## 5 Cased Study
### GNSS Signal Correction
The error analysis shows that calculation results varied slightly depending on different network nodes and transfer  
Figure 15: Curve fitting of foundation settlement. functions. Logsig function was taken as the transfer function in the paper, and the 4-8-1 node grid was adopted. MAE and MAPE in the table are written as  
\[\text{MAE}= \frac{1}{N}\sum_{i=1}^{N}\left|x_{i}-x_{i}^{\prime}\right|;\quad \text{MAPE}=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{x_{i}-x_{i}^{\prime}}{x_{i}^{ \prime}}\right|. \tag{7}\]  
Based on the two groups of parameters \(a\) and \(b\) obtained through the fitting of settlement curve, the nonlinear mapping relationship between the two groups of data is established by the learning and verification process through the method of BP neural network, and the settlement data obtained through the GNSS signal measured in real time are corrected. Figure 16 and Table 3 present the comparison of the real-time data of foundation settlement at the four observation stations of the bearing platform by the proposed algorithm, empirical mode decomposition, and wavelet decomposition. As shown in the figure, the black solid line indicates the results of precision leveling; the dotted green line shows the results processed with empirical mode decomposition and the solid blue line for the results of wavelet decomposition. The dotted red line represents the real-time subsidence data at the points processed by the proposed method: the multipath errors and random noise of the data were erased by stationary wavelet transform and empirical mode decomposition, and the data were rectified. It should be noted that, the signal-to-noise ratios and mean-square errors in the table represent the result errors between the three methods and precise leveling. According to data, the errors between the GNSS signals corrected by the proposed method with the precise leveling were all less than 1 mm, which proves that the proposed method in the paper can better rectify GNSS signal errors, and hence have higher accuracy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Neural Networks** & **Transfer Function** & **MAE (mm)** & **MAPE** & **RMSE (mm)** \\ \hline
4-7-1 & Tangent Sigmoid & 0.57 & 0.015 & 0.68 \\
4-7-1 & Logarithmic Sigmoid & 0.51 & 0.015 & 0.66 \\
4-7-1 & Line & 0.64 & 0.017 & 0.73 \\
4-8-1 & Tangent Sigmoid & 0.67 & 0.018 & 0.77 \\
4-8-1 & Logarithmic Sigmoid & 0.47 & 0.012 & 0.43 \\
4-8-1 & Line & 0.64 & 0.017 & 0.74 \\
4-9-1 & Tangent Sigmoid & 0.67 & 0.018 & 0.77 \\
4-9-1 & Logarithmic Sigmoid & 0.50 & 0.013 & 0.61 \\
4-9-1 & Line & 0.60 & 0.016 & 0.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error indexes of the proposed method forecasting results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**SNR (dB)**} & \multicolumn{3}{c}{**RMSE (mm)**} \\ \hline  & **EMD** & **DWT** & **Proposed Method** & **EMD** & **DWT** & **Proposed Method** \\ \hline Point A & 9.8 & 17.7 & 37.5 & 4.9 & 4.2 & 0.4 \\ Point B & 13.0 & 10.3 & 27.8 & 9.1 & 12.4 & 0.7 \\ Point C & 16.7 & 17.4 & 36.8 & 4.7 & 4.3 & 0.4 \\ Point D & 11.9 & 13.2 & 35.8 & 8.0 & 6.9 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the proposed method with traditional methods.