## 4 Results and Discussion
### Performance of Each Time Series Model  
#### 4.1.1 Performance of ARIMA Model  
The error series of the Donglin station rate periodically was first tested by the augmented Dickey-Fuller (ADF) test (ADF less than 0.05 is considered as stationary series) [51; 52] and plotted with autocorrelation, see Figure 5a in which it can be seen that the correlation coefficient of the original series is greater than zero for a long time, indicating a strong long-term correlation between the series. However, the \(p\) value corresponding to the ADF test statistic of the original sequence was 0.99, which was significantly greater than 0.05, indicating that the series is a non-stationary series (non-stationary series must not be a white noise series). Next, the original error series were made a first-order difference, followed by ADF detection and white noise detection. The results show that the autocorrelation plot has a solid short-term correlation, and the \(p\) values of both ADF and white noise test statistics are less than 0.05, so the error series after the first-order difference is a stationary non-white noise series.  
By observing the ACF (autocorrelation function) and PACF (partial autocorrelation function) plots after differencing (Figure 5b,c), we can see that both ACF and PACF show smearing, the PACF cut-off is at lag 6, and ACF decay tends to zero after lag 1, so it is ARMA (6, 1) [53]. Finally, ARIMA (6, 1, 1) can be determined by combining first-order differences. Next, the error series of the testing period was simulated, the predicted results were fitted to the true values, and the residual series were analyzed (Figure 5d-f). The residual series in the figure is within the confidence interval above and below the mean 0, and the residual series passes the white noise test [54]. The result of the D-W test [55] on the residuals is 2.002, which is close to 2, indicates that there is no autocorrelation in the residuals predicted by the model, and the sample quantile distribution in the residual Q-Q plot is similar to the theoretical quantile distribution. All of this shows that the fitted model has good prediction.  
Figure 6 is the runoff simulation diagram obtained using the ARIMA model to correct the simulated runoff from the BTOP model. It can be seen that the modified curve of the ARIMA model effectively reduces a certain amount of runoff in the low flow periods to make it closer to the observed flow, which reduces the error to a certain extent and improves the simulation accuracy of the model. However, the modification of the ARIMA model in both the low-flow and high-flow periods is used to reduce runoff based on the original BTOP simulation trend. This is because the ARIMA model is, after all, only a model with statistical significance, the simulation of the hydrological process lacks physical cause analysis, and the simulation of the error series is only based on the information input before this period; thus, it may be difficult to deal with the turning point of the maximum value in the period of high flow. Therefore, measures should be taken in the practical application [56]. In general, the ARIMA model relies on the time-series dependence of forecast errors so that the errors will accumulate rapidly in the extrapolation process, and it is not suitable for larger basins or situations requiring longer forecasting periods [57].  
Figure 5: Prediction process of ARIMA model. (**a**) Original sequence autocorrelation, (**b**) autocorrelation and (**c**) partial autocorrelation after difference, (**d**) residual autocorrelation and (**e**) partial autocorrelation, (**f**) residual Q-Q graph.