2.1 Intervention time model

The model presented here includes the time it takes for a controller to notice a problem, send a message to the aircraft and resolve the issue. This is called the intervention or action time and is denoted $\tau$. This is modelled as either a discrete set of times and proportions ($\tau _i, p_i$) or as a continuous distribution $f(\tau )$. Hence, a conflict projected 20 min in the future will have negligible risk, whereas a conflict 30 s in the future will allow insufficient time for intervention.

In oceanic airspace, with well-defined satellite communications (controller–pilot data link communications – CPDLC), the intervention times, $p_i$, $\tau _i$, can be accurately modelled. These are based on event sequence diagrams representing documented and legal requirements for surveillance and communications: required surveillance performance (RSP) and required communication performance (RCP) (ICAO, 2013). For example, pilot response times are required to occur in 60 s 95% of the time and the actual communication technical performance (ACTP) is required to be completed in 120 s 95% of the time. With all the components considered, the intervention time was modelled with proportions $p = [0{\cdot }9025, 0{\cdot }4075, 0{\cdot }05]$ and times $\tau = [4, 10{\cdot }5, 13{\cdot }5]$ min (Anderson, Reference Anderson2000a, Reference Anderson2000b, Reference Anderson2005).

Within domestic airspace (with push-to-talk VHF communications), this intervention time is less well known, since the VHF communication system is not conducive to large-scale monitoring and deciphering. However, there are sufficient studies to make a reasonable intervention model (an exponential distribution starting at 45 s with a 45 s scale parameter) (Cardosi, Reference Cardosi1993a, Reference Cardosi1993b; Morrow et al., Reference Morrow, Lee and Rodvold1993; Prinzo, Reference Prinzo1993; Burki-Cohen, Reference Burki-Cohen1995; Cardosi et al., Reference Cardosi, Buerki-Cohen, Boole, Hourihan and Mengert1995, Reference Cardosi, Brett and Han1997a, Reference Cardosi, Brett and Han1997b, Reference Cardosi, Brett and Han1998; Gonda et al., Reference Gonda, Saumsiegle, Blackwell and Longo2005; Prinzo and McClellan, Reference Prinzo and McClellan2005; Chen, Reference Chen2010; Nickelson, Reference Nickelson2018).

The ability of the controller to notice a conflict and intervene will vary according to many factors, including airspace complexity, workload, training and so on. This is taken into account by the distribution in the model. Hence, the longer tails of the distribution model situations where a controller is slower in resolving conflicts. Where appropriate, it is easy for an implementer of the model to change the currently used values of 45 s, in line with their airspace. This paper is not prescribing a particular intervention distribution, but demonstrating how the risk model can be used with a reasonable intervention model. Other distributions for the intervention model can be used without loss of generality.

The key components of the intervention times can be summarised as follows (with mean and standard deviation (st.dev) taken from Cardosi et al. (Reference Cardosi, Buerki-Cohen, Boole, Hourihan and Mengert1995, Reference Cardosi, Brett and Han1997a, Reference Cardosi, Brett and Han1998) and Cardosi (Reference Cardosi1993a, Reference Cardosi1993b). These are the times, in seconds:

• • for the controller to notice a deviation and begin voice communications ($\approx$15–30);

• • for the controller to deliver the message ($\approx$5–10, – mean 4$\,\cdot\,$8, st.dev 2$\,\cdot\,$3);

• • between the controller ending the message and the pilot initiating a response (mean 3$\,\cdot\,$3, st.dev 4$\,\cdot\,$8);

• • for the pilot to talk (mean 2$\,\cdot\,$6, st.dev 1$\,\cdot\,$8);

• • for possible readback errors, miscommunications and repeating of instructions (of the order of 1–3% of communications: Morrow et al., Reference Morrow, Lee and Rodvold1993; Cardosi et al., Reference Cardosi, Brett and Han1997a);

• • for the pilot to initiate a manoeuvre ($\approx$2–5);

• • for the aircraft to deviate to avoid the conflict ($\approx$15–75).

The Cardosi papers indicate a total communication time (not including the aircraft manoeuvring) of (9, 17, 23, 40) s with cumulative percentiles (50%, 90%, 95%, 99%). Cardosi noted that the times from the controller initiating communications to the final speed acknowledgement had a mean of 18 s with a range between 3 and 71 s. This study also noted the pilot response times having a mean of 10$\,\cdot\,$8 s, (minimum 4, maximum 40 s, st.dev 5$\,\cdot\,$9 s) and percentiles (5, 10, 50, 90, 95)% in (5, 6, 9, 17, 22$\,\cdot\,$5) s.

An analysis of terminal operations gave results for ‘pilot–controller discourse’ with mean 20 s and st.dev 5 s (Rantanen et al., Reference Rantanen, McCarley and Xu2004). Recent SASP work (Nickelson, Reference Nickelson2018) noted communication times of 3$\,\cdot\,$6–6$\,\cdot\,$3 s, and the time for the aircraft to start a trajectory change (16 s, st.dev 10 s, 95% in 30 s). This paper noted a response climb rate of 4$\,\cdot\,$1 s (95% in 9$\,\cdot\,$6 s) per 100 ft, giving a time to clearance (500 ft) of the order of 21–48 s.

In summary, the modelling of intervention time in a VHF-voice environment is hence not precise compared to oceanic airspace, with a moderate range of values reported in the literature. Here, we use a simple exponential distribution with scale factor and location both 45 s (further mathematical description in Section 3.1). Hence, no intervention is possible within 45 s of a deviation, and 50% of deviations would be fully resolved within 90 s: these values include the full intervention time, from an aircraft beginning to deviate, through to the aircraft manoeuvring to be safely separated. While it is possible to generate more exotic distributions, such as a Gamma, the exponential distribution is simple and is long-tailed, allowing for known occurrences where controllers do not detect the conflict, because pilot–controller communications are inadequate or because pilot responses are incorrect.

2.2 Aldis model

The Aldis model is a variation of the Anderson model in which the temporal integral limits become infinite (Aldis and Barry, Reference Aldis and Barry2015). Here, we disregard the uncertainty on the velocities of the two aircraft and the separation distance, which speeds up calculations significantly by removing three integrals. Typically, these integrals would be used for strategic evaluation of collision risk to account for the probability distributions of the speeds of all aircraft at the crossing, as well as the distribution of possible separation distances between pairs of aircraft. However, the adaptations made to use the model for evaluating aircraft encounters based on data means that the velocities of the two aircraft at each position are known, as well as their separation in terms of a crossing track projection, and as such these uncertainties are discarded.

The collision risk for two aircraft on two infinitely long crossing tracks is given by

(2.1)\begin{equation} \text{CR}^\infty = N_p \int_{-\infty}^{\infty} \text{HOP}(t\,|\,V_1, V_2) \; P_z(h_z) \left(\frac{2V_r}{\pi\lambda_{xy}} + \frac{\overline{|\dot{z}|}}{2\lambda_z}\right) d t, \end{equation}

with $V_1$, $V_2$ the speeds of both aircraft, $|\dot {z}|$ the mean relative vertical speed (usually considered 1$\,\cdot\,$5 knots for aircraft on the same flight level), $V_r$ is the relative speed between the aircraft, $h_z$ is the vertical separation, $P_z$ is the vertical overlap probability, $N_p$ is the number of pairs per flight hour, $\lambda _{xy}$ is the mean aircraft horizontal size, $\lambda _z$ is the aircraft mean vertical size and the horizontal overlap probability is denoted $\text {HOP}$. The last terms in the brackets are often called the kinematic terms.

Denote the initial distances of the two aircraft from the intersection point $d_{10}$ and $d_{20}$. The horizontal overlap probability is then given by

(2.2)\begin{align} \text{HOP} & = 2\pi\lambda_{xy}^2\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} [ f_2^A(\xi)\,f_2^C(\eta) \,f_1^A(\xi\cos\theta - \eta\sin\theta - D_x(t)) \nonumber\\ & \quad \cdot f_1^C(\xi\sin\theta + \eta\cos\theta - D_y(t))] \,d \xi\, d \eta, \end{align}

where $\theta$ is the angle between the aircraft tracks, the functions $f(\cdot )$ are the position uncertainty distributions in the along and cross-track directions for aircraft 1 and 2, and

\begin{align*} D_x(t) & = (V_1 - V_2 \cos\theta) t - d_{10} + d_{20} \cos\theta, \\ D_y(t) & =(d_{20} - V_2 t) \sin\theta. \end{align*}

Equation (2.2) calculates the horizontal overlap probability by convolving the two position error densities of the two aircraft to find the density describing their separation. The density of one aircraft needs to be rotated and shifted so that it is in the coordinate reference frame of the other aircraft. The probability of overlap can then be found from the separation density by integrating over the area of the two aircraft around the origin, i.e. capturing the probability of the separation being less than the combined aircraft dimensions which defines a collision. To simplify this calculation, rather than integrating the result of the 2D convolution, the result is evaluated at zero and multiplied by the area of the two aircraft to approximate the probability of collision.

It is assumed that the kinematic terms and the vertical overlap probability in Equation (2.1) are constant, which allows evaluating them outside the integral. The order of the time $t$ and space integrals $\xi$ and $\eta$ can be reversed, which yields

(2.3)\begin{align} \text{CR}^\infty & = 2N_P \left(\frac{2V_{r}}{\pi\lambda_{xy}}+\frac{\overline{|\dot{z}|}}{2\lambda_z}\right) P_z(h_z) I_1^*, \end{align}

(2.4)\begin{align} I_1^*(V_1, V_2)& = \pi\lambda_{xy}^2 \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_2^A(\xi)\,f_2^C(\eta) I_2^*(\xi, \eta\mid V_1, V_2)\,d\xi\, d\eta, \end{align}

(2.5)\begin{align} I_2^*(\xi, \eta\mid V_1, V_2) & = \int_{-\infty}^{\infty}\,f_1^A(\xi\cos\theta - \eta\sin\theta - D_x(t))\, f_1^C(\xi\sin\theta + \eta\cos\theta - D_y(t))\, dt, \end{align}

where $f_1^A$ and $f_2^A$ are the along-track position errors, and $f_1^C$ and $f_2^C$ are the cross-track errors of the two aircraft. The term $I_1^*(V_1, V_2)$ is the horizontal overlap probability for the infinitely long tracks, and once multiplied by the vertical overlap probability and the kinematic terms in Equation (2.3) will yield the collision risk.

The time integral can be further simplified. It is assumed that the along and cross track position errors of both aircraft are Laplace distributed with scale parameters $\lambda$ and $\nu$, and noting that the initial distance of aircraft 2 $d_{20} = 0$,

(2.6)\begin{equation} I_2^*(\xi, \eta\mid V_1, V_2) = \frac{1}{4\lambda\nu}\int_{-\infty}^{\infty} e^{-|\alpha|\cdot|{a}/{\alpha} + t|}\, e^{-\beta|{b}/{\beta} + t|}\, d t,\end{equation}

where

(2.7)\begin{equation} \begin{aligned} a & = (\xi\cos\theta - \eta\sin\theta + d)/\lambda,\\ b & =(\xi\sin\theta + \eta\cos\theta)/\nu,\\ \alpha & = (V_2\cos\theta - V_1)/\lambda, \\ \beta & = V_2\sin\theta/\nu, \\ d & = d_{10} - V_2 d_{20}/V_1. \end{aligned} \end{equation}

The parameters $\alpha$ and $\beta$ do not depend on $\xi$ or $\eta$. Note that the angles $0$ and $\pi$ are excluded which, along with $V_{2}\ne 0$, mean that $\beta$ is strictly positive. The parameter $\alpha$ can however have either sign.

The substitution of a new time variable $u={a}/{\alpha } + t$ into Equation (2.6) results in

(2.8)\begin{equation} I_2^*(\xi, \eta\mid V_1, V_2) = \frac{1}{4\lambda\nu} \int_{-\infty}^{\infty}e^{-|\alpha|\cdot| u|}\, e^{-\beta|\zeta + u|}\, du,\end{equation}

where

(2.9)\begin{align} \zeta & = \frac{b}{\beta} -\frac{a}{\alpha} \nonumber\\ & ={-}\frac{V_1}{V_2 (V_2\cos\theta - V_1)}\xi + \frac{V_2 - V_1\cos\theta}{V_2\sin\theta(V_2\cos\theta - V_1)}\eta - \frac{d}{V_2\cos\theta - V_1}, \end{align}

and thus $\zeta$ is linear in $\xi$ and $\eta$.

The time integral in Equation (2.8) can now be evaluated to give

(2.10)\begin{equation} I_2^*(\xi, \eta\mid V_1, V_2) = \frac{1}{2\lambda\nu(\alpha^2 - \beta^2)}(|\alpha|\,e^{-\beta|\zeta|} - \beta \,e^{-|\alpha||\zeta|}).\end{equation}

Two special cases also exist. If $\alpha =0$, the x-distance between the aircraft is constant, and the integral evaluates to

(2.11)\begin{equation} I_2^*(\xi, \eta\mid V_1, V_2) = \frac{1}{4\lambda\nu}\int_{-\infty}^{\infty}e^{-| a |}\, e^{-| b + \beta t|}\, dt = \frac{e^{-|a|}}{2\beta\lambda\nu}. \end{equation}

If $\beta =|\alpha |$, two identical Laplace distributions are convolved, which simplifies the integral to

(2.12)\begin{equation} I_2^*(\xi, \eta\mid V_1, V_2) = \frac{1}{4\lambda\nu}\int_{-\infty}^{\infty}e^{-| a + \alpha t|}\, e^{-| b + |\alpha| t|}\, d t = \frac{1}{4\lambda\nu}e^{-\beta|\zeta|}\left(|\zeta| + \frac{1}{\beta}\right). \end{equation}

The spatial integral in Equation (2.4) can now be evaluated using a time integral in Equations (2.10), (2.11) or (2.12) as appropriate, and the solution used to calculate the collision risk with Equation (2.3). The analytical solutions to Equation (2.4) are reported by Aldis and Barry (Reference Aldis and Barry2015).

The Aldis model has been numerically tested against the Anderson model and found to accurately represent the integration over infinite time. Some special cases however will be discussed further in Section 3.4.

3.1 Intervention model

The collision risk model includes a component that tries to model the time taken for an air traffic controller to notice an impending collision and intervene. In particular, the probability of the controller failing to intervene is modelled by a shifted exponential distribution over time $E(\mu, \lambda )$ with location $\mu =45$ s and scale $\lambda =45$s. If the time to CPA is less than the mean $\mu$, the probability of no intervention is set to 1 (the probability of intervention is 0). Thus, the probability of no intervention $P(\bar {I})$ is given by the piece-wise equation, where $\tau$ is the time to the CPA,

(3.1)\begin{equation} P(\bar{I}) = \begin{cases} e^{{(\mu-\tau)}/{\lambda}} & \tau >{=} \mu ,\\ 1 & \tau < \mu. \end{cases} \end{equation}

The Aldis model does not account for the time to CPA and intervention time model. By post-multiplying the Aldis model risk by Equation (3.1) however, based on the time to CPA, we can approximate the use of the Anderson model with a time integral based on the intervention time distribution.

3.2 Aircraft position uncertainty

Traditionally, the Anderson model is applied to determining the strategic collision risk at intersections of aircraft traffic. In these cases, the aircraft tracks are predetermined, and the position uncertainty of the aircraft travelling on them is described by the required navigation performance (RNP) of the routes (ICAO, 2008). In particular, the Laplace distributions used for the lateral and longitudinal navigation performance have their scale parameter $\lambda$ derived from the RNP values, such that

(3.2)\begin{equation} \lambda = \frac{\text{RNP}}{\log 20}. \end{equation}

The RNP value is part of performance based navigation (PBN) specifications with an engineering and legal requirement on an aircraft's ability to maintain a track line. The value (i.e. RNP 4) relates to 95% containment of the aircraft position within 4 NM of the centre-line along with an on-board alerting ability. In practice, the navigation performance of aircraft is significantly better, which is described by observed navigation performance (ONP). For example, aircraft with RNP 4 NM often perform with ONP equivalent to 0$\,\cdot\,$02 NM. Hence, for risk calculations to be conservative, they are done using both the legal RNP level and an ONP level, since both can be appropriate in different circumstances.

In the context of this paper, neither RNP nor ONP is appropriate however, because at each time step, the crossing aircraft tracks are extrapolated from the current known positions of each aircraft. As a result, each aircraft is by definition known to be at the centre of its track at the start, which is at odds with the typical strategic model that assumes steady-state behaviour. Therefore, at that time step, the position error of each aircraft relative to the projected track is near zero, and it must grow over time. The small level of surveillance inaccuracy in the recorded position can be discounted.

It is proposed that the position uncertainty of the aircraft both in the lateral and longitudinal directions is modelled by a random walk, of the form:

(3.3)\begin{align} X_t & = X_{t-1} + \epsilon_t, \end{align}

(3.4)\begin{align} X_0 & = 0, \end{align}

where $X_t$ is the position of the aircraft at time $t$ for the projected track and $\epsilon _t$ is a step that is chosen from a probability distribution. Here, $X_t$ has a variance of $t\sigma _\epsilon ^2$ at time $t$, where $\sigma _\epsilon ^2$ is the variance of $\epsilon$. Thus, the variance of $X_t$ grows linearly as a function of $t$.

The distribution of $X_t$ is assumed to be Laplace, as this is the assumption also used in the Aldis model. Its scale parameter can be found by equating the variance of a Laplace distribution with the variance of the random walk and solving for the scale parameter:

(3.5)\begin{equation} \begin{aligned} 2\lambda^2 & =t\sigma_\epsilon^2 \\ \lambda & = \sqrt{\frac{t}{2}}\sigma_\epsilon. \end{aligned} \end{equation}

The scale parameter of the aircraft's position uncertainty cannot however grow unbounded. After some threshold time $t_l$, the uncertainty is assumed to reach the ONP of the aircraft. Thus, the value of $\sigma _\epsilon$ is determined such that at time $t_l$, the value of $\lambda$ reaches that given by the ONP. This implies that the aircraft will deviate randomly from its starting location, and given a long enough time, will converge to the ONP distribution, i.e. its position will be independent of where it started. As such, the scale parameter $\lambda$ used in this paper is defined by

(3.6)\begin{equation} \lambda =\begin{cases} \sqrt{\dfrac{t}{2}}\sigma_\epsilon & 0 \leq t < t_l,\\ \dfrac{\text{ONP}}{\log(20)} & t \geq t_l, \end{cases} \end{equation}

where

(3.7)\begin{equation} \sigma_\epsilon = \frac{\text{ONP}}{\log(20) \sqrt{0\cdot5t_l}}. \end{equation}

The analytical versions of the Aldis model assume that the scale parameter of the position uncertainty of the aircraft is constant. As such, for the projected trajectories, the scale parameter in each case is a constant determined using the time to CPA from where the trajectory projection is initiated.

3.3 Vertical overlap probability

The vertical overlap probability in the Aldis model is assumed to be constant and equivalent to the extrapolated vertical separation at the CPA (based on climb and descent rates of the aircraft if appropriate). The rate of change of the difference in altitude of the two aircraft is set to zero if it is less than 100 ft/min, as this is generally indicative of errors in the altitude data rather than a clear climb or descent, which are usually of the order of 1000 ft/min.

If the two aircraft cross vertically during the projection, the extrapolated separation is conservatively set to zero. For example, consider one aircraft at 35,000 ft, the other climbing from 34,000 ft at 1000 ft/min and a CPA that is 2-min away. One might assume that at the CPA, the aircraft will then be safely at 35,000 and 36,000 ft. However, there is no guarantee that the climbing aircraft will not stop climbing at 35,000 ft (indeed this will often occur). Hence, to account for these situations, if the extrapolated vertical separation crosses zero feet, then it is set to zero.

The vertical overlap probability is calculated by assuming Laplace distributions for the altitude error of both aircraft, resulting in

(3.8)\begin{equation} P_z(h_z) = \int_{-\lambda_z}^{\lambda_z}\int_{-\infty}^{\infty} f(z_1)\, f(h_z + z_1 - z) \, d z_1 \, d z, \end{equation}

where $f(\cdot )$ is the probability density function of a Laplace distribution. The scale parameter of this distribution is 38 ft, unless the mean altitude of the encounter is in non-RVSM airspace, in which case it is doubled to 76 ft.

3.4 Degenerate cases

The Aldis model becomes impractical at angles near 0$^\circ$ or 180$^\circ$. As mentioned in Section 2.2, the Aldis model is strictly not defined for angles exactly equal to 0$^\circ$ or 180$^\circ$. Additionally, the Aldis model also diverges from the Anderson model by overestimating collision risk at angles close to 0$^\circ$, as the aircraft pair stays in close proximity for an increasingly longer time. This results in the $I_1$ integral increasing without bound as the angle between the aircraft drops to zero, especially if the two aircraft are flying at the same speed.

For these degenerate cases, the Anderson model must be employed. In general, the Anderson model is very computationally expensive; however, the angles of 0$^\circ$, 90$^\circ$ and 180$^\circ$ are simple analytical formulations. The Anderson model requires bounds on the time integral, which must be set. The integral is evaluated from 0 to 4 min; for cases where the actual time to CPA is greater than 4 min, the probability of intervention by ATC is extremely high and the collision risk would be negligible anyway. Therefore, for crossing track angles less than 2$\,\cdot\,$5$^\circ$, the Anderson model is employed using an angle of 0$^\circ$. The effect of this can be seen in Figure 2, where the dashed blue line represents the Aldis model for angles greater than 2$\,\cdot\,$5$^\circ$, and for smaller angles, the Anderson model evaluated at $\theta =0$ is used. The plot shows that the result remains conservative, as the Aldis model overestimates the risk given by the Anderson model. For crossing angles closer than 1$^\circ$ to 180$^\circ$, the Anderson model is also used, this time evaluated at $\theta =180$.

Figure 2. Comparison between the standard Anderson model and the infinite track variation (Aldis) around a relative angle of zero. The Aldis model becomes more inaccurate (conservatively over-estimating the risk) near angle 0, and can be replaced by the Anderson model with default 0$^\circ$

4.1 Classical case

The first case concerns two aircraft heading towards each other, as well as converging towards the same flight level, as can be seen in Figure 3. The aircraft icons are plotted every minute. In the top left plot, the black aircraft highlight the location where the aircraft were closest to each other, and the top right and middle plots highlight the time and position of maximum risk. The closest approach and the maximum risk occurred at very different times, as at the point of closest approach, the two aircraft are already pointing away from each other and thus are not likely to collide, whereas at the point of maximum risk, they are projected to come much closer to a collision. The bottom plot shows the trajectory of the horizontal separation on the $x$ axis and the vertical separation on the $y$ axis over time. The red arrow indicates the projection of the separation of the two aircraft, with the red square at the end denoting the projected minimum separation of the two aircraft from the point of highest risk (blue square). It can be seen that the aircraft were projected to likely collide had they continued along this straight-line trajectory.

Figure 3. Trajectory of the two aircraft in the first case. Top left is the trajectory with the point of closest approach highlighted by the black aircraft icons. Top right is the trajectory with the point of maximum risk highlighted by the black aircraft icons. The middle plot is the altitude over time, with the dark aircraft showing the point of maximum risk, and the bottom plot shows the trajectory of horizontal separation distance on the $x$ axis and the altitude separation in the $y$ axis. The aircraft icons and arrows are every minute, with the $x$ every 5 s

Figure 3 clearly shows the intervention, with aircraft AC 1 initiating a climb and lateral deviation, while AC 2 descends and deviates slightly. The figure also shows that the two aircraft barely avoided a LOS (5 NM, 1000 ft), indicated by the green rectangle on the bottom plot. However, the projected risk is high and if intervention had been delayed, the encounter may have been more significant.

4.2 Midair collision case

The second case was examined based on an actual midair collision between two aircraft. For confidentiality reasons, the aircraft trajectories cannot be shown. This case was considered to ensure that the model is able to accurately represent high-risk encounters.

A plot of the collision risk over time calculated by the model is shown in Figure 4 with different methods for estimating position uncertainty: the green line shows the random walk model used here, correctly having a risk that tends to 1. The blue line uses a position uncertainty based on the formal definition of RNP (4 NM). This RNP underestimates the risk, since two aircraft exactly aligned may still be 4 NM apart: whilst a reasonable assumption for modelling some separation standards, this is a poor model for modelling actual trajectory risk. The red line assumes a more realistic ONP model 0$\,\cdot\,$5 NM, but still underestimates the risk close to the conflict.

Figure 4. Collision risk over time for the midair collision case. Plotted are the collision risk estimates at each time using three different notions of the position uncertainty. The dashed horizontal line represents a collision risk of 1

The components of the collision risk using the proposed random walk approach are shown in Figure 5. The top plot shows the overall probability of collision trending correctly towards 1 (as these aircraft did collide). The second plot shows the horizontal overlap probability (also trending to 1). The third plot shows the vertical overlap probability, which has a maximum of 0$\,\cdot\,$55 since altimetry errors mean two aircraft at the same level may still not be overlapping vertically. The bottom graph illustrates the probability of no intervention, which indicates no chance for intervention from approximately 250 s into the encounter.

Figure 5. Midair collision case. Projected risk is plotted at each time, along with its constituent multiplicative components. The red dotted line identifies a probability of 1

4.3 Same track case

This case concerns two aircraft converging onto the same track, with one climbing to the flight level of the other aircraft, complicated by the difference in aircraft speeds (330 and 430 knots). The trajectories of the two aircraft are shown in Figure 6. The two aircraft merge onto the same track in the horizontal plane, but at different altitudes. The point of maximum risk then occurs around 220 s into the encounter, when the first aircraft starts climbing sharply, and its projected track is set to come very close to colliding with the second aircraft. The bottom plot shows that the projected tracks of the two aircraft at the point of highest risk will cause the two aircraft to overlap both horizontally and vertically if position error were completely neglected (shown by the red arrow that points into the small aircraft bounds box).

Figure 6. Trajectory of the two aircraft in the same track case. Top left is the trajectory with the point of closest approach highlighted by the black aircraft icons. Top right is the trajectory with the point of maximum risk highlighted by the black aircraft icons. The middle plot is the altitude over time, with the dark aircraft showing the point of maximum risk, and the bottom plot shows the trajectory of horizontal separation distance on the $x$ axis and the altitude separation in the $y$ axis

The collision risk components over time are given in Figure 7 showing that the collision risk evaluation between approximately 125–175 s and 250–290 s drops to be negligible. This drop is due to the velocity profile of the two aircraft in the horizontal plane, along with moments of descent, with the aircraft alternating places as the lead and follower aircraft due to the climbing behaviour of the first (blue) aircraft. In these time periods, the leader aircraft has a significantly higher speed and, as such, the two aircraft will never cross, meaning the risk evaluated is negligibly small. The risk of the entire encounter is evaluated by the maximum value, meaning these portions do not affect the overall evaluation of these types of encounters.

Figure 7. Same track case. Projected risk is plotted at each time, along with its constituent multiplicative components. The red dotted line identifies a probability of 1. The risk drops to negligible levels during two periods, due to changes in speed and where the aircraft stopped climbing