# Modeling the Submergence Depth of Oil Well States and Its Applications
## 3 Results
### Error Analysis
#### 3.2.3 Application Analysis of the Least Squares Curve-Fitting Method
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline
**No.** & \(\mathbf{h_{0}\pm\Delta h_{0}}\) & & & \(\mathbf{h_{i}\pm\Delta h_{i}}\) & & & & & **Relative Error \(\mathbf{C_{f}}\)** & **Relative Error \(\mathbf{h_{E}}\)** \\ \hline
1 & + & + & + & + & + & + & + & + & + & + & + & 8.56 \(\times\) 10\({}^{-11}\) & 9.88 \(\times\) 10\({}^{-03}\) \\
2 & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & 2.86 \(\times\) 10\({}^{-11}\) & 9.88 \(\times\) 10\({}^{-03}\) \\
3 & 0 & + & \(-\) & + & \(-\) & + & \(-\) & + & \(-\) & + & \(-\) & + & 7.79 \(\times\) 10\({}^{-03}\) & 9.76 \(\times\) 10\({}^{-03}\) \\
4 & 0 & \(-\) & + & \(-\) & + & \(-\) & \(-\) & + & \(-\) & + & \(-\) & 7.79 \(\times\) 10\({}^{-03}\) & 9.91 \(\times\) 10\({}^{-03}\) \\
5 & 0 & + & + & \(-\) & \(-\) & + & + & \(-\) & \(-\) & + & + & \(-\) & 1.99 \(\times\) 10\({}^{-02}\) & 1.48 \(\times\) 10\({}^{-02}\) \\
6 & 0 & + & + & + & \(-\) & \(-\) & \(-\) & + & + & + & \(-\) & 7.94 \(\times\) 10\({}^{-02}\) & 5.66 \(\times\) 10\({}^{-02}\) \\
7 & 0 & + & + & + & + & 0 & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & 1.95 \(\times\) 10\({}^{-01}\) & 1.28 \(\times\) 10\({}^{-01}\) \\
8 & 0 & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & 0 & + & + & + & + & 1.86 \(\times\) 10\({}^{-01}\) & 1.80 \(\times\) 10\({}^{-01}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Positive and negative error distributions of the sampling points (\(N=11\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sampling Point \(N\)** & **Relative Error \(C_{f}\)** & **Relative Error \(h_{E}\)** & **Time Used (s)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 6: Calculation results for an alternating distribution of positive and negative errors at the sampling points. Figure 6: Error trends of the least squares curve-fitting method with the number of sampling points. It can be seen in Table 6 and Figure 6 that:  
(1) The \"relative error \(C\)/\" and \"relative error \(h_{E}\)\" decrease as the number of sampling points increases;  
(2) For odd numbers of sampling points, the result is better than that for the adjacent even numbers; whether the number of nodes is odd or even, the more nodes that participate in the operation, the smaller the error;  
(3) If one of the sampling points is used as the boundary to divide the positive and negative errors of the sampling points, the operation result is relatively poor. However, the more sampling points there are, the lower the probability of positive and negative error segmentation of such sampling points. In summary, it can be seen from the derivation and testing of the numerical methods:  
(1) When the measurement errors of all sampling points are the same or close, it can be seen in the characteristic function (Formula (16) or (26)) that the error of solution \(x\) and the comprehensive seepage parameter \(C_{f}\) is also smaller. However, it affects the value of \(h_{E}\), and the error is determined by the measurement error;  
(2) The more uniform the positive and negative error distributions are, the better the results of the least squares curve-fitting method are compared with those of the nonlinear-interpolation method;  
(3) When the number of nodes involved in the calculation is three, the nonlinear-interpolation method is a special case of the least squares curve-fitting method. Therefore, the least squares curve-fitting method has wider adaptability.