LAAS Ionosphere Anomaly Prior Probability Model: “Version 3.0”

14 October 2005

Sam PullenStanford University

spullen@stanford.edu

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

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 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 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)

14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 5

Key Results from Pirreg Study

Kp Occurrence Probs. Conditional Decorrelation Probs.

Resulting Pirreg for WAAS = 9.0 × 10-6 per 15 min. (calculated)= 1.2 × 10-5 per 15 min. (add margin)

WAAS Safety

Constraint

14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 6

Observed Iono. Storm Totals since Oct. 1999

Number of Days in

Database

Fraction of Days in Database

(2038)

Fraction of Days from NOAA Storm

Scale (over 11-year = 4017 day cycle)

Storm Days with Max Kp 5 ("Minor") 96 0.04711 0.22405

Storm Days with Max Kp 6 ("Moderate") 81 0.03974 0.08962

Storm Days with Max Kp 7 ("Major") 65 0.03189 0.03236

Storm Days with Max Kp 8 ("Severe") 23 0.01129 0.01494

Storm Days with Max Kp 9 ("Extreme") 9 0.00442 0.00100

Storm Days known to be threatening in CONUS (6 April 2000, 30-31 October 2003, 20 November 2003)

4 0.00196 N/A

14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 7

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 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 = 100 (1 – /2) = lower percentile of CI xxnFxnx

xxL2,2221

α

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: 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 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

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

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%

14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 13

Probability Model for Fast-Moving Storms

Probability of Threatening Storm Day (60th pct) 0.00257

Prob. over 1 day that specific CONUS airport affected 1.7847E-05(for a given airport, only 2 * 2 = 4 approach periods per day could be threatened): Pr ~ 150 * 4 / 86400 = 0.006944

Probability of Worst-Case Approach Direction (1) 1.7847E-05(1/6 = 60/360 for a given approach, but assume many approaches, at least one of which will have worst-case direction)

Probability of Worst-Case T iming for a given aircraft (0.2) 3.5694E-06(1/5 = 30 / 150 second approach)

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 14

• 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

Triangle Distribution for Slow-Speed Gradients

Slant Gradient (mm/km)100 150 200 250

PDF

atot = 150

btot = 2/150to give Atot = 0.5 atot btot = 1

aexc = 50

0044.02251

tan

exc

exc

exc

tot

tot

b

ba

ba

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 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

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)

14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 17

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 8.6806E-08(for a given airport, only 1 * 1 = 1 approach period per day could be threatened): Pr ~ 150 * 1 / 86400 = 0.001736

Probability of Worst-Case Approach Direction (1) 8.6806E-08(1/6 = 60/360 for a given approach, but assume many approaches, at least one of which will have worst-case direction)

Probability of Worst-Case T iming for a given aircraft (1.0) 8.6806E-08(threatening slow-moving front impact could last for entire approach)

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 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

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 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 20

Appendix

• Backup slides follow…

14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 21

User Differential Error vs. Front Speed

0 200 400 600 800 1000 1200 1400-6

-4

-2

0

2

4

6

8

Time (epoch)

Diff

eren

tial E

rror

(met

er)

Differential error vs. iono speed

75 m/s 90 m/s 110 m/s 200 m/s 300 m/s 500 m/s 1000 m/s

LGF impact times

14 October 2005 LAAS Ionosphere Anomaly Prior Probability Model: Version 3.0 22

Differential Error vs Airplane Approach Direction

0 200 400 600 800 1000 1200 1400-3

-2

-1

0

1

2

3

4

5

6

Time (epoch)

Diff

eren

tial E

rror (

met

er)

Differential Error vs Airplane Approaching Direction Relative to Iono Front Speed

0 degree +/-30 degree +/-60 degree +/-90 degree +/-120 degree+/-150 degree+/-180 degree

Iono front hits LGF