# Three-Dimensional Geological Modeling Method Based on Potential Vector Fields
## 1 Introduction  
Three-dimensional geological modeling uses mathematical methods to construct the geometric models of geological bodies according to geological rules (surface model) or the spatial distribution of physical property indicators (voxel model) [1, 2]. This technique plays a significant role in geological research. [PERSON] et al. [3] implemented 3D geological modeling in urban underground space planning for Shenyang City, ensuring spatial coordination across multiple dimensions. [PERSON] et al. [4] developed integrated 3D geological models for volcanic systems, achieving geometric fusion of diverse complex geological structures. [PERSON] et al. [5] established a comprehensive 3D geological modeling framework utilizing geospatial data from Singapore, accompanied by the development of a specialized model management system. The methodology demonstrates exceptional capability in providing flexible and intuitive representation of geological structures, stratigraphy, lithology, and other geological conditions [6, 7]. The 3D geological surface modeling methods primarily include explicit modeling and implicit modeling. The explicit modeling method defines each object in the model explicitly, directly obtaining the coordinates of points, lines, and faces of the geological objects to form surfaces [4], such as the cross-section method [8], Coons surfaces [9], Bezier curve/surfaces [10], etc. However, it has certain limitations: (1) it requires significant manual interaction and is inefficient for modeling complex shapes; (2) the results often contains topological errors, such as triangle intersections, which require subsequent verification and correction; and (3) different modelers may have varying skills and understandings, which can lead to uncertainty and multiple solutions in the model [11, 12]. The implicitmodeling method treats the entire stratigraphic space as a potential field, which is a scalar field defined by an implicit function. The geological interfaces are defined as isosurfaces in the potential field [13]. In implicit modeling, the implicit function is first selected, and the unknowns quantities in the function are solved using known constraints, thereby establishing the potential field. The 3D geological surface model is then obtained using surface generation algorithms. Compared with explicit modeling, implicit modeling has advantages such as faster modeling speed, no need for manual interactions, and real-time dynamic modeling [14; 15; 16]. The key to establishing a potential field lies in the interpolation method [17]. Commonly used methods include Kriging interpolation [18; 19; 20; 21], radial basis function (RBF) interpolation [3; 16; 22; 23; 24; 25], etc. Kriging interpolation requires the data to follow a normal distribution, and it is difficult for data with large variations in spatial density to meet this requirement. In contrast, RBF interpolation has lower data requirements and performs better when constructing complex surfaces [26]. Surface generation algorithms divide the entire space into basic voxel units and compute the potential field values at the vertices of these voxel units using the implicit function. Within each voxel unit, triangular facets are used to approximate the isosurface. Depending on the type of basic voxel unit, the algorithms can be divided into marching cubes (MCs) [27] and marching tetrahedra (MTs) [28]. Compared to the MT algorithm, the MC algorithm is more complex but does not require additional tracking of the grid's topological information. Additionally, when the number of voxel units is roughly the same, the MC algorithm offers higher approximation accuracy [29]. The potential field function is continuous throughout the space and cannot represent unconformable stratigraphic interfaces, such as pinch-outs and erosions. Many researchers have chosen to post-process modeling results to extract unconformable stratigraphic interfaces [26], but this compromises the consistency of the model and consequently affects its quality.