I'll divide my comments into two sections. The first is changes I think should be made. The second, in the attached handwritten long-form discussion, are things the authors could consider to improve the manuscript but which are not essential. Note that the 'essentials' summarized here also occur in the handwritten notes, which also refer to page and line numbers. 

P4 L17: Range of knowledge. This could really benefit from an example to give explanatory context. 

Figure 1. I don't object to the figure, but I believe it needs a discussion in the body of how these errors happen. I realize that they are often 'black box application' errors, but was any work done to see if some are simply geometric? Can some be avoided by model construction methods? A specific typology is not needed, but perhaps a few words here to give non-specialists a sense of why good tools produce errors? 

P5 21-25. This is weak. This is well discussed in the literature. I'd either contract this to a sentence or do it justice by expanding it. 

P6 4,5 Geological 'area' - The term is vague. Domain? Area of common history? Think about reasonable geological cases where one would or would not put an 'area' boundary.

P6, 18 I'd argue perhaps as an aside that some mapping is intended to identify contradictions or gaps in extant geological theory. In this case a contradiction is desirable in the short term. 

22-26 There is a big conceptual jump here. 'These relations.' I think this really needs a bit of expansion and clarification as it is very important to your idea development. 

P7 13-17 Give an example of such a global/local disparity for context/clarity?

24 These vectors point roughly... not exactly. You eventually refer to them as normals. I think you need to make it clear that these are rough directions not precise vectors? 

27-28 not a volcanologist, but if you are in a vent of a volcano are the rocks extrusive or intrusive? 

P8 10-17  I find the discussion of metamorphism here tricky and perhaps even problematic. Your earlier vectors are grounded in locally observable phenomena. This might not be the case for metamorphism. I expand on this in the handwritten notes but perhaps the paper would be better by not worrying about the complexities introduced by metamorphism (other than to say a unit can be identified as 'metamorphic.'). Like fold volumes, these ?superposed? 4d volumes are going to be messy...

18 Some structural geologists would object and say they map fault volumes. Picky point but...

Erosion surface - see in the handwritten my notes to ongoing work in the terrain simulation field on what erosion surfaces are. Probably out of scope for this paper but... an area of interesting parallel development in CS/graphics.

P9 4, 9  Fabric versus elements. A fold axis is not necessarily a fabric???

15-16 In your intro you cite RMH and KLB a lot. I'd say KLB did a LOT of work thinking about fabrics and representation and you don't mention this in the intro. For example, Burns 1969, 1975 both discuss fabric chronologies...

20-21 This paragraph makes huge leaps. Perhaps 1-2s talking about topological relations like 'meets' so set up your examples? 

P17 31 Cute math. Out of context here. Either expand or remove. 

Figure 13 - give approximate scale please

P31 L13-14 I assume that this problem area is inside the volume of the model not near the boundary. If so, you can make the point that the chance of it being 'noticed' is very small, so such methods as you develop are essential. 'Looking' is not a practical alternative. 

I'd like to see a short appendix (extension to your appendix) that provides some more info on development, tools etc. I realize this is probably available beyond the link but links have a way of disappearing. 

I do think that tightening up the generate versus test versus communicate angle would make your intro and conclusions stronger. In particular, while I 'get' your tables after a lot of time looking at them, they are not inherently useful as communication devices - a novice is going to struggle with them. This is tangential to your paper but in my view understanding is part of correcting. 

Finally, probably a preprint issue but in your figures like 1, 2 I'd work to make the small blocks as large as possible for clarity. Some are quite small and you are not using the page.

Thank you for the chance to review this. RMH 2024

Review of “Consistency-Checking 3D Geological Models”

This manuscript presents an interesting and relevant prototype approach for checking if 3D geological models are topologically consistent with fundamental geological principals. I see many applications of the proposed approach (if it is developed further and accessibly implemented), and suggest that it will be of interest to the readership of GMD. The overall approach is conceptually sound, though I suggest that several aspects should be better explained, or slightly adjusted. Hence, I recommend that the paper could be accepted after the following minor revisions:

1. It is necessary to better define the concept of “material sharing” (perhaps a more appropriate term can be defined?) earlier in the manuscript, and discuss the resulting division of geobody types: those that cannot occupy the same space (e.g., intrusions, sedimentary units, etc.) and those that overprint and so co-exist with older geobodies (e.g., metamorphism, alteration, deformation). Also, as mentioned in a subsequent point, the important topological differences between “material sharing” and “volume occupying” objects should be specifically clarified.
2. I note that hydrothermal alteration is not mentioned in the types of geo-objects listed, and suggest that it needs to be included, given its general importance for many applications (e.g., mineral exploration). Either the “metamorphic” geo-object could be renamed to “metasomatic” (or a similar term) or, if necessary, a new geo-object type included. Similarly, I suggest that “Fold volume” is a misleading term for an (apparently) broad geo-object class (e.g., I presume regional tilting, widespread shearing or other deformation, e.g., a fold and thrust belt, would be represented using this class?). Please consider renaming it to e.g., “deformation volume” or “tectonic domain”. 
3. The Linear and Planar fabric geo-objects do not seem to fit well in this categorization, and are not discussed anywhere in the manuscript after being defined. I would suggest either expanding the explanation of these two geo-objects (and including them in the subsequent topological analyses), or defining them somehow as a different category of “geo-object” (i.e., as partially noted in the text, all of the previously defined geo-objects can be associated with linear and planar fabrics; hence fabric seems an inherent property of a geo-object rather than a distinct entity?).  
4. While I understand the temptation of “completeness”, consider defining / introducing (and presenting in your figures) only half of the topological relationships (Table 1 and 2), rather than including each relationship and its converse. As stated later in the manuscript, only one of each pair of (opposite) relations are needed: I.e., if “A contains B” is defined, then the “B is contained by A” relation becomes redundant. I would suggest either dropping these converse relations or presenting them clearly together (as for the temporal topology relations outlined in Table 2).
5. It is unclear from the text (and from Table 1) how the 9-intersection (9I) model treats material sharing. E.g., If A contains B, do A and B share (some of) the same space, or not? Given the distinction between material-sharing and material-defining (for want of a better term) geo-objects, I think this is an interesting and important distinction in the geological context (which, if I understand correctly, the 9I model is fully capable of capturing, if properly explained). To clarify this distinction, a short example might be given in the text: e.g., if a sill (B) intrudes into host-rock (A), does A contain B? And what if some localized alteration (C) overprints host-rock A such that they locally occupy the same space (i.e., material sharing); does A also contain C? I suggest that these two geometries are topologically distinct in important ways, but that Table 1 does not make it clear which relation is appropriate in each case.
6. The spatial relationships associated with faults and unconformities should also be explicitly addressed, as this has big implications for the subsequent model validation approach. While I understand the need to model these as 2D (embedded) surfaces, I suggest that (for the needs of this paper) they are better treated as arbitrarily thin volumes. E.g., If older and overturned sedimentary unit A is juxtaposed against a non-overturned sedimentary unit B across fault F, is there an A meets B relation? Or just A meets F and B meets F (such that the fault is treated as an arbitrarily thin volume). In the former case, the A meets B relation would lead to an incorrect invalidation of this geological model following the Truth Table (Table 3), while in the latter this situation is elegantly avoided. Equivalent arguments could also be applied to unconformities.
7. While the limitations of representing polarity as a single vector are correctly acknowledged, I suggest that this could be done so earlier in the manuscript. Furthermore, it might be worth mentioning that, given tendency is a vector field (often equal to the gradient of a scalar field used for interpolations), it could be better defined at every voxel or mesh element in a model. If implemented like this, alignment or opposition could be defined at every discrete location in the model, along with the local topology (i.e. based on the adjacent voxels / mesh elements), and the truth-table applied to identify local geological consistency or inconsistency. A model would then be considered globally consistent if all its parts are locally consistent, and any inconsistencies could be quickly located.

Other minor comments:

Page 1, Line 17: What is “natural storage”? I suggest “geological storage” or just “storage”.

Page 1, Line 19: “as well as minimize user problems” – this is a very vague statement. Can the authors be more precise?

Page 1, Line 22: Consistency with what? Please clarify.

Page 1, Line 23: “Space of consistent and inconsistent geological situations” – does this approach define a “space”? I suggest instead that the truth-table is a set of rules. If “space”, then what are the dimensions?

Page 2, Line 1-8: Implicit interpolator quirks (e.g., bubbles) should also be mentioned here, as they seem like a likely source of inconsistencies.

Page 2, Line 8: Why equiprobable? Surely a set of model realizations will contain many probable models, as well as some less probable ones (that are still consistent with the data)? I.e. samples drawn from a posterior distribution won’t be equiprobable?

Page 2, Line 10: ditto. (Line 14 also)

Page 2, Line 12: Lyell, 1833 and 2022 – that’s an impressive career!

Page 2, Line 15: Clarify what is meant by “hypothesis testing”

Page 2, Line 18: I would suggest that the number of models is not an issue; their quality (and validity) is. Rephrase.

Figure 1f: How is the topography relevant? I would remove this.

Page 4, Line 4: Note also that the visualization of complex geomodels is a significant challenge, making manual validation difficult, time-consuming and likely to miss problems.

Page 5, Line 1: Important why? Please explain.

Page 6, Line 3: Is topology (e.g., unit A intrudes unit B, foliation C crosscuts foliation D) knowledge or data under this definition? I would suggest that the observed relation is data, but it’s implications for timing is knowledge. Given the importance of spatial topology for this work, and its link to timing, I would suggest clarifying this in the text (perhaps with a small example).

Page 7, Line 19: “age direction vector” – I would suggest that this is a (directed) temporal topology relation, not a vector.

Page 15, Line 2: Use appropriate term from Table 1 (meets?)

Page 17, Line 23: “covers” and “covering” – distinguish these from the “cover” relationship defined in Table 1?

Page 19, Line 3: Material sharing has not yet been defined?

Page 19, Line 8: “Relata” and “relatum”; why not “relations” and “relation”?

Figure 4: What is BREP?

Page 22, Line 15: Note the strong link between spatial and temporal topology (e.g., xcutting relationships etc.; Thiele et al. 2016, many publications of K. Burns, etc. )

Page 23, Line 5: This would arguably be much simpler / easier in a discretized (gridded) form? As topologies could be rapidly computed for cell neighborhoods and then aggregated.

Page 23, Line 15-21: Consider using the “structural topology” and “lithological topology” terms from Thiele et. al., (2016) to more clearly explain this distinction.

Page 24, Line 1: Can the “Equivalent to” temporal relationship be used to resolve this issue, by building a graph of the more detailed structural topology and expanding the temporal relationships to fit this by adding “equivalent to” edges between all structurally distinct volumes of the same lithology?

Figure 14: What is the y-axis in this figure?

Page 38, Line 19: What makes a model “ideal”?

Page 39, Line 19: What is “parthood”?

Further minor language suggestions are included in the attached pdf.

Kind regards,

Sam Thiele

INTENTION OF THE PAPER

The paper addresses an issue of particular importance: checking the validity of multiple equiprobable geo-models possibly constructed by using automated methods. It is a pioneer work that examines how the application of formalized geological rules can help deciding which models are valid and which are not.

In order to let properly appreciate the interest of the paper and its possible limitations or defects, I will first operate a critical analysis of the text. I will then formulate a general advice and some research suggestions.

TEXT CRITICAL ANALYSIS

Part 2 Geological Consistency Checking Framework

Considered geological entities (“geo-objects”)

The term “geo-object “is confusing because these entities have different natures:

• 3D Objects: Depositional / Eruptive: Intrusive Units
• Surfaces: Fault Surface, Erosion Surface. Stratigraphic Boundaries are not considered although they constitute essential element in many 3D earth models. The authors do not think necessary considering these surfaces since they are just the boundaries of Depositional Units.
• Geological Structures (as defined in GeoSciML, GeoCore[1]) : Folds, Linear or Planar Fabrics. Deciding whether Metamorphic Units are Geological Objects or Geolgical Structures could be an object of discussion.

Polarities

• The gross polarity between two “geo-objects” is merely an age relationship (older to younger)
• The internal gross polarity with a geo-object is a growth direction vector which indicates the direction in which the geological process associated with the geo-object has progressed. Geologically this makes sense for most of the considered geological entities with some exceptions (faults, not depositional planar fabrics). In some cases, the internal gross polarity may be difficult to characterize. This is the case in  an  intrusive in which  local polarities correspond to the cooling directions. The gross polarity is related to the field of local polarities in a way that the authors do not specify.

The authors specify how gross internal polarities can be used in practice only in the cases of depositional units or intrusive. In the cases of metamorphic units, folds and not depositional fabrics, the interest of their use is not demonstrated. 

Spatial relations

They are considered only in the case of “spatial objects” i.e. 3D /2D / 1D entities (geological volumes / geological surfaces / lineations).  

Truth tables

• Some of the general principles exposed in page 17 suffer exceptions:
• Lateral continuity: heterochronous geological units are not uncommon.
• Paleontological identity: geological units are homochromous only if they bear the same association of stratigraphic fossils.
• Principle of inclusion: not applicable to geodes for instance.
• The authors mention that 45 truth tables can be established corresponding to “45 valid pairwise combinations of objects” but the example they give in table 3 only corresponds to the simplest of these combinations: the combination of two depositional units. I am not convinced that they could illustrate as well any of the 44 remaining cases.
• There is a punctual mistake in table 3. The case corresponding to the cell of column 1, line 3: Depositional Unit A spatially meeting Depositional Unit B but temporally preceding Unit B with aligned polarities for the two units, is valid.  The temporal interval of time between the depositions of units A and B may simply correspond to a period of non-deposition
• Material sharing: the material sharing rules adopted introduce severe limitations. For instance, the space-time association between a stratigraphic unit U1 of rank 1 and a unit U2 of rank 2 part of U1, will be considered invalid.
• Spatial relation matrix: the authors warn that “the entities related are the whole objects” and not fragments of them. This simplification is operated for practical computation reasons but the authors admit that it is problematic.  They indicate that, in the case of an object A having multiple spatial relations with an object B, “the most dominant relation is selected”.   But they do not specify how this selection is operated.  This introduces another severe limitation in their method.

Part 5 Discussion

The point about spatial relations mentioned in page 37 lines 4-7 is of special importance. In my opinion, it may be easier to consider surface rather than volume representations as relata in the special relations considering that:

• geological surfaces belong to two categories: polarized (horizons, erosion surfaces, intrusive external boundaries) and not polarized (faults, thrust surfaces),
• there exist only three types of relations between two geological surfaces: disjoint, interrupts /interrupted_by , splits /split_by,
• a majority of geo-modelers rest on BREP representations.

GENERAL APPRECIATION AND DISCUSSION  

As I mentioned before, the paper addresses an issue of paramount importance. In its principle, the validation approach presented is interesting but it is extremely ambitious. The authors intend to establish general validity rules resting on only three kinds of relations (spatial, temporal and polarity relations) and to apply them to a large variety of geological entities: geological units (objects), erosion surfaces, faults (surfaces), metamorphic assemblage, folds, fabrics (geological structures). The use of validation tables helps making this approach operational.

I see some difficulties in such a general approach. Internal polarities are defined as the directions in which some geological processes develop. This applies to sedimentation, erosion, magmatic intrusions and extrusions, contact and regional metamorphism and folding. However, the authors only consider the practical use of internal polarities in the case of sediment deposition. I see two difficulties in using internal polarities in the other cases for several reasons:

• In the cases of magmatic intrusions or contact metamorphism local internal polarities, are elements of a gradient. Defining a gross internal polarity, in these cases, may be problematic.
• In some other cases like not depositional 3D fabrics, no internal polarity can be defined.
• The authors give no examples on how gross internal polarities could be used in cases other that those related to sediment deposition. I hardly imagine for instance how fold vergences could be used as a validity criterion.

Another difficulty is due to the fact that the authors consider together 1D, 2D and 3D entities and put a special attention on geological volumes. As mentioned above, this put severe limitations to their approach due to:

1/ the issue of material sharing,

2/ the fact that only whole units are considered and not their partitions in entities of higher stratigraphic rank or their splitting into geological blocks due to faulting.

The authors claim that their paper presents a proof of concepts. But the concepts that they actually prove, cover but a small part of the ambitious issue that they address. The paper then consists for a good part in a list of interesting but yet unsolved research issues. For these reasons, it doesn’t fully meet, in my eyes, the requirements of a classical research paper.

RESEARCH SUGGESTIONS

Some additional research efforts may be necessary for presenting a fully convincing research paper. Let me tentatively suggest two research issues that could be studied in priority.

1/ As I mentioned above, a solution for overcoming the difficulties related to material sharing and unit partition, would consist in basing validity checking mostly on geological surfaces, whose spatial, temporal and possibly polarity relationships can easily be characterized.

2/ Fully investigating the case of regional metamorphism could be an interesting approach for demonstrating the applicability of the proposed method in a case other than sediment deposition. In this case, polarities are aligned with the temperature/pressure trajectories. Polarity inversion or gaps in the metamorphic facies succession will signal tectonic discontinuities.  

Conversely, I suggest to simply ignore, for the time being, the cases of geological entities like folds and not sedimentary 3D fabrics. These deserve a deep geological analysis and might be object of deep investigation in a second stage.      

MINOR REMARKS

Some figures represent geology in an unusual way and possibly confusing way.

• Figure 2

2b, 2c: the “roots” of the extrusion/ intrusion are not represented,

2d: metamorphism generally decreases from bottom to top; in the figure the heat source to be “suspended” between bottom and top,

2i: doesn’t represent a fabric but a single surface.

• Figure 3:

In case a (up left), the stratigraphic succession is represented in a reverse position (correct but unusual)

• Figure 9 is difficult to understand and seems to be in contradiction with the event history presented in figure 10:
• Horizon B is supposed to be anterior to the two faults but seems not to be split by these faults
• The intrusive unit is represented in a funny way (external surface or volume?); it seems to stop on the green fault while it is supposed to be posterior to this fault.
• Figure 13; the pattern transparencies make the figure confusing and do not help understanding the details of the geology.