图1 维诺图及其对应的德劳内三角化((a)点pi及其相应的维诺图单元ci，二维维诺图由所有单元的边缘构成；(b)点pi作为顶点的德劳内三角化)

Fig. 1 Voronoi diagram and its corresponding Delaunay triangulation ((a) Point pi and its corresponding Voronoi cell ci, the 2D Voronoi diagram is composed of the edges of all cells; (b) The Delaunay triangulation with points pi)

图2 方法流程((a)输入为不同岩性的柱状钻孔剖面；(b)点数据离散化后构建维诺图，提取异色相邻面；(c)在提取面上添加特征点作为拉普拉斯变形控制点；(d)求解线性方程确定地层边界，得到最终变形结果)

Fig. 2 Method work flow ((a) Input is a cylindrical borehole profile with different lithologies; (b) After discretizing the borehole data into point data, a Voronoi diagram is constructed to extract faces where adjacent sites have different colors; (c) Feature points are identified and added to the extracted faces as control points for Laplacian deformation; (d) Layer boundaries are determined by solving a system of linear equations to obtain the deformation result)

图3 钻孔数据的重采样(左图：原始钻孔数据。点Pi (X)代表第i个钻孔上地层X底部的边界点。右图：重采样后。边界两侧的采样点与其距离相同，距离为 ?)

Fig. 3 Resampling of drill core data (Left image: original drill core data. Point Pi (X) represents the boundary point at the bottom of stratum X on the i-th drill core. Right image: after resampling. The sampling points on both sides of the boundary are equidistant, with a distance of ?)

图4 设置约束点的规则如下：本方法在维诺图上指定2种类型的顶点作为约束点。类型一包括原始钻孔上的地层边界点(如V1)，类型二包括与3个或更多区域等距离的顶点(如V2)

Fig. 4 The rules for setting constraint points are as follows: This method specifies two types of vertices as constraint points on the Voronoi diagram. Type one includes the stratum boundary points on the original drill cores (e.g., V1), and type two includes the vertices equidistant from three or more regions (e.g., V2)

图5 二维情况下的实验((a)输入网格由162个顶点构成，顶点间两两相邻，包含有4个红色约束点。为了突出表示顶点的对应关系，在其中选择了一些顶点放大显示；(b)非约束点的位置权重为0；(c)非约束点的位置权重为1e-3)

Fig. 5 Experiments in 2D ((a) The input mesh consists of 162 vertices, with every pair of vertices being adjacent, and contains four red constraint points. To highlight the correspondence of vertices, some vertices are magnified; (b) The position weight of non-constraint points is 0; (c) The position weight of non-constraint points is 1e-3)

图6 三维钻孔建模结果((a)包含40根钻孔的输入数据可视化；(b)对离散后的钻孔点数据建立维诺图；(c)对维诺图进行细分；(d)构建并求解线性方程组，更新网格的顶点位置得到分界面结果)

Fig. 6 3D drill core modeling results ((a) Visualization of the input data containing 40 drill cores; (b) Construction of the Voronoi diagram from the discretized drill core point data; (c) Subdivision of the Voronoi diagram; (d) Construction and solution of the system of linear equations to update the vertex positions of the mesh, resulting in the boundary interface)

图7 三维建模结果((a)输入包含9根钻孔；(b)孤立区域导致建立维诺图后，分界面上出现非流形边结构；(c)进行变形前，将非流形边上顶点设置为约束点；(d)变形后得到了平滑的尖灭结构)

Fig. 7 3D Modeling results ((a) Input consisting of nine boreholes; (b) Non-manifold edge structures emerging on interfaces due to isolated regions after Voronoi diagram construction; (c) Vertices on non-manifold edges are designated as constraint points prior to deformation; (d) Smoothed pinch-out structure obtained after deformation)

图8 从二维上对方法的拓扑结构保留能力做了实验((a)~(d)对分层结构、尖灭结构、透镜体结构和断层结构选取具有代表性的二维截面来进行建模)

Fig. 8 Experiments were conducted to test the method’s ability to preserve topological structures in 2D ((a)~(d) Representative 2D cross-sections of layered structures, pinch-out structures, lens structures, and fault structures were selected for modeling)

图9 建模效果对比((a)Kriging插值方法；(b)本文方法)

Fig. 9 Comparison of modeling results ((a) Kriging interpolation method; (b) Ours)