On Safety Metrics related to Aircraft Separation

Abstract

ICAO safety standards require low probabilities of collision of the order of 5×10⁻⁹ per hour. Such low probabilities are virtually impossible to confirm through simulations, implying the need for alternative and related safety metrics, which are easier to use. Several such alternative safety metrics are discussed in this paper. For example, it is shown that if the r.m.s. position error σ is less than about one-tenth L/σ ∼ 10 of the minimum separation distance L, then the probabilities of collision will be less than the ICAO standard, in the case of aircraft flying in opposite directions on parallel tracks. This result is derived from an analytical formula based on the statistical application of a Gaussian probability distribution to a worst-case collision scenario and a rather different result L/σ ∼ 20 arises for a Laplace probability distribution. The implication is that the ICAO standard of low probability of collision can be satisfied, by checking that the r.m.s. position error does not exceed a given value, though that value is dependent on the shape of the ‘tail’ of the probability distribution. Whether the condition is met can be checked in simulations, by measuring the drift between the intended and actual trajectory, due to all factors (navigation inaccuracy, atmospheric disturbances, trajectory drift between position updates, etc.). Thus the r.m.s. deviation from the intended trajectory can be used as a safety metric, as an alternative to the probability of collision, using the formulas and tables provided. The formulas concern a variety of possible safety metrics, including the maximum and cumulative probabilities of coincidence, probabilities of overlap, collision rates and collision probabilities; the tables apply to horizontal and vertical separation in controlled and transoceanic airspace. The sensitivity of the results to the probability distribution assumed (Gaussian or Laplace) suggests the introduction of a parametric family exponential of probability distributions, of which these are particular cases. A choice of the parameter is given that could lead to a probability distribution with a ‘tail’ shape more suited to typical Air Traffic Management (ATM) scenarios than the Gaussian and Laplace distributions, while being simpler than multi-parameter distributions.