The manuscript by Michalak et al. introduces a new computational model to predict fault geometry in data-sparse environments. I have the following concerns, which do not necessarily preclude acceptance of the manuscript:

1.    The model is restricted to dip-slip faults w/out elevation uncertainties.

2.    This technique does not differentiate between normal and reverse dip-slip faults.

3.    There are no real-world case studies to validate the model.

4.    The use of Python would be more advantageous for the growing geomodeling community, especially since existing tools like GemPy have already established.

Despite these concerns, I believe the manuscript fits well with the scope of the journal, and I would recommend it for publication.

Overview

The authors propose a method to statistically characterize a fault segment orientation in data-sparse environments. The method relies on a direct triangulation of a faulted horizon, and a statistical analysis of the dip direction of the set of triangles that can be formed using one triangle edge on one of the fault walls and all the triangulation vertices on the other wall. In addition, the authors provide mathematical evidences showing that - under some restrictive hypotheses – their statistical analysis yields exact / robust predictions of fault dip direction. Overall, this manuscript seems to be an improvement on the work of Michalak et al., 2021 (see reference in the paper) that aims at explaining and solving some of the counterintuitive results found by the authors.

To me, the presented method is a useful tool to assess fault geometry in the absence of direct fault observations (it is entirely based on displaced horizon observations), and the mathematical details provide interesting insights to understand why the method works and what its potential caveats are. As such, I consider it can be of interest to the audience of Solid Earth and deserves to be published. However, I also  have several major concerns that would require revision of the manuscript.

Note: despite the concerns listed below, I would like to acknowledge the effort made by the authors to provide all the necessary details and information necessary to understand their work and reproduce it.

General concerns

Most of my concerns come from the fact that the presented method and the related mathematical "proofs" rely on two extremely restrictive assumptions:

• a globally horizontal horizon (i.e., only local variations/noise around a constant mean depth)
• a vertical fault with one single segment (i.e., whatever the subset of the fault surface you consider, it will systematically have the same orientation)

Main concerns:

• I would talk about "mathematical evidences" rather than formal "mathematical proofs" throughout the manuscript (starting with the abstract), and only keep the term "Proof" in the appendices where it effectively corresponds to the proofs of mathematical propositions
• Although the authors defend this choice of such ideal conditions for the sake of mathematical investigations, I would appreciate it if they discussed the expected results of the method in "not so ideal" conditions (e.g., dipping/folded horizons, normal/reverse faults, non-planar fault surfaces, ...). This would help to develop the discussion which is otherwise short.
• I would love to see an example of the application of the method on a real dataset. It could be applied on bathymetric data, as the authors already present it as one of the most direct use cases and discuss the pitfalls associated with such data
• A more detailed point: all the hypotheses are clearly stated in the proofs (in the appendices), but it is not so clear in the main body of the manuscript. Typically, I did not find the information that we assume a horizontal (constant Z) horizon. Please add a paragraph before stating the propositions, to clearly state all the hypotheses, as we can find in the appendices

Minor concerns

• One point is still unclear to me after reading the manuscript: I feel like it is somehow mandatory to know in advance the position of the fault relatively to the data points to apply the method as presented here. This does not sound like a very realistic use case, and it seems quite straightforward to think about applying the method to try to identify unknown faults from horizon data. I would like the authors to clarify this point.
• An important part of the "mathematical evidences" proposed by the authors relies on statistics ("the expectations of the coordinates", ...) whereas the method is presented for data-sparse studies. I agree that using the combinatorial strategy proposed by the authors provides more samples for computing statistics, but to me there remains a bias due to the limited number of “genetic triangles” edges. Again, it would be nice to have some additional discussion on this point

Detailed remarks

See the annotated PDF