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, analyse and compare subsurface structures based on sparse data, as our framework can be used for an initial structural approximation prior to modeling, for the setup of the interpolation method and for the falsification of probabilistic model realizations after interpolation.