Air Traffic Complexity Map Based on Linear Dynamical Systems

Round 1

Reviewer 1 Report

The paper proposes an approximation of a metric already available in the literature to measure the complexity of an airspace. The paper is well written and easy to follow.  Before recommending the paper for publication, the authors should address the following remarks.

a) A few words explaining why the nonlinear dynamical system modeling is to be preferred w.r.t. the other mentioned approaches should be given. What do the authors mean for “quite efficient” (line 145)? 

b) Lines 151-153 are already in 145-148.

c) Lines 181: speed measures are V_i, non X_i

d) Section 3. I’m not very comfortable with sections where there is only one subsection which in turn is further dived into subsubsections. Here, we have 3.1, 3.1.1, 3.1.2, etc. Either there is an error (e.g., 3.1.2 should be 3.2, instead, etc.), or just divide the section as as 3.1, 3.2, etc.

e) Line 208. What SVD stands for?Please extend all acronyms and add a list of acronyms at the end

f) Line 249: not ‘four’ aircraft: only there are shown in Figure 4!

g) The paper considers (see, e.g., Figure 4) only very regular flow patterns, where all the aircraft either converge or diverge, etc. The reality is more disordered: with some aircraft converging, others diverging, etc. How can the proposed approach deal with such circumstances, and to what extent this was an issue in the French case study - and thus how reliable are the results shown in Section 4.2?

h) I’m not sure figures 7 and 8 that show the vector field with uncertainties are so meaningful. Which addition information w.r.t. to the real traffic situation should we get?

i) Line 307. Would it better to start a new subsection just below Figure 9?

j) A few more words on the results obtained in the real case (Section 4.2) are necessary. The reader is just left with a map (Figure 17) that is hard to understand, and no analysis of results in provided.

k) Have the authors tried to solve the problem by means of the “classical” nonlinear dynamical system? In such a case, which are the advantages to use the approach proposed in this paper?

l) Line 412. Please provide details on the funding: project number; etc

m) Line 415: The sentence is incomplete. 

Author Response

Reviewer 1:

a) A few words explaining why the nonlinear dynamical system modeling is to be preferred w.r.t. the other mentioned approaches should be given. What do the authors mean for “quite efficient” (line 145)?

This approach represents more faithfully the traffic's structure.

b) Lines 151-153 are already in 145-148.

Lines 151-153 removed

c) Lines 181: speed measures are V_i, non X_i

Rectified

d) Section 3. I’m not very comfortable with sections where there is only one subsection which in turn is further dived into subsubsections. Here, we have 3.1, 3.1.1, 3.1.2, etc. Either there is an error (e.g., 3.1.2 should be 3.2, instead, etc.), or just divide the section as as 3.1, 3.2, etc.

Error: subsections indexed correctly

e) Line 208. What SVD stands for?Please extend all acronyms and add a list of acronyms at the end

SVD stands for Singular Value Decomposition

List of all acronyms added at the end.

f) Line 249: not ‘four’ aircraft: only there are shown in Figure 4!

There are 4 traffic situations with each 3 aircraft.

g) The paper considers (see, e.g., Figure 4) only very regular flow patterns, where all the aircraft either converge or diverge, etc. The reality is more disordered: with some aircraft converging, others diverging, etc. How can the proposed approach deal with such circumstances,

 Those patterns illustrate the basic situations with respect to different type of singular values in order to illustrate the metric but it can be apply to any traffic situation in order to identify "contraction" in the trajectory patterns.

and to what extent this was an issue in the French case study - and thus how reliable are the results shown in Section 4.2?

For the French situation, the metric is able to clearly identify traffic situation for which relative distances between aircraft is decreasing (convergence situations).

h) I’m not sure figures 7 and 8 that show the vector field with uncertainties are so meaningful. Which addition information w.r.t. to the real traffic situation should we get?

The uncertainty fields are slightly different. On those figures the different too small to see the difference.

We are not sure of the aircraft's position. Hence, the vector field might change depending of the real aircraft's position. The extension with uncertainties captures the possible vector field changes.

i) Line 307. Would it better to start a new subsection just below Figure 9?

Subsection added

j) A few more words on the results obtained in the real case (Section 4.2) are necessary. The reader is just left with a map (Figure 17) that is hard to understand, and no analysis of results in provided.

A sentence has been added in section 4.2.

k) Have the authors tried to solve the problem by means of the “classical” nonlinear dynamical system? In such a case, which are the advantages to use the approach proposed in this paper?

As mentioned in section 2, non-linear dynamical systems could have been computed for complexity measurement (Lyapunov exponents computation ref [10]) but such computation is much more time consuming and it can be extended with uncertainties (which is possible with Linear approximation).

l) Line 412. Please provide details on the funding: project number; etc

The grant number has been added.

m) Line 415: The sentence is incomplete.

Forgot to comment it in the tex file.

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Reviewer 2 Report

The paper propose a method to quantify the complexity of an airspace based on the eigenvalues of a dynamic system. The approach is straightforward and clear, and potentially very useful for the field.

Comments

• In Section 2. the part describing Ref. 12 (Mondoloni) seems to be overly detailed and long. It will be better to add in Ref. 10 (Delahaye) since this work seems to be closely related to Ref. 10.
• \vec{B}, basically the input to the dynamic system and the eigenvector B between line 212 and 213 can be confusing. Better to use a different symbol.
• In addition, the meaning of the matrix A is well explained in the paper, but the explanation for \vec{B} needs to be added.
• In Figure 8, (a) and (b) do not show noticeable difference. It may be better to find and example with noisier track data to clearly show the difference.
• It is unclear whether this metric is calculated at each time step within a given area, or if it is calculated centered on a particular aircraft. It should be clearly stated.
• Figures 13, 14, 15, and 16 are missing axis labels and units.
• The jump from toy example to the French airspace is too drastic. It will be better to have an example of a single sector before the large area.
• The 'accumulated complexity' in line 390 needs better explanation.

Some of the (possible) typos found. There may be more. A thorough inspection is required.

• Line 10, adapted --> adaptable?
• Line 64, modelizes --> models?
• Below line 181 --> '=' missing
• Equations in between line 203 and 215 have periods instead of \cdot for matrix and vector multiplications
• Line 284, X --> matrix A?
• Line 293, eigenvalues\lambda
• Line 323, FL4110

Author Response

Reviewer 2:

• In Section 2. the part describing Ref. 12 (Mondoloni) seems to be overly detailed and long. It will be better to add in Ref. 10 (Delahaye) since this work seems to be closely related to Ref. 10.
• Comments on Ref 12 have been reduced and extra explanations have been added for ref 10.
• \vec{B}, basically the input to the dynamic system and the eigenvector B between line 212 and 213 can be confusing. Better to use a different symbol.
• Changed to U matrix
• In addition, the meaning of the matrix A is well explained in the paper, but the explanation for \vec{B} needs to be added.
• A sentence has been added in section 3.2 to explain the meaning of \vec{B}.
• In Figure 8, (a) and (b) do not show noticeable difference. It may be better to find and example with noisier track data to clearly show the difference.
• The figure shows mainly the positions and speeds which will be used for the LMS regression. It is normal that the associated filed will not be so much different.
•  

• It is unclear whether this metric is calculated at each time step within a given area, or if it is calculated centered on a particular aircraft. It should be clearly stated.
• The metric is calculated at each time step around a reference aircraft .
• Figures 13, 14, 15, and 16 are missing axis labels and units.
• Those figures are dimensionless so no label have been included.
• The jump from toy example to the French airspace is too drastic. It will be better to have an example of a single sector before the large area.
• France has been chosen in order to validated the performance in terms of computation time (10 milli seconds) for more than 2 million trajectory sample.
• The final objective of this work is to be able to plug this metric computation in an optimization loop.
• For sure the metric could be also computed at the sector scale.
•  
• The 'accumulated complexity' in line 390 needs better explanation.
• A 2D point at (x,y) on the map represents all 4D points at those 2D coordinates (x,y).

• Line 10, adapted --> adaptable?
• Appropriate
• Line 64, modelizes --> models?
• Rectified
• Below line 181 --> '=' missing
• Rectified
• Equations in between line 203 and 215 have periods instead of \cdot for matrix and vector multiplications
• Rectified
• Line 284, X --> matrix A?
• The size of matrix X is n x 3. Without uncertainty it is "number of aircraft" x 3 and with uncertainties it is "number of aircraft x 5" x 3.
• The size of matrix A is 2x3. It does not change with respect to the number n.
  
• Line 293, eigenvalues\lambda
• Rectified
• Line 323, FL4110
• Rectified