Collision Probabilities, Aircraft Separation and Airways Safety

1. Introduction

The steady growth of air traffic at a rate of 3-7% per year over several decades has placed increasing demands on capacity that must be met with undiminished safety (Vismari & Júnior, 2011). The trend is in fact to improve safety, while meeting more stringent requirements for environment impact, efficiency and cost. The traditional method of safety assurance in Air Traffic Management (ATM) is the setting of separation rules (Houck & Powell, 2001). The separation distances are determined by: (i) wake vortex effects on approach to land and take-off queues at runways at airports (FAA, 2011; International Civil Aviation Organization [ICAO], 2007; Rossow, 1999); (ii) collision probabilities for the in-flight phases of aircraft operations (Campos & Marques, 2002; Reich, 1966; Yuling & Songchen, 2010). Only the latter aspect is considered in the present chapter.

A key aspect of ATM in the future (Eurocontrol, 1998) is to determine (i) the technical requirements to (ii) ensure safety with (iii) increased capacity. The concepts of ‘capacity’, ‘safety’ and ‘technology’ can be given a precise meaning (Eurocontrol, 2000) in the case of airways with aircraft flying on parallel paths with fixed lateral/vertical (Figure 1), or longitudinal (Figure 2) separation: (i) the ‘capacity’ increases for smaller separation L; (ii) navigation and flight ‘technology’ should provide a reduced r.m.s. position error $$\sigma$$ ; (iii) the combination of L and $$\sigma$$ should be such that the probability of collision (ICAO, 2006) does not exceed ICAO Target Level of Safety (TLS) of $$5\times {10}^{-9}$$ per hour (ICAO, 2005). Thus the key issue is to determine the relation between aircraft separation L and position accuracy $$\sigma$$ , which ensures that the ICAO TLS is met. Then the technically achievable position accuracy $$\sigma$$ specifies L, viz. the safe separation distance (SSD). Conversely, if an increase in capacity is sought, the separation L must be reduced; then the ICAO TLS leads to a position accuracy $$\sigma$$ which must be met by the ‘technology’. The position accuracy $$\sigma$$ includes all causes, e.g. navigation system (Anderson, 1966) error, atmospheric disturbances (Campos, 1984, 1986; Etkin, 1981), inaccuracy of pilot inputs (Campos, 1997; Etkin & Reid, 1996; Etkin & Etkin, 1990), etc.

Figure 1.

Aircraft flying always at minimum lateral/vertical separation distance L.

Figure 2.

Aircraft flying always at minimum longitudinal separation distance L.

The two main ATM flight scenarios are: (i) parallel paths (Figure 1) with fixed separations in flight corridors typical of transoceanic flight (Bousson, 2008); (ii) crossing (Figure 3) and climbing/descending (Figure 4) flight paths typical of terminal flight operations (Shortle at al., 2010; Zhang & Shortle, 2010). Since aircraft collisions are rare, two-aircraft events are more likely and this the case considered in the present chapter.

Figure 3.

Geometry of crossing aircraft.

Figure 4.

Geometry of climbing/descending aircraft.

The methods to calculate collision probabilities (Reich, 1966) have been applied to Reduced Vertical Separation Minima (RSVM), to lateral separation (Campos, 2001; Campos & Marques, 2002), to crossing aircraft (Campos & Marques, 2007, 2011), to free flight (Barnett, 2000) and to flight in terminal areas (Shortle et al., 2004). The fundamental input to the models of collision probabilities, is the probability distribution (Johnson & Balakrisshann, 1995; Mises, 1960) of flight path deviations; since it is known that the Gaussian distribution underestimates collision probabilities, and the Laplace distribution though better (Reich, 1966) is not too accurate, the generalized error distribution (Campos & Marques, 2002; Eurocontrol, 1988), and extensions or combinations have been proposed (Campos & Marques, 2004a). It can be shown (Campos & Marques, 2002) that for aircraft on parallel flight corridors (Figure 1) an upper bound to the probability of collision is the probability of coincidence (PC). Its integration along the line joining the two aircraft leads to the cumulative probability of coincidence (CPC); the latter has the dimensions of inverse length, and multiplied by the airspeed, gains the dimensions of inverse time, i.e., can be compared to the ICAO TLS. Alternatively the ICAO TLS can be converted to collision per unit distance, which is directly comparable to the CPC. Since most commercial aircraft fly no faster than $$V_0=625kt$$ , the ICAO TLS of $$P_0\le 5\times {10}^{-9}$$ /h, is met by $$Q_0=P_0/V_0\le 8\times {10}^{-12}$$ /nm. The latter can thus be used as an Alternate Target Level of Safety (ATLS).

In the present chapter the CPC is calculated (Section 2) for comparison with the ICAO ATLS of $$8\times {10}^{-12}$$ probability of collision per nautical mile; three probability distributions are compared (Section 2.1) and discussed in detail: the Gaussian (Section 2.2); the Laplace (Section 2.3); a generalized error distribution (Section 2.4), which is less simple but more accurate, viz. it has been shown to fit aircraft flight path deviations measured from radar tracks (Campos & Marques, 2002, 2004a; Eurocontrol, 1988). The comparison of the CPC with the ATLS, is made (Section 3) for four typical cruise flight conditions: (i/ii) lateral separation $$L_a=50nm$$ in uncontrolled (e.g. oceanic) airspace (Section 3.1) and $$L_b=5nm$$ in controlled airspace (Section 3.2); (iii/iv) standard altitude separation $$L_c=2000ft$$ used worldwire (Section 3.3) and RVSM L_d = 1000 ft introduced (figure 5) by Eurocontrol (1988) to increase capacity at higher flight levels (FL290 to FL410). Longitudinal separation along the same flight path could be considered to the limit of wake vortex effects (Campos & Marques, 2004b; Spalart, 1998). In each of the four cases: (i) the CPC is calculated for several position accuracies $$\sigma$$ , to determine the minimum which meets the safety (ATLS) standard; (ii) the Gauss, Laplace and generalized distributions are compared for the collision probabilities of two aircraft with similar position errors $$\sigma$$ ; (iii) the case of aircraft with dissimilar position errors $${\sigma }_1$$ and $${\sigma }_2$$ is considered from the beginning, and analysed in detail for the most accurate probability distribution. The discussion (Section 4) summarizes the conclusions concerning airways capacity versus position accuracy, for an undiminished safety.

Figure 5.

RVSM between flight levels (FL) 290 and 410 inclusive.

2. Comparison of probability distributions for aircraft flight path

An upper bound for the probability of collision of aircraft on parallel flight tracks (Section 2.1) is calculated using Laplace (Section 2.2), Gaussian (Section 2.3) and generalized (Section 2.4) probability distributions, for aircraft with generally dissimilar r.m.s. position errors.

2.1. Comparison of three probability distributions for flight path deviations

Consider two aircraft flying at: (i) either constant lateral or altitude separation L in parallel flight paths (Figure 1), (ii) or at constant longitudinal separation L on the same flight path (Figure 2). In the case of vertical separation there may be an asymmetry in the probability distributions, which has been treated elsewhere (Campos & Marques, 2007); in the case of longitudinal separation wake effects need to be considered as well (Campos & Marques, 2004b; Spalart, 1998). Apart from these effects, a class of probability distributions (Johnson & Balakshishnan, 1995; Mises, 1960) relevant to large aircraft flight deviations (Campos & Marques, 2002; Eurocontrol, 1998), which are rare events (Reiss & Thomas, 2001; Nassar et al., 2011), is the generalized error distribution (Campos & Marques, 2004a), viz.:

$$F_k\left(x;\sigma \right)=A\exp \left(-a{\left|x\right|}^k\right),$$ E1

where $$k$$ is the weight. The constant a is determined by the condition of unit total probability:

$$A=k{a}^{1/k}/\left[2\Gamma \left(1/k\right)\right],$$ E2

where $$\Gamma \left(\alpha \right)$$ is the Gamma function of argument $$\alpha$$ . The constant a can be related by:

$$a^{2/k}={\sigma }^{-2}\left[\Gamma \left(3/k\right)/\Gamma \left(1/k\right)\right],$$ E3

to the r.m.s. position error $$\sigma$$ or variance $${\sigma }^2$$ . The case of weight unity in (2), viz.:

$$k=1:a=\sqrt{2}/\sigma ,A=1/\left(\sigma \sqrt{2}\right),$$ E4

corresponds by (1) to the Laplace probability distribution:

$$F_1\left(x;\sigma \right)=\left[1/\left(\sigma \sqrt{2}\right)\right]\exp \left(-\sqrt{2}\left|x\right|/\sigma \right);$$ E5

the case of weight two in (2), viz.:

$$k=2:a=1/\left(2{\sigma }^2\right),A=2/\left(\sigma \sqrt{2\pi }\right),$$ E6

leads by (1) to the Gaussian probability distribution:

$$F_2\left(x;\sigma \right)=\left[1/\left(\sigma \sqrt{2\pi }\right)\right]\exp \left[-x^2/\left(2{\sigma }^2\right)\right];$$ E7

the best approximation to large aircraft flight path deviations (Campos & Marques, 2002, 2007; Campos, 2001) corresponds approximately to weight one-half, so that (2):

$$k=1/2:a^4=120/{\sigma }^2,A=\sqrt{15/2}/\sigma ,$$ E8

substituted in (1) leads to:

$$F_{1/2}\left(x;\sigma \right)=\left(\sqrt{15/2}/\sigma \right)\exp \left(-\sqrt[4]{120}\left|x/\sigma \right|{}^{1/2}\right),$$ E9

which may be designated for brevity the ‘generalized’ distribution. For any probability distribution, it can be shown (Campos & Marques, 2002) that an upper bound for the probability of collision is the probability of coincidence, which (Figure 6) implies a deviation for the first aircraft, with r.m.s. position error $${\sigma }_1$$ ; a deviation L-x for the second aircraft error $${\sigma }_2$$ .

Figure 6.

Aircraft flying on parallel paths: a coincidence will occur if position errors are x (aircraft 1) and L-x (aircraft 2).

For statistically independent aircraft deviations, the probability of coincidence at position x the product:

$$P_k\left(x;L,{\sigma }_1,{\sigma }_2\right)=F_k\left(x;{\sigma }_1\right)F_k\left(L-x;{\sigma }_2\right).$$ E10

Its integral over all positions along the line joining the two aircraft is the CPC, viz.:

$$Q_k\left(L;{\sigma }_1,{\sigma }_2\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{P}_k}\left(x;L,{\sigma }_1,{\sigma }_2\right)dx={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{F}_k}\left(x;{\sigma }_1\right)F_k\left(L-x;{\sigma }_2\right)dx,$$ E11

and, in particular, for aircraft with equal r.m.s. position errors:

$$\sigma \equiv {\sigma }_1={\sigma }_2:Q_k\left(L;\sigma \right)\equiv Q_k\left(L;\sigma ,\sigma \right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{F}_k}\left(x;\sigma \right)F_k\left(L-x;\sigma \right)dx.$$ E12

The CPC has the dimensions of inverse length. The ICAO TLS of $$5\times {10}^{-9}$$ /h (12) can be converted for a maximum airspeed $$V_0=625kt$$ in (12) to a ATLS given:

$${\overline{Q}}_0=5\times {10}^{-9}{h}^{-1},V_0\le 625kt,Q\le Q_0={\overline{Q}}_0/V_0\le 8\times {10}^{-12}n{m}^{-1},$$ E13

which is an upper bound for the CPC. The safety criterion (12) is applied next to the Laplace (Section 2.2), Gaussian (Section 2.3) and generalized (Section 2.4) probability density functions.

2.2. Laplace distributions for the dissimilar aircraft

The ATLS (12) is the upper bound for the CPC (10) calculated for aircraft whose position errors follow the Laplace probability distribution (4), with dissimilar r.m.s. position errors for the two aircraft:

$$Q_0\ge Q_1\left(L;{\sigma }_1,{\sigma }_2\right)=\left[1/\left(2{\sigma }_1{\sigma }_2\right)\right]{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\exp \left[-\sqrt{2}\left(\left|x\right|/{\sigma }_1+\left|L-x\right|/{\sigma }_2\right)\right]}dx.$$ E14

The appearance of modulus in the argument of the exponential in (13), requires that the range of integration $$-\infty ,+\infty$$ be split in three parts. The first part corresponds to coincidence at $$0\le x\le L$$ between the flight paths of the two aircraft:

$$\begin{array}{l}2{\sigma }_1{\sigma }_2{Q}_{11}={\displaystyle {\int }_0^L\exp \left[-\sqrt{2}\left(x/{\sigma }_1+\left(L-x\right)/{\sigma }_2\right)\right]}dx=\\ =\exp \left(-\sqrt{2}L/{\sigma }_2\right){\displaystyle {\int }_0^L\exp }\left[-\sqrt{2}x\left(1/{\sigma }_1-1/{\sigma }_2\right)\right]dx,\end{array}$$ E15

and involves an elementary integration:

$$2{\sigma }_1{\sigma }_2{Q}_{11}=\exp \left(-\sqrt{2}L/{\sigma }_2\right)\left\{1-\exp \left[-\sqrt{2}L\left(1/{\sigma }_1-1/{\sigma }_2\right)\right]\right\}{\left[\sqrt{2}\left(1/{\sigma }_1-1/{\sigma }_2\right)\right]}^{-1},$$ E16

and simplifies to:

$$Q_{11}=\left[2\sqrt{2}\left({\sigma }_2-{\sigma }_1\right)\right]{}^{-1}\left[\exp \left(-\sqrt{2}L/{\sigma }_2\right)-\exp \left(-\sqrt{2}L/{\sigma }_1\right)\right],$$ E17

and should be the main contribution to (13). To evaluate (13) exactly, the remaining contributions, besides, are also considered the coincidence at a point $$x\ge L$$ outside the path of second aircraft:

$$2{\sigma }_1{\sigma }_2{Q}_{12}={\displaystyle {\int }_L^{\infty }\exp \left\{-\sqrt{2}\left[x/{\sigma }_1+\left(x-L\right)/{\sigma }_2\right]\right\}}dx,$$ E18

leads to an elementary integral:

$$\begin{array}{l}2{\sigma }_1{\sigma }_2{Q}_{12}=\exp \left(\sqrt{2}L/{\sigma }_2\right){\displaystyle {\int }_L^{\infty }\exp \left\{-\sqrt{2}x\left(1/{\sigma }_1+1/{\sigma }_2\right)\right\}}dx\\ =\exp \left(\sqrt{2}L/{\sigma }_2\right){\left[\sqrt{2}\left(1/{\sigma }_1+1/{\sigma }_2\right)\right]}^{-1}\exp [-\sqrt{2}L\left(1/{\sigma }_1+1/{\sigma }_2\right)],\end{array}$$ E19

which simplifies to:

$$Q_{12}={\left[2\sqrt{2}\left({\sigma }_1+{\sigma }_2\right)\right]}^{-1}\exp \left(-\sqrt{2}L/{\sigma }_1\right);$$ E20

the coincidence $$-\infty x0$$ outside the flight path of the first aircraft:

$$\begin{array}{l}2{\sigma }_1{\sigma }_2{Q}_{13}={\displaystyle {\int }_{-\infty }^0\exp \left\{\sqrt{2}\left[x/{\sigma }_1-\left(L-x\right)/{\sigma }_2\right]\right\}}dx\\ =\exp \left(-\sqrt{2}L/{\sigma }_2\right){\displaystyle {\int }_0^{\infty }\exp \left\{-\sqrt{2}x\left(1/{\sigma }_1+1/{\sigma }_2\right)\right\}}dx,\end{array}$$ E21

is again an elementary integral:

$$Q_{13}=\left[2\sqrt{2}\left(\sigma {}_2+{\sigma }_1\right)\right]{}^{-1}\exp \left(-\sqrt{2}L/{\sigma }_2\right).$$ E22

The sum of (21), (19) and (16) specifies the CPC where:

$$\begin{array}{l}{Q}_1\left(L;{\sigma }_1,{\sigma }_2\right)={\left[2\sqrt{2}\left({\sigma }_2-{\sigma }_1\right)\right]}^{-1}\left[\exp \left(-\sqrt{2}L/{\sigma }_2\right)-\exp \left(-\sqrt{2}L/{\sigma }_1\right)\right]\\ +{\left[2\sqrt{2}\left({\sigma }_2+{\sigma }_1\right)\right]}^{-1}\left[\exp \left(-\sqrt{2}L/{\sigma }_1\right)+\exp \left(-\sqrt{2}L/{\sigma }_2\right)\right],\end{array}$$ E23

for the Laplace distribution:

$$Q_1\left(L;{\sigma }_1,{\sigma }_2\right)=Q_{11}+Q_{12}+Q_{13}\le Q_0=8\times {10}^{-12}n{m}^{-1},$$ E24

and hence (12) the safety criterion. Of the preceding expressions, only (16) breaks down for $${\sigma }_2-{\sigma }_1=0$$ , i.e., aircraft with the same r.m.s. position error $${\sigma }_1={\sigma }_2\equiv \sigma$$ . In this case the probability of coincidence is given between the flight paths of the two aircraft, instead of (14-16) by:

$${\sigma }_1={\sigma }_2\equiv \sigma :{\overline{Q}}_{11}=\left(2{\sigma }^2\right){}^{-1}{\displaystyle {\int }_0^L\exp \left(-\sqrt{2}L/\sigma \right)}dx=\left[L/\left(2{\sigma }^2\right)\right]\exp \left(-\sqrt{2}L/\sigma \right);$$ E25

outside the flight path of the second aircraft (17-19) is replaced by:

$$\begin{array}{l}{\sigma }_1={\sigma }_2\equiv \sigma :{\overline{Q}}_{12}={\left(2{\sigma }^2\right)}^{-1}\exp \left(\sqrt{2}L/\sigma \right){\displaystyle {\int }_L^{\infty }\exp }\left(-2\sqrt{2}x/\sigma \right)dx\\ ={\left(4\sqrt{2}\sigma \right)}^{-1}\exp \left(-\sqrt{2}L/\sigma \right);\end{array}$$ E26

outside the flight path of the second aircraft (20-22) is replaced by:

$$\begin{array}{l}{\sigma }_1={\sigma }_2\equiv \sigma :{\overline{Q}}_{13}={\left(2{\sigma }^2\right)}^{-1}\exp \left(-\sqrt{2}L/\sigma \right){\displaystyle {\int }_{-\infty }^0\exp }\left(2\sqrt{2}x/\sigma \right)dx\\ ={\left(4\sqrt{2}\sigma \right)}^{-1}\exp \left(-\sqrt{2}L/\sigma \right).\end{array}$$ E27

The sum of (23), (24) and (25) specifies:

$$Q_1\left(L;\sigma \right)=\exp \left(-\sqrt{2}L/\sigma \right){\left(2\sigma \right)}^{-1}\left(L/\sigma +1/\sqrt{2}\right),$$ E28

as the safety criterion:

$${\sigma }_1={\sigma }_2\equiv \sigma :Q_1\left(L;\sigma \right)={\overline{Q}}_{11}+{\overline{Q}}_{12}+{\overline{Q}}_{13}\le Q_0=8\times {10}^{-12}n{m}^{-1},$$ E29

for Laplace probabilities with equal r.m.s. position errors for both aircraft.

2.3. Gaussian distribution with distinct variances

The ATLS (12) is the upper bound for the CPC (10) calculated next for aircraft whose flight path deviations satisfy the Gaussian probability distribution (6) for aircraft with dissimilar variances of position errors:

$$Q_0\ge Q_2\left(L;{\sigma }_1,{\sigma }_2\right)={\left(2\pi {\sigma }_1{\sigma }_2\right)}^{-1}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\exp \left\{-\left[{\left(x/{\sigma }_1\right)}^2+\left((L-x\right)/\sigma {{}_2)}^2\right]/2\right\}}dx.$$ E30

The integral in (27) does not need splitting to be evaluated, e.g. in the case of equal variances:

$$\begin{array}{l}{\sigma }_1={\sigma }_2\equiv \sigma :Q_0\ge Q_2\left(L;\sigma \right)={\left(2\pi {\sigma }^2\right)}^{-1}{\displaystyle \underset{}{\overset{+\infty }{\int }}\exp }\left\{-\left[x^2+{\left(L-x\right)}^2\right]/\left(2{\sigma }^2\right)\right\}dx,\\ ={\left(2\pi {\sigma }^2\right)}^{-1}\exp \left[-L^2/2{\sigma }^2\right]{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\exp }\left\{-\left(x^2-xL\right)/{\sigma }^2\right\}dx,\end{array}$$ E31

the change of variable (29):

$$y=\left(x-L/2\right)/\sigma :{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\exp \left(-y^2\right)}dy=\sqrt{\pi },$$ E32

leads to a Gaussian integral (29), viz.:

$$Q_2\left(L;\sigma \right)={\left(2\pi {\sigma }^2\right)}^{-1}\exp \left[-L^2/\left(2{\sigma }^2\right)\right]{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\exp \left(-y^2+L^2/4{\sigma }^2\right)}dy;$$ E33

using (29) in (30) leads to:

$$Q_2\left(L;\sigma \right)={\left(2\sqrt{\pi }{\sigma }^2\right)}^{-1}\exp \left[-{\left(L/2\sigma \right)}^2\right]\le Q_0=8\times {10}^{-12}n{m}^{-1},$$ E34

as the safety criterion.

In the more general case (27) of aircraft with dissimilar r.m.s. position errors:

$$Q_2={\left(2\pi {\sigma }_1{\sigma }_2\right)}^{-1}\exp \left(-L^2{\sigma }_2^{-2}/2\right){\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\exp \left\{-\left[\left(x^2/2\right)\left({\sigma }_1{}^{-2}+{\sigma }_2^{-2}\right)-xL{\sigma }_2^{-2}\right]\right\}}dx,$$ E35

the change of variable:

$$y=\left[x\sqrt{{\sigma }_1^{-2}+{\sigma }_2^{-2}}-L{\sigma }_2^{-2}/\sqrt{{\sigma }_1^{-2}+{\sigma }_2^{-2}}\right]/\sqrt{2},$$ E36

leads again to a Gaussian integral (29), viz.:

$$Q_2={\left(2\pi {\sigma }_1{\sigma }_2\right)}^{-1}\exp \left(-L^2{\sigma }_2^{-2}/2\right)\exp \left\{L^2{\sigma }_2^{-4}/\left[2\left({\sigma }_1^{-2}+{\sigma }_2^{-2}\right)\right]\right\}{\displaystyle {\int }_{-\infty }^{+\infty }\exp \left(-y^2\right)}dy,$$ E37

which simplifies the safety condition to:

$$Q_0\ge Q_2\left(L;{\sigma }_1,{\sigma }_2\right)={\left(2\sqrt{\pi }{\sigma }_1{\sigma }_2\right)}^{-1}\exp \left\{-\left(L^2/2\right)/\left[{\left({\sigma }_1\right)}^2+{\left({\sigma }_2\right)}^2\right]\right\}.$$ E38

This reduces to (31) in the case of equal r.m.s. position errors.

2.4. Generalized error or Gaussian distribution

The safety condition (12) for (10) the more accurate (8) generalized probability distribution:

$$\begin{array}{l}c\equiv \sqrt[4]{120}:\\ Q_0\ge Q_3\left(L;{\sigma }_1,{\sigma }_2\right)=\left[15/\left(2{\sigma }_1{\sigma }_2\right)\right]{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\exp }\left\{-c\left[{\left|x/{\sigma }_1\right|}^{1/2}+{\left|\left(L-x\right)/{\sigma }_2\right|}^{1/2}\right]\right\}dx,\end{array}$$ E39

requires again a split in the region of integration as for the Laplace distribution (Section 2.2), with the difference that the evaluation of integrals is not elementary. The contribution to the cumulative probability of coincidence of the position between the flight paths of the two aircraft is:

$$$$

$$Q_{31}=\left[\left(15/2{\sigma }_1{\sigma }_2\right)\right]{\displaystyle \underset{0}{\overset{L}{\int }}\exp \left\{-c\left[\sqrt{x/{\sigma }_1}+\sqrt{\left(L-x\right)/{\sigma }_2}\right]\right\}}dx,$$ E40

$$=\frac{15}{2{\sigma }_1{\sigma }_2}{\displaystyle \sum _{n=0}^{\infty }\frac{{(-)}^n{c}^n}{n!}}{{\displaystyle {\int }_0^L\left\{\sqrt{x/{\sigma }_1}+\sqrt{\left(L-x\right)/{\sigma }_2}\right\}}}^{n}dx,$$ E41

where the exponential was expanded in power series, and binomial theorem:

$$\left[\sqrt{x/{\sigma }_1}+\sqrt{\left(L-x\right)/{\sigma }_2}\right]{}^n={\displaystyle \sum _{m=0}^n\left\{n!/\left[m!\left(n-m\right)!\right]\right\}}{\left(x/{\sigma }_1\right)}^{m/2}{\left[\left(L-x\right)/{\sigma }_2\right]}^{\left(n-m\right)/2},$$ E42

can also be used:

$$Q_{31}=\frac{15}{2{\sigma }_1{\sigma }_2}{\displaystyle \sum _{n=0}^{\infty }{\displaystyle \sum _{m=0}^n\frac{{(-)}^n{c}^n}{m!\left(n-m\right)!}}}{\sigma }_1^{-m/2}{\sigma }_2^{-\left(n-m\right)/2}{{\rm I}}_{n,m},$$ E43

and $${{\rm I}}_{n,m}$$ denotes the integral:

$${{\rm I}}_{n,m}\equiv {\displaystyle {\int }_0^L{x}^{m/2}}{\left(L-x\right)}^{\left(n-m\right)/2}dx,$$ E44

which can be reduced to an Euler’s Beta function. The Beta function (40) is defined (Whittaker & Watson, 1927) by:

$$B\left(\alpha ,\beta \right)\equiv {\displaystyle {\int }_0^1{y}^{\alpha -1}}{\left(1-y\right)}^{\beta -1}dy=\Gamma \left(\alpha \right)\Gamma \left(\beta \right)/\Gamma \left(\alpha +\beta \right),$$ E45

and can be evaluated (40) in terms of Gamma functions (Goursat, 1950). The integrals (39) are evaluated in terms of the Beta function via a change of variable.

$$\begin{array}{l}y\equiv x/L:\\ L^{-1-n/2}{{\rm I}}_{nm}={\displaystyle {\int }_0^1{y}^{m/2}}{\left(1-y\right)}^{\left(n-m\right)/2}dy=B\left(1+m/2,1+\left(n-m\right)/2\right)\\ =\Gamma \left(1+m/2\right)\Gamma \left(1+\left(n-m\right)/2\right)/\Gamma \left(2+n/2\right).\end{array}$$ E46

Substitution of (41) in (39) yields:

$$Q_{31}=\frac{15L}{2{\sigma }_1{\sigma }_2}{\displaystyle \sum _{n=0}^{\infty }{\displaystyle \sum _{m=0}^n}\frac{{(-)}^n{c}^n}{m!\left(n-m\right)!}}{\left(\frac{L}{{\sigma }_1}\right)}^{m/2}{\left(\frac{L}{{\sigma }_2}\right)}^{\left(n-m\right)/2}\frac{\Gamma \left(1+m/2\right)\Gamma \left(1+n/2-m/2\right)}{\Gamma \left(2+n/2\right)},$$ E47

as the first contribution to (36).

Since (42) may be expected to be the main contribution to (36), we seek upper bounds for the two remaining contributions. The second contribution to (36) concerns coincidence outside the path of the second aircraft:

$$Q_{32}=\left[15/\left(2{\sigma }_1{\sigma }_2\right)\right]{\displaystyle {\int }_L^{\infty }\exp }\left\{-c\left[\sqrt{x/{\sigma }_1}+\sqrt{\left(x-L\right)/{\sigma }_2}\right]\right\}dx;$$ E48

an upper bound is obtained by replacing $$x\ge L$$ by $$L$$ in the first exponential:

$$Q_{32}\le \left[15/\left(2{\sigma }_1{\sigma }_2\right)\right]\exp \left(-c\sqrt{L/{\sigma }_1}\right){\displaystyle {\int }_L^{\infty }\exp \left[-c\sqrt{\left(x-L\right)/{\sigma }_2}\right]}dx,$$ E49

the change of variable (44) leads:

$$y=c\sqrt{\left(x-L\right)/{\sigma }_2},Q_{32}\le \frac{15}{{\sigma }_1{c}^2}\exp \left(-c\sqrt{L/{\sigma }_1}\right){\displaystyle {\int }_0^{\infty }{e}^{-y}}ydy,$$ E50

to an integral (44) which is evaluated in terms (Whittaker & Watson, 1927; Goursat, 1950) of the Gamma function:

$${\displaystyle {\int }_0^{\infty }{e}^{-y}}{y}^{n}dy=\Gamma \left(n+1\right)\equiv n!$$ E51

using (45) in (44) leads to the upper bound for the second contribution to (36), viz.:

$$Q_{32}\le \left[15/\left({\sigma }_1{c}^2\right)\right]\exp \left(-c\sqrt{L/{\sigma }_1}\right).$$ E52

The third contribution to (36) corresponds to coincidence outside the flight path of the first aircraft:

$$Q_{33}=\left[15/\left(2{\sigma }_1{\sigma }_2\right)\right]{\displaystyle \underset{-\infty }{\overset{0}{\int }}\exp }\left\{-c\left[\sqrt{-x/{\sigma }_1}+\sqrt{\left(L-x\right)/{\sigma }_2}\right]\right\}dx,$$ E53

$$=\left[15/\left(2{\sigma }_1{\sigma }_2\right)\right]{\displaystyle {\int }_0^{\infty }\exp \left(-c\sqrt{x/{\sigma }_1}\right)}\exp \left(-c\sqrt{\left(L+x\right)}/{\sigma }_2\right)dx;$$ E54

an upper bound is obtained by replacing in the second exponential $$L+x\ge L$$ by L:

$$Q_{33}\le \left[15/\left(2{\sigma }_1{\sigma }_2\right)\right]\exp \left(-c\sqrt{L/{\sigma }_2}\right){\displaystyle {\int }_0^{\infty }\exp \left(-c\sqrt{x/{\sigma }_1}\right)}dx.$$ E55

The last integral is evaluated via a change of variable:

$$y=c\sqrt{x/{\sigma }_1}:Q_{33}\le \left[15/\left({\sigma }_2{c}^2\right)\right]\exp \left(-c\sqrt{L/{\sigma }_2}\right){\displaystyle {\int }_0^{\infty }{e}^{-y}}ydy,$$ E56

leading by (45) to:

$$Q_{33}\le \left[15/\left({\sigma }_2{c}^2\right)\right]\exp \left(-c\sqrt{L/{\sigma }_2}\right).$$ E57

If the upper bounds (45) and (47) are small relative to the first contribution (42) to (36), viz.:

$$Q_{31}\left(15/c^2\right)\left[{\sigma }_1^{-1}\exp \left(-c\sqrt{L/{\sigma }_1}\right)+{\sigma }_2^{-1}\exp \left(-c\sqrt{L/{\sigma }_2}\right)\right]\ge Q_{32}+Q_{33},$$ E58

then (46) alone can be used in the safety criterions (12), viz.:

$$8\times {10}^{-12}n{m}^{-1}=Q_0\ge Q_{31},$$ E59

with an error whose upper bound is specified by the ratio of the r.h.s. to l.h.s. of (48). If the latter error is not acceptable, then (43) and (46) must be evaluated exactly. Concerning the second contribution (43) to (36), the change of variable (49):

$$x=L{\cosh }^2\alpha ,x-L=L{\sinh }^2\alpha ,$$ E60

implies (49), and transforms (43) to:

$$Q_{32}=\left[15L/\left({\sigma }_1{\sigma }_2\right)\right]{\displaystyle {\int }_0^{\infty }d\alpha \cosh \alpha \sinh \alpha \exp }\{-c\sqrt{L}\left({\sigma }_1^{-1/2}\cosh \alpha +{\sigma }_2^{-1/2}\sinh \alpha \right).$$ E61

Concerning the third contribution (46) to (36) the change or variable (50):

$$x=L{\sinh }^2\alpha ,x+L=L{\cosh }^2\alpha ,$$ E62

implies (50), and leads to:

$$Q_{33}=\left[15L/\left({\sigma }_1{\sigma }_2\right)\right]{\displaystyle {\int }_0^{\infty }d\alpha }\sinh \alpha \cosh \alpha \exp \left\{-c\sqrt{L}\left({\sigma }_1^{-1/2}\sinh \alpha +{\sigma }_2^{-1/2}\cosh \alpha \right)\right\},$$ E63

which is similar to (49) interchanging $${\sigma }_1$$ with $${\sigma }_2$$ . The integrals (49) and (50) can be evaluated numerically, and coincide in the case of equal r.m.s. position errors:

$${\sigma }_1={\sigma }_2\equiv \sigma :Q_{32}=Q_{33}=\frac{15L}{4{\sigma }^2}{\displaystyle {\int }_0^{\infty }\exp \left(-c\sqrt{L/\sigma }{e}^{\alpha }\right)}\left(e^{2\alpha }-e^{-2\alpha }\right)d\alpha .$$ E64

A further change of variable (51) yields:

$$\begin{array}{l}y=c\sqrt{L/\sigma }{e}^{\alpha }:\\ Q_{32}+Q_{33}=\frac{15L}{2{\sigma }^2}{\displaystyle {\int }_{c\sqrt{L/\sigma }}^{\infty }{e}^{-y}}\{\left[\sigma /\left(c^{2}L\right)\right]y-\left(c^{2}L/\sigma \right)y^{-3}]dy.\end{array}$$ E65

The exponential integral of order n is defined (Abramowitz & Stegun, 1965) by:

$$E_n\left(z\right)={\displaystyle {\int }_z^{\infty }{y}^n}{e}^{-y}dy,$$ E66

and allows evaluation of (51), viz.:

$$Q_{32}+Q_{33}=\left[15L/\left(2{\sigma }^2\right)\right]\{\left[\sigma /\left(c^{2}L\right)\right]E_1\left(c\sqrt{L/\sigma }\right)-\left[\left(c^{2}L\right)/\sigma \right]E_{-3}\left(c\sqrt{L/\sigma }\right)\}.$$ E67

The sum of the three contributions (42) plus (49) and (50) or (52), specifies:

$$8\times {10}^{-12}n{m}^{-1}=Q_0\ge Q_3\left(L;{\sigma }_1,{\sigma }_2\right)=Q_{31}+Q_{32}+Q_{33},$$ E68

as the safety condition.

3. Application to four ATM scenarios

The preceding safety-separation criteria are applied to the four major airway scenarios, viz. lateral separation in uncontrolled (Section 3.1) and controlled (Section 3.2) airspace and standard (Section 3.3) and reduced (Section 3.4) vertical separation.

Probability distribution		Laplace	Gauss	Generalized
quantity	$${\sigma }_a$$	$$Q_{1a}$$	$$Q_{2a}$$	$$Q_{3a}$$
Unit	nm	-	-	-
10	nm	2,42E-04	5,45E-06	3,80E-04
5	nm	7,72E-07	1,57E-13	3,58E-05
4	nm	3,47E-08	1,91E-19	1,28E-05
3	nm	1,68E-10	2,17E-32	2,75E-06
2	nm	2,84E-15	9,77E-70	1,92E-07
1	nm	4,95E-30	1,04E-272	3,88E-10
0,5	nm	3,84E-60	0,00E-00	4,70E-14

Table 1.

Lateral a CPC for the Laplace, Gaussian and generalized probabilities.

3.1 Lateral separation in oceanic airspace

The lateral separation in oceanic airspace is (53):

$$L_a=50nm,{\sigma }_a=0.5,1.0,2.0,3.0,4.0,5.0,10,nm,$$ E69

and the r.m.s. position error is given the values (53) in Table 1, where the CPC are indicated for the Laplace, Gaussian and generalized probabilities. Taking as reference the generalized probability distribution, that is the most accurate representation of large flight path deviation considerably underestimates the risk of collision, and the Laplace distribution although underestimating less is still not safe. For example the ICAO ATLS of $$8\times {10}^{-12}$$ /nm is met according to the generalized probability distribution for a r.m.s. position deviation $${\sigma }_a\le 1$$ nm; the Laplace distribution would give $${\sigma }_a\le 3$$ nm and the Gaussian $${\sigma }_a\le 5$$ nm. The latter are both unsafe, because for $${\sigma }_a=3$$ nm the generalized distribution gives a collision probability $$2.75\times {10}^{-6}$$ /nm and for $${\sigma }_a=5$$ nm it gives $$3.58\times {10}^{-5}$$ /nm and both significant exceed the ICAO ATLS.

3.2. Lateral separation in controlled airspace

In controlled airspace the lateral separation (53) is reduced to (54):

$$L_b=5nm,{\sigma }_b=0.05,0.1,0.2,0.3,0.4,0.5,1.0nm,$$ E70

and the r.m.s. position errors considered (54) are correspondingly smaller than (53). Again the generalized distribution meets the ICAO ATLS for a r.m.s. deviation $${\sigma }_b\le 0.05$$ nm smaller than predicted by the Laplace ( $${\sigma }_b\le 0.2$$ nm) and Gaussian ( $${\sigma }_b\le 0.5$$ nm) distributions. For the safe r.m.s deviation $${\sigma }_b=0.05$$ nm the Gaussian probability of collision is negligible.

Probability distribution		Laplace	Gauss	Generalized
quantity	$${\sigma }_b$$	$$Q_1{}_b$$	$$Q_2{}_b$$	$$Q_3{}_b$$
Unit	nm	-	-	-
1,0	nm	2,42E-03	5,45E-04	3,80E-03
0,5	nm	7,72E-06	1,57E-11	3,58E-04
0,4	nm	3,47E-07	1,91E-17	1,28E-04
0,3	nm	1,68E-09	2,17E-30	2,75E-05
0,2	nm	2,84E-14	9,77E-68	1,92E-06
0,1	nm	4,95E-29	1,04E-270	3,88E-09
0,05	nm	3,84E-59	0,00E-00	4,70E-13

Table 2.

Lateral b CPC for the Laplace, Gaussian and generalized probabilities.

3.3. Vertical separation in oceanic airspace

The probabilities of vertical separation can be less upward than downward, due to gravity, proximity to the service ceiling, etc.; apart from this correction (Campos & Marques, 2007, 2011), the preceding theory can be used with the standard vertical separation (55):

$$L_c=2000ft,{\sigma }_c=40,50,100,200,300ft,$$ E71

and r.m.s. deviations (55). The r.m.s. height deviation that meets the ICAO ATLS is about 40 ft according to the generalized distribution, with larger and unsafe predictions for the Laplace (100 ft) and Gaussian (200 ft) distributions.

Probability distribution		Laplace	Gauss	Generalized
Quantity	$${\sigma }_c$$	$$Q_{1c}$$	$$Q_{2c}$$	$$Q_{3c}$$
Unit	ft	-	-	-
300	ft	9,88E-07	4,68E-11	4,03E-06
200	ft	1,93E-08	9,79E-17	8,76E-07
100	ft	5,39E-14	1,05E-48	2,11E-08
50	ft	1,10E-25	2,16E-178	8,12E-11
40	ft	1,24E-31	6,49E-276	8,21E-12

Table 3.

Vertical a CPC for the Laplace, Gaussian and generalized probabilities.

3.4. Reduced vertical separation

The RSVM (Figure 5) introduced by Eurocontrol in upper European air space halves the vertical separation (56) to (58):

$$L_d=1000ft,{\sigma }_d=15,50,100,150ft,$$ E72

and the r.m.s. position errors are correspondingly reduced from (56) to (58) in Table 4.

Probability distribution		Laplace	Gauss	Generalized
quantity	$${\sigma }_d$$	$$Q_{1d}$$	$$Q_{1d}$$	$$Q_{3d}$$
Unit	ft	-	-	-
150	ft	1,98E-06	1,87E-10	8,05E-06
100	ft	3,86E-08	3,92E-16	1,71E-06
50	ft	1,08E-13	4,20E-48	4,04E-08
15	ft	2,55E-41	0,00E-00	6,86E-13

Table 4.

Vertical b CPC for the Laplace, Gaussian and generalized probabilities.

Taking as reference the generalized distribution to meet the ICAO ATLS: (i) the RVSM from 2000 ft (Table 3) to 1000 ft (Table 4) requires a reduction in r.m.s. altitude error from 40 ft to 15 ft; (ii) the reduction of lateral separation from 50 nm in transoceanic (Table 1) to 5 nm in controlled (Table 2) airspace required a reduction of r.m.s. deviation from 0.5 to 0.05 nm.

4. Discussion

The separation-position accuracy or technology-capacity trade-off was made for an air corridor ATM scenario with aircraft flying along the same flight path (Figure 2) or on parallel flight paths (Figure 1) with a constant separation. The generalized probability distribution leads to lower values of the r.m.s. deviation to meet the ICAO TLS, than the Laplace and Gaussian. Although the latter distributions are simpler, they underestimated the collision risk, and do not yield safe predictions. Using simultaneously lateral and vertical separations leads to much lower collision probabilities, and allows reducing each separation for the same overall safety. In the case of aircraft flying on parallel tracks, it is desirable to use alternate directions of flight (Figure 5), because: (i) adjacent flight paths correspond to aircraft flying in opposite directions, which spend less time close to each other, reducing the collision probability (Campos & Marques, 2002; Eurocontrol, 1988; Reich, 1966); (ii) the aircraft which spend more time ‘close’ by are on a parallel track at twice the separation 2L, thus allowing a larger r.m.s. position error $$\sigma$$ for the same safety. If the aircraft have both altitude and lateral separation, and the two position errors are statistically independent, the ICAO ATLS is $$\sqrt{8\times {10}^{-12}}/nm\approx 2.8\times {10}^{-6}/nm$$ in each direction. For transoceanic flight this is met by a lateral r.m.s. deviation $${\sigma }_l\le 3$$ nm; for flight in controlled airspace with RVSM the ICAO ATLS wold be met with lateral $${\sigma }_l\le 0.2$$ nm and altitude $${\sigma }_h\le 150$$ ft r.m.s. deviations. Using also along track or longitudinal separation adds a third dimension, requiring a smaller ICAO ATLS $$\sqrt[3]{8\times {10}^{-12}}/nm=2\times {10}^{-4}/nm$$ and allowing larger r.m.s. deviations in three directions.