On the reconstruction of three-dimensional complex geological objects using Delaunay triangulation

Abstract

In the past four decades, huge amount of geologic data has been accumulated. Now it is time to integrate these geologic data with other sources of data, such as the digital elevation model (DEM) and remote sensing data, to do various analyses, 3D modelling, and real-time interactive visualisation to extract more meaningful results. Reconstruction of 3D geological objects is very important for geological research and mining. Traditional geologic maps that show structures and materials at the earth’s surface no longer are sufficient for storing, displaying, and transmitting geoscience information. Fully three-dimensional geologic maps are needed to provide users with geologic information of the quality, accuracy, and detail necessary to solve problems related to natural hazard mitigation, resource management, mineral and petroleum exploration, contaminant dispersion, and other issues. Although some data models and algorithms have been developed, there are few works on reconstruction of real 3D complex geological objects. In this paper, we apply Delaunay triangulation and tetrahedron for 3D reconstruction of geological structures with sudden changes. Since this technique is mainly used for 3D reconstruction of medical objects, follows that we provide a new field of application.

Introduction

Computer-based representations of areal geology extended into the subsurface as three-dimensional geologic maps can now be developed to provide continuous quantitative 3D geologic information for a variety of practical needs. Such 3D databases will allow even the inexperienced user to figuratively ‘walk around’ in the earth to examine the data and extract needed information. One important application unique to 3D geologic maps is predictive process modelling of geologic, tectonic, and hydrologic processes needed for land-use planning, hazard mitigation, and resource management. Examples of immediate applications of 3D maps include ground shaking estimation, refined earthquake relocation, fault segmentation analysis for probabilistic earthquake forecasting, resource exploration, contaminant source and dispersion pathway definition, ground water flow modelling for resource management, and landslide modelling.

Traditional geologic maps, which show the distribution and orientation of geologic structures and materials at the ground surface, for decades have served as effective tools for storing and transmitting geologic information. The introduction of Geographic Information Systems enhanced traditional geologic maps in terms of ease of use and communication of surface geologic information. However, these maps, even enhanced with GIS capabilities, are no longer sufficient for storing and transmitting subsurface information, information that is critical in the role of the map as a window into the subsurface. Fortunately, advances in computer hardware and geologic modelling and visualisation software now provide us the opportunity to construct 3D geologic maps that retain all the information in a traditional geologic map while explicitly and quantitatively extending this information throughout the subsurface [18], [22]. There are also some commercial software packages which can be used for oil and mining industries [12]. The simulation rather than the representation of real 3D objects are used in packages. For example, 3D triangulation mesh is used to represent the surface of ore bodies in MINTEC from American and cuboids are used to represent the spatial distribution and capacities of ore bodies. In seismology, using regular mesh to represent the ore bodies has been studied. Although it is easy to analyse data, it is difficult to visualise and to rotate ore bodies because of the huge amount of data.

Reconstruction of three-dimensional geological objects is very important for geological research and mining. In order to represent the real complex geological objects in mining, oil industries and geological research, it is important to develop novel data models and algorithms to use real measurement data for the reconstruction of bodies. Jessell [12] reviewed different ways potential-field modelling systems represent the 3D structure of the earth’s crust. 3D structures may be represented by discrete objects, voxels, surfaces or as a kinematic history and each technique provides the user with a different path through the model-building process and different abilities to honour the available geological and geophysical observations. At present, no one system is capable of supporting the range of uses for which potential-field data are collected, and one emphasis for future development is the ability to translate between the different representations that are used.

Many data models based on tetrahedron mesh have been developed to represent the complex objects in 3D GIS [4], [5], [11], [15]. Most research works focused on common 3D GIS data models to represent 3D objects and their relationships. The Integration and visualisation study by de Kemp [7] shows that complex fold geometries can be more rigorously constructed using constraining structural field data, but that current 3D technologies are still very cumbersome for hard rock applications. Future development of field based case studies that will help validate and communicate the benefit of these methods is much needed, and will hopefully better articulate specific requirements to software developers. Professional 3D software developers are encouraged to work towards the implementation of these types of programs in order to move the regional mapping community beyond 2D and 21/2D GIS-based modelling. National surveys and explorationists who are responsible for the collection, management, and archiving of geoscience data need to be diligent in maintaining their structural field data. This will be especially critical as 3D visualisation and modelling techniques become available to the bedrock mapping community.

Lattuada and Raper [25] developed a common method for the reconstruction of a geometric figure given a set of sample points using a triangulation algorithm to connect the points and to find the convex hull. They used Delaunay triangulation procedures. The use of 3D Delaunay triangulations is particularly suited when we do not want to face any constrains on the set of points to be connected. Besides, Delaunay triangulations have some interesting properties as optimal equiangularity and uniqueness (2D).

Delaunay triangulation [8] and its application to 3D objects reconstruction is well-developed methodology. The theory, computations, and applications of Delaunay triangulations and Voronoi diagrams [20] have been described intensively in the literature [9], [10], [17]. Aurenhammer [1] provided a detailed survey and bibliography on the Voronoi diagram and related structures, emphasising the unified exposition of their mathematical and algorithmic properties. Because the Delaunay triangulation and the Voronoi diagram are geometric duals, it is desirable to discuss briefly the theory and properties of both data structures. For standard and detailed sources, see [1], [16].

Courrioux et al. [6] proposed an automatic method based on the use of Voronoi diagrams for the 3D volumetric reconstruction of geological objects. It enables volumes to be constructed starting from the combined use of geological maps and cross-sections defined in multiple directions, and can also take into account incomplete information on geological interfaces and faults. The method is suitable for global modelling of a set of geological objects, it gives a consistent 3D volumic partition of space according to geological information contained in the maps and the cross-sections. The method was applied to modelling the main domains of the cadomian collisional orogenic belt of Panafrican age in northern Brittany (France). Thanks to the contrasting densities of the main units involved in the collisional process, the gravimetric effect of the volumic model can be calculated. Although some improvements appear to be necessary, they concluded that the method could make geological modelling more efficient through providing the possibility of rapidly testing many hypotheses.

Lindenbeck et al. [14] developed an efficient technique to cut polygonal meshes as a step in the geometric modelling of topographic and geological data. In boundary represented models of outcropping strata and faulted horizons polygonal meshes often intersect each other. TRICUT determines the line of intersection and re-triangulates the area of contact. Along this line the mesh is split in two or more parts which can be selected for removal. The user interaction takes place in the 3D model space. The intersection, selection and removal are under graphic control. The visualisation of outcropping geological structures in digital terrain models is improved by determining intersections against a slightly shifted terrain model. Thus, the outcrop line becomes a surface which overlaps the terrain in its initial position. The area of this overlapping surface changes with respect to the strike and dip of the structure, the morphology and the offset. Some applications of TRICUT on different real data sets are shown. TRICUT is implemented in C++ using the Visualisation Toolkit in conjunction with the RAPID and TRIANGLE libraries. The program runs under LINUX and UNIX using the MESA OpenGL library. This work gives an example of solving a complex 3D geometric problem by integrating available robust public domain software.

Delaunay triangulation developed mainly for reconstruction of medical object is used in this paper. We apply the above approach to a new type of objects (geological structures with sudden changes) considering geological attributes attached to each point. In 3D reconstruction of medical objects previous information about the geometrical shape usually exists, while many of the geological structures, subject of 3D reconstruction and visualisation, are unknown and have very complicated shape with sudden changes. The latter makes 3D reconstruction and visualisation of the geological objects a challenging problem.

In this paper, we developed a new tetrahedron mesh algorithm to calculate isolines and reconstruct real complex geological objects based on 3D constrained Delaunay triangulation algorithm. Much less spatial discrete data points are used. The attributes of spatial discrete data points, such as attributes of rocks and ore bodies, are used to construct the spatial attribute of the geological object. Following the description of 3D Delaunay triangulation algorithm, the application of Delaunay triangulation for the reconstruction of 3D complex geological objects is addressed. We discussed the different developments of 3D geological objects reconstruction with linear and sudden change attributes. Finally, conclusion and further developments are discussed.