On the Probability of Collision Between Climbing and Descending Aircraft

Abstract

The probability of coincidence (which is a upper bound for the probability of collision) is calculated for aircraft on arbitrary straight flight paths with constant speed: either aircraft may be climbing, descending, or in level flight and they may cross at any angle. Gaussian statistics with equal or distinct rms deviations for the two aircraft are used to calculate the probability of coincidence as a function of time in the first instance, with a correction factors for 1) the more accurate generalized error distribution; 2) asymmetry in vertical position errors. The time and distance of closest approach are used to calculate the position for maximum probability of coincidence. The cumulative probability of coincidence over all time is calculated, and expressed as a probability of coincidence per unit distance and per unit time: the latter is compared with the International Civil Aviation Organization target level of safety. Examples are given of the effect on coincidence probabilities of the seven parameters of the problem: velocities and glide slopes of each aircraft, crossing angle in horizontal projection, vertical separation, and rms position error.