1. Introduction
The steady growth of air traffic at a rate of 37% 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 takeoff queues at runways at airports (FAA, 2011; International Civil Aviation Organization [ICAO], 2007; Rossow, 1999); (ii) collision probabilities for the inflight 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
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, twoaircraft events are more likely and this the case considered in the present chapter.
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
In the present chapter the CPC is calculated (Section 2) for comparison with the ICAO ATLS of
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.:
where
where
to the r.m.s. position error
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 onehalf, 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 error
For statistically independent aircraft deviations, the probability of coincidence at position x the 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
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
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
leads to an elementary integral:
which simplifies to:
the coincidence
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 for
outside the flight path of the second aircraft (1719) is replaced by:
outside the flight path of the second aircraft (2022) is replaced by:
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:
(31) 
the change of variable (29):
leads to a Gaussian integral (29), viz.:
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
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.
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
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
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
A further change of variable (51) yields:
The exponential integral of order n is 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 safetyseparation 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.
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
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
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.
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 distribution  Laplace  Gauss  Generalized  
quantity 




Unit  ft       
150  ft  1,98E06  1,87E10  8,05E06 
100  ft  3,86E08  3,92E16  1,71E06 
50  ft  1,08E13  4,20E48  4,04E08 
15  ft  2,55E41  0,00E00  6,86E13 
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 separationposition accuracy or technologycapacity tradeoff 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