# Estimating CPT Parameters at Unsampled Locations Based on Kriging Interpolation Method
## 3 Kriging Interpolation  
### Set Up Semivariogram  
The core theory of the Kriging method is the semivariogram, which is established to reflect the relationship between distance \(d\) and semivariance \(\gamma_{d}\) between two sampled points. Generally speaking, the semivariogram is an increasing function. The larger the semivariance, the smaller the spatial dependence and mutual influence between the two sample points. According to the different forms of the semivariogram, different models are selected for fitting. Figure 7 shows some commonly used semivariogram models in the literature, such as the spherical model, exponential model, and Gaussian model. [PERSON] [18] used semivariogram to analyze the spatial variability of CPT data performed in alluvial deposits of sand and gravel in the Mississippi River flood plain, and the results showed that the spherical model was more proper for fitting the actual curve compared with other models. In addition, previous studies also demonstrated that the spherical model can fit well to the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & E3 & G4 & I5 & D6 & F7 & J8 \\ \hline
**Shift distance (mm)** & 0 & 90 & 230 & 0 & 115 & 130 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Shift distance of the six boreholes by CCF.  
Figure 6: CCF of borehole F7.  
semivariogram of various soils [19]. So in this study, we also chose a spherical model to fit the semivariogram.  
The semivariogram is defined as follows:  
\[\gamma_{d}=\frac{1}{2}Var[Z_{x}-Z_{x+d}] \tag{1}\]  
where: \(Z_{x}\) is the variable at location, \(x\);  
\(Z_{x+d}\) is the variable at location, \(x+d\);  
\(d\) is the distance between the sample pairs;  
\(Var[x]\) refers to the variance of \(x\).  
Equation (1) implies that the nature of the semivariogram function is half the variance of \(Z_{x}\) and \(Z_{x+d}\), sample pairs separated by \(d\), and for a certain semivariogram, the variance is dependent only on \(d\). This hypothesis, however, is hard to be satisfied for soil, for example, \(Z_{x1}\), \(Z_{x2}\), and \(Z_{x3}\) are samples at different depths in the same CPT borehole, and the distance from \(x_{2}\) to \(x_{1}\) and \(x_{3}\) is the same. The hypothesis means that \(x_{2}\) should have the same effect on the other two points, but if \(x_{2}\) and \(x_{1}\) are in the same layer and \(x_{3}\) is in another layer, then \(x_{2}\) will obviously have different effects on the other two points. Although the hypothesis is difficult to meet, the Kriging interpolation is still a reasonable estimation method under the condition that the information of the soil layer at the estimation point is unknown.  
The variables used to plot the semivariogram should satisfy the following equation:  
\[E[Z_{x}-Z_{x+d}]=0\quad\forall x,d \tag{2}\]  
where \(E[x]\) refers to the expectation of \(x\).  
As shown in the above equation, all the variables within the study area should have the same expectations, which are always called stationary variables. Figure 8 shows the \(q_{\text{c}}\) profiles of borehole E3, and it is clear from Figure 8 that CPT data are not stationary but with a specific trend. In order to satisfy Equation (2), ordinary least squares regression is performed to estimate the trend. After that, the trend component will be removed from the original CPT data so that the residuals would be stationary, and the semivariogram could be established upon it. The process is shown in Figure 8.  
Figure 7: Commonly used semivariogram models.  
Considering Equation (2), Equation (1) can be simplified as:  
\[\gamma_{d}=\frac{1}{2}E\Big{[}(Z_{x+d}-Z_{x})^{2}\Big{]} \tag{3}\]  
In practice, the semivariogram must be estimated from the available discrete data, i.e., the theoretical expectations could be estimated by sample mean:  
\[\gamma_{d}^{*}=\frac{1}{2N}\sum_{i=1}^{N}\Big{[}(Z_{i+d}-Z_{i})^{2}\Big{]} \tag{4}\]  
where \(N\) is the number of point pairs that are separated by distance \(d\).