# Research Status of and Trends in 3D Geological Property Modeling Methods: A Review
## 3 Interpolation Methods for 3D Property Modeling
### Multiple-Point Geostatistics Methods  
The two-point geostatistical method uses the spatial correlation between any two points in the space or uses this correlation as a simple variogram of the spatial continuity model for interpolation. This method relies only on the relationship between two points, but in practice, the correlation of multiple points at once is desired. The formation process of geological heterogeneity is very complex and cannot be expressed by a two-point variogram model. Some complex geological phenomena, such as meandering river systems, cannot be described by traditional random function models. As shown in Figure 4, there are three different spatial structures. For the variogram East-West, situation 1 and situation 2 are almost the same, while the spatial structures are totally different. Meanwhile, for the variogram North-South, the 3 spatial structures are alike. The variation function cannot fully reflect the spatial structure. Therefore, multiple-point geostatistics are introduced to simulate this highly complex geological feature [72]. The two-point geostatistical method differs from the object-based method and is more likely to be faithful to valid data. The idea of multiple-point geostatistics is to use more than two points to infer unknown random variables. Traditional methods consider a stochastic model between two points; that is, the spatial continuity is assumed to be linear. Multiple-point geostatistics uses a structure of more than two points for statistical inference; consequently, this method can reproduce nonlinear spatial correlations. However, in practical applications, it is difficult to find enough points, especially in the case of sparse data, thus not allowing direct inference of subsurface information or the establishment of a random function model that can describe the spatial structure. Therefore, it is necessary to use the conceptual image of geometric properties and geological features (which may be built using the interpreted geological data) for multiple-point geostatistical inference. Such a conceptual image is called a training image. Figure 3: A case study of the meandering stream facies modeling using object-based method: the grey color refers to the background facies, and the yellow color refers to the river channel of the meandering stream facies. The training image can be established from geological phenomena, such as the simulation of river sedimentation. The corresponding inference does not come from real data but from the simulated data depicted by the training image. In geological work, by simulating geological processes and even making outcrop photographs, one can depict waterways or lobes and can make multiple-point geostatistical inference with photographs. Since the training image covers multiple points, its spatial continuity pattern is more complex and comprehensive than that of the variogram. Therefore, the training image is closely related to the sedimentary conditions. However, it is not faithful to any specific data but is simply a conceptual image that characterizes the subsurface space. Therefore, it is appropriate to use the object-based method to generate a training image that meets the requirement. In 1993, [PERSON] [72] studied the algorithmic implementation of multiple-point geostatistics. This implementation first assumes an n-dimensional template \((h_{1},h_{2},\ldots,h_{n})\) and then scans the training image by using the template for all \(n\) random variables \(Z_{1}=z(u+h_{1}),\ldots,Z_{n}=z(u+h_{n})\) to derive the experimental conditional distribution of \(Z(u)\) at point \(u\). Finally, a sample value \(Z(u)\) is taken from this distribution and assigned to the current grid to be estimated. A sequential-like method is used here to simulate all grid points randomly. As show in Figure 5, a 2 \(\times\) 2 scan frame is applied. For a target training image, the scan frame is employed to scan all the parts to obtain different data events. Different data events have different probabilities.