Local Predictability of the Performance of an Ensemble Forecast System

1. Local Predictability of the Performance of an Ensemble Forecast System Liz Satterfield and Istvan Szunyogh University of Maryland, College Park Institute for Physical Science and Technology & Department of Atmospheric and Oceanic Science AMS Annual Meeting Phoenix, AZ January 2009

2. Definition of the Problem • Ensemble prediction systems account for the influence of spatio-temporal changes in predictability on forecasts • Performance of an ensemble prediction system is flow dependent

3. Goals • To illustrate and assess this flow dependence • To lay the theoretical foundation of a practical approach to predict spatio-temporal changes in the performance of an ensemble prediction system

4. Experiment Design • The Local Ensemble Transform Kalman Filter (LETKF) of the University of Maryland is used: • to obtain imperfect state estimates (analyses) by assimilating noisy simulated observations of the atmospheric state • to provide initial conditions (analysis perturbations) for an ensemble of forecasts

5. Experiment Design • Model: T62L28 resolution version of the NCEP GFS • Observations: • Simulated Observations in Random Location: 2000 randomly placed vertical soundings that provide 10% coverage of model grid points (Kuhl et al. 2007, JAS). • Simulated Observations at the Location of Conventional Observations: Observational noise added to “true states”, location and type taken from conventional observations • Conventional Observations of the Real Atmosphere: Observations used to obtain the type and location for simulated observations (excludes satellite radiances) • Time Period: 1 January 2004 0000 UTC to 15 February 2004 0000 UTC

6. An Interesting Result for Simulated Observations in Random Locations 0ºS 160ºW Digital Filter Wipes Out Semi-diurnal Tidal Signal

7. Explained Variance: A measure of ensemble performance in the local region • b: true error • a: projection of the true error Explained Variance = ||a||/||b|| Fraction of forecast error contained in the space spanned by the ensemble true state Maximum value of 1 when the ensemble correctly captures the space of uncertainty X(1) b X(3) a Xa,f Plane of ensemble perturbations for the local state vector X(2)

8. E-Dimension:A local measure of complexity based on eigenvalues of the ensemble-based error covariance matrix in the local region Variance confined to a single spatial pattern of uncertainty Variance is evenly distributed between N independent spatial patterns of uncertainty Three perturbations in one plane Three orthogonal perturbations of unequal magnitude Three equal magnitude orthogonal perturbations E-dimension=1 2 < E-dimension < 3 E-dimension=3 Introduced in Patil et al. 2001, PRL; discussed in detail in Oczkowski et al., 2005, JAS

9. Local Regions defined for each model grid point • The local region is an atmospheric column • For computation of E-dimension and explained variance, we consider grid point values of two horizontal components of wind, temperature and surface pressure • Forecast error is computed for the meridional component of wind at 500 hPa 10 hPa 1000km 500 hPa Surface Pressure

10. Joint Probability Distribution Function (JPDF) for Explained Variance and Forecast Error shown for NH extratropics Simulated observations in realistic locations Observations of the real atmosphere Colors show mean E-dimension Colors show mean E-dimension For high forecast errors the ensemble does a good job of capturing the space of uncertainties Lower E-Dimension Strong Instabilities Higher Forecast Error

11. Joint Probability Distribution Function (JPDF) for Explained Variance and Forecast Error shown for conventional observations of the real atmosphere for NH extratropics Linear space provides an increasingly better representation of the space of uncertainty up to 120 hours Increased forecast lead time

12. Local Relative Nonlinearity a measure of linearity in the local regions  = || xa,f-xa,f|| / ___(1/k)||xa,f(k)|| Time mean of globally averaged values for conventional observations Modified from Gilmour et al (2001) Standard deviations of values computed using localization show a high degree of variability Local Regions Global

13. Correlation between relative nonlinearity and explained varianceshown for conventional observations High values of explained variance at the 120 hour lead time are not due to strong linearity

14. Conclusions • The approach of using a digital filter in an Ensemble Kalman Filter may need to be reconsidered (filter increments instead of the full field) • In the extratropics, the ensemble does a better job of capturing forecast error when forecast error is high. We find this result to hold for both perfect model and the real atmosphere (we explained this behavior by using the E-dimension diagnostic) • The linear space spanned by the ensemble perturbations provides an increasingly better representation of the space of uncertainty up to the 120 hour lead time (although the forecast evolution becomes more nonlinear)