## 4 Results and Discussion
### Performance of Each Time Series Model
#### 4.1.3 Performance of Improved Prophet Model  
Several parameter rates were determined, and Figure 9 shows the analysis results of the error time series. This is an overall forecast result graph in which only the first 80% of the time series infer the point of change so that there is enough runway to predict the direction of the trend and avoid over-fitting the fluctuation at the end of the time series. In the evaluation of the results, the blue curve is used as the basis for prediction. The light blue confidence region is used for reference to improve the model. Looking at the trend analysis in Figure 10, it can be seen that the variation trend of the error series starts to decline in May, and then increases significantly from September to November, that is, the error in high-flow periods is bigger than in low-flow periods, which is closely related to the simulation of the BTOP model in-high flow and low-flow periods. The trend in the predicted value dramatically correlates with the change in the historical value and the current error situation [46].  
Figure 11 is the runoff simulation diagram obtained using the Prophet model to correct the BTOP model. From the curves of the observed and simulated values therein, it can be seen from the observed and simulated curves that the BTOP model has achieved a good daily runoff simulation effect in the study area, the base simulated flow is close to the observed flow, and the flood peak simulation time is accurate [62]. Careful analysis of the error sources shows that the simulated flow of BTOP is generally slightly higher than the observed flow during the low-flow periods, while the simulated flood peak flow is much smaller than the observed flow in high-flow periods, which is the result of the transmission of BTOP model operation error from upstream to downstream [61]. Next, by looking at  
Figure 8: Hydrographs from Donglin Station in 1985 with observed, simulated, and LSTM correction runoff.  
the Prophet correction curve, it can be found that the corrected runoff curve is closer to the observed flow in the low-flow periods. It also has a good correction effect in high-flow periods, but is not as significant as in low-flow periods. This indicates that the revision of the Prophet model is mainly reflected in reducing runoff through \(g_{(t)}\) and \(s_{(t)}\) in low-flow periods, and increasing runoff through \(f_{(t)}\) in the maximum high-flow periods. However, for the few events where the simulated BTOP values are too large during the high-flow periods, the Prophet model only slightly reduces the runoff and is not effectively corrected (20 August). This is also the focus of further improvement of the model.  
Figure 10: Trend prediction analysis of Prophet model. The blue line shows the trend that the model fits from the test data, and the light blue shade shows the predicted trend.  
Figure 9: Error forecast of the Prophet model. The black dots represent the historical input data, the blue line represents the predicted trend line after model fitting, the light blue area above and below the blue curve represents the confidence interval, and the gray dotted-line reflects the position of the trend mutation point of the sequence.  
In general, the correction of runoff errors for BTOP using the Prophet model is effective.