## 3 Airborne Balloon Sensitivity Tests
### Airborne balloon Density Sensitivity Test
As the density of the airborne balloons increases, the MAE of the vertical trajectory simulated by FLEXPART increases likewise. However, the RMSE in the vertical direction is minimized when the density of the balloon matches that of the atmosphere at the specific altitude layer where it is positioned. In summary, the motion trajectory of airborne balloons simulated by FLEXPART is highly sensitive to the input density. With the increase in the density of the airborne balloon, the trajectory bias increases accordingly, especially in the horizontal manifestation, where there is relatively small, simulated bias of 16.21 km in the RMSE of the maximum. In the vertical direction, although certain conditions may lead to an increase in MAE conditions (e.g., when the density equals atmospheric density), the RMSE reaches a minimum value of 218.72 m. When the airborne balloon is simplified to a point mass, the FLEXPART simulation yields minimal horizontal trajectory bias and the smallest MAE in the vertical direction. Although the RMSE may be larger, it still falls within 100 m from the minimum bias difference. Therefore, in practical studies, treating the airborne balloon as a point mass without the consideration of other characteristic factors (such as density and radius) results in more accurate motion trajectories, especially in the horizontal direction. Even with increased bias vertically, it remains within an acceptable range of 100 m.  
In this section, a series of sensitivity tests examining the density and radius of airborne balloons were conducted. The following conclusions can be drawn: (i) The radius of the balloon has a relatively negligible effect on the trajectory. Since the movement of the balloon is primarily influenced by wind, the resistance caused by changes in the balloon's volume is minor compared to the force exerted by wind speed. Thus, their impact on the trajectory is relatively insignificant. (ii) Balloon density significantly affects the flight trajectory. Combining Tables 2 and 3, the sensitivity simulations reveal that the vertical bias produced by the change in the balloon radius ranges from 9.43 to 10.50 km, and the density bias ranges from 9.43 to 12.89 km. Similarly, horizontal bias ranges from 147.96 to 155.75 m for radius, and from 147.96 to 195.88 m for density. The fluctuation in the density bias is larger than the radius bias in all directions, with simulation results indicating that, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test Name** & **Bias Classification** & **1 h** & **2 h** & **3 h** & **4 h** & **RMSE** & **MAE** \\ \hline \multirow{2}{*}{Test 1} & Horizontal/km & 4.36 & 10.00 & 14.81 & 17.98 & 11.50 & 9.43 \\  & Vertical/m & 20.06 & 5.93 & 5.23 & 708.59 & 317.04 & 147.96 \\  & Horizontal/km & 4.42 & 10.62 & 16.04 & 20.24 & 12.64 & 10.26 \\  & Vertical/m & 16.48 & 79.76 & 107.30 & 559.23 & 257.25 & 152.55 \\  & Horizontal/km & 4.58 & 11.24 & 17.52 & 22.59 & 13.89 & 11.19 \\  & Vertical/m & 55.38 & 158.30 & 227.21 & 399.29 & 218.72 & 168.04 \\  & Horizontal/km & 4.82 & 12.01 & 19.21 & 25.48 & 15.40 & 12.31 \\  & Vertical/m & 101.47 & 251.38 & 369.37 & 209.31 & 225.27 & 186.31 \\  & Horizontal/km & 4.88 & 12.48 & 20.10 & 27.01 & 16.21 & 12.89 \\  & Vertical/m & 125.69 & 300.26 & 444.00 & 109.45 & 251.03 & 195.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Density sensitivity test results for airborne balloon. except for the plasma simulation (which does not take into account the characteristics of the balloon itself), the density bias surpasses the radius bias, i.e., the change in the density of the balloon has a more pronounced effect on the trajectory. This may be attributed to the fact that the density directly affects the buoyancy of the balloon, thereby altering the vertical trajectory of the balloon. As the density increases, the simulation bias also increases. (iii) The simulation bias of the horizontal motion trajectory is minimized when simplified into a mass point.