## 3 Airborne Balloon Sensitivity Tests
### Airborne balloon Density Sensitivity Test  
It can be hypothesized that the airborne balloons maintain equilibrium at any altitude layer of the atmosphere, ensuring a balance between their buoyancy and gravity [32]. According to the formula \(m=\rho V\) (where \(m\) is the mass of the airborne balloon, \(V\) is the volume of the airborne balloon, and \(\rho\) is the atmospheric density of the height layer where the airborne balloon is located), if there are no appendages mounted on the airborne balloon, then its gas density should be equal to that of the surrounding atmosphere at its current altitude. To investigate how changes in airborne balloon density affect trajectory simulations conducted by FLEXPART, we chose airborne balloon density as a sensitivity factor for testing purposes. The latitude, longitude, altitude, and atmospheric density associated with each specific balloon are inputted into the FLEXPART input file. (1) Setup  
The airborne balloon process selected for the test is identical to that used in the airborne balloon radius (volume) sensitivity test. Considering the potential occurrence of the airborne balloons at various altitudes, a wider range of density values was established, leading to the design of five groups of tests. The specific configurations of the tests are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \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 & 4.36 & 10.08 & 15.05 & 18.37 & 11.70 & 9.57 \\  & Vertical/m & 12.96 & 20.26 & 16.61 & 679.62 & 304.22 & 145.89 \\  & Horizontal/km & 4.52 & 10.77 & 16.44 & 20.77 & 12.95 & 10.50 \\  & Vertical/m & 24.45 & 95.79 & 131.69 & 526.84 & 246.85 & 155.75 \\  & Horizontal/km & 4.36 & 10.08 & 15.05 & 18.37 & 11.70 & 9.57 \\  & Vertical/m & 12.97 & 20.26 & 16.61 & 679.62 & 304.22 & 145.89 \\  & Horizontal/km & 4.36 & 10.24 & 15.35 & 18.97 & 12.00 & 9.78 \\  & Vertical/m & 4.42 & 37.53 & 42.93 & 644.73 & 289.46 & 145.92 \\  & Horizontal/km & 4.52 & 10.77 & 16.44 & 20.77 & 12.95 & 10.50 \\  & Vertical/m & 24.45 & 95.79 & 131.70 & 526.83 & 246.85 & 155.75 \\ \hline \end{tabular}
\end{table} 
Table 2: Airborne balloon radius (volume) sensitivity test results. \begin{table}
\begin{tabular}{c c c} \hline
**Test Name** & **Sensitizer** & **Input Parameter** \\ \hline \multirow{3}{*}{Test 1} & Treating the airborne balloon \\  & as a mass, disregarding its \\  & density and radius \\  & Set the density of airborne \\  & balloon to 0.3 kg/m\({}^{3}\) \\  & Set the density of the airborne \\  & balloon to the atmospheric \\  & density of 0.49 kg/m\({}^{3}\) at that \\  & altitude layer \\  & Set the density of airborne \\  & balloon to 1.0 kg/m\({}^{3}\) \\  & Set the density of airborne \\  & Set the density of airborne \\  & balloon to 1.29 kg/m\({}^{3}\) of \\  & set the density of airborne \\  & balloon to 1. 2. Results  
Table 4 gives the results of the airborne balloon density sensitivity tests. Regarding simulation biases in the horizontal direction, there are variations in both MAE and RMSE values for different density conditions of the balloon trajectories. In general, Test 1 exhibits the smallest bias of the 4 h horizontal trajectory at a distance of 9.43 km, whereas Test 5 shows the largest one at 12.89 km, with a difference of 4.71 km between them. Furthermore, as the density of the airborne balloons increases, there is an observed synergistic increase in biases within their simulated horizontal trajectories by FLEXPART. For the vertical direction, the results also evidence a significant effect of density on the trajectory simulation. The smallest bias of the 4 h vertical trajectory of the airborne balloon simulated by the five groups of tests is 147.96 m (Test 1) and the largest is 195.88 m (Test 5). The smallest RMSE is 218.72 m (Test 3) and the largest is 317.04 m (Test 1).