## 2 Mathematical Model
### Construction of Uncertainty Model  
During the cruise phase of a hydrogen hybrid UAV, suitable wind uncertainty can reflect the basic statistical characteristics of the wind field and meet the requirements for trajectory integration applications, which is crucial for the capacity configuration of long-endurance hydrogen hybrid UAVs. In the local analysis of UAV trajectories that need to maintain intervals, capturing the spatial and temporal correlations of the wind field is key. Typically, in this scenario, assuming that the wind field is steady, homogeneous, and isotropic is reasonable. However, for mid- to long-range trajectory prediction that supports Air Traffic Control (ATC) and Decision Support Tools (DSTs), the model needs to reflect the non-stationarity, non-homogeneity, weather-dependency, and potential non-Gaussian characteristics of the wind field. This model satisfies the requirements for long-endurance hydrogen hybrid UAV capacity configuration. A convenient method for modeling wind uncertainty is to effectively utilize a large amount of historical RUC (Rapid Update Cycle) data, which are provided by the National Weather Service's numerical weather prediction system [20]. However, since RUC data are only valid at discrete network points and specific time points, appropriate interpolation schemes are required to obtain wind uncertainty samples for general locations and times while preserving the spatial and temporal correlations of wind error fields. Wind uncertainty samples for a general location and forecast time period are obtained by weighted summation of historical RUC wind errors from nearby locations. These neighboring points are selected within the spatial range of the current location. For each location, historical RUC wind error data are derived from delayed RUC forecasts during the same time period in the same season under similar conditions. The forecast time interval is within the temporal correlation time range of the specified prediction time. The weight selection in the model is inversely proportional to the square of the relative distance and the absolute value of the forecast time interval difference. Additionally, the model includes two purely random correction terms to account for other sources of uncertainty. \[\Delta W_{G,j}(r,\tau)=-\sum\limits_{k}L_{k}(\tau)\sum\limits_{i}C_{i}(r_{i})[ \Delta W_{G,j}^{RUC}(r_{i},\tau_{k})+\varepsilon_{G,j}^{m}(r_{i},\tau_{k})]+ \varepsilon_{G,j}^{r} \tag{13}\]  
where \(j=1\), \(\ldots\), \(N_{s}\) represents the sample number of wind uncertainty, \(\Delta W_{G,j}(r,\tau)\) represents the position vector denoted as \(r\), and the predicted wind uncertainty at the forecast lead time \(\tau\) is represented. \(\Delta W_{G,j}^{RUC}\) represents the wind components provided by RUC. \(\varepsilon_{G,j}^{m}(r,\tau_{k})\) represents the pure random variable where the position vector is \(r\), the sample index of wind uncertainty is \(j\), and the forecast lead time is \(\tau_{k}\). \(\varepsilon_{G,j}^{r}\) represents the pure random variable where the position vector is \(r\) and the sample index of wind uncertainty is \(j\). \(r\) and \(r_{i}\) represent the position vector and the position vector at grid point index \(i\). \(\tau\) and \(\tau_{k}\) represent the forecast lead time, where \(\tau_{k}=0\), \(1\), \(\ldots\), \(9\). \(m\) represents the mass of the aircraft. And \(C_{i}(r)\) and \(L_{k}(\tau)\) are as follows:  
\[C_{i}(r)=\frac{|r-r_{i}|^{-p}}{\sum\limits_{k=1}^{N_{R}}|r-r_{k}|^{-p}},L_{k}( \tau)=\prod\limits_{k\
eq j}^{N_{t}}\left(\frac{\tau-\tau_{k}}{\tau_{j}-\tau_ {k}}\right) \tag{14}\]  
In the equation, \(C_{i}(r)\) and \(L_{k}(\tau)\) both represent weights, where \(C_{i}(r)\) is related to position, and \(L_{k}(\tau)\) is related to the forecast lead time. Due to the variations in different geographical regions and between different predictions in each RUC forecast, the resulting model is naturally non-stationary, non-homogeneous, non-Gaussian, and weather-dependent.