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NLR-CR-2010-427

Collision risk assessment of 50 nm interseCting-traCk lateral separation in north atlantiC mnps airspaCe

G. Moek

NLR-CR-2010-427

Executive summary

UNCLASSIFIED

Report no. NLR-CR-2010-427 Author(s) G. Moek Report classification UNCLASSIFIED Date August 2010 Knowledge area(s) Safety & security Descriptor(s) collision risk lateral separation intersecting tracks North Atlantic MNPS airspace

COLLISION RISK ASSESSMENT OF 50 NM INTERSECTING-TRACK LATERAL SEPARATION IN NORTH ATLANTIC MNPS AIRSPACE Problem area Previous work on lateral collision risk modelling indicated that, based on the assumptions made, it would not be possible for the North Atlantic (NAT) to use 50 NM lateral separation of Minimum Navigation Perform-ance Specification (MNPS) aircraft on intersecting tracks. As a possible way forward, it was suggested to utilise the equipage mix of aircraft operating in NAT MNPS airspace. In response to this suggestion, this report presents the results of calculations of the lateral collision risk on intersecting tracks for a mixed aircraft population with MNPS, RNP 10, RNP 4 and GNSSS navigation performance, where GNSS navigation performance was treated as a free parameter. The calculations are based on a protected airspace concept where the protected airspace of an aircraft comprises a width of 50 NM either side of its nominal track.

Description of work A previously used collision risk model was extended in two steps. The model was first extended to the case of a mixed aircraft population and was subsequently made specific to the protected airspace concept and the lateral separation method. The thus extended collision risk model was used to calculate maximum and average lateral collision risk on intersecting tracks in NAT MNPS airspace assuming a 50 NM lateral separation minimum. Results and conclusions Compared with a “full MNPS” population, a mixed aircraft population with MNPS, RNP 10, RNP 4, and “GNSS as MNPS” performance gave a reduction of the maximum collision risk (averaged over a range of intersection angles from 5 to 175 degrees) of approximately 66%. When “GNSS as RNP 10” performance was used, the reduction of the maximum risk (averaged over the range of

NLR-CR-2010-427

UNCLASSIFIED

NLR Air Transport Safety Institute Anthony Fokkerweg 2, 1059 CM Amsterdam, P.O. Box 90502, 1006 BM Amsterdam, The Netherlands Telephone +31 20 511 35 00, Fax +31 20 511 32 10, Web site: http://www.nlr-atsi.nl

intersection angles) was approximately 78% of the “full MNP” risk. Assuming “GNSS as RNP 4” performance showed a marginally larger average reduction of 79% compared with the “full MNPS” risk. Application of a “larger than 1% of the peak value” averaging approach to the mixed aircraft population with “GNSS as RNP 10” performance gave, on average over the intersection angle, a reduction of the maximum risk by a factor of approximately three. An attempt to assess maximum and average collision risk in an absolute sense was hampered by two factors, namely that currently a Target Level of Safety (TLS) specific to the lateral

collision risk on intersecting tracks in NAT MNPS airspace does not (yet) exist, and that an estimate of the number of aircraft pairs passing an intersection in this airspace per flight hour was not readily available. Assuming a TLS of 9105 accidents per flight hour and one aircraft pair passing an intersection per flight hour, the average lateral collision risk was found to be less than the TLS for all intersection angles between 20 and 155 degrees inclusive. The TLS was exceeded by up to a factor of five for angles between 5 and 15 degrees inclusive and by up to a factor of seven for angles between 160 and 175 degrees inclusive.

NLR-CR-2010-427

COLLISION RISK ASSESSMENT OF 50 NM INTERSECTING-TRACK LATERAL SEPARATION IN NORTH ATLANTIC MNPS AIRSPACE

G. Moek No part of this report may be reproduced and/or disclosed, in any form or by any means without the prior written permission of the owner.

Customer ISAVIA Contract number ATSI/1181/6058 Owner National Aerospace Laboratory NLR Division Air Transport Distribution Limited Classification of title Unclassified August 2010 Approved by: Author

Reviewer Managing department

NLR-CR-2010-427

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CONTENTS

1 INTRODUCTION 7

2 LATERAL SEPARATION METHOD FOR INTERSECTING TRACKS 9

3 THE COLLISION RISK MODEL 13 3.1 Baseline collision risk model 13 3.2 Collision risk model for current application 15

4 RESULTS 21 4.1 Introduction 21 4.2 Maximum risk 21 4.3 Average collision risk. 27 4.3.1 Background 27 4.3.2 Results 38

5 CONCLUSIONS 43

6 REFERENCES 45

6 NLR-CR-2010-427 August 2010

ABBREVIATIONS

ATS Air Traffic Services FPL Flight Plan ft Feet GNSS Global Navigation Satellite System ICAO International Civil Aviation Organisation MNPS Minimum Navigation Performance Specification NAT North Atlantic NM Nautical Mile PANS-ATM Procedures Air Navigation Services - Air Traffic Management RNP Required Navigation Performance RNAV Area Navigation SARSIG Safety and Reduced Separation Implementation Group SASP Separation and Airspace Safety Panel TLS Target Level of Safety

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1 INTRODUCTION

It has been proposed by Iceland to allow RNP 10 approved aircraft to operate in NAT MNPS airspace and to use 50 NM lateral separation between any mix of MNPS and RNP 10 approved aircraft (Ref. 1). A safety case supporting the reduction of the current lateral separation minimum of 60 NM in the NAT to 50 NM would include, inter alia, a collision-risk-model-based assessment of the collision risk due to the loss of lateral separation for aircraft on parallel tracks as well as for aircraft on intersecting tracks. The former type of assessment is beyond the scope of this report, but the latter is the subject of the underlying report. Some work on lateral separation minima for aircraft on intersecting tracks has been carried out by the ICAO SASP. See references 2 and 3. Reference 1 has summarised the conclusion of the SASP calculations as follows: a) 50 NM lateral separation on intersecting tracks can be used for RNP 10

aircraft based on a Double Exponential distribution. b) Based on 100% MNPS equipage (i.e. no aircraft equipped to a more stringent

requirement), maximum collision risk and a Double Exponential distribution 50 NM lateral separation on intersecting tracks cannot be used for MNPS aircraft. A Gaussian distribution would however support the use of a 50 NM lateral separation on intersecting tracks for MNPS aircraft.

Reference 1 then noted that it was clear that those results do not enable the NAT to use 50 NM lateral separation of MNPS aircraft on intersecting tracks, but that there are other factors that need to be taken into account when investigating the suitability of using the 50 NM separation for MNPS aircraft on intersecting tracks, notably the equipage mix of aircraft operating in NAT MNPS airspace. Based on an analysis of flight plans filing MNPS capability for the period of time from 1 January – 30 June 2009, reference 1 concluded that only 28.9% of MNPS aircraft in the Reykjavik area only have the basic MNPS capability. Reference 1 recommended, therefore, that collision risk calculations similar to the SASP-WG/WHL-WP/28 calculations be carried out based on the surveyed equipage mix rather than for a full basic MNPS capability only.

8 NLR-CR-2010-427 August 2010

This report presents the results of such calculations based on an extended survey of the equipage mix of aircraft operating in NAT MNPS airspace for the period of time from 1 May 2009 – 30 April 2010. The extended survey comprised 136,321 flight plans for MNPS approved aircraft (X in FPL field 10) and showed that 24.3% of the MNPS aircraft surveyed had only the basic MNPS capability. Notice that this percentage is approximately 4.5 percentage points smaller than the 28.9% found in the initial survey. The key difference between the SASP calculations (Refs. 2 and 3) and the current ones concerns the probability distributions for the along-track and cross-track navigational errors. Whereas the SASP calculations assumed that these distributions were single distributions representing a single navigation specification, the current calculations assume that the along-track and cross-track navigational error distributions are mixture distributions made up of four different navigation specifications, namely MNPS, RNP 4, RNP 10, and GNSS. The weighting factors for the individual navigation-specification distributions are based on the extended equipage mix survey. In line with SASP practise, the navigational error probability distributions for MNPS, RNP 4, and RNP 10 aircraft are taken as Double Exponential distributions with the standard deviations based on the pertinent accuracy requirements (95%). GNSS navigational error distributions will also be assumed to be Double Exponential distributions, but the standard deviations will not be fixed in advance. It should be noted that the current use of mixture navigational-error-distributions based on different navigation specifications still comprises a form of using required performance rather than of using actual performance, since it is likely that the actual performance of the aircraft approved to a particular navigation specification is better than the required performance.1 Section 2 of the report recalls the lateral separation method for intersecting tracks utilised in the SASP calculations of references 2 and 3. Section 3 provides some background of the SASP collision risk model and describes the extension of the model to account for the navigational-error mixture-distributions. Results and conclusions then follow in the sections 4 and 5.

1 Using actual performance as another step has also been suggested in reference 1 but has not been explored in this report.

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2 LATERAL SEPARATION METHOD FOR INTERSECTING TRACKS

The lateral separation method from references 2 and 3 is based on a protected airspace concept as follows. Consider two aircraft on intersecting tracks as shown in figure 2.1. Aircraft 1 has a protected airspace of width S NM either side of it. Aircraft 2 enters aircraft 1’s protected airspace after vertical separation is established as shown in figure 2.2. Lateral separation of a pair of aircraft on intersecting tracks is then defined to exist if aircraft 2 is outside the protected airspace of aircraft 1 OR aircraft 2 is outside the protected airspace of aircraft 1. Consequently, lateral separation between the two aircraft does not exist when both aircraft are simultaneously inside each other’s protected airspace, i.e. when they are simultaneously within the red dash-lined rhombus. (The midpoints of the sides of the rhombus are the points 13A , 23A , 15A , and 25A .) The along-track distance L at which an aircraft may lose lateral separation is given by

sin1324262415242423SAAAAAAAAL (2.1)

The above concept of lateral separation corresponds with that described in paragraph 5.4.1.2.1.5, RNAV operations (where RNP is specified) on intersecting tracks or ATS routes, and illustrated in figure 5-5, of the PANS-ATM (Ref. 4). It is different, however, to the concept of lateral separation based on non-overlapping protected airspace utilized in reference 5. Having defined the lateral separation concept, the next step of the lateral separation method is to calculate the lateral collision risk for a specified value of the lateral separation minimum S , here 50S NM, and to compare the calculated risk value with an appropriate Target Level of Safety (TLS). The risk is calculated before, during, and after the climb-descent manoeuvre of aircraft 2 over the track of aircraft 1 shown in figure 2.2. Let jt2 , 7,...1j , denote the points in time at which aircraft 2 passes the points

jA2 and similarly for aircraft 1. Suppose a controller expects that lateral separation will not be able to be maintained between the pair of aircraft in figure 2.2 during their passing of the intersection and requires aircraft 2 to be vertically

10 NLR-CR-2010-427 August 2010

Figure 2.1 Plan view of the protected airspaces about aircraft 1 and aircraft 2 on tracks intersecting at an angle θ

Figure 2.2 Sketch of two aircraft on intersecting tracks showing aircraft 2 achieving vertical separation upon entering aircraft 1’s protected airspace and ending vertical separation upon leaving aircraft 1’s protected airspace

S

S

S

S θ

23A

25A

13A

15A

ac1

ac2

S

S θ

23A

25A

13A 15A

ac1

ac2

22A21A

26A27A

24A

NLR-CR-2010-427

August 2010 11

separated by the point 23A at time 23t . Aircraft 1 may then be anywhere between 13A and 15A , and the lateral collision risk for the aircraft will depend on the

location, say, )( 231 tx , of aircraft 1 at time 23t . The lateral collision risk as a function of )( 231 tx can be calculated as

)(;,0)(;,)( 23127252312321231 txttCRtxttCRtxCR (2.2) where the first term in the right-hand side of eq. (2.2) represents the lateral risk prior to achieving vertical separation at time 23t , the middle “zero” term expresses that there is no lateral collision risk when the aircraft are vertically separated during the time interval ],[ 2523 tt , and the last term in the right-hand side of eq. (2.2) represents the lateral risk after ending vertical separation at time

25t . The collision risk model used in references 2 and 3 for the two risk terms

)(;, 2312321 txttCR and )(;, 2312725 txttCR is given by

23

21

21221123121

2312321

)()(2

2))(())(;,|(2

))(;,(t

t zxy

relzz dVdtdVVfVf

zVthPtxVVtHOP

txttCR

(2.3)

27

25

21221123121

2312725

)()(2

2))(())(;,|(2

))(;,(t

t zxy

relzz dVdtdVVfVf

zVthPtxVVtHOP

txttCR

(2.4) The model calculates the expected number of accidents per aircraft pair passing an intersection. The factor of 2 upfront in the right-hand side of eqs. (2.3) and (2.4) accounts for the conversion of collisions into accidents. Notice that eqs. (2.3) and (2.4) differ only with regard to the range of the time integration. Since the form of the collision risk model in eqs. (2.3) and (2.4) may not be familiar to the SARSIG, some background information will be provided in section 3.

12 NLR-CR-2010-427 August 2010

Rather than calculating the lateral collision risk )( 231 txCR explicitly as a function of )( 231 tx , reference 2 calculated a conservative estimate by maximising over all possible locations )( 231 tx of aircraft 1. The same maximisation approach was used in reference 3, but )( 231 txCR was also examined as a function of the location of aircraft 1 for some cases in order to calculate an average collision risk value.

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3 THE COLLISION RISK MODEL

Like the collision risk model used in references 2 and 3, the collision risk model used in this report is a select version of the collision risk model based on reliability theory that allows for unequal navigational accuracy from reference 6. The reference 6 model is well known within the ICAO SASP, but is probably less well-known within the SARSIG. Therefore, a few mathematical details of the model are included in this section for the benefit of the mathematician participants of the SARSIG. Following the description of the reference 6 model, the modifications necessary for the current assessment will be presented.

3.1 BASELINE COLLISION RISK MODEL The collision risk model of reference 6 calculates the expected number of accidents per aircraft pair passing an intersection under an angle . See figure 3.1.

Figure 3.1 Nominal and true positions of the aircraft as a result of along-track and cross-track navigational errors The model assumes that each aircraft has along-track and cross-track navigational errors and that the true aircraft speeds 1V and 2V have probability densities )( 11 Vf and )( 22 Vf respectively. The key of the model is an integration

θ

A2ac2

ac1

x

y

C2

C1

A1

True position of aircraft 1

Nominal position of aircraft 1

True position of aircraft 2

Nominal position of aircraft 2

14 NLR-CR-2010-427 August 2010

over a suitable time interval ],[ 10 tt during which the aircraft are exposed to the risk of the loss of lateral separation on intersecting tracks. Equation (3.1) shows the baseline collision risk model.

1

0

2122112110 )()(22))((),|(2),(

t

t zxy

relzz dVdtdVVfVf

zVthPVVtHOPttCR (3.1)

The integrals in the right-hand side of eq. (3.1) include an instantaneous probability of horizontal overlap, ),|( 21 VVtHOP , an instantaneous probability of vertical overlap ))(( thP zz , and a kinematic factor (expression in square brackets) representing the two ways in which a collision can develop. Notice that the effect of the aircraft speed errors is represented by the outer two integrals in eq. (3.1) and that the probability of horizontal overlap is conditional on the aircraft speeds. The probability of horizontal overlap at time t further depends on the navigational errors of the two aircraft as

222 )()(121222112211

22221

)()()()(

})()({Pr),|(

xytytx

AACCCCCCAAAA

xy

ddddffff

tytxobVVtHOP

(3.2)

where )(xf Ai and )(xf ci denote the probability density functions for along-track and cross-track navigational errors, respectively, for aircraft 2,1i and )(tx and )(ty denote the true differences in x - and y -coordinates at time t given

1V and 2V . The intersection angle appears in the relative speed relV between the two aircraft,

cos2 212

22

1 VVVVVrel (3.3) and in the true differences )(tx and )(ty in x - and y -coordinates

sincos)()()()( 22121CAA

x tDtxtxtx (3.4)

cossin)()()()( 22121CAC

y tDtytyty (3.5)

NLR-CR-2010-427

August 2010 15

cosˆˆcos)( 020121 ddtVVtDx (3.6)

sinˆ)( 202 tVdtDy (3.7) The quantities )(tDx and )(tDy denote the nominal differences in x - and y -coordinates at time t given 1V and 2V and the nominal distances 01̂d and 02d̂ between the aircraft and the intersection at time 0t (Ref. 6). Since the region 222 )()( xytytx is small, the probability of horizontal overlap ),|( 21 VVtHOP can be approximated by

0)(,0)(121222112211

221

)()()()(

}0)(,0)({Pr),|(

tytx

AACCCCCCAAAA

xy

ddddffff

tytxobVVtHOP

(3.8)

The two conditions 0)( tx and 0)( ty can now be used to reduce the four-fold integral of eq. (3.8) to a double integral by eliminating, for example, A1 and C1 . Thus,

A C

ACCCy

CACAAx

CAAxy ddftDfftDf

VVtHOP

2 2

2222221222212

21

)())(cossin()())(sincos(

),|(

(3.9) The double integral in eq. (3.9) has been evaluated in reference 6 for two cases, namely for either all the four probability densities being single Double Exponential probability densities, or all the four probability densities being single Gaussian probability densities. See reference 6 for details of that evaluation.

3.2 COLLISION RISK MODEL FOR CURRENT APPLICATION For the current application, the select collision risk model is obtained in two steps. Firstly, the baseline collision risk model is extended to the case of a mixed aircraft population. Secondly, the thus extended model will be made specific for the lateral separation method outlined in section 2.

16 NLR-CR-2010-427 August 2010

Mixed aircraft population Thus, the first step is to account for the different navigation specifications to which the NAT MNPS aircraft population has been approved, i.e. MNPS only, MNPS and RNP 10, MNPS and RNP 4, and MNPS and GNSS. To this end it is necessary to return to the approximation for the probability of horizontal overlap

),|( 21 VVtHOP in eq. (3.8) and to notice that the integrand is the product of the joint along-track and across-track probability densities of aircraft 1 and aircraft 2. In the case of a mixed aircraft population, each aircraft’s joint along-track and cross-track navigational error probability density becomes a mixture density, i.e.

GNSSC

GNSSA

GNSSRNPC

RNPA

RNP

RNPC

RNPA

RNPMNPSC

MNPSA

MNPSCA

yfxfyfxfyfxfyfxfyxf

)()()()(

)()()()(),(

1141414

1011011011,

1

(3.10)

and, similarly,

GNSSC

GNSSA

GNSSRNPC

RNPA

RNP

RNPC

RNPA

RNPMNPSC

MNPSA

MNPSCA

yfxfyfxfyfxfyfxfyxf

)()()()(

)()()()(),(

2242424

1021021022,

2

(3.11)

with

1410 GNSSRNPRNPMNPS (3.12) Putting

MNPSji 0 (3.13)

101 RNPji (3.14)

42 RNPji (3.15)

GNSSji 3 , (3.16) eqs. (3.11) and (3.12) can be expressed in a slightly more convenient form as

3

011

,1 )()(),(

ii

Ci

Ai

CA yfxfyxf (3.17)

NLR-CR-2010-427

August 2010 17

3

022

,2 )()(),(

jj

Cj

Aj

CA yfxfyxf (3.18)

and the product of the two left-hand side probability densities can be written as

3

0

3

02211221122

,211

,1 )()()()(),(),(

i jj

Ci

Cj

Ai

Aji

CACA yfyfxfxfyxfyxf (3.19)

Equation (3.19), with the appropriate arguments for 2121 and ,,, yyxx needs to be substituted into eqs. (3.8) and (3.9) for the probability of horizontal overlap

),|( 21 VVtHOP . Interchanging the order of the double integration and the double summation gives

3

0

3

0,2121 ),|(),|(

i jjiji VVtHOPVVtHOP (3.20)

where

A C

ACj

CCiy

CACj

AAix

CAA

xyji

ddftDfftDf

VVtHOP

2 2

222222122221

2,21

)())(cossin()())(sincos(

),|(

(3.21) Substitution of eq. (3.20) for the probability of horizontal overlap ),|( 21 VVtHOP into the baseline collision risk model of eq. (3.1) and changing the order of the integrations and summations gives the following collision risk model for the mixed aircraft population

3

0

3

0212211,2110

1

0

)()(2

2))((),|(2),(i j

t

t zxy

relzzjiji dVdtdVVfVf

zVthPVVtHOPttCR

(3.22)

or

3

0

3

0,1010 ),(),(

i jjiji ttCRttCR (3.23)

with

18 NLR-CR-2010-427 August 2010

1

212211,21,10

0

)()(2

2))((),|(2),(t

t zxy

relzzjiji dVdtdVVfVf

zVthPVVtHOPttCR

(3.24) Lateral separation method Recall that the collision risk model for the lateral separation method described in section 2 was given by eqs. (2.2) – (2.4). Specific to the method was the split up of the time integration into three parts, or, effectively, two non-trivial parts. Eqs. (2.2) – (2.4) also used a slightly refined notation for the probability of horizontal overlap at time t , viz. ))(;,|( 23121 txVVtHOP to emphasise the dependence of this probability on the location of aircraft 1. Like the baseline model of section 3.1, the model of eqs. (2.2) – (2.4) is valid for a uniform aircraft population with the navigational error probability densities being single probability densities. The collision risk model of eqs. (3.21), (3.23), and (3.24) for a mixed aircraft population can now be further extended to account for the lateral separation method from section 2 by splitting up the time integration and making the dependence on the location of aircraft 1 explicit. The result is the following model.

)(;,0)(;,)( 23127252312321231 txttCRtxttCRtxCR (3.25)

3

0

3

0,23123212312321 ))(;,())(;,(

i jjiji txttCRtxttCR (3.26)

3

0

3

0,23127252312725 ))(;,())(;,(

i jjiji txttCRtxttCR (3.27)

23

21

212211,23121

,2312321

)()(2

2))(())(;,|(2

))(;,(t

t zxy

relzzji

ji

dVdtdVVfVfzVthPtxVVtHOP

txttCR

(3.28)

NLR-CR-2010-427

August 2010 19

27

25

212211,23121

,2312725

)()(2

2))(())(;,|(2

))(;,(t

t zxy

relzzji

ji

dVdtdVVfVfzVthPtxVVtHOP

txttCR

(3.29) with jitxVVtHOP ,23121 ))(;,|( given by eq. (3.21). The location )( 231 tx of aircraft 1 enters into jitxVVtHOP ,23121 ))(;,|( through 01d̂ in

)(tDx of eqs. (3.6) and (3.4). In summary, the collision risk model for the current application is given by eq. (3.21) together with eqs. (3.25) – (3.29). Remaining parameters A few parameters of the collision risk model remain to be specified. Based on the extended survey of the equipage mix of aircraft operating in NAT MNPS airspace for the period of time from 1 May 2009 – 30 April 2010, the mixed aircraft population proportions are

321,136118,33MNPS (3.30)

321,136790,8210 RNP (3.31)

321,136749,14 RNP (3.32)

321,136664,18GNSS (3.33) The (average) aircraft dimensions have been taken from the NAT MNPS Risk Quick Reference Guide (Ref. 7) as

03108.0xy NM (188.8 ft) (3.34)

00892.0z NM (54.2 ft) (3.35) The reference 3 calculations were based on (maximisation over) three nominal aircraft speeds, namely 300, 480, and 600 kts and the same values will be used as starting point in this report. The middle value is equal to the average aircraft

20 NLR-CR-2010-427 August 2010

speed of 480 kts listed in reference 7. Assuming a speed range of Mach 0.74 to Mach 0.86 gives a range of approximately 420 kts to 540 kts for the nominal aircraft speeds. Experience suggests (see e.g. reference 3) that, when maximised over the nominal aircraft speeds, the largest risk is found when aircraft 1 has the maximum nominal speed and aircraft 2 has the minimum nominal speed. Calculations will also be performed, therefore, for the following triple of nominal aircraft speeds: 420, 480, and 540 kts. The speed probability densities )( 11 Vf and )( 22 Vf have been taken as Double Exponential densities, centred at the nominal speeds with a scale parameter equal to 82.5v kts (Ref. 6). For aircraft in level flight, 5.1z kts is assumed (Refs. 2, 3, and 7) and the climb and descent rates for aircraft 2 are taken as 1000ft/min (9.8747 kts) (Refs. 2 and 3). The last collision risk model parameter to be discussed is the instantaneous probability of vertical overlap ))(( thP zz . In the same manner as in references 2 and 3, this is calculated as a function of the nominal height difference between two aircraft, and a model originally developed in reference 6 based on a European paired data sample of aircraft height deviations scaled so that

55.0)0( zP . The model was acknowledged to be conservative.

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4 RESULTS

4.1 INTRODUCTION As set out in section 3, the collision risk model accounts for speed errors through the probability densities )( 11 Vf and )( 22 Vf of the actual aircraft speeds (nominal speeds plus speed errors). The effect of speed errors on the calculated risk values, however, has been found to be small though at the cost of very significant increases in calculation time (Refs. 2 and 3). It was desirable, therefore, to examine the feasibility of not using the speed error option of the collision risk model. To this end, maximum collision risk was calculated for three intersection angles, namely 15, 50, and 90 degrees, for a mixed aircraft population (with GNSS as RNP 10) with and without speed errors. The results were found to be virtually the same and all the remaining calculations in sections 4.2 and 4.3 have been carried out without speed errors. In the same manner as in references 2 and 3, all calculations are based on a 2000 ft level change for aircraft 2 at a climb/descent rate of 1000 ft/minute. Section 4.2 presents results on maximum risk and section 4.3 results on average risk.

4.2 MAXIMUM RISK Table 4.1 shows maximum collision risk in fatal accidents per flight hour as a function of track intersection angle for various aircraft populations and fixed speeds, i.e. no speed errors. The results are valid for the case of 1 aircraft pair passing an intersection per flight hour. Results for the general case of NP aircraft pairs passing an intersection per flight hour can be obtained by multiplying the table 4.1 entries by the factor NP (pairs/flight hour). For each track intersection angle, the table shows collision risk maximised over both the location of aircraft 1 and the nominal aircraft speeds

600 ,480 ,300 , 21 VV kts, i.e. for each combination of aircraft speeds the risk is maximised over the aircraft 1 location and then the largest of the maxima over the nine aircraft speed combinations is taken. For each aircraft population and intersection angle, the maximum collision risk occurred for the largest nominal speed of aircraft 1 (600 kts) and the smallest nominal speed of aircraft 2 (300

22 NLR-CR-2010-427 August 2010

kts). The speed combination producing the maximum risk but one was 480 kts for aircraft 1 and 300 kts for aircraft 2.

θ (degrees)

Aircraft population

Mixed aircraft population Full MNPS GNSS as

MNPS GNSS as RNP 10

GNSS as RNP 4

5 2.17E-07 7.71E-08 5.25E-08 4.70E-08 10 1.42E-07 5.00E-08 3.38E-08 3.04E-08 15 9.23E-08 3.23E-08 2.17E-08 1.96E-08 20 5.93E-08 2.07E-08 1.38E-08 1.26E-08 25 3.82E-08 1.33E-08 8.82E-09 8.09E-09 30 2.57E-08 8.89E-09 5.87E-09 5.42E-09 35 1.84E-08 6.35E-09 4.17E-09 3.88E-09 40 1.42E-08 4.90E-09 3.22E-09 3.00E-09 45 1.17E-08 4.05E-09 2.66E-09 2.49E-09 50 1.02E-08 3.54E-09 2.32E-09 2.17E-09 55 9.59E-09 3.30E-09 2.15E-09 2.02E-09 60 9.60E-09 3.31E-09 2.16E-09 2.01E-09 65 1.03E-08 3.55E-09 2.32E-09 2.15E-09 70 1.17E-08 4.01E-09 2.62E-09 2.42E-09 75 1.38E-08 4.63E-09 3.01E-09 2.77E-09 80 1.61E-08 5.32E-09 3.46E-09 3.17E-09 85 1.80E-08 5.89E-09 3.83E-09 3.49E-09 90 1.89E-08 6.19E-09 4.02E-09 3.66E-09 95 1.88E-08 6.15E-09 4.00E-09 3.65E-09

100 1.77E-08 5.84E-09 3.80E-09 3.47E-09 105 1.62E-08 5.37E-09 3.49E-09 3.20E-09 110 1.47E-08 4.92E-09 3.19E-09 2.94E-09 115 1.36E-08 4.57E-09 2.97E-09 2.74E-09 120 1.33E-08 4.43E-09 2.88E-09 2.66E-09 125 1.37E-08 4.56E-09 2.97E-09 2.75E-09 130 1.50E-08 5.03E-09 3.28E-09 3.03E-09 135 1.74E-08 5.89E-09 3.85E-09 3.56E-09 140 2.16E-08 7.29E-09 4.77E-09 4.40E-09 145 2.85E-08 9.67E-09 6.35E-09 5.83E-09 150 4.01E-08 1.37E-08 9.05E-09 8.27E-09

NLR-CR-2010-427

August 2010 23

155 5.95E-08 2.04E-08 1.35E-08 1.23E-08 160 8.96E-08 3.08E-08 2.05E-08 1.85E-08 165 1.31E-07 4.56E-08 3.06E-08 2.75E-08 170 1.86E-07 6.53E-08 4.42E-08 3.96E-08 175 2.57E-07 9.12E-08 6.21E-08 5.54E-08

Table 4.1 Maximum collision risk in fatal accidents per flight hour as a function of track intersection angle θ for one aircraft pair passing an intersection per flight hour and various aircraft populations. Nominal aircraft speeds 600 ,480 ,300 , 21 VV kts, no speed errors. Values less than a TLS of 9105 fatal accidents per flight hour shaded light yellow Consider the “Full MNPS” aircraft population first. The numbers in the sceond column of table 4.1 may be compared with those for double exponential position errors in table 11.1 of reference 3. The current numbers are smaller by a factor of approximately 7. This reduction is the result of two factors, namely the assumed number of aircraft pairs passing an intersection per flight hour, and the dimensions of a typical aircraft. Reference 3 conservatively assumed that 5NP aircraft pairs would pass an intersection per flight hour and also, conservatively, utilized the Airbus A380 aircraft dimensions. None of the “Full MNPS” maximum collision risk values in table 4.1 meets a TLS of 9105 fatal accidents per flight hour, i.e. the threshold that was used in references 2 and 3 to compare the calculated risk against. The remaining three columns show the effect of a mixed aircraft population made up of four sub-populations as set out in section 3.2, namely MNPS approved only, MNPS and RNP 10 approved, MNPS and RNP 4 approved, and MNPS approved only and carrying GNSS. For the first three sub-populations, the navigation performance was taken as MNPS, RNP 10, and RNP 4 respectively. For the “MNPS approved only and GNSS carrying” sub-population, three different navigation performance levels were assumed, namely MNPS, RNP 10, and RNP 4. The results in the third column of table 4.1, “GNSS as MNPS”, show the effect of the improved navigation performance of the RNP 10 and RNP 4 sub-populations. The (maximum) collision risk values in the third column are, on average, 66% smaller than their “Full MNPS” counterparts. The (maximum) collision risk is now smaller than the TLS of 9105 fatal accidents per flight hour for a number of

24 NLR-CR-2010-427 August 2010

intersection angles, namely for 7540 degrees and for 125110 degrees. The results in the fourth column of table 4.1, “GNSS as RNP 10”, show the effect of assuming better than MNPS navigation performance for the sub-population of “MNPS-approved-only aircraft carrying GNSS”. Assuming the navigation performance of this sub-population to be the same as that of RNP 10 aircraft results in (maximum) collision risk values in the fourth column that are now, on average, 78% smaller than their “Full MNPS” counterparts. The (maximum) collision risk is now also smaller than the TLS of 9105 fatal accidents per flight hour for a continuous range of intersection angles, namely for 14035 degrees. Finally, the results in the fifth column of table 4.1, “GNSS as RNP 4”, show the effect of assuming the navigation performance of the sub-population of “MNPS-approved-only aircraft carrying GNSS” to be the same as that of RNP 4 aircraft. The (maximum) collision risk values in the fifth column are, on average, 79% smaller than their “Full MNPS” counterparts, i.e. marginally smaller than those in the fourth column. The range of intersection angles for which the (maximum) collision risk is smaller than the TLS of 9105 fatal accidents per flight hour is also the same as for the “GNSS as RNP 10” case. This suggests that some law of diminishing returns is active, i.e. the overall collision risk is dominated by aircraft pairs involving “MNPS approved only” aircraft. This is illustrated in table 4.2 for an intersection angle of 55 degrees. Recall eqs. (3.13) – (3.16) and eqs. (3.25) – (3.29). In the table, “Risk term

jiCR ,(...) ” denotes the sum of jiCR ,(...) defined in eqs. (3.28) and (3.29) and similarly for the “Weighted risk term ji jiCR ,(...) ” . The contribution of two sub-populations i and j to the overall collision risk for a given intersection angle is determined by the product of their proportions i and j (Table 4.2, column 3) and the (conditional) risk jiCR ,(...) due to their navigation specifications (table 4.2, columns 4 and 9) and are shown in the fifth and eighth columns of table 4.2 for the “GNSS as RNP 4” and “GNSS as RNP 10” cases respectively. The entries involving the GNSS sub-population ( 3i or 3j ) are highlighted in light yellow. Although the (weighted) risk terms for 3i or 3j are seen to be (sometimes considerably) smaller for the “GNSS as RNP 4” case than for the “GNSS as RNP 10” case, this has little effect on the cumulative

NLR-CR-2010-427

August 2010 25

weighted risk, particularly due to the interaction between MNPS and RNP 10 aircraft as reflected in the first two rows of table 4.2.

GNSS as RNP 4 GNSS as RNP 10

i j

Weight ji

Risk term jiCR ,(...)

Weighted risk term

ji

jiCR ,(...)

Cumula-tive

weighted risk

Cumula-tive

weighted risk

Weighted risk term

ji

jiCR ,(...)

Risk term jiCR ,(...)

Weight ji

0 0 5.9E-02 9.59E-09 5.66E-10 5.66E-10 5.66E-10 5.66E-10 9.59E-09 5.9E-020 1 1.5E-01 6.62E-09 9.77E-10 1.54E-09 1.54E-09 9.77E-10 6.62E-09 1.5E-010 2 3.1E-03 4.66E-09 1.45E-11 1.56E-09 1.56E-09 1.45E-11 4.66E-09 3.1E-030 3 3.3E-02 4.66E-09 1.55E-10 1.71E-09 1.78E-09 2.20E-10 6.62E-09 3.3E-021 0 1.5E-01 1.01E-09 1.49E-10 1.86E-09 1.93E-09 1.49E-10 1.01E-09 1.5E-011 1 3.7E-01 3.46E-10 1.28E-10 1.99E-09 2.05E-09 1.28E-10 3.46E-10 3.7E-011 2 7.8E-03 1.72E-10 1.34E-12 1.99E-09 2.06E-09 1.34E-12 1.72E-10 7.8E-031 3 8.3E-02 1.72E-10 1.43E-11 2.00E-09 2.08E-09 2.88E-11 3.46E-10 8.3E-022 0 3.1E-03 2.82E-10 8.79E-13 2.01E-09 2.09E-09 8.79E-13 2.82E-10 3.1E-032 1 7.8E-03 4.96E-12 3.86E-14 2.01E-09 2.09E-09 3.86E-14 4.96E-12 7.8E-032 2 1.7E-04 5.24E-21 8.63E-25 2.01E-09 2.09E-09 8.63E-25 5.24E-21 1.7E-042 3 1.8E-03 5.24E-21 9.20E-24 2.01E-09 2.09E-09 8.71E-15 4.96E-12 1.8E-033 0 3.3E-02 2.82E-10 9.38E-12 2.01E-09 2.12E-09 3.36E-11 1.01E-09 3.3E-023 1 8.3E-02 4.96E-12 4.12E-13 2.02E-09 2.15E-09 2.88E-11 3.46E-10 8.3E-023 2 1.8E-03 5.24E-21 9.20E-24 2.02E-09 2.15E-09 3.02E-13 1.72E-10 1.8E-033 3 1.9E-02 5.24E-21 9.82E-23 2.02E-09 2.15E-09 6.49E-12 3.46E-10 1.9E-02 Table 4.2 Comparison of (maximum) collision risk when navigation performance of MNPS-approved-only aircraft carrying GNSS is assumed to be RNP 4 and RNP 10. Track intersection angle 55 degrees. No speed errors It should be remarked that the collision risk values in table 4.1 are conservative in that they are maximum values, obtained by maximizing over both the location of aircraft 1 and the nominal aircraft speed combinations. Concerning the latter, it was found that for each intersection angle , the largest risk was obtained for

6001 V kts and 3002 V kts. This suggests that the maximum speed difference between the two aircraft may have a significant effect on the value of the collision risk. Similar calculations as in table 4.1 have been carried out, therefore, for a speed range of 420 kts to 540 kts, which is more in line with the range of mach speeds utilized in NAT MNPS airspace. The results of these

26 NLR-CR-2010-427 August 2010

calculations are shown in table 4.3. For each aircraft population, the risk values in table 4.3 are, on average over the intersection angle , 46% smaller than their table 4.1 counterparts, with a maximum reduction of 65% and a minimum reduction of 19% (for 175 degrees). The average reduction is 14 percentage points smaller than the nominal speed range reduction of 60%. Concerning the maximisation over the location of aircraft 1, it should be noted that, as explained in reference 3, testing the maximum collision risk against a TLS is an efficient way of proving the safety of a procedure, when it works, but it is a stricter test than necessary. Therefore, consistent with earlier SASP work (see reference 3 and the references therein), collision risk averaged over the location of aircraft 1 has also been examined and is reported in section 4.3 below.

θ

(degrees) Aircraft population

Mixed aircraft population Full MNPS GNSS as

MNPS GNSS as RNP 10

GNSS as RNP 4

5 1.36E-07 4.88E-08 3.33E-08 2.99E-08 10 9.26E-08 3.16E-08 2.12E-08 1.90E-08 15 5.85E-08 1.89E-08 1.24E-08 1.13E-08 20 3.40E-08 1.11E-08 7.21E-09 6.62E-09 25 1.88E-08 6.41E-09 4.19E-09 3.88E-09 30 1.31E-08 4.87E-09 3.23E-09 3.07E-09 35 8.54E-09 2.83E-09 1.83E-09 1.73E-09 40 5.52E-09 1.95E-09 1.28E-09 1.22E-09 45 4.45E-09 1.56E-09 1.02E-09 9.68E-10 50 3.82E-09 1.33E-09 8.65E-10 8.21E-10 55 3.49E-09 1.20E-09 7.79E-10 7.39E-10 60 3.41E-09 1.17E-09 7.59E-10 7.16E-10 65 3.58E-09 1.23E-09 7.97E-10 7.47E-10 70 4.06E-09 1.38E-09 8.93E-10 8.32E-10 75 4.81E-09 1.60E-09 1.03E-09 9.55E-10 80 5.78E-09 1.87E-09 1.20E-09 1.11E-09 85 6.64E-09 2.12E-09 1.35E-09 1.24E-09 90 7.12E-09 2.26E-09 1.44E-09 1.32E-09 95 7.04E-09 2.24E-09 1.43E-09 1.32E-09

100 6.57E-09 2.12E-09 1.36E-09 1.25E-09

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August 2010 27

105 5.93E-09 1.94E-09 1.25E-09 1.15E-09 110 5.40E-09 1.79E-09 1.15E-09 1.07E-09 115 5.10E-09 1.70E-09 1.09E-09 1.02E-09 120 5.15E-09 1.71E-09 1.10E-09 1.02E-09 125 5.57E-09 1.85E-09 1.19E-09 1.11E-09 130 6.44E-09 2.15E-09 1.39E-09 1.30E-09 135 7.97E-09 2.68E-09 1.73E-09 1.62E-09 140 1.05E-08 3.53E-09 2.29E-09 2.13E-09 145 1.49E-08 5.00E-09 3.26E-09 3.02E-09 150 2.25E-08 7.65E-09 5.01E-09 4.60E-09 155 3.63E-08 1.24E-08 8.16E-09 7.45E-09 160 6.03E-08 2.06E-08 1.36E-08 1.23E-08 165 9.78E-08 3.36E-08 2.24E-08 2.02E-08 170 1.53E-07 5.32E-08 3.58E-08 3.21E-08 175 2.32E-07 8.19E-08 5.56E-08 4.97E-08

Table 4.3 Maximum collision risk in fatal accidents per flight hour as a function of track intersection angle θ for one aircraft pair passing an intersection per flight hour and various aircraft populations. Nominal aircraft speeds 540 ,480 ,420 , 21 VV kts, no speed errors. Values less than a TLS of 9105 fatal accidents per flight hour shaded light yellow

4.3 AVERAGE COLLISION RISK. There are various ways to calculate an average collision risk value and one method proposed in reference 3 was to average over all initial locations of aircraft 1 for which the risk is larger than 1% of the maximum collision risk. Before presenting average collision risk results, it is useful to gain some insight into the dependence of the risk on the initial location of aircraft 1 first.

4.3.1 BACKGROUND Consider aircraft 2 passing the intersection. The lateral separation method will be applied each time that aircraft 1 is expected to be anywhere between the points 13A and 15A of its nominal track during the time interval in which aircraft 2 is between the points 23A and 25A on aircraft 2’s nominal track. Cf. figure 2.2. Thus, the procedure will be executed when

28 NLR-CR-2010-427 August 2010

sin,

sin)(1

SStx (4.1)

for any point in time 2523, ttt . This defines the range of initial locations )( 231 tx of aircraft 1 for which the lateral separation procedure will be applied to aircraft 2. Assuming constant aircraft speeds, it follows that

sin,21

sin)(,)(

2

1231231

SVVStxtx RL (4.2)

Clearly, this range of initial locations is larger the larger 1V and/or the smaller

2V . The speed ratio has a significant effect on the lower end of the range of initial locations of aircraft 1, e.g. an overall factor of five for 6001 V kts and

3002 V kts. Figure 4.1 shows the range of aircraft 1 initial locations as a function of the intersection angle for three combinations of nominal aircraft speeds, i.e.

6001 V kts and 3002 V kts, 4801 V kts and 4802 V kts, and 3001 V kts and 6002 V kts. Notice that there is only a single curve for the upper limit of the interval defined by eq. (4.2). The upper limit of the interval for the initial location )( 231 tx of aircraft 1 corresponds with aircraft 1 leaving the lateral conflict area upon aircraft 2 entering it whereas the lower limit corresponds with aircraft 1 entering the lateral conflict area when aircraft 2 is about to leave it. These two situations will involve only very little risk of collision because the two aircraft are relatively far apart. Figure 4.2 shows the same quantities for the smaller range of nominal aircraft speeds of 420, 480, and 540 kts. Notice that the pink curves in figures 4.1 and 4.2 for equal aircraft speeds are the same. Because the ratio between the largest and smallest speeds in figure 4.2 is approximately 65% smaller than in figure 4.1, the potential range of initial locations of aircraft 1 to be considered is much smaller for the smaller range of nominal aircraft speeds. Nonetheless, in both cases, these ranges are impractically large, particularly for very small and very large intersection angles. Indeed, as noted in reference 3, the particular lateral separation method is not well suited for (very) small or large intersection angles.

NLR-CR-2010-427

August 2010 29

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000

0 20 40 60 80 100 120 140 160 180 200

theta (degrees)

x1(t2

3)L,

x1(

t23)

R (N

M)

600, 300480, 480300, 600all

Figure 4.1 Range of initial locations )( 231 tx of aircraft 1 to be considered for collision risk assessment as a function of intersection angle when aircraft 2 changes level on passing the intersection. Nominal aircraft speeds of 300, 480, and 600 kts

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000

0 20 40 60 80 100 120 140 160 180 200

theta (degrees)

x1(t2

3)L,

x1(

t23)

R (

NM

)

540, 420420, 420420, 540all

Figure 4.2 Range of initial locations )( 231 tx of aircraft 1 to be considered for collision risk assessment as a function of intersection angle when aircraft 2 changes level on passing the intersection. Nominal aircraft speeds of 420, 480, and 540 kts

30 NLR-CR-2010-427 August 2010

The lateral collision risk will be relatively large when aircraft 1’s initial location is in or around the intervals formed by the projection of aircraft 2’s entry and exit intervals ],[ 2322 AA and ],[ 2625 AA (cf. figure 2.2) on the nominal track of aircraft 1. The projections are given by

cos

602cos

sin)(

2

1222231 VVVStx R (4.3)

cos

sin)( 23231

Stx R (4.4)

2

125231

2cossin

)(VVStx L

(4.5)

cos

6022cos

sin)(

2

12

2

126231 V

VVVVStx L (4.6)

The factor 602 in eqs. (4.3) and (4.6) refers to the time in hours necessary for the level change of aircraft 2, i.e. 2 minutes. Figure 4.3 shows the various quantities defined by eqs. (4.3) – (4.6) as a function of the intersection angle for nominal aircraft speeds of 6001 V kts and

3002 V kts. The (small) ranges around which the maximum risk occurs for each angle are within the dark blue and pink curves and between the yellow and turquoise curves. For intersection angles less than 90 degrees, the two aircraft have a “same direction” component of movement and the initial locations of aircraft 1 for which maximum risk occurs are both negative and well within the range of initial locations of figure 4.1 for which the lateral separation procedure is applied. For each intersection angle, the less negative initial locations of aircraft 1 refer to the two aircraft being relatively proximate when aircraft 2 enters the lateral conflict area and the more negative value refers to the aircraft being relatively proximate when aircraft 2 leaves the lateral conflict area. The latter effectively means that the faster aircraft 1 is catching up on the slower aircraft 2.

NLR-CR-2010-427

August 2010 31

V1 = 600 kts, V2 = 300 kts

-3500.00

-3000.00

-2500.00

-2000.00

-1500.00

-1000.00

-500.00

0.00

500.00

1000.00

0 20 40 60 80 100 120 140 160 180 200

theta (degrees)

x1(t2

3)...

(N

M)

22R23R25L26L

Figure 4.3 Initial locations of aircraft 1 for which maximum collision risk is expected as a function of intersection angle. Nominal aircraft speeds of 600 kts and 300 kts

V1 = 480 kts, V2 = 480 kts

-3500.00

-3000.00

-2500.00

-2000.00

-1500.00

-1000.00

-500.00

0.00

500.00

1000.00

0 20 40 60 80 100 120 140 160 180 200

theta (degrees)

x1(t2

3)...

(N

M)

22R23R25L26L


32 NLR-CR-2010-427 August 2010

For intersection angles larger than 90 degrees, the two aircraft have an “opposite direction” component of movement and one of the initial locations of aircraft 1 for which maximum risk occurs is positive and the other negative. For each intersection angle, the positive initial locations of aircraft 1 refer to the two aircraft being relatively proximate when the “opposite direction” aircraft 2 enters the lateral conflict area and the negative value refers to the aircraft being relatively proximate when aircraft 2 leaves the lateral conflict area. Figure 4.4 shows the same quantities for nominal aircraft speeds of 4801 V kts and 4802 V kts. Figure 4.4 looks like figure 4.3 but for two things. Firstly, for intersection angles larger than 90 degrees, the aircraft 1 initial locations for which maximum risk occurs in figure 4.4 are much smaller in absolute value than in figure 4.3. This is consistent with the overall ranges in figure 4.1 and is a direct consequence of the ratio 21 VV being equal to one. Secondly, the two ranges of maximum risk are much closer together in figure 4.4 for intersection angles less than 90 degrees, particularly for very small intersection angles. This is also a consequence of 121 VV as can easily be verified from eqs. (4.3) – (4.6). For example, when the intersection angle is small, eqs. (4.3) and (4.4) take nearly the same value and similarly eqs. (4.5) and (4.6). Moreover, the first factor in brackets in eqs. (4.5) and (4.6) becomes equal to approximately minus one, making them approximately equal to eqs. (4.3) and (4.4). Operationally, this means that when the nominal aircraft speeds are the same, an aircraft pair that is proximate when aircraft 2 enters the area will also be proximate when aircraft 2 leaves the lateral conflict area. Finally, Figure 4.5 shows the same quantities for nominal aircraft speeds of

3001 V kts and 6002 V kts. Figure 4.5 also looks like figure 4.3 but for two things. Firstly, for intersection angles larger than 90 degrees, the aircraft 1 initial locations for which maximum risk occurs in figure 4.4 have decreased further in absolute value compared to figure 4.3. This is again consistent with the overall ranges in figure 4.1 and is a direct consequence of the ratio 21 VV being equal to one half. Secondly, the range defined by eqs. (4.5) and (4.6) starts off around zero for very small intersection angles. This reflects that the slower aircraft 1 must be ahead of the faster aircraft 2 when aircraft 2 enters the area, to be at approximately the projection of the points 25A and 26A when aircraft 2 is leaving the area. In other words, the faster aircraft 2 is then catching up on the slower aircraft 1.

NLR-CR-2010-427

August 2010 33

V1 = 300 kts, V2 = 600 kts

-3500.00

-3000.00

-2500.00

-2000.00

-1500.00

-1000.00

-500.00

0.00

500.00

1000.00

0 20 40 60 80 100 120 140 160 180 200

theta (degrees)

x1(t2

3)...

(N

M)

22R23R25L26L


V1 = 540 kts, V2 = 420 kts

-3500.00

-3000.00

-2500.00

-2000.00

-1500.00

-1000.00

-500.00

0.00

500.00

1000.00

0 20 40 60 80 100 120 140 160 180 200

theta (degrees0

x1(t2

3)...

(N

M)

22R23R25L26L


34 NLR-CR-2010-427 August 2010

V1 = 420 kts, V2 = 540 kts

-3500.00

-3000.00

-2500.00

-2000.00

-1500.00

-1000.00

-500.00

0.00

500.00

1000.00

0 20 40 60 80 100 120 140 160 180 200

theta (degrees0

x1(t2

3)...

(N

M)

22R23R25L26L

Figure 4.7 Initial locations of aircraft 1 for which maximum collision risk is expected as a function of intersection angle. Nominal aircraft speeds of 420 kts and 540 kts Figures 4.6 and 4.7 above are the equivalents of figures 4.3 and 4.5 for the smaller range of nominal aircraft speeds of 420, 480, and 540 kts. The main difference concerns the interval of initial locations defined by eqs. (4.5) and (4.6) corresponding to aircraft 1 being proximate with aircraft 2 when aircraft 2 leaves the lateral conflict area. Figure 4.6, for example, shows that due to the smaller speed range (540 – 420 kts), the slower aircraft 1 must be closer to the intersection to be able to be proximate with aircraft 2 when aircraft 2 is leaving the lateral conflict area. Figures 4.8 – 4.13 show the calculated lateral collision risk as a function of the initial location of aircraft 1 for various acute and obtuse intersection angles for the following three nominal aircraft speed combinations: 6001 V kts and

3002 V kts, 4801 V kts and 4802 V kts, and 3001 V kts and 6002 V kts.

NLR-CR-2010-427

August 2010 35

0

0.00000005

0.0000001

0.00000015

0.0000002

0.00000025

-2000 -1500 -1000 -500 0

x1(t23) (NM)

CR

(x1(

t23)

) (a

ccid

ents

per

airc

raft

pair)

theta = 5theta = 15theta = 50theta = 80theta = 90

Figure 4.8 Collision risk as a function of the initial location of aircraft 1 for a right and four acute intersection angles. Nominal aircraft speeds 6001 V kts and 3002 V kts

0

0.00000005

0.0000001

0.00000015

0.0000002

0.00000025

0.0000003

-3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000

x1(t23) (NM)

CR

(x1(

t23)

) (a

ccid

ents

per

airc

raft

pair)

theta = 100theta = 130theta = 165theta = 175

Figure 4.9 Collision risk as a function of the initial location of aircraft 1 for four obtuse intersection angles. Nominal aircraft speeds 6001 V kts and

3002 V kts

36 NLR-CR-2010-427 August 2010

0

0.00000002

0.00000004

0.00000006

0.00000008

0.0000001

0.00000012

-800 -700 -600 -500 -400 -300 -200 -100 0 100 200

x1(t23) (NM)

CR

(x1(

t23)

) (a

ccid

ents

per

airc

raft

pair)

theta = 5theta = 15theta = 50theta = 80theta = 90


0

0.00000005

0.0000001

0.00000015

0.0000002

0.00000025

-3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000

x1(t23) (NM)

CR

(x1(

t23)

) (a

ccid

ents

per

airc

raft

pair)



4802 V kts

NLR-CR-2010-427

August 2010 37

0

0.00000002

0.00000004

0.00000006

0.00000008

0.0000001

0.00000012

0.00000014

0.00000016

-800 -700 -600 -500 -400 -300 -200 -100 0 100 200

x1(t23) (NM)

CR

(x1(

t23)

) (a

ccid

ents

per

airc

raft

pair)

theta = 5theta = 15theta = 50theta =80theta = 90


0

0.00000005

0.0000001

0.00000015

0.0000002

0.00000025

-1500 -1000 -500 0 500 1000

x1(t23) (NM)

CR

(x1(

t23)

) (a

ccid

ents

per

airc

raft

pair)



6002 V kts

38 NLR-CR-2010-427 August 2010

The locations of the peaks may be compared with those predicted in figures 4.3 – 4.7. For example, figures 4.8 and 4.9 for 6001 V kts and 3002 V kts show two peaks for each intersection angle, consistent with figure 4.3. Figure 4.10 for equal nominal aircraft speeds 4801 V kts and 4802 V kts shows only a single peak for intersection angles of 5 and 15 degrees, consistent with the coinciding ranges in the left-hand part of figure 4.4. For the same speed combination, figure 4.11 shows two peaks for each obtuse intersection angle as suggested by figure 4.4. Based on the course of the lateral collision risk as a function of the initial location of aircraft 1 in diagrams similar to figures 4.8 – 4.13, an averaging approach proposed in reference 3 was to average over all initial locations for which the risk is larger than 1% of the peak value. Results obtained with this approach are presented in the next sub-section.

4.3.2 RESULTS Table 4.4 shows the results of the “larger than 1% peak value” averaging strategy for the case where GNSS performance is assumed to be RNP 10 performance. Recall from table 4.1 that assuming GNSS performance to be RNP 4 performance gave only a very moderate additional reduction to the risk values. Maximum risk shown before in table 4.1 is included in table 4.4 for comparison. Figure 4.14 shows the ratio between maximum and average risk as a function of the track intersection angle. This ratio varies between 3 and 3.5, except for the four smallest angles and the two largest ones. The range of angles for which the calculated risk is smaller than a TLS of 9105 accidents per flight hour now extends from 20 degrees to 155 degrees. It should be recalled that the calculated risk is based on the assumption of one aircraft pair passing an intersection per flight hour.

Mixed aircraft population

GNSS as RNP 10 θ

(degrees) Maximum risk Average risk

5 5.25E-08 2.53E-8 10 3.38E-08 1.39E-8 15 2.17E-08 8.02E-9 20 1.38E-08 4.81E-9 25 8.82E-09 2.98E-9

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30 5.87E-09 1.93E-9 35 4.17E-09 1.32E-9 40 3.22E-09 9.53E-10 45 2.66E-09 7.64E-10 50 2.32E-09 6.93E-10 55 2.15E-09 6.75E-10 60 2.16E-09 6.95E-10 65 2.32E-09 7.50E-10 70 2.62E-09 8.40E-10 75 3.01E-09 9.53E-10 80 3.46E-09 1.08E-9 85 3.83E-09 1.19E-9 90 4.02E-09 1.25E-9 95 4.00E-09 1.26E-9 100 3.80E-09 1.21E-9 105 3.49E-09 1.12E-9 110 3.19E-09 1.03E-9 115 2.97E-09 9.53E-10 120 2.88E-09 9.08E-10 125 2.97E-09 9.05E-10 130 3.28E-09 9.62E-10 135 3.85E-09 1.12E-9 140 4.77E-09 1.42E-9 145 6.35E-09 1.90E-9 150 9.05E-09 2.64E-9 155 1.35E-08 3.93E-9 160 2.05E-08 6.23E-9 165 3.06E-08 1.05E-8 170 4.42E-08 1.88E-8 175 6.21E-08 3.52E-8

Table 4.4 Maximum and average collision risk in fatal accidents per flight hour as a function of track intersection angle θ for one aircraft pair passing an intersection per flight hour. Mixed aircraft population with GNSS performance modelled as RNP 10. Nominal aircraft speeds

600 ,480 ,300 , 21 VV kts, no speed errors

40 NLR-CR-2010-427 August 2010

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 20 40 60 80 100 120 140 160 180 200

intersection angle theta (degrees)

Series1

Figure 4.14 Ratio between maximum and average risk as a function of the track intersection angle θ for mixed aircraft population with GNSS performance modelled as RNP 10 performance A small reduction in the calculated risk values of approximately 10% will be possible when GNSS performance is modelled as RNP 4 and a further reduction by a factor of approximately 1.5 when the nominal speed range can be assumed to be 420 to 540 kts rather than 300 to 600 kts. It should also be noted that the “larger than 1% peak value” averaging approach is conservative in that it takes only a limited range of potential aircraft 1 initial locations into account. This can be explained as follows. The second column in table 4.5 shows the size of the lateral conflict area for aircraft 1 as a function of the intersection angle . Aircraft 2 will execute the level change procedure from section 2 for each initial location )( 231 tx of aircraft 1 in the interval 1513, AA (actually over a larger range, see eq. (4.2)). The risk could potentially be averaged over all those locations. The last two columns of table 4.5 show the smallest and largest initial locations over which the averaging actually takes place for the “larger than 1% peak value” averaging approach. Figures 4.15 and 4.16 illustrate the potential range (based on eq. (4.2)) of aircraft 1 initial locations and the two ranges of the actual averaging based on the “larger than 1% peak value” averaging approach. The averaging stops at the

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tails of the two peaks in figure 4.15, more clearly visible on the logarithmic scale of figure 4.16. Averaging over the full range of aircraft 1 initial locations would clearly result in a lower average lateral risk value, but goes beyond the methodology of reference 3.

Intersection angle θ

(degrees)

1513 , AA

sin50,

sin50

(NM)

Smallest aircraft 1 initial location

)( 231 tx over which risk is

averaged (NM)

Largest aircraft 1 initial location

)( 231 tx over which risk is

averaged (NM)

5 ± 573.7 -595 * -505 * 10 ± 287.9 -310 * -220 * 15 ± 193.2 -216 * -126 * 20 ± 146.2 -526 -79 25 ± 118.3 -443 -50 30 ± 100.0 -390 -29 35 ± 87.2 -355 -13 40 ± 77.8 -332 0 45 ± 70.7 -316 13 50 ± 65.3 -305 24 55 ± 61.0 -297 32 60 ± 57.7 -290 39 65 ± 55.2 -285 44 70 ± 53.2 -281 48 75 ± 51.8 -278 51 80 ± 50.8 -276 53 85 ± 50.2 -277 56 90 ± 50.0 -280 60

Table 4.5 Some characteristics related to the calculation of average collision risk for acute and right intersection angles. (Mixed aircraft population with GNSS performance modelled as RNP 10. Nominal aircraft speeds 600 1 V kts and 003 2 V , no speed errors.) Legend: * right-most peak only

42 NLR-CR-2010-427 August 2010

0.00E+00

1.00E-09

2.00E-09

3.00E-09

4.00E-09

5.00E-09

6.00E-09

-500 -400 -300 -200 -100 0 100

x1(t23) (NM)

CR

(x1(

t23)

) (a

ccid

ents

per

airc

raft

pair)

potential actual actual

Figure 4.15 Collision risk as a function of potential and actual ranges of aircraft 1 initial locations for the benefit of collision risk averaging. 30 degree intersection angle, nominal aircraft speeds 6001 V kts and 3002 V kts, no speed errors

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0-500 -400 -300 -200 -100 0 100

x1(t23) (NM)

10 lo

g C

R(x

1(t2

3))

(acc

iden

ts p

er a

ircra

ft pa

ir)

Figure 4.16 Logarithm (base 10) of collision risk as a function of potential and actual ranges of aircraft 1 initial locations for the benefit of collision risk averaging. 30 degree intersection angle, nominal aircraft speeds

6001 V kts and 3002 V kts, no speed errors

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5 CONCLUSIONS

Lateral collision risk per aircraft pair passing an intersection has been calculated for intersection angles between 5 and 175 degrees inclusive using a protected airspace concept. For an aircraft under consideration, the lateral collision risk depends on the location of the other aircraft when it is passing the intersection. This dependence has been accounted for in two ways, namely maximisation and averaging. Based on previous SASP work, a “larger than 1% of the peak value” averaging approach was used. A major objective was to examine the effect on the risk of a mixed aircraft population made up of four sub-populations with different levels of navigation performance, i.e. MNPS, RNP 10, RNP 4, and “MNPS approved only and carrying GNSS”. Successively improved levels of navigation performance were assumed for the last sub-population, namely MNPS, RNP 10, and RNP 4. Compared with a “full MNPS” population, a mixed aircraft population with MNPS, RNP 10, RNP 4, and “GNSS as MNPS” performance gave a reduction of the maximum collision risk, averaged over the range of intersection angles between 5 and 175 degrees inclusive, of approximately 66%. When “GNSS as RNP 10” performance was used, the reduction of the maximum risk, averaged over the range of intersection angles, was approximately 78% of the “full MNP” risk. Assuming “GNSS as RNP 4” performance showed a law of diminishing returns effect, viz. an average reduction of 79% compared with the “full MNPS” risk. Collision risk was calculated for various combinations of nominal aircraft speeds, particularly 300, 480, and 600 kts and subsequently maximised over the speed combinations. As a function of the nominal aircraft speeds, collision risk was found to be maximal for a speed of 600 kts for aircraft 1 and a speed of 300 kts for aircraft 2. The maximal risk was found to decrease when the difference between the maximum and minimum speeds for aircraft 1 and 2 respectively decreased. The “larger than 1% of the peak value” averaging approach was applied to the mixed aircraft population with “GNSS as RNP 10” performance and gave, on average over the intersection angle, a reduction of the maximum risk by a factor of approximately three.

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In addition to the above relative results, some attempt was made towards an absolute assessment. This was hampered by two factors. The first factor was the fact that currently a Target Level of Safety (TLS) specific to the lateral collision risk on intersecting tracks in NAT MNPS airspace does not (yet) exist. To be able to provide some guidance concerning the calculated risk values, a value of

9105 accidents per flight hour has been used as a substitute. The second factor was that an assumption had to be made concerning the number of aircraft pairs passing an intersection in the NAT MNPS airspace per flight hour. In the absence of further information, it was assumed that one aircraft pair would pass an intersection per flight hour. Maximum and average collision risk was then compared with the TLS. The average lateral collision risk was found to be less than the TLS for all intersection angles between 20 and 155 degrees inclusive. The TLS was exceeded by up to a factor of five for angles between 5 and 15 degrees inclusive and by up to a factor of seven for angles between 160 and 175 degrees inclusive.

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6 REFERENCES

1. RNP 10 and 50 NM lateral separation in MNPS airspace, NAT-SARSIG/10-WP/-8, Paris, 12-16 October, 2009.

2. Application of SASP-WG/WHL/13-WP/49 methodology to RNP 4, RNAV 2 and RNAV 10 (RNP 10) intersecting tracks, Geoffry Aldis and Geert Moek, SASP-WG/WHL/14-WP/44, Paris, 13 to 24 October, 2008.

3. Lateral separation on intersecting tracks in a protected airspace context, Geoffry Aldis and Geert Moek, SASP-WG/WHL/15-WP/28, Paris, 25 May to 05 June, 2009.

4. Procedures for Air Navigation Services, Air Traffic Management, Doc 4444 ATM/501, Fifteenth edition – 2007, including Amendment 2 (19/11/09).

5. Update to SASP-WG/WHL/13-WP/26: Collision risk modelling for protected airspace: lateral separation for intersecting RNP 4 tracks, Geert Moek, SASP-WG/WHL/14-WP/42, Paris, 13 to 24 October, 2008.

6. A collision risk model based on reliability theory that allows for unequal navigational accuracy, David Anderson, SASP-WG-WHL/7-WP/20 REVISED, Montreal, 9 – 20 May, 2005.

7. North Atlantic MNPS Risk Quick Reference Guide, NAT MWG/46-IP/02, Paris, 23 – 30 April 2010.

1 Introduction2 Lateral separation method for intersecting tracks3 The collision risk model3.1 Baseline collision risk model3.2 Collision risk model for current application

4 Results4.1 Introduction4.2 Maximum risk4.3 Average collision risk.4.3.1 Background4.3.2 Results

5 Conclusions6 References2010-427-cr.pdf1 Introduction2 Lateral separation method for intersecting tracks3 The collision risk model3.1 Baseline collision risk model3.2 Collision risk model for current application


5 Conclusions6 References

2010-427-cr.pdf1 Introduction2 Lateral separation method for intersecting tracks3 The collision risk model3.1 Baseline collision risk model3.2 Collision risk model for current application


5 Conclusions6 References

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