Map comparisons in three-dimensional space (3D) and 3D time series (4D) are becoming a necessity with increased availability of multidimensional data and model simulation outputs. Therefore, this research study extends the two-dimensional (2D) map comparison methods with the aim to propose a suite of 3D approaches such as 3D Kappa, 3D Fuzzy, and 4D Fuzzy Kappa coefficients specifically designed to perform with voxel data. These proposed approaches can account for fuzziness where small categorical differences in 3D space or space-time are given a degree of similarity instead of a binary similarity value. The developed approaches are tested using different voxel data sets: (a) hypothetical with two and four classes to confirm the methods produce expected results, (b) voxelized LiDAR data to demonstrate the comparison of real 3D data sets, (c) soil horizon voxel data sets to conduct a sensitivity analysis of 3D voxel window sizes, (d) 4D outputs from an agent-based forest-fire smoke model to demonstrate the 4D Fuzzy Kappa coefficient and a sensitivity analysis of 4D voxel window size. The obtained results indicate that 3D and 4D voxel data comparisons are feasible allowing for further work on comparison of 3D data and evaluation of multidimensional models.

中文翻译：

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三维空间和时间体素数据的地图比较方法

随着多维数据和模型模拟输出的可用性增加，三维空间 (3D) 和 3D 时间序列 (4D) 中的地图比较正成为必要。因此，本研究扩展了二维 (2D) 地图比较方法，旨在提出一套 3D 方法，例如专门设计用于处理体素数据的 3D Kappa、3D Fuzzy 和 4D Fuzzy Kappa 系数。这些提出的方法可以解释模糊性，其中 3D 空间或时空中的小类别差异被赋予一定程度的相似性，而不是二进制相似性值。开发的方法使用不同的体素数据集进行了测试：（a）假设有两个和四个类以确认方法产生预期的结果，（b）体素化的 LiDAR 数据以展示真实 3D 数据集的比较，(c) 土壤层位体素数据集，用于对 3D 体素窗口大小进行敏感性分析，(d) 基于代理的森林火灾烟雾模型的 4D 输出，以证明 4D 模糊 Kappa 系数和 4D 体素窗口大小的敏感性分析. 所得结果表明，3D 和 4D 体素数据比较是可行的，可以进一步开展 3D 数据比较和多维模型评估工作。