Safety Assessment Method for Parallel Runway Approach Based on MC-EVT for Quantitative Estimation of Collision Probability

Abstract

The construction of parallel runways is an effective solution to address the constraints of urban land resources and mitigate flight delays caused by the increasing volume of air traffic. To ensure the safety of parallel approach operations and further enhance operational efficiency, this study proposes a quantitative safety risk assessment method for parallel approaches based on Monte Carlo simulation (MCS) and extreme value theory (EVT). Taking a parallel runway at a major airport in Southwest China as a case study, historical Automatic Dependent Surveillance-Broadcast (ADS-B) trajectory data were processed and analyzed to derive traffic flow characteristics and the actual distribution of approach performance. Subsequently, we developed a collision probability estimation model for parallel approaches based on Monte Carlo–extreme value theory (MC-EVT). Monte Carlo simulation was employed to conduct simulation experiments on the parallel approach process, and the collision risk was quantitatively assessed by integrating experimental data with an analysis based on extreme value theory. Finally, taking the parallel runways of a major airport in southwest China as a case study, experiments were conducted under various parallel approach scenarios to quantitatively assess the collision risk between aircraft. The experimental results indicate that the MC-EVT-based safety risk assessment method for parallel approaches reduces the reliance on traffic flow assumptions. Compared to the conventional Monte Carlo method, it achieves a faster convergence rate, significantly reduces computational workload, and improves computational efficiency by a factor of ten, thus demonstrating that the proposed method is capable of accurately and effectively quantifying low-probability collision risks. Furthermore, the findings reveal a strong correlation between parallel runway width and collision risk. The approach risk under a mixed-aircraft-type configuration is higher than that of a single-aircraft-type configuration, while offset approaches can enhance approach safety. This study can provide valuable references for the construction of parallel runways and the development of regulatory frameworks for parallel approach operations in China.

1. Introduction

The global civil aviation industry remains in a phase of rapid growth, with the continuous increase in air traffic volume leading to intensified airspace congestion and frequent delays. To accommodate the continuously increasing air traffic volume while considering land resource constraints, the surrounding airport environment, and overall development trends, the construction of parallel runways has become a key trend in civil aviation airport development worldwide [1]. Meanwhile, to ensure the safe operation of parallel runways, regulatory bodies such as the Civil Aviation Administration of China (CAAC) and the International Civil Aviation Organization (ICAO) have established operational regulations for the simultaneous instrument operations of parallel runways [2,3].

However, these regulations are, to some extent, “static” and cannot dynamically adapt to specific operational adjustments required under varying airport conditions. Most existing rules remain relatively conservative. With the rapid advancement of communication, navigation, and surveillance (CNS) systems and related aviation technologies [4], parallel runway approach operations have become increasingly complex. While advanced CNS technologies enhance aircraft positioning accuracy and optimize air traffic management (ATM) efficiency, they also introduce new operational challenges, including increased airspace utilization, reduced separation standards, and higher traffic density, thereby imposing more stringent safety requirements on parallel approach operations. Globally, the construction of parallel runways has emerged as a key strategy for alleviating airport capacity constraints and improving operational efficiency. Consequently, there is a pressing need to develop a comprehensive safety risk assessment model for parallel approaches, integrating multi-factor analyses and quantitative methods to evaluate collision risks and identify safety margins and key influencing factors under diverse operational conditions. Such an approach is not only crucial for enhancing the safety and efficiency of parallel runway operations but also provides a scientific foundation for the formulation and refinement of future parallel approach operational standards.

Collision risk assessment (CRA) is a critical factor in enhancing air traffic operational efficiency and ensuring aviation safety. It plays a pivotal role in both maintaining safe aircraft operations and optimizing air traffic management. Prior to the implementation of new aircraft separation standards or the introduction of novel operational concepts, a rigorous risk assessment is essential to evaluate potential hazards, validate feasibility, and ensure operational safety. Scholars worldwide have extensively studied parallel runway operations and the safety risk assessment of parallel approaches. In terms of operational modes, particular focus has been placed on independent parallel approaches for widely spaced parallel runways and paired approaches for closely spaced parallel runways [5]. Current safety risk assessment methodologies can be broadly classified into three categories: direct modeling methods, statistical analysis methods, and computer-based simulation techniques.

In the domain of geometric modeling, an improved version of the classical Route Collision Model has been applied to risk assessment in the approach phase. One approach involves modeling and analyzing collision risk in dual-runway operations under the independent parallel approach framework, exploring the impact of different airspace configurations and traffic flow patterns on risk levels [6]. Another study employed the Fault Tree Analysis (FTA) method to quantitatively assess potential risk factors in paired approach operations and identified key variables affecting the safety of parallel approach operations [7]. Furthermore, there are studies that investigated simultaneous approach operations under yaw conditions, analyzing the effects of lateral meteorological disturbances on aircraft separation and collision probability [8]. In the domain of statistical analysis methods, researchers have attempted to model collision risk in parallel approach operations using inferential statistics. For instance, a study conducted a statistical analysis of the safety of approaches to closely spaced parallel runways and developed a collision probability estimation model based on historical data [9]. Another study performed a quantitative assessment of collision risk for parallel runways under instrument approach conditions and investigated the impact of varying weather conditions, traffic flow patterns, and navigation accuracy on risk levels [10]. Furthermore, a study leveraging the differential games theory proposed an algorithm for independent operational conflict alerting to enhance the safety of parallel approaches [11]. In recent years, computer simulation technology has been increasingly applied to civil aviation safety risk assessment [12]. Traditional Monte Carlo simulation (MCS) has been widely employed in the safety analysis of parallel approaches where collision risk is estimated through large-scale random sampling and probabilistic statistical methods [13]. Through Monte Carlo simulation, one study drew the conclusion that the runway capacity is 60, 78, and 52 aircrafts/hour for each operation mode [14]. Simultaneously, there are research findings showing that the speed difference between two paired aircraft has the most significant impact on separation safety [15].

These studies suggest that while geometric analysis models are intuitive and computationally efficient, their applicability is constrained, as they struggle to accurately capture the dynamic uncertainties inherent in complex operational environments. Statistical methods effectively utilize historical operational data for risk modeling; however, they exhibit high computational complexity when handling high-dimensional, nonlinear, or multivariable interaction effects. Additionally, they may encounter convergence issues, limiting their applicability in large-scale, complex operational scenarios. The Monte Carlo method encounters significant computational costs and slow convergence when analyzing low-probability events, which limits its applicability in efficiently assessing low-probability collision risks. Optimizing computational approaches to enhance simulation accuracy while reducing computational costs is essential. Historical studies indicate that the aircraft approach and descent process is a complex, dynamic, and stochastic system. Conducting a risk analysis based on a single or a limited set of influencing factors is insufficient to fully characterize the approach dynamics. Existing risk assessment methodologies have limitations in capturing the interactions within the dynamic approach system and its temporal evolution. Moreover, aircraft collision events are rare, with extremely low probabilities, requiring a substantial number of simulations to observe collision cases. Otherwise, the estimation accuracy may be significantly compromised.

To address the limitations of existing collision risk models based on Monte Carlo simulation, this paper proposes a novel approach that integrates Monte Carlo simulation with extreme value theory (EVT) to assess the probability of mid-air collision risk. This method addresses the limitation of the traditional Monte Carlo approach, which struggles to provide explicit confidence intervals, particularly when rare events occur infrequently, leading to significant estimation errors. EVT facilitates the extrapolation of low-probability events through the Peaks Over Threshold (POT) method, which fits a generalized Pareto distribution (GPD) to extreme values. EVT has already been applied in various domains, such as financial modeling to simulate extreme market fluctuations [16] and in the medical field to assess surgical risks [17], among other rare event scenarios.

The main contributions of this paper are as follows:

• The proposed approach utilizes ADS-B trajectory data to extract approach deviation performance and quantify the uncertainty of errors under the performance-based navigation (PBN) operational mode. This uncertainty is assumed to follow a two-dimensional Gaussian distribution, allowing the simulation to better capture its variability.
• This study proposes a quantitative collision risk assessment method for parallel approaches, integrating Monte Carlo simulation with extreme value theory (MC-EVT). This method enables the rapid simulation of approach risk scenarios and utilizes Bayesian conditional probability to estimate risk levels.
• Experimental validation is performed to conduct an in-depth analysis of key risk factors, with the goal of further improving the efficiency of parallel approach operations.
• In the approach risk scenarios constructed in this study, the proposed method exhibits superior effectiveness and efficiency compared to conventional Monte Carlo simulation approaches.

The rest of this paper is organized as follows. We first describe the studied parallel approach operation and risk scenario problem in Section 2. Then, Section 3 introduces in detail the proposed parallel approach collision risk assessment method based on MC-EVT. Next, simulation experiments are conducted in Section 4. Finally, Section 5 concludes the research findings.

2. Parallel Approach Operation Risk Scenario

2.1. Scene Description

During parallel approaches, two aircraft on adjacent flight paths may conduct either independent or dependent parallel approaches. If one aircraft unintentionally deviates from its intended flight path due to factors such as human error or equipment malfunction—a phenomenon known as a blunder [18]—it forms a deviation angle relative to the nominal approach trajectory. In such cases, the ground-based air traffic controller responsible for the deviating aircraft detects the anomaly via radar monitoring and immediately issues an instruction for the aircraft to return to its planned flight path. Simultaneously, the other aircraft on the adjacent parallel approach path receives an avoidance instruction from the controller and executes an evasive maneuver.

If the deviating aircraft fails to comply with the controller’s instructions due to communication failure or other external environmental factors and continues its unintended trajectory towards the adjacent flight path, a severe blunder scenario arises. Once an unintended deviation occurs, a hazardous approach scenario is already established, and a collision risk exists regardless of whether the pilot follows the controller’s instructions. According to the Federal Aviation Administration (FAA) 8020 report, a collision risk incident is defined as occurring when the separation between two aircraft falls below the minimum safe distance of 500 feet (153 m) [19]. The collision risk under blunder-induced deviations during approach is the primary focus of this study.

Figure 1 illustrates the process of collision risk during parallel approaches. The operational characteristics of parallel runway systems are largely determined by the runway separation distance. Approach scenarios are classified based on approach procedures and can be divided into two categories: independent parallel approaches for widely spaced parallel runways and paired parallel approaches for closely spaced parallel runways. Figure 2 presents the risk scenarios associated with independent parallel approaches. This study examines two distinct scenarios: (1) the deviating aircraft adheres to the air traffic controller’s instructions and returns to its planned trajectory, and (2) the deviating aircraft fails to comply and continues its unintended deviation. Given the potentially severe consequences of such rare deviations, rare-event simulation has become an essential tool for risk quantification in air traffic systems [20].

Figure 3 illustrates the risk scenarios associated with paired approaches. Approach modes are classified into two types: parallel paired approaches and offset paired approaches.

2.2. Target Level of Safety (TLS)

When evaluating collision risk, the probability of mid-air collision is typically compared against an acceptable risk threshold, referred to as the Target Level of Safety (TLS). The overall TLS for an entire flight operation within a given airspace is set at 1 × 10⁻⁷. This overall TLS is further partitioned based on different flight phases. Specifically, for en route airspace, the TLS for mid-air collision risk is 1 × 10⁻⁸, while for the approach phase, different organizations have established varying safety thresholds: the FAA sets the TLS at 4 × 10⁻⁸ [19], whereas MITRE establishes a stricter threshold of 1 × 10⁻⁹ [21]. In this study, a conservative value of 5 × 10⁻⁹ is adopted. The blunder rate P(B), can be derived from historical data. According to the FAA’s aviation performance metrics database and approach incident reports, the latest estimated blunder probability is P(Blunder) = 1/24,000. A fixed deviation probability simplifies the overall calculation of collision risk [22].

3. Parallel Approach Collision Risk Assessment Model Based on MC-EVT

3.1. Extreme Value Theory

Extreme value models are typically classified into two primary categories. The first category is generalized extreme value (GEV) theory, which is based on the Block Maxima Approach (BMA). This approach involves extracting the maximum values from multiple independent samples drawn from the same distribution. The second category is represented by the Peaks Over Threshold (POT) method, which models all observations exceeding a predefined high threshold. As the POT method directly models extreme values that exceed a predefined high threshold, and achieve a higher data utilization efficiency, it is regarded as more suitable for real-world applications [23].

• Block Maxima Approach

Extreme value analysis involves estimating the frequency of extreme events to construct a probabilistic model for such occurrences. Consequently, it is closely related to the distribution of the maximum value, $M_n=\max (X_1,X_2,\dots ,X_n)$. Let $X_1,X_2,\dots ,X_n$ be a sequence of independent and identically distributed (IID) random variables, following a cumulative distribution function (CDF) F. In practical applications, $X_i$ represents the measured value at time ii based on a predefined time scale, while M_n denotes the maximum observed value over n time units. Given that $X_1,X_2,\dots ,X_n$ are IID random variables, the following relationship holds [24].

$$\begin{array}{ll}pr(M_n& \le x)=pr\left\{X_1\le x,\dots X_n\le x\right\}\\ & =pr\left\{X_1\le x\right\}\times \dots \times pr\left\{X_n\le x\right\}\\ & ={\left\{F(x)\right\}}^n\end{array}$$

(1)

According to extreme value theory, ${\left\{X_i\right\}}_{i=1}^n$ are independent and identically distributed random variables with cumulative distribution function F. Most of the time, the distribution function F is unknown, but according to the Fisher–Tippett–Gnedenko theorem [25], it provides an asymptotic result if there exist sequences of real numbers $a_n>0$ and $b_n\in ℝ$ such that the following limit converges to a non-degenerate distribution function $G(x)$:

$$\underset{n\to \infty }{\lim }P\left({\displaystyle \frac{\max \left\{X_1,\dots X_n\right\}-b_n}{a_n}}\le x\right)=G(x)$$

(2)

The limiting distribution $G(x)$ belongs to the generalized extreme value (GEV) family. Actually, if the normalized maxima converges to a non-degenerate distribution G, then the limiting distribution must be a location-scale transformed GEV distribution.

• Peaks Over Threshold (POT)

Compared to traditional methods, the Peaks Over Threshold (POT) approach utilizes more information from the dataset. Unlike the Block Maxima Approach (BMA), which considers only block-wise maxima, POT analyzes the entire time series by extracting all values that exceed a predefined threshold and using them to fit an extreme value distribution. This approach enhances both feasibility and practical applicability in statistical analysis [26,27].

Therefore, the Peaks Over Threshold method enables a more rational selection of events that meet the criteria for being extreme [28]. In this approach, observations that exceed a given threshold are referred to as exceedances over a threshold. The POT method has been widely applied in estimating the return levels of significant wave height. For instance, the POT method has been applied in studies on hurricane-induced damage [29], the effects of traffic loads on bridges [30], and pedestrian safety on roadways [31]. Previous research has demonstrated that for a sequence of independent and identically distributed (IID) random variables $X_1,\dots ,X_n$, and a sufficiently large threshold u, the conditional excess distribution function F_u(y) = P(X-u ≤ y | X > u) can be well approximated by the generalized Pareto distribution $H_{\xi ,\beta (u)}$ [32,33].

$$P(X-u\le y|X>u)=H_{\xi ,\beta (u)}(y)=\left\{\begin{array}{l}1-{\left(1+{\displaystyle \frac{\xi y}{\beta (u)}}\right)}^{-1/\xi }\mathrm{for}\xi \ne 0\\ 1-e^{-y/\beta (u)}\mathrm{for}\xi =0\end{array}\right.$$

(3)

In the equation, β is the scale parameter; if the shape parameter $\xi$ is $\xi \ge 0$, $H_{\xi ,\beta (u)}$ is y ∈ [0, ∞], and y ∈ $[0,-\beta (u)/\xi ]$ otherwise.

In various research domains, studies related to risk management are dedicated to understanding and analyzing extreme events. The objective of this study was to model the minimum separation distance between aircraft, with a particular focus on $\xi <0$ scenarios. Under $\xi <0$ conditions, the distribution follows a Pareto-type heavy-tailed distribution, and the estimation of its parameters involves determining the values of $\xi$ and $\beta$ that maximize the log-likelihood function under the given $u=\beta /\xi$ constraints.

• Threshold selection

In the application of the Peaks Over Threshold (POT) method, selecting an appropriate threshold u is crucial. When the chosen threshold u is too high, it results in fewer exceedances, leading to increased variance. On the other hand, selecting a threshold that is too low may introduce bias into the estimation. Various methods can be employed to determine the optimal threshold, among which graphical approaches are the most commonly used. One such method is the mean excess function plot (MEFP), which provides a visual representation of the mean excess function relative to the threshold level u. The empirical mean excess function is defined as follows:

$$e_n(u)={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(X_i-u)I_{\{X_i>u\}}}}{{\displaystyle \sum _{i=1}^n{I}_{\{X_i>u\}}}}}u\le X_{(n)}$$

(4)

In the equation, $X_1,\dots ,X_n$ are independent and identically distributed samples; I is the indicator function; if X_i > u, then I{X_i > u} = 1; otherwise, I{X_i > u} = 0. The mean excess plot consists of points {(X_(i), e_n(X_(i)):1 ≤ I ≤ n−1}, where X_i represents the ith statistical sample, and n denotes the total sample size. Typically, the threshold is selected as the smallest value at which the mean excess function plot exhibits an approximately linear behavior. Such linearity suggests that the generalized Pareto distribution provides a suitable approximation for the tail distribution exceeding the chosen threshold.

3.2. Collision Risk Probability Estimation Based on MC-EVT

3.2.1. Aircraft Model Representation

To enhance the accuracy of collision risk estimation, this study represents an aircraft as a sphere, where the diameter corresponds to the wingspan of the aircraft, as illustrated in Figure 4. Using a spherical representation is a more conservative approach compared to traditional methods, where aircraft are typically modeled as rectangular prisms or cylinders. The spherical model encompasses a larger volume and allows for a reasonable estimation of the sphere’s diameter based on the aircraft’s average wingspan.

3.2.2. Kinematic Model of Aircraft Motion

As illustrated in Figure 5, assume there is an aircraft p (where p_b is the offset aircraft and p_e is the avoidance aircraft) moving with uniform deceleration during its final approach. Equations (5) and (6) describe the velocity relationship at time t = i and the motion parameter v_i. The motion parameters v_i and u_i of the two aircraft represent their velocities in the horizontal and vertical directions, respectively. The parameters v_xi and v_yi denote the two velocity components, while α represents the track angle. l denotes the horizontal distance, h represents the altitude, and s is the distance between the parallel runways. Since the runways are not staggered, the coordinates of the two Final Approach Fix (FAF) points are (0, l, h) and (s, l, h), respectively. At time t = i, the aircraft’s coordinates are (x_pi, y_pi, z_pi).

$$v_{pi}=v_{pi-1}+a\times \delta t_i$$

(5)

$$u_{pi}=u_{pi-1}+a\times \delta t_i$$

(6)

The range of velocity and deceleration depends on the aircraft type; the turn radius r is related to the velocity; the turn rate is denoted as $\psi$, with $\delta \psi$ representing the rate of change in the turn rate; and the bank angle is given as $\phi$ = 25°.

$${\alpha }_i={\psi }_i+\delta {\psi }_i$$

(7)

$${\psi }_i={\displaystyle \frac{562\tan \phi }{v_i}}$$

(8)

$$r={\displaystyle \frac{v^2}{g\cdot \tan \phi }}$$

(9)

$$x_{pi}=x_{pi-1}+[v_{xi-1}\cdot \cos ({\alpha }_{i-1})+v_{yi}\cdot \sin ({\alpha }_{i-1})]\delta t$$

(10)

$$y_{pi}=y_{pi-1}+[v_{xi-1}\cdot \sin ({\alpha }_{i-1})+v_{yi}\cdot \cos ({\alpha }_{i-1})]\delta t$$

(11)

$$z_{pi}=z_{pi-1}+[u_{pzi}-u_{pzi-1}]\delta t$$

(12)

$$\begin{array}{l}{d}_i=‖(x_b,y_b,z_b)-(x_e,y_e,z_e)‖\\ =\sqrt{{(x_{bi}-x_{ei})}^2+{(y_{bi}-y_{ei})}^2+{(z_{bi}-z_{ei})}^2}\end{array}$$

(13)

3.2.3. Monte Carlo Simulation

Based on the previous analysis of aircraft kinematic performance, a Monte Carlo simulation is conducted to model the flight trajectories during the parallel approach process. The acquisition of simulated flight trajectories relies on performance data from the BADA database, characteristics of approach traffic flow, and airspace information. In each Monte Carlo simulation run, the flight trajectories of two aircraft are selected. As the simulation progresses, these trajectories are replayed, allowing for the extraction of the minimum observed three-dimensional Euclidean distance, this minimum distance is referred to as the Closest Point of Approach (CPA) distance, denoted as d_CPA. By conducting multiple simulation runs, a distribution of CPA distances can be obtained. The Monte Carlo simulation workflow is illustrated in Figure 6.

3.2.4. Collision Risk Probabilities

Since the Monte Carlo simulations are performed under identical approach scenarios and each run is independent of the others, the resulting statistics are independently and identically distributed (i.i.d.). Therefore, according to the Central Limit Theorem, we have

$${\displaystyle \frac{{\displaystyle \sum _{i=1}^n{X}_i-n\mu }}{\sqrt{n}\sigma }}\sim \mathcal{N}(0,1)$$

Based on the obtained distance data, the Peaks Over Threshold (POT) method is employed to derive the distribution of CPA distances. Traditional extreme value theory (EVT) primarily focuses on modeling the characteristics of maxima. In contrast, this study applies EVT to model minima, specifically the minimum distance between two aircraft observed in each Monte Carlo simulation run. Ultimately, this approach enables the estimation of the probability that the CPA distance falls below the minimum safe separation threshold, indicating a collision event between the two aircraft. To achieve this, the following mathematical expression is utilized:

$$\min (X_1,\dots ,X_n)=-\max (-X_1,\dots ,-X_n)$$

Therefore, d_CPA needs to be transformed into a negative value, and the risk is defined as: −d_CPA ≥ −λ

To select an appropriate threshold u, the mean excess plot (MEP) method is applied. As previously described, the mean excess function plot is generated for various threshold values, and the threshold is chosen at the point where the curve begins to exhibit a linear trend.

Once the threshold u is determined, extracting situations where the CPA interval exceeds the threshold u and satisfies GPD. Subsequently, the maximum likelihood estimation (MLE) technique is employed to estimate the GPD parameters. It is assumed that complete overlap (d_CPA = 0) is possible, implying that the GPD range lies between u and 0, with the following relationship: ξ < 0, β = −ξ u, Therefore, only ξ needs to be estimated. With the estimated GPD parameters, the risk probability can be computed. Given that the CPA distance is below the threshold u, the probability of collision can then be derived using Equation (14).

$$\begin{array}{ll}P(PAC& |-d_{\mathrm{CPA}}>u)=1-H_{\xi ,\beta (u)}(u-\lambda )\\ & ={\left(1+{\displaystyle \frac{\xi (u-\lambda )}{\beta (u)}}\right)}^{-1/\xi }\\ & ={\left(1+{\displaystyle \frac{\xi (u-\lambda )}{-\xi u}}\right)}^{-1/\xi }\\ & ={\left({\displaystyle \frac{\lambda }{u}}\right)}^{-1/\xi }\end{array}$$

(14)

By applying the chain rule, the probability of a collision can be formulated as follows:

$$\begin{array}{ll}P(PAC)& =P(PAC|-d_{\mathrm{CPA}}>u)\cdot P(-d_{\mathrm{CPA}}>u)\\ & ={\left({\displaystyle \frac{\lambda }{u}}\right)}^{-1/\xi }\cdot {\displaystyle \frac{N}{n}}\end{array}$$

(15)

where n is the number of simulations, N is the number of times the selected threshold is exceeded, and $\lambda$ represents the diameter of the aircraft sphere.

As Equation (15) is the product of two probabilities that are both estimated with a certain degree of uncertainty, involves the product of two estimators, and the resulting quantity is a random variable. The estimated value of $P(-d_{\mathrm{CPA}}>u)$, denoted as $\widehat{P}(-d_{\mathrm{CPA}}>u)$, is derived from a Bernoulli trial, where the trial criterion is defined as $-d_{\mathrm{CPA}}>u$. Therefore, the estimate of the probability of success can be described thanks to the CLT with a random variable following a normal distribution, where

$$\widehat{P}(-d_{\mathrm{CPA}}>u)\sim \mathcal{N}\left({\displaystyle \frac{N}{n}},{\displaystyle \frac{N(n-N)}{n^3}}\right)$$

(16)

Furthermore, the maximum likelihood estimate of ξ is a random variable with an associated variance, and thus subject to estimation uncertainty. The standard error (SE) associated with the MLE can be computed as the square root of the diagonal elements of the inverse of the observed Fisher information matrix $I({\widehat{\xi }}_{\mathrm{MLE}})$. The Fisher information matrix is derived from the negative Hessian of the log-likelihood function. Since the parameter ξ is a scalar, its variance corresponds to the relevant element of the inverse of the observed Fisher information matrix. Under standard regularity conditions, the maximum likelihood estimator (MLE) of the GPD shape parameter ξ is asymptotically normal [34]. The log-likelihood function for the exceedances x₁, x₂,…,x_k over a threshold u, assuming $\xi \ne 0$, is given by

$$\log L(\xi ,\beta )=-k\log \beta -(1+{\displaystyle \frac{1}{\xi }}){\displaystyle \sum _{i=1}^k\log (1+\frac{\xi (x_i-u)}{\beta })}$$

(17)

The MLE of ξ is obtained by maximizing this function numerically, and when the sample size is sufficiently large, the estimator $\widehat{\xi }$ is asymptotically normally distributed:

$${\widehat{\xi }}_{\mathrm{MLE}}\sim N\left({\widehat{\xi }}_{\mathrm{MLE}},{[I({\widehat{\xi }}_{\mathrm{MLE}})]}^{-1}\right)$$

(18)

Equation (18) denotes the observed Fisher information $I({\widehat{\xi }}_{\mathrm{MLE}})$, computed as the negative second derivative of the log-likelihood function with respect to ξ. These results provide a foundation for constructing standard errors and confidence intervals for $\widehat{\xi }$ in large-sample settings.

By sampling the random variables $\widehat{P}(-d_{\mathrm{CPA}}>u)$ and $\widehat{P}(-d_{\mathrm{CPA}}>u)$ a sufficient number of times, the final distribution $\widehat{P}(PAC)$ of PAC probability estimation can be obtained through Monte Carlo simulation.

$$\widehat{P}(PAC)={\left({\displaystyle \frac{\lambda }{u}}\right)}^{-1/{\widehat{\xi }}_{\mathrm{MLE}}}\cdot \widehat{P}(-d_{\mathrm{CPA}}>u)$$

(19)

4. Experimental Analysis

4.1. Analysis of Characteristics of Approaching Traffic Flow Based on ADS-B

Taking Chengdu ZUUU Airport as an example, where performance-based navigation (PBN) flight procedures are implemented, a schematic diagram of the precision approach procedure within the framework of PBN for Runway02L is shown in Figure 7. Based on ADS-B arrival data from four months in 2022, a total of 5000 arrival trajectories for Runway02R were identified and analyzed, as illustrated in Figure 8. The trajectory data contain various operational parameters of the aircraft, including flight number, longitude, latitude, altitude, speed, heading, aircraft type, and timestamp. Since the raw trajectory data contain noise and lacks standardized formatting, preprocessing is necessary to ensure a more precise analysis of sector traffic flow characteristics. The preprocessing steps primarily involve trajectory data selection, the removal of duplicate trajectories, and the filtering of erroneous data points.

4.1.1. Aircraft Type

As described above, an aircraft is represented as a sphere with a diameter of λ. To determine the value of λ, an analysis was conducted on the aircraft types operating within the sector and their corresponding wingspans. The aircraft type data were extracted and analyzed, and the proportional distribution of different aircraft types is shown in Figure 9. The A320 accounts for 26%, the A330 for 22%, and the A321 for 18%. The fleet is predominantly composed of medium-sized aircraft, which constitute 97% of the total, while heavy and light aircraft account for a smaller proportion. Categorizing the aircraft into three classes—heavy, medium, and light—their respective proportions are 1.5%, 97%, and 1.5%.

Based on the acquired aircraft speed data during the approach phase, the speed and deceleration characteristics were classified according to aircraft type. Assuming uniform deceleration during the final approach phase, the corresponding speed and deceleration values are summarized in

Table 1. Aircraft types and speed parameters.

Type	Speed		Deceleration −m/s2
	kn	m/s
Heavy	140–165	72–85	0.06
Medium	120–140	61–72	0.05
Light	91–120	47–61	0.03

.

4.1.2. Arrival Distribution of Traffic Flow

The hourly arrival traffic throughout the day is illustrated in Figure 10 below. Assuming that the time intervals between consecutive aircraft on each runway follow a negative exponential distribution, the number of aircraft arrivals per unit time follows a Poisson distribution [35]. A Poisson fit was applied to the hourly arrival counts, and the results indicate that the number of arriving aircraft per hour follows a Poisson distribution with a parameter of 20. Figure 10b presents the empirical histogram of hourly aircraft arrivals along with the fitted Poisson distribution curve.

4.1.3. Analysis of Approach Deviation Performance

The collision risk of an aircraft during the approach phase is related to its actual position relative to the nominal glide path. The actual flight trajectory is influenced by various factors, including navigational and human factors. Therefore, the trajectory generated by simulation must account for approach deviation performance. Trajectory cross-sections are established at 1000 m intervals along the approach path, extending in the opposite direction from the runway threshold. At the final approach phase, all trajectory points within a given section are identified based on the intersection between the actual trajectory and the cross-section. Outliers detected using the Grubbs test criterion are removed. The lateral and vertical deviations are determined by comparing the identified trajectory points with the corresponding nominal trajectory points in each cross-section. Figure 11 illustrates the deviation of trajectory points from the nominal trajectory at the 6000 m cross-section. A study conducted by TU Dresden indicates that a normal distribution function is the most appropriate for describing the statistical characteristics of approach and departure procedures [36]. Therefore, the deviations are assumed to follow a normal distribution.

The data points are fitted to a normal distribution to determine the mean and standard deviation of both the lateral and vertical deviations across all cross-sections, as shown in

Table 2. Distribution of the deviation parameters for each section.

Coss Section Window	Threshold Distance [m]	Lateral		Vertical
		${\mathit{\mu }}_{\mathit{L}}$	${\mathit{\sigma }}_{\mathit{L}}$	${\mathit{\mu }}_{\mathit{V}}$	${\mathit{\sigma }}_{\mathit{V}}$
Section 1	0 m	1.156	7.343	1.035	6.479
Section 2	1000 m	9.847	10.932	1.119	6.255
Section 3	2000 m	23.654	13.181	1.594	7.153
Section 4	3000 m	18.611	13.746	2.005	7.718
Section 5	4000 m	19.893	17.951	3.653	9.031
Section 6	5000 m	10.107	15.433	3.325	9.126
Section 7	6000 m	−1.124	15.991	2.649	8.891
Section 8	7000 m	−7.473	19.135	1.809	10.611
Section 9	8000 m	−10.667	19.012	1.825	14.003
Section 10	9000 m	−17.698	24.173	2.192	14.543
Section 11	10,000 m	−16.295	25.981	1.461	16.912
Section 12	11,000 m	−14.473	30.736	−1.014	19.235
Section 13	12,000 m	−17.512	35.841	1.521	20.012

.

The distribution of cross-sections is illustrated in Figure 12. It can be observed that trajectory deviations are correlated with the distance from the runway threshold—the farther the distance, the larger the standard deviations in both lateral and vertical directions. Additionally, in the lateral direction, flight trajectories exhibit minor oscillations around the nominal glide path. In the vertical direction, the mean value of the cross-section is higher than that of the nominal trajectory. This phenomenon occurs because, during the approach phase, aircraft tend to maintain a slightly higher altitude than the nominal glide path as a safety precaution.

The distribution pattern of cross-sections at any given distance is determined using linear interpolation. The coefficient of determination test and F-test are employed for validation, leading to the derivation of the relationship between the standard deviation and distance in the horizontal direction through linear fitting.

$${\sigma }_{\mathrm{Lateral}}=0.0015x+5.621$$

(20)

The linear relationship in the vertical direction is expressed as follows:

$${\sigma }_{\mathrm{Vertical}}=0.00107x+4.34$$

(21)

where x represents the distance from the runway threshold (in meters), and $\sigma \left(x\right)$ denotes the standard deviation. The linear increase in both lateral and vertical standard deviations with distance from the runway threshold is consistent with diffusion-based models, where positional variance tends to increase with spatial displacement due to the cumulative nature of uncertainty in such processes.

4.2. Monte Carlo Simulation Under Typical ICAO Error Scenarios

The typical error scenario parameter values specified by ICAO are presented in

Table 3. ICAO typical scene parameters.

Parameter	Values
Velocity	150 kn
Deviation Angle	30°
Avoidance angle	45°
Error deviation rate	1/24,000
Reaction and Communication Time	8.5 s
NTZ width	610 m
Runway centerline spacing	1036 m
Collision distance limit	153 m

[2]. Although this static assumption with fixed parameters does not capture the stochastic nature of the human–machine system, it serves as a reference for model validation.

For each simulated aircraft pair, the Peaks Over Threshold (POT) method was employed to obtain the three-dimensional Euclidean distance at the Closest Point of Approach (CPA). Figure 13 (left) presents the histogram of the negative CPA separation distances (in meters) for 5 w aircraft pairs. No collisions were observed.

Following the CPA computation, an extreme value analysis of the CPA separation distances was conducted. This study specifically examines the tail behavior of the CPA separation distribution. Figure 13b provides a zoomed-in view of this distribution tail. Notably, no collisions were observed, and the closest recorded CPA separation was −168 m.

To perform the extreme value analysis, this study adopts the POT method. The first step involves selecting an appropriate threshold value u. Figure 14 illustrates the sample mean excess plot of CPA separation distances.

The threshold was selected as the minimum value at which the curve begins to exhibit linear behavior, specifically at −525 m, with 53 samples exceeding this threshold. The threshold of −525 m corresponds to the lowest value where the empirical mean excess function plot (Figure 14) begins to exhibit linear behavior. This aligns with standard EVT practice in threshold selection. After determining the threshold, the CPA distances beyond u were fitted to a generalized Pareto distribution (GPD). Estimate the shape parameter value ξ of the generalized Pareto distribution (GPD) using maximum likelihood estimation (MLE) to obtain ${\widehat{\xi }}_{\mathrm{MLE}}$. Additionally, $\widehat{\beta }=-{\widehat{\xi }}_{\mathrm{MLE}}\times u$, and the histogram below illustrates the distribution of CPA distances exceeding the selected threshold.

4.3. Collision Probability

Finally, calculate the probability of collision risk:

$$\begin{array}{ll}P(PAC& |Blunder)\approx {\displaystyle \frac{N}{n}}\cdot {\left({\displaystyle \frac{\lambda }{u}}\right)}^{-1/\xi }\\ & \approx {\displaystyle \frac{53}{50000}}\cdot {\left({\displaystyle \frac{60}{525}}\right)}^{-1/\xi }\\ & \approx 2.827\times {10}^{-5}\end{array}$$

(22)

Calculate the probability of a blunder:

$$P(Blunder)={\displaystyle \frac{1}{24000}}$$

Calculate the probability of a parallel approach collision:

$$\begin{array}{ll}P(PAC)& =P(PAC|Blunder)\cdot P(Blunder)\\ & =2.827\times {10}^{-5}\cdot {\displaystyle \frac{1}{24000}}\\ & =1.178\times {10}^{-9}\end{array}$$

(23)

It can be seen that the obtained results meet the minimum safety level, which proves the rationality of the analysis. The probability result obtained above is not the distribution of a single CPA, but an estimate of the probability of multiple CPA events. It can be said that this probability is the mean of n independent Bernoulli random variables, so we have

$$\begin{array}{c}\widehat{P}(-d_{\mathrm{CPA}}>u)\sim \mathcal{N}({\displaystyle \frac{N}{n}},{\displaystyle \frac{N(n-N)}{n^3}})\\ \sim \mathcal{N}({\displaystyle \frac{53}{50000}},{\displaystyle \frac{53(50000-53)}{{50000}^3}})\\ \sim \mathcal{N}(1.06\times {10}^{-3},2.12\times {10}^{-8})\end{array}$$

(24)

After conducting 30,000 Monte Carlo simulations, the histogram of the probability distribution is shown in the following Figure 15. It can be seen that for a 95% confidence interval,

$$2.13\times {10}^{-9}\le P(PAC)\le 2.45\times {10}^{-10}$$

We can observe that the confidence interval obtained through EVT is smaller than those generated by other methods. To demonstrate this, we analyzed the results of 50,000 Monte Carlo simulations. Since no collisions were observed, we applied the Rule of Three [37], which states that for an event with n samples and no observed occurrences, the 95% confidence interval for the event’s probability ranges from 0 to 3/n. Applied to this study, the confidence interval for the probability of a collision is

$$0\le P(PAC|\mathrm{Blunder})\le 6\times {10}^{-5}$$

At the same confidence level, the result of Rule of Three is three times that of EVT:

$$P{(PAC|\mathrm{Blunder})}_{\mathrm{EVT}}\le 2\times {10}^{-5}$$

Compared with traditional Monte Carlo simulation methods, the method proposed in this study has higher computational efficiency. If traditional Monte Carlo simulation is used, due to the low probability of observed collisions, at least 50–60 w simulations are required using the Wilson Score confidence interval. The method proposed in this paper only uses 50,000 simulations, resulting in a tenfold increase in efficiency.

As is shown in the following Figure 16. where u = −525, ${\widehat{\xi }}_{\mathrm{MLE}}=-0.519$, and $\widehat{\beta }=272$.

4.4. Analysis of Other Factors

The research model was utilized to analyze additional factors affecting the approach process, simulation experiments were conducted to investigate the impact of various influencing parameters, including runway spacing, response time, and aircraft type composition. The experimental parameters are presented in

Table 4. Scenario parameters.

Properties	Detailed Properties
The ratio of heavy, medium, and light aircraft	①1:0:0; ②1.5%:97%:1.5%; ③0:0:1
Speed and deceleration	Refer to Table 1
Runway centerline spacing	400 m~700 m, 1036 m~1500 m
NTZ width	610 m (widely spaced parallel runway)
The final approach segment	600 m high and 12 km long from the runway threshold
Arrival law	Poisson distribution, 20/per hour
Deviation angle	α~(30°, 52), αmax~35°, αmin~25°
Error deviation rate	1/24,000
Departure point	FAF to any point 2 km before the runway threshold
Avoidance angle	α~(45°, 52), αmax~50°, αmin~40°
Avoidance acceleration	0.514 m/s2
The relative position	Located at 0~3 km in the longitudinal direction behind the deviation (widely spaced runway)
Response time	From the aircraft deviation order to the pilot receiving the command: 8~12 s
Safety zone	445 m–3891 m (short runway) [15]
The actual approach deviation	The probability distribution is shown in Table 4
Collision distance limit	153 m
Safety target level	5 × 10−9

, an the simulated experimental scenario is shown in the Figure 17.

For widely spaced parallel runways with a No Transgression Zone, a simulation experiment was conducted for runway centerline spacings ranging from 1100 to 1700 m under an NTZ width of 610 m. The results are presented in

Table 5. Runway centerline spacing and collision risk.

Runway Centerline Spacing	Collision Risk Probability
1100	1.667 × 10−10
1300	8.333 × 10−11
1500	4.167 × 10−11
1700	1.25 × 10−12

.

According to the simulation results in Figure 18, as the runway spacing increases, the collision risk decreases. The NTZ with a width of 610 m provides a margin for further reduction. An additional experiment was conducted to evaluate different NTZ widths ranging from 500 to 650 m for a runway centerline spacing of 1300 m. The results indicate that under the safety target level, the NTZ width can be reduced to 530 m.

Considering the typical aircraft fleet composition, comprising 1.5% heavy aircraft, 97% medium aircraft, and 1.5% light aircraft—an analysis was performed for different fleet compositions. As is shown in

Table 6. Aircraft fleet composition and collision risk.

The Proportion of Heavy, Medium, and Light Aircrafts	Collision Risk Probability
1:0:0	4.167 × 10−10
1.5%:97%:1.5%	1.05 × 10−9
0:0:1	8.333 × 10−10

, the collision risk is highest when heavy and medium aircraft are mixed, lowest when all aircraft are heavy, and slightly higher when all aircraft are light.

Furthermore, the deviation point was set to occur at fixed intervals of 1000 m from the runway threshold instead of being randomly assigned for each experiment. An experiment was conducted for a runway centerline spacing of 1350 m with an aircraft fleet composition of 1.5% heavy, 97% medium, and 1.5% light aircraft. The deviation angle was set to 30°, and the avoidance angle to 45°. The avoidance aircraft was positioned within a range of 0 to 1 km behind the deviating aircraft.

The experimental results are presented in Figure 19, which illustrates that the closer the deviation point is to the runway threshold, the lower the associated risk. This finding is generally consistent with the characteristics of approach deviation characteristics.

The results of Figure 20a indicate that the offset approach significantly enhances approach safety. Specifically, for a runway centerline separation of 365 m, the minimum safety threshold under the 3° offset paired approach was 400–450 m.

Moreover, the speed difference between the lead and following aircraft is another critical factor that must be considered. To analyze its effect, simulation experiments were conducted for three different speed difference ranges: 0 to 5 knots, 10 to 15 knots, and 15 to 20 knots.

As shown in Figure 21, a comparative analysis of runway spacing and the speed difference between the leading and following aircraft indicates that, in terms of the contribution to the collision safety threshold, the speed difference between the leading and following aircraft is the primary influencing factor, followed by the runway centerline spacing.

5. Conclusions

This study proposes a Monte Carlo–extreme value theory (MC-EVT)-based risk assessment model for parallel runway approaches. The model integrates Monte Carlo simulations with extreme value theory to efficiently estimate collision risk under various operational conditions and analyze risk-influencing factors.

The MC-EVT-based parallel approach safety risk assessment method reduces reliance on traffic flow assumptions and achieves a convergence speed that is ten times faster than conventional Monte Carlo methods, demonstrating its effectiveness in accurately quantifying low-probability collision risks. By deeply mining ADS-B data, the traffic flow distribution characteristics are derived, making the analysis more aligned with real-world conditions and further enhancing the rationality of the experiment.

The simulation results reveal the impact of runway centerline separation and aircraft type composition on collision risk. The findings indicate that approaches involving a homogeneous aircraft type composition exhibit lower collision risks compared to those with a mixed aircraft composition. Regarding contributions to the collision safety threshold, the speed differential between the lead and following aircraft is the dominant factor, followed by the runway centerline separation.

Although the proposed MC-EVT-based safety risk assessment model for parallel runway approaches represents an important advancement, there are a few limitations worth addressing. The model assumes idealized operational conditions, such as fixed aircraft types and environmental parameters, which may not always reflect real-world complexities. In future work, it would be beneficial to extend the model to account for more dynamic, real-time factors, such as changing weather conditions and varying traffic patterns. Addressing these limitations will allow for the further refinement of the model and potentially its deployment in more complex operational settings.