## 4 Practical Application and Result Analysis
### Comparative Analysis of Model Prediction Effect  
The quantitative prediction results are shown in Table 4. It can be observed that the KELM (radial basis function, RBF) model has better prediction accuracy than the ELM model, indicating that the addition of the kernel function can improve the prediction effect. We can also observe that the prediction accuracy of the HKELM model is better than that of KELM (RBF) model, which indicates that the addition of a hybrid kernel function can also improve the prediction ability. The quantitative prediction results show the higher accuracy of the BO-HKELM model compared to the ELM, KELM (RBF) and HKELM models. It can be interpreted that the BO-HKELM model adopts the hybrid kernel and the Bayesian optimization algorithm to obtain the optimal parameters, which improves the generalization ability and prediction accuracy.  
The comparison between measured and estimated RHOB in Well A2 by the trained models (ELM, KELM (RBF), HKELM, and BO-HKELM) in Well A1 is shown in Figure 7. It can be observed that when the change in the logging curve is relatively stable, the output results of the BO-HKELM model and other models show little difference. Meanwhile, when the log curve has a local abrupt change (the green frame), the prediction result of the BO-HKELM model is closer to the measured value than the other models, indicating that the BO-HKELM model can extract the spatial characteristics of the log curve more. Therefore, compared with the ordinary machine learning model, the BO-HKELM model is more suitable for missing logging curve prediction.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **RMSE** & **MAE** & **R-Square** \\ \hline ELM & 0.1133 & 0.0798 & 0.7916 \\ KELM & 0.1055 & 0.0638 & 0.8178 \\ HKELM & 0. 0901 & 0.0623 & 0.8874 \\ BO-HKELM & 0.0767 & 0.0613 & 0.9029 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The outcomes of estimating RHOB in Well A2 using the trained models from Well A1.  
Figure 6: The fitness reduction rate in different iterations of Bayesian optimization algorithm for training HKELM.  
The cross-plots of original RHOB vs predicted RHOB using ELM, KELM (RBF), HKELM, and BO-HKELM in Well A2 are shown in Figure 8. It can be observed that the predicted results of BO-HKELM are closer to the actual RHOB than the other models, indicating that the BO-HKELM is better than other models in fitting multiple local details of DTS. In addition, it can be observed from the quantitative prediction results in Table 4 that compared with other models, the BO-HKELM model has lower errors and higher accuracy, which also reflects its advantages.  
Figure 8: _Cont._  
Figure 7: Visual results of the RHOB prediction results of Well A2.  
To compare the prediction ability of the BO-HKELM model and other models in the case of local abrupt changes in the logging curves, the prediction results of Well A2 are locally amplified and analyzed. The comparison outputs are shown in Figure 9. The black curve is the measured RHOB, and the red, orange, purple and green curves represent the prediction results of BO-HKELM, HKELM, KELM and ELM, respectively. It can be found from the Figure 9 that when the logging curve smoothly changes, the output results of the BO-HKELM model and other models show little difference. While the log curve has the local mutation, the prediction result of the BO-HKELM model is closer to the measured value than other models. The reason is that the BO-HKELM model adds a hybrid kernel function and adopts a Bayesian algorithm to optimize the hyper-parameters, which can fully extract the spatial characteristics of the log curve and enhance the prediction ability of the model.  
In conclusion, the BO-HKELM model proposed in this research has an excellent prediction effect overall and reveals a local abrupt change in the logging curves, clearly demonstrating its advantages in predicting the missing well logs.