# An Automated Mapping Method of 3D Geological Cross-Sections Using 2D Geological Cross-Sections and a DEM

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Input Data

#### 2.1. Study Area

#### 2.2. Input Data

## 3. Methodology

#### 3.1. Matching the Corresponding Nodes and Generating Feature Lines

- (1)
- Loading data. Reading any 2D GC and its corresponding GCL, then saving the strata surfaces in the GC into $\mathrm{S}=\left\{{s}_{o}|o=1,2,\dots ,ON\right\}$, where $\mathrm{o}$ means the index of each stratum surface, $ON$ is the number of strata surfaces and the GCL is $\mathrm{SL}$, respectively;
- (2)
- Matching corresponding node pairs. Firstly, merging all strata surfaces in the set $\mathrm{S}$ to obtain a complete polygon $\mathrm{P}$, then obtaining node ${P}_{l}=\left({x}_{l},{y}_{l}\right)$ with the smallest X value and the largest Y value in $\mathrm{P}$ and node ${P}_{v}=\left({x}_{v},{y}_{v}\right)$ with the largest X value and the largest Y value, respectively, creating a node ${P}_{r}=\left({x}_{r},{y}_{r}\right)$ with an X value equal to ${x}_{v}$ and a Y value equal to ${y}_{l}$. Finally, obtaining the starting point ${P}_{f}=\left({x}_{f},{y}_{f}\right)$ and ending point ${P}_{t}=\left({x}_{t},{y}_{t}\right)$ of the GCL and constructing corresponding node pairs with node ${P}_{l}$ and node ${P}_{r}$;
- (3)
- Generating feature lines. Taking ${P}_{l}$ as the starting node and ${P}_{r}$ as the ending node to construct a straight line, which is called the transform line, $\mathrm{TL}$. Dividing the outer boundary of polygon $\mathrm{P}$ at nodes ${P}_{l}$ and ${P}_{r}$ and segmenting the boundary starting from node ${P}_{l}$ and ending at node ${P}_{v}$ as the ground line, $\mathrm{GL}$.

#### 3.2. Computing Affine Transformation Parameters and Generating Matrix

#### 3.3. Generating 3D GC

- (1)
- Generating intersection points. Obtaining any node ${{P}_{s}}^{\prime}=\left({{x}_{\mathrm{s}}}^{\prime},{{y}_{\mathrm{s}}}^{\prime},{{z}_{\mathrm{s}}}^{\prime}\right)$ on the stratum surface ${{s}_{o}}^{\prime}$ boundary line to create a straight line perpendicular to the direction of the GCL $\mathrm{SL}$. It intersects with the line $\mathrm{GL}$ and the GCL $\mathrm{SL}$, respectively. The intersection points were recorded as ${P}_{GL}=\left({x}_{GL},{y}_{GL}\right)$ and ${P}_{SL}=\left({x}_{SL},{y}_{SL}\right)$;
- (2)
- Calculating distance. Calculating the Euclidean distances from node ${{P}_{s}}^{\prime}$ to point ${P}_{SL}$, point ${P}_{GL}$ to point ${P}_{SL}$, and node ${{P}_{s}}^{\prime}$ to point ${P}_{GL}$, respectively, and recording them as ${d}_{1}$, ${d}_{2}$, and ${d}_{3}$;
- (3)
- Calculating elevation difference. The elevation values at point ${P}_{SL}$ and node ${{P}_{f}}^{\prime}({{x}_{f}}^{\prime},{{y}_{f}}^{\prime})$ (the node ${P}_{f}$ transformed in Section 3.1) were obtained from the DEM and recorded as ${z}_{SL}$ and ${{z}_{f}}^{\prime}$, and the difference’s absolute value ${d}_{4}$ between them was calculated;
- (4)
- Calculating 3D coordinates of nodes. The node ${{P}_{f}}^{\prime}$ is recorded as a 3D node ${{P}_{s}}^{\u2033}=\left({{x}_{s}}^{\u2033},{{y}_{s}}^{\u2033},{{z}_{s}}^{\u2033}\right)$ after 3D transformation. In theory, ${d}_{4}$ and ${d}_{4}$ are equal if the current terrain is consistent with the terrain when making the GCs. However, the terrain is dynamically changing. This study classifies different nodes according to their positions, then determines their elevation calculation equations by comparing the elevation of point ${P}_{SL}$ and node ${{P}_{f}}^{\prime}$ and the difference between ${d}_{4}$ and ${d}_{2}$. Through the experiment, it is found that all nodes are mainly distributed in four situations (Figure 5a), where ${{x}_{s}}^{\u2033}={x}_{SL}$ and ${{y}_{s}}^{\u2033}={y}_{SL}$. The calculation method of the specific elevation value ${{z}_{s}}^{\u2033}$ is shown in Table 1;
- (5)
- Generating a 3D stratum surface. Cycling steps (1)–(3), executing 3D transformation for all boundary nodes of stratum surface S and closing all 3D nodes to obtain a 3D stratum surface.

#### 3.4. Handling the Bent GC

#### 3.5. Quantitative Tests of 3D Transformation Effect

## 4. Experimental Results

^{2l}) at the intersection can be joined without dislocation. On the one hand, it is proved that the current DEM is consistent with the DEM when the two GCs are produced, and the ground undulation has not changed significantly. On the other hand, it is also well proved that the proposed method for transforming stratigraphic geometry is faithful and effective.

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^{3y/5}in the cross-section could completely fit with these in the planar geological map. However, one point to note in this study area is the following: in the 2D GC, the Quaternary strata are not distinguished by age and are uniformly replaced by Q; the planar geological map distinguishes and gives different colors to the Quaternary strata of different ages, so the two will have color differences after fitting, but the result is correct.

## 5. Discussion and Application

#### 5.1. Factors Affecting the Method

#### 5.1.1. DEM at Different Periods

#### 5.1.2. DEM Resolution and Number of Geometric Nodes in the GC

#### 5.1.3. Mapping Quality of GCs

#### 5.1.4. Effects of Different 3D Transformation Strategies

- (1)
- Geometric Shape Maintenance Strategy

- (2)
- Surface Fitting Maintenance Strategy

#### 5.2. Applicability of the Method

- (1)
- The boundary lines on the left and right sides of the geological cross-section are not vertical. It is an essential prerequisite for the proposed method that the boundary lines on both sides are vertical. With non-vertical lines, it is difficult to effectively match corresponding points and generate feature lines, which affects the accuracy of the 3D transformation;
- (2)
- Vertical or horizontal exaggeration. General geological cross-sections will be mapped with the same vertical and horizontal scales (true-scale cross-sections) [8]. When constructing large regional or semi-regional cross-sections. Typically, it is the vertical scale that is exaggerated. This kind of geological cross-section needs to be transformed to the actual scale and then adopts the proposed method to obtain an accurate result.

#### 5.3. Application Cases of the Transformed 3D GCs

#### 5.3.1. 3D Geological Modeling

#### 5.3.2. 3D Presentation on the Digital Earth

#### 5.3.3. Parameter Setting of 3D Geological Models of GCs Display Effect

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) The location of the study area and the distribution of GCLs. (

**b**) Schematic diagram of two typical GCs. The GC M-M’ is straight, and the GC B-B”-B’ is bent.

**Figure 3.**Schematic diagram of corresponding nodes matching and feature lines generation. x-y and x’-y’ represent the GC and the GCL in different coordinate systems.

**Figure 4.**Schematic diagram of affine transformation parameter calculation. x-y (the Cartesian coordinate system) and x’-y’ (the projection coordinate system) represent the transform line $\mathrm{TL}$ and the GCL $\mathrm{SL}$ in different coordinate systems. ①~⑤ represents the sequence of different transformation operations.

**Figure 5.**(

**a**) Schematic diagram of different situations of nodes on the stratum surface, where the Vertical Line is perpendicular to the Section Line. (

**b**) The 3D transformation effect of the straight GC M-M’.

**Figure 6.**(

**a**) The affine transformation effect of two branch GCs from the GC B-B”-B’ divided into two parts at the turning point B”. In order to avoid overlap, this paper presents them in different directions. (

**b**) The merged result after 3D transformation of the two branch GCs displayed with 3D coordinate axis.

**Figure 7.**Fourteen 3D GCs in Nanjing City. (

**a**) The global view. (

**b**) Intercross-section of two 3D GCs. (

**c**,

**d**) The 3D GC fits to the DEM. (

**e**) The 3D GC fits to planar geological map. Note: In the 2D GC, the Quaternary strata are not distinguished by age and are uniformly replaced by Q. The planar geological map distinguishes and gives different colors to the Quaternary strata of different ages, so the two will have color differences after fitting.

**Figure 8.**Two other 3D transformation strategies are applied to transform the GC B-B”-B’: (

**a**) Geometric Shape Maintenance Strategy. (

**b**) Surface Fitting Maintenance Strategy. The transformed results are displayed with 3D coordinate axis.

**Figure 9.**The 3D solid models constructed with two parallel 3D GCs and displayed with 3D coordinate axis. (

**a**) 3D solid model of the GC M-M’. The strip distance is 200 m, 100 m on both side. (

**b**) 3D solid model of the GC B-B”-B’. The strip distance is 50 m and the lower right corner is a partial enlarged view.

**Figure 10.**(

**a**) The 3D presentation effect of fourteen 3D models of all GCs in Nanjing City on Google Earth. (

**b**) For convenient observation, we exaggerated the Z-axis of all 3D models by three times the elevation and moved it up to the ground. (

**c**) The local looking-up view for observing the intersection of two 3D models. (

**d**) The local looking-down view for observing the fitting of the 3D model with the ground.

**Figure 11.**Take the three scenarios in Figure 10a–d as examples to show the optimal observation scheme in different regions and different perspectives. (

**a**) When looking up in the global view, the Z-axis exaggeration can be more intuitive. (

**b**) When looking down in the local view, the 3D geological model can be moved up to observe the stratigraphic geometric shape and distribution. (

**c**) When looking up in the local view, the best display effect can be achieved by rotating the viewing positions and angles.

Type | Situation | Condition | Formula $\left({{\mathit{z}}_{\mathit{s}}}^{\u2033}\right)$ | Description |
---|---|---|---|---|

① | ${d}_{2}={d}_{1}$ | - | ${z}_{SL}$ | The node ${{P}_{s}}^{\u2033}=\left({{x}_{s}}^{\u2033},{{y}_{s}}^{\u2033},{{z}_{s}}^{\u2033}\right)$ is on the ground line. |

② | ${d}_{1}={d}_{2}+{d}_{3}$ | ${{z}_{f}}^{\prime}>{z}_{SL}\cap {d}_{4}<{d}_{2}$ | ${z}_{SL}-{d}_{3}$ | The elevation at node ${{P}_{f}}^{\prime}$ is still higher than the elevation at node ${P}_{SL}$, and ${d}_{4}<{d}_{2}$. |

${{z}_{f}}^{\prime}>{z}_{SL}\cap {d}_{4}>{d}_{2}$ | ${{z}_{f}}^{\prime}-{d}_{1}$ | The elevation at node ${{P}_{f}}^{\prime}$ is still higher than the elevation at node ${P}_{SL}$, and ${d}_{4}>{d}_{2}$. | ||

${{z}_{f}}^{\prime}<{z}_{SL}$ | ${z}_{SL}-{d}_{3}$ | The elevation at node ${{P}_{f}}^{\prime}$ changes to be lower than the elevation at node ${P}_{SL}$. | ||

③ | ${d}_{2}={d}_{1}+{d}_{3}$ | ${{z}_{f}}^{\prime}>{z}_{SL}$ | ${z}_{SL}$ | The elevation at node ${P}_{SL}$ changes to be lower than the elevation at node ${{P}_{f}}^{\prime}$. |

${{z}_{f}}^{\prime}<{z}_{SL}$ | ${z}_{SL}-\left(\frac{{d}_{3}\times {d}_{4}}{{d}_{2}}\right)$ | The elevation at node ${P}_{SL}$ is still higher than the elevation at node ${{P}_{f}}^{\prime}$, but ${d}_{4}\ne {d}_{2}$ | ||

④ | ${d}_{3}={d}_{1}+{d}_{2}$ | ${{z}_{f}}^{\prime}>{z}_{SL}$ | ${{z}_{f}}^{\prime}-{d}_{1}$ | The elevation at node ${P}_{SL}$ changes to be lower than the elevation at node ${{P}_{f}}^{\prime}$. |

${{z}_{f}}^{\prime}<{z}_{SL}\cap {d}_{4}>{d}_{2}$ | ${z}_{SL}-{d}_{3}$ | The elevation at node ${{P}_{f}}^{\prime}$ is still lower than the elevation at node ${P}_{SL}$, and ${d}_{4}>{d}_{2}$. | ||

${{z}_{f}}^{\prime}<{z}_{SL}\cap {d}_{4}<{d}_{2}$ | ${{z}_{f}}^{\prime}-{d}_{1}$ | The elevation at node ${{P}_{f}}^{\prime}$ is still lower than the elevation at node ${P}_{SL}$, and ${d}_{4}<{d}_{2}$ |

GC | A-A’ | B-B’’-B’ | C-C’ | D-D’ | E-E’ | F-F’ | G-G’ | H-H’ | I-I’ | J-J’ | K-K’ | L-L’ | M-M’ | N-N’ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

NNG | 183 | 241 | 132 | 143 | 75 | 164 | 137 | 97 | 153 | 159 | 178 | 127 | 91 | 47 |

NNG_Error (m) | 1.35 | 1.53 | 1.32 | 1.27 | 1.21 | 1.37 | 1.24 | 1.17 | 1.33 | 1.36 | 1.41 | 1.24 | 0.95 | 1.22 |

NNC | 2411 | 3362 | 1347 | 1337 | 636 | 1823 | 1327 | 807 | 1626 | 1670 | 2130 | 1280 | 806 | 474 |

NNC_Error (m) | 5.45 | 13.48 | 4.16 | 4.57 | 3.32 | 5.21 | 3.79 | 3.53 | 5.75 | 6.31 | 4.74 | 3.39 | 2.46 | 1.37 |

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## Share and Cite

**MDPI and ACS Style**

Shang, H.; Shen, Y.-G.; Li, S.; Li, A.-B.; Zhang, T. An Automated Mapping Method of 3D Geological Cross-Sections Using 2D Geological Cross-Sections and a DEM. *ISPRS Int. J. Geo-Inf.* **2023**, *12*, 147.
https://doi.org/10.3390/ijgi12040147

**AMA Style**

Shang H, Shen Y-G, Li S, Li A-B, Zhang T. An Automated Mapping Method of 3D Geological Cross-Sections Using 2D Geological Cross-Sections and a DEM. *ISPRS International Journal of Geo-Information*. 2023; 12(4):147.
https://doi.org/10.3390/ijgi12040147

**Chicago/Turabian Style**

Shang, Hao, Yan-Gen Shen, Shuang Li, An-Bo Li, and Tao Zhang. 2023. "An Automated Mapping Method of 3D Geological Cross-Sections Using 2D Geological Cross-Sections and a DEM" *ISPRS International Journal of Geo-Information* 12, no. 4: 147.
https://doi.org/10.3390/ijgi12040147