# Advanced collision risk estimation in terminal manoeuvring areas using a disentangled variational autoencoder for uncertainty quantification

Timothé Krauth, Jérôme Morio, Xavier Olive, Benoit Figuet

► To cite this version:

Timothé Krauth, Jérôme Morio, Xavier Olive, Benoit Figuet. Advanced collision risk estimation in terminal manoeuvring areas using a disentangled variational autoencoder for uncertainty quantification. Engineering Applications of Artificial Intelligence, 2024, 133, pp.108137. 10.1016/j.engappai.2024.108137. hal-04633480

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Research paper

# Advanced collision risk estimation in terminal manoeuvring areas using a disentangled variational autoencoder for uncertainty quantification

Timothé Krauth<sup>a,b,*</sup>, Jérôme Morio<sup>b</sup>, Xavier Olive<sup>b</sup>, Benoit Figuet<sup>a</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

- Air traffic management
- Rare event probability estimation
- Deep generative models
- Variational autoencoder
- Uncertainty quantification

## ABSTRACT

Air Traffic Management aims at ensuring safety during aircraft operations, particularly within Terminal Manoeuvring Areas where traffic density is high. The challenge lies in balancing safety and efficiency by closely managing the likelihood of mid-air collisions regarding the airport movements. Traditional models like the Reich and Anderson-Hsu models have been influential, but they fall short in representing the complex reality of Terminal Manoeuvring Areas. Data-driven approaches are emerging, with Monte Carlo simulations offering a more flexible methodology for collision risk estimation. This paper introduces a framework for assessing Mid-Air Collision likelihood resulting from Terminal Manoeuvring Area procedures by combining the field of Deep Generative Modelling using a Variational Autoencoder with the domain of rare event statistics through Subset Simulation. By incorporating disentanglement into the Variational Autoencoders model, we create a latent space that aligns dimensions with distinctive trajectory traits. Then, Subset Simulation is employed to gauge Mid-Air Collision probability, utilizing latent representations as input. Finally, sensitivity analysis reveals pivotal factors influencing collision risk, correlated with trajectory attributes via disentanglement. The methodology is applied to traffic around Zurich Airport: it evaluates the risk arising from go-around and take-off procedures using Automatic Dependent Surveillance-Broadcast data.

## 1. Introduction

Air Traffic Management (ATM) aims at ensuring safety during aircraft operations, e.g. within Terminal Manoeuvring Areas (TMAs) where the traffic density is particularly high. These areas located around airports are crucial as planes perform complex manoeuvres like takeoffs, landings, and go-arounds. Civil Aviation Authorities (CAA) and Air Navigation Service Providers (ANSP) face the challenge of balancing safety, as defined by the Target Level of Safety (TLS), with the need for efficient airport movements. This involves carefully managing the likelihood of mid-air collisions (MAC) by making informed operational decisions for various flight procedures.

The *Reich model* (Reich, 1966) and the *Anderson-Hsu model* inspired by Anderson and Lin (1996) and Hsu (1981) have been influential in advancing collision risk assessment. They both assume air traffic along predefined routes, and account for uncertainty related to aircraft positions and speeds to quantify the likelihood of a MAC occurring. While these analytical models are effective for en-route scenarios with well-structured traffic flows, they often fall short in addressing the complexity of traffic patterns typically found in TMAs, where aircraft experience frequent variations in altitude, speed, and heading (Netjasov

and Janic, 2008). With the increasing availability of flight data, data-driven methods have begun to emerge to fill in the gaps in analytical models (Henry et al., 2013; Blom et al., 2006; Morio et al., 2012). Aircraft behaviours and uncertainty sources are inherently present within the underlying distribution of the data, which allows relaxing a multitude of simplifying assumptions. To estimate the risk of MAC in a data-driven manner, Monte Carlo simulations appear as one of the most appropriate frameworks to monitor the probability of MAC occurrences. However, they suffer from a number of shortcomings: 1. the computational cost to estimate rare event probabilities can be prohibiting; 2. considering the volume of simulations needed, the available pool of observed trajectories could fall short. Gathering a million take-off trajectories at a mid-size European airport necessitates a period of over 20 years (Krauth et al., 2023a). This extended time frame is compounded by the fact that procedures undergo modifications throughout this duration.

To tackle these difficulties, Krauth et al. (2023a) demonstrated that the Variational Autoencoder (VAE) developed in Krauth et al. (2023b) can be used to estimate the underlying distribution of observed trajectories in TMAs, which makes it possible to apply Monte Carlo

<footnote><sup>a</sup> Zurich University of Applied Sciences, Centre for Aviation, Winterthur, Switzerland</footnote>

<footnote><sup>b</sup> ONERA/DTIS, Université de Toulouse, F-31055 Toulouse, France</footnote>

<footnote>* Correspondence to: Technikumstrasse 71, 8400 Winterthur, Switzerland. E-mail address: timothe.krauth@zhaw.ch (T. Krauth).</footnote>
Abbreviations

<table><tr><td>ADS-B</td><td>Automatic Dependent Surveillance-Broadcast</td></tr><tr><td>ANSP</td><td>Air Navigation Service Providers</td></tr><tr><td>ATC</td><td>Air Traffic Controller</td></tr><tr><td>ATM</td><td>Air Traffic Management</td></tr><tr><td>CAA</td><td>Civil Aviation Authorities</td></tr><tr><td>CRM</td><td>Collision Risk Model</td></tr><tr><td>DGM</td><td>Deep Generative Modelling</td></tr><tr><td>GAN</td><td>Generative Adversarial Network</td></tr><tr><td>LSF</td><td>Limit State Function</td></tr><tr><td>MAC</td><td>Mid-Air-Collision</td></tr><tr><td>PCA</td><td>Principal Component Analysis</td></tr><tr><td>TLS</td><td>Target Level of Safety</td></tr><tr><td>TMA</td><td>Terminal Manoeuvring Area</td></tr><tr><td>VAE</td><td>Variational Autoencoder</td></tr><tr><td>UQ</td><td>Uncertainty Quantification</td></tr></table>

simulation on samples from the estimated distribution rather than observed trajectories. This allows for obtaining an accurate collision risk estimate solely based on recent observed trajectories. The method yields satisfactory results, but it also operates as a black-box, making it challenging to discern the specific factors that contribute to an increased collision risk. Furthermore, basic Monte Carlo simulations prove inefficient for exceedingly low probabilities.

The aim of this paper is to establish a method for assessing the likelihood of MAC resulting from TMA procedure design, independently of other safety measures, by combining the frameworks of VAE and Uncertainty Quantification (UQ). Additionally, it facilitates the identification of key factors that significantly impact collision risk. We approach the research objective through four key points:

1. representing the intricate high-dimensional distribution of aircraft trajectories by a lower-dimensional distribution through the VAE latent space to enable the use of UQ,

2. incorporating disentanglement into the VAE model inspired by Chen et al. (2018), constructing a latent space where each dimension aligns with a distinctive trait of the modelled trajectories,

3. employing Subset Simulation, we gauge the probability of MAC. This involves utilizing the latent representations of trajectories within the VAE's latent space as input,

4. utilizing First-Order Sobol' Indices, we perform sensitivity analysis to discern the pivotal factors influencing collision risk. These factors are subsequently correlated with trajectory attributes by leveraging the disentanglement approach.

We implement the methodology to calculate the risk arising from the design of both the go-around procedure for runway 14 and the take-off procedure for runway 16 at Zurich Airport, by using Automatic Dependent Surveillance-Broadcast (ADS-B) data from the OpenSky Network (Schäfer et al., 2014). The code corresponding to this study can be found on the following Github repository.<fnref n="1" />

The reminder of this paper is structured as follows: in Section 2 the literature about data-driven Collision Risk Models. Section 3 offers foundational insights into disentangled VAEs and its combined used in an Uncertainty Quantification framework. The case study and the approach employed are exposed in Section 4. Section 5 lays out the acquired results, while Section 6 contextualizes them, highlights their

significance, and addresses the model's limitations. Concluding the paper, Section 7 encapsulates the findings and outlines potential future research works.

## 2. Literature review

This section brings together the literature relevant for the paper. Section 2.1 sets out the various methods that can be used to quantify the risk of collision between aircraft, whereas Section 2.2 explains how the distribution of aircraft trajectories can be estimated.

### 2.1. Collision Risk Models

Collision Risk Models (CRM) were first developed in the 60's to assess the probability of collision between aircraft in regions with low navigation surveillance; typically trans-oceanic traffic. The first widely used analytic CRM model is the Reich model (Reich, 1966), which suppose that the nominal trajectories of each aircraft are known and subject to uncertainty on position and speed. Moreover, it considers that blunders, deviations, equipment failures and operational errors do occur, and that communication and surveillance might fail. By modelling each of the sources of uncertainty by a probability density, the risk of collision is obtained by calculating the probability of overlap of the aircraft collision volumes. The Reich model was mainly designed for parallel airways, with constant speed and altitude, and with a non-time dependent error. Later on, the Anderson-Hsu model (Anderson and Lin, 1996) supplements the Reich's model by allowing for crossing flight tracks with a time-dependant error. Fujita (2013) extended those frameworks by getting rid of the simplifying assumptions of constant altitude and heading.

The three above-mentioned analytic models are mainly designed for en-route scenarios, and rely on a great number of assumptions. Alternatively to analytic CRMs, it exists Monte Carlo simulation-based CRMs. Netjasov and Janic (2008) state that these data-driven approaches capitalize on the inherent presence of sources of uncertainty in aircraft behaviour within the underlying data distribution. This allows to relax numerous modelling decisions and restrictive assumptions as there is no need to identify and model uncertainty sources separately. The work presented in Blom et al. (2006) illustrates the advantageous synergy of employing both a simulation model of air traffic and Monte Carlo methods for risk estimation. This approach facilitates the separation of the specific case study from the risk estimation methodology itself, enabling the application of the modelling technique to a wide array of scenarios. The approach from Eckstein (2010), and refined by Henry et al. (2013) uses the Principal Component Analysis (PCA) to simulate rectilinear trajectories to determine the probability of collision in intersecting runway configuration. Morio et al. (2012) estimate the risk of collision in uncontrolled airspace by using the importance splitting algorithm on randomly sampled observed trajectories. The method enables to deal effectively with the problem of rare events, but the study relies on 170 observed trajectories, which is probably too few to simulate such diversified air traffic. Jacquemart and Morio (2013) use Markov chains to simulate en-route trajectories with uncertainty, having constant altitude and speed. Wang et al. (2020) use as inputs for crude Monte Carlo simulations a three-dimensional path prediction for unmanned air traffic in order to assess the collision risk between drones and aircraft operation in the TMA of Singapore Changi airport. In their study, Figuet et al. (2023b) employed Monte-Carlo simulations in combination with extreme value theory to estimate collision probabilities. Their approach involves extrapolating the tail of the distribution that characterizes the spacing between two aircraft; however, it should be acknowledged that extrapolation has inherent limitations that may restrict its applicability.

Compared to traditional analytical methods, Monte Carlo-based CRMs do not require the identification of all sources of uncertainty, nor to make simplifying modelling choices. Monte Carlo methods can be

<footnote n="1">¹ https://github.com/kruuZHAW/MAC_estimation_disentangled_VAE.</footnote>
used directly on observed trajectories, but depending on the magnitude of the probability to be estimated, the amount of data available may be too small (Morio et al., 2012). As seen in above, one approach to circumvent this issue is by constructing a model for the distribution of observed trajectories. Generating a synthetic trajectory that is identically distributed can be as straightforward as drawing samples from a recognized statistical distribution. However, accurately estimating a high-dimensional distribution that characterizes air traffic poses significant challenges, due to the curse of high dimensions. In the subsequent section, we delve into the body of literature that addresses techniques for estimating trajectory distributions in the context of air traffic flows.

## 2.2. Density estimation for aircraft trajectories

Historically, aircraft trajectory generation methods relied on flight dynamic equations. Such models can be traced back as early as 1982 with Quanbeck (1982), and have been improved with aircraft performance models (Nuic et al., 2010). However, in practice, obtaining all the necessary information about an aircraft and its environment can be challenging. Many of these variables are either hidden, subject to sudden changes, or not publicly available. In recent years, researchers have predominantly focused on developing data-driven trajectory generation methods. Considering the trajectory generation as an optimization problem appears to be a widely used solution (Soler et al., 2010; Patel and Goulart, 2011; Kamgarpour et al., 2010; Koyuncu et al., 2016; Delahaye et al., 2014). These methods are efficient at generating a particular trajectory subject to precise constraints. However, they do not give an expression of the distribution of observed trajectories, which makes their use in UQ impossible.

Air traffic flows are described by high-dimensional statistical distributions, where variability can be high due to air traffic control, weather or even human error. In the literature, many papers have focused on estimating these distributions with the aim of improving ATM. Henry et al. (2013) reduce the complexity of the problem by using a PCA to simulate altitude and ground speed profiles for strictly rectilinear trajectories. Although dimensionality reduction is an interesting idea, PCA does not offer the statistical guarantees needed to present a reliable estimation method. Morio et al. (2012) approximate the distribution of flight traffic in uncontrolled airspace by its empirical distribution based on observed trajectories. However, this method requires a large number of observations and is applied to a limited database of only 170 samples. Jacquemart and Morio (2013) model air traffic flows with stochastic processes such as Markov chains. It enables to model statistically the deviations from the nominal straight-line trajectory caused by wind, position tracking, navigation, or control by estimating the transition between two successive positions.

Further research has been carried out to apply classical parametric density estimation methods to air flows. For instance, Murça and de Oliveira (2020) applied a Gaussian mixture model to represent the arrival trajectories at São Paulo airport. Each multivariate Gaussian distribution portrays a flow of approaching trajectory. Yet, they belong to an elementary family of distributions, with few degrees of freedom, and do not appear to be sufficiently developed to represent reality. In Krauth et al. (2021), we focused on statistical copulas to estimate the distribution of go-around trajectories at Zurich Airport. Even though they allow for more complex representations, they also come with numerous parameters leading to a costly estimation process, which limits their use to distribution with no more than a few dozen of dimensions. Dimensionality reduction methods are then required, but the one developed in Krauth et al. (2021) can only represent lateral shapes of trajectories. Figuet et al. (2023a) extended this idea by employing 3-dimensional density estimation to describe air traffic flows in the TMA of Zurich airport to calculate a risk of collision. However, the density estimation only takes into account the 3-dimensional spatial distribution, and information about speeds is lost.

With the expansion of Deep Generative Modelling (DGM) in recent years, new estimation methods appeared to model high-dimensional distributions with complex dependencies. Jarry et al. (2019) explored the use of Generative Adversarial Network (GAN) to artificially generate simple one-dimensional trajectories. GAN are known to be state-of-the art methods when it comes to generate synthetic samples, but they offer very little explainability about their generation process, and are unstable to train because of their MinMax optimization program. Alternatively, the VAE is a more explainable framework, and can be suitable to estimate the distribution of complex 4-dimensional trajectories, as shown in Krauth et al. (2023b). In Krauth et al. (2023a), we have demonstrated that they are reliable enough to estimate the collision risk between arrivals and departures in TMAs. The reminder of this paper focuses on improving the explainability of the method by introducing disentanglement (Chen et al., 2018) in the VAE training and performing sensitivity analysis to unravel intricate factors impacting collision risk.

# 3. Methodology

## 3.1. Disentangled VAE for density estimation

As demonstrated by Krauth et al. (2023a), Monte Carlo simulations for mid-air collision risk assessment is complicated by a number of issues: (i) an extremely large amount of simulations must be carried out to accurately estimate mid-air collision probability, (ii) the number of relevant observed trajectories is usually not sufficient compared to the number of simulations required because air traffic procedures are constantly evolving, (iii) reducing the number of simulations with variance reduction methods cannot be applied as the distribution of the inputs should be known.

To address these limitations, Krauth et al. (2023b) developed a generator of synthetic aircraft trajectories based on the VAE that provides an estimator for the statistical distribution of observed trajectories. First introduced by Kingma and Welling (2013), the VAE framework is a generative artificial intelligence algorithm which estimates the statistical distribution of the underlying training data $p^*(x)$ by using unobserved latent variables $z$. Based on the idea of Bayesian Variational Inference, it can be decomposed into three main components:

* the probabilistic encoder is a neural network which maps every input $x$ into a learned posterior distribution of smaller dimension $q_\phi(z | x)$.

* the latent space is the projection space of the inputs, and is distributed according to the aggregated approximate posterior distribution:

$$q(z) = \frac{1}{\mathrm{Card}(\mathcal{X})} \sum_{x \in \mathcal{X}} q_{\phi}(z | x), \qquad (1)$$

where $\mathcal{X}$ is the training dataset and $\mathrm{Card}(\mathcal{X})$ its cardinality. One of the VAE objective is to make the posterior distributions as close as possible to a prior distribution $p(z)$ which can be fixed, or learned.

* the probabilistic decoder maps each latent variable $z$ taken from the latent space into the reconstruction distribution $p_\theta(x | z)$. If $z \sim q_\phi(z | x)$, the VAE reconstructs an observed sample, and if $z \sim p(z)$, the VAE generates a non-observed synthetic sample.

The VAE training aims to maximize the Evidence Lower Bound (ELBO) averaged over a batch of inputs of size $M$ (with $\beta = 1$):

$$\mathcal{L}_{\beta-VAE} = \frac{1}{M} \sum_{m=1}^{M} \underbrace{\mathbb{E}_{z \sim q_{\phi}(z|x)} [\log p_{\theta}(x_m | z)]}_{\text{A: Decoder Likelihood}} - \beta \underbrace{D_{KL} [a_{\phi}(z | x_m) || p(z)]}_{\text{B: KL Divergence}}, \qquad (2)$$

where $D_{KL}$ is the Kullback-Leibler (KL) divergence. Optimizing ELBO takes into account two phenomena: (A) maximizing the decoder likelihood which ensures a reliable reconstruction of the observed data,
(B) minimizing the KL divergence between the posterior and the prior which makes sure that the latent space is distributed according to the prior. The β-VAE from Higgins et al. (2016) is a modification of the VAE objective that introduces an adjustable β parameter: β > 1 means that more importance is put on the prior regularization than the reconstruction.

The VAE framework is part of the field of Representation Learning, which focuses on creating a relevant representation of the input data in the latent space. Two similar inputs are encoded with similar posteriors, while two very different ones are associated with distant posteriors. However, it is often impossible to identify a particular dimension of the latent space with a realistic characteristic of the input. To improve the explainability of the generative models, researchers have looked at Disentanglement Learning. Bengio et al. (2013) define it as a representation where a change in one dimension corresponds to a change of one underlying explanatory factor.

According to Kim and Mnih (2018), Chen et al. (2018) and Mathieu et al. (2019), the necessary prerequisite for disentanglement is a factorized aggregated posterior distribution. Let d the number of dimensions in the latent space, and z = (z₁, ..., z₉):

$$q(z) = \prod_{j=1}^{d} q_j(z_j), \qquad (3)$$

where $q_j(z_j)$ the jth marginal of q. This criterion implies choosing a factorized prior distribution $p(z)$. Burgess et al. (2018) show that higher values of β in Eq. (2), coupled with a standard Gaussian prior, can result in more disentangled representations of z. However, the β-VAE objective leads to a trade-off, and increasing β ultimately results in a decrease in the fidelity of the reconstruction.

In this paper, we focus on the work of Chen et al. (2018) which manipulates the KL divergence from Eq. (2) to isolate the terms influencing disentanglement:

$$\mathbb{E}_{x_m \sim p(x)} D_{KL} [q_{\phi}(z | x_m) \| p(z)] = \underbrace{D_{KL} [q(z, x_m) \| q(z) p(x_m)]}_{\text{Mutual Information}} + \underbrace{D_{KL} \left[ q(z) \| \prod_{j=1}^{d} q_j(z_j) \right] + \sum_{j=1}^{d} D_{KL} [q_j(z_j) \| p_j(z_j)]}_{\text{Total Correlation}} \underbrace{\text{Dimension-wise KL}}_{\text{Dimension-wise KL}} \qquad (4)$$

The Mutual Information describes the statistical dependence between the initial input $x_m$ and its encoding in the latent space z. The Total Correlation of $q(z)$ describes the difference between the actual aggregated posterior $q$ and its equivalent with independent marginals. For an efficient disentanglement, this term should be as low as possible. The dimension-wise KL divergence enforces the marginals of the aggregated posterior $q$ to be as close as possible to the marginals of the prior $p$. The β-VAE penalises all these terms equally, but Chen et al. (2018) state that only penalizing the Total Correlation improves significantly the disentanglement property with little reduction in the fidelity of the reconstruction. Chen et al. (2018) propose then the β-TCVAE, whose objective is:

$$\mathcal{L}_{\beta-TC} = \frac{1}{M} \sum_{m=1}^{M} \mathbb{E}_{z \sim q_{\phi}(z | x)} [\log p_{\theta}(x_m | z)] \\ -\alpha D_{KL} [q(z, x_m) \| q(z) p(x_m)] - \beta D_{KL} \left[ q(z) \| \prod_{j=1}^{d} q_j(z_j) \right] \\ -\gamma \sum_{j=1}^{d} D_{KL} [q_j(z_j) \| p_j(z_j)], \qquad (5)$$

where α, β and γ are adjustable hyperparameters.

## 3.2. Uncertainty Quantification

Uncertainty Quantification (UQ) deals with the characterization of uncertainties in unknown systems. It is particularly important in

situations where models are used to make decisions or predictions, but where there are inherent uncertainties in the input data, model parameters, or even the model itself. UQ plays a crucial role in enhancing the reliability and credibility of predictions and decisions. Such methods generally require knowledge of the distribution of inputs, but in some cases, it is too complex to guarantee their success. Resorting to a disentangled VAE to represent the input distribution in a low-dimensional latent space extends our range of options:

* The intricate input distribution is transformed by the encoder into a simpler low-dimensional prior distribution, with each marginal becoming independent, and whose likelihood can be evaluated.

* Disentanglement allows the assignment of interpretable attributes to each marginal of the latent space distribution, rendering the latent representation of the inputs as meaningful as their original form.

* Once trained, the VAE decoder acts as a deterministic “black-box” function, as there is no need to understand its internal mechanism. However, since it consists of a sequence of matrix multiplications, it is highly efficient to evaluate and does not require a huge computational budget.

The complexity due to the high-dimensional input distribution with complex correlations between its marginals is transferred to the encoder and decoder networks, while the associated prior distribution is more convenient to work with. This constitutes a much more favourable framework for UQ, and it is therefore possible to use traditional tools such as sensitivity analysis and subset simulation.

Let us consider a trained disentangled VAE comprising an encoder $E_{\phi}$ and a decoder $D_{\theta}$. Here, $Z = [Z_1, ..., Z_d] \in \mathbb{R}^d$ represents a random vector from the latent space, where d is the dimension of the latent space. The fundamental property of disentanglement in the VAE ensures that the aggregated posterior distribution in the latent space possesses independent marginals. Consequently, the prior distribution $p(z)$, used for generating new samples Z, is specifically chosen to have independent marginals as well. Now, let consider the deterministic function:

$$D: \mathbb{R}^d \to \mathbb{R} \\ z \mapsto Y = \psi(D_{\theta}(z)), \qquad (6)$$

where $\psi$ is a deterministic transformation employed to achieve the desired output $Y \in \mathbb{R}$ for UQ study. In the context of CRM, $\psi$ stands for the calculation of the minimal distance between aircraft, and Y is this minimal distance. D is then associated to the limit state function (LSF) of the problem. The primary objective of the study is to estimate the probability of the rare event $p_c = \mathbb{P}(Y < \delta)$, where $\delta$ is the collision threshold, but this framework can be extended to other UQ scenarios.

### 3.2.1. Sensitivity analysis: Sobol' indices

Sobol' indices introduced by Sobol (2001) are one of the most widely used methods in sensitivity analysis, and are applied to break down the variance of the output Y into fractions that can be attributed to each input $Z_i$, $i = 1, ..., d$. In this study, we consider the First-Order Sobol' Index $S_i$ which measures the sensitivity of the output variance Var[Y] explained by the effect of $Z_i$ alone. Sobol' Indices of higher order can also be defined to quantify the effect of a group of input variables on the output. However, they will not be considered in this paper. Because $p(z)$ has independent marginals, the Sobol' indices are guaranteed to be easily interpretable. $S_i$ is expressed as:

$$S_i = \frac{\mathrm{Var}[\mathbb{E}[Y | Z_i]]}{\mathrm{Var}[Y]}. \qquad (7)$$

Despite the availability of tools tailored for more complex scenarios, classic Sobol' indices remain valuable due to their robust statistical properties. They demand a factorized input distribution of Z, which is ensured by the prior distribution of the VAE, even if the original distribution of the pairs of trajectories is overly intricate for Sobol'
indices. Moreover, the disentanglement property of VAE ensures that meaningful characteristics can be linked to the dimensions of the latent space. Consequently, sensitivity analysis can not only pinpoint influential dimensions within the latent space but also associate them with understandable features through the disentanglement property of the VAE.

### 3.2.2. Monte Carlo simulation and subset simulation

Monte Carlo simulations aim to approximate a statistical expectation with the empirical mean of N independent and identically distributed random variables. Given a threshold $\delta \in \mathbb{R}$, Monte Carlo simulations can be used to estimate the failure probability $p_c = P(Y < \delta) = E[1_{Y<\delta}]$ of the considered system, where $1_{[\cdot]}$ stands for the indicator function. $p_c$ can be approximated by enumerating failure events occurring over a round of N simulations. Let $z_1, z_2, \dots, z_N, N$ realizations of Z sampled from the VAE prior distribution $p(z)$, and $y_1 = D(z_1), y_2 = D(z_2), \dots, y_N = D(z_N)$. The Monte Carlo estimator $\hat{p}_c$ is given by:

$$ \hat{p}_c = \frac{1}{N} \sum_{i=1}^{N} 1_{y_i < \delta}. \qquad (8) $$

Achieving a satisfying accuracy on the estimate $\hat{p}_c$ can be computationally expensive if $p_c$ is particularly low. As highlighted in Krauth et al. (2023a), conventional Monte Carlo simulations are not always suitable to estimate small probabilities such as the risk of MAC. Nevertheless, the sampling distribution $p(z)$ in the presented framework is completely defined, which enables the use of variance reduction methods such as Subset Simulation.

Originally proposed by Au and Beck (2001), Subset Simulation modifies crude Monte Carlo simulations to better estimate rare event probability while reducing computational costs. Its core idea consists in splitting the rare event $C = \{Y \le \delta\}$ into more common events. Let consider a set of m intermediate nested events such that $C = C_m \subset C_{m-1} \subset \dots \subset C_1$, where $C_s = \{Y \le \delta_s\}$ for $s = 1, \dots, m$ and $\delta_1 > \delta_2 > \dots > \delta_m = \delta$. The chain rule for conditional probabilities gives:

$$ \begin{aligned} p_c &= P(C_m) \\ &= P(C_m | C_{m-1}) P(C_{m-1}) \\ &= \prod_{s=1}^{m} p_s, \end{aligned} \qquad (9) $$

where $p_1 = P(C_1)$ and $p_s = P(C_s | C_{s-1})$ for $s = 2, \dots, m$. The Subset Simulation estimator is then the product of the Monte Carlo estimator for each term:

$$ \begin{aligned} \hat{p}_c &= \hat{p}_1 \prod_{s=2}^{m} \hat{p}_s, \\ \hat{p}_1 &= P(C_1) = \frac{1}{N} \sum_{i=1}^{N} 1_{y_i < \delta_1}, \quad s = 1, \end{aligned} \qquad (10) $$

$$ \hat{p}_s = P(C_s | C_{s-1}) = \frac{1}{N} \sum_{i=1}^{N} 1_{(y_i < \delta_s) | (y_i < \delta_{s-1})}, \quad s > 1. $$

Even though the core idea of Subset Simulation is elementary, it relies on two difficulties: (i) finding adequate intermediate threshold $\delta_s$ such as the conditional probabilities $P(C_s | C_{s-1})$ are not too small nor too large, (ii) being able to generate in the conditional density $P(z | C_{s-1})$. To deal with (i), the threshold $\delta_s$ correspond to the empirical quantile of the considered sample $\{z_s^{(j)}\}_{j=1}^N$ so that the conditional probability $\hat{p}_s$ is equal to 0.2 (in our case). The second issue (ii) is traditionally solved by using Markov Chain Monte Carlo to generate conditional samples. To that extend, Au and Beck (2001) propose a version of Subset Simulation which uses the modified Metropolis-Hastings sampler, the solution we use in this study. Although the Subset Sampling itself does not impose specific assumptions, Au and

<caption>Fig. 1. Collected observed take-offs from runway 16 (orange) and go-around on runway 14 (blue) at Zurich Airport. The study is limited to the red box, and grey parts of trajectories are trimmed.</caption>

Beck (2001) modified Metropolis-Hastings algorithm, utilized for high-dimensional data, necessitates a factorized distribution of Z. In many real-world applications, this requirement is not met by the original distribution of the inputs. However, its replacement in the VAE latent space, $p(z)$, is specifically tailored to meet all the assumptions required to ensure smooth operation of Subset Simulation.

## 4. Case study

In Section 3, we explored the merging of the disentangled VAE framework with UQ, creating a powerful approach to tackle complexities in input distribution. This method's practical application is then showcased in estimating MAC risk within the Zurich Airport TMA.

### 4.1. The GATO 14/16 problem

While the approach can be employed across a variety of situations involving conflicting aircraft movements, our study primarily centres on assessing the risk associated with MAC resulting from the take-off procedure on runway 16 and the go-around procedure on runway 14 at Zurich Airport. In instances where an aircraft approaching runway 14 needs to execute a go-around, there exists a significant risk of collision with an aircraft concurrently taking off from runway 16. To manage and minimize this risk, Air Traffic Controllers (ATC) are unable to operate independently runways 14 and 16; departures are temporarily halted until the approaching aircraft has safely landed.

A comprehensive dataset encompassing 8124 instances of aircraft take-offs from runway 16 between January 1st, 2022 to December 31st, 2022 was collected. This dataset was derived from ADS-B data procured from the OpenSky Network (Schäfer et al., 2014). Furthermore, we specifically selected 685 instances of go-around events for runway 14, spanning from January 1st, 2017 to December 19th, 2022. The collected trajectories are displayed on Fig. 1. To build the training dataset for our VAE, we randomly select pairs of take-off/go-around within the datasets of collected observed trajectories. We choose 20,000 instances of pairs from a total of 8124 × 685 = 5,564,940 potential ones.

<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Method workflow for estimating the MAC probability of the GATO 14/16 case in an explainable way.</figcaption></figure>

In our investigation, the first observation point of the go-around trajectory is located 6 kilometres before runway 14 as aircraft are still descending and the go-around manoeuvre is generally initiated later in the approach. Coincidentally, precisely at this moment, an aircraft undergoing take-off crosses the terminal threshold of runway 16. Trajectories are trimmed within the red bounding box displayed on Fig. 1 and have been resampled to contain exactly 50 observation point. For each observation point, values for the track angle, the ground speed, the altitude, and the cumulative time from the start are kept. As a result, the training dataset for the VAE is a tensor of size 20,000×8×50, whose each instance is a matrix in R$^{8×50}$ describing a multivariate time-series representing a pair of trajectories. Each column of the matrix corresponds to one observation point which has 8 features: track angle, ground speed, altitude and time from start for both the take-off and the go-around of the considered pair.

In this study, mitigation measures such as ATC actions, TCAS alerts or pilot interventions are not taken into account. Finally, a MAC event is defined as a situation where the aircraft centres of mass are separated by less than 55 m (the average wingspan).

## 4.2. Outline of the approach

In this section, we detail the sequential stages that constitute the methodology introduced in this paper. The objective of this methodology is to estimate and explain the potential risk associated with MAC incidents between aircraft take-offs and go-arounds at Zurich Airport. The methodology is summarized in Fig. 2 and can be described with the following steps:

* A VAE is trained in a disentangled manner to model the distribution of the pairs take-off/go-around for the GATO 14/16 problem. The intricate high-dimensional distribution describing the pairs of trajectories is encoded by the lower-dimensional VAE prior distribution $p(z)$ which has independent marginals.

* Once the VAE trained, the simpler distribution $p(z)$ allows for applying UQ algorithm, as explained in Section 3.2. The latent random variable $Z \sim p(z)$ is used as input. The LSF $D: \mathbb{R}^d \to \mathbb{R}$ decodes $Z$ through the VAE decoder into a pair take-off/go-around, and calculates the minimal distance between the two aircraft.

* From here, Subset Simulation algorithm can be applied to estimate the probability of MAC between the considered procedures.

* Concurrently, we calculate the First-Order Sobol' Indices to find the marginals of the input distribution $p(z)$ that have the greatest influence on the minimal distance between aircraft.

* Finally, the marginals in the VAE latent space that are identified as the most influential are linked with tangible attributes of the trajectory pair using the property of disentanglement.

## 4.3. VAE training

This section outlines the VAE model employed for generating a take-off/go-around pair from a unified latent space. The architecture we propose draws significant inspiration from the work of Krauth et al. (2023b), with the key adaptation being the accommodation of a trajectory pair instead of a single trajectory. The take-off and the go-around of a same pair are encoded separately by two parallel Temporal Convolutional Networks (Bai et al., 2018) before being concatenated to form the posterior distribution $q(z|x)$. The posterior is taken to be Gaussian, and the parameters $\mu$ and $\sigma$ are estimated by two dense layers. A point $z$ in the latent space is decoded by two parallel branches of Temporal Convolutional Networks: one builds the take-off, the other the go-around.

To ensure that the aggregated posterior $q(z)$ has independent marginals as disentanglement suggests, we use a factorized prior distribution $p(z)$. The idea is similar to the VampPrior presented by Tomczak and Welling (2018), but instead of having a Gaussian mixture where each component is d-dimensional, we construct independently a one-dimensional Gaussian mixture per marginal. Its expression is given by:

$$p(z) = \prod_{i=1}^{d} \sum_{j=1}^{n_c} \mathcal{N}(z | \mu_{i,j}(x), \sigma_{i,j}(x)) \quad (11)$$

where $d$ is the number of dimensions in the latent space, and $n_c$ the number of components of the Gaussian mixture for each marginal. The overall architecture of the VAE is presented on Fig. 3.

The VAE is then trained on the dataset presented Section 4.1 for 1000 epochs, and with a batch size of 1000. We use the ADAM optimizer with a learning rate of 0.001 which is halved every 200 epochs. The dimension of the latent space is $d = 10$, and the factorized prior $p(z)$ has $n_c = 2$ Gaussian components per marginal. Finally, the hyperparameters from Eq. (5) are set to $\alpha = 1$, $\beta = 20$, and $\gamma = 20$.
<figure><img src="image_6.png" /><figcaption>Fig. 3. VAE architecture to generate a pair of aircraft trajectories from a single latent space in a disentangled manner.</figcaption></figure>

# 5. Results

Initially, we delve into an analysis of the disentanglement property of the VAE trained using take-off/go-around trajectory pairs. Following that, we display the results of a sensitivity analysis concerning the minimal separation distance between aircraft. Lastly, we scrutinize the estimated MAC probability. We intentionally selected a case study with a comparatively high probability estimation to facilitate a comparison between our methodology's outcomes and those yielded by traditional Monte Carlo simulations. In this advantageous context, Monte Carlo simulations are deemed robust and reliable, thus serving as a benchmark against which we can evaluate the efficacy of our proposed method.

## 5.1. Disentanglement

As explained in Section 4.3, the prior distribution of our VAE has independent marginals, where each of them is a Gaussian mixture with 2 components. The calculated marginals are displayed on the left side of Fig. 4, whereas the right side quantifies how a change in one marginal corresponds to a change in the trajectory pair characteristics: from a randomly sampled reference pair (in bold), we draw randomly in a single marginal while keeping the other constants.

Fig. 4 shows that marginals 2, 4 and 8 have small variances, meaning that they encode little to no information: modifying the dimensions corresponding to these marginals while keeping the other constants will result in little change to the trajectory pair. The effect can be clearly seen on the corresponding pairs, where we observe no variations for those marginals.

In the opposite way, marginals 3, 5, 9 and 10 have large variances, and induce significant variations on the trajectories characteristics. Thanks to disentanglement, we can associate specific characteristics to each of those marginals. For instance, marginal 9 seems to encode the point initiating the first turn of the go-around. Marginal 3 seems to have an influence on the position of the end point of the go-around in the considered box. As for marginal 5, it seems to encode the direction of the take-off (east or west) which is clearly represented by the 2 modes distribution. Finally, marginal 10 seems to have the same effect as marginal 9 on the lateral shape of the trajectories, but also clearly encodes the altitude and ground speed profiles of the go-around, as seen on Fig. 5.

Marginals 1, 6 and 7 have variances that are neither large nor small. They have an influence on the characteristics of trajectories, but it is more complicated to associate them with a specific phenomenon.

## 5.2. Sensitivity analysis

We determine the dimensions of the latent space that have the most influence on the minimal distance between aircraft of a pair by computing the normalized First-Order Sobol' indices. We run the simulation on a sample size of 10 000, and Fig. 6 displays the relative influence of each marginal on the output distance. Marginal 5, which stands for the direction of the take-off (east or west), accounts for 93.4% of the variability of the distance between aircraft. Marginal 10 also plays a non-negligible role in the explanation of the distance, but its influence is down to only 4.2%. All the other marginals are considered as non-significant.

To be more specific, the Sobol' indices computed in this analysis reveal the impact of each marginal of the prior distribution on the separation distance between the two aircraft, rather than directly on the collision risk itself. For instance, it is conceivable that marginal 5 exerts a substantial influence on the distance between the two aircraft without necessarily affecting the collision risk. To verify this finding, we conducted CMC simulations, segregating take-offs by direction and randomly sampling pairs among observed trajectories. In the case of eastbound take-offs, we observed a collision probability of $6.6 \times 10^{-5}$ for 2,289,270 simulations (cv = 0.08), whereas for westbound take-offs, the probability was $3.9 \times 10^{-4}$ for 3,275,670 simulations (cv = 0.03). As established, marginal 5 not only significantly impacts the distance but also influences the probability of collision. It is important to note that this conclusion is in line with current operations at Zurich airport: mitigation measures are more restrictive when take-off is expected to be to the west. We can thus confirm that our method of identifying the parameters influencing the risk of collision is reliable.

## 5.3. Probability estimation

We performed 100 Subset Simulation runs with 10 000 samples per intermediate event $C_s$, a conditional probability between consecutive thresholds $p_s = 0.2$, and a spread in the Metropolis-Hastings sampler of 0.01. The probability estimate is the average of the runs, the empirical coefficient of variation is the ratio between the standard deviation and the mean, and the computational cost is the average number of evaluations of the limit state function per run. To ensure that the Subset Simulation is working properly, we also performed a hundred times a crude Monte Carlo with an equivalent computational cost, ie 50 000 simulations each. The probability estimate is the average over the 100 runs. Finally, to ensure that the bias induced by the distribution
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Marginal distributions of the VAE prior (left), and their influence on the lateral shape of trajectories (right). Yellow stands for a small standard deviation, which means the marginal has little influence on the trajectories characteristics. Purple stands for a large standard deviation, which means the marginal has a great influence on the trajectories characteristics.</figcaption></figure>

estimation provided by the VAE on the probability calculation is reasonable, we performed a hundred times a crude Monte Carlo on pairs of observed trajectories. Results are shown in Table 1.

Firstly, when comparing Subset Simulation and crude Monte Carlo using the same type of inputs (VAE prior), it is evident that Subset Simulation proves to be as reliable as crude Monte Carlo for this case study. It demonstrates that the Metropolis-Hastings algorithm identifies the regions in the latent space that are likely to produce colliding

trajectories. This result is confirmed by Fig. 7. Within this range of probability values, crude Monte Carlo exhibits slightly greater accuracy than Subset Simulation, with a lower empirical coefficient of variation for comparable computational costs. Nevertheless, Subset Simulation remains applicable in scenarios where the risk of MAC is much rarer, while crude Monte Carlo would demand an impractical number of simulations. The proposed methodology is more versatile than crude Monte Carlo and can be employed in various case studies. However, it
<figure><img src="image_8.png" /><figcaption>Fig. 5. Variations on the altitude and ground speed profiles for marginal 10.</figcaption></figure>

## Sobol' indices for the minimal distance

<figure><img src="image_9.png" /><figcaption>Fig. 6. Contribution of each marginal to the minimal distance between the take-off runway 16 and the go-around runway 14.</figcaption></figure>

<table><caption>**Table 1**
Estimated risk of collision between take-off and go-around procedures at Zurich Airport. We performed 100 runs per method.</caption><thead><tr><td>Method</td><td>Input</td><td>Probability estimate</td><td>Coefficient of variation</td><td>Computational cost</td></tr></thead><tbody><tr><td>Subset simulation</td><td>VAE prior</td><td>3.75 × 10<sup>-4</sup></td><td>0.29</td><td>53 000 simulations</td></tr><tr><td>Crude Monte Carlo</td><td>VAE prior</td><td>4.70 × 10<sup>-4</sup></td><td>0.20</td><td>50 000 simulations</td></tr><tr><td>Crude Monte Carlo</td><td>Observed</td><td>2.52 × 10<sup>-4</sup></td><td>0.28</td><td>50 000 simulations</td></tr></tbody></table>

proves to output similar results in scenarios where crude Monte Carlo are proven to be robust.

Secondly, when comparing Subset Simulation with the VAE prior and crude Monte Carlo with observed trajectories, the estimated probability with our proposed methodology aligns reasonably well with this ground truth. The bias introduced by using a complex approximation of the real underlying distribution with the VAE seems relatively low. Once more, the versatility of the proposed methodology extends beyond the straightforward application of crude Monte Carlo to observed trajectories and can be employed in situations where observed trajectories alone are inadequate for estimating collision risk. However, our demonstration on a simple case study indicates that, utilizing the VAE prior distribution as input does not appear to introduce bias.

Fig. 7 illustrates the failure samples obtained from both the Subset Simulation and crude Monte Carlo algorithms along dimensions 5 and 10, which were identified as having the most influence on the distance between aircraft. Two distinct regions in the latent space, where the density of pairs leading to a collision is higher, can be clearly identified for both Subset Simulation and Monte Carlo algorithms. In our case, the limit state is likely highly complex, with regions leading to collisions consisting of numerous point-like, relatively isolated clusters. This complexity explains why the positions of failure samples may vary slightly from one run to another. Regardless of the form of the limit state, crude Monte Carlo is always capable of finding the failure region, provided a

sufficient number of simulations are conducted. The fact that the results of Subset Simulation align with those of crude Monte Carlo suggests that our methodology is successful in identifying hazardous regions in the VAE latent space.

Lastly, Fig. 8 exhibits sets of trajectories reconstructed from the failure domains outlined earlier. The majority of the trajectory pairs generated within the Subset Simulation failure space exhibit a noteworthy level of realism. This observation indicates that the latent space of our VAE has been effectively structured, enabling it to decode points originating from the VAE's prior distribution while conditioned on a collision event. This reinforces the validity of our research, demonstrating that the resultant aircraft separation distances are meaningful and align with physical realism.

## 6. Discussion

### Limitations of the disentangled VAE.
Achieving perfect disentanglement as defined in Section 3.1 on aircraft trajectories appears to be an utopian dream as aircraft data are entangled by nature. For instance, altitude and ground speed are heavily correlated, and a change in the former inevitably leads to a change in the latter regarding the flight dynamics equations. Managing to find a latent space where each dimension corresponds to exactly only one feature does not seem to be a realistic goal. According to Locatello et al. (2019), unsupervised
<figure><img src="image_10.png" /><figcaption>Fig. 7. Failure samples given by Subset Simulation and crude Monte Carlo algorithms for two randomly picked runs.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Examples of conflicting trajectories decoded from the failure domain of the Subset Simulation algorithm.</figcaption></figure>

disentanglement learning is not possible without inductive bias on both models and datasets. Having no a priori knowledge of the features to be encoded by the latent space marginals is very limiting, and there is a risk of obtaining dimensions that are not as meaningful as desired. As seen in Section 5.1, only few dimensions can be considered as well disentangled.

Fig. 8 illustrates that while trajectories originating from the failure domain are not entirely implausible, certain points within these trajectories might be inaccurately positioned, resulting in a disruption of the trajectory's smoothness and overall shape. According to Table 1, this limitation seems to have a limited effect on overall probability estimate, but the lack of physical constraints in the estimation of the trajectory distribution and in the generation of the synthetic trajectories is problematic in terms of getting stakeholders to accept the model.

To construct a sufficiently extensive training dataset for the VAE, we adopted a method of randomly selecting pairs of take-off/go-around trajectories from the observed dataset. It is important to note that the observed trajectories within a same pair might have been subject to distinct atmospheric conditions as they were flown on different days. However, our VAE does not factor in meteorological data, resulting in the generation of trajectories of a same pair that could potentially be influenced by different atmospheric conditions.

**Limitations of the Sobol' indices.** The application of First-Order Sobol' indices serves as a straightforward tool for sensitivity analysis, offering a simple analysis of aircraft distance variability. For our specific goals, the use of First-Order indices is adequate, aligning with our disentangled framework's focus on individual dimension influence rather than collective consideration of dimension groups. It is important to note that these indices solely depict the influence on the distance between aircraft, not directly on the probability of a MAC. Given that

we have identified only two influential dimensions for distance, we believe these dimensions also hold the most significant influence on the collision probability. We demonstrated the significant impact of take-off direction (represented by marginal 5, the dimension with the greatest influence on distance according to the Sobol' indices) on the MAC probability using CMC. However, for more comprehensive studies, employing more refined tools like (Chabridon et al., 2021) would be wise to precisely quantify each dimension's influence on the probability of collision itself.

**Limitations of the probability estimation method.** Estimating the distribution of pairs of trajectories using a lower-dimensional latent space with the aid of a VAE undeniably improves the explicability of the model. However, like any approximation, the estimation of the density contains approximation errors that propagate throughout the methodology right up to the estimation of the MAC probability. As a result, the probability estimate depends on the choice of the VAE architecture. It could be interesting to quantify the propagation of uncertainty by varying the density estimation method (latent space size, prior and posterior definitions, other frameworks, ...).

As the Sobol' indices and Fig. 7 show, only 2 out of the 10 dimensions of the latent space have a real influence on the distance between aircraft, one being largely dominant. Within this projection space, the limit state has a very complex shape, probably composed of a multitude of almost point-like failure regions. This greatly complicates the work of the Subset Simulation algorithm, and the Metropolis-Hastings sampler settings may be difficult to find. The VAE latent space was constructed completely independently of the aim of estimating the risk of collision. Its purpose is to have the best shape to be able to generate new pairs, not to generate pairs that lead to collision.
## 7. Conclusion and outlook

In this study, we showcased the effectiveness of a disentangled VAE in combination with the Subset Sampling algorithm to assess the risk of MAC, all while identifying the pivotal factors that exert a notable influence on collision risk. We established associations between marginal distributions of the prior distribution and tangible characteristics inherent to trajectory pairs. Additionally, our method successfully pinpoints regions within the latent space where the risk of collision holds greater significance. Furthermore, we demonstrated the proximity between the calculated MAC probability and the ground truth, which is derived from observed trajectory pairs. Through this straightforward case study where crude Monte Carlo can be applied, we have highlighted the robustness of our method, instilling confidence in its applicability to more intricate scenarios where the risk is substantially lower and traditional crude Monte Carlo methods may be less effective.

The VAE is entirely built to generate synthetic pairs of trajectories that are identically distributed to the observed ones, which appears to be successful, as highlighted by Table 1. However, there are no physical constraints to ensure that the newly formed trajectories comply with the flight dynamics equations. Table 1 suggests that the bias induced by this lack of physical realism is contained, as the estimated probability is closed to the ground truth. However, we are convinced that guaranteeing the strict physical realism of the trajectories generated is an essential prerequisite for future validation of the model by players in the air traffic management sector. Future work will focus on the development of a hybrid VAE model that takes into account the estimation of the statistical distribution of trajectories and the flight dynamics equations by using physics informed machine learning. In addition, it seems essential to introduce meteorological data into the model to ensure that the trajectories of the same pair fly in the same weather conditions.

The latent space showed encouraging signs of disentanglement. Yet, Locatello et al. (2019) state that achieving perfect disentanglement learning is not possible in a completely unsupervised manner. Enforcing a factorized aggregated posterior does not necessarily guarantee disentanglement, and a good choice of hyperparameters on a regular VAE seems to be more effective than fancy objective functions. On top of that, there are no consistent strategy for hyperparameters choice. The natural next step for future work would be to explore semi-supervised disentanglement learning, where prior knowledge is used to build the VAE latent space. Furthermore, we have seen that the VAE latent space was not optimized to specifically generate colliding trajectories, resulting in only two latent dimensions being significantly influential in calculating the risk of collision. One potential direction is to consider shaping the latent space during the training of the VAE, aligning it more effectively with our specific objectives. By integrating sensitivity analysis within the VAE training process, we can ensure that each dimension of the latent space possesses comparable Sobol' indices, further enhancing interpretability. Another interesting direction could involve incorporating regularization into the VAE loss function. This regularization could be designed to ensure that adjacent pairs within the latent space maintain similar distances between aircraft. Such an approach could contribute to a more coherent and realistic arrangement of the synthetic trajectories.

### CRediT authorship contribution statement

**Timothé Krauth:** Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. **Jérôme Morio:** Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Writing – review & editing. **Xavier Olive:** Conceptualization, Formal analysis, Investigation, Methodology, Resources, Supervision, Visualization, Writing – review & editing. **Benoît Figuet:** Formal analysis, Investigation, Resources, Writing – review & editing.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper

### Data availability

Data will be made available on request.

### Acknowledgements

This work was supported by the Swiss Federal Office of Civil Aviation, grant number BAZL SFLV 2018-037.

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