the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Aug 2025
Quantitative comparison of three-dimensional bodies using geometrical properties to validate the dissimilarity of a standard collection of 3D geomodels
Abstract. The quantification of 3D structural shapes is a central goal across multiple scientific disciplines, serving purposes such as image analysis and the precise geometric characterization of objects. This study proposes a methodology for the shape quantification based on a set of geometrical parameters in 2D sections of 3D geological shapes and establishes a set of synthetic regular geometries as benchmark models in 3D geomodeling approaches. The proposed methodology is demonstrated on a number of simple geometric bodies and the benchmark models to assess their geometrical dis-/similarity. The dimensions of the structures are measured perpendicular and vertically to their horizontal main axes on a fixed amount of cross sections. Furthermore, gradient and curvature measurements on these cross sections are conducted. A subsequent multi-step data analysis provides insight into the main geometrical characteristics of the structures and visualizes differences between various datasets: Analysis of extension measurements reveals the anisotropy of structures, the existence of overhangs and the character of the top surface of an investigated structure. Analyzing the gradients and curvatures offers information on the slopes of the lateral walls of the structure and its sphericity as well as top surface. Kullback-Leibler divergence is utilized to quantitatively compare individual parameter distributions. Dimensionally reduced cluster analysis groups and systematizes input structures based on the combined statistical parameters and serves for the identification of benchmark models showing large geometrical similarity. It is expected that the methodology and set of benchmark models will aid in advances to model and compare subsurface structures based on sparse data.
Competing interests: The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Friedrich Carl, Jian Yang and Marlise Colling Cassel are funded by the German Federal Company for Radioactive Waste Disposal (BGE).
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-3203', Mark Lindsay, 14 Sep 2025
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RC2: 'Comment on egusphere-2025-3203', Mark Jessell, 14 Oct 2025
Major comments:
This is undoubtably a challenging and interesting topic, however I think the authors could help their cause by better framing the problem and restructuring the paper so that we do not go through two whole cycles of method and results.
Line 33: The introduction begins with the observation: "Accurate shape quantification independent of the objects’ orientation is essential for applications such as geological modelling, resource management and structural analysis, where understanding the geometric properties of objects can inform decision making and enhance predictive capabilities"
This is key, as the choice of model comparison method fundamentally depends on the use to which the models are to be put. A model for constraining a geophysical inversion has to retain distinct petrophysical domains, but could merge others with similar properties. A model to calculate transport properties (fluid flow, electrical conductivity... is concerned both with rock physics and adjacent body topology. A structural interpretation needs to understand age relationships, not just geometry.
It is not at all clear to me what specific scientific question is being addressed by this method, that is limited to the study of isolated convex hulls with no branching structures? Why do we want to compare two models of bodies around salt domes, when with the 3D seismic we actually have a detailed description of the geometry of the object? What do we learn from comparing it with another potentially similar object? All this needs to be addressed to underpin this work. I suggest that the paper by Parquer et al 2025 (https://gmd.copernicus.org/articles/18/71/2025/) will provide some interesting insights into this topic.
Section 4.3 Given this analysis, it seems to me this should be a part 2 of the methods and results, and not placed in the discussion as we are essentially revisiting the entire methodology. I suggest that the authors introduce the full and restricted model sets from the beginning, compare the methods and results and then discuss this, rather than the structure of the ms currently used.
Line 274: If we have a method that, when applied to the simplest possible shape (a sphere) tells us "...the most similar models regarding the respective distributions of the six parameters are the “sheet(cylindric_rounded)” for the horizontal data of the first direction, the “prism” for the orthogonal horizontal data, the “batholithV3” for the combined horizontal data, the “anticline_wall (rounded)_batholithV1” for the vertical data, the “phacolith” model for the gradients and the “ellipsoid” for the curvatures".
Is this really a selling point for the method? It sounds a bit like the apocryphal sightless committees' description of an elephant. It raises the question as to why a sphere is not the best descriptor of a sphere, and if not is this really a useful method?
Minor comments:
Line 84: "Since all clastic sediments are initially deposited conformably onto the underlying strata" not sure why this has to be true, can’t clastic sediments deposit on erosional surfaces?
Line 104 "Their current shape depends not only on the geometry of the original rock body but also on the specific mineral assemblage of the protolith" Why does the currently geometry depend on the mineral assemblage? Similarly, why does it depend on the "the pressure‑temperature conditions experienced during metamorphism"
Line 106 "Overall, most high‑grade metamorphic rock bodies in the German subsurface are bounded by either plutonic intrusions or fault zones" Presumably the shallower ones, of interest to this study, are mostly bound on their top-most surface by unconformities?
Figure 1. This is actually a part of the methodology, not the introductory material in my opinion and should appear there, together with the description.
Figure 1. Not sure if lateral stratigraphic pinching out of horizons is accounted for in this scheme, but occurs in Fig 2 so maybe I am missing something?
Figure 2. Some of the shapes (e.g. anticline_wall) are difficult to understand in terms of their 3D geometry as much of the body is hidden, perhaps consider semitransparent front halves for some of these models?
Figure 2. What about more complex fold geometries such as refolded folds, or even simple saddles?
Line 167 "the input file format has to be changed if it is not .vtk" Maybe just say: “The system only supports .vtk file formats”.
Line 170 "which first rasters" should be "which first discretises"
Line 173: "Orientation of sections normal to the longitudinal axis of the structure (first direction)" What if the structure doesn't have a simple geometry that you can assign a longitudinal axis to? Does this system allow bounding boxes that do not have x & y axes parallel to real-world horizonal planes?
Line 176" "Extensional measurements are conducted on each cross section at 5 equidistant transects" Why 5, did you perform studies to show this was optimal?
Line 177: "Since the very first and last cross section of both directions are excluded from the measurements as they would (undesirably) slice irregular polygons several times" I don't follow this logic, if the body is really irregular any section may intersect it multiple times?
Line 179: "20 intervals are considered for every input structure" Why 20, did you perform studies to show this was optimal?
Line 191: The assertions in this paragraph are hard to understand, maybe it needs a figure (even in sup materials would do?)
Section 3.3 This section would benefit from some sub-structure as it is hard to digest for the uninitiated.
Fig. 4c, d, f, g, h, i, j text way too small to read.
Citation: https://doi.org/10.5194/egusphere-2025-3203-RC2
Data sets
Carl-et-al._in-review-2025 - raw data & standard models Friedrich Carl https://doi.org/10.5281/zenodo.15795851
Model code and software
Carl-et-al._in-review-2025 - python code Friedrich Carl https://doi.org/10.5281/zenodo.15795851
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- 1
This paper describes a method to take 3D digital geo-objects and quantify their shape. The shape quantities are then compared with a set of standard geometries that represent various geo objects as defined by the author. Different statistical methods are used to achieve this via a sophisticated workflow.
The significance of this work is not adequately made until the paper is almost over. I do appreciate the slightly dramatic approach of saving the best for last, but I really think that the authors can be very clear in the abstract and introduction what the method offers in terms of scientific rigour and how it can address subjective bias in geomodelling. That said, the work is important and should be disseminated. I will elaborate on this further into my review.
The structure needs more work. Some of the sections are hard to follow. The figure quality needs to be improved as well. Access to Carl et al. 2023 is key (see below). Perhaps you can ask RING nicely and add the paper to an open repository.
A major omission is a geological description and figure of the Altenbruch -Beverstedt structure. It’s an essential subject of the manuscript. It is shown (at least in parts) in Figure 4, but this doesn’t let us see what the object looks like in traditional 3d view (e.g. in Paraview or a commercial package) nor help us understand its geological significance and why it is a good example to test your method.
The writing quality is good, but parts seem rushed. The manuscript would benefit from a thorough edit.
I make comments on each section below, with a few minor suggestions after that.
Abstract
The abstract will benefit from examples of application. Who is this aimed at? The second sentence of the introduction is a good example that would help here.
Intro
I admit to being a bit lost in the first few paragraphs of the introduction. My understanding of “shape quantification” starts very simply, with volumes, surface areas, aspect ratios, elongation, flattening, and so on. 3D modelling software can do all this, so I wondered why things like cross sections, CNNs and Transformers were raised so early in the manuscript. I gathered that the intent is to quantify from multi-modal sources, such as sections, images, maps, etc. That is, not in a constructed 3D model itself. This is why you cite many sophisticated methods that are totally unnecessary if you have a 3D model, but are if they are static 2D representations of a 3D object (e.g. Multi-view approaches). Some 3D model-related studies are then cited (L61-65), so it remains confusing. I suggest opening the introduction with a clear description of shape quantification and the media through which it will be conducted. (I completely understand now I have read through to the method section. Please rewrite the introduction around a similar description).
From what I can gather, the intent is to quantify the shape so it can be compared to a set of standard geometries, and then the modelled object can be given a label (e.g. wall(highly.anisotropic_hourglass-shape_rounded)). Once you know the label, then you can make some interpretation of the geological history? Usually when you build a model, the geologist has a good idea (i.e. conceptual model) about what the intended object should look like. Obvisouly this has subjective bias behind it. Thus, what you method does is to check whether the desired object is close to what it should look like. If the object respects the data, but not the conceptual model, then it could more questions about what the geological history could be. If this is what you are doing this for, that’s great, and good science. Please make that clear in the abstract and introduction. (I didn’t come to that realisation until the results and discussion)
Carl et al 2023 is cited and useful to read (and view, as there is a video), however is not available from RING without the necessary login credentials. You need to summarise this paper given it describes more fully the concept of “standard geometries” (at least in this context). Admittedly I only found one version, and may have missed any open-access.
The introduction to topology with conformable, unconformable, concordant and discordant is good, however it’s not clear why you have introduced it with respect to shapes. If you are making a point about how geological history -> topology -> modelled shape and that can be quantified, that’s great, but you need to be quite explicit about that.
This section, especially the description of various halite geometries would benefit from reference to figure 2. It’s quite hard to follow without a visual representation, and geologists like pictures. Also, some field photos would be nice, but not critical.
What are the four thin and unlabelled objects at the end of figure 2?
Method
The approach needs a figure showing the entire workflow from the initial vtk to the final computation of gradients and curvatures. You could add the figure to the pseudo code in Fig 3 by running down alongside it. (reading on) an improved version of figure 4 would do (see additional comments below).
Fig. 4. The text is too small, and quality of the images not adequate for publication. Screen shots are okay sometimes, but if there is text, one needs to be able to read it (e.g. the coordinates, the key, legends etc)
Results
You are comparing a generic sphere with a model of Altenbruch-Beverstedt. So you need a section in the introduction describing Altenbruch-Beverstedt otherwise we have no point of reference to know whether your results are meaningful given the structure we would expect to be quantified. You also need an image of the geological model, or the structure you have picked out from it for the analysis.
PCA – I’d be careful about interpreting too much from anything beyond PC6. You have stopped there, but the number of PCs to get to 90% indicates a pretty complex and high-dimensional data set. Two things:
L83 Are you introducing grain size as you’ll be quantifying their shapes? It doesn’t seem to have much to do with the rest of the paragraph.
L86 “tilting and folding *of flat-lying structure* can result in a range of geometries that remain generally conformable.”
L100 “Crystalline rocks considered are both plutonic and high‑grade metamorphic rocks (migmatites and gneisses).” Crystalline rock is a catch-all term meaning the basement to an overlying sedimentary basin. Technically, high-grade plutonic rocks end up as gneisses, but are not “both”. Also crystalline rock can be extrusive, low and medium grade metamorphic, just depends on where you are.
L146 What kind of clustering? Eg. KNN? Or something else
L149 “Our method cannot be used to quantitatively compare implicit representations of structures.” so not directly from a scalar field (e.g. Geomodeller or Leapfrog) – if so I’d be clear about that, because it reads like you can’t use your method on any object rendered from an implicit method, while I’m sure you can!
L216 Assuming these moments are centred around the mean?