Assessing Predictability with a Local Ensemble Kalman Filter

1. Assessing Predictability with a Local Ensemble Kalman Filter Istvan Szunyogh “Chaos-Weather Team” University of Maryland College Park SAMSI DA Workshop, October 5, 2005

2. 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)

3. 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

4. E-dimension A local measure of complexity Illustration in 2D model grid space A spatio-temporally changing scalar value is assigned to each grid point Based on the eigenvalues of the ensemble based estimate of the local covariance matrix: Introduced by Patil, Hunt et al. (2001) Studied in details by Oczkowski et al (2005) Complexity: E-dimension 1 Number of Ensemble Members-1 The more unevenly distributed the variance in the ensemble space, the lower the E-dimension

5. Explained (Background) Error Variance Illustration for a rank-2 covariance matrix (3-member ensemble) True state Eigenvector 1 Explained Variance:  be2/  b2 b Projection onto the plane of the eigenvectors be Background mean Eigenvector 2 A perfect explained variance of 1 implies that the space of uncertainties is correctly captured by the ensemble, but it does not guarantee that the distribution of the variance within that space is correctly represented by the ensemble

6. 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

7. Depth of Local Cubes Lower stratosphere Upper troposphere Dimension of Local State Vector ~1,700 Mid-troposphere Lower troposphere

8. Time evolution of errors surface pressure Observational error Rms analysis error analysis cycle (time) The error settles at a similarly rapid speed for all variables 15-days (60 cycles) is a safe upper bound estimate for the transient

9. Evolution of the Forecast Errors As forecast time increases the extratropical storm track regions become the regions of largest error 45-day mean D. Kuhl et al.

10. Evolution of the E-dimension The E-dimension rapidly Decreases in the storm Track regions The error growth and the decrease of the E-dimension is closely related D. Kuhl et al.

11. Evolution of the Explained Variance The explained variance is the largest in the storm track regions and it increases with time Large error growth, low E-dimension, and large explained variance are closely related There seems to exist a ‘local analogue’ to the unstable subspace D. Kuhl et al.

12. The scatter plots confirm the increasingly close correspondence between low E-dimensionality and high explained variance (improving ensemble performance) E-dimension Explained Variance D. Kuhl et al.

13. Time Mean Evolution of the Forecast Errors (exponential growth) (linear growth) Curves fitted for First 72 hours The error doubling time in the extratropics is about 35-37 hours D. Kuhl et al.

14. 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

15. 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