the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 Jan 2025
Computational modelling and analytical validation of singular geometric effects in fault data using a combinatorial algorithm
Abstract. This study analyzes the directional properties of geological faults using triangulations to model displaced horizons. We investigate two scenarios: one without elevation uncertainties and one with such uncertainties. Through formal mathematical proofs and computational experiments, we explore how triangular surface data can reveal geometric characteristics of faults. Our formal analysis introduces four propositions of increasing generality, demonstrating that in the absence of elevation errors, duplicate elevation values lead to identical dip directions. For the scenario with elevation uncertainties, we find that the expected dip direction remains consistent with the error-free case. These findings are further supported by computational experiments using a combinatorial algorithm that generates all possible three-element subsets from a given set of points. The results offer insights into predicting fault geometry in data-sparse environments and provide a framework for analyzing directional data in topographic grids with imprecise elevation data. This work has significant implications for improving fault modeling in geological studies, particularly when dealing with limited or uncertain data.
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Status: closed
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CC1: 'Comment on egusphere-2024-3327', Giacomo Medici, 14 Feb 2025
General comments
Very good geo-modelling research with a focus on representation of fault geometries. Please, follow my specific comments to improve the manuscript.
Specific comments
Line 16. “Geometrical” better than “directional” for an abstract.
Line 32. Add other applications in the growning fields of geo-sciences of CO2 storage and geothermal energy. Please, insert the following references for the importance of faults in these two geo-energy fields:
Geothermal energy: Medici, G., Ling, F., Shang, J. 2023. Review of discrete fracture network characterization for geothermal energy extraction. Frontiers in Earth Science, 11, 1328397.
CO2 storage: Nicol, A., Seebeck, H., Field, B., McNamara, D., Childs, C., Craig, J., Rolland, A. 2017. Fault permeability and CO2 storage. Energy Procedia, 114, 3229-3236.
Line 50. Clearly state the 3 to 4 specific objectives of your geo-modelling research by using numbers (e.g., i, ii, and iii).
Page 6. I can see several equations without numbers associated with.
Lines 281-294. This part of the discussion shows paucity of literature. I suggest to back-up your statements with supporting literature.
Line 362. Add a “take home message” for the researchers working in the field.
Figures and tables
Figure 3. You can make the four diagrams closer, gain space and enlarge the overall image. The four blocks are difficult to analyse.
Figure 4c. This is a conceptually different image. It should represent a separate Figure 5.
Figure 6c and d. Same issue here. These are very different images. They should represent a separate figure.
Figure 7c. Improve the graphical resolution of the Figure 7c which is a stereonet.
Citation: https://doi.org/10.5194/egusphere-2024-3327-CC1 - AC3: 'Reply on CC1', Michal Michalak, 23 May 2025
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RC1: 'Comment on egusphere-2024-3327', Anonymous Referee #1, 10 Mar 2025
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.Citation: https://doi.org/10.5194/egusphere-2024-3327-RC1 - AC1: 'Reply on RC1', Michal Michalak, 23 May 2025
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RC2: 'Comment on egusphere-2024-3327', Anonymous Referee #2, 07 May 2025
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
- AC2: 'Reply on RC2', Michal Michalak, 23 May 2025
Status: closed
-
CC1: 'Comment on egusphere-2024-3327', Giacomo Medici, 14 Feb 2025
General comments
Very good geo-modelling research with a focus on representation of fault geometries. Please, follow my specific comments to improve the manuscript.
Specific comments
Line 16. “Geometrical” better than “directional” for an abstract.
Line 32. Add other applications in the growning fields of geo-sciences of CO2 storage and geothermal energy. Please, insert the following references for the importance of faults in these two geo-energy fields:
Geothermal energy: Medici, G., Ling, F., Shang, J. 2023. Review of discrete fracture network characterization for geothermal energy extraction. Frontiers in Earth Science, 11, 1328397.
CO2 storage: Nicol, A., Seebeck, H., Field, B., McNamara, D., Childs, C., Craig, J., Rolland, A. 2017. Fault permeability and CO2 storage. Energy Procedia, 114, 3229-3236.
Line 50. Clearly state the 3 to 4 specific objectives of your geo-modelling research by using numbers (e.g., i, ii, and iii).
Page 6. I can see several equations without numbers associated with.
Lines 281-294. This part of the discussion shows paucity of literature. I suggest to back-up your statements with supporting literature.
Line 362. Add a “take home message” for the researchers working in the field.
Figures and tables
Figure 3. You can make the four diagrams closer, gain space and enlarge the overall image. The four blocks are difficult to analyse.
Figure 4c. This is a conceptually different image. It should represent a separate Figure 5.
Figure 6c and d. Same issue here. These are very different images. They should represent a separate figure.
Figure 7c. Improve the graphical resolution of the Figure 7c which is a stereonet.
Citation: https://doi.org/10.5194/egusphere-2024-3327-CC1 - AC3: 'Reply on CC1', Michal Michalak, 23 May 2025
-
RC1: 'Comment on egusphere-2024-3327', Anonymous Referee #1, 10 Mar 2025
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.Citation: https://doi.org/10.5194/egusphere-2024-3327-RC1 - AC1: 'Reply on RC1', Michal Michalak, 23 May 2025
-
RC2: 'Comment on egusphere-2024-3327', Anonymous Referee #2, 07 May 2025
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
- AC2: 'Reply on RC2', Michal Michalak, 23 May 2025
Data sets
Computational modeling and analytical validation of singular geometric effects in fault data using a combinatorial algorithm - Input and processed data Michał Michalak https://doi.org/10.5281/zenodo.13986509
Model code and software
michalmichalak997/3GeoCombine2: v. 1.0 - Initial release Michał Michalak https://doi.org/10.5281/zenodo.13974878
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