# Research Status of and Trends in 3D Geological Property Modeling Methods: A Review
## 3 Interpolation Methods for 3D Property Modeling
### Traditional Geostatistical Methods
#### 3.1.1 Kriging Method  
The kriging method was proposed by [PERSON] [49] in 1951 for estimating unknown variables. The method uses the linear least-squares estimation technique and relies on variables from sparse and valid hard data for interpolation. This method was initially applied to mineral deposits under the assumption of continuity of measured mineral gradesusing a variogram. As show in Figure 2, the variogram has several main components: range, nugget, arch height, and sill. The range refers to the range where regionalized variables have spatial correlation. Within the variable range, the data are correlated, while outside the variable range, the data are not correlated with each other; the observed values outside the variable range will not affect the estimation results. If the variation function is discontinuous at the origin, this is called the nugget effect in geostatistics, which shows that it has large spatial variability in a very short distance. The spatial variability is the nugget. It should be noted that it can be caused by geological variability or measurement error. The arch height refers to the difference between the sill and the nugget. Sill is the total spatial variability of the whole variables. When the value of the variation function at distance is greater than the range, the sum of the nugget value and arch height is the sill. The kriging method uses linear combinations of known data to obtain values at unsampled locations without assuming any statistical distribution. The weights for these linear combinations need to be calculated to minimize the estimated variance in a least-squares sense, which can be expressed by the following least-squares variance equation:  
\[\lambda_{a}=\left[z(u)-z^{*}(u)\right]^{2} \tag{1}\]  
where \(\lambda_{a}\) represents the value of the linear combination weight and \(z^{*}\) represents the estimated value of the random variable \(z\) at location \(u\); the linear estimate of \(z^{*}\) is obtained as follows:  
\[z^{*}(u)-m(u)=\sum_{\alpha=1}^{n}\lambda_{\alpha}\cdot\left[z(u_{\alpha})-m(u _{\alpha})\right] \tag{2}\]  
where \(m\) denotes the mean at the unsampled location \(u\) and \(\lambda_{\alpha}(\alpha=1,2,\ldots,n)\) denotes the weight of \(n\) measured variables. In two-point geostatistics, the structural function that characterizes the spatial variability of a variable is called the variogram, which is the spatial correlation between any two points in the space. That is, as the distance between two points in the space increases, the variogram gives the change in the spatial correlation between them. This is the simplest way to correlate the value of an unsampled point with its neighboring observations. Under stationary conditions, the variogram is defined in probability theory as follows:  
\[2\gamma(h)=E\Big{\{}[z(u)-z(u+h)]^{2}\Big{\}} \tag{3}\]  
Figure 2: Schematic diagram of variation function (edited from [5; 50; 51; 52]). where \(\gamma(h)\) denotes the variogram value calculated based on the pairing of the available data for a particular lag distance; \(E\) denotes the expected squared error of the estimation point, \(z\) denotes the random variable, and \(h\) denotes the relative distance from location \(u\). The kriging method can describe only relatively simple geological features, and the simple variogram fails to apply to some geological phenomena with high variability. To address this problem, many scholars have conducted research and improvements. Depending on the interpolation model and application environment, a variety of kriging methods (e.g., simple kriging and ordinary kriging) have emerged [5, 50, 51]. The kriging method is characterized by underestimating high values and overestimating low values, limiting its ability to simulate nonlinear spatial structures. However, in some cases, what is important is not the estimated value of the variable itself but the variable's nonlinear function, such as estimating the mineral grade greater than a certain threshold.