1. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 1 Preliminary Analysis of ABFM Data WSR 11 x 11-km Average Harry Koons 30 October 2003

2. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 2 Scatter Plot of dBZ vs Emag WSR 11 x 11-km Average

3. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 3 Approach Objective is to determine the probability of an extreme electric field intensity for a given radar return Use the statistics of extreme values to estimate the extreme electric field intensities –Reference: Statistical Analysis of Extreme Values, Second Edition, R. -D. Reiss and M. Thomas, Birkhäuser Verlag, Boston, 2001 As an example analyze WSR 11x11- km Average data Use both Peaks over Threshold (POT) and Maximum out of Blocks (MAX) methods Determine extreme value distribution functions for two-dBZ wide bins –For example the 0-dBZ bin is defined to be the range:  1 < dBZ < +1

4. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 4 Extreme Value Methods Parametric models –  n,  n,  n,  n Parameters strictly hold only for the sample set analyzed Basic assumption is that the samples, x i, come from independent, identically distributed (iid) random variables –In our experience the techniques are very robust, as verified by the Q-Q plots, when this assumption is violated Analysis Methods –Peaks Over Threshold (POT) Take all samples that exceed a predetermined, high threshold, u. Exceedances over u fit by Generalized Pareto (GP) distribution functions –Maxima Out of Blocks (MAX) Take the maximum value within a pass, Anvil, etc. Tail of distribution is fit by extreme value (EV) distribution functions

5. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 5 Generalized Pareto (GP) Distribution Functions (Peaks-over-Threshold Method) Exponential (GP0,  = 0): Pareto(GP1,  > 0): Beta (GP2,  < 0): The  – Parameterization:

6. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 6 Standard EV Distribution Functions (Maximum out of Blocks Method) Gumbel (EV0,  = 0): Fréchet (EV1,  > 0): Weibull (EV2,  < 0): The  – Parameterization:

7. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 7 Tutorial Example Based on a Manufactured Gaussian (Normal) Distribution Function Select 100 random samples uniformly from a normal distribution –Mean value,  = 5 –Standard deviation,  = 1 Maximum Likelihood Estimates for a normal model are the sample mean and sample standard deviation –  = 4.81789 –  = 1.05631 In general you maximize the likelihood function by taking appropriate partial derivates

8. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 8 Distribution function, F, of a real- valued random variable X is given by F (X) = P{ X  x } Histogram The density function, f, is the derivative of the distribution function –Sample Density –Model Density –Kernel Density Distribution and Density Functions

9. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 9 Quantile Functions and the Q-Q Plot

10. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 10 T-Year Values T-year level is a higher quantile of the distribution function T-year threshold, u(T), is the threshold such that the mean first exceedance time is T years –u(T) = F -1 (1-1/T) –1-1/T quantile of the df The T-year threshold also is exceeded by the observation in a given year with the probability of 1/T

11. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 11 Scatter Plot of dBZ vs Emag WSR 11 x 11-km Average; 0 dBZ bin

12. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 12 Cumulative Frequency Curve for Emag –1 < dBZ < +1

13. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 13 Sample Statistics -1 < dBZ < +1 Sample Size, N = 271 Minimum = 0.02 kV/m Maximum = 2.41 kV/m Median = 0.6 kV/m Mean = 0.67 kV/m Choose Peaks Over Threshold (POT) Method for Extreme Value Analysis u = 1.0 kV/m (high threshold) n = 57 (samples over threshold) k = 32 (mean value of the stability zone)

14. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 14 Gamma Diagram Point of stability is on The plateau between 30 and 40

15. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 15 Kernel Density Function and Model Density Function Peaks Over Threshold Method (POT) –N = 271 samples between –1 and +1 dBZ –u = 1.0 kV/m (high threshold) –n = 57 (samples over threshold) MLE (GP0) – k = 32 (mean zone of stability) –  = 0.0 –  = 1.00457 (~left end point) –  = 0.30387

16. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 16 Sample and Model Distribution Functions

17. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 17 Sample and Model Quantile Functions

18. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 18 Q – Q Plot Sample Model

19. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 19 T-Sample Electric Field Intensity -1 < dBZ < +1 T, SamplesProbability, p Quantile,  p kV/m 1000.011.74 1,0000.0012.43 10,0000.00013.12 100,0000.000013.81 1,000,0000.0000014.50 10,000,0000.00000015.20

20. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 20 Extend Analysis to Other Bins Use bins centered at –2, 0, +2, and +4 dBZ Kernel Density Plot Sample Statistics Model Parameters T-Sample Electric Field Intensity

21. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 21 Sample Kernel Densities for Exceedances Blk: -2 dBZ Red: 0 dBZ Grn: +2 dBZ Blu: +4 dBZ Emag kV/m

22. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 22 Sample Statistics and Model Parameters -2 dBZ0 dBZ+2 dBZ+4 dBZ u, kV/m0.751.0 N173271369418 k24325784  0.0  0.811.001.061.09  0.150.300.280.37

23. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 23 T-Sample Electric Field Intensity, kV/m T-Sample-2 dBZ0 dBZ+2 dBZ+4 dBZ 1001.211.741.822.20 1,0001.562.432.473.05 10,0001.913.123.113.90 100,0002.263.813.754.76 1,000,0002.614.504.395.61 10,000,0002.965.205.056.46

24. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 24 Maximum Out of Blocks Method #1 One Block is One Pass Analyze the 0 dBZ bin 38 Blocks (unique pass numbers) Maximum Emag for each block shows a slight dependence on sample-count per block –We will ignore this MLE (EV) Model Parameters  = 0.138  = 0.634  = 0.322

25. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 25 Sample Count vs. Pass Number Max for Passes

26. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 26 Number of Occurrences vs. Sample Count Max for Passes

27. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 27 Plot of Maximum Emag vs. Sample Count MAX for Passes

28. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 28 Density Functions MAX for Passes

29. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 29 Q-Q Plot MAX for Passes

30. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 30 T-Pass Electric Field Intensity -1 < dBZ < +1 TEmag, kV/m 502.30 1002.70 2003.14 5003.80 1,0004.35 2,0004.96 5,0005.85 10,0006.61

31. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 31 Maximum Out of Blocks Method #2 One Block is One Anvil Analyze the 0 dBZ bin 21 Blocks (unique anvil numbers) MLE (EV) Model Parameters  = -0.044  = 0.830  = 0.431 Right End Point = 10.5 kV/m

32. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 32 Density Functions MAX for Anvils

33. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 33 Distribution Functions MAX for Anvils

34. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 34 Q-Q Plot MAX for Anvils

35. For information contact H. C. Koons E-Mail: hkoons@aero.org 30 October 2003 35 T-Anvil Electric Field Intensity -1 < dBZ < +1 TEmag, kV/m 502.37 1002.62 2002.86 5003.17 1,0003.39 2,0003.61 5,0003.89 10,0004.09