# Research Status of and Trends in 3D Geological Property Modeling Methods: A Review
## 3 Interpolation Methods for 3D Property Modeling
### Multiple-Point Geostatistics Methods
#### 3.2.2 Iterative Methods  
Simulated Annealing (SA)  
SA is a global optimization method that can reduce specific objective functions. The first application of SA in space was introduced by [PERSON] [78]. This algorithm generates a model by iteratively minimizing the objective function, which characterizes the degree of mismatch between the desired statistic and the spatial continuity of the a priori model. The process starts by simulating a random spatial distribution of values on the grid. In each iteration, the node values on the grid are perturbed, and a new mismatch between the current statistic and the target statistic is quantified. This perturbation is retained only when the degree of mismatch is reduced, i.e., when the objective function is reduced. However, to reach the global minimum without falling into a local minimum, the SA method sometimes discards the perturbations with low mismatches or even accepts unfavorable perturbations in a so-called \"annealing scheme\". The grid is updated continuously and iteratively in this process to achieve the multiple-point statistics of the training image. This iterative method has been applied to the simulation of categorical variables [79] or the simulation of the physical properties of continuous rocks [80]. The optimization of the objective function in SA relies on the subjective selection of optimization parameters and perturbation mechanisms for efficient simulation. Many studies have attempted to select the initial temperature [81] or update the objective function and the perturbation mechanism in SA optimization [82]. However, in general, these parameters strongly depend on the spatial continuity model and need to be set through trial and error. Later, [PERSON] [83] applied the SA method in parallel computing to reduce the computational cost. Despite these efforts, the simple visual effect comparison of geometric features in the implementation shows that considering only the low-order statistics of the spatial model (because of the computational limitation of the objective function) is insufficient to construct a desired subsurface model. The iterative method of SA also allows for slight inconsistencies in the implementation. In addition, the increase in the type and order of multiple-point statistics causes CPU and storage-related problems. The choice of which multiple-point statistic to retain in the objective function is also an open question in the SA method. As shown in Figure 6, compared with the traditional geostatistics and multiple-point geostatistics methods, the multiple-point geostatistics methods can better model the original spatial models. The Sequential Gaussian Simulation and Sequential Indicator Simulation belong to the Tradition Geostatistics and Indicator-based Methods, and Simulated Annealing belongs to Multiple-Point Geostatistics. Neural networks are based on the way the brain performs computations and are used to fit nonlinear functions and identify patterns in the data. [PERSON] [84] was the first to apply neural networks in multiple-point geostatistics. The neural network is trained by the training image to determine the local conditional probabilities. A neighborhood template is selected, and then all the multiple-point neighboring values are mapped by the network to the one-dimensional output, which represents the conditional probabilities of the nodes centered on the template. In the learning phase, the training image statistics are used to adjust the network parameters. Figure 6: Comparison of Traditional Geostatistics and Multiple-Point Geostatistics Methods for the same model: the Sequential Gaussian Simulation and Sequential Indicator Simulation belong to the Tradition Geostatistics and Indicator-based Methods, and Simulated Annealing belongs to Multiple-Point Geostatistics (edited from [78]). Similar to all iterative methods, the simulation starts with a random initialization of the grid.