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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RC1: 'Comment on egusphere-2025-3203', Mark Lindsay, 14 Sep 2025
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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:
- Check if your metrics are dependent. Use an SPLOM to find out and include in the manuscript. Strong dependency can explain why need 12 components.
- I wouldn’t bother showing anything beyond a comparison of PC5 and 6 in Figure 9. You don’t present these in the results.
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?
Citation: https://doi.org/10.5194/egusphere-2025-3203-RC1
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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