ENRI International Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2009).

Frequency of Rare Event Occurrences 
M. Fujita**ATM Department
Electronic Navigation Research Institute (ENRI)
Tokyo, Japanm-fujita@enri.go.jp
Abstract: Safety is achieved through a continuous risk management process. Risk assessment is an important part of
the risk management process. In the risk assessment process, the level of risk in the current/new circumstances is
estimated. We introduce the risk assessment methodology developed for the determination of separation minima by theInternational Civil Aviation Organization (ICAO). It is called the Collision Risk Model (CRM) and estimates the
expected number of midair collision occurrences in a given time period or the expected number of fatal accidents per
flight hour due to loss of a given separation. Many factors such as aircraft navigation performance and the frequency of
position reports are considered to be random variables in CRM. Estimation of the frequency of extremely rare
deviations of such random variables is extremely important because a midair collision is considered to be caused bysome extremely rare deviations of these random variables. We discuss how to model these random variables so as not
to underestimate the frequency of rare event occurrence. We treat the route separation of the parallel route system as an
example.
Keywords: separation, route spacing, ICAO collision risk model
Risk assessment is an important part of the risk
management process. In the risk assessment process,the level of risk in the current/new circumstances is
estimated. The results of risk assessment support
decision making, indicating whether or not risk
mitigation action is required and sometimes proposes
which remedial action will be effective.

1. RISK ASSESSMENT
The word “safety” has different connotations
depending on one’s perspective. For most of thetraveling public, it means zero (serious) accidents.
However, zero accidents are not achievable.
Statistically speaking, even the probability of thereader being injured due to a falling stone is not zero.

In the course of wide implementation of a safety
management system, the author believes that risk
assessment should be conducted much more frequently,
and that risk assessment technology should be
improved in quality.

ICAO Safety Management Manual[1], which defines a‘Safety Management System’ and overviews how to
implement it, defines the word “safety” as follows:
Safety is the state in which the risk of harm topersons or of property damage is reduced to, andmaintained at or below, an acceptable level througha continuing process of hazard identification andrisk management.

2. COLLISION RISK MODELIn air traffic control, spatial and time-dimensional
separation between aircraft is established to reduce therisk of aircraft colliding. Separation minima are
determined by ICAO and member states to meet
specific safety objective.

The manual remarks that the concept of ‘safety’ does
not mean zero accidents but means that the risk is at or
below an acceptable level in the ICAO definition.

Two methods of evaluating safety are proposed in [3].
The first method is comparison with a reference system.
A new separation minimum is considered to be safe if
the collision risk under this separation minimum isproved not to be larger than one of a reference system
which has been proved to be safe historically. The
second method is the evaluation of risk against athreshold. This threshold is called the target level ofsafety (TLS). The value of 5.0x10-9[fatal accidents per

The air traffic control environment keeps changing –fleet populations and traffic patterns change year by
year, and we might implement a new system or a newprocedure to meet new users’ demands or to mitigate
excessive risk. We should continuously monitor
current risk level and take remedial action if necessary.
Readers should note that safety is achieved throughcontinuous effort.

131

M. Fujita& &Ψ (t ) = ³³ ³³³ f t ( X , V )(n ⋅ V ) + dV dS

flight hour] is often utilized. This value wasdetermined based on the historical accident rate.

∂Ω

)

(2)

&

The notation ∂Ω denotes the boundary of Ω and n is
the normal vector of the boundary surface ∂Ω . The
notation ft(X,V) is the probability density function on
the phase space (X,V) at the given time instance t. The
symbol (⋅)+ is the function given by

The value of 5.0x10-9[fatal accidents per flight hour] isfairly small and it is impossible to conclude whether
the current separation minimum meets the TLS bycounting the number of actual collisions. Not only
post-implementation safety assessment but also preimplementation safety assessment should be conducted.
Of course, counting the number of actual collisions is
not applicable for pre-implementation safetyassessment. Hence, a mathematical model for
estimation of aircraft collision risk has been developed.

(x )+ = ­®

x

x>0

(3)

¯0 elsewhere.

3. EXAMPLE OF CRM: PARALLEL ROUTE
If we apply the Rice formula to the parallel route caseand accept some technical assumptions, we can get thefollowing Reich-type formula[6].

ICAO models for collision risk estimation are called
collision risk models (CRMs). ICAO CRMs have along history. The first CRM is found in the papers
published by Reich in 1966[2]. The Reich CRM has
been improved and applied for risk estimation ofparallel routes (Appendix 4 in [3]) and reduced verticalseparation minimum (RVSM)[4]. In the Reich CRM,
the position error of aircraft is assumed to be timeinvariant. In some cases, we should consider timevariant cases such as collision risk estimation of
longitudinal separation in an ADS-C environment[5].
At present, CRM is extended to the time-variant case[6].

§ λ y λ z ·N ay = N x Py (S y ) Pz (0 )¨1 + x+ x ¸¨ λ y x λ z x ¸
©
¹

(4)

The notations utilized in this formula are given in
Table 1.
Table 1 CRM Parameters

The proposed fundamental formula in [6] is called the
Rice formula. This formula gives the probability that apair of aircraft will collide. In this formula, aircraft areconsidered to be points of mass and collision occurs if
the relative position of aircraft enters into a volume Ω.
In the safety assessment of route spacing, we consider
an aircraft as a cuboid with length = λx (average
aircraft length), depth = λy (average wing span) andheight = λz (average aircraft height) for simplicity. In
this case, Ω is the cuboid with length = 2λx, depth =2λy and height = 2λz and its center is the origin. When
we estimate collision risk of crossing track, an aircraft
is assumed to be a cylinder.

λz
λx

(

λy

Notation

Description

Nay

Collision risk defined as the expected
number of fatal accidents due to loss oflateral separation per flight hour

NxPassing frequency. twice the number ofpassing events divided by total flighthours

Py(Sy)

Lateral overlap probability. Probability
that a pair of aircraft being nominallyand laterally separated by Sy NM
overlap laterally. (Figure 2)

Pz(0)Verticaloverlap
probability.
Probability that a pair of aircraft flying
at the same flight level overlap
vertically. (Figure 2)

λx, λy, λz

Average aircraft length, average wing
span and average aircraft heightx , y , z

Average longitudinal, lateral andvertical relative velocity of a pair ofaircraft which are about to collide.

Figure 1 Simple Representation of Aircraft ShapeThe Rice formula gives the collision probability of agiven pair of aircraft during a given time interval asfollows:t1

Pr{collision during [t0 , t1 ]} = ³ Ψ (u )duFigure 2 Lateral and Vertical Overlap(1)
We derive the collision risk formula for a special caseunder the assumption that all aircraft on the same routeat the same flight level cruise at the same speed. We

t0

Here Ψ(t) is given by132

ENRI International Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2009).

also assume that aircraft enter into the route system atthe same flow rate on each route at each flight level.
Consider the case where an N parallel route system isobserved for H hours. The notations ni,j and Vi,j denotethe flow rate (the number of aircraft entering into theroute over a unit of time) and the aircraft speed on theroute Ri at the flight level Hj. (The variable Vi,j may be
a negative value.)

2×

ni , j ni +1, j | Vi +1, j − Vi , j |=| Vi , j || Vi +1, j |Δt ×

ni +1, j L| Vi +1, j |=ni , j ni +1, j | Vi +1, j − Vi , j || Vi , j || Vi +1, j |ni , j | Vi +1, j | + ni +1, j | Vi , j |Here Si,i+1 denotes the route spacing between Ri and
Ri+1. Hence the total number of expected collisions in
the whole route system during the observation period
is given by
N −1 KHn i , j n i +1, j | Vi +1, j − Vi , j || Vi , j || Vi +1, j |Ldt =N −1 K= LH ¦¦
i =1 j =1| Vi , j || Vi +1, j |

i

i +1

LHni +1, ji, ji +1, jj

2ni , j ni +1, j | Vi +1, j − Vi , j || Vi +1, j || Vi , j ||). (10)Py ( Si ,i +1 ) Pz (0)§·v y /(2λ y )vz /( 2λz )¸× ¨1 ++¨ |V| Vi +1, j − Vi , j | /( 2λｘ) ¸¹
i +1, j − Vi , j | /( 2λｘ)©

Since the total flight hours in this route system are
N −1 K§ni , j·i =1 j =1

©

i, j

¹

¦¦ ¨¨ L × | V | × H ¸¸ ,

(5)

(11)

the collision risk in the whole route system is given as
follows:§ N −1 K 2n n
|V
− Vi , j |N ay = ¨ ¦¦ i , j i +1, j i +1, j
Py ( Si , i +1 ) Pz (0)¨ i =1 j =1
| Vi +1, j || Vi , j |
©
(12)
··
§
v y /(2λ y )
v
/(
2
)
λ
z
z
¸¸
× ¨1 ++¨ | Vi +1, j − Vi , j | /( 2λｘ) | Vi +1, j − Vi , j | /( 2λｘ) ¸ ¸
¹¹
©

LΔt

n i , j n i +1, j | Vi +1, j − Vi , j |LHni , j¦¦ N ( R , R ; H （) | V | + | V§ N −1 K ni , j | Vi +1, j | + ni +1, j | Vi , j | ·¸¨ ¦¦¸¨ i =1 j =1
| Vi +1, j || Vi , j |¹©

Hence, the total number of passing events during theobservation period H is as follows:0

Py ( Si ,i +1 ) Pz (0)§·v y /(2λ y )vz /(2λz )¸× ¨1 ++¨ |V| Vi +1, j − Vi , j | /(2λｘ) ¸¹
i +1, j − Vi , j | /( 2λｘ)
©

(6)

³

)
| Vi +1, j | . (8)(9)

2ni , j ni +1, j | Vi +1, j − Vi , j |The number of aircraft on Ri+1 is L×(density) =L×ni,j/|Vi,j|, and the total number of passing events ofaircraft flying on the adjacent routes Ri and Ri+1 at
flight level Hj is as follows:| Vi , j |LHni +1, j

N ay ( Ri , Ri +1; H j ) =We will find the total number of passing events ofaircraft flying on the adjacent routes Ri and Ri+1 at
flight level Hj. Consider an aircraft on the route Ri+1.
The flight distance of the aircraft relative to an aircrafton Ri for a sufficiently small duration Δt is given by|Vi+1,j−Vi,j|×Δt. Since the aircraft density is ni,j/|Vi,j| on
Ri, the expected number of aircraft passing the given
individual aircraft on Ri+1 is given by the relative
flight distance times aircraft density. Hence, it isni , j × | Vi +1, j − Vi , j |+ni , j | Vi +1, j | + ni +1, j | Vi , j |i =1 j =1

Δt .

| Vi , j |2ni , j ni +1, j | Vi +1, j − Vi , j |ay| Vi , j |LHni , j

The collision risk Nax(Ri,Ri+1;Hj) for the aircraft pairs
flying on the adjacent routes Ri and Ri+1 at flight level
Hj is given by the following formula by Equations (4)and (8).

We fix a route Ri and flight level Hj in consideration.
We first find the aircraft density (the number of aircrafton the portion of a given route with a unit length at agiven flight level). Consider an aircraft entering into
the route system at time 0. This aircraft proceeds|Vi,j|×T in distance for duration T. The number ofaircraft entering the route system is ni,j×T. The ni,j×T
aircraft entering into the route system during theduration T lie on the segment with length = |Vi,j|×T attime T. Hence the aircraft density is (ni,j×T) /(|Vi,j|×T)= ni,j/|Vi,j| and the average longitudinal distance
between aircraft is given by |Vi,j|/ni,j.

ni , j × | Vi +1, j − Vi , j |

LH （
/

LH

It should be remarked that route spacing might be
reduced due to the strategic lateral offset procedure(SLOP, lateral offset 1NM or 2NM to the right, Figure3). Aircraft can apply SLOP without informing air
traffic control. We cannot know whether aircraft
deviate 1NM intentionally (SLOP) or not fromsurveillance data. Hence, it is conservative to assume
that the values of route spacing of parallel routes are
28NM, 30NM and 26NM in the case of Figure 3,
unless we get reliable data on the proportion of lateral
offset application.

(7)
On the other hand, the total flight time of aircraft on Ri
at Hj is given by the average number of aircraft at each
instance times observation period, namely,
L×( ni,j/|Vi,j|)×H. Hence, the passing frequency for theaircraft pairs flying on the adjacent routes Ri and Ri+1
at flight level Hj is given by133

M. Fujita

same direction

opposite direction

opposite direction

28NM

30NM

26NM

1NM 1NM

1NM 1NM

1NM 1NM

1NM 1NM

1NM 1NM

1NM 1NM

Figure 3 Effect of SLOP to Route Spacing (30NM nominal route spacing)

The Performance Based Navigation (PBN) concept
specifies RNAV system performance requirements
in terms of accuracy, integrity, availability,
continuity and functionality needed for the
proposed operation in the context of a particularairspace concept, when supported by theappropriate navigation infrastructure.

4. DISTRIBUTION MODEL
An air traffic controller gets position information
through a surveillance system. It is in principal
impossible to know the exact position of an aircraft atarbitrary time instance without latency. So aircraft cancollide even if the separation among aircraft estimatedfrom the surveillance data is sufficient.

In the PBN concept, RNAV-X and RNP-X aircraft aredesigned. The expression ‘X’ refers to the lateralnavigation accuracy of X nautical miles (NM) that is
expected to be achieved at least 95 percent of the flighttime by the population of aircraft operating within theairspace, route and procedure. RNP-X aircraft have amonitoring and alerting function that ensures that theprobability that the total system error (Figure 4) ofeach aircraft exceeds 2 times X without annunciation isless than 10-5.
Total System Error (TSE)

If a potential collision is predicted, the air traffic
controller gives some instruction to the relevant
aircraft through the communication system. However,
if it takes time to transmit the instruction from theground ATC system to the aircraft, or if the ATC
instruction is not transmitted, the aircraft might collide
before receiving the instruction.
Aircraft try to fly as initially planned or instructed byATC. However, the navigation performance of aircraft
also has some limitation and a collision due to poornavigation performance is possible.
Air traffic controllers and pilots are human beings andhuman beings make errors even if they are welldisciplined.

Desired Path
Path Definition Error (PDE)Defined Path

Flight Technical Error (FTE)
Estimated Position
Navigation System Error (NSE)
True Position
Figure 4 Lateral Navigation Error[7]Let fTSE(u) be the probability density function of total
system errors (TSEs), then the lateral overlap
probability is given by

Collision risk is determined by the performance ofcommunication, navigation, surveillance, humanfactors, and so on. All factors are quantified for theestimation of collision risk in CRM. They are often
expressed as random variables. What kind of
distribution model is appropriate for this purpose? We
introduce some methodologies to find a distribution
model from a given data set or given requirements.

Py ( S y ) = ³

S y + λ y +∞³ f

S y −λ y −∞+∞TSE

( y + u ) f TSE (u )dudy(13)≅ 2λ y ³ f TSE ( S y + u ) f TSE (u )du−∞

where Sy is the route spacing and λy is the average
wing span.
4.1 Estimation from Requirement
The implementation of Performance-Based Navigation(PBN) separation is being promoted. The PBN concept
is defined as follows[7]:134ENRI International Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2009).

Table 2 Convolution for X=4
+∞

³ f
−∞

( S y + u ) f TSE (u )du
2X

X

³f

Satisfied
Condition

TSE³f( x)dx = 0.95

−X

TSE

( x)dx = 1 − 10 −5

−2 X

DE

Distribution TypeSy (NM)

TSE

Gaussian
-8

-19

DE

Gaussian-17

2.43×10-24

26

1.31×1027

6.40×10-9

1.37×10-20

1.53×10-18

3.95×10-26

28

3.13×10-9

5.05×10-22

3.41×10-18

5.45×10-28

29

1.53×10-9

1.65×10-23

7.58×10-19

6.56×10-3030

7.45×10-10

4.77×10-25

1.68×10-19

6.70×10-32

3.30×10

There is a significant gap in convolution values
between the DE and Gaussian cases, and theconvolution value decreases more rapidly in theGaussian case than the DE case. Convolution value(and lateral overlap probability) is more than 107
smaller in the Gaussian case than the DE case. If we
accept the Gaussian distribution satisfying Equation(17) as the TSE distribution, the collision risk of
30NM route spacing for RNP-4 aircraft is expected to
be much less than TLS even in unrealistically heavy
traffic cases.

ICAO SASP (Separation and Airspace Safety Panel)
recommends the double exponential (DE) distribution
and the Gaussian distribution as the distribution
models for RNAV-X and RNP-X aircraft,respectively[8]. For the double exponential case, theprobability density function is given by

f TSE ( y ) =exp(− | y | / λ )2λ6.86×10(14)

where λ = X/3. The parameter λ is determined tosatisfy the following equation:
X

³f

TSE

( x)dx = 0.95

The problem we have is whether the Gaussian
distribution is appropriate to use as the TSE
distribution model of RNP-X aircraft. We do not haveany observation data set on lateral navigation
performance of RNP aircraft. It takes a very long time
to collect sufficient data to model the type of TSE
distribution. At present, we do NOT have any evidence
that the TSE distribution does NOT follow theGaussian distribution, but we do NOT have any
evidence that it DOES follow Gaussian distribution.

(15)

−X

On the other hand, for the Gaussian case, theprobability density function is given by

f TSE ( y ) =exp(− y 2 /(2σ 2 ))
2π σ

(16)

where σ = X/2.23. The parameter σ is also determinedto satisfy the following equation:
2X

³f

TSE

( x)dx = 1 − 10 −5

The choice of distribution type affects the estimation of
the risk more than the estimation of flow rate. The
estimated collision risk will be approximately halved if
the flow rate is mistakenly underestimated by half.
(See Equation (12)) On the other hand, if we select theGaussian distribution in place of the DE distribution,
the estimated collision risk becomes more than 107
smaller. (See Equation (4) or (12))

(17)

−2 X

Table
(

2

+∞

³ f
−∞

TSE

shows

the

valuesofconvolution( S y + u ) f TSE (u )du ) for X=4 1 . The left

two columns show the convolutions of the probability
density function of TSEs satisfying Equation (15). The
right two columns show the convolutions of thatsatisfying Equation (17).

Should we accept an assumption in a case where no
positive evidence on the validity of the assumption isavailable, and where this assumption may lead asignificant underestimation of the risk? The Gaussian
distribution
possiblyleadsasignificant
underestimation of the risk.

1

RNP4 is a required navigation performance for 30NM routespacing in oceanic/remote airspace.

135

M. Fujita

on the type of distribution if we wait for a sufficient
data set. What we should do is to make an assumption
on the type of distribution and conduct a safetyassessment based on this assumption. This assumption
should seem to be sufficiently conservative. After the
implementation of a new separation minimum orprocedure, we should continue monitoring to check
whether a new hazard has been found and whether theassumption made is truly conservative.

The neighborhood of the peak of a unimodal
distribution is called the ‘core’ of the distribution andthe extremities, which are far from the peak, are called‘tails’ (See Figure 5).

core (neighborhoodof the peak)f(x) : probability
distribution function

4.2 Estimation from Data Set
We introduce methodologies to check whether the
initial assumption is satisfied based on extreme value
theory. Extreme value theory is also applied for theestimation of the longitudinal speed prediction error
distribution of ADS-C aircraft[10] in the safetyassessment of ADS-C longitudinal separations.

x0
tail (far from the peak)Figure 5 Core and Tail of Distribution

We consider the TSE distribution here. NOPAC (North
Pacific) route is one of the most congested routesystems in Fukuoka FIR. 50NM route spacing for
RNP10 2 aircraft has been implemented in NOPAC.
TSE was assumed to have exponential tails in thesafety assessment of this route system.

A collision occurs because of a significant size of TSE.
Hence the tail of the TSE distribution dominates thecollision risk rather than the core. It is the case foralmost all other parameters.
Extreme value theory[9] claims that the conditional
probability Pr{Y<y|Y>u} of distributions satisfying
certain technical assumptions approximately follows ageneralized Pareto distribution (GPD) when u is large
enough. More precisely, for any distribution which isin the domain of attraction, Pr{Y<y|Y>u} weakly
converges to the GPDs as u →∞. The cumulative
distribution function of a generalized Pareto
distribution is given byy·§
H ( y ) = 1 − ¨1 + ξ ¸
σ¹©

KUSHIRO
ARSR−1 / ξ

, 1 + ξy / σ > 0 . (18)When the shape parameter ξ<0, the GPDs are Beta
distributions. They are exponential distributions andPareto distributions in the case where ξ=0 and ξ>0,respectively. Extreme value theory is also introduced
in [6].

Figure 6 NOPAC route system

The segment between NUBDA and NANNO is within
the radar coverage of Kushiro ARSR by inches andaircraft are expected to follow remote procedure there.
Figure 7 shows one-minus-cumulative function of theTSE distribution at the middle point between NUBDA
and NANNO on ATS route R220 estimated from
Kushiro ARSR data and FDPS data. These data werecollected from October 2006 to September 2007. The
tail of the TSE distribution seems to be exponential
rather than Gaussian.

Since we do not have any information on the type oftail distribution, we should assume that the taildistribution approximately follows a GPD. It should beremarked that GPD is an exponential distribution incases where the original distribution is Gaussian.
Hence, in the TSE case, we should choose the DE
distribution rather than the Gaussian distribution for
the estimation of collision risk.
DE is not a perfect model either because there is noevidence that TSE distribution follows a DE
distribution. However, we cannot make any decision2

Note that RNAV10 in oceanic/remote airspace in the PBN
definition is usually called RNP10 for historical reasons.

136

ENRI International Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2009).

1.19

log10(1-cummulative)

1
0

1

2

3

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)Figure 7 TSE observed at the middle point between NUBDA and NANNO (Oct. 2006 − Sep. 2007)

Since the value of ξ is approximately equal to 0, theTSE distribution seems to have an exponential tail.
However, the estimated parameter has a large standard
deviation and some dots are somewhat distant from the
diagonal line in the QQ-plot diagram because of thesmall size of data set. Therefore, we cannot get any
conclusion from this analysis.

The author analyzed the tail of Figure 7 by POT (Peakover Threshold). R-package extRemes[11] was utilized
for the analysis. First, we set the threshold u = 5
considering the stability of estimated shape parameterξ and scale parameter σ. The number of excesses ofthresholds is 44 (0.186% of the whole data set). By the
maximum likelihood method (MLM), we found ξ =−0.103 and 95% confidence interval is [−0.518, 0.480].
σ = 2.92 and its standard error is 0.765. Figure 8 and
Figure 9 show the QQ-plot and the density plot of thismodel, respectively. If the empirical distribution
follows the distribution model, the dots in QQ-plot areon a straight line.

5. SUMMARYICAO has developed mathematical models which areutilized for quantitative safety assessment in thedetermination of separation minima. They are called
collision risk models (CRMs). We introduce two
fundamental collision risk formulae, namely the Reich
formula and the Rice formula. The Reich formula isapplicable only for time-invariant cases but the Rice
formula is an extended version of the Reich formulaand is applicable for both time-invariant and timevariant cases. We developed a collision risk formulafor a parallel route system in the case where flow rate
and aircraft speed on each route at each flight level isconstant.
Collision risk is determined by the performance ofcommunication, navigation, surveillance, humanfactors, and so on. Regardless of the CRM utilized, allfactors are quantified for the estimation of collisionrisk. They are often expressed as random variables. We
discussed what kind of distribution model isappropriate for this purpose. We considered a special
case − the distribution of total system error (TSE), but
the same argument is applicable in many cases. The
double exponential (DE) distribution is compared with
the Gaussian distribution. We should choose aconservative model so as not to underestimate the risk.
The author believes that the DE distribution is betterfor the model of the TSE distribution. However, theFigure 8 QQ-plot of TSE distribution

Figure 9 Density plot of TSE distribution137

M. Fujita[5] Fujita, M, “Safety Assessment prior to 30NM

DE distribution model is not a perfect model, either.
We should continue monitoring to check whether a
new hazard is found and whether the assumptions on
the distribution model are truly conservative enoughfor risk estimation.

Longitudinal Separation Minimum under ADS-C
Environment”, ICAO SASP/WG/WHL/13 IP/8,
Montreal, May 2008.
[6] “A unified framework for collision risk modellingin support of the manual on airspace planning
methodology for further application” ICAO Circ
319-AN/181, advance edition.

The Generalized Pareto distribution (GPD) was also
introduced and applied to the TSE data set. The GPD isa good tool to investigate the shape of tails if a data setof sufficient size is obtained.

[7] “Performance-Based Navigation Manual”, ICAO

Doc. 9613-AN/937 (3rd ed.), 2008.
[8] “Report of the Mathematicians Sub Group meetingat the SASP WG/WHL/10 meeting”, ICAO
SASP/WG/WHL/10, Gold Coast, November 2006.

6. ACKNOWLEDGMENTS[9] Coles, S, “An Introduction to Statistical Modeling

The author would like to thank the Japan Civil
Aviation Bureau and Sapporo ACC for the provisionof data sets.

of Extreme Values”, Springer, 2001.
[10] Fujita, M, “Distribution of Longitudinal Speed

Prediction Error of ADS-C System”, 3rdInternational Conference on Research in Air
Transportation (ICRAT2008), p.143-150.

REFERENCES[1] “Safety Management Manual”, ICAO Doc 9859-[11] http://www.assessment.ucar.edu/toolkit/

AN/460, 2006
[2] Reich, P. G., “Analysis of long-range air traffic

Copyright Statement

system: separation standards – I, II, III”, Journal of
the Institute of Navigation, vol. 19 (1966), p.88-98,
p.169-186, 331-347.

The author confirms that he, and/or his company or
institution, hold copyright of all original material
included in their paper. He also confirms he has
obtained permission, from the copyright holder of any
third party material included in their paper, to publish
it as part of their paper. The author grants full
permission for the publication and distribution of his
paper as part of the EIWAC 2009 proceedings or as
individual
off-prints
fromtheproceedings.

[3] “Manual on Airspace Planning Methodology for

the Determination of Separation Minima”, ICAODoc 9689-AN/953, 1998.
[4] “Manual on Implementation of a 300m (1 000ft)

Vertical Separation Minimum Between FL 290
and FL 410 Inclusive”, ICAO Doc. 9574-AN/934(2nd ed.), 2002.

138