## 2 Materials and Methods
### Kriging Interpolation Method  
The story number of buildings is preferentially obtained using deep net data, but the coverage of deep net data is limited, and there are still some buildings with unknown story number. In order to obtain the story number of all buildings, it is assumed that story number of buildings follows the first law of geography. Based on the assumptions, story number of the unknown building is deduced from story number of the surrounding buildings using ordinary Kriging interpolation, with the following basic idea:  
The story number of a building is denoted by \(s_{i}\), which is a smooth random function. Assuming that the story number of all buildings in a certain range is \(\{s_{1},s_{2},\ldots,s_{n}\}\), the estimate of the number of unknown story number \(s_{0}\) is  
\[s_{0}^{\prime}=\sum\limits_{i=1}^{n}\lambda_{i}s_{i} \tag{7}\]  
where \(\lambda_{i}\) is a weighting factor indicating the degree of contribution of the story number of the ith building to the unknown layer value of the building. Solving for \(\lambda_{i}\) requires that the estimate of the unknown story number of building \(s_{k}^{\prime}\) is an unbiased estimate of the true value \(s_{k}\) and is optimal, i.e., the mathematical expectation of the deviation is 0 and the variance is minimized, as expressed in Equations (8) and (9). From the unbiased constraint, it can be deduced that \(\sum\limits_{i=1}^{n}\lambda_{i}=1\). \[E\big{(}s_{0}^{\prime}-s_{0}\big{)}=0 \tag{8}\]  
\[\min\big{(}Var\big{(}s_{0}^{\prime}-s_{0}\big{)}\big{)}=min\Big{(}E\big{(}s_{0 }^{\prime}-s_{0}\big{)}^{2}\Big{)} \tag{9}\]  
Therefore, the estimated value of \(s_{0}^{\prime}\) for the number of unknown story number is obtained, which is transformed into a weighting factor \(\lambda_{i}\) that seeks the minimum variance of the estimated value \(s_{0}^{\prime}\) from the true value \(s_{k}\), and the solution steps are as follows:  
1. Calculate the semi-covariance \(\gamma_{ij}\) of the distance \(d_{ij}\) between two buildings and the story number of building, as shown in Equation (10) and Equation (11), respectively: \[\gamma_{ij}=\frac{\big{(}s_{i}-s_{j}\big{)}^{2}}{2}\] (10) \[d_{ij}=d\big{(}z_{i},z_{j}\big{)}=d\big{(}\big{(}x_{i},y_{i}\big{)},\big{(}x _{j},y_{j}\big{)}\big{)}=\sqrt{\big{(}x_{i}-x_{j}\big{)}^{2}+\big{(}y_{i}-y_{ j}\big{)}^{2}}\] (11) where \((x_{i},y_{i})\) and \(\big{(}x_{j},y_{j})\) are the position coordinates of the building \(z_{i}\) and \(z_{j}\), respectively;
2. Fitting the relationship between the building distance \(d_{ij}\) and the semi-covariance of the story number \(\gamma_{ij}\): \(\gamma_{ij}=f\big{(}d_{ij}\big{)}\);
3. According to the fitted model in step 2, the semi-variance of any building story number in a certain range and the unknown building story number are found;
4. Construct the Lagrangian function according to Equation (9) and \(\sum\limits_{i=1}^{n}\lambda_{i}-1=0\). The text continues here;
5. Let the first-order partial derivatives of \(F(\lambda_{k},\mu)\) with respect to \(\mu\) and \(\lambda\) be equal to zero to obtain the system of Kriging equations in Equation (12), which can also be written in the matrix form of Equation (13). Solve to obtain the optimal weight coefficients \(\lambda_{i}\); \[\left\{ \begin{aligned} \sum\limits_{i=1}^{n}\lambda_{i}=1\\ \sum\limits_{i=1}^{n}\gamma_{ij}\lambda_{i}-\mu=\lambda_{i0},j=1,2,3, \ldots,n\end{aligned} \right.\] (12)\[ \begin{bmatrix}\gamma_{11}&\gamma_{12}&\dots&\gamma_{1n}&1\\ \gamma_{21}&\gamma_{22}&\dots&\gamma_{2n}&1\\ \dots&\dots&\dots&\dots\\ \gamma_{n1}&\gamma_{n2}&\dots&\gamma_{mn}&1\\ 1&1&\dots&1&0\end{bmatrix}\begin{bmatrix}\lambda_{1}\\ \lambda_{2}\\ \dots\\ \lambda_{n}\\ -\mu\end{bmatrix} = \begin{bmatrix}\gamma_{10}\\ \gamma_{20}\\ \dots\\ \gamma_{n0}\\ 1\end{bmatrix}  \tag{13}\]  
\(6\) Substitute the \(\lambda_{i}\) solved in step 5 into Equation (7) to obtain the unknown building layer value.