## 4 Dynamic Runway Overrun Risk Model
### Establishment of the Risk Assessment Model  
Based on the idea of the risk assessment model described in Section 4.1, for a specific flight, we first extract the automatic deceleration rates to plot the theoretical risk boundary curve, as shown by the purple curve in Figure 8. The green vertical line corresponds to the touchdown moment, and the red vertical line corresponds to the moment when the ground speed of the aircraft drops to 30 knots (which is regarded as a safe speed). In the risk model described in Section 4.1, each flight is represented by a single point on the figure, corresponding to the ground speed and remaining runway distance at touchdown, and the risk of runway overrun can be assessed based on the distance from that point to the theoretical curve. The main drawback of this assessment method is that the influence of the pilot's deceleration operations after touchdown on the risk is not considered. To address this problem, we propose a dynamic risk assessment model. Specifically, we use the modified trajectory points to calculate the remaining runway distance and plot the dynamic ground speed with respect to the remaining runway distance, as shown by the blue scattered points in Figure 8. Then, the distance from each point to the theoretical curve is calculated, and the dynamic risk of runway overrun is measured based on this distance. We select the phase from the touchdown moment to the moment when the ground speed decreases to 30 knots, and we use a weight function of the distance between the ideal curve and the actual aircraft trajectory as the evaluation indicator for the runway overrun risk, which is calculated as follows:  
\[risk=\frac{\sum_{t=t_{1}}^{t_{2}}w(v_{t})\cdot f(d_{t})}{\sum_{t=t_{1}}^{t_{2} }w(v_{t})} \tag{6}\]  
where \(t_{1}\) and \(t_{2}\) represent the touchdown moment and the moment when the speed drops to 30 knots, respectively. The weight and risk calculation functions are as follow:  
\[w(v_{t})=1-(\frac{1}{e})^{\frac{v_{t}}{a}} \tag{7}\]  
\[f(d_{t})=(\frac{1}{e})^{\frac{d_{t}}{b}} \tag{8}\]  
Figure 7: Distribution of automatic deceleration rate. Equation (7) is a velocity-based weight function, which indicates that as the speed \(v\) increases the weight function value also increases, and the confidence level of the calculated risk is higher. This is because when the aircraft's speed is higher it is usually a critical phase for the pilot to perform deceleration operations, and the calculated risk indicator at this time is more reliable. However, when the aircraft's speed drops to a certain level (such as 40-50 knots), the pilot may intentionally maintain speed to quickly exit the runway, and the confidence level of the risk assessment is lower. Here, \(a\) is a control parameter determined by experiments, which is set to 85 in this case. Equation (8) is the risk calculation function, which indicates that when the distance from a point to the theoretical curve is negative, i.e., the actual trajectory point is to the left of the ideal curve, the risk value is greater than 1. When the distance is positive, i.e., the actual trajectory point is to the right of the ideal curve, the risk value is less than 1. The greater the distance, the lower the risk. Here, \(b\) is a control parameter determined by experiments, which is set to 500 in this case. Figure 9 shows the variation in the risk and weight functions over time for the flight corresponding to Figure 8, from the touchdown moment to the moment when the ground speed decreases to 30 knots. With the original automatic deceleration rate, the aircraft cannot exit from the middle of the runway, and the aircraft needs to turn around at the end of the runway. Therefore, the pilot needs to lightly tap the brakes to deactivate the autobrake and maintain a certain speed to taxi to the end of the runway. Then, the pilot can perform a manual brake to turn around at the end of the runway.