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

2. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 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

3. 14 October 2005 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 P MD 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.

4. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 4 “P irreg ” Prior Prob. Model used in WAAS Cited by Bruce – used in GIVE verification in WAAS “P HMI document” (October 2002) –“P irreg ” formerly known as “P storm ” –Examines probability of transition from “quiet” to “irregular” conditions in given time interval –Upcoming GIVE algorithm update does not need it (can assume P irreg = 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)

5. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 5 Key Results from P irreg Study Kp Occurrence Probs. Conditional Decorrelation Probs. Resulting P irreg for WAAS = 9.0 × 10 -6 per 15 min. (calculated) = 1.2 × 10 -5 per 15 min. (add margin) WAAS Safety Constraint

6. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 6 Observed Iono. Storm Totals since Oct. 1999

7. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 7 Severe Kp State Probability Comparison P irreg Model (1932-2000) NOAA Storm Scale (one solar cycle) Observed Since October 1999 Kp = 8 (“severe”) 0.00260.014940.01129 Kp = 9 (“extreme”) 0.00040.00100.0044 P irreg 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

8. 14 October 2005 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)  P TS –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  = 100 (1 –  /2) = lower percentile of CI

9. 14 October 2005 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: P TS = s / n = 0.00196 –60 th percentile estimate: 1  L(x).4 = P TS 60th = 0.00257 –80 th percentile estimate:1  L(x).2 = P TS 80th = 0.00330 For slow-moving storms: –s  0; n = 2038; x = n – s = 2038 –ML (“point”) estimate: P TS = s / n = 0 –Point est. “bound” for s = 1: P TS_bnd = s / n = 4.91 × 10 -4 –60 th percentile estimate: 1  L(x).4 = P TS 60th = 4.50 × 10 -4 –80 th percentile estimate:1  L(x).2 = P TS 80th = 7.89 × 10 -4

10. 14 October 2005 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 N max = 10 satellite ionosphere pierce points (IPP’s) –Satellites below 12 o 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

11. 14 October 2005 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

12. 14 October 2005 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%

13. 14 October 2005 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

14. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 14 For slow-moving storms, both point estimate bound and 60 th -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 Triangle Distribution for Slow-Speed Gradients Slant Gradient (mm/km) 100 150 200 250  PDF a tot = 150 b tot = 2/150 to give A tot = 0.5 a tot b tot = 1 a exc = 50  A exc = “threatening” fraction of PDF = 0.5 a exc b exc = 1/9 = 0.1111

15. 14 October 2005 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

16. 14 October 2005 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)

17. 14 October 2005 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

18. 14 October 2005 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 P MD for fast-moving anomalies is 0.111 / 7.14 = 0.01555 (K MD = 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

19. 14 October 2005 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 P irreg ) –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

20. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 20 Appendix Backup slides follow…

21. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 21 User Differential Error vs. Front Speed LGF impact times

22. 14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 22 Differential Error vs Airplane Approach Direction Iono front hits LGF