ANNEX A - Minister for Transport, Tourism and Sport · ANNEX A Calculation of Third Party...

ANNEX A

Calculation of Third Party Individual Risk: Determining Public Safety Zones for Airports

CONTENTS

A1 INTRODUCTION A1

A1.1 DESCRIPTION OF CALCULATION METHOD A1 A1.2 SUMMARY OF CALCULATION METHOD A1

A2 AIRCRAFT CRASH RATE A3

A2.1 AIRCRAFT CLASSIFICATION A3 A2.2 ALL CLASSES MOVEMENT WEIGHTED AVERAGE CRASH RATE A4 A2.2.1 Crash Rate (Crashes per Year) A5 A2.2.2 Average Crash Rate (Crashes per Movement) A5 A2.2.3 Weighted Average Destroyed Area A6 A2.2.4 Type of Crash A6 A2.3 ANNUAL CRASH RATE FOR EACH CRASH MODE A7 A2.4 LONGITUDINAL AND LATERAL DISTANCE CALCULATION A9

A3 IMPACT PROBABILITY CALCULATION - LARGE AIRCRAFT A12

A3.1 PROBABILITY DENSITY FUNCTIONS A12 A3.1.1 Take-off Overruns - Wreckage Location A13 A3.1.2 Landing Overruns - Wreckage Location A14 A3.1.3 Take-off crash (non overrun) A14 A3.1.4 Landing crash (non overrun) – Impact Location A15

A4 EVALUATION OF CRASH PROBABILITIES A17

A4.1 INDIVIDUAL RISK CALCULATION METHOD – ‘LARGE’ AIRCRAFT A17 A4.1.1 Take-off Overruns – Wreckage Location A21 A4.1.2 Landing Overruns – Wreckage Location A22 A4.1.3 Take-off Crashes (non overruns) – Impact Location A23 A4.1.4 Landing Crashes (non overruns) – Impact Location A23 A4.2 INDIVIDUAL RISK CALCULATION METHOD – ‘LIGHT’ AIRCRAFT A24 A4.3 METHOD FOR DETERMINING THE DIMENSIONS OF THE INDIVIDUAL RISK

TRIANGLES A28

ENVIRONMENTAL RESOURCES MANAGEMENT DT & DOEHLG 7608 7-FEBRUARY- 2005

A1

A1 INTRODUCTION

This annex describes the calculation of the individual risk to persons in the vicinity of airports as a result of crashing aircraft. The individual risk values calculated are used to determine Public Safety Zones (PSZs). The purpose of PSZs is to protect the public on the ground from the small, but real possibility that an aircraft might crash in a populated area. Essentially, a PSZ is used to prevent inappropriate use of land where the risk to the public is greatest.

A1.1 DESCRIPTION OF CALCULATION METHOD

The calculation method is based on a model produced for the UK Department of the Environment, Transport and the Regions (DETR) (1) and subsequently reported by the National Air Traffic Services (NATS) (2). Environmental Resources Management’s (ERM’s) implementation of the model, as described in this annex, has been independently validated by a mathematician at the University of Manchester Institute of Science and Technology (UMIST), England (3). The NATS model considers aircraft with Maximum Take-off Weight Authorised (MTWA) of 4 tonnes and above (i.e. ‘large’ aircraft). Many airports have significant proportions of flights by ‘light’ aircraft (i.e. <4 tonnes MTWA). It could be assumed that the distribution of crashes involving light aircraft would be the same as for larger aircraft. However, this could result in inaccuracies, as light aircraft and their use differ significantly from large aircraft and hence their crash location distributions are different. Therefore, light aircraft crashes are modelled using a separate model specifically developed for light aircraft (see Section A4.2).

A1.2 SUMMARY OF CALCULATION METHOD

The calculation of risks upon which to determine Public Safety Zones (PSZs) in Ireland involved the following stages: 1. identifying the number of annual movements (i.e. landings and take-offs)

with respects to aircraft types/classes (4); (1) Evans, A. W., Foot, P. B. et al. Third Party Risk Near Airports and Public Safety Zone Policy. June 1997. National

Air Traffic Services Limited. R&D Report 9636. RDD File Reference 8CS/091/03/10. (2) Cowell, P. G., Foot, P. B. et al. A Methodology for Calculating Individual Risk Due to Aircraft Accidents Near

Airports. January 2000. National Air Traffic Services Limited. R&D Report 0007. R&DG File Reference 8RD/07/002/11.

(3) Muldoon, M. (28-Jan-02). Validation of ERM’s Public Safety Zone (PSZ). Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD.

(4) The extent of PSZs is related to the number of aircraft movements and aircraft types. To minimise the need to periodically revise zone extents the number of aircraft movements for each runway has been set as either (a) the runway’s movement capacity, or (b) the expected maximum number of movements. Similarly, aircraft types have been categorised as either ‘large’ or ‘light’, and the proportion of both set to provide a good representation of the expected split.


A2

2. calculating an ‘all classes’ movement-weighted average crash rate (crashes per million movements). This is done by using crash rates for each aircraft class (crashes per million movements) and multiplying it by the proportion of movements for that class, and summing the individual products;

3. calculating average crash areas (within which persons ‘on the ground’ are

assumed to be fatally injured) for ‘large’ aircraft and ‘light’ aircraft. These are calculated by determining the average Maximum Aircraft Weight (MAW) for each class, multiplying the average crash area by the proportion of annual crashes for that class, and summing the individual products;

4. calculating the probability that crashing aircraft impact a specified

location. For ‘large’ aircraft, this is performed by integrating probability density functions over the calculated average crash area. A similar calculation is performed for ‘light’ aircraft;

5. calculating the annual frequency that crashing aircraft impact a specified

location (i.e. the individual risk). This is performed by multiplying the annual probability of a crash for the specified location by the appropriate average crash rate and associated number of movements (landings and take-offs) for each runway end;

6. using the individual risk results to determine ‘best fit’ zones representing

specified annual individual risks (e.g. 1 in 100,000 per year and 1 in one million per year for the proposed inner and outer PSZs). The shape of each contour (extending away from the runway end) is very similar to that of a triangle. Therefore, to provide a simple geometric area that can be readily defined and easily reproduced on maps and plans, the risk contours are represented by zones alongside and parallel to the runway and triangular zones extending away from the runway ends.


A3

A2 AIRCRAFT CRASH RATE

The numbers of annual movements for the airport must be provided, according to aircraft type. This information can be collated from airport movement records for a recent year or derived from the estimated future, or maximum capacity.

A2.1 AIRCRAFT CLASSIFICATION

Examples of the Boeing aircraft classes are summarised in the DETR report (see footnote 1, page A1) and are detailed in Table A2.1.

Table A2.1 Examples of Aircraft Grouped According to Boeing Class

Boeing Class Aircraft Type I Aerospatiale Caravelle BAe Comet Boeing 707/720 General Dynamics CV880 General Dynamics CV990 McDonnell Douglas DC-8

II BAe (BAC) ‘One-Eleven’ BAe (HS) Trident BAe (Vickers) VC-10 Boeing 727 Boeing 737 100/200 Dassault Mercure Fokker F28 McDonnell Douglas DC-9 VFW 614

III Airbus Industrie A300 BAe/Aerospatiale Concorde Boeing 747 Lockheed Tristar McDonnell Douglas DC-10

IV Airbus Industrie A310 Airbus Industrie A320/321 Airbus Industrie A330 Airbus Industrie A340 BAe 146 Boeing 737 300/400/500 Boeing 757 Boeing 767 Boeing 777 Canadair Regional Jet Fokker 70 Fokker 100 McDonnell Douglas MD11 McDonnell Douglas MD80

Turboprop T1 Aerospatiale ATR 42 Aerospatiale ATR 72 BAe ATP BAe Jetstream 31 BAe Jetstream 41


A4

Boeing Class Aircraft Type DeHavilland Dash 7 DeHavilland Dash 8 Dornier 228 Dornier 328 Embraer Brasilia - EMB110 Embraer Bandeirante - EMB120 Fokker 50 SAAB 340 SAAB 2000 Shorts 330 Shorts 360

Turboprop T2 BAe (HS) 748 BAe (Vickers) Vanguard BAe (Vickers) Viscount Convair 540/580/600/640 Handley Page Dart Herald DeHavilland Twin Otter Fairchild F27 Fairchild FH227 Fairchild Metro Fokker F27 Gulfstream 1 Lockheed Hercules Lockheed Electra Shorts Skyvan

Executive Jets Learjet 35/60 Eastern Jets Tupolev 134/154

Light Cessna 172/177 Notes 1. Aircraft grouped in classes I-IV are Western airliner jets. 2. Aircraft grouped in class ‘Turboprop T1’ are turboprop aircraft designed and first delivered

after 1970. 3. Aircraft grouped in class ‘Turboprop T2’ are turboprop aircraft designed and first delivered

prior to 1970. 4. Aircraft grouped in class ‘light’ are those with a maximum take-off weight of less than

4 tonnes. The method requires a distinction to be made between, aircraft of less than 4 tonnes Maximum Take-off Weight Authorised (MTWA) and aircraft greater than 4 tonnes MTWA. Therefore, Classes I-IV, Turboprops T1 and T2, Executive and Eastern Jets are referred to as ‘large’ aircraft and others (less than 4 tonnes MTWA) referred to as ‘light’.

A2.2 ALL CLASSES MOVEMENT WEIGHTED AVERAGE CRASH RATE

Crash frequency data are summarised in Table A2.2 and explained below.


A5

Table A2.2 Crash Frequency Summary

Boeing Class Number of movements

per year

Crash rate (crashes per movement)

Crash rate (crashes per

year)

Average MTWA

(kg)

Destroyed area

(hectare) NATS MODEL Class I jets NBC,1 RBC,1 RBC,1NBC,1 WBC,1 ABC,1

Class II-IV jets (PAX) NBC,2 RBC,2 RBC,2NBC,2 WBC,2 ABC,2

Class II-IV jets (NP) NBC,3 RBC,3 RBC,3NBC,3 WBC,3 ABC,3

Eastern jets NBC,4 RBC,4 RBC,4NBC,4 WBC,4 ABC,4

Executive jets NBC,5 RBC,5 RBC,5NBC,5 WBC,5 ABC,5

Turboprops T1 (PAX) NBC,6 RBC,6 RBC,6NBC,6 WBC,6 ABC,6

Turboprops T1 (NP) NBC,7 RBC,7 RBC,7NBC,7 WBC,7 ABC,7

Turboprops T2 NBC,8 RBC,8 RBC,8NBC,8 WBC,8 ABC,8

Miscellaneous NBC,9 RBC,9 RBC,9NBC,9 WBC,9 ABC,9

TOTAL N

AVERAGE R A

AEA LIGHT AIRCRAFT MODEL Piston engine NBC,10 RBC,10 RBC,10NBC,10 WBC,10 ABC,10

Notes 1. Total movements per year for the Airport. 2. MTWA - maximum take-off weight authorised. 3. PAX - passenger aircraft movements. 4. NP - non passenger aircraft movements (e.g. freight).

A2.2.1 Crash Rate (Crashes per Year)

Each aircraft group has an associated crash rate (crashes per million movements) as detailed in the NATS report (1). The crash rate (crashes per year) for each group is calculated from the total movements per year and the crashes per movement.

A2.2.2 Average Crash Rate (Crashes per Movement)

An average crash rate (crashes per movement) for large aircraft is calculated accounting for the number of movements of each aircraft class. The total crash rate (crashes per year) is divided by the total number of annual movements as, summarised below. The total number of movements is calculated as the sum of the movements for the individual Boeing Classes:

Equation A2.1

9

1,9,2,1, ...

i

iiBCBCBCBC NNNNN

The average crash rate (per movement) is calculated as a movement weighted average:

(1) See footnote 2, page A1.


A6

Equation A2.2

N

NR

N

NRNRNRR

i

iiBCiBC

BCBCBCBCBCBC

9

1,,

9,9,2,2,1,1, ...

A2.2.3 Weighted Average Destroyed Area

The average maximum take-off weight authorised (MTWA) for each Boeing Class is determined by analysis of the detailed airport movement data. The destroyed area is calculated for each Boeing Class from the following relationship detailed in the NATS report (1):

Equation A2.3

iBCiBC WA ,, ln474.016.610000lnln

where ABC,i area destroyed (m²) WBC,i MTWA (kg) The weighted average destroyed area is weighted according to the number of crashes per year for each aircraft class:

Equation A2.4

9

1,,

9

1,,,

9,9,1,1,

9,9,9,1,1,1,

...

...i

iiBCiBC

i

iiBCiBCiBC

BCBCBCBC

BCBCBCBCBCBC

RN

ARN

RNRN

ARNARNA

A2.2.4 Type of Crash

In both the NATS and the DETR reports, the proportions of the four types of crash were estimated as follows: take-off crashes from flight, 20%; take-off overruns, 8%; landing crashes from flight, 52%; and landing overruns, 20%. The average crash rate per movement is apportioned to the four types of crash as summarised in Table A2.3. The model for light aircraft does not distinguish between take-off and landing crashes, therefore a combined (i.e. take-off and landing) crash rate is used. (1) Note that the term ln(10000) is included such that the units of area destroyed are m² as opposed to hectares, since

there are 10000 m² in a hectare.


A7

Table A2.3 Summary of Crash Rates (All Aircraft Classes)

Crash rate per million movements Crashes Overruns Total NATS model Landings 0.52 R 0.2 R 0.72 R Take-offs 0.2 R 0.08 R 0.28 R AEA light aircraft model Total N/A N/A 3.27 Notes 1. N/A - not applicable. The light aircraft model does not distinguish between crashes and

overruns.

A2.3 ANNUAL CRASH RATE FOR EACH CRASH MODE

The annual crash rate is related to the number of movements on a given runway. The numbers of annual movements for a runway can be described in the general form as summarised in Table A2.4 and shown schematically in Figure A2.1.

Table A2.4 Summary of Runway Movements

Runway 1 2 NATS model Landings ML1 ML2

Take-offs MT1 MT2 AEA light aircraft model

Landings + take-offs MLight1 MLight2

Figure A2.1 General Schematic of Large Aircraft Movements Associated with an Airport Runway

Landingsin Direction 1, ML1

End 1 End 2

Take-offsin Direction 1, MT1

Take-offsin Direction 2, MT2

Landingsin Direction 2, ML2


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Figure A2.2 General Schematic of Light Aircraft Movements Associated with an Airport Runway

For a specified point at End 35, the following crash rates are calculated (1) (2): Take-off overrun (A)

Take-off overrun crash rate (crashes per movement)

× Number of movements

= Crash rate per year

0.08 R × MT1 = 0.08 RMT1 Landing overrun (A)

Landing overrun crash rate (crashes per movement)



0.20 R × ML1 = 0.20 RML1 Take-off crash (non overrun) (A) Take-off non overrun crash rate

(crashes per movement) × Number of

movements = Crash rate per

year 0.2 R × MT1 = 0.2 RMT1

Take-off crash (non overrun) (B) Take-off non overrun crash rate



year 0.2 R × MT2 = 0.2 RMT2

Landing crash (non overrun) (A) Landing non overrun crash rate



year 0.52 R × ML1 = 0.52 RML1

Landing crash (non overrun) (B) Landing non overrun crash rate



year 0.52 R × ML2 = 0.52 RML2

(1) Crash rates associated with movements where the calculation point is after and before the runway (relative to the

movement) are indicated by the letter ‘A’ and ‘B’, respectively. (2) For individual risk calculation other than alongside the runway, take-off and landing overruns are only

considered when the direction of landing or take-off is towards the calculation point. Therefore, only crash rates indicated by ‘A’ are detailed.

End 1 End 2

Landings in Direction 1, MLight1

Landings in Direction 2, MLight2


A9

Light aircraft (Direction 1) Take-off overrun

crash rate (crashes per movement)



(3.27 ÷ 1,000,000) × MLight1 = 3.27 × 10-6(MLight1) Light aircraft (Direction 2)

Take-off overrun crash rate (crashes

per movement)



(3.27 ÷ 1,000,000) × MLight2 = 3.27 × 10-6(MLight2)

A2.4 LONGITUDINAL AND LATERAL DISTANCE CALCULATION

Consider the calculation area for a specified point at End 2, as illustrated in Figure A2.3. The longitudinal displacement from the runway ends, used to calculate the Probability Density Function (PDF), is dependent on the direction of take-off or landing. Taking the specified point marked on Figure A2.3, there is a probability that a crash during take-off in both Direction 1 and 2 will impact this point (although the greater likelihood is associated with Direction 1). However, the longitudinal parameter ‘y’ is dependent on the take-off direction. For example, when take-off is in Direction 1, the specified point is assigned a positive value (Y0 metres) of the longitudinal parameter ‘y’. Conversely, when take-off is in Direction 2, the specified point is assigned a negative value (-L - Y0 metres) of the longitudinal parameter ‘y’. Similarly, the longitudinal parameter ‘y’ is also dependent on the landing direction. Therefore, for take-off and landing crashes (i.e. non overruns), PDFs must be calculated for two sets of longitudinal parameters as follows (using the specified point illustrated in Figure A2.3 as an example): Take-off non overrun (A) - relative to the movement, the calculation point

is after the runway, take-off in Direction 1 (y>0) Take-off non overrun (B) - relative to the movement, the calculation point is

before the runway, take-off in Direction 2 (y<0) Landing non overrun (A) - relative to the movement, the calculation point

is after the runway, landing in Direction 1 (y>0) Landing non overrun (B) - relative to the movement, the calculation point is

before the runway, landing in Direction 2 (y<0) Note that, for individual risk calculation other than alongside the runway, take-off and landing overruns are only considered when the direction of landing or take-off is towards the calculation point. Therefore, only crash rates indicated by ‘A’ are detailed. The runway threshold is the first point on a runway upon which an aircraft can safely land. Typically, this is taken as the ‘end of the tarmac’. Some runway ends may feature ‘displaced thresholds’ for landing movements. Displaced thresholds are defined to help facilitate obstacle clearance during


A10

landings. Therefore, the displaced threshold is only accounted for when considering landing movements. In practice, an aircraft pilot will aim to land the aircraft in the ‘touchdown zone’. For large aircraft at major airports, the touchdown zone is taken by convention to be between 150-300 m beyond the runway threshold (either the ‘end of tarmac’ or the displaced threshold). In an accident report, if an aircraft is reported to have impacted on the touchdown zone, then, unless otherwise stated, the y value for the impact location is taken as 225 m. In an accident report, distances stated with respect to the touchdown point are assumed to be 225 m different from distances with respect to the runway threshold. It is understood that this has been accounted for in the derivation of the PDFs. Therefore, the origin for y coordinates is either: the ‘end of the tarmac’ for runway ends without a declared displaced

threshold; or the displaced threshold for runway ends with a declared displaced

threshold.

Figure A2.3 Representative Runway Diagram

Notes 1. Some runways may have a displaced threshold for landings. The displaced threshold at

Ends 1 and 2 are shown by the dimensions Loffset1 and Loffset2, respectively. For runways with no displaced threshold, Loffset = 0.

For the specified point at End 2, illustrated in Figure A2.3, the values of x and y are defined for each crash mode as summarised in Table A2.5.

Runway of length L metres1 2

Take-off Direction1

Take-off Direction2

y +vey = 0

y = - Ly -ve

y -vey = - L

y = 0y +ve

Point of interest

Runway of length L metres1 2

Landing Direction1

Landing Direction2

y -ve

y = 0 y -ve

y +vey = 0

y = L-Loffset2y +ve

x +ve

x +ve

x +ve

x +ve

Loffset2

Loffset1

y = L-Loffset1

Y0 X0

Point of interest

Y0 X0


A11

For example, the longitudinal distance of the specified point, for a landing overrun is:

Runway length (m)

- Offset distance (m)

+ Distance from runway threshold (m)

= Longitudinal distance (m)

L - LOffset1 + Y0 Similarly, for a landing crash (non overrun) (y<0), the longitudinal distance of the specified point, for a landing overrun is:

- Offset distance (m)

- Distance from runway threshold (m)

= Longitudinal distance (m)

-LOffset2 - Y0

Table A2.5 Summary of Lateral and Longitudinal Coordinates

Crash mode Lateral, x (m) Longitudinal, y (m) Take-off overrun (A) X0 Y0

Landing overrun (A) X0 Y0 + L - LOffset1

Take-off non overrun (A) X0 Y0

Take-off non overrun (B) X0 -(Y0 + L)

Landing non overrun (A) X0 Y0 + L - LOffset1

Landing non overrun (B) X0 -Y0 - LOffset2


A12

A3 IMPACT PROBABILITY CALCULATION - LARGE AIRCRAFT

Impact probability is calculated at a specified point using the Probability Density Functions (PDFs) and aircraft crash rates (see Section A2 for description of the calculation).

A3.1 PROBABILITY DENSITY FUNCTIONS

The PDFs are described in detail in the NATS report (1). The PDFs for a specified point are functions of the perpendicular distance (x) from the (extended) runway centreline and the longitudinal distance (y) from the threshold of the appropriate runway end. The distributions are PDFs in the form:

Equation A3.1

yxhygyxf ,, where g(y) a function representing the longitudinal location along the

direction of the extended runway centreline h(x, y) lateral distribution perpendicular to the runway centreline The function g(y) is derived from y coordinate data. The function h(x, y) is derived from x coordinate data, for which the corresponding y coordinate is known. The PDFs are based on the Gamma and Weibull distributions. The Gamma distribution for parameters z, and is:

zzzf exp

1,, 1

The Weibull distribution for parameters z, and is:

z

zzf exp,, 1

(1) See footnote 2, page A1.


A13

The following PDFs are calculated at the specified point: Take-off overrun (A); Landing overrun (A); Take-off crash (non overrun) (A); Take-off crash (non overrun) (B); Landing crash (non overrun) (A); Landing crash (non overrun) (B); Light aircraft (Direction 1); Light aircraft (Direction 2). The PDFs for each crash mode are detailed in the following sections, based on the general description of a runway for the specified point illustrated in Figure A2.3. See Section A2.4 for an explanation of the longitudinal and lateral parameters x and y.

A3.1.1 Take-off Overruns - Wreckage Location

For take-off overruns beyond the departure end of the runway (y>0), the PDF of the wreckage location is calculated as:

Equation A3.2

y

ypygTO exp1 for y > 0

where = 1.336 = 342.6 p = 0.761 (fraction of take-off overruns with y > 0) and

Equation A3.3

ccTO yx

xyyxh

exp2

1,

1 for y > 0, x 0

where c = 0.354 = 0.684 = 74.37 Equation A3.2 and Equation A3.3 are based on the location data for 35 accidents.


A14

A3.1.2 Landing Overruns - Wreckage Location

For landing overruns (y > 0), the PDF of the wreckage location is calculated as:

Equation A3.4

yyyg LO exp

1 1 for y > 0

where = 4.906 = 392.1 and

Equation A3.5

ccLO yx

xyyxh

exp2

1,

1 for y > 0, x 0

where c = 0.778 = 0.831 = 10313 Equation A3.4 and Equation A3.5 are based on data for 79 accidents.

A3.1.3 Take-off crash (non overrun)

The PDF of the impact location for take-off crashes from flight (non-overrun crashes) is calculated as:

Equation A3.6

y

ypygTC exp1 for y > 0

where = 0.687 = 2863.7 p = 0.630 (fraction of take-off crashes with y > 0) and


A15

Equation A3.7

yypyg TC exp

11

1 for y < 0

where = 1.968 = 570.62 p = 0.630 (fraction of take-off crashes with y > 0)

Equation A3.8

ccTC yx

xyyxh

exp2

1,

1 for y 0, x 0

where c = -0.617 for y > 0 = 0.668 for y > 0 = 4.705 for y > 0 and c = 0.211 for y < 0 = 0.485 for y < 0 = 589.91 for y < 0 Equation A3.6 to Equation A3.8 are based on the locations of 51 accidents with a positive y coordinate and the locations of 30 accidents with negative y coordinate.

A3.1.4 Landing crash (non overrun) – Impact Location

The PDF of the impact location for landing crashes from flight (non-overrun crashes) after the runway threshold is calculated as:

Equation A3.9

yypyg LC exp

1 1 for y > 0

where = 0.283 = 6441.9 p = 0.307 (fraction of landing crashes with impact y > 0) and


A16

Equation A3.10

y

ypyg LC exp11 for y < 0

where = 0.567 = 3609.0 p = 0.307 (fraction of landing crashes with impact y > 0) and

Equation A3.11

ccLC yx

xyyxh

exp2

1,

1 for y 0, x 0

where c = -0.877 for y > 0 = 0.427 for y > 0 = 0.213 for y > 0 and c = -0.952 for y < 0 = 0.507 for y < 0 = 0.158 for y < 0 Equation A3.9 to Equation A3.11 are based on the locations of 68 accidents with a positive y coordinate and the locations of 160 accidents with negative y coordinate.


A17

A4 EVALUATION OF CRASH PROBABILITIES

The following method can be used to calculate individual risk at a point (X0, Y0) which is a distance (1): ‘Y0’ from the end of tarmac, along the runway extended centreline; and ‘X0’ perpendicular to the runway extended centreline. The method described has been implemented in a Visual Basic (VB) program. The VB program has been validated and checked using Excel spreadsheets. Furthermore, the VB program and the method has been independently checked by an ERM mathematician using the calculation and computation tool MathCadTM.

A4.1 INDIVIDUAL RISK CALCULATION METHOD – ‘LARGE’ AIRCRAFT

The crash area is represented by a square, the length of whose sides are ‘a’. If a crash occurs centred at any point within the shaded square area, the point indicated in Figure A4.1, would be impacted.

Figure A4.1 Individual Risk Calculation Schematic

Therefore, the individual risk at the point indicated in Figure A4.1 is the frequency with which a crash would occur within the shaded square. This is evaluated by the following integral, over two dimensions.

Equation A4.1

2

2

2

2

21

21

11

arg

0

0

0

0

.

,2,2

,2,2

,2,2a

X

aX

aY

aY

LCLLC

LCLLC

TCTTC

TCTTC

LOLLO

TOTTO

eL dydx

yxfMRLyxfMR

LyxfMRyxfMR

LyxfMRyxfMR

IR

(1) Note that the distance Y0 relates to the distance from the ‘end of tarmac’ regardless of whether a displaced

threshold is defined.

End 1 End 2

MT2 ML2

MT1

(X0, Y0)

a

a

ML1

Y0

X0

L


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which can be expanded to:

Equation A4.2

2

2

2

2

22

2

2

2

1

2

2

2

2

22

2

2

2

1

2

2

2

2

12

2

2

2

1arg

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.,2.,2

.,2.,2

.,2.,2

aX

aX

aY

aY

LCLLC

aX

aX

aY

aY

LCLLC

aX

aX

aY

aY

TCTTC

aX

aX

aY

aY

TCTTC

aX

aX

aY

aY

LOLLO

aX

aX

aY

aY

TOTTOeL

dydxyxfMRdydxLyxfMR

dydxLyxfMRdydxyxfMR

dydxLyxfMRdydxyxfMRIR

where, MT1 Number of take-offs per year in Direction 1 MT2 Number of take-offs per year in Direction 2 ML1 Number of landings per year in Direction 1 ML2 Number of landings per year in Direction 2 RTO Take-off overrun frequency per movement RLO Landing overrun frequency per movement RTC Take-off crash (non overrun) frequency per movement RLC Landing crash (non overrun) frequency per movement fTO(x, y) Probability density function for take-off overruns fLO(x, y) Probability density function for landing overruns fTC(x, y) Probability density function for take-off crashes (non overruns) fLC(x, y) Probability density function for landing crashes (non overruns) a Length of side for square crash area x Distance perpendicular to the extended runway centreline y Distance along the extended runway centreline The maximum extent of the contour occurs on the extended runway centreline (i.e. x = 0). Therefore, to calculate the maximum extent of the individual risk contours it is important to calculate the individual risk at x = 0. However, the probability density functions (PDFs) exhibit singularities (i.e. are undefined or infinite) for x = 0. These singularities can be eliminated by mathematically transforming the PDFs according to the following scheme. All the PDFs can be constructed from the following base functions where ‘’, ‘‘ and ‘c’ are as defined in the report produced by NATS (1) and detailed in Section A3.1:

Equation A4.3

yyyGamma exp

)(

1),,( 1

(1) Cowell, P. G., Foot, P. B. et al. A Methodology for Calculating Individual Risk Due to Aircraft Accidents Near

Airports. January 2000. National Air Traffic Services Limited. R&D Report 0007. R&DG File Reference 8RD/07/002/11.


A19

Equation A4.4

y

yyWeibull exp),,( 1

Equation A4.5

cc yxx

ycyxq exp

2

1,,,,

1

The function, q has a singularity at x = 0 for < 1, so it is split into a non-singular part (q1) and a singular part (q2) such that (q = q1 + q2).

Equation A4.6

1exp

2

1,,,,1

1

cc yxx

ycyxq

Equation A4.7

1

2

1,,,,2

x

ycyxq

c

The singular part, q2 can be integrated analytically. Providing X0 is positive and is greater than a/2, the integral of q2 from x = X0-a/2 to x = X0+a/2 is:

Equation A4.8

222

1,,,,2 000

aX

aX

ycyXQ

c

for X0 > a/2 > 0

For calculation on, or close to the runway extended centerline, the integration must be treated differently, since Q2 is a function of the modulus of x expressed as x . However, if X0 is positive (or zero) and is less than a/2, part

of the crash area occupies the region where x is less than zero (x<0), as illustrated in Figure A4.2.


A20

Figure A4.2 Illustration of the Case of Individual Risk Calculation Close to the Runway Extended Centreline

Notes 1. The crash area is represented by a square whose side is length ‘a’. 2. The runway extended centreline represents x=0. 3. The region of the crash area where x<0 is indicated by ‘cross hatching’. The region of the

crash area where x>0 is indicated by solid shading.

The integral of q2 from x = X0-a/2 to x = X0+a/2 is considered as follows:

Equation A4.9

2

0

0

2

2

2

0

0

0

0

,,,,2,,,,2,,,,2a

X

aX

aX

aX

dxcyxqdxcyxqdxcyxq

Since q2 is a function of the modulus of x, the following is true, by definition: q2(-x, y, , , c) = q2(x, y, , , c) Therefore, since X0-a/2 is negative,

Equation A4.10

0

0

2

0

0

2

,,,,2,,,,2X

a

aX

dxcyxqdxcyxq

This can then be substituted in Equation A4.9 to give,

Equation A4.11

2

0

2

0

2

2

000

0

,,,,2,,,,2,,,,2a

XXaa

X

aX

dxcyxqdxcyxqdxcyxq

Therefore, providing X0 is positive (or zero) and is less than a/2, the integral of q2 from x = X0-a/2 to x = X0+a/2 is:

X0

a/2-

X0

(X0, Y0)

Extended runway centreline

x=X0-a/2

x=X0+a/2

a

a


A21

Equation A4.12

222

1,,,,2 000

aXX

aycyXQ

c

for a/2 > X0 0

Note that for X0=a/2 and 0, Equation A4.8 and Equation A4.12 are equivalent.

A4.1.1 Take-off Overruns – Wreckage Location

For take-off overruns, the wreckage location PDF (y>0) is:

Equation A4.13

354.0,37.74,684.0,,2

354.0,37.74,684.0,,16.342,336.1,761.0,

yxq

yxqyWeibullyxf TO

where 0.761 is the fraction of take-off overruns with y > 0. Therefore,

Equation A4.14

2

2

2

2

2

2

2

2

0

0

0

0

0

0

0

0

.354.0,37.74,684.0,,2

354.0,37.74,684.0,,16.342,336.1,761.0

.,

aX

aX

aY

aY

aX

aX

aY

aY

TO

dydxyxq

yxqyWeibull

dydxyxf

and

Equation A4.15

2

2

2

2

2

2

2

2

2

2

2

2

0

0

0

0

0

0

0

0

0

0

0

0

.354.0,37.74,684.0,,26.342,336.1,761.0

.354.0,37.74,684.0,,16.342,336.1,761.0

.,

aX

aX

aY

aY

aX

aX

aY

aY

aX

aX

aY

aY

TO

dydxyxqyWeibull

dydxyxqyWeibull

dydxyxf


A22

Equation A4.16

2

2

2

2

2

2

2

2

2

2

2

2

0

0

0

0

0

0

0

0

0

0

0

0

354.0,37.74,684.0,,26.342,336.1,761.0

.354.0,37.74,684.0,,16.342,336.1,761.0

.,

aY

aY

aX

aX

aX

aX

aY

aY

aX

aX

aY

aY

TO

dydxyxqyWeibull

dydxyxqyWeibull

dydxyxf

Equation A4.17

2

2

0

2

2

2

2

2

2

2

2

0

0

0

0

0

0

0

0

0

0

354.0,37.74,684.0,,26.342,336.1,761.0

.354.0,37.74,684.0,,16.342,336.1,761.0

.,

aY

aY

aX

aX

aY

aY

aX

aX

aY

aY

TO

dyyXQyWeibull

dydxyxqyWeibull

dydxyxf

To evaluate this integral on the extended runway centerline, X0 = 0. Since the function q2 involves the modulus of x, the function is symmetrical about the line representing x = 0. Therefore:

Equation A4.18

2

0

2

2

354.0,37.74,684.0,,22354.0,37.74,684.0,,2aa

a dxyxqdxyxq

Both terms of the integral defined in Equation A4.17 can be integrated numerically. Note that for x = 0:

Equation A4.19

0354.0,37.74,684.0,,01 yq The principle for the other crash distributions is similar to that described for landing overruns. Therefore, a similar approach is employed for the other crash distribution functions. The functions are summarised in Sections A4.1.2 to A4.1.4.

A4.1.2 Landing Overruns – Wreckage Location

For landing overruns, the wreckage location PDFs (y>0) is:


A23

Equation A4.20

778.0,10313,831.0,,2

778.0,10313,831.0,,11.392,906.4,,

yxq

yxqyGammayxf LO

A4.1.3 Take-off Crashes (non overruns) – Impact Location

For take-off crashes, the impact location PDFs are:

Equation A4.21

617.0,705.4,668.0,,2

617.0,705.4,668.0,,17.2863,687.0,630.0,

yxq

yxqyWeibullyxf TC

for y > 0, and

Equation A4.22

211.0,91.589,485.0,,2

211.0,91.589,485.0,,162.570,968.1,630.01,

yxq

yxqyGammayxf TC

for y 0, where 0.630 is the fraction of take-off crashes with impact y > 0.

A4.1.4 Landing Crashes (non overruns) – Impact Location

For landing crashes, the impact location PDFs are:

Equation A4.23

877.0,213.0,427.0,,2

877.0,213.0,427.0,,19.6441,283.0,307.0,

yxq

yxqyGammayxf LC

for y > 0, and

Equation A4.24

952.0,158.0,507.0,,2

952.0,158.0,507.0,,13609,567.0,307.01,

yxq

yxqyWeibullyxf TC

for y 0, where 0.307 is the fraction of landing crashes with impact y > 0.


A24

A4.2 INDIVIDUAL RISK CALCULATION METHOD – ‘LIGHT’ AIRCRAFT

Light aircraft crashes are modelled using a method developed by SRD based on USA and Canadian light aircraft data (1). The model was produced for the distribution of airport related crashes for aircraft less than 2.3 tonnes MTWA. The model has been adopted for aircraft with MTWA less than 4 tonnes because it is judged that the traffic patterns for such aircraft are similar to those of aircraft with MTWA less than 2.3 tonnes. The model employs a PDF which represents the probability of an airport related crash in relation to a given end of a runway. However, the model does not distinguish between take-off and landing crashes (i.e. the combined effect of take-off and landing crashes are considered). A polar coordinate system is used with coordinates represented by (r, ), where r is the distance in metres (m) from the end of the runway (i.e. the intersection of the centreline and the threshold nearest the end of the runway in question) and is the positive angle in radians from the extended centreline pointing away from the runway (see Figure A4.4 and Figure A4.5). In the model, the units of probability density are probability per m². The crash area is represented by a square, the length of whose sides are ‘b’. If a crash occurs centred at any point within the shaded square area, the point indicated in Figure A4.3, would be impacted.

Figure A4.3 Individual Risk Calculation Schematic – ‘Light’ Aircraft

Therefore, the individual risk at the point indicated in Figure A4.1 is the frequency with which a crash would occur within the shaded square. This is evaluated by the following integral, over two dimensions.

(1) Phillips D.W. (July 1987). Criteria for the Rapid Assessment of the Aircraft Crash Rate onto Major Hazards

Installations According to Location. SRD/HSE/R435.

End 1 End 2

(X0, Y0)

b

b

MLight1

Y0

X0

L

MLight2


A25

Equation A4.25

2

2

2

2

22

11

0

0

0

0

.,,b

X

bX

bY

bY

LightLightLight

LightLightLightLight dydxyxfMRyxfMRIR

The crash distribution model adopted for aircraft less than 4 tonnes MTWA is:

Equation A4.26

3

exp2500

exp108),( 8 rrf Light

The coordinate system depends on the end of the runway at which individual risk calculation is to be performed (see Figure A4.4 and Figure A4.5).

Figure A4.4 Illustration of Light Aircraft Coordinate System for Individual Risk Calculation at End 1

The values of 1 and 2 are expressed in terms of Cartesian coordinates as shown in Equation A4.27 and Equation A4.28. The standard implementations of the inverse tangent operation (tan-1), including for example in Microsoft Excel and similar calculation tools, determine values for tan-1() in the range

22

. Therefore, domains are given for which the calculation of 1 and

2 are applicable.

Equation A4.27

y

x11 tan for

22 1

y

X0

2

r1

Grid point, of interestwith cartesian

cordinates (X0,Y0)

Y0

1

r2

L

End 1 End 2


A26

and

Equation A4.28

Ly

x12 tan for

2

3

2 2

By definition, for the calculation performed at a point beyond runway End 1 (as illustrated in Figure A4.4), the values of 1 and 2 are in the range

20 1

and 22

, respectively. Therefore, since these domains lie in

the domains for which the equations are applicable, Equation A4.29 and Equation A4.30 are suitable for calculation of 1 and 2. Therefore, for calculation of individual risk at End 1, the probability density functions for light aircraft are expressed in terms of the Cartesian coordinates as:

Equation A4.29

y

x

yxyxf Light

1

2281

tan3

exp2500

exp108),(

and

Equation A4.30

Ly

x

Lyxyxf Light

1

2282

tan3

exp2500

exp108),(


A27

Figure A4.5 Illustration of Light Aircraft Coordinate System for Individual Risk Calculation at End 2

Similarly, the values of 1 and 2 are expressed in terms of Cartesian coordinates as follows.

Equation A4.31

Ly

x11 tan for

2

3

2 1

and

Equation A4.32

y

x12 tan for

22 2

Similarly, for calculation of individual risk at End 2, the probability density functions for light aircraft are expressed in terms of the Cartesian coordinates as:

Equation A4.33

Ly

x

Lyxyxf Light

1

2281

tan3

exp2500

exp108),(

and

2

r 2

Grid point, of interestwith cartesiancordinates (X0,Y0)

Y0

X0

1

r1

L

End 1 End 2y


A28

Equation A4.34

y

x

yxyxf Light

1

2282

tan3

exp2500

exp108),(

The integration of the probability density functions for light aircraft is performed numerically in a similar way to that performed for large aircraft, as described in Section A3. However, since there is not a singularity for calculation on the extended runway centerline (i.e. the PDF can be evaluated for x = 0), a mathematical transformation is not required. The total individual risk at a specified point is the sum of the individual risks associated with large and light aircraft:

Equation A4.35

LighteargL IRIRIR

A4.3 METHOD FOR DETERMINING THE DIMENSIONS OF THE INDIVIDUAL RISK TRIANGLES

In general, the shape of each individual risk (IR) contour (extending away from the runway) is very similar to that of a triangle. Therefore, to provide a simple geometric area that can be readily defined and easily reproduced on maps and plans, the risk contours are represented by zones alongside and parallel to the runway and triangular zones extending away from the runway ends. For a given IR contour, the point furthest away from the runway end, along its extended centreline, is taken as the ‘sharp point’ of the triangle. The ‘base’ side of the triangle is perpendicular to the runway centre line, with its mid point at the runway ‘end of tarmac’. The triangle is drawn such that the point of maximum width of the contour is encompassed by the triangle. The dimensions of each triangle are determined as follows: determine the maximum distance (Ymax) along the extended runway

centreline at which the individual risk of interest is calculated; and determine the maximum distance (Xmw) perpendicular to the extended

runway centreline at which the individual risk of interest is calculated and record the distance (Ymw) along the extended runway centreline where the ‘maximum width’ occurs.

The triangle base-width is calculated as illustrated below and as given in Equation A4.36 based on the following scheme.


A29

Figure A4.6 Triangle Width Calculation Scheme

Equation A4.36

mw

mw

YY

XYX

max

maxmax

The triangle base width is double Xmax.

(Xmw, Ymw)

Extended runway centrelineX

max

Ymax

1

CONTENTS

A1 INTRODUCTION 1

A1.1 DESCRIPTION OF CALCULATION METHOD 1 A1.2 SUMMARY OF CALCULATION METHOD 1

A2 AIRCRAFT CRASH RATE 3

A2.1 AIRCRAFT CLASSIFICATION 3 A2.2 ALL CLASSES MOVEMENT WEIGHTED AVERAGE CRASH RATE 4 A2.2.1 Crash Rate (Crashes per Year) 5 A2.2.2 Average Crash Rate (Crashes per Movement) 5 A2.2.3 Weighted Average Destroyed Area 6 A2.2.4 Type of Crash 6 A2.3 ANNUAL CRASH RATE FOR EACH CRASH MODE 7 A2.4 LONGITUDINAL AND LATERAL DISTANCE CALCULATION 9

A3 IMPACT PROBABILITY CALCULATION - LARGE AIRCRAFT 12

A3.1 PROBABILITY DENSITY FUNCTIONS 12 A3.1.1 Take-off Overruns - Wreckage Location 13 A3.1.2 Landing Overruns - Wreckage Location 14 A3.1.3 Take-off crash (non overrun) 14 A3.1.4 Landing crash (non overrun) – Impact Location 15

A4 EVALUATION OF CRASH PROBABILITIES 17

A4.1 INDIVIDUAL RISK CALCULATION METHOD – ‘LARGE’ AIRCRAFT 17 A4.1.1 Take-off Overruns – Wreckage Location 21 A4.1.2 Landing Overruns – Wreckage Location 22 A4.1.3 Take-off Crashes (non overruns) – Impact Location 23 A4.1.4 Landing Crashes (non overruns) – Impact Location 23 A4.2 INDIVIDUAL RISK CALCULATION METHOD – ‘LIGHT’ AIRCRAFT 24 A4.3 METHOD FOR DETERMINING THE DIMENSIONS OF THE INDIVIDUAL RISK

TRIANGLES 28

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