A Triangulated Irregular Network Constrained Ordinary Kriging Method for Three-Dimensional Modeling of Faulted Geological Surfaces

Three-dimensional (3D) modeling of geological surfaces, such as coal seams and strata horizons, from sparsely sampled data collected in the field, is a crucial task in geological modeling. Interpolation is a common approach for this task to construct continuous geological surface models. However, this problem becomes challenging considering the impact of the faults on geological surfaces. Existing methods tend to solve this problem through three steps, including interpolating stratum and fault surface, applying a fault modeling method to modify the geological surface, and optimizing the modified surface to pass sample points fallen into the fault displacement zone. This paper presents a more concise method to generate a faulted geological surface, in which 1) a constrained Delaunay triangulated irregular network (CD-TIN) is constructed to facilitate the neighborhood search process of the ordinary kriging (OK) interpolation, 2) the CD-TIN is also directly constrained by horizon cut-off lines formed from theoretical fault displacement profiles, and 3) subsequently, neighbors of the location to be estimated are selected effectively in the CD-TIN considering the fault topology. The proposed method significantly improves the time efficiency of the OK interpolation by utilizing the CD-TIN and incorporates fault effects directly into the interpolation process by inserting fault horizontal cut-off lines into CD-TIN. Moreover, by integrating the fault effects directly into the interpolation process, the surface modeling process is accomplished in a single stage instead of two separate stages of interpolation first and then modifying the surface in the fault area. By this strategy, the proposed method significantly improves the time efficiency of the OK interpolation algorithm and achieves more accurate modeling of the faulted geological surface. Experiments were designed to compare the performance of our method with several commonly used approaches, and the results indicate that the proposed TIN-constrained OK method achieves better accuracy and efficiency in modeling faulted geological surfaces than other methods. This method could also be used in geospatial interpolation studies, such as meteorological data interpolation.