## 2 Methodology
### Construction of Model  
Cokriging interpolation greatly improves the accuracy of the interpolation by introducing one or more covariates (also referred to as auxiliary variables) that are closely related to the main variables, which is an improved method of kriging interpolation. This method is based on the theory of co-regionalized variables and carries out spatial interpolation according to it. The cokriging interpolation formula is as in Equation (2). \[Z_{0}^{*}=\sum\limits_{i=1}^{n}\lambda_{i}Z_{i}+\sum\limits_{j=1}^{\text{k}} \sum\limits_{i=1}^{n}\mu_{ij}Y_{\text{i}}, \tag{2}\]  
where \(Z_{1}\), \(Z_{2}\),..., \(Z_{n}\) and \(Y_{j1}\), \(Y_{j2}\),..., \(Y_{jn}\) are \(n\) sample data for the main variables and covariates, respectively; \(j\) is the number of covariates; \(\lambda_{1}\), \(\lambda_{2}\),..., \(\lambda_{n}\) and \(\mu_{j1}\), \(\mu_{j2}\),..., \(\mu_{jn}\) are the cokriging weighting coefficients to be determined; \(Z_{0}^{*}\) is the repaired value of the random variable at 0. Taking the second order as an example, if there are two regionalized values \(Z_{\text{i}}(x)\) and \(Y_{\text{i}}(x)\) (\(i\) = 1, 2,..., \(n\)), which are related to the attribute in a region, then \(Z_{0}^{*}=\sum\limits_{i=1}^{n}\lambda_{i}Z_{i}+\sum\limits_{i=1}^{n}\mu_{i}Y_ {i}\). In order to satisfy the unbiased condition of the estimator, the sum of weights of main variables should be equal to 1, and the sum of weights of covariates should be equal to 0, as seen in Equation (3). \[ \begin{cases}\sum\limits_{i=1}^{n}\lambda_{i}=1\\ \sum\limits_{i=1}^{n}\mu_{i}=0\end{cases} . \tag{3}\]  
According to the irreversible bias of the estimated value and the least square method, the Lagrange multiplication is used to solve the cokriging equation, as seen in Equation (4). \[ \begin{cases}\sum\limits_{i=1}^{n}\lambda_{j}C_{Z_{i}Z_{j}}+\sum\limits_{i=1} ^{m}\mu_{j}C_{Z_{i}Y}-\eta_{1}=C_{Z_{0}Z_{i}}\\ \sum\limits_{i=1}^{n}\mu_{i}C_{Y_{i}Y_{j}}+\sum\limits_{i=1}^{m}\lambda_{i}C_{Z _{i}Y_{j}}-\eta_{2}=C_{Z_{0}Y_{j}}\\ \sum\limits_{i=1}^{n}\lambda_{i}=1\\ \sum\limits_{i=1}^{n}\mu_{i}=0\end{cases} . \tag{4}\]It is expressed in matrix form as shown in Equation (5). \[ \begin{pmatrix}\text{C}_{Z_{0}Y_{m}}&\cdots&\text{C}_{Z_{1}Z_{m}}&\text{C}_{Z_{1}Y }&\cdots&\text{C}_{Z_{1}Y_{m}}&1&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ C_{Z_{m}Z_{1}}&\cdots&\text{C}_{Z_{m}Z_{m}}&\text{C}_{Z_{n}Y_{1}}&\cdots& \text{C}_{Z_{n}Y_{m}}&1&0\\ \text{C}_{Y_{1}Z_{1}}&\cdots&\text{C}_{Y_{1}Z_{m}}&\text{C}_{Y_{1}Y_{1}}& \cdots&\text{C}_{Y_{1}Y_{m}}&0&1\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\ C_{Y_{m}Z_{1}}&\cdots&\text{C}_{Y_{m}Z_{m}}&\text{C}_{Y_{m}Y_{1}}&\cdots& \text{C}_{Y_{m}Y_{m}}&0&0\\ 1&\cdots&1&0&\cdots&0&0&0\\ 0&\cdots&0&1&\cdots&1&1&1\end{pmatrix}\begin{bmatrix}\lambda_{1}\\ \vdots\\ \lambda_{n}\\ \mu_{1}\\ \vdots\\ \mu_{m}\\ -\xi_{1}\\ -\xi_{2}\end{bmatrix} = \begin{bmatrix}\text{C}_{Z_{0}Z_{1}}\\ \vdots\\ \text{C}_{Z_{0}Z_{n}}\\ \vdots\\ \text{C}_{Z_{0}Y_{1}}\\ \vdots\\ \text{C}_{Z_{0}Y_{m}}\\ 0\end{bmatrix} . \tag{5}\]  
The cokriging method is based on the theory of co-regionalized variables. It uses cross-covariance and cross-semi-variance function modeling between multiple regionalized variables to characterize the correlation between them to obtain unbiased and optimal estimates of the regionalized variables. Because each environmental component has different effects on the effect variables of different parts, the environmental measurement value cannot be directly used as covariates to participate in the construction of the model. In this paper, by using the duration sequence of the measured value of each measuring point and the water pressure, temperature, rainfall, and aging, the non-linear fitting method was adopted to search for the most suitable non-linear expression of the measured value of each measuring point about each environmental variable in the global range.