Operator Dynamics Approach to Short-Arc Orbital Prediction Based on the Wigner Distribution

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

All my comments are provided in the attached document.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents a phase-space-based Wigner–Kalman filtering and prediction framework, in which position and velocity errors in orbit determination are treated as the marginals of a Wigner quasi-probability distribution in classical phase space. By introducing a generalized Koopman–von Neumann (KvN) equation derived from operator dynamical modeling (ODM), the temporal evolution of the error distribution is described. The authors further define an effective uncertainty parameter κ to characterise the continuous compression of uncertainty during the filtering update process. Overall, the topic of the paper is novel, the theoretical framework is complete, and the structure is clear. The work exhibits a certain degree of innovation in introducing phase-space operator dynamical concepts into short-arc orbit prediction. The manuscript is generally well organised, the numerical experiments are systematically designed, and the results are reasonably convincing.

However, prior to publication, the following issues still need to be clarified and addressed by the authors in a revised manuscript:

(1) In the experimental design, the orbital inclination is only considered up to 45°. It is commonly recognised that near-polar or polar-orbiting satellites exhibit significantly different characteristics compared with low- and medium-inclination satellites. The rationale for this limitation should therefore be clarified.

(2) Since non-conservative forces are considered in the paper, the satellite geometry and surface properties (e.g., shape, panels, and cross-sectional areas) adopted in the simulation and processing should be appropriately described.

(3) For perturbations such as solar radiation pressure and atmospheric drag, the level of solar activity can significantly affect the modelling accuracy. Higher solar activity generally leads to stronger solar radiation pressure effects and increased atmospheric density, which may in turn result in larger modelling errors. The authors should clarify whether such conditions have been considered in their analysis.

(4) One concern relates to the orbit prediction problem itself. In the presented experiments, it appears that only a single observation update (a 3-minute arc) and a single orbit prediction (a 10-minute arc) are performed, and that the initial state is assumed to be free of orbit state errors, i.e., the reference epoch is treated as perfectly known. This assumption is not realistic, as in practical orbit determination—both during the launch phase and on-orbit operations—the estimated state always differs from the true state. The authors may therefore consider evaluating the effectiveness of the proposed algorithm under different initial state error conditions.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I can see that the authors have addressed several of my comments. That said, my major concerns (i.e., those that would have required more substantial work) are not actually resolved in the current submission, but are instead postponed to future work (see their Responses #2, #4, and #11).

Without going through each of them individually, my overall view at this stage is the following. If we want to see the paper as a mainly mathematical, conceptual methodological contribution, then it could be acceptable to publish it in its current form, viewing the work as an incremental step that may eventually lead to a more complete and operationally validated tool. If, on the other hand, the algorithm is meant to be presented as a practical method that brings concrete advantages in operational settings, the paper falls short of providing convincing evidence. In particular, given that one of the main motivations for introducing the Wigner/Koopman-based machinery is to move beyond a purely Gaussian assumption, a direct benchmark against an EKF/UKF under the same measurement model, noise levels, and scenario settings would have been a natural and necessary comparison. Without such a benchmark, the reader has no concrete way to assess what is gained in practice by leaving the Gaussian framework. More importantly, even from a purely methodological perspective, the framework developed by the authors (while in principle applicable to non-Gaussian uncertainties) is constrained in such a way that, as the authors themselves acknowledge, it ultimately behaves like a standard linear estimator.

To be fair, the authors have sufficiently toned down their claims so that the scope of the paper is now more aligned with the former, more conceptual interpretation, and in this sense it could perhaps be published as such. It is not breakthrough research, but it may still be considered a decent incremental contribution. Yet, I must admit that after reading the paper I am left with the impression that extending the framework to genuinely accommodate non-Gaussian uncertainties is not that straightforward.

Minor comments:

• I would personally suggest adding to the abstract that the proposed framework lays the analytical foundation for a future treatment of non-Gaussian features, in order to give incoming readers a clearer sense of the potential practical usefulness of the method.
• The authors should once again fix some remaining typos in the text (e.g., “establishe”, or the missing capital letter at line 764 in “. in contrast”).

Comments for author File: Comments.pdf

Author Response

Comments 1：I would personally suggest adding to the abstract that the proposed framework lays the analytical foundation for a future treatment of non-Gaussian features, in order to give incoming readers a clearer sense of the potential practical usefulness of the method. Response 1:Thank you for this constructive suggestion. We fully agree that emphasizing the forward-looking value of our framework is important for readers. Accordingly, we have added the following sentence to the abstract: “Although the potential is presently retained only to second order—so that both propagation and update preserve Gaussian form and permit direct Kalman recursion—the framework itself lays the analytical foundation for a future treatment of non-Gaussian features.” This revision is highlighted in earthyellow and can be found in the revised manuscript at lines 8–12 .

Comments 2: The authors should once again fix some remaining typos in the text (e.g., “establishe”, or the missing capital letter at line 764 in “. in contrast”)

Response 2: Thank you for pointing this out. We agree with this comment and apologize for the oversight. We have carefully proofread the manuscript and corrected the spelling and capitalization errors at the specific locations you mentioned. The changes are as follows:

1.  Corrected "establishe" to "established" on Page 3, Line 114.
2.  Capitalized "we" to "We" on Page 12, Line 432.
3.  Capitalized "Flattering" to "Filtering" on Page 16, Line 550.
4.  Capitalized "in contrast" to "In contrast" on Page 28, Line 767.