## 4 Airborne Balloon Movement Simulation
### Assessment of the Simulated Forward Trajectory Accuracy  
Figure 3 shows the comparison between the observed trajectory and the forward forecast trajectory of the airborne balloon, and Table 5 provides the forecast bias for the forward motion trajectory of the airborne balloon. In Figure 3, it is noted that the forecasting trajectory of the airborne balloon closely aligns with the observed trajectory in terms of the horizontal direction. When utilizing forecast data in the Lagrangian model, the minimum horizontal trajectory bias of the airborne balloon forecasted by FLEXPART within 11 h is 8.3 km, whereas the maximum bias is as high as 69.1 km. Both the RMSE and MAE lie within 50 km (Table 5). These results indicate an enlargement of the horizontal motion trajectory bias along with the forecast time. This is possibly because of the cumulative effect of wind speed bias [33], which is consistent with FLEXPART's outcomes. The motion trajectory bias of the airborne balloon is relatively small in the vertical perspective. According to the data in Table 5, FLEXPART predicts a minimum vertical trajectory bias of 2.1 m and a maximum bias of 779.9 m within an 11 h period. The RMSE and the MAE both fall within a range of 400 m. Unlike the horizontal direction, there are no increasing treads observed for the vertical trajectory biases as forecast time progresses. This observed phenomenon can be attributed to the physical properties of the balloon and its vertical movement mechanism. As the airborne balloon ascends to a certain altitude layer, it reaches an equilibrium state due to the inter-balance of its own gravity and buoyancy. Even if there is a vertical force disrupting this balance, causing the balloon to ascend or descend, it ultimately returns to the equilibrium altitude layer under the combined influence. Nevertheless, the bias in the vertical trajectory does not significantly increase due to changes in vertical wind speed, which is identical to the forecast results of FLEXPART. Wind speed plays a crucial role in determining the trajectory of airborne  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Times** & **Bias Classification** & **1 h** & **2 h** & **3 h** & **4 h** & **5 h** & **6 h** & **7 h** & **8 h** & **9 h** & **10 h** & **11 h** & **RMSE** & **MAE** \\ \hline Forecast-ing tests & Horizontal/km & 8.3 & 17.7 & 24.8 & 27.8 & 30.8 & 45.2 & 57.3 & 62.2 & 64.9 & 62.1 & 69.1 & 45.4 & 39.2 \\ Vertical/m & 293.8 & 313.2 & 447.6 & 688.7 & 417.4 & 141.2 & 419.0 & 779.9 & 218.1 & 157.9 & 2.1 & 398.9 & 323.2 \\ \hline \hline \balloons; however, its influence differs between the horizontal and vertical directions [34]. Movement in the horizontal direction primarily depends on wind speed; hence, inaccuracies in wind speed prediction directly led to increased biases in horizontal trajectory movement. Conversely, movement in the vertical direction is primarily influenced by the physical properties of the balloon itself and remains unaffected in the changes in wind speed. Furthermore, given that the vertical component of wind speed is relatively small in the atmosphere, fluctuations in vertical wind speed have limited influence on the balloon's trajectory once it reaches an equilibrium altitude layer. Therefore, when using FLEXPART for forecasting airborne balloon trajectories, particular attention should be paid to accurately simulating and predicting horizontal wind speeds to enhance forecast accuracy. Wind speed is important in determining the trajectory of balloons, both horizontally and vertically. Figure 4 illustrates the horizontal and vertical bias curves for predicting the forward trajectory of the airborne balloon. Notably, it is evident that the horizontal trajectory bias is greater than the vertical trajectory bias during the balloon's movement, exhibiting a fluctuating upward trend. After 11 h, the bias reaches its maximum value, increasing by 60.8 km, with a corresponding RMSE of 45.4 km.