Use of WAAS for LAAS Ionosphere Threat Status Determination

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LAAS Ionosphere Anomaly Prior Probability Model
Version 3.0
Sam Pullen Stanford University spullen_at_stanford.ed
u
14 October 2005
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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
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.

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 gt 1)

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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)
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Observed Iono. Storm Totals since Oct. 1999
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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

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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

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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

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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 (IPPs)
• 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 IPPs 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

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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

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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

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Probability Model for Fast-Moving Storms
gt Resulting fast-moving-storm prior prob. for a
single airport is 7.14 10-7 per approach
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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., gt 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 atot btot 1
atot 150
aexc 50
100
200
150
250
Slant Gradient (mm/km)
? Aexc threatening fraction of PDF 0.5 aexc
bexc 1/9 0.1111
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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

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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)

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Probability Model for Slow-Moving Storms
gt Resulting slow-moving-storm prior prob. for a
single airport is 1.74 10-8 per approach
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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

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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

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Appendix

• Backup slides follow

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User Differential Error vs. Front Speed
LGF impact times
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Differential Error vs Airplane Approach Direction
Iono front hits LGF