Preliminary Analysis of ABFM Data WSR 11 x 11-km Average

1. Preliminary Analysis of ABFM Data WSR 11 x 11-km Average Harry Koons 30 October 2003

2. Scatter Plot of dBZ vs EmagWSR 11 x 11-km Average

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. Extreme Value Methods • Parametric models • n, n, n, n • Parameters strictly hold only for the sample set analyzed • Basic assumption is that the samples, xi , 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. Generalized Pareto (GP) Distribution Functions(Peaks-over-Threshold Method) • Exponential (GP0,  = 0): • Pareto(GP1,  > 0): • Beta (GP2,  < 0): • The  – Parameterization:

6. Standard EV Distribution Functions(Maximum out of Blocks Method) • Gumbel (EV0,  = 0): • Fréchet (EV1,  > 0): • Weibull (EV2,  < 0): • The  – Parameterization:

7. Tutorial Example Based on a ManufacturedGaussian (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. Distribution and Density Functions • 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

9. Quantile Functions and the Q-Q Plot

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. Scatter Plot of dBZ vs EmagWSR 11 x 11-km Average; 0 dBZ bin

12. Cumulative Frequency Curve for Emag–1 < dBZ < +1

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) Methodfor Extreme Value Analysis • u = 1.0 kV/m (high threshold) • n = 57 (samples over threshold) • k = 32 (mean value of the stability zone)

14. Gamma Diagram Point of stability is on The plateau between 30 and 40

15. Kernel Density Functionand 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) • g = 0.0 • m = 1.00457 (~left end point) • s = 0.30387

16. Sample and Model Distribution Functions

17. Sample and Model Quantile Functions

18. Q – Q Plot Sample Model

19. T-Sample Electric Field Intensity-1 < dBZ < +1

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. Sample Kernel Densities for Exceedances Blk: -2 dBZRed: 0 dBZGrn: +2 dBZBlu: +4 dBZ Emag kV/m

22. Sample Statistics and Model Parameters

23. T-Sample Electric Field Intensity, kV/m

24. Maximum Out of Blocks Method #1One 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 g = 0.138 m = 0.634 s = 0.322

25. Sample Count vs. Pass NumberMax for Passes

26. Number of Occurrences vs. Sample CountMax for Passes

27. Plot of Maximum Emag vs. Sample CountMAX for Passes

28. Density FunctionsMAX for Passes

29. Q-Q PlotMAX for Passes

30. T-Pass Electric Field Intensity-1 < dBZ < +1

31. Maximum Out of Blocks Method #2One Block is One Anvil • Analyze the 0 dBZ bin • 21 Blocks (unique anvil numbers) • MLE (EV) Model Parameters g = -0.044 m = 0.830 s = 0.431 Right End Point = 10.5 kV/m

32. Density FunctionsMAX for Anvils

33. Distribution FunctionsMAX for Anvils

34. Q-Q PlotMAX for Anvils

35. T-Anvil Electric Field Intensity-1 < dBZ < +1