Assessing Precipitation Extremes in the COLA C20C Simulations Using L-Moments

1. Assessing Precipitation Extremes in the COLA C20C Simulations Using L-Moments L. Marx and J. L. Kinter III Center for Ocean-Land-Atmosphere Studies Third C20C Workshop - ICTP, Trieste, Italy 20 April 2004

2. Outline • Are precipitation distributions, including means and extremes, in C20C integrations comparable to the observed? • Introduction to L-moments • An alternative to assuming a normal distribution • Estimating the moments • Fitting a standard distribution and testing the fit with Monte Carlo simulation • Application to the COLA AGCM and analyzed precipitation observations

3. COLA C20C AGCM v2.2 • T63L18 resolution • NCAR CCM3 spectral dynamics with semi-Lagrangian moisture transport • Simple SiB land surface model • Relaxed Arakawa-Schubert convection; Tiedtke shallow convection • Harshvardhan longwave radiation; Lacis and Hansen shortwave radiation • Mellor-Yamada 2.0 turbulence closure • Predicted supersaturation and convective clouds • Gravity wave drag

4. COLA C20C Experiment • 10 ensemble members • Nov1948 - Nov2003 (NCEP/NCAR reanalysis initial states from 00Z, 10-19 Nov 1948) • Hadley Centre HadISST1 SST and sea ice • Climatological initial snow depth, soil moisture and temperature; predicted thereafter • Land vegetation values from satellite-based climatology • CAMS precipitation analysis from land-based gauges used as verification data

5. December Means CAMS COLA ensemble mean Observed and 10-member COLA AGCM C20C ensemble mean December mean Europe precip are comparable … what about extremes?

6. December Top Decile CAMS COLA ensemble mean Top 10% averages: Is this comparison meaningful, given the fact that the model result is an ensemble average while Nature has only one realization?

7. Brief Intro to L-Moments The L-moments of a sample distribution of n elements, sorted in ascending order as: X1:n, X2:n, … Xn:n are defined as  E (X1:1) 21/2E (X2:2 - X1:2) 31/3E (X3:3 - 2 X2:3 + X1:3) 41/4E (X4:4 - 3 X3:4 + 3 X2:4 - X1:4) L-moments defined in this way maintain certain mathematical and statistical properties analogous to the uniform distribution Hosking and Wallis, 1997 Regional Frequency Analysis: An Approach Based on L-Moments

8. Intro to L-Moments (cont.) The first L-moment, 1is obviously the mean of the distribution. Statistics analogous to other moments of the normal distribution may be defined as follows: L-CV = 2/ 1 (coefficient of L-variation) 3 = 3/ 2 (L-skewness) 4 = 4/ 2 (L-kurtosis)

9. Estimating L-moments If we define a quantity br as br = n-1i [(i-1)(i-2)•••(i-r)] / [(n-1)(n-2) •••(n-r)] Xi:n then the L-moments can be estimated as follows: b0 22b1 - b0 36b2 - 6b1+ b0 420b3 - 30b2 + 12 b1 -b0 L-CV, 3 and 4 can then be defined based on these estimates.

10. Properties of L-moments • Due to linearity, all L-moments higher than the first are more robust than their normally-defined counterparts • The L-CV is bounded between 0 and 1, and 3 and 4 are bounded between -1 and 1, unlike the normally-defined skewness and kurtosis which are unbounded • For a purely normally distributed variable: • =  • 2=    • 3= 0 • 4= 0.1226

11. Fitting Distributions • Plotting distributions in 3 - 4 space makes it possible to see how they differ • Two-parameter distributions (e.g. normal) are represented as points in this space; more general distributions are curves • Fitting particular distributions to sample data and choosing the best fit is accomplished by a modification of the Hosking and Wallis “index flood” procedure that involves pooling data and Monte Carlo simulation to estimate goodness of fit Hosking and Wallis, 1997

12. A word about sample size … N = 20: N = 80: Hosking and Wallis, 1997

13. One advantage of L-moments: Hosking and Wallis, 1997

14. One advantage of L-moments: better separation of distributions … Hosking and Wallis, 1997

15. CAMS • COLA

16. CAMS • Nearly all points in Europe clustered • between Normal & Gumbel distributions Normal Gumbel Normal Gumbel • COLA • Much broader distribution

17. CAMS • Nearly all points in Europe clustered • between Normal & Gumbel distributions • Almost no points with L-skewness < 0 Normal Gumbel Normal Gumbel Negative Skewness • COLA • Much broader distribution • Many points in Europe have • L-skewness < 0

18. EM 0.03 OBS 0.14 EM -0.09 OBS 0.18 M9 -0.13 M2 -0.05 M6 0.10 M7 0.07 (2E,48N) (2E,60N) Selected points show: Ensemble mean is poor representation of distribution Some members do a credible job of matching the obs 3 < 0 indicates lack of representation of upper end of distribution

19. Relative Errors of Fit Normal Distribution Gumbel Distribution GEV Distribution Lowest 20% 2nd quintile 3rd quintile 4th quintile Highest 20%

20. (2) CAMS COLA-EM

21. (2) Ensemble mean suppresses variability determined in this way CAMS COLA-10 COLA-EM

22. L-CV Similar to (2) except that mean bias reduces amplitude CAMS COLA-10 COLA-EM

23. L-Skewness CAMS COLA-EM

24. L-Skewness Ensemble mean exagerrates negative skewness problems CAMS COLA-10 COLA-EM

25. L-Kurtosis Sample size probably too small, even with 10 ensemble members CAMS COLA-10 COLA-EM

26. Conclusions • L-moments analysis is a promising technique for quantifying precipitation distributions • Provides a method for distinguishing among distributions  can help diagnose model errors • In particular, 3< 0 indicates that high value extremes are missing • Provides more robust estimates of characteristics of variability  can be used in place of more traditional statistical measures • Ensemble averaging masks considerable richness in the variability and distribution of precipitation