LAAS Ionosphere Anomaly Prior Probability Model: “Version 3.0”

1. LAAS Ionosphere Anomaly Prior Probability Model: “Version 3.0” Sam Pullen Stanford University spullen@stanford.edu 14 October 2005

2. 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 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

3. 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. LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

4. “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) LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

5. Key Results from Pirreg Study Kp Occurrence Probs. Conditional Decorrelation Probs. WAAS Safety Constraint Resulting Pirreg for WAAS = 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

6. Observed Iono. Storm Totals since Oct. 1999 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

7. Severe Kp State Probability Comparison • 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 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

8. 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) • Where 100 a = 100 (1 – g/2) = lower percentile of CI LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

9. 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 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

10. “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 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

11. “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 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

12. 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% LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

13. Probability Model for Fast-Moving Storms => Resulting fast-moving-storm prior prob. for a single airport is 7.14 × 10-7 per approach LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

14. 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 PDF q btot = 2/150 to give Atot = 0.5 atotbtot = 1 atot = 150 aexc = 50 200 100 150 250 Slant Gradient (mm/km)  Aexc = “threatening” fraction of PDF = 0.5 aexcbexc = 1/9 = 0.1111 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

15. “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 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

16. 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 solar-ionosphere 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) LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

17. Probability Model for Slow-Moving Storms => Resulting slow-moving-storm prior prob. for a single airport is 1.74 × 10-8 per approach LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

18. 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 • 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

19. 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 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

20. Appendix • Backup slides follow… LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

21. Differential error vs. iono speed 8 6 4 2 Differential Error (meter) 0 75 m/s 90 m/s -2 110 m/s 200 m/s 300 m/s -4 500 m/s 1000 m/s -6 0 200 400 600 800 1000 1200 1400 Time (epoch) User Differential Error vs. Front Speed LGF impact times LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0

22. Differential Error vs Airplane Approach Direction Iono front hits LGF LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0