Statistical Forecasting of Lightning Cessation II

1. H-1
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.

2. H-2
STATISTICAL FORECASTING OF LIGHTNING CESSATION
James Ervin Glover, Ph.D.
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 Force’s 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 stormstrike-rates over each unit
time interval
Sample Years
Year Number of Storms
1999 12
2000 13
2001 14
2002 7
2003 12

3. H-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 anotherflash
 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 randomvariable 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:

4. H-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
0:02
0:08
0:14
0:20
0:26
0:32
0:38
0:44
0:50
0:56
1:02
1:08
1:14
1:20
1:26
1:32
1:38
1:44
1:50
1:56
2:02
2:08
2:14
2:20
2:26
2:32
2:38
2:44
Time(Z)
#CGStrikes
CG Strikes
Figure 1. Cloud to Ground Strike Rates

5. H-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

6. H-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

7. H-7
Survivability/Reliability of Strike Rates
y = -0.3085Ln(x) + 1.7599
R2
= 0.8642
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 50 100 150 200 250
Time (Minutes)
Reliability
y
Log. (y)
Figure 6. Reliability Curve for Strike Rates
Survival/Reliability Curve for Composite Storm
y = 5E-14x6
- 2E-11x5
- 1E-09x4
+ 1E-06x3
- 0.0002x2
+ 0.0004x + 0.9909
R2
= 1
0
0.2
0.4
0.6
0.8
1
1.2
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222
# Minutes
P(#Minutes)
Series1
Poly. (Series1)
Figure 7. Reliability Curve for Composite Storm

8. H-8
PDF for STRIKE RATES of STORM 1y = 0.8551e-0.5876x
R2
= 0.9681
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1 2 3 4 5 6 7
# Strikes / Min
P(#Strikes/Min)
Series1
Expon. (Series1)
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-time determined
 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.

9. H-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

10. H-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)

11. H-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.

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

13. H-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)

14. H-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.

15. H-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.)
==================================================================

16. H-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.
==================================================================

17. H-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.
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======================== 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.
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