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Collision Probabilities, Aircraft Separation and Airways Safety

Written By

Luís Campos and Joaquim Marques

Submitted: November 14th, 2010 Reviewed: May 23rd, 2011 Published: September 12th, 2011

DOI: 10.5772/21468

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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 errorσ; (iii) the combination of Land σshould be such that the probability of collision (ICAO, 2006) does not exceed ICAO Target Level of Safety (TLS) of 5×109per hour (ICAO, 2005). Thus the key issue is to determine the relation between aircraft separation L and position accuracyσ, which ensures that the ICAO TLS is met. Then the technically achievable position accuracy σspecifies L, viz. the safe separation distance (SSD). Conversely, if an increase in capacity is sought, the separation Lmust be reduced; then the ICAO TLS leads to a position accuracy σwhich must be met by the ‘technology’. The position accuracy σincludes 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 distanceL.

Figure 2.

Aircraft flying always at minimum longitudinal separation distanceL.

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 thanV0=625kt, the ICAO TLS of P05×109/h, is met by Q0=P0/V08×1012/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 8×1012probability 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 La=50nmin uncontrolled (e.g. oceanic) airspace (Section 3.1) and Lb=5nmin controlled airspace (Section 3.2); (iii/iv) standard altitude separation Lc=2000ftused worldwire (Section 3.3) and RVSM Ld = 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 accuraciesσ, 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 errorsσ; (iii) the case of aircraft with dissimilar position errors σ1and σ2is 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 Lin parallel flight paths (Figure 1), (ii) or at constant longitudinal separation Lon 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.:


where kis the weight. The constant ais determined by the condition of unit total probability:


where Γ(α)is the Gamma function of argumentα. The constant acan be related by:


to the r.m.s. position error σor varianceσ2. The case of weight unity in (2), viz.:


corresponds by (1) to the Laplace probability distribution:


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


leads by (1) to the Gaussian probability distribution:


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


substituted in (1) leads to:


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 errorσ1; a deviation L-xfor the second aircraft errorσ2.

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 xthe product:


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


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


The CPC has the dimensions of inverse length. The ICAO TLS of 5×109/h (12) can be converted for a maximum airspeed V0=625ktin (12) to a ATLS given:


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:


The appearance of modulus in the argument of the exponential in (13), requires that the range of integration ,+be split in three parts. The first part corresponds to coincidence at 0xLbetween the flight paths of the two aircraft:

2σ1σ2Q11=0Lexp[2(x/σ1+(Lx)/σ2)]dx=              = exp(2L/σ2)0Lexp[2x(1/σ11/σ2)]dx,E15

and involves an elementary integration:


and simplifies to:


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


leads to an elementary integral:


which simplifies to:


the coincidence x0outside the flight path of the first aircraft:


is again an elementary integral:


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


for the Laplace distribution:


and hence (12) the safety criterion. Of the preceding expressions, only (16) breaks down forσ2σ1=0, i.e., aircraft with the same r.m.s. position errorσ1=σ2σ. In this case the probability of coincidence is given between the flight paths of the two aircraft, instead of (14-16) by:


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

σ1=σ2σ:         Q¯12=(2σ2)1exp(2L/σ)Lexp(22x/σ)dx                                    =(42σ)1exp(2L/σ);E26

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

σ1=σ2σ:Q¯13=(2σ2)1exp(2L/σ)0exp(22x/σ)dx                                     =(42σ)1exp(2L/σ).E27

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


as the safety criterion:


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:


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

σ1=σ2σ:       Q0Q2(L;σ)=(2πσ2)1+exp{[x2+(Lx)2]/(2σ2)}dx,                          =(2πσ2)1exp[L2/2σ2]+exp{(x2xL)/σ2}dx,E31

the change of variable (29):


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


using (29) in (30) leads to:


as the safety criterion.

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


the change of variable:


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


which simplifies the safety condition to:


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:


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:


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


can also be used:


and Ιn,mdenotes the integral:


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


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.

yx/L:L1n/2Ιnm=01ym/2(1y)(nm)/2dy=B(1+m/2,1+(nm)/2)                                         =Γ(1+m/2)Γ(1+(nm)/2)/Γ(2+n/2).E46

Substitution of (41) in (39) yields:


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:


an upper bound is obtained by replacing xLby Lin the first exponential:


the change of variable (44) leads:


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


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


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


an upper bound is obtained by replacing in the second exponential L+xLby L:


The last integral is evaluated via a change of variable:


leading by (45) to:


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


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


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):


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


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


implies (50), and leads to:


which is similar to (49) interchanging σ1withσ2. The integrals (49) and (50) can be evaluated numerically, and coincide in the case of equal r.m.s. position errors:


A further change of variable (51) yields:


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


and allows evaluation of (51), viz.:


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


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 distributionLaplaceGaussGeneralized

Table 1.

Lateral aCPC for the Laplace, Gaussian and generalized probabilities.


3.1 Lateral separation in oceanic airspace

The lateral separation in oceanic airspace is (53):


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 8×1012/nm is met according to the generalized probability distribution for a r.m.s. position deviation σa1nm; the Laplace distribution would give σa3nm and the Gaussian σa5nm. The latter are both unsafe, because for σa=3nm the generalized distribution gives a collision probability 2.75×106/nm and for σa=5nm it gives 3.58×105/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):


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 σb0.05nm smaller than predicted by the Laplace (σb0.2nm) and Gaussian (σb0.5nm) distributions. For the safe r.m.s deviation σb=0.05nm the Gaussian probability of collision is negligible.

Probability distributionLaplaceGaussGeneralized

Table 2.

Lateral bCPC 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):


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 distributionLaplaceGaussGeneralized

Table 3.

Vertical aCPC 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):


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

Probability distributionLaplaceGaussGeneralized

Table 4.

Vertical bCPC 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 σfor the same safety. If the aircraft have both altitude and lateral separation, and the two position errors are statistically independent, the ICAO ATLS is 8×1012/nm2.8×106/nmin each direction. For transoceanic flight this is met by a lateral r.m.s. deviation σl3nm; for flight in controlled airspace with RVSM the ICAO ATLS wold be met with lateral σl0.2nm and altitude σh150ft r.m.s. deviations. Using also along track or longitudinal separation adds a third dimension, requiring a smaller ICAO ATLS 8×10123/nm=2×104/nmand allowing larger r.m.s. deviations in three directions.


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Written By

Luís Campos and Joaquim Marques

Submitted: November 14th, 2010 Reviewed: May 23rd, 2011 Published: September 12th, 2011