LAAS Ionosphere Anomaly Prior
Probability Model: “Version 3.0”
Sam Pullen
Stanford University
spullen@stanford.edu
14 October 2005
Proposed Iono. Anomaly Models for LAAS
• “Version 1.0” (November 2002 – proposed to FAA)
– Fundamentally based on average or “ensemble” risk over all
approaches
– Insufficient data to back up assumed probability of threatening
storm conditions
• “Version 2.0” (May 2005 – internal to SU)
– Uses enlarged database of iono. storm days to estimate
probability of threatening conditions
– Considers several options for “threshold” Kp above which
threat to LAAS exists
• “Version 3.0” (October 2005) – details in this briefing
– Two results: one for fast-moving wave-front anomalies
(detectable by LGF) and one for slow-moving (potentially
undetectable) anomalies
– Establishes basis for averaging over both storm-day
probabilities and over “hazard interval” within a storm day
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
2
Two Cases for this Study
• For fast-moving storms: prior probability
of potentially-hazardous fast-moving
storm prior to LGF detection, but
including “precursor” credit
– Result sets PMD for relevant LGF monitors
• For slow-moving storms: prior probability
of slow-moving (and thus potentially
undetectable by LGF) storm, including
“precursor” credit
– Feasible mitigation is included in prior prob.
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
3
“Pirreg” Prior Prob. Model used in WAAS
• Cited by Bruce – used in GIVE verification in
WAAS “PHMI document” (October 2002)
– “Pirreg” formerly known as “Pstorm”
– Examines probability of transition from “quiet” to
“irregular” conditions in given time interval
– Upcoming GIVE algorithm update does not need it (can
assume Pirreg = 1)
• Uses a pre-existing model of observed Kp
occurrence probabilities from 1932 - 2000
• Each Kp translates into a computed conditional
risk of unacceptable iono. decorrelation for GIVE
algorithm (decorr. ratio > 1)
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
4
Key Results from Pirreg Study
Kp Occurrence Probs.
Conditional Decorrelation Probs.
WAAS
Safety
Constraint
Resulting Pirreg for WAAS
14 October 2005
= 9.0 × 10-6 per 15 min. (calculated)
= 1.2 × 10-5 per 15 min. (add margin)
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
5
Observed Iono. Storm Totals since Oct. 1999
Storm Days with Max Kp
 5 ("Minor")
Storm Days with Max Kp
 6 ("Moderate")
Storm Days with Max Kp
 7 ("Major")
Storm Days with Max Kp
 8 ("Severe")
Storm Days with Max Kp
 9 ("Extreme")
Storm Days known to be
threatening in CONUS (6
April 2000, 30-31 October
2003, 20 November 2003)
14 October 2005
Number of
Days in
Database
Fraction of Days
in Database
(2038)
Fraction of Days
from NOAA Storm
Scale (over 11-year
= 4017 day cycle)
96
0.04711
0.22405
81
0.03974
0.08962
65
0.03189
0.03236
23
0.01129
0.01494
9
0.00442
0.00100
4
0.00196
N/A
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
6
Severe Kp State Probability Comparison
Pirreg Model
(1932-2000)
NOAA Storm
Scale (one
solar cycle)
Observed
Since October
1999
Kp = 8
(“severe”)
0.0026
0.01494
0.01129
Kp = 9
(“extreme”)
0.0004
0.0010
0.0044
•
Pirreg model has ~ 5x lower probs. than more recent numbers
•
Observations since 10/99 are conservative since they cover
the worst half of a solar cycle
•
Appears reasonable to use actual fraction of days potentially
threatening to CONUS: 4 / 2038 = 0.00196
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
7
Confidence Interval for Probability of
Threatening Storms (1)
• Use binomial(s,n) model to express confidence
interval (CI) for Pr(threatening storm)  PTS
– i.e., observed s threatening storm days over n total days
(x  n – s = number of non-threatening days)
– Analog to Poisson continuous-time model
– CI needed since s = 0 for slow-moving storms
• More conservative lower tail limit 1 - L(x): (Martz
and Waller, Bayesian Reliability Analysis, 1991)
L x  
x
x  n - x  1 Fα 2 n - 2 x  2, 2 x 
– Where 100 a = 100 (1 – g/2) = lower percentile of CI
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
8
Confidence Interval for Probability of
Threatening Storms (2)
• For fast-moving storms:
– s  4; n = 2038; x = n – s = 2034
– ML (“point”) estimate:
PTS = s / n = 0.00196
– 60th percentile estimate: 1 - L(x).4 = PTS60th = 0.00257
– 80th percentile estimate: 1 - L(x).2 = PTS80th = 0.00330
• For slow-moving storms:
– s  0; n = 2038; x = n – s = 2038
– ML (“point”) estimate: PTS = s / n = 0
– Point est. “bound” for s = 1:
PTS_bnd = s / n = 4.91 × 10-4
– 60th percentile estimate: 1 - L(x).4 = PTS60th = 4.50 × 10-4
– 80th percentile estimate: 1 - L(x).2 = PTS80th = 7.89 × 10-4
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
9
“Time Averaging” over Course of One Day
• For all non-stationary events, anomalous
ionosphere gradient affects a given airport for a
finite amount of time
• Model each airport as having Nmax = 10 satellite
ionosphere pierce points (IPP’s)
– Satellites below 12o elevation can be ignored, as max.
slant gradient of 150 mm/km is not threatening
– Conservatively (for this purpose) ignore cases of multiple
IPP’s being affected simultaneously
• For both cases, determine probability over time
(i.e., over one threatening day) that a given airport
has an ionosphere-induced hazardous error
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
10
“Time Averaging” for Fast-Moving Storms
• Fast-moving storms are detected by LGF during
rapid growth of PR differential error right after LGF
is impacted by ionosphere wave front
– SU IMT detects within ~ 30 seconds of being affected
– Thus, for each satellite impacted, only worst 30-second
period represents a potential hazard
• Assume EXM excludes all corrections once two
different satellites are impacted
– Based on two-satellite “Case 6” resolution in SU IMT EXM
– Fast motion of front prevents recovery between impacts
• Assume two fast-moving fronts (rise then fall, or
vice-versa) can occur in one day
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
11
Modeling “Precursor Event” Probabilities
• Ionosphere anomalies are typically accompanied by
amplitude fading, phase variations, etc. that make reliable
signal tracking difficult
– CORS data usually shows L1 and (particularly) L2 losses of lock during
time frame of ionosphere anomalies
– This fact makes searching CORS data for verifiable ionosphere
anomalies quite difficult
– LGF receivers and MQM should be more sensitive to these transients
than CORS receivers
• Multiple gaps in data render over 80% of CORS station pairs
unusable for gradient/speed estimation during iono. storms
• Therefore, pending further quantification, conservatively
assume that 80% of threatening ionosphere fronts are
preceded by “precursor” events that make the affected
satellites unusable
– Actual probability is likely above 90%
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
12
Probability Model for Fast-Moving Storms
Probability of Threatening Storm Day (60th pct)
0.00257
Prob. over 1 day that specific CONUS airport affected
(for a given airport, only 2 * 2 = 4 approach periods per day
could be threatened): Pr ~ 150 * 4 / 86400 = 0.006944
1.7847E-05
Probability of Worst-Case Approach Direction (1)
(1/6 = 60/360 for a given approach, but assume many
approaches, at least one of which will have worst-case direction)
1.7847E-05
Probability of Worst-Case Timing for a given aircraft (0.2)
(1/5 = 30 / 150 second approach)
3.5694E-06
Probability of No Early LGF (i.e. Precursor) Detection (0.2)
7.1389E-07
(conservative precursor credit based on > 80% data rejection during iono. anomalies)
> Resulting fast-moving-storm prior prob. for a
single airport is 7.14 × 10-7 per approach
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
13
Triangle Distribution for Slow-Speed Gradients
• For slow-moving storms, both point estimate
bound and 60th-pct bound seem too conservative
– no gradients large enough to be threatening (i.e., > 200
mm/km) have been observed at all
• To address expected rarity of slow-moving and
threatening gradients, a triangle distribution is
proposed
– Linearly decreasing PDF as slant gradient increases
– Assume practical maximum of 250 mm/km
atot
btot
tan q  
PDF
q
btot = 2/150
to give Atot =
0.5 atot btot = 1
100
> bexc
atot = 150
150
200


1
225
aexc = 50
aexc
bexc
 0.0044
250
Slant Gradient (mm/km)
 Aexc = “threatening” fraction of PDF = 0.5 aexc bexc = 1/9 = 0.1111
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
14
“Time Averaging” for Slow-Moving Storms
• Slow-moving storms may not be detected by LGF
during worst-case approach, but would be
detected soon afterward
– Thus, for each satellite impacted, one 150-second
approach duration represents the hazard interval
• Slow-moving (linear-front) storms can only affect
one satellite at a time
– Very wide front might affect multiple satellites, but
gradient would not be hazardous
– Slow motion of front prevents recovery between impacts
• Assume only one slow-moving front event can
occur in one day
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
15
Possibility of Truly “Stationary” Storms
• Time averaging for slow-moving storms assumes a
minimum practical speed of roughly 20 m/s
– Below this speed, a hazardous gradient could persist for more than one
approach (indefinitely for zero speed)
• We have seen no suggestion of storms with zero
velocity (relative to LGF) in CORS data
• Even if an event were stationary relative to the solarionosphere frame, it would be “moving” relative to LGF
due to IPP motion
– In other words, “stationary” relative to LGF implies motion in iono.
frame “cancelled out” by IPP motion
• Recommendation is to presume some risk of “truly
stationary” that is a fraction of slow-speed risk and
can be allocated separately within “H2” (see slide 18)
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
16
Probability Model for Slow-Moving Storms
Probability of Slow-Speed Storm Day (60th pct)
0.000450
Probability Storm Day has threatening gradients (from triangle dist)
0.000050
Prob. over 1 day that specific CONUS airport affected
(for a given airport, only 1 * 1 = 1 approach period per day
could be threatened): Pr ~ 150 * 1 / 86400 = 0.001736
8.6806E-08
Probability of Worst-Case Approach Direction (1)
(1/6 = 60/360 for a given approach, but assume many
approaches, at least one of which will have worst-case direction)
8.6806E-08
Probability of Worst-Case Timing for a given aircraft (1.0)
(threatening slow-moving front impact could last for entire approach)
8.6806E-08
Probability of No Early LGF (i.e. Precursor) Detection (0.2)
1.7361E-08
(conservative precursor credit based on > 80% data rejection during iono. anomalies)
> Resulting slow-moving-storm prior prob. for a
single airport is 1.74 × 10-8 per approach
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
17
Observations from these Results
• Feasible CAT I (GSL C) sub-allocation from “H2”
integrity allocation is as follows:
– Total Pr(“H2”)  1.5 × 10-7 per approach (from MASPS)
– Allocate 20% (3.0 × 10-8) to all hazardous iono. anomalies
– 58% of this (1.74 × 10-8) must be allocated to slow-moving iono.
anomalies
– Reserve an additional 5% of this (7.5 × 10-9) for the possibility of
“truly stationary” iono. anomalies
– Then, 37% of allocation (1.11 × 10-8) remains for fast-moving
ionosphere anomalies
– Implied PMD for fast-moving anomalies is 0.111 / 7.14 = 0.01555
(KMD = 2.42)
• Given a threatening iono. event, implied probability
that threat is from slow-moving storm is roughly 0.174 /
7.14 = 0.024
14 October 2005
– This makes sense given apparent rarity of (non-threatening)
slow-moving storms
in CORS data sets
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
18
Summary
• A feasible prior probability model has been developed
to support CAT I (GSL C) LAAS
• The key “probability averaging” steps are:
– Averaging over probability of threatening iono-storm days
(used by WAAS for Pirreg)
– Time averaging based on fraction of time that a given airport
would face a potential hazard
– Triangle distribution for probability of slow-speed iono.
gradients large enough to be threatening
• Some probabilities used here depend on magnitude of
hazardous gradient
– Need to iterate between prior model and mitigation analysis
• For extension to CAT III (GSL D), additional (airborne?)
monitoring is needed against slow-speed events
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
19
Appendix
• Backup slides follow…
14 October 2005
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
20
User Differential Error vs. Front Speed
Differential error vs. iono speed
8
Differential Error (meter)
6
4
2
0
75 m/s
90 m/s
110 m/s
200 m/s
300 m/s
500 m/s
1000 m/s
LGF impact times
-2
-4
-6
0
14 October 2005
200
400
600
800
Time (epoch)
1000
1200
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
1400
21
Differential Error vs Airplane Approach
Direction
Differential Error vs Airplane Approaching Direction Relative to Iono Front Speed
6
0 degree
+/-30 degree
+/-60 degree
+/-90 degree
+/-120 degree
+/-150 degree
+/-180 degree
5
Differential Error (meter)
4
3
Iono front
hits LGF
2
1
0
-1
-2
-3
14 October 2005
0
200
400
600
800
Time (epoch)
1000
1200
LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0
1400
22