## 4 Dynamic Runway Overrun Risk Model
### Automatic Extraction of the Deceleration Rate  
To plot the risk curve mentioned above, it is necessary to determine the deceleration rate of the aircraft under different conditions. Since the QAR data used in our study do not include the runway or weather condition information, the focus here is mainly on the deceleration rates under different autobrake levels. Specifically, after touchdown, the autobrake system is activated, and then the aircraft enters the autobrake phase and maintains a relatively constant deceleration rate before manual braking is applied. Therefore, by identifying the autobrake phase, the autobrake level and deceleration rate can be determined based on the average deceleration rate of this phase. Based on the above considerations, we design an algorithm to extract the deceleration rate during the autobrake phase, which not only identifies the autobrake level but also determines the deceleration rate. The algorithm flowchart is shown in Figure 6, and the specific steps of the algorithm are as follows:  
_Step 1_: Identify the aircraft's touchdown point based on the landing gear state parameters and determine the time of manual brake intervention based on the brake pedal parameters. In addition, when the parameter value of the brake pedal is particularly small, there may be situations where the autobrake release cannot be triggered. Therefore, in this paper, the time of manual brake intervention is defined as any value of the brake pedal parameters greater than or equal to 4. Figure 5: Two-dimensional diagram of landing distance and landing speed. Step 2: Starting from the touchdown point, find the time point \(t_{start}\) when the engine power N11 first reaches a maximum value of 90%. Choosing 90% instead of the moment of the maximum speed ratio improves the stability of the algorithm. Step 3: Starting from \(t_{start}\), find the first time point when the deceleration rate is stable and replace \(t_{start}\) with this time point. The stability of the deceleration rate at any time \(t\) is defined as follows. Starting from time point \(t\), take the two preceding deceleration rate values and the three subsequent deceleration rate values and add the deceleration rate value at time point \(t\) itself to form an array [\(a_{t-2}\), \(a_{t-1}\), \(a_{t}\), \(a_{t+1}\), \(a_{t+2}\), \(a_{t+3}\)]. Calculate the standard deviation \(\sigma(t)\) of this array, and when it satisfies the condition \(\sigma(t)\leq\theta\) the deceleration rate is considered stable; otherwise, it is considered unstable. Here, \(\theta\) is a stability threshold parameter, which is set to 0.15 in this paper. Step 4: Determine whether \(t_{mkk}>t_{start}\), i.e., whether manual braking has already been applied before the deceleration rate becomes stable. Here, \(t_{mkk}\) is the time point when manual braking begins. If manual braking has already been applied, end the program; otherwise, go to Step 5. Step 5: Traverse each point starting from \(t_{start}\) until a time point with an unstable deceleration rate is found, and mark this point as \(t_{end}\). Step 6: Mark [\(t_{start}\), \(t_{end}\)] as an autobrake phase. Step 7: Calculate the average value of ground speed (GS) during [\(t_{start}\), \(t_{end}\)] as the automatic deceleration rate. Figure 6: Automatic deceleration extraction algorithm for aircraft. _Step 8_: Determine whether there is a next deceleration phase. If so, let \(t_{start}=t_{end}+1\) and return to _Step 3_ to find the next deceleration phase; otherwise, proceed to _Step 9_. _Step 9_: Output all [\(t_{start}\), \(t_{end}\)] as automatic deceleration phases. We apply the above autobrake level extraction algorithm to the 180 flight samples and record the distribution of automatic deceleration rates for each flight, as shown in Figure 7. From Figure 7, it can be observed that the automatic deceleration rates are concentrated at approximately 1.6 m/s\({}^{2}\) and 2.9 m/s\({}^{2}\), which aligns with the experience of flight experts.