GLOVER_NFFP_04_Tech_Rpt

1. 2004 NASA FACULTY FELLOWSHIP PROGRAM JOHN F. KENNEDY SPACE CENTER UNIVERSITY OF CENTRAL FLORIDA STATISTICAL FORECASTING OF LIGHTNING CESSATION James Ervin Glover, Ph.D. Associate Professor Department of Computer Science and Mathematics Oral Roberts University KSC Colleague: Francis J. Merceret CCAFS/45WS Colleague: William P. Roeder ABSTRACT This study developed new methods for applying statistical curve fitting to slowing lightning flash rates in decaying thunderstorms. The curves are then utilized to estimate the probability of one or more lightning flashes occurring by observing cloud-to-ground lightning in selected thunderstorms within a ten nautical mile radius of Cape Canaveral Air Force Station. H-1

2. STATISTICAL FORECASTING OF LIGHTNING CESSATION Jim E. Glover 1. INTRODUCTION Problem/Objective: Forecasting the end of lightning is one of the most important operational weather challenges at Cape Canaveral Air Force Station (CCAFS) and NASA Kennedy Space center (KSC). The Air Forces 45th Weather Squadron (45 WS) provides comprehensive meteorological services to operations at these locations. There is little objective guidance on how to predict the end of lightning, so 45 WS must be conservative in ending their lightning advisories to ensure personnel safety. Unfortunately, this costs a lot of money and delays processing to meet space launch schedules. One of the 45 WS strategic goals is to end lightning advisories more quickly, while maintaining safety. The 45 WS teamed with KSC to begin the Statistical Forecasting of Lightning Cessation project, which was funded under the NASA Faculty Fellowship Program of 2004. Phase-1 of the project had two goals: 1) create a climatology of the times between last and next-to-last lightning flashes, and 2) determine if curve fitting to lightning flash rates in the decaying phase of thunderstorms could be used to predict the time when the probability of another flash drops below some low operational threshold. The first goal was designed to provide some immediate object guidance to the forecasters on ending lightning advisories. The second goal was designed as a proof-of-concept for future techniques. The long-term vision is a system that will automatically analyze lightning flashes from individual storms in real-time, identify which are decaying and display the time until the probability of another flash falls below some low operational threshold, flagging which ones have already met that threshold, so that their lightning advisories should be considered for cancellation. 2. Project Summary 45 WS needs to end lightning advisories sooner, while maintaining safety. This is a very tough, technical challenge. For safety, 45WS has to be conservative in ending lightning advisories. This proves to be costly to CCAFS/KSC ground processing operations and solving the problem is one of the strategic goals of 45WS for 2004. Long-Term Vision: Development of software that will automatically analyze lightning flash-rates for local storms in real-time and flag the storms that are decaying and the storms whose advisories should be considered for canceling. 3. Methodology Restricted analysis to CGLSS data (only cloud-to-ground lightning was considered) 58 convective season storms were analyzed A 59th composite storm was created, consisting of the mean of all 58 storm strike-rates over each unit time interval Sample Years Year Number of Storms 1999 12 2000 13 2001 14 2002 7 2003 12 H-2

3. Two Approaches 1. Climatological distribution of times between last flash and next to last flash 2. Curve fitting of slowing flash rates in the decaying phase of a thunderstorm Climatological Distribution Method Times between last two flashes Generates empirical chart Provides probability that most recent flash is last one, given time history Also gives time interval corresponding to a specified probability of another flash Forecaster can use to determine when to cancel Phase II Curve-Fitting Method Slowing flash rates in decaying phase Derive probability that there will be at least one more flash in a specified time Derive probability that most recent flash is the last one 4. Theoretical Background Let the continuous random variable T with probability density function f(t) represent the time to failure (cessation) of lightning strikes in a storm. The reliability at time t, R(t) of the storm is the probability that it will continue to last (strike) for at least a specified time. ),(1 )(1)( )()( 0 tF dttfdttf tTPtR t t = == >= (1) where F(t) is the cumulative distribution of T, with F(t)=f(t). )( )()( )( tR tFttF ttimetosurvivedstormtheift]t[t,inceasewillstormP + =+ (2) The failure rate, Z(t), is given by: H-3

4. . )( )( )(1 )( )( )(' )( 1)()( lim )( )()( lim 0 0 tR tf tF tf tR tF tRt tFttF t tR tFttF Z(t) t t = = = + = + = (3) This expression shows the failure rate in terms of the distribution of the time to failure. Since R(t)=1-F(t), and R(t)= F(t), we get the differential equation ( ))(ln )( )(' )( tR dt d tR tR tZ == . (4) Solving by integration yields: ,)( )( ln)()(ln )( ln)( = = += + dttZ cdttZ ectR etR cdttZtR (5) where R(0)=1, F(0)=1-R(0)=0. Hence, knowledge of either the failure rate or the density function uniquely determines the other. CG Strike Flash Rate - 30 Jul 03 0 1 2 3 4 5 6 7 8 9 Time(Z) CG Strikes Figure 1. Cloud to Ground Strike Rates H-4

5. KSC FACULTY FELLOWSHIP SUMMER 2004 STATISTICAL FORECASTING OF LIGHTNING (CESSATION) (TLast T2nd Last) NumberofEvents Best-Fit Curve (Family, Coefficients, Goodness of Fit) % Chance Of Another Flash 25% 10% 5% 01% Average Time (integrate best-fit curve) 10 min 15 min 20 min 25 min % Chance Of Another Flash 25% 10% 5% 01% Average Time (integrate best-fit curve) 10 min 15 min 20 min 25 min Hypothetical Integration form of best-fit equation t-to-infinity Figure 2. Hueristic Time Displacements PDF of DELTA-t y = -0.0469Ln(x) + 0.1493 R2 = 0.7509 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 5 10 15 20 25 DELTA-t P(DELTA-t) Series1 Log. (Series1) Figure 3. Probability Density Function for Real-Time Displacements H-5

6. Z-Score of Ln(Displacement) y = 0.7604x - 1.3254 R2 = 0.997 -1.5 -1 -0.5 0 0.5 1 1.5 0 1 2 3 4 Ln( Displacment ) Z-Score Z-Score Linear (Z-Score) Figure 4. Linearization of Time Displacements Curve-Fitting Method Slowing flash rates in decaying phase Derive probability that there will be at least one more flash in a specified time Derive probability that most recent flash is the last one time flashrate 1% Chance Of >= One More Flash (integrate best-fit curve) Time to consider canceling advisory (%-threshold TBD) Current Time When should I cancel Lightning Advisory Figure 5. Hueristic Decaying Flash Rates H-6

7. Figure 6. Reliability Curve for Strike Rates Figure 7. Reliability Curve for Composite Storm H-7

8. Figure 8. Probability Density Function for Strike Rates 5. Summary 58 local thunderstorms analyzed, leading to a composite storm Climatological distributions of Delta-t developed Available for immediate operational use Probability thresholds vs. wait-timedetermined Preliminary proof-of-concept for curve-fitting of flash rates in decaying thunderstorms to predict probabilities and timing of last flash 6. Future Plans Test techniques on independent cases Increase sample size Filter better for individual thunderstorms and their decay phase Include complex multi-cellular organized storms Expand study to utilize LDAR (all types of lightning, including lightning aloft, not just cloud-to-ground lightning as detected by CGLSS) Hope for Phase II and Phase III in summers of 2006 and 2007. H-8

9. Table 1. Matrix of reliabilities with corresponding times and mean times to failure. R(t) = 1 - F(t) = P( T > t ) for the random variable T, where T is the time to failure, or storm cessation. TMIN is the total # of minutes per event. ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- R(t) --> 50 % 45 % 40 % 30 % 20 % 15 % 10 % 5 % 3 % 1 % TMIN ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- EVENT # TIME t TO FAILURE for each R(t) ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- 1 ). 40 44 48 55 64 76 83 88 89 93 --> 98 2 ). 52 56 61 97 127 145 199 237 246 249 --> 260 3 ). 104 107 110 114 120 124 130 148 155 164 --> 208 4 ). 35 38 40 44 48 51 59 61 63 65 --> 65 5 ). 54 59 63 67 74 76 82 87 88 94 --> 104 6 ). 40 41 43 47 54 108 121 185 187 214 --> 236 7 ). 50 51 53 58 66 69 71 74 76 79 --> 88 8 ). 64 69 73 85 97 106 121 141 147 153 --> 177 9 ). 45 48 49 52 56 57 61 64 65 67 --> 74 10 ). 66 70 79 87 94 99 103 108 111 116 --> 125 11 ). 54 60 65 71 75 84 88 109 110 112 --> 114 12 ). 108 111 113 118 121 124 131 141 156 230 --> 242 13 ). 37 37 39 44 48 52 58 61 64 69 --> 70 14 ). 53 55 57 62 67 69 74 78 81 89 --> 92 15 ). 82 85 88 93 98 101 104 113 116 132 --> 157 16 ). 65 69 71 75 80 84 88 93 97 100 --> 125 17 ). 69 71 73 76 80 82 85 87 91 94 --> 101 18 ). 33 35 38 43 48 52 57 74 84 100 --> 150 19 ). 32 35 37 42 48 50 53 56 59 73 --> 100 20 ). 43 48 50 55 62 65 70 79 82 85 --> 86 21 ). 40 43 46 58 64 69 81 94 100 113 --> 117 22 ). 56 59 63 70 78 82 146 163 182 201 --> 214 23 ). 18 19 20 23 26 28 31 36 39 39 --> 40 24 ). 78 82 86 92 96 99 103 109 115 122 --> 148 25 ). 71 73 76 81 87 93 100 115 120 123 --> 150 26 ). 84 86 89 93 98 100 104 108 112 117 --> 139 27 ). 120 124 129 135 140 145 147 155 162 196 --> 203 28 ). 31 33 35 37 39 41 44 64 65 66 --> 70 29 ). 41 44 45 52 68 76 83 92 96 139 --> 166 30 ). 31 32 35 39 44 46 49 55 57 61 --> 70 31 ). 98 101 103 106 109 110 113 115 118 123 --> 124 32 ). 44 47 50 58 79 83 86 88 90 92 --> 117 33 ). 65 69 73 79 87 93 99 107 114 123 --> 165 34 ). 124 131 140 148 157 163 174 211 233 237 --> 244 35 ). 31 32 34 41 46 48 51 57 61 81 --> 127 36 ). 55 57 59 68 83 87 96 106 109 132 --> 168 37 ). 38 41 44 51 56 60 66 71 74 92 --> 106 38 ). 51 52 53 56 60 64 71 78 83 88 --> 97 39 ). 24 26 28 33 37 41 43 60 65 77 --> 104 40 ). 63 76 81 91 97 103 108 114 118 121 --> 160 41 ). 86 88 89 92 96 98 101 105 109 111 --> 113 42 ). 67 70 73 80 88 91 97 108 115 174 --> 188 43 ). 66 68 69 72 79 82 86 96 100 104 --> 108 44 ). 39 49 51 53 56 58 59 64 66 72 --> 73 45 ). 51 53 55 59 64 66 69 73 78 82 --> 90 46 ). 69 70 71 74 78 80 84 88 92 97 --> 100 47 ). 162 166 170 177 185 190 197 209 218 248 --> 276 48 ). 159 163 167 176 190 196 201 211 217 226 --> 239 49 ). 193 196 199 207 214 218 224 236 242 347 --> 376 50 ). 94 97 99 106 113 116 119 125 126 133 --> 151 51 ). 57 66 69 74 79 81 83 89 92 96 --> 115 52 ). 20 20 21 22 25 27 30 33 45 45 --> 53 53 ). 95 97 99 103 108 111 114 121 127 132 --> 145 54 ). 57 58 60 62 65 66 67 70 71 72 --> 72 55 ). 63 65 67 71 76 79 82 86 90 98 --> 103 56 ). 35 37 40 43 48 52 58 63 66 68 --> 75 57 ). 112 119 122 130 140 146 246 256 259 270 --> 303 58 ). 40 44 48 55 64 76 83 88 89 93 --> 98 59 ). 75 82 89 108 139 160 179 199 205 215 --> 219 ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- MEAN t --> 65 68 71 77 84 90 99 109 113 125 --> 141 ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- StdDev --> 43 44 44 46 47 47 54 55 55 77 ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- R(t) --> 50 % 45 % 40 % 30 % 20 % 15 % 10 % 5 % 3 % 1 % TMIN ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- Mean Duration of Storms = mu = 139 Minutes Mean Total Number of Strikes per Storm = 339 Strikes H-9

10. REFERENCES [1] Holle, Ronald L., Murphy, Martin, Lopez, Raul E, ,Distances and Times Between Cloud-to-Ground Flashes in a Storm, KMI 103-79, (2003) [2] Kedem, B and Pfeiffer, R.,Short, D.A.,Variability of Space-Time Mean Rain Rate ((1997) [3] Walpole, R.E., Myers, R.H., Probability and Statistics for Engineers and Scientists, Fourth Edition, MacMillan Publishing Company (1989) H-10

11. Appendix A: Weibull Distribution (content found at http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm) Probability Density Function The formula for the probability density function of the general Weibull distribution is (A1) where is the shape parameter, is the location parameter and is the scale parameter. The case where = 0 and = 1 is called the standard Weibull distribution. The case where = 0 is called the 2-parameter Weibull distribution. The equation for the standard Weibull distribution reduces to (A2) Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the Weibull probability density function. Figure A1. The Weibull probability density function. Cumulative Distribution Function The formula for the cumulative distribution function of the Weibull distribution is (A3) The following is the plot of the Weibull cumulative distribution function with the same values of as the pdf plots above. H-11

12. Figure A2. Plot of the Weibull cumulative distribution function. H-12

13. Appendix B: Lognormal Distribution (content found at http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm) Probability Density Function A variable x is lognormally distributed if y = ln(x) is normally distributed with ln denoting the natural logarithm. The general formula for the probability density function of the lognormal distribution is (B1) where is the shape parameter, is the location parameter and m is the scale parameter. The case where = 0 and m = 1 is called the standard lognormal distribution. The case where equals zero is called the 2-parameter lognormal distribution. The equation for the standard lognormal distribution is (B2) Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the lognormal probability density function for four values of . Figure B1. The lognormal probability density function. Cumulative Distribution Function The formula for the cumulative distribution function of the lognormal distribution is (B3) H-13

14. where is the cumulative distribution function of the normal distribution. The following is the plot of the lognormal cumulative distribution function with the same values of as the pdf plots above. Figure B2. Plot of the lognormal cumulative distribution function. The formulas below are with the location parameter equal to zero and the scale parameter equal to one. Mean Median Scale parameter m (= 1 if scale parameter not specified). Mode Range Zero to positive infinity Standard Deviation Skewness Kurtosis Coefficient of Variation H-14

15. Appendix C: Developed Software Descriptions ======================== PROGRAM A-STAR ======================== A-STAR READS A SEQUENCE OF NUMBERS OF STRIKES AT A SEQUENCE OF TIMES, t, IN UNIT INCREMENTS OF MINUTES AND DETERMINES STRIKE RATES AND TIME DISPLACEMENTS BETWEEN THE (2ND-TO-LAST AND LAST) STRIKES AND BETWEEN THE (3RD-TO-LAST AND 2ND-TO-LAST) STRIKES, TO GENERATE A CLIMATOLOGY FOR TIME DISPLACEMENTS BETWEEEN STRIKES, AND TO DETERMINE OPTIMAL TIMES FOR CANCELLING ADVISORIES, BASED UPON EMPIRICALLY COMPUTED RELIABILITIES DERIVED FROM STORM SIGNATURES. THE ROUTINE GENERATES A CUMULATIVE DISTRIBUTION F(t) AS A FUNCTION OF TIME t AND A CORRESPONDING RELIABILITY FUNCTION R(t) FOR THE RANDOM VARIABLE T, WHERE T IS THE EXPECTED OR MEAN TIME TO FAILURE (E.G., CESSATION OF LIGHTNING STRIKES) IN A MEASURED STORM. THE RELIABILITY AT TIME t IS GIVEN BY: R(t) = 1 - F(t) = P( T > t ). THE ANALYST IS QUERIED FOR A DESIRED MAXIMUM PROBABILITY (%) FOR THE EXPECTED OCCURRENCE OF ONE OR MORE LIGHTNING STRIKES AND THE ROUTINE DETERMINES THE TIME t CORRESPONDING TO THE RELIABILITY R(t). A SUMMARY FILE IS GENERATED INTO A-STAR.OT0, CONSISTING OF: STORM #, DATE OF STORM, DURATION TIME PER STORM, AND CUMULATIVE NUMBER OF STRIKES PER STORM. ================================================================== ======================== PROGRAM B-STAR ======================== B-STAR READS A SEQUENCE < t > OF VALUES OF SOME RANDOM VARIABLE T (e.g., LIGHTNING DISPLACEMENTS IN MINUTES BETWEEN 2ND-TO-LAST AND LAST STRIKES OR BETWEEN 3RD-TO-LAST AND 2ND-TO-LAST STRIKES, GENERATING A CLIMATOLOGY FOR TIME DISPLACEMENTS BETWEEEN STRIKES AND DETERMINING OPTIMAL TIMES FOR CANCELLING ADVISORIES, BASED UPON EMPIRICALLY COMPUTED RELIABILITIES DERIVED FROM STORM SIGNATURES.) THE ROUTINE GENERATES FREQUENCY/CUMULATIVE FREQUENCY DISTRIBUTIONS, BASED UPON TACIT SORTING, BINNING, AND NORMALIZATION. A RELIABILITY DISTRIBUTION R(t) IS GENERATED FOR THE RANDOM VARIABLE T, WHERE t IS THE EXPECTED OR MEAN DISPLACEMENT TIME TO FAILURE ( e.g., THE OCCURRENCE OF AT LEAST ONE MORE STRIKE TIME DISPLACEMENT HAVING SIZE EXCEEDING g(T) ) FOR SOME FUNCTION g. MOREOVER, THE ANALYST IS PROMPTED FOR A DESIRED MAXIMUM PROBABILITY (%) FOR THE EXPECTED OCCURRENCE OF AT MOST ONE MORE VALUE OF g(T) FOR WHICH T EXCEEDS t AND THE ROUTINE DETERMINES THE VALUE t CORRESPONDING TO THE RELIABILITY R(t). (NOTE, E.G., THAT IS IS NATURALLY EXPECTED THAT DECREASING NUMBERS OF STORMS WILL HAVE INCREASING NUMBERS OF TIME DISPLACEMENTS BETWEEN SUCCESSIVE LIGHTNING STRIKES.) ================================================================== H-15

16. ======================== PROGRAM C-STAR ======================== C-STAR READS A SEQUENCE OF BLOCKS OF EVENTS, EACH CONSISTING OF A SEQUENCE OF NUMBERS OF LIGHTNING STRIKES AT A SEQUENCE OF TIMES, t, IN UNIT INCREMENTS OF MINUTES AND DETERMINES STRIKE RATES AND TIME DISPLACEMENTS BETWEEN THE (2ND-TO-LAST AND LAST) STRIKES OR BETWEEN THE (3RD-TO-LAST AND 2ND-TO-LAST) STRIKES, TO GENERATE A CLIMATOLOGY FOR TIME DISPLACEMENTS BETWEEEN STRIKES, AND TO DETERMINE OPTIMAL TIMES FOR CANCELLING ADVISORIES, BASED UPON EMPIRICALLY COMPUTED RELIABILITIES DERIVED FROM STORM SIGNATURES. A RELIABILITY DISTRIBUTION IS ALSO GENERATED FOR THE RANDOM VARIABLE T, WHERE T IS THE EXPECTED OR MEAN TIME TO FAILURE (CESSATION OF LIGHTNING STRIKES) IN EACH MEASURED STORM. THIS PROCESS IS CYCLICALLY REPEATED FOR EACH STORM IN THE SEQUENCE. THE ROUTINE THEN GENERATES A MATRIX OF TIMES t CORREESPONDING TO A DESIRED SEQUENCE OF RELIABILITIES, R(t), AS WELL AS THE AVERAGE VALUE OF t ASSOCIATED WITH EACH VALUE OF R(t), WHERE R(t) = 1 - F(t) = P( T > t) AND F(t) IS THE CUMULATIVE DISTIBUTION OF T. R(t), THE RELIABILITY OF T AT TIME t, IS THE PROBABILITY (%) OF THE EXPECTED OCCURRENCE OF ONE OR MORE LIGHTNING STRIKES AFTER TIME t. ================================================================== ======================== PROGRAM D-STAR ======================== D-STAR READS A SEQUENCE OF PMAX STORM EVENTS CONSISTING OF Q0 VARYING LIGHTNING STRIKE RATES AS A FUNCTION OF TIME t IN UNIT INCREMENTS OF MINUTES AND DETERMINES A COMPOSITE STORM, CONSISTING OF V0 COMPUTED MEAN STRIKE RATES AT EACH TIME t , WHERE THE TIME DURATION OF STORMS VARIES. THE ROUTINE THEN DETERMINES TIME DISPLACEMENTS BETWEEN THE (2ND-TO-LAST AND LAST) STRIKES AND BETWEEN THE (3RD-TO-LAST AND 2ND-TO-LAST) STRIKES,GENERATING TIME DISPLACEMENTS BETWEEEN STRIKES FOR THE COMPOSITE STORM AND DETERMINING OPTIMAL TIMES FOR CANCELLING ADVISORIES, BASED UPON EMPIRICALLY COMPUTED RELIABILITIES DERIVED FROM THE STORM SIGNATURE. THE ROUTINE GENERATES A CUMULATIVE DISTRIBUTION F(t) AS A FUNCTION OF TIME t AND A CORRESPONDING RELIABILITY FUNCTION R(t) FOR THE RANDOM VARIABLE T, WHERE T IS THE EXPECTED OR MEAN TIME TO FAILURE (E.G., CESSATION OF LIGHTNING STRIKES) IN A MEASURED STORM. THE RELIABILITY AT TIME t IS GIVEN BY: R(t) = 1 - F(t) = P( T > t ). THE ANALYST IS QUERIED FOR A DESIRED MAXIMUM PROBABILITY (%) FOR THE EXPECTED OCCURRENCE OF ONE OR MORE LIGHTNING STRIKES AND THE ROUTINE DETERMINES THE TIME t CORRESPONDING TO THE RELIABILITY R(t). THE ANALYST MAY ENTER A DESIRED MAXIMUM VALUE FOR THE INTEGER-VALUED DURATION TIMES IN MINUTES FOR EACH STORM TO BE SELECTED FROM D-STAR.INP AND ANALYZED. THE MAXIMUM DIMENSIONS OF P(N,M) P(10,400), WHERE N IS THE NUMBER OF STORMS AND M IS THE NUMBER OF MINUTES OF DURATION. A FILE OF THE STORM WITH MEAN VALUES OF NUMBER OF STRIKES AT TIME t IS GENERATED INTO D-STAR.OT8. D-STAR IS A VARIANT OF A-STAR. ================================================================== H-16

17. ======================== PROGRAM E-STAR ======================== E-STAR READS A SEQUENCE OF PMAX STORM EVENTS CONSISTING OF Q0 VARYING LIGHTNING STRIKE RATES AS A FUNCTION OF TIME t IN UNIT INCREMENTS OF MINUTES AND DETERMINES A COMPOSITE STORM, CONSISTING OF V0 COMPUTED MEAN STRIKE RATES AT EACH TIME t , WHERE THE TIME DURATION OF STORMS VARIES. THE ROUTINE THEN DETERMINES TIME DISPLACEMENTS BETWEEN THE (2ND-TO-LAST AND LAST) STRIKES AND BETWEEN THE (3RD-TO-LAST AND 2ND-TO-LAST) STRIKES,GENERATING TIME DISPLACEMENTS BETWEEEN STRIKES FOR THE COMPOSITE STORM AND DETERMINING OPTIMAL TIMES FOR CANCELLING ADVISORIES, BASED UPON EMPIRICALLY COMPUTED RELIABILITIES DERIVED FROM THE STORM SIGNATURE. THE ROUTINE GENERATES A CUMULATIVE DISTRIBUTION F(t) AS A FUNCTION OF TIME t AND A CORRESPONDING RELIABILITY FUNCTION R(t) FOR THE RANDOM VARIABLE T, WHERE T IS THE EXPECTED OR MEAN TIME TO FAILURE (E.G., CESSATION OF LIGHTNING STRIKES) IN A MEASURED STORM. THE RELIABILITY AT TIME t IS GIVEN BY: R(t) = 1 - F(t) = P( T > t ). THE ANALYST IS QUERIED FOR A DESIRED MAXIMUM PROBABILITY (%) FOR THE EXPECTED OCCURRENCE OF ONE OR MORE LIGHTNING STRIKES AND THE ROUTINE DETERMINES THE TIME t CORRESPONDING TO THE RELIABILITY R(t). THE DETERMINES THE MAXIMUM STORM DURATION, DMAX, AND UTILIZES THIS DURATION TIME IN MINUTES FOR EACH STORM TO BE SELECTED FROM E-STAR.INP AND ANALYZED. THE MAXIMUM DIMENSIONS OF P(N,M) P(40,400), WHERE N IS THE NUMBER OF STORMS AND M IS THE NUMBER OF MINUTES OF DURATION. A CUMULATIVE FILE OF STORMS WITH MEAN VALUES OF NUMBER OF STRIKES AT TIME t IS GENERATED INTO E-STAR.OT9. E-STAR IS A VARIANT OF D-STAR. ================================================================== ======================== PROGRAM F-STAR ======================== F-STAR READS A SEQUENCE OF PMAX STORM EVENTS CONSISTING OF Q0 VARYING LIGHTNING STRIKE RATES AS A FUNCTION OF TIME t IN UNIT INCREMENTS OF MINUTES AND FILTERS THOSE STORMS FROM THE SEQUENCE FOR WHICH TM LIES IN THE DESIRED INTERVAL [ T1,T2 ] AND CS LIES IN THE DESIRED INTERVAL [ CS1,CS2 ] , WHERE TM IS THE TOTAL DURATION OF EACH STORM (IN MINUTES) AND CS IS THE CUMULATIVE NUMBER OF STRIKES OF EACH STORM. FILTERED STORMS ARE PRINTED IN F-STAR.OT2 WITH HEADERS FOR FURTHER ANALYSIS. ================================================================== ======================== PROGRAM G-STAR ======================== G-STAR READS A SINGLE STORM EVENT CONSISTING OF Q0 VARYING LIGHTNING STRIKE RATES AS A FUNCTION OF TIME t IN UNIT INCREMENTS OF MINUTES AND PRINTS THE MEASUREMENTS AS EITHER A-STAR.INP OR B-STAR.INP. ================================================================== H-17

GLOVER_NFFP_04_Tech_Rpt

Documents

lightning advisories

end of lightning

ground lightning

lightning flashes

lightning flash rates

lightning flashrates

ws strategic goals

decaying thunderstorms