Probability of Potential Collision for Aircraft Encounters in High Density Airspaces

The Reich model uses information related to the probabilistic distributions of aircraft's 15 lateral and vertical position, traffic flows of the routes, aircraft's relative velocities and 16 aircraft dimensions to generate estimation of collision risk. However, this model does not 17 cover adequately situations where ground controllers monitor the air traffic through radar 18 surveillance and provide tactical instructions to the aircraft crews. Furthermore, the problem 19 of collision risk modeling in the analysis of “high traffic density” ATC scenarios is different 20 to that of “procedural scenarios”, which have been developed by Reich [4] and Brooker [5], 21 amongst others. This is mainly due to the active role of Controllers in the first case. In this 22 case positive control is used extensively to modify the planned aircraft route. This requires 23 the inclusion in the model of “human factor response” behavior. 24


Introduction
Collision risk estimation in airspace and mathematical modeling of mid-air collisions have 7 been carried out for over more than 40 years [1]. During this period there has been a 8 development of mathematical models for processes leading to possible collisions of aircraft 9 flying nearby in order to estimate the risk of collision. 10 B. L. Marks [2] developed the principles in which a collision risk model could be developed 11 in the early 1960s. Marks' work was modified and enhanced by P. Reich [3] and that model, 12 later called the Reich model, has been the basis for many of the important developments in 13 this field. 14 The Reich model uses information related to the probabilistic distributions of aircraft's 15 lateral and vertical position, traffic flows of the routes, aircraft's relative velocities and 16 aircraft dimensions to generate estimation of collision risk. However, this model does not 17 cover adequately situations where ground controllers monitor the air traffic through radar 18 surveillance and provide tactical instructions to the aircraft crews. Furthermore, the problem 19 of collision risk modeling in the analysis of "high traffic density" ATC scenarios is different 20 to that of "procedural scenarios", which have been developed by Reich [4] and Brooker [5], 21 amongst others. This is mainly due to the active role of Controllers in the first case. In this 22 case positive control is used extensively to modify the planned aircraft route. This requires 23 the inclusion in the model of "human factor response" behavior. 24 These "collision risk models" were initially applied in the 60s to determine safe separation 25 standards between pairs of aircraft flying at the same altitude on parallel courses over the 26 North Atlantic Ocean [6]. Since then, new models have been developed and continually 27 refined and improved. They have been applied for different geographic regions (USA [7], 28 European airspace [8]), for oceanic or radar [9] environments, and different flight regimes 29 (for example, high-altitude cruise and landing on close [10,11,12] and ultra close spaced 1 runways [13,14]), for specific flight phases [15] (focused for example on the separation 2 between aircraft on final approach and landing, when flight risks are greater than during 3 any other phase of flight), for different types of separation (vertical, longitudinal and lateral) 4 and also for current and future operational concepts [16], such as free flight [17], airborne 5 self separation [18],…. 6 Most of these models, amongst them the formula proposed by Brooker [19] for mid-air 7 collision risk, involve the aggregation of terms comprising different factors related to: initiating 8 events which produce defective flight paths; the probability of safety defenses correcting these 9 defective flight plans; and traffic and kinematic scalers. But, as he indicates: "it does no more 10 than spell out the mechanisms by which collisions logically have to occur. The hard problem is 11 how to populate the parameters in the formulation with sensible numbers". 12 Risk models have also been developed for the estimation of conflict probability (understood 13 as the probability that the distance between a pair of aircraft becomes smaller than some 14 specified minimum separation value). Paielli  problems of evaluating conflict probability. 18 The main point of conflict probability is its clear relation to a well known safety criterion in 19 civil aviation: the separation minimum, which puts a requirement on the air traffic 20 management system; not to let aircraft come closer to each other than a certain minimum 21 distance. In addition to minimum separation values, ICAO (International Civil Aviation  22 Organization) has also defined limiting criteria for acceptable risk levels of fatal accident, 23 and in particular, for the risk of mid air collision [24 ]. The allowed probability values for 24 such events are of the order of one mid-air collision or physical crossing per 10^9 flight hour. 25 Furthermore , some effort has been also devoted to the problem of aircraft conflict detection. 26 An excellent survey of the different conflict detection and resolution schemes has been 27 carried out by Kuchar [25,26], where the conflict detection schemes are classified according 28 to the modeling method used for projecting the aircraft position in the future. 29 According to Brooker  avoidance maneuvers in order to maintain the prescribed minimum separation standards 21 among aircraft [34,35]. 22 The previous considerations give an idea of the complexity of using stored aircraft tracks, 23 within a given scenario and time frame, to infer safety level, collision risk probability and 24 associated system weaknesses. In most high density airspace scenarios recorded tracks can 25 be obtained for all aircraft flying in it, for example, from Radar Data Processing systems 26 (RDP). In fact, this provides us with a robust data source, which could be used for safety 27 analysis. This could include indirect information which is closely related to the "human 28 factor response". Despaite its importance not much effort have been devoted to the 29 development of risk and collision models based upon the analysis of the stored aircraft 30 tracks. 31 Furthermore, it has to be considered that the distribution of aircraft position errors over 32 their intended tracks is one of the most important factors in determining route safety, and 33 consequently it has been broadly studied.

Fundamentals behind probability of collision estimation
1 Jaroslav Krystul [39] defines the risk as the probability of a particular adverse event 2 occurring during a stated period of time. Usually, this is an event occurring when the 3 system reaches a particular critical state. These events with a very small probability of 4 occurrence are called rare events. Applying this definition to an ATC scenario, it is accepted 5 that risk is closely related to those situations in which two aircraft are on conflict course and 6 would not only pass closer than the prescribed horizontal and vertical separation minima 7 but which would, in fact, collide. 8 The work presented here was originally inspired by the principle stated in [2] by B. L. 9 Marks: "… the task of relating collision risk to a traffic configuration can be taken in two parts: 10 parts: 11 1. Determining the frequency with which aircraft are exposed to risk by passing close 12 together; and 13 2. Determining what chance of collision is inherent in the passing". 14 According to this idea, the probability of aircraft collision can be expressed as: 15 FeR, Frequency of exposition to Risk, here is considered as the relative frequency that 18 an aircraft would potentially violate the separation standards defined for the particular 19 situation, here referred to as potential conflict. It is easily seen that this value increases 20 with the traffic density. 21  P(pot.coll/pot.conf) is the conditional probability of a potential collision (pot.coll) 22 between two aircraft that have previously violated the separation standards (pot. conf). 23 Its value depends on the encounter kinematics and uncertainties associated to predicted 24 positions. It represents the intrinsic severity of the encounter and it is independent of 25 the traffic density. 26  P(coll/pot.coll ) is the conditional probability of collision among potential collisions 27 having failed all the safety barriers (ATC, TCAS) which are in place to mitigate the risk. 28 A time horizon is established within which all aircraft positions are projected to explore 29 existence of "potential conflicts". In the following discussion 10 minutes look ahead time has 30 been considered. Accordingly, the relative frequency of potential collisions among potential 31 conflicts F(pot.coll/pot.conf) could be expressed as: 32 where Num.pot.collisions is the number of aircraft that are about to collide (and will do if all 34 safety barriers fail). 35 An initial expectation for probability of potential collision among potential conflicts, E(Pa), 1 could be obtained as the relative frequency that two aircraft, on a conflict course, would not 2 only pass closer than the prescribed horizontal and vertical separation minima, but would in 3 fact collide. This expression provides an expected, or global, value and does not assess the 4 severity of each individual potential encounter itself. This chapter proposes an approach to 5 estimate the severity of the encounter using the conditional probability of a potential 6 collision Pa for each particular aircraft encounter. This proposed approach aims at 7 improving the previous works by: 8


Providing an individual probability of collision for each individual encounter based 9 on the: (1) geometry and kinematics of the encounter, (2) the minimum predicted 10 lateral separation at the CPA, and (3) the minimum predicted vertical separation at 11 the CPA. 12  Taking into consideration the radar data errors and the segmentation errors. 13 separations. When two aircraft are closer than these distances the ATC system is considered 28 to have failed. These values (R, H) allow us to use another cylinder shaped protection model 29 for all aircraft which should be free of any other aircraft to fulfil this separation minima (see 1 figure 3). This volume will be called the "conflict cylinder" as it is considered that two 2 aircraft potentially violating these separations are exposed to risk. During the en route phase of flight, for example, the conflict cylinder would be 5 nm in radius 6 and 2,000 ft in height. However, these current minimum separation standards were 7 determined many years ago and the method by which they were calculated is not well

15
A summary of the modelling cylinders defined so far is presented in the following 16 table. 17 When civil aircraft are climbing or descending, it is considered that pitch angles are small 1 and so, vertical and horizontal dimensions have small changes. Therefore, all the "modelling 2 cylinders" will be considered as horizontal, as indicated on figure 4. 3 As all the cylinders are considered parallel, the longitudes and surfaces ratios among them 4 will be constant when they are projected onto any plane. 5 6 Cylinder Diameter Height Aircraft representation   Considering the changes in the CPA coordinates due to radar and radar data segmentation 5 errors, the probability of potential collision for an intruder aircraft that has violated the 6 separation standards and whose projection consequently hits within the conflict area can be 7 calculated as: 8 This equation provides and individual probability of collision based on: 1  geometry and kinematics of the encounter (SPCOL), 2  the predicted minimum lateral separation at the CPA (y1p), and 3  the predicted minimum vertical separation at the CPA (z1p). 4 This takes into consideration the two probability density functions stating segmentation 5 lateral and vertical errors and the projection lateral and vertical errors characterization. 6 As a result, the bi-dimensional probability density function of the CPAs can be derived 7 from previous equation as: 8  Figure 7. Changes in the CPA coordinates due to projecting errors.. Both expressions estimate the probability of potential collision, having a potential separation 1 violation (potential conflict), for each aircraft encounter, provided that uncertainties in the 2 projection of segmented trajectories and in the segmentation process have been 3 characterised by associated pdfs, f1 and f2, respectively. 4

5
The previous mathematical formulation is supported by the previously mentioned ad-hoc 6 software, which has been developed by the authors for Eurocontrol in the framework of 7 the 3D-CRM programme. This software is intended to measure the collision risk in high 8 density ATC en route airspace, based on an analysis of the stored aircraft tracks that have 9 flown in it within a given time frame. 10 With the purpose of evaluating the mathematical expressions to estimate the probability of 11 collisions, the previously mentioned software tool has been applied to a radar data sample from 12 the

Pa estimation for each aircraft encounter
This equation provides and individual probability of collision based on: 8  kinematics of the encounter (ratio vz,to,vx), 9  the predicted minimum lateral separation at the CPA (yp), and 10  the predicted minimum vertical separation at the CPA (zp). 11 It also takes into consideration the segmentation of lateral and vertical errors (f2y and f2z). 12 A result for Pa estimation for leveled flight encounter is shown in the upper part of figure 8. 13 In this case when CPAp coordinates (yp,zp) are very close to the reference aircraft (ACi), Pa 14 estimated value reaches 3*10-2. This value has a magnitude of two orders higher than the 15 empirical expected result (5.4·10 -4 ), but strongly decreases when predicted CPAp lays apart 16 from ACi, resulting in values much lower than the empirical one. In the lower part of this 17 figure, the graphs show when one or both aircraft are climbing/descending but having vz/vr 18 ratio close to zero, it could be seen that regardeless the decrease of the maximum value of Pa 19 (7*10-3), It is still greater than the empirical expected result for Pa. Furthermore, the 20 probability of collision for CPAp for which yp coordinates close to zero but zp coordinates 21 values for Pa are different in both cases (9*10-3 for vz/vx=0.1, and 2*10-2 for vz/vx=20), 8 showing that Pa maximum values for CPAp close to reference aircraft (ACi) has a 9 decreasing trend when vz/vr ratio increases. The following table summarises the results  10 obtained from empirical and estimated Pa for the worst case, that is to say Pa for predicted 11 CPAp=(0,0). 12 The results clearly shows that it is unrealistic to assign the same probability for potential 13 collisions to all potential conflicts, independently of the predicted coordinates for CPA, no 14 matter how these coordinates have been derived.  6 When a collision risk analysis is applied to a representative aircraft population, using 7 segmentation of their stored radar tracks, a 2D histogram of projected horizontal and 8

Expected Pa estimation for a given scenario and traffic sample
vertical separations at the CPA can be obtained, as it is shown in figure 8. This histogram 9 provides a first approach for expected Pa using equation (4), which is the way we used to 10 obtain E[Pa]= 5.4*10-4, (this value can taken as reference value for Pa) . If the histogram 11 exhibits a close to uniform distribution, it can be understood that any "generic" potential 12 conflict would became a potential collision with the same probability. It is also possible 13 to propose a different approach to establish the expected value for Pa in a given scenario 14 and for a given aircraft population, discussed below. 15 Where Pa(yji,zji,rji) is the individual probability of each potential collision where: 17  rji=vz/vx the between vertical and horizontal relative speeds, 18  f2zji the probability density function applied to each aircraft encounter (between each  19 pair of aircraft, i and j). 20 When this equation is applied to previous MUAC data sample, expected value for Pa results 21 8.2*10-4, which is slightly higher than the empirical results. 22

Conclusions
1 This chapter analyse in detail the inherent collision risk involved for each aircraft proximity 2 event by assessing the conditional probability Pa of a potential collision between aircraft 3 that are exposed to risk, that is to say, they are potentially going to violate the separation 4 standards defined for a specific airspace if no corrective action is taken. The proposed 5 approach allows the determination of the severity of each aircraft encounter as the 6 probability of potential collision for each individual aircraft encounter in high density ATC 7 en route airspace, based on an analysis of the stored aircraft tracks that have flown within a 8 given time frame. The authors propose a mathematical formulation to characterise the 9 severity of each aircraft proximity event using the convolution of the bi-dimensional 10 probability density function of the predicted Closest Point of Approach between the aircraft 11 involved and the distribution of lateral and vertical error in the projected position of the 12 aircraft.The presented work aims to provide an individual probability of collision based on 13 the geometry and kinematics of the encounter and the minimum lateral separation and the 14 minimum vertical separation at the predicted Closest Point of Approach or CPA. The 15 formula takes into consideration uncertainties introduced by the radar data error and the 16 segmentation error. The results of this chapter shows that there is not the same severity for 17 all the proximity events on which aircraft pass closer than the prescribed horizontal and 18 vertical separation minima, and also that the expected severity for given a scenario and 19 traffic sample can also vary depending on the kinematic characteristics of aircraft involved 20 within this scenario. It is also considered that collision risk for high density of air traffic can 21 be analysed from the estimation of three different factors: 22


Relative frequency of exposition to risk (FeR Probability of potential collision among encounters exposed to risk, Pa or its expected value 36 E(Pa) for the same sample, oscillates between 8.2*10-4(expected) and 2*10-2(worst case). 37 Previous results demand a probability of "safety barrier failure" lower than 0.4*10-5 and 38 1.7*10-7 respectively, to reach the ATM en route target level of safety of TLS=10-9. This last 39 value is normally the one used as TLS. For instance, in reference (Eurocontrol, 2006) mid-air 40 collision given as accident frequency (per flight) is 5.4*10-09, specifying that, among them, 1 the frequency of fatal accident, directly caused by ATC (per flight), is 3.5*10-09.