalnay, E. Ott, and J. A. Yorke, 2005: Assessing a local ensemble Kalman filter: Perfect model experiments with the NCEP global model. Tellus 57A, 528-545.

• Oczkowski, M., I. Szunyogh, and D. J. Patil, 2005: Mechanisms for the development of locally low dimensional atmospheric dynamics. J. Atmos. Sci., 1135-1156.

• Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, J. A. Yorke, 2004: A local ensemble Kalman Filter for atmospheric data assimilation.Tellus 56A , 415-428.

• Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, and J. A. Yorke, 2004: Estimating the state of large spatio-temporally chaotic systems. Phys. Lett. A., 330, 365-370.

• Patil, D. J., B. R. Hunt, E. Kalnay, J. A. Yorke, and E. Ott, 2001: Local low dimensionality of atmospheric dynamics, Phys. Rev. Let., 86, 5878-5881.

• Reprints and preprints of papers by our group are available at http://keck2.umd.edu/weather/weather_publications.htm