Uncertainty in ATM

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This section addresses the research theme of uncertainty in ATM. The main research challenge is to understand and characterize all sources of uncertainty that affect the ATM system and propagate through it (at several scales), so that its performance can be improved, as well as to identify the relevant tools for the analysis.

Five different research lines are considered, four of which are linked to the scales that have been identified in the problem; they are the following:

  • ;
  • ;
  • ;

Within each of these research lines, different research challenges are identified and analysed. To complete the analysis of the research theme, some case studies are defined, which can be considered as applied research topics; they are the following:

With this structure, the research lines are focused on fundamental research and the case studies on application-oriented research.

General objectives

The topic, 'Uncertainty in ATM' is not completely new; some of the components of ATM have been using the concept of uncertainty for decades. For instance, when analysing errors in air navigation systems it is necessary to formulate some concept of uncertainty. The location of the aircraft is determined not just as a point, but rather as a region of confidence (a volume of space where one expects with very high probability that the aircraft lies). This region evolves in time and can shrink or grow depending on the navigation system type and quality. Other types of uncertainty have always been present in ATM, and the system has been able to deal with it more or less successfully. For example, ATCOs never know the exact position of aircraft at a given time, but rather their approximate position. To accommodate these uncertainties in position, ATCOs have used rules that incorporate safety buffers.

Recently, researchers in ATM have begun to recognize the importance of uncertainty, not only for navigation systems or safety buffers, but also for many other ATM components and aspects. The future ATM envisioned by SESAR requires improved prediction and optimization mechanisms to meet future capacity and air safety needs. To address these challenges, new concepts such as uncertain 4-D trajectories, uncertain ATM networks, or stochastic conflict resolution are arising.

However, ATM researchers working in uncertainty at different levels are using different approaches and lack a common language that facilitates not only dissemination of their work but also collaboration towards common objectives. Thus, the general objective of this section is to provide a framework in which to study how uncertainty affects the ATM system. This framework must be as general as possible to cover all scales of the system and to encompass the many possible approaches to study uncertain systems.

Other specific objectives are:

  • Define with clarity the concept of uncertainty, and differentiate among the various types of uncertainty that affect the ATM system;
  • Provide a set of relevant research challenges that can guide researchers interested in the study of uncertainty in ATM;
  • Give real-world examples (case studies) in which the inclusion of uncertainty can be useful in improving (in a measurable way) the performance of the ATM system.


There are several possible definitions of uncertainty. From a psychological point of view, a statement that an individual does not know with total certitude to be either true or false, is uncertain. A more useful definition is usually given for complex socio-technical systems (such as ATM). There, uncertainty i defined as the condition of being partially or totally unknown or in doubt; typically this is applied to quantitative values (such as the cartesian position of an aircraft at a given time, or an estimated time of departure/arrival, quantities which are seldom known exactly).

When speaking of uncertainty in a system, it is important to make a distinction between objective uncertainty and subjective uncertainty. In the case of objective uncertainty, the system is intrinsically non-deterministic, i.e. it does not evolve in a deterministic way (for instance, quantum systems). Thus, it is inherently impossible to perfectly know its state. On the other hand, in the case of subjective uncertainty, the system itself is deterministic, but still its state is not perfectly known. This might be due to several reasons, for example when the inputs that affect the dynamics of the system are not perfectly known or measured (for example due to measurement errors caused by sensor noise), or when the mathematical laws of evolution of the system are not exactly known or determined (for example when human factors play a significant role in the system). Another classification of uncertainty is into aleatory and epistemic. Uncertainties are characterized as epistemic if there is a possibility to reduce them by gathering more data or by refining models, and as aleatory if the possibility of reducing them is not foreseen.

Most complex systems present subjective uncertainty (called simply uncertainty from now on in this document), due to the presence of numerous agents, complex interactions between them, human factors, imperfectly known dynamics, or measurement errors, among other factors. In particular, subjective uncertainty is key to modelling and properly controlling the ATM system. The sources of uncertainty present in ATM are different, and in this document they are classified into the following types:

  • Data uncertainty. This type of uncertainty source exists when there is data known but with some level of uncertainty, when data is unknown and has to be guessed, and/or imperfect models. Thinking about aircraft trajectories, examples are the position given by a GPS, the aircraft take-off weight, and simplified aircraft models.
  • Operational uncertainty. The decisions taken by humans (pilots, ATCOs, or managers) have a very significant influence on the flight, but they are difficult to predict (even if they are based on perfectly known rules). Thus they introduce a degree of uncertainty, which can affect for example, the taxi times or the departure or arrival times. Clearly human factors are a key issue in this type of uncertainty.
  • Equipment uncertainty. This type of uncertainty refers to problems in equipment, such as aircraft or vehicle breakdown, or system failure.
  • Weather uncertainty. Meteorological conditions are a wide group of sources of uncertainty. These include wind velocity, fog, snowfall and thunderstorm regions and time intervals, or temperatures which require deicing. Meteorology conditions are predictable for a short time horizon; therefore, flight plans have to be based on estimations which oftentimes are far away from reality. In particular, adverse weather can lead to changes in the scenarios which introduce high levels of uncertainty, not only in a particular trajectory, but also in the air traffic as a whole. Strategies to deal with adverse weather may include re-routing or cancelling flights. Adverse weather poses a problem quite different from those posed by the other types of uncertainty, with a clear link to resilience.

Note that catastrophic events are not considered here, and, hence, they are not a part of the analysis.

These types of sources of uncertainty pose different research challenges, the solutions to which require different approaches and methods, as will be shown in the next sections.

All the uncertainties that affect a flight make the flight itself uncertain. This is called flight uncertainty. These uncertainties are present at the different stages of the flight: strategic, pre departure, gate-to-gate (this, in turn, includes ground and airborne stages), and post arrival. Of particular interest is the analysis of the uncertainties that affect the aircraft trajectory (gate-to-gate), which make the trajectory itself uncertain. This is called trajectory uncertainty.

Flight uncertainty and in particular trajectory uncertainty are sources of uncertainty for other problems in which the flight is a basic element, for example traffic problems or network problems.


The research theme 'Uncertainty in ATM' potentially has a very wide scope, since it can refer to the many sources of uncertainty that one can find in the ATM system, which are described in the previous section. To clarify the scope of the research theme, the analysis of uncertainty is linked to the different scales of the problem, which in general are affected by the various sources of uncertainty, and also in different ways. There are three clearly differentiated scales depending on the level of detail and aggregation, as defined in Section 0.2.2:

  • Microscale – A single flight. At this smallest scale one must analyse all the uncertainty sources that affect the flight, at its different stages, and precisely characterize the flight uncertainty. As indicated before, of particular interest within this scale is the study of the aircraft trajectory. At this subscale, one must consider the dynamics of the aircraft and the changing environment through which the aircraft moves (e.g. the atmosphere, runway, etc.); one must analyse all the uncertainty sources that affect the trajectory and define what is meant as trajectory uncertainty.
  • Mesoscale – Air traffic: This is an intermediate scale that allows us to focus on a given area that contains many individual aircraft that interact among themselves following a given set of rules; for instance, a TMA or a sector. The analysis of flow management problems are framed within this scale, because, even though they affect the air transport network, they are not considered network problems. Mesoscopic models exploit probabilistic methods to account for details of the microscopic scale without completely losing the macroscopic and strategic view of the system. This scale still considers individual vehicles, but describes their activities and interactions based on aggregate relationships. The mesoscopic scale allows us, for example, to assess of the potential impact of routing rules and conflict resolution strategies on aircraft delay, or to study propagation patterns and performance degradation.
  • Macroscale – Air transport network: Air transport can be considered at the level of regional, national, or trans-national network, or even at the level of the whole ATM system. This scale integrates the state of the various ATM elements and allows focus on the network properties, giving a high-level view of the system. It is important to study how uncertainty in flights and air traffic (microscopic/mesoscopic scale) propagates to affect the macro-scale, even though some authors consider the aircraft trajectory perfectly known (without any uncertainty) at this scale. Other relevant problems include the study of how operational uncertainty and strong disturbances affect the air transport network, (in particular adverse weather).

Within each of these scales, one can find two types of problems related to the time horizon under analysis, namely:

  • Estimation of the present state, that defines a short-term time horizon, at which the main concern is to enforce safety. Data uncertainty is the main uncertainty source types that affect this problem. Information sharing and filtering techniques can be used to reduce uncertainty.
  • Prediction of the future state, that defines medium and long-term time horizons. The main concern now is efficient planning. This problem must consider flight plans, weather forecasts and predicted traffic. Uncertainty propagation and “domino effects” have to be minimised while optimizing system performance. All uncertainty source types affect this problem; for long-term planning, data uncertainty is unavoidable, while for the medium-term, strong disturbances can have a considerable impact, causing unexpected planning changes or even cancellations.

Another set of temporal scales, related to the stages of the flight, is frequently used in uncertainty analysis:

  • Strategic, covering the timeframe months before the flight up to two hours before the off-block time.
  • Pre-departure, from slot allocation to the aircraft off-block time.
  • Gate-to-gate, including ground stage and airborne stage.
  • Post-arrival, which commences once the aircraft is on-block.

Research lines

The analysis of the research theme ‘Uncertainty in ATM’ is divided into five research lines, four of which are linked to the scales that have been identified in the previous subsection; they are the following:

  • Airborne Trajectory uncertainty;
  • Flight uncertainty;
  • Traffic uncertainty;
  • Network uncertainty;
  • Weather uncertainty.

Airborne trajectory uncertainty

In this research line the airborne stage of the gate-to-gate trajectory is considered. The aircraft flight trajectory is a very important part of the overall flight, the study of which is of interest not only because of its contribution to the flight uncertainty, but also by itself. In the following, for simplicity, it is called just trajectory uncertainty. The ground stage of the trajectory is analyzed in Research Line 2, along with the other flight stages.

Problem Statement

Trajectory uncertainty quantifies deviations of the aircraft trajectory from its nominal (preferred) trajectory. This deviation has to be unknown to qualify as uncertain; trajectory changes due to the pilot purposefully deviating from its flight plan are not considered trajectory uncertainty (even though they can be seen as uncertain in other scales). It is also necessary to consider 4-D trajectories since uncertainty in the time variable will finally manifest itself as flight delay. Trajectory uncertainty can be present (the present position of a flying aircraft is seldom perfectly known) or future (uncertainty propagated along a predicted trajectory). While both situations are different, they are studied using the same concepts and techniques.

When dealing with uncertainty, the most adequate framework to use is that of probability and statistics. Sources of uncertainty such as the initial mass are usually modelled by random variables which are described by distribution functions, and their uncertainty can be quantified by the value of the covariance of the random variable. Then the aircraft position at a given time can be described in a similar way; a very useful construction is that of a region of confidence, whose centre lies at the expected value of the position and includes around it a region of space where the aircraft could be located with a certain degree of probability (the volume of a region of confidence depends fundamentally on the degree of uncertainty, which is measured by the covariance matrix). However, to describe an uncertain trajectory, one needs to include a more advanced concept of probability: a stochastic process, in which a quantity (e.g. the aircraft position) is described by a random variable that changes with time. This allows the consideration of not only the uncertainty at a given time but also how uncertainty propagates into the future. The trajectory of an aircraft is described by well-known models using differential equations. However, if the initial conditions of these equations are uncertain or/and the differential equations contain uncertain terms, the equations become stochastic differential equations, whose solution is an uncertain trajectory (with uncertainty changing with time as dictated by the equations). Trajectory uncertainty analysis requires the study of the sources of uncertainty that affect the trajectory, their mechanism of propagation, and also the mechanisms of mitigation that the aircraft might have. Among these sources there are the following: uncertainty in the initial conditions, uncertainty in the aircraft performance models, wind uncertainty, navigational errors, FMS errors and operational uncertainty.

Literature Review

The basic concepts that define the framework chosen for the study of trajectory uncertainty can be found in textbooks and classical references: probability theory and statistics are covered in Feller (1968), the theory of stochastic process in Grimmet (2001), and the field of stochastic differential equations in Oksendal (2003). One of the first applications of this framework in ATM was in the field of navigation systems. The objective of a navigation system is to use the available sensors to compute the most accurate possible estimation of the present position of the aircraft. Navigation systems can be autonomous (depending only on internal measurements, such as inertial navigation systems) or depending on external signals (such as GPS satellites or radio aids like DME). The former use their measurements to solve the differential equations of the trajectory, and the latter uses a rather sophisticated type of triangulation to obtain the position (see Grewal et al. (2000)). However, since the signals and measurements are always corrupted by some degree of noise, the resulting navigational solution is therefore uncertain, with a degree of uncertainty that can be computed using the theory of stochastic differential equations. To visualize this uncertainty, it is best to represent the position of an aircraft as an uncertainty ellipsoid, centred in the computed position, whose size depends on the uncertainty; the more uncertain the navigational solution, the larger the ellipsoid. The real position of the aircraft can lie anywhere within this ellipsoid, which will change size with time as the navigational solution changes its quality. This framework allows one to quantify and compare different navigation systems, and is one of the elements used to design RNAV/RNP procedures (see EUROCONTROL (1999)). In general, for an uncertain trajectory the confidence regions can take other forms (spheres, parallelepipeds, or other irregular shapes). When the uncertainty region moves through space, one actually obtains a tube of uncertainty, that will be thicker at some parts and thinner at others; with very high probability an aircraft will be contained in this tube along its real trajectory. This is perhaps the best way to visualize an uncertain trajectory. The methods used to study trajectory uncertainty and uncertainty propagation can be classified into two main groups:

  • Monte Carlo methods (Hastings (1970)): These are computational methods that rely on repeated random sampling to compute their results. That is, one randomly selects a value for all the uncertain elements that affect the trajectory, and then computes it in a deterministic way. This is repeated as many times as necessary until the resulting trajectories are a representative sample of the uncertain trajectory itself. The main advantage of these types of methods is that they can be used with all types of uncertainty and do not require any complicated computation beyond solving the equations of flight mechanics many times. However it is a very expensive method in computational terms (suffering from “the curse of dimensionality”; if there are many sources of uncertainty one needs many sample points and therefore it is not implementable in practice) and requires random sampling of the sources of uncertainty, which is not always easy. More advanced Monte Carlo variants, such as the sequential Monte Carlo (Liu and Chen (1998)), use more advanced ideas (carefully selecting the samples of the sources) to improve the convergence and reduce the required number of sample points.
  • Non-Monte Carlo methods. To avoid the computational problem of Monte Carlo methods, other techniques have been proposed to study uncertainty propagation in dynamical systems. Halder and Bhattacharya (2011) classify those methods into two categories:
  • Parametric: in which one evolves in time the statistical moments, such as mean and covariance, but the probability distribution remains unknown). An example is the Generalized Polynomial Chaos method (see Prabhakar et al. (2010)).
  • Non-parametric: in which the probability density function itself is evolved. An example is the method presented in Halder and Bhattacharya (2011) which finds the probability density function by solving a stochastic Liouville equation.
  • Another family of methods which do not fall under any of the previous two categories uses Differential Algebra (Berz et al., 1999) to study the evolution in time of a neighbouring set of initial conditions. They can be used to efficiently compute the statistics of the propagated state (Valli et al., 2013). Differential algebra has been mostly used in orbital mechanics problem with uncertain initial conditions, however they are suitable to be applied in the setting of uncertain aircraft trajectories.

The above methods help to analyse and describe uncertain trajectories and uncertainty propagation along the trajectory. However, there are also methods that deal with the problem of uncertainty mitigation, i.e., reduction in the uncertainty of the trajectory. This is usually only possible if new information is obtained, for example by sensors or communication with other agents; this new information carries its own uncertainty, and the question arises of how to update the uncertain trajectory to incorporate the new information, in a way that the obtained trajectory decreases its uncertainty as much as possible. The algorithms that deal with this issue are called filtering algorithms, and the most widely known is the Kalman Filter, see Anderson and Moore (1979). Although the Kalman filter was initially developed to solve a navigation system problem (how to fuse information from inertial-type sensors with information from other sensors to obtain the most accurate possible navigational solution), it can be applied in any situation in which one has an estimate of the uncertainty of the trajectory and new data (with also a degree of uncertainty) is obtained. While stochastic differential equations have been used for decades to assess the accuracy of navigation systems, its application to trajectories is rather recent, and the literature dealing directly with trajectory uncertainty and uncertainty propagation is scarce. Some works in this field, are reviewed next. The method of polynomial chaos is used in the works of Prabhakar et al. (2010) and Dutta and Bhattacharya (2010) to study, respectively, uncertainty propagation and trajectory estimation, for hypersonic flight dynamics with uncertain initial data. Vazquez and Rivas (2011) develop methods to study the impact of initial mass uncertainty on fuel consumption during the cruise flight phase. Zheng and Zhao (2011) develop a statistical model of wind uncertainties and apply it to stochastic trajectory prediction in the case of straight, level aircraft flight trajectories. Weather uncertainty is considered in Matthews et al. (2009), Yen et al. (2003), and Nilim et al. (2001). Navigational errors and FMS errors are studied in Grewal et al. (2000) and Kim et al. (2009), respectively. Crisostomi et al. (2009) combine Monte Carlo methods with deterministic worst-case methods to analyse trajectory prediction for air traffic control. Some of these method can also be useful in trajectory optimisation problems; for instance, Fisher and Bhattacharya (2010) use polynomial chaos to solve optimal trajectory generation problems with probabilistic uncertainty in system parameters, and Li et al. (2014) apply the same method to robustly solve problems such as supersonic aircraft short-time climb with uncertainties in the aerodynamic data. 

More details on uncertainty propagation for aircraft trajectories can be found here.

Research Challenges

Within the research line of trajectory uncertainty, the following research challenges have been identified:

Characterization of uncertainty sources. A careful characterization of the different sources of uncertainty that affect the aircraft trajectory is an important challenge in this research line. This characterization may have different components: to identify all the relevant sources, to determine their statistical properties, to identify the dependence structure between different sources, to characterize and quantify their effects (do they affect the whole trajectory, or just a flight phase?, in 4D trajectories, do they affect both space and time, or just one of the two?). A very important question is how to discriminate negligible sources of uncertainty. A sound sensitivity analysis should be performed, which could be carried out, for instance, by using trajectory prediction (TP) tools. It would be necessary to define a 'standardized' sensitivity study (with a clear methodology to be followed) that would allow to compare (homogeneously for all TPs used) the sensitivity results obtained, and, therefore, conclude what uncertainty sources can be removed from the analysis.

Analysis of uncertainty propagation. To define the trajectory uncertainty, one must analyse the propagation of the different sources of uncertainty along the trajectory. To do that, one must find the right stochastic models that allow the description of uncertain trajectories with sufficient accuracy while being tractable enough. The first element to consider is what kind of aircraft motion model (AMM) is to be used for trajectory prediction (6DoF vs. 3DoF/ ODE vs. DAE). This has a big impact on the inputs required for computing a trajectory, and therefore, on what sources of uncertainties have to be considered. Although each approach presents its own casuistic, there are some invariant inputs (sources of uncertainty) that are common to all of them: the initial conditions (IC), the aircraft performance model (APM), the weather model (WM) and the Earth model (EM). This four inputs are required by all trajectory predictors regardless the implementation used for their development. It would be interesting to analyze the influence of these inputs. Another important outcome of this analysis is to know whether the different uncertainty sources amplify or decay along the trajectory.

Analysis of the effect of closed-loop control Detailed knowledge about the flight guidance algorithms and closed loop controls inside the FMS is not usually accessible. In classic aircraft trajectory prediction, the aircraft states are propagated over time, starting with intitial conditions by solving the differential equations and propagating uncertainties by common means of error propagation. The uncertainties of the aircraft states usually grow with time, with the amount depending on the "process noise" entering the aircraft dynamic model. The only way to reduce uncertainty is by obtaining updated data, i.e. updated measurements. However, if a closed-loop control is active and therefore applies to some of the state variables, certaint assumptions about the associated uncertainties can be made, given the capability of the closed-loop control to keep the uncertainties bounded. An important model required for this approach is the aircraft performance model (APM) that directly affects the extent to which uncertainties can be mitigated by closed loop control, i.e. the FMS in conjunction with the autopilot.

Development of metrics for uncertain trajectories. The challenge is to define a set of performance metrics that incorporates uncertainty in its definition. As trajectory uncertainty is a source of uncertainty at other scales, these metrics can play the role of “macroscopic properties” of the trajectory, to be taken as input for the study of uncertainty at other scales (traffic or network uncertainty).

Validation of trajectory prediction tools. A validation process for stochastic prediction tools that include distribution functions or uncertainty thresholds is required, to prove that the simulated data is appropriate for modeling parts of the real ATM process. This can be achieved in two ways: By comparing with real flight data or by comparing with other well known prediction tools. When it comes to 4D trajectories as envisaged by SESAR, a new challenge arises that is the lack of real input data to compare with. Apart from a very few flight trials that performed first initial 4D tracking operation, there is no flight data of real aircraft, doing real 4D trajectory tracking. It is not representative to compare 4D TP with real flight data without an active 4D tracking algorithm.

Flight uncertainty

The flight lies at the smallest scale of the problem, and its study is of key importance because it is the core element of ATM.

Problem Statement

Flight uncertainty encompasses all the uncertainties present at the different stages of the flight. The term ‘stage’ has been used to avoid confusion with ‘phase’ (which is normally reserved for a phase of flight per se and is not wholly appropriate for the ‘strategic’ and ‘post-arrival’ stages). The stages of the flight are the following:

  • Strategic, covering the timeframe months before the flight up to two hours before the off-block time. This includes the filing of flight plans but not the ATFM slot allocation process.
  • Pre-departure, which coincides with slot allocation (commencing two hours beforehand) and continues up to the aircraft off-block time.
  • Gate-to-gate, including ground stage (taxi-in and taxi-out) and airborne stage.
  • Post-arrival, which commences once the aircraft is on-block.

While the spatial uncertainty affects mainly to safety issues (loss of separation), the temporal uncertainty manifest itself as delays. Flight delay is a very important phenomenon that affects other scales (traffic and network). In the following, flight delay is discussed, and, in particular, how it is caused and propagated at the different stages of the flight.

Uncertainty by phase.bmp

Figure 1.1 Evolution of uncertainty for a flight.

Figure 1.1 sets the discussion of uncertainty in the context of a typical IFR flight. It shows how the uncertainty associated with a flight evolves as a function of time, from the strategic stage (which could be several months before the flight), through flight plan filing, active flight, and on to the passenger arriving at their (final) destination. The figure is drawn approximately to scale. The unit of the horizontal axis is time, albeit with several discontinuities indicated so that the stages fit into one diagram. The vertical axis is standard deviation of time (one measure of uncertainty) for each stage of the flight. Data for 2008 are used in the figure (and will be used subsequently). Each stage is discussed in turn, next.

(a) Strategic planning stage

Airport slots, the strategically planned time for take-offs or landings, can change from season to season. IATA’s Schedules Conference is held twice a year (in advance of the summer and winter schedules). Uncertainty at this stage may be important when ATFM and airport slots are not compliant (see later, [c]), and when too many flights are scheduled to depart or arrive during a time window (it is not unusual to have up to ten flights scheduled to depart or arrive at the same time, to be resolved through operational practice at the airport and ATC). When airlines fill flight plans far in advance, flight details are uncertain. For example, airlines rely on IFPS to complete route information when final processing occurs 20 hours prior to the flight. Clearly, predictions of sector congestion are also less reliable, and weather data are only known with reasonable reliability a very short time period before the flight. Airlines ‘direct filing’ flight plans to IFPS (five days before the flight onwards) have the opportunity to take advantage of more dynamic information as it becomes available. Even so, potential slot delay is still an unknown.

(b) Pre-departure stage

Around 20% of the 10.1 million flights in 2008 were regulated by ATFM slots – the CTOT (Calculated Take-Off Time) determined via receipt of a SAM (Slot Allocation Message) – with ATFM delayed flights having an average delay of around 20 minutes. Although not all ATFM slots result in a delay (almost half of the regulated flights in 2008 suffered no delay due to ATFM measures), slot allocation and the management of slots contribute to uncertainty. For example, an allotted CTOT can be revised or an initial SAM can be issued late, close to the off-block time. Uncertainty relating to other aspects of a flight can result in inefficient slot use – for instance, rather than delaying a regulated flight in order to rectify a ground-based problem, an airline might retain the slot in the hope that the problem can be overcome and the slot achieved, although this may lead to the slot being wasted when it is too late to re-allocate the ATM resources efficiently.

(c) Gate-to-gate stage

The gate-to-gate stage covers the ground stage (taxi-in and taxi-out) and the airborne stage. Uncertainty impacts the take-off time. Actual taxi time can differ greatly from the planned taxi time. Standard taxi times can be modified at short notice in response to changing airport conditions, and congestion, or the lack of it, can change expected taxi times dynamically. Uncertainty once the aircraft is airborne has been studied in the previous section (airborne trajectory uncertainty). As with the pre-departure stage, arrival time is with respect to the airline’s last-filed Estimated Off-Block Time (EOBT) and is thus an underestimate of the full variability, as airlines may re-file to manage delay.

(d) Post-arrival stage The post-arrival stage commences once the aircraft is on-block. The point estimate ‘pax a’ in Figure 1.1 represents uncertainty relating to the arrival time of passengers, with ‘pax f’ representing uncertainty for arriving passengers with connecting flights. The corresponding temporal uncertainty for ‘pax f’ is more difficult to estimate. Even with a relatively short arrival delay, passengers could miss onward connections if held up by security checks or baggage transfer problems.

(e) Delay costs and further resources

Most airline costs are incurred at the arrival and post-arrival stages, through the costs of delayed passengers and those of reactionary delays in the rest of the network. In Europe, it is estimated that ATFM delays alone cost airlines EUR 1 450 million in 2011, although these only comprised around 11.5% of all departure delay minutes, with an established, strong relationship between departure and arrival delay (EUROCONTROL, 2012). Indeed, the Single European Sky initiative, a paradigm shift in the design and function of European airspace, was launched by the European Commission in 1999 specifically in response to increasing delays. There are important trade-offs between strategic and tactical delay costs which are governed by their uncertainties (see below).

  • For a full review of these costs, and how they arise, see Cook and Tanner (2011).
  • For a discussion on propagation of delay in the network, see under Resilience.
  • For an analysis of the trade-off between strategic and tactical costs, see Cook et al. (2010).
  • For detailed reporting on delay causes and trends in Europe, see the Central Office for Delay Analysis website [1]
  • For reporting on delay costs and their impacts in Europe, see the Performance Review Unit (EUROCONTROL) website [2]

Literature Review

Flight uncertainty, and in particular the analysis and prediction of flight delay, has been frequently studied in the literature. There is a huge amount of data from past years, and delays at the different stages of the flight have been compared. For instance, based on intra-European flights in 2008 (see EUROCONTROL (2009)), the standard deviation of delay for all causes during the pre-departure stage was 17 minutes. However, this figure is with respect to the airline's last-filed Estimated Off-Block Time (EOBT) and is thus an underestimate of the full variability, as airlines may re-file to manage delay. On the other hand, regarding the gate-to-gate stage, the standard deviation of delay was found to be 19 minutes (typically due to sequencing, weather and other local disruption factors). Post-arrival delay, which refers to the delay experienced by travelling passengers, can be higher than aircraft delay; for instance, delay experienced by US passengers was found to be 1.6 times higher when compared with the delay of aircraft (Cook et al. (2010)).

While a 15-minute take-off slot tolerance is available for ATC departure sequencing purposes (see EUROCONTROL (2010)), which can be -5 minutes to +10 minutes in relation to CTOT, 18.5% of regulated flights in 2008 took-off outside their 15 minute ATFM slot window, of which 9.8% were early and 8.7% late. EU Regulation No 255/2010, in force since September 2011, reiterates the requirement for slots to be adhered to, with a requirement for non-compliance monitoring and action to be taken at an airport if 80% or less of flights depart within their allotted slots. In the Commission’s 2011 White Paper, its ‘roadmap to a Single European Transport Area’ for 2050, it lists as an initiative, the revision of the (airport) slot regulation, “to favour more efficient use of airport capacity”.

Another cause of delay is airport ground handling (AGH), which refers to the planning, scheduling, and control of all aircraft turnarounds at an airport. During turnaround, an aircraft must undergo a whole set of ground handling activities consisting of disembarking/embarking passengers, unloading/loading luggage, maintenance checks, fuelling, cleaning, catering, and so on. Independent ground service providers handle many of these activities. The interest of the service providers need not be in line with the one of aircraft. This makes scheduling the activities in AGH a challenging task. Efficient procedures to reduce this source of delay have been developed; for example, Mao et al. (2009) develops an algorithm for scheduling of airport ground handling services, including disturbances, using a heterogeneous multi-agent scheduling frame and on-line scheduling to cope with uncertainty, considering cooperative and a non-cooperative settings. Prediction of delay is of great interest. It is of particular interest to study the dynamics that generate delay at hub airports, which present many challenges dues to its many simultaneous operations. For instance, in Andersson et al. (2000), three separate models are proposed to capture the dynamics at busy hub airports. The first two models consist on simple queuing models that are introduced to capture the taxi-out and taxi-in processes, and the third is an integer programming model aimed at representing airline decision-making (attempting to capture the dynamics of the aircraft turnaround process). However, in Balakrishna et al. (2010), it is noted that taxi-out time prediction is difficult due to the airport dynamically changing environment, and introduces more complex models, developing a prediction method using nonparametric reinforcement learning (RL) set in the probabilistic framework of stochastic dynamic programming. A case study of Tampa International Airport (TPA) is presented, with good matches between the predictions and the real data. Other types of delay model are helpful to identify the key factors that contribute to delay. In this direction, Wonga and Tsai (2012) examine flight delay propagation involving a Taiwanese domestic airline using Cox regression analysis (proportional hazards model). A departure and arrival delay model are developed that show how flight delay propagation can be formulated through repeated chain effects in aircraft rotations. The hazard ratios obtained provide measures of the chances of recovering from flight delays under a variety of situations and the effects that individual contributing factors of flight delays have on airline schedule reliability. In particular, Cox regression analysis reveals that the key contributing factors of departure delays include ‘turnaround buffer time’, ‘aircraft type’, ‘cargo and mail handling’, ‘technical and aircraft equipment’, ‘passenger and baggage handling’, and ‘weather’, whilst the key contributing factors of arrival delays include ‘block buffer time’ and ‘weather’. When predicting delay, the time horizon of prediction will change the accuracy of the estimates. Ohsfeldt et al (2011) compares the pre-departure delay estimates with in-flight delay estimates by using trajectory crossing time uncertainty at an U.S. Oceanic Flight Information Region (FIR) boundary (entry or exit), which are metering points for tracks and congestion points for flights off the track system. After analysing real data from six months of operations and comparing the actual crossing times with the flight planned estimates, linear regression methods are used to model the delay. It is found that, compared to the in-flight (shorter horizon) uncertainty, which is typically assumed to be a normal distribution, the uncertainty for pre-departure planning (longer prediction horizon) behaves less like a normal distribution and it is less represented with linear regression models.

Research Challenges

Analysis of flight delay Flight delay is the final result of the propagation of the different sources of uncertainty along the different stages of the flight. To be able to analyse and predict flight delays, first it would be necessary to develop probabilistic models that include the sources of uncertainty as inputs and the delay as an output. Adequate metrics should also be generated to better quantify not only flight delay but also its impact on different parts of the ATM system.

Analysis of uncertainty in turnaround operations The turnaround of an aircraft is a complex process, given the involved stakeholders and restrictions due to technical, legal or operational aspects. Turnaround operations consist of several parallel and sequentially running processes which partly depend on each other or the successful completion of other running process. Uncertainty in turnaround operations is mostly characterized by variations from the predicted off-block time. Analysed field data shows that turnaround performance is highly depend on the operational flight parameters, e.g. passenger amount, aircraft type or flight distance and can cause high process variations. Furthermore, external factors may contribute uncertainty to turnaround performance. Preliminary analysis has shown that the most influential factors are:

  1. Delay: process start times show a direct correlation to the inbound delay of a flight
  2. Airport category and corresponding skill level of ground staff: Depending on the airport category, significant variations in process performance can be observed. This correlates with the level of standardization and skills of the ground staff.
  3. Weather: Similar to the flight segment, weather may impact ground operations significantly; e.g. thunderstorms located above the airport require ground operations to be interrupted.

Traffic uncertainty

The previous and next research lines take a microscopic and a macroscopic view of the ATM system, respectively. While the points of view of the trajectory (microscopic view) and the air transport network (macroscopic view) are of great importance, there is another key, intermediate scale (the mesoscopic scale), which appears at the level of the TMA or an air traffic sector. In this scale, microscopic uncertain objects (trajectories) interact with management procedures and separation rules of macroscopic objects (nodes of the network), and the uncertain atmospheric conditions. This generates a dynamic, rapidly changing environment where air traffic control has to take decisions (such as re-routing or holding) trying to fulfil the conflicting objectives of optimization (increasing capacity, following user-preferred trajectories, minimising delays) and safety (avoiding incidents and especially accidents). Recall that, as indicated in Section 1.1.3, the analysis of flow management problems are framed within this mesoscopic scale.

Problem Statement

When studying a scenario at the mesoscopic scale (TMA/sectors), one finds many sources of uncertainty, which affect the scenario:

  • Uncertain trajectories, which need to be predicted by ATCs with a lot of unknown data (making the predictions even more uncertain);
  • Operational uncertainty, which are partly due to human factors, but also due to the interaction between sometimes ambiguous or conflicting management rules;
  • Discrepancies between the weather forecast and the real weather, which are specially significant in the case of adverse weather.

The two main types of scenarios that appear at this scale are the TMA and the air traffic sector:

  • The TMA is often the most complex scenario, due to traffic converging to the runway (incoming flights) interacting among themselves and with outgoing flights, even though at the same time there are many available sensors and flight aids which help decrease uncertainty to a more manageable level. The main concern is safety, followed by delays and capacity;
  • An air traffic sector typically presents a lesser level of complexity, as traffic in general is not convergent, but at the same time the level of uncertainty is higher due to decreased sensor coverage and more uncertainty in the weather forecast. Safety has to be enforced while at the same time capacity needs to be maximized and the deviations from user-preferred trajectories minimised.

The possibility of having aircraft conflicts is an important problem in any traffic scenario, which requires the use of conflict detection and resolution (CDR) algorithms. These algorithms perform three different processes: prediction that a conflict is going to occur in the future, communication of the detected conflict to a human operator, and assistance in the resolution of the conflict. These three processes can be affected in different ways by uncertainties: in the estimation of the present state of the aircraft (observable states, sensor errors), in the prediction of their future states (trajectory prediction), in the communication to the human operator (system malfunction, operator misinterpretation), and in the resolution process (stochastic resolution, negotiation process). Nowadays, uncertainty is handled with deterministic algorithms (traffic flow management, conflict detection, conflict resolution, etc.) where the level of uncertainty is just guessed and used as a buffer (minimum separation distance). If capacity is to be maximised and user-preferred trajectories implemented, while still enforcing safety, it is necessary to develop stochastic algorithms that include uncertainty models in their formulation. These algorithms should take into account all different sources of uncertainty present in the problem and compute a solution. The solution would be stochastic in nature, for instance, and will maximize the probability of having a capacity as large as possible (rather than directly maximizing a value of capacity).

Literature Review

When studying air traffic under uncertainty, one key problem that becomes very difficult to solve is the problem of flow management. This problem was first considered in Odoni (1987), who considered that congestion could arise at any point in the trajectory: at the airport of origin or destination or en route (at a waypoint or sector). It was understood from the beginning that the flow management problem is stochastic in nature, as it involves, among other things, a prediction (forecast) of weather; however, to simplify the problem most authors have considered that the aircraft trajectories are deterministic.

Ground holding programs were the first approaches to solve the problem. Vranas et al. (1994) consider a multi-airport ground holding program taking into account propagation of delay and solve the problem using a heuristic approach. The inclusion of both ground holding and en route decisions is first addressed in Helme (1992), but still deterministic models are used. Alonso et al. (2000) first include uncertain factors in both the airport and airspace, and Mukherjee et al. (2009) make use of more realistic weather scenarios where decisions are dynamically taken and reviewed as a function of updated weather forecasts. Another approach (Nilim et al. (2004)) models the problem as a Markov decision process with the weather evolving as a Markov chain. Chang et al. (2010) considers the use of ground delay, cancellation, and cruise speed for each flight on the ground and air holding and diversion recourse actions for each flight in the air, then determining how aircraft are sent toward a sector under the uncertainty of weather; the problem is solved by using rolling horizon methods. Kim et al. (2009) derive service time distributions for different flight phases and assess traffic flow efficiency using queuing network models. Clarke et al. (2009) develop a methodology to study the capacity of a volume of airspace in the presence of weather uncertainty (stochastic capacity) and formulate a stochastic dynamic programming algorithm for traffic flow management; they present a stochastic routing algorithm that provides guidance for routing aircraft in the presence of the uncertainties of adverse weather. Knorr and Walter (2011) analyse the impact of trajectory uncertainty on sector complexity and workload, considering the following sources of uncertainty: trajectory data quality, model quality, and operational procedures. Sengupta et al. (2014) consider the problem of air traffic flow management under airspace capacity uncertainty arising from weather or environmental effects, and use stochastic programming to develop risk-hedge decisions resulting in the least delay at a specified level of acceptable variance.

Recently, a SESAR WP-E ComplexWorld PhD project proposed by the University of Erlangen (Heidt (2010)) addresses the problem of uncertainty and robustness in the context of ATM planning. In this project different time horizons are considered (strategic, pre-tactical and tactical planning), taking into account the impact level of uncertainty, because a source of uncertainty which plays a crucial role for one planning horizon can be irrelevant for others. And another SESAR WP-E ComplexWorld PhD project proposed by the University of Hanover (Hauf (2010)) addresses the need of modelling in an intelligent way the impact of weather on ATM performance.

In the CDR context, Kuchar and Yang (2000) identify two fundamental methods to handle uncertainty in trajectory prediction:

  • Worst-case method. It is assumed that an aircraft will perform any [one?] xxx of a range of manoeuvres. If any one of these manoeuvres could cause a conflict, then a conflict is predicted. This method is conservative in the sense that they trigger conflict alerts whenever there is any possibility of a conflict within the definition of the worst-case trajectory model. For example, Durand and Alliot (1997) consider the effect of ground and vertical speeds uncertainties, while Tomlin et al. (1998) determine the minimal unsafe operating region for each aircraft;
  • Probabilistic method. The uncertainties are modelled to describe potential variations in the future trajectory of the aircraft. This is usually done in one of two ways. In CTAS, for example, a position error is added to a nominal trajectory, from which the conflict probability can be derived (Isaacson and Erzberger (1997), Vela et al. (2009)). A second approach is to develop a complete set of possible future trajectories, each weighted by a probability of occurring. The trajectories are then propagated into the future to determine the probability of conflict. An example of this second approach is described by Prandini et al. (2000).

In conflict resolution, uncertainties can be found if stochastic processes are used to obtain the resolution manoeuvres, as is done by Durand and Alliot (1997) who use genetic algorithms. Another example of the existence of uncertainties can be found in distributed algorithms, which usually require a negotiation process where the aircraft agree a coordinated solution. This negotiation process is commonly affected by uncertainties, as delays in the communications, ignorance of the algorithms used by the other aircraft, etc. However, these uncertainties can disappear by means of replacing the negotiating process by some mathematical criteria that the resolution manoeuvres must meet, as those proposed by Narkawicz and Muñoz (2011). An example of a conflict solver that includes uncertainty can be found in CATS (Complete Air Traffic Simulator), which is a general purpose simulator for en route traffic that includes uncertainty in ground speed and in climbing and descending rates, see Alliot et al. (1997). Because the problems of conflict resolution and weather avoidance can arise simultaneously, some authors develop integrated algorithms to solve them. For instance, Lauderdale and Erzberger (2014) propose a unified solution in which three algorithms are applied sequentially to solve the problems of separation conflicts, arrival sequencing, and weather-cell avoidance in en-route airspace; safety buffers are included to reduce the effect of aircraft performance uncertainties and efficiently handle complex weather environments.

For both traffic flow management and conflict resolution it is important to explicitly take into account the costs of delay and delay propagation. In Cook et al. (2009), the concept of dynamic cost indexing is introduced to include passenger delay costs, delay recovery decision windows and air traffic control cooperation, as well as environmental impacts. In Cook and Tanner (2011), the cost of delay propagation is studied both from the point of view of the airline (flight-centric) and also from the point of view of the passenger (passenger-centric metrics). The SESAR WP-E project POEM studies this type of passenger-centric metric, investigating also their statistical distribution, linking the metrics to predictability. The cost of delay can be also used as a basis for decision-making; for instance, Dunbar et al. (2012) study the airline scheduling problem, introducing a new approach to calculate and minimise the cost of propagated delay in a framework that integrates aircraft routing and crew pairing.

Another outstanding question for both traffic flow management and conflict resolution is how much traffic can be safely accommodated. To perform safety risk assessment, Monte Carlo simulations can be used. Blom et al. (2006) explain the key issues in this type of simulations: first, one has to develop an appropriate simulation model, and a sound way to speed up the simulation. Also, one has to validate the simulation model versus the real operation, and the simulation-supported approach has to be embedded within the safety risk assessment of the total operation. Given that safety-critical events are rare occurrences, advanced techniques have to be used. Blom and Bakker (2011) perform an accident risk assessment for a self-separation concept of operations, using agent-based modelling and rare event Monte Carlo simulations. Their agent-based modelling is based on a stochastic hybrid model: dynamically coloured Petri Nets (see Everdij et al. (2006)). In Andrews et al. (2006) another approach is presented; to perform a fault analysis of a highly automated separation assurance system. The growth and decay of risk during a failure of the system is evaluated using fault tree methods that integrate risk over time.

Research Challenges

Safety assessment for uncertain traffic. While it is evident that uncertainty has a significant impact in safety assurance levels, there have been few studies directly relating the effect of uncertainty (and its propagation) and safety levels. However, if safety is to be greatly increased, it is necessary to improve the actual knowledge of this relationship. In particular, it would be of great interest to investigate what types of uncertainty most impact the safety of air traffic, that is, to perform a sensitivity analysis with respect to uncertainties. To carry out this analysis one could think of safety Key Performance Indicators (KPIs) as functions of the degree of uncertainty; this functional dependence could be obtained through a simulation study (using, for instance, rare-event Monte Carlo techniques) so that one would be able to quantify the variation of the safety KPIs when the levels of uncertainty change. Such a study would allow one to determine which uncertainties are most detrimental to safety and are worth mitigating (if possible at all) to improve safety levels.

Development of metrics for uncertain traffic. The challenge is to develop a set of stochastic metrics that provide measures of capacity, airspace complexity, congestion, predictability, etc., under uncertain conditions. These metrics must provide guidance as to how to decrease adverse effects (for example delays) by reducing uncertainty.

Air traffic robustness under strong disturbances. The challenge is to increase the robustness of air traffic under strong disturbances (such as adverse weather or congested air transport network). Adverse weather represents a challenging limiting factor of the capacity of the ATM system. In practice, it requires traffic managers to reroute all flights affected. Today’s methods for rerouting traffic are such that the reroute alternatives provided are limited, leading to high air traffic congestion. An open research question is how and which complexity science tools can be used to widen the set of operationally acceptable reroutes (selected according to a set of metrics that must be defined), so that the available airspace capacity is maximized, while maintaining a safe airspace throughput, taking into account the inherent uncertainty that characterizes adverse weather. Another strong disturbances to be considered is the effect of a congested air transport network (the network scale affecting the traffic scale!).

Development of trajectory management systems for uncertain air traffic. One of the greatest challenges that the future ATM system faces in the next decades is the integration of new airspace users and the continuous increase in delegating capacity and safety critical traffic management functions to automated systems. The accommodation of these new airspace users, which will have to coexist with conventional users, a widely reorganized airspace and the increased level of automation will necessarily need a paradigm shift with regard to the trajectory management functions. It is necessary to develop essential trajectory management functions to efficiently manage heterogeneous traffic considering the increasing presence of autonomous ATM systems, in uncertain conditions. The development of such systems has to focus on how to deal with uncertainty sources and their propagation along the trajectory and define requirements and algorithms to synchronize trajectory data and ensure safe management of merging traffic with different capabilities in an extended terminal manoeuvring area (ETMA).

Network uncertainty

To analyse uncertainty at the scale of the air transport network (be it at a regional, national, trans-national or at a global level), it is best to abstract and integrate the various complex and heterogeneous ATM elements in a way that allows us to assess uncertainty and other properties of interest without needing to include too much detail (which would be impractical or even impossible if dealing with the whole ATM system). A good framework that can help to develop these models and analyse uncertainty is complex network theory, which has proven to be very useful in applications, since very different systems often share similar characteristics when investigated at a network level, suggesting the possible existence of universal mechanisms of organization.

Problem Statement

When looking at an air transport network from a large-scale perspective and ignoring particular details that belong to smaller scales, it is clear that the network is fundamentally composed of airports and flights. Thus it is very natural to model it as a (directed) graph with the airports as nodes and the flights as (directed) links between these nodes. This network representation has some peculiar characteristics:

  • The number of nodes is rather limited (from less than a hundred to a few thousand at most), while the number of links is very large;
  • The network structure under normal conditions is relatively stationary with respect to time and space (the links can have slight changes but the changes in the nodes are extremely slow);
  • It has bidirectional, weighted links (flights) with (under normal conditions) somewhat fluctuating frequency (fundamentally due to trajectory uncertainty propagating to this larger scale);
  • Strong disturbances (such as adverse weather) can produce sudden, almost-instantaneous changes in the network as the ATM system tries to accommodate such occurrences.

Complex network theory studies those networks whose structure and relations are irregular, uncertain, and dynamically evolving in time. It is particularly well suited to study the air transport network, especially under strong-disturbance conditions, which will help study robustness and resilience of the system. Moreover, even under normal operating conditions, the delay due to trajectory uncertainty of several flights can accumulate, and, if above a certain threshold, cause congestion in the system, which again will produce a sudden change in the network. These sudden changes result in unexpected configurations due to complex interaction among many agents and the rules of operation. It is of great interest to study their impact on overall performance of the system: capacity, predictability, stability, robustness, etc.

For this network model, the trajectories are usually assumed to be deterministic. Trajectory uncertainty can then be summarized in macroscopic properties (the identification of these is a clear research challenge).

Literature Review

Network theory has been a subject of study for centuries; see for instance Newman (2010) for an up-to-date review. In particular, complex networks (see Albert and Barabási (2002), Bocaletti et al. (2006)) and evolving networks (see Dorogovtsev and Mendes (2002)) have attracted considerable attention and work from the mathematics and physics community. Modelling of the ATM system with network theory to study its properties has been done in the past decade both at a worldwide level and at the national level. In these works, the airports are the nodes of the network whereas the flights are the links connecting these nodes. Some examples are:

  • Bagler (2004) investigates the topology of the airport network of India and finds that the traffic is mostly accumulated on interconnected groups of airports.
  • Li and Cai (2004) study the airport network of China finding some statistical properties of the airport degree and flight frequency distribution.
  • Wu et al. (2006) model the global airport system as a complex network with weighted nodes to study several properties such as robustness and optimal paths. Guimera and Amaral (2004) have modelled the global airport network, and reported findings such as that the network is of a small-world type with power-law decaying degree, and that the most connected cities are typically not the most central cities.

The idea of using these ATM network models to study topics such as such as efficiency, safety, or flexibility is very recent; some examples are the following:

  • Bonnefoy and Hansman (2007) study light jet operations from the point of view of the airport and route network to understand how airports can attract the use of Very Light Jets (VLJs).
  • Lillo et al. (2009) study STCA (Short Term Conflict Alert) networks (i.e. alerts linked by the co-presence of at least one aircraft) to identify their topological characteristics and their geographical distribution, and use these results to analyse ATM safety.
  • Reggiani et al. (2010) analyse indicators found in the regional network of Lufthansa flights to study changing patterns in the network configuration and to find whether airline strategies can be revealed using network theory.
  • Kotegawa et al. (2010) investigates the evolution of the network to find which parameters identify unconnected pairs of airports likely to be connected in the future.
  • Paleari et al. (2010) studies the connectivity of the airport networks in China, Europe and US. The main objective is to understand which network topology is best (from the point of view of the final passenger) in several possible terms. The Chinese network is found to be the quickest (given its smaller size), the American network is the most coordinated for indirect connections, and the European network provides the most homogeneous level of service.

These references give diverse examples of how network theory tools can be applied to diverse ATM problems. However, there seems to be a lack of works where the uncertainty is taken into account, which clearly identifies a research challenge at this macroscopic scale.

In complex networks, uncertainty has been mainly understood as the presence of errors in measurements. For instance, when analysing the protein-protein interaction networks, nodes represent the protein of a cell, and a link between two of them is created when a correlation in their expression level exists. When measuring such correlations, there is always an error: thus, non-existent correlation may be detected, and existing correlation may be neglected; see von Mering et al. (2002). To solve this problem, several algorithms have been developed, which try to assess the "validity" of links in the network (see Clauset et al. (2008), Zhou et al. (2009), or Guimerà and Sales-Pardo (2009)). Other models of uncertain networks have been used in the study of communication systems (Kelly and Yudovina, 2014).

The concept of uncertain links (i.e., links that may or may not exist) has not yet been exhaustively studied. In Ahnert et al. (2007), the concept of an uncertain complex network has been used to study weighted networks. Specifically, the authors suppose that the weight associated with each link (between 0 and 1) is equivalent to the probability of that link existing. Then, complex weighted networks can be represented as an ensemble of random networks, whose analysis can then be performed by using standard metrics. In a similar line, Zanin (2011) proposed the interpretation of link weights as a measure of the uncertainty of such links, i.e., of their probability of existing. By transforming the adjacency matrix of the network, it was possible to associate a probability of existence to each path inside the graph.

A different approach is considering that the structure of the network is deterministic, but that, from time to time, a link (or a node) suffers a failure or an attack. The first one is random, while in the second case an external agent tries to weaken the network by suppressing the most vulnerable (or the most important) element. It has been shown that the topology of the network defines its vulnerability (see, for instance, Albert et al. (2000) or Holme et al. (2002)). Following this line of research, Cardillo et al. (2013) analyse the resilience of the European Air Transport Network against random flight failures. A multiplex network formalism (with the set of flights of each airline as an interdependent network) is used to solve the passenger's rescheduling problem. A comparison with the single-plex approach shows that the multiplexity strongly affects the robustness of the European Air Network.

Recently, two SESAR WP-E projects addressed network problems, specifically studying ATM system uncertainty at the network level. The ELSA project (Lillo et al. (2011)) investigates how airport network topology is related to the statistical properties of flight delays and how it affects the propagation of flight delays from an airport to another. On the other hand, the NEWO project (Sanchez et al. (2011)) studies the link between specific prioritisation rules applied to flights in cases of capacity shortfalls at nodes and the network behaviour and stability, both locally and at network-wide scales.

Research Challenges

Network analysis under trajectory and traffic uncertainty. The challenge is to consider micro and mesoscale uncertainties in the analysis of the structure and behaviour of the air transport network. To do so, network theory tools must be adapted to take into account those types of uncertainty. From a general point of view, this challenge may also include the definition of network metrics that take into account uncertainty at the inner scales. More specifically, a particular challenge would be the study of how trajectory uncertainty affects the network; how the propagation of delays, for example, affect the network capacity, and eventually leads to the congestion of the system (a phase change), and also how to modify operational procedures to minimise the probability of this happening. How to model the behaviour of the network under a strong disturbance? Agent-based modelling? What would be the best rules to follow to minimise the impact on the network? Stochastic modelling?

Network vulnerability Networks can be broadly described according to their topology. In the air transport context, hubs are relatively common, in comparison with a random network, albeit small in absolute number. Thus, it is more likely that many connections will use a hub – this dependence is an important factor governing the behaviour of the network under degradation, e.g. when and how any hubs fail. Air transport systems, may be more sensitive than a random network, if the hubs are more likely to fail. Hubs are obviously particularly susceptible to act as delay multipliers in the ATM network, often operating at, or near to, capacity and with multiple dependencies between flights, which may be in contrast to other complex networks where hubs are more readily protected or otherwise resilient. The challenge is to explore such vulnerabilities with respect to different metrics in real air transport networks.

Weather uncertainty

In this section, uncertainty for atmospheric processes is analyzed at different scales, and its relationship with ATM is identified.

Problem Statement

When discussing weather uncertainty, one needs to distinguish three types of uncertainty which are present in metereological phenomena. These are:

  1. Fundamental uncertainty (as it is referred to by the corresponding research community, although also known as 'objective uncertainty');
  2. Hidden determinism;
  3. Preliminary uncertainty.

Fundamental uncertainty (or objective uncertainty) is an inherent feature of certain processes and is found e.g. in quantum physics, while playing dice, in radioactive decay but also in atmospheric physics. Here are some examples for the latter. The first two, cyclogenesis and convection, emerge from instabilities in atmospheric dynamics, e.g. from baroclinic instability which is the origin of cyclogenesis, or from convective instability, which leads to convection ranging from small fair weather clouds up to large thunderstorms. In both cases, energy flows from smaller to larger scales at typical time and spatial scales. For mid-size convection the time scale is 20 minutes to one hour and the spatial scale is on the order of several kilometers to few deca-kilometers. A minimum distortion thus leads to a much large scale response, where the latter represents a mathematically unstable but not unbound solution. A cyclone emerging from a small spatial pressure gradient discontinuity will not grow further than several hundred kilometers in diameter. Given a baroclinic situation, it is known that any such arbitrary distortion will generate a cyclonic development, but the onset of that instability in space and time is not deterministic, neither certain characteristics of the cyclone itself, as for instance, the precise rain rate at any location within, or the central pressure minimum at a given time. All those quantities, however, fall into certain value ranges. A pressure minimum of an ordinary cyclone will never drop below 900 mb, for instance. Baroclinic instability and cyclogenesis are well covered by numerical weather prediction models (NWP). If, however, numerical experiments are repeated with slightly different, but physically equivalent initial conditions, typically solutions remain in close neighborhood till a baroclinic instability occurs and solutions then diverge without bound. This can be observed in so-called ensemble solutions. The differences in initial conditions might be as small as a difference of a hundredth of a degree in temperature at an arbitrary location, while all other initial conditions might be equal. Ensemble solutions as depicted for instance by isolines or contour lines of pressure, temperature or humidity thus remain for a certain time and within a certain region and thus are bounded. After the onset of instability no such tube can be determined.

Similar statements hold for convective instability in an unstable stratified environment. Many trigger mechanisms are known here such as horizontal temperature or soil humidity gradients, low level flow convergence, upper level flow divergence and uplifting, hills or mountains, boundary layer waves and others as explained e.g. by Wilson et al. (1998, 2004), Ebert et al. (2004). If we want to handle convection in a deterministic way, we have to model and simulate the trigger mechanism. That objective, however, is hard to achieve, as many processes remain obscured, such as the gravity wave motion in the inversion topping the convective boundary layer. Again, onset and further development of convection is in many aspects stochastic in nature, though there are certain bounds which determine the later developing overall structure. One of such governing parameters is the vertical lapse rate profile. Small buoyancy differences between a convective updraft and its environment over a large height scale lead to slow growing towering cumulus clouds, as observed typically over the ocean, while large differences are accompanied by explosive growth with related down bursts, gust fronts, and even tornados, as typically observed over the US great plains. The evolution of convection and cyclogenesis, therefore, appears as a fundamental stochastic process which has to be described in a probabilistic rather than a deterministic way. Certain characteristics of both processes, however, are governed by external and/or larger scale parameters which can be simulated in a deterministic way. Weather prediction models, in general, capture those parameters and the related forcing well and thus e.g. region, onset time, mean propagation direction and maximum height of thunderstorms can be predicted reliably well. The location of the first storm and its track, in contrast, cannot be forecasted sufficiently. No rule without exception. If, however, a triggering mechanism is dominant, for instance a mountainous island, then this feature might be captured by NWPs as well.

In summary, some characteristics of cyclogenesis and convection can be simulated well in a deterministic way, others are of stochastic nature and ensemble simulations will reveal, first of all, that basic feature, and, secondly, give an estimate of the expected range of results. Due to the hyperbolic nature of atmospheric Navier-Stokes equations, uncertainty and errors will grow in a conical phase space shape. At a fixed point the error or uncertainty amplitude will grow like a 2D-cone around the true value with time. Spatially, the uncertainty will spread like a cone with time around the mean wind speed.

A third stochastic atmospheric phenomenon is lightning. Number of cloud-to-cloud lightning strokes, their onset, and their magnitude in terms of transported charge show a fundamental uncertainty. The same holds for cloud-to-ground lightning. Again, there are certain upper bounds for e.g. the current within a lightning stroke but its exact value can only be given in a probabilistic way. Also, if there are towers, masts, mountains, or even aircraft lightning may be triggered and preferably will be attracted by those features, respectively will become part of the lightning channel. Thus, there are increased probabilities that in case of a storm a potential lightning strike will hit a tower, but it will not be exactly 100%.

Another example of fundamental uncertainty in the atmosphere is the obvious fractal nature of cloud shape. Onset, growth, decay, and size of individual turbulent eddies do represent stochastic dynamical elements which may become a significant risk for aircraft. If strong cross winds come together with a strong turbulent eddy, a landing aircraft, for instance, immediately before touch down will be in trouble as occurred in Hamburg on March 1, 2008. On the microscale, for instance, hail stone growth, composition and maximum size are eluded from a deterministic description.

Forecast accuracy of NWPs depends on the weather situation, region, and quantity considered. As a rule of thumb it is limited to 7 days due to cyclogenesis. Improvements are possible in the field of cloud physics, rainfall forecast and in other areas due to continuous data assimilation, higher spatial resolution of the numerical models and integration of more observational data (such as AMDAR aircraft observations). In future the reliability of the forecast will also be determined and probability forecasts will be made taking explicitly into account the involved stochastic elements.

Due to the unavoidable uncertainty, aviation needs probability forecasts indicating the range of expected results rather than a deterministic forecast with just one number (Source).

Hidden determinism. In many cases we assume with high confidence that the underlying atmospheric processes are deterministic in nature but that they appear as chaotic of stochastic as we either do not fully understand them or have no or insufficient observation or simulation tools. We, therefore, assume that if we knew those processes better and/or if we have better satellites, or if we can make e.g. in-situ stratospheric measurements, we can remove the apparent uncertainty.

One example is the impact of soil humidity on maximum rain fall rates. In the latter case, we expect that with a high spatial resolution (~ 10 m) of a soil humidity cadaster and NWPs with identical or even smaller numerical grid size we can forecast the rain distribution and its strength in a much better way as today. Currently, there is sometimes a surprising effect of heavy rain fall which in a posteriori analysis can be traced back to abundant soil humidity. Thus we assume that the seemingly chaotic occurrence of heavy rainfall can be understood, simulated and also forecasted, if we had a better knowledge of the water stored in the soil.

Another example was the sudden experienced shear effect on the glide path towards Hong Kong International Airport caused by horizontal vortices generated on the leeward side of Lantau island. In the past those events seemed to be erratic. Recent research has revealed the true nature of the event and current Lidar monitoring detects the shear zones and even warns the cockpit. Supercooled large drops (SLD) are a serious weather hazard to aircraft. SLD occur very often unexpected and in patches. One can imagine that in future with an airborne forward looking light-weighted remote sensing system such as a radar or a LIDAR or a combination of both, the pilot can be warned prior to the first ice accretion and that respective measures can be taken in advance (exiting, heating).

Whether an oceanic earthquake causes a tsunami or not depends on the rapid vertical displacement of the oceanic crust. The correlation of a tsunami with an observed earthquake of a certain magnitude seems to be stochastic. With a dense ocean bottom monitoring network and/or with atmospheric sensors detecting tsunami related side-effects, like ionospheric gravity waves caused by low frequency tsunami-wave uplift and detectable in GPS signals, it is hoped that the existence or non-existence of a tsunami after an initial earthquake can be confirmed within 15 minutes.

Preliminary uncertainty. Before discussing the preliminary uncertainty, it is worth to discuss the time and spatial scales of stochastic or partially stochastic atmospheric processes. Because of its relevance for ATM, we will focus on convection. We consider a situation with an unstable lapse rate allowing deep penetrative convection within the next hour up to the tropopause. This statement can be made with sufficient accuracy based on conventional parcel methods. Deep convection is visible to the pilot by a cloud in contrast to e.g. shallow dry convection. Once the onset occurred and a cloud became visible, the uncertainty has been removed with respect to time and location. There still exists an uncertainty concerning the further growth as not each small cloud becomes a deep cloud.

If, however, the cloud growth continues to a cumulus congestus, then it is almost certain, that this cloud will become a cumulonimbus cloud within a couple of minutes. As growing clouds have typical growth rates there are also typical time scales of 10 – 30 minutes and spatial scales of 1 – 50 km involved. Due to these scales the associated uncertainty becomes smaller with decreasing spatial distance and time. Once the stochastic event has started, the onset time is known and, therefore, removed as an uncertain quantity. The event becomes visible and its spatial dimensions become lesser and lesser uncertain while approaching the object. If the observer is at the object, respectively the pilot at the cloud edge, all uncertainty has been removed. Thus we have a narrowing uncertainty cone.

Change of uncertainty dependent on the proximity (in time and space) to any event.

We may compare it with a circling ball in a roulette wheel during its last turn. The number of reachable final numbers becomes smaller and smaller during the last few seconds till uncertainty finally becomes certainty. If, therefore, as for convection in the atmosphere, the stochastic process has a certain lifetime, the priori existing uncertainty vanishes with time and approaching distance. That is why we may refer to that phenomenon as preliminary uncertainty. If we have, in contrast, a process with a very short lifetime such as a lightning strike (~ 1 s), no preliminary uncertainty exists. The lightning strike becomes existent within 1 – 10 milliseconds and lives for ~ 1 seconds and with that realization the stochastic experiment is finished. In the approach by Nilim and coworkers (2003, 2004, 2009) it is exactly that feature which enables a less conservative navigation through a storm field.

Literature Review

In the literature we can find a number of works that study the effects of weather uncertainty on air traffic.

Nilim et al. (2003,2004) model the evolution of convection as Markov chain and develop a stochastic model for N aircraft under the constraints of maximum sector capacity and conflict avoidance among aircraft. This approach reduces the overall delay of the system compared with assuming convective areas as deterministic no-go zones. The weather uncertainty model assumes a deterministic knowledge of a convective storm 0-15 minutes in future, respectively a distance away from it. The evolution of a storm is modeled in time steps of 15 minute, where the transition from one stage to the other follows a Markovian chain with multiple outcomes. A storm can either exist or not, leading to a very simplified, binary treatment of convective weather. Matthews et al. (2009) studied the avoidance behavior of pilots with air corridors impacted by convection and used that as a metric for convective weather forecasts. Both, the pilot behavior and the convective weather are subject to stochastic elements but the verification of the new forecast metric with conventional ones shows a good correlation. Similar as in Nilim et al. (2004), Clarke et al. (2009) try to calculate the stochastic capacity of a given airspace in the presence of convective weather where the latter is considered as stochas-tic in nature. They also provide specific guidance for routing aircraft based on the calculated capacity. Based on available probabilistic convective weather forecasts a relation between the forecast and the related maximum capacity is generated. The forecast blends RUC model forecasts with radar observations and gives a probability for disrupting convection at a given location and given forecast time. Forecast probabilities are provided every 15 minutes and the nominal spatial resolution of the RUC model is 13 km. The works by Wilson et al. (2004), Ebert et al. (2004), and Wilson et al. (1998) show the limits of forecasting and nowcasting convective storms. Major reasons for the forecast limit of 30 minutes to 1 hour are (1) the lack of knowledge and inability to handle external forcing mechanisms, (2) that the predictability of convection is partially dependent on the predictability of the forcing or triggering mechanism. Unknown or un-observed triggering and forcing mechanisms lead to an apparent stochastic behavior of convective cells. At present it is not foreseeable that improved modeling techniques, stochastic models, observing techniques will allow e.g. the forecast of a new convective cell which might be caused by a congruence of boundary layer convergence zone with a local soil humidity patch. Existing mature storms and isolated storms in a forcing-free environment might be longer forecasted as others. In Dallavalle et al. (2004), the authors explain how the model output statistic system (MOS) is applied to aviation forecast problems.

An interesting problem is is the modeling of weather (atmospheric conditions and wind) as a source of uncertainty. Given that deterministic weather forecasts are not accurate, probabilistic models are necessary. One of today’s trend is to use Ensemble Prediction Forecasts (EPF), which attempt to characterize and quantify the inherent prediction uncertainty based on ensemble modeling; see, for instance, Hacker et al. (2003). Ensemble forecasting is a prediction technique that aims to generate a representative sample of the possible future states of the atmosphere. An ensemble forecast is a collection of typically 10 to 50 weather forecasts which may be obtained in different ways based on time-lagged, multi-model, and/or multi-initial conditions approaches (see Arribas et al., 2005, or Lu et al., 2007, for a description of some of these approaches). Different national Met offices provide weather EPFs, which can be used to model weather uncertainty, be it at the flight, traffic, or network scale.

Research Challenges

Within the research line of weather uncertainty, the following research challenges have been identified:

Recognition of stochastic processes in the atmosphere As explained above, there are stochastic processes in the atmosphere which have to be recognized as such. Their existence imply that there is objective (inherent) uncertainty related with these processes. Therefore, if the uncertainty is of fundamental nature, it cannot be avoided and one has to accept the consequences in terms of unavoidable risk and/or delay.

Identification of realistic ATM accuracies In view of the atmospheric uncertainties, either in forecast or fundamental, arrival accuracies of the order of one minute are unrealistic from a meteorological point of view. Realistic accuracies have to be determined for a range of weather situations in parallel with the introduction of new ATM procedures. It also makes no sense to postulate accuracies and set requirements for MET which are not feasible.

Removal of hidden determinism If the stochastic nature of a metereological process is only temporary, research and development has to be focused on identifying the deterministic parts of the process. For instance, continuous 3D wind and turbulence monitoring of the whole airspace above an airport with high temporal and spatial resolution will remove any ambiguity associated and help to detect shear layers, severe gusts, eddies, downdrafts, and strong crosswind components which today still lead to unexpected atmospheric behavior which could be predicted if this information was available.

Joint R&E for ATM In order to remove as much as possible atmospheric uncertainty impact on ATM, more coordinated research on that subject has to be done. All proposed operations, including the 4D-trajectory concept, have to be investigated taking a-priori into account the atmospheric uncertainty. This implies on the MET side a transition from conventional “deterministic” forecasts to ensemble forecasts and to deliver probabilities to ATM. It also requires to identify fundamental uncertainties and to incorporate them into operations as e.g. proposed by Nilim and coworkers. Simultaneous simulations of air traffic and weather have to be fostered as the complexity of 4D development of both weather and air traffic with stochastic elements is only resolvable by simulations.

Case studies

Information sharing protocols to reduce uncertainty

The ATM system is composed of many players, who interact among themselves. These interactions follow certain rules, which are frequently based on estimations of the present state of the agents, or on predictions of the future evolution of the agents. Due to the uncertainties present in the system, estimations always contain some degree of error; this is even truer in the case of future predictions, since frequently the uncertainty propagates in time degrading the accuracy of future predictions. Also prediction tools require data which is often not available (for instance, aircraft mass) and has to be guessed, which impacts on the quality of the predictions. Even though eliminating all uncertainty in present estimation and predictions of future evolution is not possible in practice, it is possible to mitigate it by implementing information-sharing protocols that make available to the relevant actors as much real-time data as possible. For instance, to mitigate trajectory uncertainty, aircraft could inform neighbouring air traffic and ATCOs of their present GPS position at all times; information about performance models and initial mass could be shared prior to take-off to facilitate trajectory prediction and thus also mitigate air traffic uncertainty. Real-time weather information should be available at all times to all agents involved. This would allow not only ATCOs but also individual aircraft to have an accurate model of the present state and future evolution of the TMA or airspace sector, for example. If all this data is shared, there would be possibly redundant information and thus a need for sensor fusion and integration algorithms to obtain a coherent solution that minimizes uncertainty as much as possible, by giving more importance to accurate sources of information than to other less-trusted sources. There are already suitable algorithms that could be adapted for this task, such as Kalman filters or particles filters, which are well known and already in use in air navigation systems design.

Stochastic decision support tools

This case study addresses the need for the design of decision support tools (DSTs) robust to uncertainty.

  • Stochastic Traffic Flow Management. Deterministic traffic flow management (TFM) decisions often result in “lost” airspace capacity, because uncertainties make it difficult to determine the number of aircraft that can be safely accommodated in a volume of airspace during a given period of time. Hence, there is a need to design TFM algorithms for improved decision making in the presence of uncertainties, especially weather and trajectory uncertainties. An important problem to tackle is the impact of uncertainty on airspace capacity, or, from other point of view, to improve capacity under given levels of uncertainty. In this context, the term stochastic capacity is used in the literature.
  • Stochastic Conflict Detection and Resolution. In this application the objective is to design conflict detection and resolution algorithms robust to the sources of uncertainty that affect the air traffic, namely, trajectory uncertainty and operational uncertainty. An important issue to take into account is the study of how uncertainty affects the saturation of the resolution method (measured by the occurrence on unsolved conflicts).
  • The theory of robust control. This deals with optimizing performance in a partially unknown system with external disturbances. Therefore, since the DSTs implicitly optimize some performance objectives, that theory can be used in the development of such stochastic tools, which must be used in uncertain conditions.

The design of DSTs requires trajectory prediction (TP) and it would be necessary to define a formal structure that can be used by them. This would allow to manage the uncertainty prediction independent the TP. However, it would be necessary to develop a system that accommodates the prediction uncertainties generated by different TPs minimizing the impact on the outputs of the DSTs. Another issue that needs to be addressed is the identification of the most useful set of data to reduce prediction uncertainties, when air-ground synchronization is available.

The main state variables that would be used by the DSTs (generally speaking) are 4D position. In future TBO, the Reference Business Trajectory (RBT) will be agreed in advance and followed as much as possible. It is necessary to determine when the update or review process is triggered, how it is influenced by the prediction uncertainties, and how a standard procedure can be defined that avoids triggering updates unnecessarily.

Stochastic flight management systems

While conventional Flight Management Systems (FMS) have been quite successful in the past, the future ATM scenario in which 4-D preferred trajectories have to be implemented in an uncertain environment will require a guidance and control system capable of dealing with uncertain trajectories and uncertain air traffic in a safe and optimal way. Such an FMS should be able to accept uncertain external and internal variables (modelled for instance by using statistics) and generate guidance commands which follow as close as possible the preferred trajectory while preserving safety in all possible situations. Also, the guidance system should try to minimize its own trajectory uncertainty as much as possible, thus rejecting external disturbances. As in the previous case study, since robust control theory deals with optimizing performance in a partially unknown system, this theory can be adapted to develop stochastic FMSs with the required capabilities. However, verification and certification of such an FMS would not be an easy task, since it would be necessary to make sure that it works properly not only in certain nominal conditions, but for certain “nominal uncertainty”. As a consequence of being able to work with uncertain inputs, and of having uncertainty minimization as a performance objective, stochastic FMSs would have the possibility of performing on-board uncertainty mitigation, which could be considered as an important task in the future ATM scenario.

Stochastic safety assessment tools

The future ATM system will demand a considerable improvement in safety levels; to obtain a significant increase in safety, new unconventional mechanisms will have to be developed. Moreover, the transition from the actual ATM paradigm to the future ATM system devised by SESAR will pose great challenges by itself that will require the creation of new tools to assess and ensure safety levels. One potential area of improvement in conventional safety assessment tools is the inclusion of uncertainties that are present in the ATM system. In this way, once the mechanisms of uncertainty propagation throughout the ATM system are well understood, then control and management rules can be developed to avoid amplification of uncertainties that would greatly decrease the safety of the system.

Recent Developments

This section is devoted to describing recent research results that are relevant to the research theme of uncertainty in ATM. If you have any related results that you wish to contribute, please feel free to add your contribution as a subsection below. Also consider linking your results with relevant portions of the main text of the article and/or other articles (e.g. related research lines) to increase its visibility and help give it a context inside the research theme.

Systemic Delay Propagation in the US Airport Network

The following presentation is a short summary of our current work on delay propagation dynamics - Download it or view it in your browser:
File:Data driven modeling of systemic delay propagation.pdf

Or view it in page here:

For more information please visit Analysis of air transportation using complex networks.

A 4D micro-model framework to enhance an integrated view of the ATM for test-bed research

The Test-bed Platform for ATM Studies (TPAS) is a computer-based test-bed environment based on a global 4D micro-scale perspective of the full European ATM (dynamic) system; see Ruiz et al. (2013a). The tool has been designed to provide a framework for the fast-prototyping, development and evaluation of new ATM tools as well as the verification of new ATM concepts that may contribute to a better understanding of the ATM system.

TPAS Architecture.

TPAS can efficiently manage, represent and store ATM information of micro-level objects (such as 4D trajectories either in SBT or RBT format, airspace data and weather information) with a global and n-dimensional view. Therefore, this framework shall enable a common overall sight of the ATM current and predicted states, thus allowing the anticipated observation and management of the potential emergent dynamics that may appear in the network as a consequence of the local decision-making processes that are based on a narrowed perspective of the network. One example can be found in some conflict resolution procedures of Air Traffic Control (ATC) that do not consider the effects of a trajectory resolution maneuver downstream in other sectors; see Bilimoria et al. (2000) and Krozel et al. (2001).

The TPAS platform has been evaluated in several European scenarios to test a Strategic De-confliction (SD) decision support tool, provided a (simplified) microscopic 4D trajectory model of the traffic flows; see Ruiz et al. (2014) and Ruiz (2013). This SD tool has been based on the technologies included in TPAS and is able of processing thousands of 4D trajectories at the European ATM level and with a global optimization approach in a few seconds. Some types of uncertainty were also introduced in these simulations performed with TPAS, thus allowing the observation and study of how the uncertainty affects and modifies the results with respect the nominal scenarios.

Ruiz (2013) presented some strategies for tackling different types of ATM uncertainty, all of them described either as a potential implementation or as an already-implemented service in TPAS:

  • Spatial safety buffers:
Navigational imprecision and tracking errors can be tackled by adding uncertainty buffers to the SBTs/RBTs. The sources of trajectory uncertainty can be either the error in the input data measurement, the error during the data processing (e.g., due to the limitations of the model) or the error in the operational practices of the airspace user, thus altogether forming the total system error (TSE) that can be fitted within a certain tolerance or buffer.
Some simulations have been performed by Ruiz (2013) setting the SD algorithms of TPAS with different sizes of spatial buffers and applied to the conflict detection and resolution process with different combinations, i.e., applied to conflict detection process only, to conflict resolution process only, or to both of them simultaneously.
Note that the use of extra safety distance (i.e., buffer) to give tolerance for the navigational and tracking errors with a buffer-size “bigger than necessary”, can also contribute to partially give system tolerance to delays and trajectory deviations of certain dimensions (the bigger the buffer the bigger the dimensions tolerable), thus achieving a more stable flight route configuration. In this case, the trade-off between the robustness and the capacity of the system must be taken into account.
Results suggest that the application of 10NM of safety distance minimum separation (i.e., 5NM of spatial buffer applied to both the CD and CR processes) among all the 4D trajectories at the pre-tactical phase and for the whole duration of each flight may reduce significantly the traffic de-synchronization effects (i.e., conflicts) derived from some sources of uncertainty, i.e., the wind prediction errors and the flight delays, while the losses in ATM capacity and average flight efficiency might be notably compensated by the benefits of a more robust traffic synchronization.

  • Temporal looseness:
The concept of temporal/longitudinal looseness, λ, gives information about how much time a trajectory can be advanced or delayed without entering in conflict with another (nominal) trajectory (whilst preserving the speed profile defined in the flight plan).
Concept of temporal looseness: in the example of the figure the current SBT/RBT could be advanced 3 units of time or delayed 5 units of time without causing any conflict in the network.
TPAS can be set for calculating λ, which can help to identify trajectories that are more sensitive to delays (i.e., less robust) and give this relevant network information to the airport controllers in order to prioritize the departures according to such information, thus trying to preserve the planned departure time for those flights that could generate extra workload to ATC officers.

  • Temporal safety buffers:
The introduction of a temporal buffer extent the temporal utilization of a nominal trajectory for the spatial resources used in the nominal flight. Temporal buffers can be used to give some tolerance to the longitudinal dimension of the trajectories, mostly affected by flight uncertainties (e.g., execution errors and takeoff delays) and weather uncertainties (e.g., wind prediction errors). The concept of contract tube proposed by the PHARE project (see Meckiff (1998)) is a good choice to easily represent the degree of temporal buffer given to a trajectory.
Note that the bigger the temporal buffer, the bigger is the spatial separation among nominal trajectories. Thus, the introduction of temporal buffers to mitigate the impact of delays has direct consequences on the airspace capacity, and therefore there is a trade-off between the ATM capacity loss due to the lack of ATM plan robustness and the airspace capacity loss due to the application of larger time-space buffers.
TPAS provides with some interesting modules that shall allow the introduction of a temporal buffer for each trajectory that can be parameterized according to the statistical distribution of delays in each origin airport. Thus it is possible to control the level of robustness desired for each trajectory and for the network, while being flexible to different policies to deal with the trade-off between flight efficiency and ATM capacity.

  • 4D tubes:
The concept of 4D tube has been explored in many literature references. One of the most relevant references in the context of ATM is given by the PHARE project (see Meckiff (1998)), in which a 4D tube is defined as “series of 4d windows that are used to describe airspace volumes around a trajectory”.
The 4D tubes can be defined taking into account the minimum spatial safety distances plus a buffer and the minimum temporal safety distances plus a buffer.
TPAS has got already implemented the algorithms to build and locate 4D tubes around a 4D trajectory. This 4D tubes have been used by Ruiz et al. (2013b) to model the 4D airspace areas in which the turbulence contrails generated by aircraft can be probabilistically confined during descent approach and landing operations in a Terminal Maneuvering Area, in the context of a Medium Term Conflict Detection and Resolution tool that works under the paradigm of time-based separation operations.
Construction, rotation and translation of a discretized 4D tube over a 4D trajectory.

  • Risk-of-deviation cones:
In the ATM system, when a flight does not follow the expected nominal trajectory (i.e., RBT), within a certain spatio-temporal buffer, it is called a deviation. In such cases, the ATC usually generates control actions (indirectly executed by the pilots, who are expected to execute the control consignees) in order to correct the system deviation and avoid possible conflicts or collisions.
Modern automated CD systems such iFACTS constantly evaluate the probability of collision in a look-ahead time-window of 20 minutes and give this information to tactical controllers so they can be aware of the possible consequences of any potential trajectory deviation. Tactical controllers take action (usually between 6 and 10 minutes before) when the risk associated to a potential collision is considered too high, even when the nominal trajectories may not be in conflict.

Ruiz (2013) proposes that a Strategic De-confliction nominal algorithm can be extended with a risk-of-deviation model based on probabilistic 4D tubes that can be computed for every time-instant of the nominal trajectories, and projecting the uncertainty along a certain time-window look-ahead (typically 20 minutes) after which the evolution of uncertainty can be truncated (controllers are expected to take action before that time in case a deviation is detected). Similar concept was also considered by EUROCONTROL (1997).
A risk-of-deviation model that could be considered during strategic flight planning to reduce the number of tactical amendments.

The parameterization of those probabilistic 4D tubes (i.e., the size and the associated probability values) can be obtained through off-line TP studies and customized for every type of aircraft and weather conditions.
Thus, at every moment in which a distance comparison is performed between two trajectories, which potentially may share a certain 4D coordinate or space, the CD algorithm should multiply the probability values (i.e., intersection of probabilities) in which those aircraft may cross such airspace region. Given a threshold (considered safe-enough), the CD can determine if the risk of collision is too high and thus consider it as a conflict if necessary (thus the CR will provide resolution alternatives).

  • Microscopic trajectory model and uncertainty propagation:
Any disturbances originated at the microscale (e.g., a flight delay) may propagate through the system and disturb aviation not only on the mesoscale but also on the macroscale. Therefore, it is important to study how uncertainty in flights and air traffic (microscopic/mesoscopic scale) propagates to affect the macro-scale and also to study how the operational uncertainty and strong disturbances may affect the air transport network (e.g., adverse weather) at the level of traffic and flight trajectories.
Note that only with a global network perspective of the current and future predicted states it is possible to observe the potential emerging dynamics derived from the individual decisions made or disturbances occurred at local level, e.g. a resolution trajectory generated for solving a conflict between two trajectories or a delay applied to a flight could generate new interactions (i.e., downstream conflicts) and/or new delays that previously did not exist in the network.
In fact, according to SESAR, any new trajectory proposition, revision or update will be made in due consideration of the complete trajectory still to be flown and not only at sector level, taking due account of the wider impact on other flights’ concerned trajectories as well as on the network operations (i.e., domino effects/emergent dynamics).
To observe and anticipate the propagation of any local decision or disturbance across the network, the TPAS software includes the technology of Spatial Data Structures (SDSs) that have the capability to efficiently store spatial data (e.g. 4D trajectory information), which together with the efficient database access methods have been a key factor for the development of new tools to analyze the entire ATM state-space information under a global/macroscopic scope. This enables for instance a complete and precise identification of the emergent dynamics that new a disturbance may cause in the network, since all the processed trajectories remain stored as a “4D snapshot” of the ATM system (see Ruiz (2013) and Ruiz et al. (2013a)).

  • Flexibility (or fast adaptability to network changing conditions)
In SESAR, flexibility is the ability to adapt the ATM planning to unexpected network changes.
Note that in the current ATM model the flight planning processes are conducted by the airlines off-line and prior to the execution phase and expressed through the FPLs. Thus, under this framework there is little flexibility (almost null) on either re-planning a flight or reconfiguring the airspace structure (e.g., routes) during the tactical ATFCM and/or the ATC procedures at execution phase.
The introduction of modern communication, navigation and surveillance technologies combined with the development of specific ATM procedures is intended to provide traffic managers with a greater degree of flexibility in dynamically reconfiguring airspace to adapt to changing conditions (e.g., convective weather disruptions, Flexible Use of Airspace or any other unforeseen event) and to user-preferred routing; see Zelinski et al. (2012), Kopardekar et al. (2007) and EUROCONTROL (2013).
The fast updating rate of the (4D/nD) ATM state-space micro-scale model provided in TCAS can be a contributor to support different decision support processes in real-time applications, which in turn it may also contribute to deal with some sources of uncertainties that often cause unexpected changes in the network (e.g., delays, weather, wind prediction errors, trajectory deviations, and so on). In Ruiz et al. (2013a) it is presented the (4D/nD) ATM state-space micro-scale model used in TPAS whereas in Ruiz et al. (2013b) and Ruiz et al. (2014) are respectively presented a MTCD and SD tools which benefit of the high flexibility provided by this software framework.

  • Probabilistic 4D Trajectories:
The concept of RBT is used in SESAR as the optimal reference trajectory that considers the airspace user preferences and the network restrictions from the origin airport up to the destination airport. In Ruiz (2013) is proposed, as part of a future work, to extent this concept with the introduction of probabilistic trajectories (or perhaps probabilistic trajectory segments).
Probabilistic SBTs/RBTs can be helpful to tackle some sources of ATM uncertainty (e.g., weather uncertainties).

Note that the information about the probability of occurrence of a certain event (e.g., a severe thunderstorm) is more precise the closer is the prediction with respect the potential time of occurrence of the event. Thus, different probabilistic trajectories should be considered in those cases in which a certain event could happen with a certain probability and may affect the normal execution of a nominal RBT. Therefore, the definition of “optimal RBT” shall be reconsidered with regards to such probabilistic information of the event. For instance, see the figure above in which a particular flight is taking off at time t0 and a certain event (i.e., a severe thunderstorm) is predicted (at time t0) to likely happen with a probability p=0.5 at time t0+60’. Let us consider that due to the nature of the event, the prediction/probability of the event will be perfectly known 30 minutes before the event, thus in t0+30’ it will be perfectly known if the event will actually happen (p=1) or not (p=0). If the event would not happen, the preferred trajectory for that flight would be trajectory a in the figure, whereas if the event would finally happen the preferred trajectory would be trajectory d. Thus, the optimal decision in t0 with the information available should be to fly to an intermediate point at t0+30’ (ideally calculated taking into account the optimization logic of the Airspace User for that flight and weighing up the probabilistic information about the occurrence of the uncertain event) thus flying the first segment of trajectories c and d. In time t0+30’, when the information about the actual occurrence of the event will become more precise, the rest of the RBT segment will be decided, which could be the segments of trajectory c (in case that p=1 at t0+30’) or trajectory b (if p=0 at t0+30’).
Note that (in this case) at t0 both trajectories b and c have the same probability of finally being flown, thus both trajectories should be considered as probabilistic RBTs and the potential interactions of these trajectories with other flights should be weighed up with their respective probability values associated.
The conflict detection and resolution platform included in TPAS can be adapted to support this kind of probabilistic models in which the probabilistic information of the trajectories can be stored as extra n-dimensional information in the SDS. Also note that the high updating rate of the state space content and of the strategic de-confliction process is a key factor to take better decisions as soon as the probabilistic information associated to certain ATM events becomes more precise with the pass of the time.

  • Summary table of the TPAS strategies presented and their potential applications to tackle ATM uncertainty
TPAS strategies to tackle uncertainty Trajectory uncertainty Flight uncertainty Traffic uncertainty Network uncertainty Weather uncertainty
Spatial Buffers -Trajectory execution errors
-Tracking errors
-Minimize risk of conflicts
Temporal looseness -Measure of longitudinal robustness -Measure of longitudinal robustness
Temporal buffers -Longitudinal execution errors
-Tracking errors
-Can tackle little/moderate delays -Robust synchronization of traffic
-Minimize risk of conflicts
-Tackle little/moderate wind prediction errors
4D tubes -Trajectory execution errors
-Longitudinal execution errors
-Tracking errors
-Can absorb little/moderate delays -Robust synchronization of traffic
-Minimize risk of conflicts
-Tackle little/moderate wind prediction errors
-Modeling of stochastic behavior of aircraft wake vortex
Risk-of-deviation cones -Minimize risk of collision and the risk of conflicts
-May reduce the ATC officers' workload (if the 4D trajectories are strategically planned taking into consideration the risk of deviation present during the flight execution)
-More stable ATM desing may be achieved if the ATC officers do not observe tactical potential conflicts during flight execution
Microscopic model and uncertainty propagation -Anticipation of domino effects -Anticipation of trajectory, flight and traffic uncertainties and their propagation across the network
Flexibility -Fast adaptation to trajectory changes -Fast adaptation to flight changes -Fast adaptation to traffic changes -Fast adaptation to network changes -Fast adaptation to weather changes
Probabilistic 4D trajectories -To tackle unexpected traffic events -To tackle unexpected network events -To tackle unexpected weather events


  • Albert R. and Barabási A.-L., 2002. Statistical mechanics of complex networks, Review of Modern Physics, 74, pp. 47-97.
  • Albert, R., Jeong, H. and Barabási, A.-L., 2000. Error and attack tolerance of complex networks. Nature 406, pp. 378-382.
  • Ahnert, S. E., Garlaschelli, D., Fink, T. M. A. and Caldarelli, G., 2007. Ensemble approach to the analysis of weighted networks. Phys. Rev. E 76, 016101.
  • Alliot, J.M., Bose, J.F., Durand, N. and Maugis, L., 1997. CATS: A Complex Air Traffic Simulator, Proceedings IEEE 1997, pp. 8.2-30-37.
  • Alonso, A., Escudero, L. F. and Ortuno, M. T., 2000. A stochastic 0-1 program based approach for the air traffic flow management problem, European Journal of Operational Research, 120, pp. 47-62.
  • Anderson B. and Moore J., 1979. Optimal Filtering, Prentice-Hall.
  • Andersson K., Carr F., Feron E. and Hall W.D., 2000. Analysis and Modeling of Ground Operations at Hub Airports, 3rd ATM Seminar.
  • Andrews J.W., Erzberger H. and Welch J.D., 2006. Safety analysis for advanced separation concepts. ATC Quarterly 14, pp. 5-24.
  • Arribas A., Robertson K.B., and Mylne K.R., 2005. Test of a poor man’s ensemble prediction system for short-range probability forecasting, Monthly Weather Review, 133 (7): 1825-1839.
  • Balakrishna P., Ganesan R. and Sherry L., 2010. Accuracy of reinforcement learning algorithms for predicting aircraft taxi-out times: A case-study of Tampa Bay departures, Transportation Research Part C, vol. 18, pp. 950–962.
  • Bagler G., 2004. Analysis of the Airport Network of India as a complex weighted network, Physica A 387, pp. 2972-2980.
  • Berz M., Makino K., Shamseddine K., and Wan W., 1999. Modern Map Methods in 
Particle Beam Physics, Elsevier Science.
  • Blom H.A.P. and Bakker G.J., 2011. Safety of advanced airborne self separation under very high en-route traffic demand, 1st SESAR Innovation Days.
  • Blom H.A.P., Stroeve S.H. and De Jong H.H., 2006. Safety risk assessment by Monte Carlo simulation of complex safety critical operations, Proc. 14th Safety-critical Systems Symposum, Bristol, UK, Eds: F. Redmill and T. Anderson, Springer.
  • Bocaletti S., Latora V., Moreno Y., Chavez M. and Hwang D., 2006. Complex Networks: Structure and Dynamics, Physics Reports, 424, pp. 175-308.
  • Bonnefoy F. and Hansman R.J., 2007. Potential Impacts of Very Light Jets on the National Airspace System, Journal of aircraft 44, pp. 1318-1326.
  • Cardillo, A., Zanin, M., Gómez-Gardeñes, J., Romance, M., del Amo, A. J. G., Boccaletti, S., 2013. Modeling the multi-layer nature of the European air transportn etwork: resilience and passengers re-scheduling under random failures, The European Physical Journal Special Topics, 215, 23–33.
  • Chang Y.-H., Clarke J.-P. and Johnson E.L., 2010. Stochatic programming approaches to air traffic flow management under uncertainty of weather, 12th International Conference on Stochastic Programming.
  • Clarke, J., Solak, S., Chang, Y., Ren, L. and Vela, A., 2009. Air Traffic Flow Management in the Presence of Uncertainty, Proceedings of the 8th USA-Europe ATM Seminar ATM2009.
  • Clauset, A., Moore, C. and Newman, M. E. J., 2008. Hierarchical structure and the prediction of missing links in networks. Nature 453, pp. 98-101.
  • Cook A., Tanner G., Williams V. and Meise, G., 2009. Dynamic cost indexing: managing airline delay costs, Journal of Air Transport Management 15, pp. 26-35.
  • Cook A., Tanner G. and Enaud P, 2010. Quantifying airline delay costs - the balance between strategic and tactical costs. 14th Air Transport Research Society (ATRS) World Conference, Porto, July 2010.
  • Cook A. and Tanner G., 2011. Modelling the airline costs of delay propagation, 2011 AGIFORS Airline Operations Conference.
  • Cook A. and Tanner G., 2011. European airline delay cost reference values, Commissioned by EUROCONTROL Performance Review Unit, http://www.eurocontrol.int/documents/european-airline-delay-cost-reference-values
  • Cook A., Tanner G., Perez D. and Cristobal S., 2011. POEM Project: Passenger-Oriented Enhanced Metrics, 1st SESAR Innovation Days.
  • Crisostomi E., Lecchini-Visintini A. and Maciejowski J., 2009. Combining Monte Carlo and worst-case methods for trajectory prediction in air traffic control: A case study, Automatic Control in Aerospace, 2, 1-14.
  • Dallavalle, J. P., M. C. Erickson, and J. C. Maloney III, 2004. Model output statistics (MOS) guidance for short-range projections. Preprints. 20th Conf. on Weather Analysis and Forecasting/16th Conf. on Numerical Weather Prediction, Seattle, WA, Amer. Meteor. Soc., 6.1.
  • Dorogovtsev S.N. and Mendes J.F.F., 2002. Evolution of networks, Advances in Physics, 51, 1079-1187.
  • Dunbar M., Froyland G., Wu C.-L., 2012. Robust Airline Schedule Planning: Minimizing Propagated Delay in an Integrated Routing and Crewing Framework, Transportation Science, Volume 46, Issue 2, 204-216.
  • Durand, N. and Alliot, J.M., 1997. Optimal Resolution of En Route Conflicts, 1st USA/EUROPE ATM R&D Seminar.
  • Dutta, P. and Bhattacharya, R., 2010. Nonlinear Estimation of Hypersonic State Trajectories in Bayesian Framework with Polynomial Chaos, Journal of Guidance, Control and Dynamics, 33, 1765-1778.
  • Ebert, E. E., L. J. Wilson, B. G. Brown, P. Nurmi, H. E. Brooks, J. Bally, and M. Jaeneke, 2004. Verification of Nowcasts from the WWRP Sydney 2000 Forecast Demonstration Pro-ject. Wea. Forecasting, 19, pp. 73–96.
  • EUROCONTROL, 1997. PHARE highly interactive problem solver version 4 API definitions.
  • EUROCONTROL, 1999. Guidance Material for the Design of Terminal Procedures for DME/DME and GNSS Area Navigation.
  • EUROCONTROL, 2010, ATFCM User Manual. Ed.14.0.
  • EUROCONTROL, 2009. Performance Review Report 2008: An Assessment of Air Traffic Management in Europe during the Calendar Year 2008, EUROCONTROL Performance Review Commission, Brussels, May 2009.
  • EUROCONTROL, 2012. Performance Review Report 2011: An Assessment of Air Traffic Management in Europe during the Calendar Year 2011, EUROCONTROL Performance Review Commission, Brussels, May 2012.
  • EUROCONTROL, 2013. SESAR concept of operations Step 2.
  • European Commission, 2010, Commission Regulation No 255/2010 of 25 March 2010 laying down common rules on air traffic flow management.
  • European Commission, 2011. White Paper: Roadmap to a Single European Transport Area – Towards a competitive and resource efficient transport system.
  • Everdij M.H.C., Klompstra M.B., Blom H.A.P. and Obbink B.K., 2006. Compositional specification of a multi-agent system by stochastically and Dynamically Coloured Petri Nets. Eds: H.A.P. Blom, J. Lygeros. Stochastic Hybrid Systems: Theory and Safety Critical Applications, LNCIS series, Springer, pp. 325–350.
  • Feller W., 1968. An Introduction to Probability Theory and Its Applications: v.1, Wiley.
  • Fisher, J. and Bhattacharya, R., 2011. Optimal Trajectory Generation with Probabilistic System Uncertainty Using Polynomial Chaos, Journal of Dynamic Systems, Measurement, and Control, 133, 014501-1-6.
  • Grewal M.S., Weill L.R. and Andrews A.P., 2000. Global Positioning Systems, Inertial Navigation, & Integration, John Wiley & Sons.
  • Grimmet G., 2001. Probability and Random Processes, OUP Oxford.
  • Guimera R. and Amaral L.A.N., 2004. Modeling the world-wide airport network, Europena Physics Journal B 38, pp. 381-385.
  • Guimera, R. and Sales-Pardo, M., 2009. Missing and spurious interactions and the reconstruction of complex networks. Proc. Nat. Acad. Sci. USA 106 (52), pp. 22073-22078.
  • Hacker J.P., Krayenhoff E.S., and Stull R.B., 2003. Ensemble experiments on numerical weather prediction error and uncertainty for a North Pacific forecast failure, Weather and Forecasting, 18 (1): 12-31.
  • Halder, A. and Bhattacharya, R., 2011. Dispersion Analysis in Hypersonic Flight During planetary Entry Using Stochastic Liouville Equation, Journal of Guidance, Control and Dynamics, 34, 459-474.
  • Hastings W., 1970. Monte-Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 97-109.
  • Hauf, T., 2010. Intelligent Modelling the Impact of Unpredictable Adverse Weather on ATM Performance, SESAR WP-E ComplexWorld Ph.D. Proposal.
  • Heidt, A., 2010. Uncertainty Models for Robust and Optimal ATM Schedules, SESAR WP-E ComplexWorld Ph.D. Proposal.
  • Helme, M. P., 1992. Reducing air traffic delay in a space-time network, in IEEE international conference on systems, man, and cybernetics, pp. 236-242.
  • Holme, P., Kim, B. J., Yoon, C. N. and Han, S. K., 2002. Attack vulnerability of complex networks. Phys. Rev. E 65, 056109.
  • Isaacson, D. and Erzberger, H., 1997. Design of a Conflict Detection Algorithm for the Center/TRACON Automation System, in Proceedings of the 16th Digital Avionics Systems Conference, Irvine, CA, pp. 9.3-1 - 9.3-9.
  • Kelly F. and Yudovina E., 2014. Stochastic Networks, Cambridge University Press.
  • Kim, J., Tandale, M. and Menon, P.K., 2009. Air-Traffic Uncertainty Models for Queuing Analysis, AIAA Aviation Technology, Integration and Operations Conference (ATIO), paper AIAA 2009-7053.
  • Knorr D. and Walter L., 2011. Trajectory Uncertainty and the Impact on Sector Complexity and Workload, 1st SESAR Innovation Days.
  • Kotegawa T., DeLaurentis D.A. and Sengstacken A., 2010. Development of network restructuring models for improved air traffic forecasts, Transportation Research Part C, 18, pp. 937-949.
  • Kopardekar., P., Bilimoria, K., and Sridhar, B., 2007. Initial concepts for dynamic airspace configuration. In AIAA aviation technology, integration, and operations Conference; 7th, AIAA aviation technology, integration, and operations.
  • Kuchar, J.K. and Yang, L.C., 2000. A review of conflict detection and resolution modelling methods, IEEE Transactions on Intelligent Transportation Systems, 1, 179-189.
  • Lauderdale T.A., and Erzberger H., 2014. Automated Separation Assurance with Weather and Uncertainty, in Air Traffic Management and Systems, Springer Japan, pp. 35-47.
  • Li W. and Cai X., 2004. Statistical analysis of the airport network of China. Physical Review E 369, 16. 046106.
  • Li X., Nair P.B., Zhang Z., Gao L., and Gao C., 2014. Aircraft Robust Trajectory Optimization Using Nonintrusive Polynomial Chaos, Journal of Aircraft, in press: pp. 1- 12, doi: 10.2514/ 1.C032474.
  • Lillo F., Micciche S., Mantegna R.N., Beato V. and Pozzi S., 2011. ELSA Project: Toward a complex network approach to ATM delays analysis, 1st SESAR Innovation Days.
  • Lillo F., Pozzi S., Tedeschi A., Ferrara G., Matrella G., Lieutaud F., Lucat B. and Licu A., 2009. Coupling and Complexity of Interaction of STCA Networks, EUROCONTROL 8th Innovative Research Workshop & Exhibition.
  • Liu J. and Chen R., 1998. Sequential Monte-Carlo methods for dynamic systems, J American Statistical Association 93, 1032-1044.
  • Lu C., Yuan H., Schwartz B.E., and Benjamin S.G., 2007. Short-range numerical weather prediction using time-lagged ensembles, Weather and Forecasting, 22 (3): 580- 595.
  • Mao X., Roos N. and Salden A., Stable Multi-project Scheduling of Airport Ground Handling Services by Heterogeneous Agents, Proc. of 8th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2009).
  • Matthews, M. P., Wolfson, M. M., DeLaura, R. A. and Evans, J. E., 2009. Measuring the Uncertainty of Weather Forecasts Specific to Air Traffic Management Operations, 89th Annual Meeting – ARAM Special Symposium.
  • Meckiff, C., Chone, R. and Nicolaon, J.P., 1998. The tactical load smoother for multi-sector planning. Presented at the Second USA/Europe Air Traffic Management Research and Development Seminar.
  • von Mering, C., Krause, R., Snel, B., Cornell, M., Oliver, S. G., Fields, S. and Bork, P., 2002. Comparative assessment of large-scale data sets of protein-protein interactions. Nature 417 (6887), pp. 399-403.
  • Mukherjee, A. and Hansen, M., 2009. A dynamic rerouting model for air traffic flow management, Transportation Research Part B, 43, pp. 159-171.
  • Narkawicz, A. and Muñoz, C., 2011. State-based implicit coordination and applications, Technical Publication NASA/TP-2011-217067.
  • Newman, M.E.J., 2010. Networks: An Introduction. Oxford University Press.
  • Nilim, A., El Ghaoui, L. and Duong, V., 2004. Algorithms for multi-aircraft routing under uncertainty, RIVF International Conference on Computing and Communication Technologies, pp. 21–32, 2004.
  • Nilim, A., El Ghaoui, L., Hansen, M. and Duong, V., 2003. Trajectory-Based Air Traffic Management (TB-ATM) under Weather Uncertainty, Proceedings of the 5th USA-Europe ATM Seminar ATM2003.
  • Odoni, A. R., 1987. The flow management problem in air traffic control, in Flow control of congested networks (Odoni, A. R., Bianco, L., and Szego, G., eds.), Springer Verlag, pp. 269–288.
  • Ohsfeldt, M., Zhu, K., and Wang J, 2011. Pre-departure flight uncertainty of U.S. Oceanic boundary crossing time, Integrated Communications, Navigation and Surveilance Conference (ICNS).
  • Oksendal, B., 2003. Stochastic Differential Equations: An Introduction with Applications, Universitext.
  • Paleari S., Redondi R. and Malighetti P., 2010. A comparative study of airport connectivity in China, Europe and US: Which network provides the best service to passengers?, Transportation Research Part E, 46, pp. 198-210.
  • Pepper, J.W., Mills, K.R. and Wojcik, L.A., 2001. Predictability and Uncertainty in Air Traffic Flow Managemen”, Proceedings of the 4th USA-Europe ATM Seminar ATM2001.
  • Prabhakar, A., Fisher, J. and Bhattacharya, R., 2010. Polynomial Chaos-Based Analysis of Probabilistic Uncertainty in Hypersonic Flight Dynamics, Journal of Guidance, Control and Dynamics, 33, pp. 222-234.
  • Prandini, M., Hu, J., Lygeros, J. and Sastry, S., 2000. A Probabilistic Approach to Aircraft Conflict Detection, IEEE Transactions on Intelligent Transportation Systems, 1, 199-220.
  • Reggiani A., Nijkamp P. and Cento A., 2010. Connectivity and Concentration in Airline Networks: A Complexity Analysis of Lufthansa’s Network, European Journal of Information Systems 19, pp. 449-461.
  • Ruiz, S., Piera, M.A., Nosedal, J. and Ranieri, 2014. A. Strategic de-confliction in the presence of a large number of 4D trajectories using a causal modeling approach. Transportation Research Part C: Emerging Technologies.
  • Ruiz, S., 2013. Strategic Trajectory De-confliction to Enable Seamless Aircraft Conflict Management, PhD thesis (director: M.A. Piera), Universidad Autónoma de Barcelona (Download link: https://cloudup.com/cZ3WuSh3jSC)
  • Ruiz, S. and Piera, M.A., 2013a. Relational time-space data structure to enable strategic de-confliction with a global scope in the presence of a large number of 4D trajectories, Journal of Aerospace Operations, IOS press.
  • Ruiz, S., Piera, M.A., and del Pozo, I., 2013b. A Medium Term Conflict Detection and Resolution system for Terminal Maneuvering Area based on Spatial Data Structures and 4D Trajectories, Journal of Transportation Research part C: Emerging Technologies, Elsevier.
  • Sanchez M., Etxebarria I. and Arranz A., 2011. Dynamic Approaches from Complexity to Manage the Air Transport Network, 1st SESAR Innovation Days.
  • Sengupta P., Tandale M.D., and Menon P. K., 2014. Risk-Hedged Traffic Flow Management Under Airspace Capacity Uncertainties, Journal of Guidance, Control, and Dynamics, Vol. 37, No. 5, pp. 1487-1500.
  • Tomlin, C., Pappas, G. and Sastry, S., 1998. Conflict Resolution for Air Traffic Management: A Study in Multi-Agent Hybrid Systems, IEEE Transactions on Automatic Control, 43, 509-521.
  • Vazquez, R. and Rivas D., 2011. Propagation of Initial Mass Uncertainty in Aircraft Cruise Flight, AIAA Aviation Technology, Integration and Operations Conference (ATIO), paper AIAA AIAA-2011-6899.
  • Vranas, P. B., Bertsimas, D. J. and Odoni, A. R., 1994. Dynamic ground-holding policies for a network of airports, Transportation Science, 28, pp. 275-291.
  • Yen J.W., Zabinsky Z.B. and Serve C.A., 2003. Incorporating Weather Uncertainty in Airport Arrival Rate Decisions, NEXTOR Conference on Air Traffic Management and Control.
  • Valli M., Armellin R., Lizia P.D., and Lavagna M.R., 2013. Nonlinear Mapping of Uncertainties in Celestial Mechanics, Journal of Guidance, Control, and Dynamics, 36: 
  • Vela A.E., Salaun E., Solak S., Feron E., Singhose W., and Clarke J.P., 2009. A Two- Stage Stochastic Optimization Model for Air Traffic Conflict Resolution Under Wind 
Uncertainty, Proc. IEEE/AIAA 28th Digital Avionics Systems Conference.
  • Wilson, J., N. A. Crook, C. K. Mueller, J. Sun, and M. Dixon, 1998. Nowcasting Thunderstorms: A Status Report. Bull. Am. Met. Soc., 27, no.10, pp. 2079 – 2099.
  • Wilson, J. W., E. E. Ebert, T. R. Saxen, R. D. Roberts, C. K. Mueller, M. Sleigh, C. E. Pierce, and A. Seed, 2004. Sydney 2000 Forecast Demonstration Project: Convective Storm Now-casting. Wea. Forecasting, 19, pp. 131–150.
  • Wonga J.-T. and Tsai S.-C., 2012. A survival model for flight delay propagation, Journal of Air Transport Management, vol. 23, pp. 5—11.
  • Wu Z., Braunstein L.A., Colizza V., Cohen R., Havlin S. and Eugene Stanley H.E., 2006. Optimal paths in complex networks with correlated weights: The worldwide airport network, Physical Review E 74, 056104.
  • Zanin, M., 2011. Uncertainty in Complex Networks. Int. J. Complex Systems in Science 1, pp. 78–82.
  • Zelinski, S and Jastrzebski, M., 2012. Defining dynamic route structure for airspace configuration, Journal of Aerospace Engineering.
  • Zhou, T., Lü, L. and Zhang, Y.-C., 2009. Predicting missing links via local information. Europ. Phys. J. B 71 (4), pp. 623-630.
  • Zheng, Q. M. and Zhao, Y. J., 2011. Modeling Wind Uncertainties for Stochastic Trajectory Synthesis, AIAA Aviation Technology, Integration and Operations Conference (ATIO), paper AIAA 2011-6858

This project has received funding from the SESAR Joint Undertaking under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 783287.