Electronic Navigation Research Institute (ENRI)

Frequency of Rare Event
Occurrence
ICAO collision risk model for
separation minima

Masato Fujita
ATM department, ENRI

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Contents
1.
2.
3.
4.

Safety & Risk assessment
Collision Risk Model (CRM)
Example of CRM: Parallel RouteDistribution Model
1. Estimation from Requirement
2. Estimation from Data Set

5. Summary
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Safety (Definition)
Safety is the state in which the risk of harm toperson or property of damage is reduced to, andmaintained at or below, an acceptable level througha continuing process of hazard identification andrisk management“Safety Management Manual”, ICAO Doc 9859-AN/953

Safety is considered to be a state as well as action.

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Safety Management Process“Safety Management Manual”,
ICAO Doc 9859-AN/953
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Separation minima

2S z

Separation minimum ofaircraft pairs for tacticalcollision avoidance.

2Sy
2S xZYX
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Risk Comparison
Risk of reference
system or absolute
threshold

Risk of newseparation
minima orprocedures

TLS (Target Level of Safety)

5.0 ×10-9 [fatal accidents
per flight hour]
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ICAO collision risk model
ICAO Collision Risk Model (CRM)
Mathematical models used in predicting risk of
mid-air collision•Reich CRM

Time-invariant error model

Reich, J., “Analysis of long-range air traffic system: separation
standard”, Journal of the Institute of Navigation vol. 19 (1996)•Rice CRM

Time-invariant & time-variant error model“A unified framework for collision risk modelling in support of themanual on airspace planning methodology for further application”,
ICAO Circ 319-AN/181
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Area of Collision
Area of Collisionλzλx

2λzλyλh

2λx
2λh

2λyλz
2λz

Simplification ofaircraft shape3/6/2009

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Rice formulaPr{collision during [t 0 , t1 ]} = ∫ Ψ (u )dut1(t0)r r +Ψ (t ) = ∫∫ ∫∫∫ f t ( X , V )(n ⋅ V ) dV dS∂ΩΩ : area of collision∂Ω : the boundary of Ωn : the normal vector of the boundary surface
ft(X,V) : the probability density function on the phase space(X,V) at the given time instance t
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Parallel Route CaseSy⎛ λ y& λ z& ⎞N ay = N x Py (S y ) Pz (0 )⎜1 + x+ x ⎟⎜ λ y x& λ z x& ⎟⎝⎠3/6/2009

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Description
Notation

Description

Nay

Collision risk defined as the expected number of fatal
accidents due to loss of lateral separation per flighthourNxPassing frequency. twice of the number of passing events
divided by total flight hourPy(Sy)

Lateral overlap probability. Probability that a pair ofaircraft being nominally and laterally separated by Sy
NM overlaps laterally.

Pz(0)Vertical overlap probability. Probability that a pair ofaircraft flying at the same flight level overlaps
vertically.

λx, λy, λz

Average aircraft length, average wing span and average
aircraft heightx& y& z&Average longitudinal, lateral and vertical relative velocity
of a pair of aircraft which are about to collide.

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Lateral overlap

Vertical overlap

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Special Case(flow rate and cruise speed are given)⎛ N −1 K 2ni , j ni +1, j | Vi +1, j − Vi , j |N ay = ⎜ ∑∑Py ( Si ,i +1 ) Pz (0)⎜ i =1 j =1
| Vi +1, j || Vi , j |⎝⎛⎞⎞v y /(2λ y )v z /(2λz )
⎜
⎟⎟
× 1++
⎜ | V − V | /(2λ ) | V − V | /(2λ ) ⎟ ⎟ｘｘ ⎠⎠i +1, ji, ji +1, ji, j⎝⎛ N −1 K ni , j | Vi +1, j | + ni +1, j | Vi , j | ⎞⎜ ∑∑⎟⎜ i =1 j =1⎟|V||V|i+1,ji,j⎝⎠Remark:Risk is approximately
twice if the flow rates
are doubledni,j : flow rate of aircraft on route Ri at flight level Hj
Vi,j : speed of aircraft on route Ri at flight level Hj
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Effect of SLOP application
SLOP
lateral offset 1NM or 2NM to the rightsame direction

opposite direction

opposite direction

28NM

30NM

26NM

1NM 1NM
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1NM 1NM

1NM 1NM

1NM 1NMENRI International Workshop for
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1NM 1NM

1NM 1NM
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Performance-based Navigation
The Performance Based Navigation (PBN)
concept specifies RNAV system performancerequirements in terms of accuracy, integrity,availability, continuity and functionality needed
for the proposed operation in the context of aparticular airspace concept, when supported by
the appropriate navigation infrastructure.
“Performance-Based Navigation Manual”, ICAO Doc 9613-AN/937

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Total System Error (TSE)

Total System Error
Desired Path
Path Definition Error (PDE)Defined Path

Flight Technical Error (FTE)
Estimated Position
Navigation System Error (NSE)
True PositionfTSE(u) : probability density function oftotal system errors (TSEs)“Performance-Based Navigation Manual”, ICAO Doc 9613-AN/937
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RNAV-X vs RNP-X
The expression ‘X’ refers to the lateral navigation accuracy of X
nautical miles (NM) that is expected to be achieved at least 95 percentof the flight time by the population of aircraft operating within theairspace, route and procedureX∫fTSE( x)dx = 0.95

−X

RNP-X aircraft has a monitoring and alerting function thatensures that the probability that the total system error ofeach aircraft exceeds 2 times X without annunciation isless than 10-5.
2X−5f(x)
dx
=
1−10
∫ TSE
−2 X

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SASP MSG TSE model
RNAV-X aircraft

exp( − | y | / λ )f TSE ( y ) =2λDouble Exponential (DE)distribution

RNP-X aircraftf TSE ( y ) =exp(− y 2 /(2σ 2 ))2π σGaussian distribution“Report of the Mathematicians Sub Group meeting at the SASP WG/WHL/10 meeting”,
ICAO SASP/WG/WHL/10, Gold Coast, November 2006
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Lateral Overlap Probability
SyPy ( S y ) = ∫S y + λ y +∞f∫
λS y − y −∞TSE

( y + u ) f TSE (u )dudy+∞≅ 2λ y ∫ f TSE ( S y + u ) f TSE (u )du−∞fTSE(u) : probability density functionof total system errors (TSEs)

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Convolution+∞∫ f−∞

SatisfiedCondition

Sy (NM)

Distribution
Type

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2627282930

TSE( S y + u ) f TSE (u )du for X=4

X

2X−X

−2 X

∫ f TSE ( x)dx = 0.95
DE−5f(x)
dx
=
1−10
∫ TSE

Gaussian

DEGaussian

1.31×10-8 3.30×10-19 6.86×10-17 2.43×10-24
6.40×10-9 1.37×10-20 1.53×10-18 3.95×10-26
3.13×10-9 5.05×10-22 3.41×10-18 5.45×10-28
1.53×10-9 1.65×10-23 7.58×10-19 6.56×10-30
7.45×10-10 4.77×10-25 1.68×10-19 6.70×10-32
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Tail of Distribution
core (neighborhoodof the peak)f(x) : probability
distribution function

0x

Collision risk is dominated mainlyby the tails of distributions

tail (far from the peak)

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Extreme Value Theory
Extreme value theory claims that the conditional probability
Pr{Y<y|Y>u} of distributions satisfying some technical assumptions
approximately follows a generalized Pareto distribution (GPD) whenu is large enough. The cumulative distribution function of ageneralized Pareto distribution is given byy⎞⎛H ( y ) = 1 − ⎜1 + ξ ⎟σ⎠⎝−1 / ξ, 1 + ξy / σ > 0

When the shape parameter ξ<0, the GPDs are Beta distributions.
They are exponential distributions and Pareto distributions in thecase where ξ=0 and ξ>0, respectively.
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DE vs Gaussian• Convolution value (and lateral overlap probability) ismore than 107 smaller in Gaussian case than DE case.
• We do NOT have any evidence that TSE distribution
does NOT follow Gaussian distribution, but we doNOT have any evidence that it DOES follow
Gaussian distribution.
• GPD is an exponential distribution in case where theoriginal distribution is Gaussian.
Conclusion

DE is better for the model of TSE.
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What we should do in case wheresufficient data set is not available• to make an assumption on the type ofdistribution and conduct a safety assessment
based on this assumption. This assumption
should seem to be sufficiently conservative• to continue monitoring to check whether a newhazard is found and whether the assumption
made is truly conservative after theimplementation of a new separation minimum
or a procedure,
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TSE distribution (observed)
1.19

log10(1-cummulative)

10123

0.1

95% line
0.01

0.436%

4

No. of data : 23,639
Max. 12.9 (NM)Min. -15.9 (NM)AVG. -0.183 (NM)STD. 0.651 (NM)
SKEW -3.17
KURT 69.6

0.001
absolute value of TSE (NM)KushiroARSR

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Peak over Threshold

Small sizeof given
data sety⎞⎛
H ( y ) = 1 − ⎜1 + ξ ⎟σ⎠⎝
3/6/2009−1 / ξ⎧ξ = −0.103⎨⎩ σ = 2.92

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Summary• ICAO developed mathematical models which areutilized for the quantitative safety assessment in thedetermination of separation minima. They are called
collision risk models (CRMs).
• Reich formula is applicable only for time-invariantcase but Rice formula is an extension version ofReich formula and it is applicable for both timeinvariant and time-variant case.
• We developed a collision risk formula for a parallel
route system in case where the flow rate and theaircraft speed on each route at each flight level isconstant.
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Summary (cont’d)• Double exponential (DE) distribution is compared with
Gaussian distribution. We should choose a conservative model
not to underestimate the risk. The author claimed that DEdistribution is better for the model of TSE distribution.
• We should continue monitoring to check whether a new hazard
is found and whether the assumption on the distribution modelis truly conservative for the risk estimation.
• Generalized Pareto distribution (GPD) is also introduced andapplied to TSE data set. GPD is a good tool to investigate theshape of tails if the sufficient size of data is obtained.

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Thank you for your attention!!

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