# Research Status of and Trends in 3D Geological Property Modeling Methods: A Review
## 3 Interpolation Methods for 3D Property Modeling
### Traditional Geostatistical Methods
#### 3.1.2 Stochastic Simulation Method  
All kriging methods derive the \"optimal linear unbiased estimator\" based on variogram models. Therefore, kriging methods are only a smooth representation of the subsurface model. Due to uncertainties, it is not enough to generate a single deterministic model. Therefore, optional stochastic reservoir modeling methods are needed to establish a series of models with different property distributions. Stochastic simulation was first introduced by [PERSON] [55] and [PERSON] [4] to correct the smoothing effect and other human factors in the kriging method. Stochastic simulation also uses the variation model to predict the spatial variability, resulting in multiple optional models of the spatial distribution of variables in the study area. The main differences between the kriging method and the stochastic simulation method [48] are that the kriging method performs the \"optimal\" linear estimation of the variables without conducting statistical analysis of the estimated values together; that is, the results of the kriging method are a local presentation of the variables, while the stochastic simulation method provides a global expression of the variables, which can reproduce the spatial continuity pattern. The stochastic simulation process is also called sequential simulation and is particularly important for numerical simulations of, for example, the control of fluid flow in a reservoir by permeability. Therefore, global stochastic simulation algorithms outperform the locally accurate kriging methods in constructing reservoir models. The smoothing effect of the kriging method introduces significant bias when dealing with nonlinear problems. In addition to providing a variety of optional results, sequential simulation can also truly characterize the spatial variability of geological phenomena. Multiple conclusions about reservoir connectivity or fluid mobility can be summarized from the results of each realization. Therefore, functions that characterize complex nonlinear structures can be obtained by calculating each simulation result and then obtaining the probability distribution of the target variables. Stochastic simulation is based on a simple mathematical principle, which mainly aims to obtain the conditional probability \(z\) of the variable \(Z(u)\) after calculating all values at the neighboring locations of \(u\). For \(N\) random variables of \(N\) valid data types \(\{z_{a}=z(u_{a}),\alpha=1,2,\ldots,N\}\), the N-dimensional conditional probability distribution function is expressed as Equation (4). \[F(Z_{1},Z_{2},\ldots,Z_{N}|(n))=\text{Prob}\{Z_{a}<z_{a},\alpha=1,2,\ldots,n|( n)\,\} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  
Using [PERSON] theorem, the N-dimensional probability distribution can be obtained by calculating the univariate conditional probability distribution function as Equation (5). \[F(Z_{1},Z_{2},\ldots,Z_{N}|(n)\,)=F(Z_{1}(n))\cdot F(Z_{2}(n+1))\cdots F(Z_{N}(n +N-1)) \tag{5}\]  
Therefore, the general process of sequential simulation is summarized as follows:  
Calculate the univariate conditional probability \(z_{1}\) of \(Z_{1}\), conditional on \(n\) original data. Calculate the conditional probability \(z_{2}\) of \(Z_{2}\), conditional on \(n\) original data and the previously simulated value \(Z_{1}=z_{1}\). Calculate the conditional probability of \(Z_{N}\), conditional on \(n\) original data and \(N-1\) previously simulated values. Sequential Gaussian simulation has been widely used because of its simplicity and flexibility due to the introduction of a Gaussian assumption. Sequential Gaussian simulation has a simple basic idea: each variable is simulated according to the normal conditional probability distribution function obtained by a simple kriging method, which selects the expected value of the variable as the estimated value of the point to be estimated.