# Dispersion Curve Interpolation Based on Kriging Method
## 3 Methods  
### Kriging and Dispersion Curves  
Kriging was created to describe the spatial relationships between geological variables. It originated in statistics, but differs from traditional statistical methods. Geological variables are not independent, and tend to be spatially related. In surface tomography, the dispersion curve represents the velocity of waves in different periods. At the same time, surface wave tomography assumes that the medium is homogeneous and layered. Although the horizontal axis of the dispersion curve is the time domain, it represents the spatial properties of the medium. Therefore, dispersion curve interpolation based on kriging is feasible. We used universal kriging and ordinary kriging to predict the dispersion curves.  
A brief account of the difference between the two methods is given here. Kriging is an interpolation technique that uses observations \(z(x_{i})\) to estimate \(z_{0}(x_{0})\), where \(x_{i}\) and \(x_{0}\) indicate the locations. Kriging assumes that the variable \(Z\) can be written as the sum of a deterministic component \(m\) and a stochastic component \(R(x)\), as follows:  
\[Z(x)=m(x)+R(x). \tag{1}\]  
In simple kriging, (1) can be written as:  
\[E[R(x)]=E[Z(x)-m(x)]=E[Z(x)]-m=0\ \ \forall x, \tag{2}\]  
Figure 2: The original ambient noise tomography results for different depths.  
where it is assumed that the mean \(m\) is constant across the field of interest (i.e., \(E[R(x)]=m\)). In contrast, in universal kriging, the mean is assumed to have a functional dependence on spatial location, and can be approximated by a suitable model with the form [25]:  
\[m(x)=\sum_{k=0}^{K}a_{k}f_{k}(x), \tag{3}\]  
where \(a_{k}\) is the \(k\)th coefficient to be estimated from the data, \(f_{k}\) is the \(k\)th function of spatial coordinates that describes the drift, and \(k\) is the number of functions used in modeling the drift. Furthermore, in ordinary kriging, \(m\) is re-estimated at each location \(x\). More detailed descriptions about different kriging methods can be found in previous studies (see, e.g., [34; 35]).  
However, different kriging methods must satisfy the conditions of unbiased estimates and minimal estimated variance [34; 36]. Briefly, we first find the covariance function based on the available data, then obtain the kriging estimates through the covariance function. The covariance function is mainly used to describe the spatial relationships of different variables. The variance function is often used, instead of the covariance function. For second-order stationary processes, the covariance function and variogram are equivalent [34]. The most popular function models include the Gaussian, exponential, and spherical models.  
We used cross-validation to determine that the spherical model was the theoretical model with the best fitting effect (Table 1); more detailed data are provided in Table 1. The criterion for evaluating the suitability of the variogram model was the R square, which indicates how accurately the data fit the model. The best-fitting model has the value closest to 1. It was determined using the'sklearn' package in Python. It can be written as:  
\[R^{2}=1-\frac{\sum_{i}(\hat{y}_{i}-y_{i})^{2}}{\sum_{i}(\overline{y}-y_{i})^{2 }}\ (i=1,\ 2,\ 3,\ldots), \tag{4}\]  
where \(\hat{y}_{i}\) is the predicted value, \(y_{i}\) is the real value, \(\overline{y}\) is the mean value, and \(\hat{y}_{i}-y_{i}\) is the residual.