# Research Status of and Trends in 3D Geological Property Modeling Methods: A Review
## 3 Interpolation Methods for 3D Property Modeling
### Markov Model
#### 3.2.3 Pattern-Based Methods  
Simpat  
The probabilistic patterns (e.g., SNESIM) in multiple-point geostatistics were replaced by the pattern-based methods of Arpat [94; 95] to circumvent the limitations of the probability-based methods (these limitations are mostly the reproduction of the training image or the lack of such). The pattern-based MPS method is called simpat (simulation of patterns). This method introduces an alternative method for multiple-point conditional probabilistic inference of random variables. This method essentially obtains probabilities from the entire multiple-point pattern constrained to the same multiple-point data event in the training image. The concept of \"pattern\" represents a set of values defined by a specific multiple-point template. For example, in 2D, the pattern can be multiple-point values extracted from a 5 * 5 squares on the training image grid. The proposed method also follows the sequential approach of the previous MPS methods. At each node location, simpat extracts the data events defined by the pattern template and then searches all patterns in the training image to find the most similar pattern. The most similar pattern is then affixed to the grid. This process continues until all nodes of the simulation grid have been visited. As patterns are searched for similarity of data events, conditional adjacency data values are basically considered to retrieve multipoint patterns. In other words, the MPS method obeys the local pattern similarity criterion rather than the local conditional distribution. Stochastic simulation is thus analogous to an image construction problem, such as trying to piece puzzles together. Simpat is demonstrated to provide appealing visual effects in the implementation. However, a main problem of this method is the enormous computational complexity associated with the similarity search, i.e., the comparison of data events with all patterns in the training image, thus resulting in poor CPU performance compared to the CPU performance of other geostatistical algorithms. The direct sampling method was proposed by [PERSON] [96]. It is a pattern-based method similar to simpat in that it follows the same pattern concept to statistically infer the same conditional distribution from the training image. In contrast to simpat, the direct sampling method claims to be independent of the predefined template size. The simulation includes sequential visits to all nodes of the grid. At each node, the direct sampling method retrieves n known neighboring nodes of the grid within the specified search radius I. Then, this method defines a data event according to the lag vector obtained from the known neighboring nodes and searches the training image for this data event. However, unlike simpat, the direct sampling method enforces some stopping criteria and stops searching in two cases: whenever an exact match is obtained or when the dissimilarity of facies is lower than a given threshold. If neither case holds in the entire training image, then the direct sampling method returns to the simpat idea when selecting the structure with the minimum dissimilarity distance. However, the multigrid concept used by the previous MPS technique to reproduce the large-scale pattern of the training image is inherent to this method. At the beginning of the simulation, the direct sampling method can cover a large-scale grid, but as the simulation progresses, n-point data events can be easily retrieved from a small spatial range [97]. The direct sampling method directly searches the training image to find the current data event. Therefore, this method is very similar to the simpat algorithm. The differences are as follows: (1) the search stops at an exact match; (2) if the distance is less than the threshold, then there is another stopping criterion; and, most importantly, (3) the data events do not conform to any predetermined shape. However, the final difference is similar to that of simpat and other pattern-based methods.