# Traffic complexity measurement via collective dynamics analysis of arrival traffic patterns

Xuhao Gui, Junfeng Zhang, Daniel Delahaye, Xinmin Tang

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Xuhao Gui, Junfeng Zhang, Daniel Delahaye, Xinmin Tang. Traffic complexity measurement via collective dynamics analysis of arrival traffic patterns. Air Transportation Research and Development Symposium 2025, Eurocontrol FAA, Jun 2025, Prague (Czech Republic), Czech Republic. hal-05128510

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# Traffic complexity measurement via collective dynamics analysis of arrival traffic patterns

**Xuhao GUI**

College of Civil Aviation

Nanjing University of Aeronautics and Astronautics

Nanjing, China

showhowgui@outlook.com

**Junfeng ZHANG**

College of Civil Aviation

Nanjing University of Aeronautics and Astronautics

Nanjing, China

zhangjunfeng@nuaa.edu.cn

**Daniel DELAHAYE**

OPTIM Lab

Ecole Nationale de l'Aviation Civile

Toulouse, France

daniel@recherche.enac.fr

**Xinmin TANG**

College of Transportation Science and Engineering

Civil Aviation University of China

Tianjin, China

tangxinmin@nuaa.edu.cn

**Abstract**—The measurement of arrival traffic operational complexity is essential in supporting decision-making processes, as it provides valuable insights into arrival traffic operations. Many studies have concentrated on measuring subjective complexity, achieving significant success in their respective areas; however, subjective complexity measurement requires the involvement of professional controllers, leading to higher costs during implementation. To reduce the cost for measuring subjective complexity, this paper analysed the collective dynamics of arrival traffic in both horizontal and vertical dimensions, and summarized several traffic patterns through a coupled analysis. Subsequently, this paper found that the complexity can be reflected in the evolution of arrival traffic patterns and developed a metric to measure the subjective complexity based on the evolution of patterns. A case study, focusing on the arrival traffic operations within the terminal area of Guangzhou Baiyun airport, is conducted to validate the proposed metric.

**Keywords**—air traffic management; terminal management area; complexity; data-driven; data smoothing, collective dynamics

## I. INTRODUCTION

The air traffic volume is predicted to grow over the long term [1], and the air traffic management system must continue to cope with the ever-increasing demand [2]. Considering that the complexity measurement, measuring the level of challenge a traffic situation poses to controllers [3], provides valuable insights into air traffic operations for decision making process, it has been an area of interest in the past two decades.

Mogford et al. [4] claimed that the complexity is a subjective construct, which is not only determined by the

source factors (e.g. air traffic pattern, sector characteristics) but also mediating factors (e.g. equipment quality, individual differences, and controller cognitive strategies).

Many intrinsic complexity metrics focus mainly on **source factors**, as they are more relevant for highly automated air traffic control systems that do not involve human factors [5], [6]. For example, Delahaye et al. developed a complexity metric for both en-route and terminal sectors based on non-linear [7] and linear [8] dynamic systems. Those approach aims to quantify the intrinsic complexity of trajectories by aligning a vector field with observed positions and speeds. These metrics can support the decision making process [9]–[11], as they offer the valuable ability to assess whether traffic is well-organized, regardless of its density [12].

Considering the terminal area is the most complex part of the airspace, Netjasov et al. [13] proposed a generic metric for measuring the complexity of a terminal area, which considering the airspace configuration, the number and length of arrival/departure trajectories, the airport(s) runway system capacity, etc. Deng et al. [14] argued that modern navigation systems provide aircraft with greater freedom and measured complexity while taking radar vectoring into account. Lim et al. [15] claimed that operational outcomes, such as radar vectoring and holding patterns, provide a more accurate reflection of real-time complexity. They then proposed factors such as total arc length, curvature, and holding duration to capture the complexity of terminal area.

To accurately estimate the workload of controllers, **mediating factors** should be considered as well since complexity is a psychological and subjective construct. Therefore, to get from the traffic situation to complexity there is a leap that cannot be made without input from controllers [16]. NASA conducted several early works [17]–[19]. Those studies proposed the dynamic density

<footnote>This work was supported by the National Natural Science Foundation of China under Grant No. 52372315, U1933117, 52072174; and grants from the Funds of China Scholarship Council No. 202306830116.</footnote>
concept, which is a function of the number of aircraft and their changing geometries in a given airspace. The coefficients can be determined by showing different traffic scenarios to several controllers. The dynamic density was further extended by adding more complexity factors [20], [21]. Considering that the linear models may not be able to capture the non-linear interactions, Andrasi et al. [16] applied artificial neural networks to estimate the complexity. Controllers were recruited to perform human-in-the-loop simulations, during which they were asked to provide real-time assessment of complexity.

Although these subjective metrics are straightforward, they necessitate the involvement of professional controllers, leading to higher human resource costs during implementation. To address this gap and make subjective complexity measurement more cost-effective, this paper uses radar trajectory data—which already captures controllers’ decision-making behaviour—to analyse controllers’ perceptions of arrival traffic complexity. Specifically, this paper infers the change in the subjective complexity level by analysing the evolution of arrival traffic patterns.

Section II details the problem to be solved in this paper and provides a framework to illustrate how to learn complexity levels. Section III presents a contour-invariant trajectory heading smoothing method, which formulates the data smoothing problem as a minimisation problem and solves it by training a neural network. Section IV proposes two approaches to capture the evolution of arrival traffic in both horizontal and vertical dimensions. Section V provides a case study to summarise patterns of arrival traffic operations through a coupled analysis and measures the complexity based on the evolution of these operational patterns. Finally, section VI concludes and proposes steps for further research.

## II. PROBLEM FORMULATION

This paper addresses the problem of estimating the subjective complexity of TMA for arrival flights by analysing the controllers’ decision-making behaviour as captured in radar trajectory data. Unlike traditional approaches that directly rely on controllers’ subjective scoring, our method indirectly infers the complexity of TMA for arrival flights through their corrective actions observed in trajectory data.

The core assumption is that controllers’ subjective perceptions of arrival traffic complexity are reflected in controllers’ decision-making behaviours, which in turn reflected in the evolution of arrival traffic patterns. Based on this, we quantify subjective complexity by analysing how these patterns change over time.

This assumption is not overly strong, as controllers’ behaviours are closely tied to their situation awareness [22]. In fact, it aligns well with real-world observations: when controllers perceive low traffic complexity, they tend to allow aircraft to follow economical profiles. Conversely, under high perceived complexity, controllers often require adherence to standard procedures, and may introduce

additional vectoring or even holding patterns to maintain safe separation.

The methodology framework, shown in figure 1, comprises four components: data source, data processing, collective dynamics analysis, and complexity measurement. Each component is essential for developing a comprehensive understanding of subjective complexity:

1. The data source is the historical radar data. This dataset provides comprehensive information on arrival aircraft trajectories, including latitude, longitude, altitude, and timestamps. Radar data offers detailed insights into real time arrival traffic operations, making it possible to analyse the evolution of arrival traffic patterns. The radar data utilised in this study covers the period from January 1, 2019 to January 31, 2019.

2. The data processing is detailed in section III. This component, including trajectory resampling and smoothing, aims to obtain a smooth representation of trajectories. As the curvature calculation performed in a subsequent step requires the second derivative of a trajectory, it is highly sensitive to noise. Thus, reducing the noise in trajectory can support the following computations.

3. The collective dynamics analysis of arrival traffic is detailed in section IV. This component aims to describe the collective dynamics of arrival traffic both horizontally and vertically. As for the evolution of horizontal dynamics, the curvature is adopted and further calibrated to quantify the number of aircraft in turn over time. As for the evolution of vertical dynamics, the Kernel Density Estimation (KDE) is applied to count the number of aircraft in level flight over time.

4. The complexity measurement is detailed in the case study section (section V), as this component is based on the previously mentioned assumption that complexity can be reflected in the evolution of arrival traffic patterns. This assumption should be validated through a coupled analysis prior to its application in the case study. Then, this component measures the complexity according to the arrival traffic pattern evolution.

The case study focusing on arrival traffic operations within the terminal area of Guangzhou Baiyun airport is conducted to validate the proposed metric. The reason for selecting this case for analysis is that Guangzhou Airport experiences a high traffic volume, with significant pressure on arrival operations. Additionally, the impact of departing flights on arriving flights is relatively low.

## III. DATA SMOOTHING

Preprocessing the data by reducing noise and bias facilitates more accurate and meaningful analysis. This section first frames the data smoothing problem as a minimisation problem. It then solves the problem by training a neural network using a first-order gradient-based optimiser.
<figure><img src="image_2.png" /><figcaption>Figure 1: Methodology framework for learning complexity values.</figcaption></figure>

The heading of trajectories often exhibits high-frequency fluctuations, despite the trajectory itself appearing smooth. Such noise is commonly observed when calculating the direction of movement between consecutive points. Therefore, it is necessary to smooth the trajectory to ensure a refined heading before proceeding with further research.

To prevent the deformation of trajectory contours caused by Runge's phenomenon [23] while smoothing the data, this paper proposes a contour-invariant trajectory heading smoothing technique, which aims to:

1) minimise the heading change of trajectory;

2) maintain the trajectory's contour.

The above mentioned two targets can be written as follows:

$$ \underset{\forall \hat{x}_i, \hat{y}_i}{\operatorname{argmin}} \left\{ \sum_{i=1}^{N} \left[ (x_i - \hat{x}_i)^2 + (y_i - \hat{y}_i)^2 \right] + \lambda \sum_{i=3}^{N-2} |\hat{\theta}_i - \hat{\theta}_{i-1}| \right\} \quad (1) $$

$$ \hat{\theta}_i = \arctan \frac{\hat{x}_{i+1} - \hat{x}_i}{\hat{y}_{i+1} - \hat{y}_i} \quad (2) $$

where $x_i$ and $y_i$ represent the real x-coordinate, y-coordinate, and heading of the $i^{-th}$ trajectory point. $\hat{x}_i$, $\hat{y}_i$, and $\hat{\theta}_i$ represent the estimated x-coordinate, y-coordinate, and heading of the $i^{-th}$ trajectory point, $\lambda$ is the weight parameter.

So far, we formulate the data smoothing as a minimisation problem. The rest of this section details how to solve this minimisation problem.

Firstly, this section introduces the assumption:

**Assumption 1.** The distance between two adjacent trajectory points is c (a constant value):

$$ \left\{ \begin{array}{l} dx = c \cdot \sin \theta \\ dy = c \cdot \cos \theta \end{array} \right. \quad (3) $$

The assumption 1 can be realised by resampling the trajectories, which will have no regular time sampling (we

can not have both regular spatial sampling and regular time sampling).

The loss function mentioned in equation (1) can be denoted as $f$. Then, the partial derivative of $f$ with respect to $\hat{x}_i$ is given by:

$$ \begin{aligned} \frac{\partial f}{\partial \hat{x}_i} = & 2(\hat{x}_i - x_i) \\ & - \lambda \left[ \operatorname{sign}\left(\frac{\alpha_i}{\beta_i}\right) - \operatorname{sign}\left(\frac{\alpha_{i-1}}{\beta_{i-1}}\right) \right] (\hat{y}_i - \hat{y}_{i-1}) \quad (4) \\ & + \lambda \left[ \operatorname{sign}\left(\frac{\alpha_{i+1}}{\beta_{i+1}}\right) - \operatorname{sign}\left(\frac{\alpha_i}{\beta_i}\right) \right] (\hat{y}_{i+1} - \hat{y}_i) \end{aligned} $$

$$ \left\{ \begin{array}{l} \alpha_i = (\hat{x}_{i+1} - \hat{x}_i)(\hat{y}_i - \hat{y}_{i-1}) - (\hat{y}_{i+1} - \hat{y}_i)(\hat{x}_i - \hat{x}_{i-1}) \\ \beta_i = (\hat{y}_{i+1} - \hat{y}_i)(\hat{y}_i - \hat{y}_{i-1}) + (\hat{x}_{i+1} - \hat{x}_i)(\hat{x}_i - \hat{x}_{i-1}) \end{array} \right. \quad (5) $$

where $\operatorname{sign}(\cdot)$ denotes the sign function. Due to space limitations, the proof of equation (4) has been omitted.

Analogously, the partial derivative of $f$ with respect to $\hat{y}_i$ is given by:

$$ \begin{aligned} \frac{\partial f}{\partial \hat{y}_i} = & 2(\hat{y}_i - y_i) \\ & + \lambda \left[ \operatorname{sign}\left(\frac{\alpha_i}{\beta_i}\right) - \operatorname{sign}\left(\frac{\alpha_{i-1}}{\beta_{i-1}}\right) \right] (\hat{x}_i - \hat{x}_{i-1}) \quad (6) \\ & - \lambda \left[ \operatorname{sign}\left(\frac{\alpha_{i+1}}{\beta_{i+1}}\right) - \operatorname{sign}\left(\frac{\alpha_i}{\beta_i}\right) \right] (\hat{x}_{i+1} - \hat{x}_i) \end{aligned} $$

where the proof of equation (6) is omitted due to space limitations.

Leveraging equations (4) and (6), the gradient information of the loss function $f$ is determined. Besides, $f$ is convex since it can be expressed as a sum of a series of convex functions. Therefore, we can exploit first-order gradient-based optimisation algorithms, such as Adam [24] or stochastic gradient descent [25], to solve the problem. Due to the favourable properties of the loss function, convergence can always be ensured.

Finally, this paper solve the minimisation problem by training a neural network model via Adam on Java. As
shown in figure 2, the model is quite simple since it only includes a linear layer. The pseudo code of the training process is given in algorithm 1. Please note that:

1) The raw trajectory is represented by a series of headings instead of a series of trajectory points. Thus, this paper does not need to specify the map projection method, thereby enhancing the general- ity of the proposed method. Representing a horizon- tal trajectory via heading still preserves the contour of the trajectory since the distance between two adjacent points is constant (assumption 1).

2) The trainable parameters can be initialized using the raw trajectory data. Consequently, even if the model has not been trained, it can output the original trajectory. This initialization significantly reduces the training time, as the loss value starts at an already low level.

**Algorithm 1: Trajectory smoothing process**

Input: Raw trajectory heading {$\theta_i | i = 1, 2, \dots, N$}

Output: Smoothed heading {$\theta_i | i = 1, 2, \dots, N$}

$x_i \leftarrow \sum_{t=0}^{i} \cos \theta_t$; $y_i \leftarrow \sum_{t=0}^{i} \cos \theta_t$;

$T \leftarrow [x_1, \dots, x_N, y_1, \dots, y_N]$ // infer trajectory;

$\Phi \leftarrow [x_1, \dots, x_N, y_1, \dots, y_N]$ // initialise ;

for $i \leftarrow 1$ to E do

Forward pass: $\hat{T} \leftarrow [1] \times \Phi$;

Compute loss based on equation (1);

Compute gradients based on equations (4), (6);

Update $\Phi$;

Compute smoothed heading based on equation (2);

return {$\theta_i | i = 1, 2, \dots, N$};

The code for smoothing trajectory headings has been uploaded to GitHub: https://github.com/ShowhowGui/ ToolHeadingSmoothing.git, in which the learning rate, training epochs, and $\lambda$ are $1e^{-3}$, 500, and 1, respectively. The determination of learning rate, training epochs, and $\lambda$ is based on empirical tuning and experimentation. It is worth noting that $\lambda$ depends on the units of the reconstruction error (the first term in Equation (1)) and the smoothness term (the second term in Equation (1)).

IV. COLLECTIVE DYNAMICS ANALYSIS

A. Horizontal dynamics analysis

Although a trajectory is displayed on the screen as a series of 2D points, it is actually executed by the aircraft through the control of heading and speed [26]. Delahaye et al. [27] emphasize the significance of heading change and asserts that curvature, which is directly related to heading change, is of primary importance in air traffic management studies. Thus, this paper aims to analysis the horizontal dynamics of arrival traffic based on the heading of trajectories.

A trajectory primarily consists of straight lines and arcs. Curvature [28], which measures how quickly a curve turns, presents itself as a promising metric for detecting

changes in an aircraft's flight state. The general formula of curvature is as follows:

$$K(t) = \frac{\|\gamma'(t) \times \gamma''(t)\|}{\|\gamma'(t)\|^3} \quad (7)$$

where $\times$ is the cross product, and $\gamma$ is the trajectory.

However, the curvature value does not directly corre- spond to the aircraft's turning rate, as it also depends on aircraft's speed. As illustrated in Figure 3, when two aircraft maintain the same turning rate but differ in speed, their respective curvatures vary accordingly. Thus, eliminating the influence of speed is crucial when using curvature to measure complexity, especially in terminal area operations, where aircraft speed often changes con- tinuously and significantly.

This paper calibrates the curvature based on the force analysis of the aircraft. According to figure 4, we have:

$$\tan \omega = \frac{\frac{v^2}{r}}{mg} = \frac{v^2}{g} \cdot \frac{1}{r} \quad (8)$$

where $\omega$ is the bank-angle, $m$ is the aircraft mass, $g$ is the gravitational acceleration, $v$ denotes the speed, and $r$ is the turning radius.

The curvature is equals to the reciprocal of turning radius [28]. So, the curvature can be calculated as follows :

$$K = \frac{1}{r} \quad (9)$$

Combining equations (8) and (9), we have:

$$\omega = \arctan \left( \frac{v^2}{g} K \right) \quad (10)$$

The bank-angle $\omega$, calculated based on the curvature $K$ and speed $v$, can be interpreted as the calibration of the original $K$ to account for $v$. Since the bank-angle preserves the properties of the curvature while eliminating the negative influence of the speed change, it can be used to analysis the turning behaviour of an arrival aircraft within the terminal area.

This paper defines that an aircraft is in turn if its bank angle (calibrated curvature) is greater than 10 degree. By aggregating the turning data of all arrival aircraft, we can gain insights into the horizontal dynamics of the arrival traffic. This process is detailed in the subsection V-B.

B. Vertical dynamics analysis

This subsection details the KDE [29] based approach to obtaining the vertical dynamics of arrival traffic.

KDE, which is a non-parametric method, is used to estimate the probability density function of aircraft's altitude in this section. Unlike traditional histograms, KDE does not rely on fixed binning but instead smooths each data point's influence using a kernel function, such as a Gaussian kernel, to produce a continuous density
<figure><img src="image_3.png" /><figcaption>Figure 2: The structure of trajectory smoothing model. The input data is an all-ones matrix $J_{1×1}$, which is multiplied (via matrix multiplication) with the training vector $\Phi_{1×2N}$ to produce the output data. Consequently, training the model will smooth the original trajectory.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: The curvature difference caused by the speed difference. Even if two aircraft have the same turning rate, the aircraft with lower speed exhibits a higher curvature. Such a situation often happens in terminal area.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: The force analysis of an aircraft.</figcaption></figure>

curve. This approach provides a more refined and intuitive density estimation, especially when the underlying distribution is unknown.

The formula of univariate KDE (Gaussians), used in this paper, is as follows:

$$ \hat{f}(x, h) = \frac{1}{n} \sum_{i=1}^{N} \mathcal{N}(x; X_i, h^2) \quad (11) $$

where $\mathcal{N}(x; \mu, \sigma^2)$ denotes the normal distribution (kernel function), $X_i$ is the $i$-th sample, $h$ is the bandwidth which controls the width of the kernel function, and $N$ is the number of samples.

The resulting KDE-based process for detecting level flight is shown in figure 5. The sub-figure in the left bottom corner displays a trajectory's vertical profile, and the sub-figure in the right bottom corner displays the density of the aircraft's altitude. When an aircraft is in level flight at a certain altitude, the corresponding density at that altitude increases, as the aircraft stays there for an extended period, resulting in a peak in the altitude density distribution.

There is a relationship between the altitude distribution and whether the aircraft implements a continuous descent operation. The more concentrated the distribution, the less likely the aircraft is to perform a continuous descent operation; conversely, the more dispersed the distribution, the more likely the aircraft is to engage in a continuous descent operation. The bandwidth $h$, an empirical parameter, is set to be 20 in this paper.

As shown in the sub-figure in the upper left of figure 5, based on the density estimation of an aircraft's altitude, we can infer whether the aircraft is in a level flight state at a given moment. For instance, if the density of the aircraft's altitude at a particular time is exceptionally high, we can conclude that the aircraft is in a level flight state at that moment.

By aggregating the level flight data of all arriving aircraft, we can gain insights into the vertical dynamics
<figure><img src="image_6.png" /><figcaption>Figure 5: KDE-based level flight detection. If an aircraft maintains level flight at a certain altitude for some time, that altitude exhibits higher density.</figcaption></figure>

of the arrival traffic. This process is detailed in subsection V-B.

# V. CASE STUDY

## A. Data smoothing results comparison

The data smoothing results are given in figure 6. The left column presents the bank angle measurements based on cubic spline interpolation [30]; and the right column presents the bank angle measurements based on the proposed minimisation-based heading smoothing method.

In the horizontal profile, the smoothed trajectory in figure 6 (a) appears smoother, albeit at the cost of a slight change in the trajectory's contour. In contrast, the smoothed trajectory in figure 6 (b) preserves the contour while smoothing, though it is not as smooth as the former. Although those two perform differently, both results are acceptable from the perspective of horizontal profile.

In the heading profile, the smoothed trajectory in figure 6 (c) exhibits both under-fitting (from 0 to 700 seconds) and Runge's phenomenon (from 1250 to 1500 seconds). In contract, the smoothed trajectory in figure 6 (d) not only gets rid of noise but also better conforms to the stepwise changes. As heading changes are more representative of a stepwise pattern, the smoothed trajectory aligns closely with real air traffic operations. In fact, the proposed smoothing method can also be applied to smooth other stepwise functions; however, since it is not relevant to the theme of this paper, we will not elaborate further.

In the bank angle profile, the result in figure 6 (e) exhibits noticeable distortion. Although the spikes in the original data are eliminated, the peaks after smoothing are significantly lower than the original ones. Since both noise and valuable information are reduced simultaneously, the signal-to-noise ratio of the data has not im-

proved, leading to the persistence of considerable noise in subsequent analyses. In contract, the result in figure 6 (f) not only eliminates most of the spikes but also preserves valuable information, significantly improving the signal-to-noise ratio of the data.

## B. Arrival traffic operational pattern evolution

This subsection aims to summary the patterns of arrival traffic operations through a coupled analysis and validate the assumption that the complexity can be reflected in the evolution of traffic patterns.

According to the horizontal dynamics analysis mentioned in subsection IV-A, one can gain insights into the arrival traffic operations in horizontal perspective by aggregating the turning information of all arrival aircraft. The evolution of the horizontal dynamics of arrival traffic over time is illustrated in figure 7 (a), which provides the estimated number of aircraft in a turning state at each time step.

According to the vertical dynamics analysis mentioned in subsection IV-B, one can gain insights into the arrival traffic operations in vertical perspective by aggregating the level flight information of all arrival aircraft. The evolution of the vertical dynamics of arrival traffic over time is illustrated in figure 7 (b), which provides the estimated number of aircraft in a level flight state at each moment.

Then, a coupled analysis is conducted to reveal the dependency between the horizontal and vertical collective dynamics of arrival traffic. As depicted in figure 7 (c), this paper integrates figures 7 (a) and (b). The curve in figure 7 (c) can be viewed as the trajectory of a particle moving over a two-dimensional plane, where the horizontal axis represents the number of aircraft in turn, and the vertical axis represents the number of aircraft in level flight. Although the particle in figure 7 (c) moves randomly, its motion exhibits transitional behaviour. At the beginning, the particle moves around the bottom-left corner. During this period, it momentarily jumps to the upper-right corner (transition I) but swiftly returns (transition II). After the third transition (transition III) to the upper-right corner, the particle continues to move around the upper-right area and does not return again.

According to the figure 7, we found a very interesting transitional phenomenon. If this phenomenon is widespread, we can classify arrival traffic operations into distinct patterns and use these patterns as a basis for measuring complexity. The remainder of this subsection validates the prevalence of this transitional phenomenon and quantitatively evaluates the transition boundary of the particles.

Firstly, this paper collects the operational data of arriving flights at Guangzhou Baiyun airport from January 1, 2019, to January 7, 2019, resulting in a total of 120,961 data points. Then, this paper analysed the arrival traffic's collective dynamics horizontally and vertically to obtain the movement of the "particle". The distribution of the particle is illustrated in figure 8,
<figure><img src="image_7.png" /><figcaption>(a) Horizontal profile (Cubic spline).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) Heading (Cubic spline).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(e) Bank angle (Cubic spline).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b) Horizontal profile (Minimisation).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(d) Heading (Minimisation).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(f) Bank angle (Minimisation).</figcaption></figure>

Figure 6: Results of different data smoothing methods. The proposed method outperforms the cubic spline interpolation approach in smoothing aircraft's heading (a stepwise function) while preserving the contour of the horizontal profile, thereby preventing the distortion of bank angle estimations. The shaded areas indicate the same time period, allowing for the bank angle value to be related to the turn in the horizontal profile.

As shown in figure 8, there are two peaks in the particle distribution. The first peak occurs near the origin ((0, 0), indicating fewer aircraft engaging in turning manoeuvrings and maintaining level flight. This peak represents periods during which arrival traffic operations remain low in complexity. The second peak occurs near the (4, 4), indicating an average of four aircraft engaging in turning manoeuvrings and maintaining level flight. This peak represents periods during which arrival traffic operations remain high in complexity.

Actually, the distribution in figure 8 can be approximated as a combination of two sub-distributions, suggesting that the movement of the particle, following different sub-distributions, exhibits distinct characteristics. The

movement of the particle from one sub-distribution to the other can be viewed as a transition.

Based on the two identified peaks in figure 8, this paper divided the distribution into three regions via two dashed lines as shown in figure 8. The first region is in the bottom left, representing periods with low complexity; the second region is in the middle, representing periods with higher complexity but the operations still exhibit a certain degree of regularity; the third region is in the upper right, representing periods with super high complexity and the operations become chaotic, without regular patterns.

In the rest of this subsection, to validate the identified transition boundaries, this paper conducted eight sets of experiments based on the arrival traffic operations in the
<figure><img src="image_13.png" /><figcaption>(a) The number of aircraft in turn over time.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(b) The number of aircraft in level flight over time.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(c) The curve describing vertical and horizontal dynamics.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 8: The distribution of the particle movement. The distribution can be approximated as a combination of two sub-distributions, indicating that the movement of the particle exhibits different characteristics. The complex situations are located as far as possible from the origin, in the upper-right corner. The transition of the particle from one sub-distribution to the other can be viewed as a transition.</figcaption></figure>

terminal area of Guangzhou Baiyun airport on January 30, 2019. The validation results are shown in figure 9.

According to figure 9 (a)~(b), when the number of arrival flight within an hour increase from 10 to 21, the particle moves within the first region and the centre of movement of the particle gradually moves towards upper right. According to figure 9 (c)~(d), the number of arrival flight within an hour increase from 21 to 45, the particle smoothly moves from the first region to the second region and stays dynamically stable within this region.

According to figure 9 (e)~(h), when the number of arrival flights within an hour is larger than 45, the particle is more likely to cross the second boundary. Once the

particle cross the second boundary, its dynamic stability is reduced and its movement is more random and close to the upper right corner.

Note that the number of aircraft is not completely correlated with the probability of exceeding the transition boundary. For example, figure 9 (f) has fewer arrival flights than figure 9 (e), but the particle in figure 9 (f) exceeds the transition boundary much more frequently.

According to the transition boundaries, as shown in figure 9, we infer that there are three types of arrival traffic operation patterns within Guangzhou Baiyun airport terminal area.

1) **Low complexity**: corresponds to the first region (the green region in figure 9). Under this level, the arrival traffic mainly focuses on economic operations;

2) **High complexity**: corresponds to the second region (the blue region in figure 9). Under this level, the arrival traffic mainly focuses on efficiency operation (make as much airspace resources as possible with the help of automation facilities and equipment).

3) **Super complexity**: corresponds to the third region (the red region in figure 9). Under this level, the arrival traffic becomes highly complex, requiring controllers to implement more temporary intervention measures (e.g. radar vectoring, holdings), to manage the air traffic. Such complexity could exacerbate disorder in air traffic operations and significantly increase the workload for air traffic controllers.

### C. Complexity measurement

This subsection applies the proposed method in real air traffic operations.

The complexity results of the proposed metric is shown in figure 10 (a). The complexity trend derived from the
<figure><img src="image_17.png" /><figcaption>Figure 9: The movement of a particle within an hour on January 30, 2019.</figcaption></figure>

metric exhibits a similar overall profile with the number of arrival flights. For instance, from 3 AM to 10 AM (there are a few arrival flights within this period), the complexity remains at a low level, while at other times, it tends to be at a higher level.

The remainder of this subsection examines four distinct periods (A, B, C, and D) to assess whether the proposed metric is able to effectively capture the complexity of air traffic operations.

1) In period A (from 5:00 to 5:30), the proposed metric claims that the complexity of arrival traffic is high. This complexity result may appear counter-intuitive, as one would typically expect fewer arrival flights during the early morning hours, but this is not the case. As shown in figure 10 (b), only one runway was available at that time, and there are fourteen arrival aircraft. Consequently, controllers had to instruct some aircraft to perform additional manoeuvres, such as dog-leg and short-cut. In other words, the proposed accurately recognise the high-complexity arrival traffic operation. Besides, it is important to note that the turns inherent in standard terminal arrival routes (STARs) are unlikely to significantly affect the complexity results, as aircraft following STARs are less likely to execute turns (or level flights) at the same time.

2) In period B (from 6:15 to 6:45), the proposed metric claims that the complexity of arrival traffic is low. As shown in figure 10 (c), there are only eight arrival aircraft while more than one runways are available. Consequently, most of arrival can fly along their planned flight routes, resulting in a low level complexity of the arrival traffic operations.

3) In period C (from 10:15 to 10:45), the proposed metric claims that the complexity of arrival traffic is super high. As shown in figure 10 (d), there

are at least two trajectories have holding patterns, indicating that the demand is closing to the capacity and some aircraft have to wait for a while before landing. The proposed metric accurately recognises this super high-complexity arrival traffic operation.

4) In period D (from 19:15 to 19:45), the proposed metric claims that the complexity of arrival traffic is high. Although the number of arrival aircraft in period D is close to that in period C, the trajectories in figure 10 (e) are more neat than that in figure 10 (d). Therefore, the complexity of period D should be higher than that of period C, and the proposed metric exhibits consistency.

The case study demonstrates that the proposed complexity measurement method is able to capture the complexity of arrival traffic operations.

## VI. CONCLUSION

This paper proposes a subjective complexity measurement method based on arrival traffic pattern evolution. Unlike previous approaches that rely on controllers' self-assessment, the proposed method infers subjective complexity indirectly by analysing the evolution of arrival traffic patterns, which reflect controllers' behaviour during air traffic management.

Firstly, a contour-invariant data smoothing approach is proposed to smooth the trajectory. Compared with the cubic spline smoothing approach, the proposed method preserves more helpful information while reducing noise. Secondly, a curvature-based manoeuvring detection method and a KDE-based level-flight detection method are proposed to measure the dynamics of arrival traffic. Thirdly, a case study is conducted to summarize the patterns of arrival traffic operations within Guangzhou Baiyun airport through a coupled analysis and measure the complexity of arrival traffic operations based on the evolution of these operational patterns.
<figure><img src="image_18.png" /><figcaption>(a) Hourly trend in complexity.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) Arrival traffic in period A.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(c) Arrival traffic in period B.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(d) Arrival traffic in period C.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(e) Arrival traffic in period D.</figcaption></figure>

<caption>Figure 10: Applying the proposed metric to real air traffic operations.</caption>

There are two drawbacks in this paper:

1) The proposed method relies on analysing air traffic controllers' decision-making behaviours to assess their perception of complexity in relation to the current air traffic situation. However, it has not yet been empirically validated through direct feedback from operational controllers.

2) The proposed metric may not be ideal for assessing departure operations, as controllers typically rely on pre-departure sequencing and scheduling, which makes level-offs and vectoring less common in departure trajectories.

In future work, we plan to gather subjective assessments from controllers during real operational settings to further validate and refine the proposed approach.

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