local predictability of the performance of an ensemble forecast system liz satterfield and istvan...
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Local Predictability of the Performance of an
Ensemble Forecast System
Liz Satterfield and Istvan SzunyoghTexas A&M University, College Station, TX
Third THORPEX International Science SymposiumMonterey California, 14-18 September 2009
Introduction Ensemble prediction systems account for the
influence of spatio-temporal changes in predictability on forecasts
Performance of an ensemble prediction system is flow dependent
The goal of our study is to lay the theoretical foundation of a practical approach to predict spatio-temporal changes in the performance of an ensemble prediction system
Experiment Design We use an implementation of the Local Ensemble
Kalman Filter (LETKF) on T62L28 resolution version of the NCEP GFS
Experiments with 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)
Linear Diagnostics calculated in local regions using energy rescaling Explained Variance Fraction of forecast error contained in the space
spanned by the ensemble Minimum value of zero when the error lies orthogonal to the space
spanned by ensemble perturbations Maximum value of 1 when the ensemble correctly captures the space
of uncertainty
E-Dimension A local measure of complexity based on eigenvalues of the ensemble-based error covariance matrix in the local region (Introduced in Patil et al. 2001)
Minimum value of 1 when the variance is confined to a single spatial pattern of uncertainty
Maximum value of N when the variance is evenly distributed between N independent spatial patterns of uncertainty
Relationship between Explained Variance, E-Dimension, and Forecast Error shown for conventional observations
Lower E-Dimension
Higher Forecast ErrorStrong Instabilities
Linear space provides an increasingly better representation of the space of uncertainty up to 120 hours
Colors show mean E-dimension
Joint ProbabilityDistribution
Local Relative Nonlinearity a measure of linearity in the local regions
= || xa,f-xa,f|| / ___(1/k)||xa,f(k)||Modified from Gilmour et al (2001)
Standard deviations of values computed using localization show a high degree of variability
Time mean of globally averaged values for conventional observations
Local Regions
Global
Distance between ensemble mean and control forecasts normalized by the average perturbation magnitude
Forecast Lead Time
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
12 24 36 48 60 72 84 96 108 120 132
Forecast Lead Time
Co
rrel
atio
n
Northern Hemisphere
Correlation between relative nonlinearity and explained variance shown for conventional observations
High values of explained variance at the 120 hour lead time are not due to strong linearity of the evolution of uncertainties
TV = Square of the magnitude of the error in the ensemble mean forecast
TVs =Portion of TV which lies in the space spanned by the ensemble perturbations
Vs = ensemble variance
Evolution of Forecast Error shown for randomly placed simulated observations
Forecast Lead Time
For a perfect ensemble, TV and TVs would equal Vs at the initial time
At initial time, TVs equals Vs therefore further inflating the variance would not improve analyses
Evolution of Forecast Error results shown for the Northern Hemisphere Extratropics
Simulated obsRealsitic location
Conventionalobs.
TV
Vs
Forecast Lead TimeForecast Lead Time
The total ensemble variance underestimates the forecast error captured by the ensemble
TVs
Spectrum of the Ratio Between Observed and Predicted Probability (d-ratio) at analysis time Modified from Ott et al (2002)
Simulated obsRandom location
Simulated obsRealsitic location
Conventionalobs.
eigen-direction
Optimal performance in this measure would be indicated by 1 for all k
Uncertainty is underestimated
Uncertainty is overestimated
dk=(xtk)2/k
Spectrum of the Ratio Between Observed and Predicted Probability (d-ratio) at 120-hour lead time
Conventional obs
Simulated obsRealsitic location
Simulated obsRandom location
By 120-hr lead time, the ensemble underestimates uncertainty in all directions
The spectrum is steepest for observations of the real atmosphere
di < 1 : ensemble overestimates
di > 1 : ensemble underestimates
eigen-direction
The leading direction of d-ratio calculated for temperature at 850hPa
For realistically placed observations, the regions of largest underestimation are those of highest observation density
Simulated, Realistically Placed Obs. of the real atmosphere
Conclusions The linear space spanned by ensemble perturbations
provides an increasingly better representation of the space of uncertainties with increasing forecast time.
The improving performance of the space of ensemble perturbations with increasing forecast time is not due to local linear error growth, but rather to nonlinearly evolving forecast errors that have a growing projection on the linear space.
At analysis time, we find that the ensemble typically underestimates uncertainty more severely in regions of high observation density than for regions of low observation density. This result indicates that implementing a spatially varying adaptive covariance inflation technique may improve analyses.
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