## 2 Mathematical Model
### Construction of Uncertainty Model
Additionally, uncertainties arising from initial measurements and the determination of the nominal wind field can also be appropriately incorporated into the model. \[-\Delta W_{G}(r,t+\tau|t)=\Delta W_{G}^{RUC}(r,t+\tau|t)+e_{G}^{m}(r,t+\tau)+ e_{G}^{m}(r,t+\tau|t) \tag{15}\]  
In the equation, \(-\Delta W_{G}(r,t+\tau|t)\) represents the wind uncertainty prediction at the position vector \(r\) and forecast lead time \(t+\tau\), based on the value at time \(t\); \(\Delta W_{G}^{RUC}(r,t+\tau|t)\) represents the time-lagged RUC data error at position vector \(r\) and forecast lead time \(\tau\), based on the value at time \(t\); \(e_{G}^{m}(r,t+\tau)\) and \(e_{G}^{m}(r,t+\tau|t)\) represent the initial measurement error and nominal wind error, respectively. Equation (15) shows how the initial measurement inputs of the RUC model introduce uncertainty. The basis for the measurements is the relationship between certain physical properties of the wind field and the wind speed and direction. Thus, the measurement errors can be divided into calibration errors of the equipment and errors caused by uncertainties in the underlying physical processes. While further research is needed, it can be assumed that the latter follows a pattern similar to wind errors. Therefore, a pure random term is added to the model to account for the correlated part of the measurement error, while another pure random term is added outside the weighted summation to represent the measurement calibration error. In contrast, the uncertainty introduced during the nominal wind field determination depends on the specific interpolation or approximation method used. When interpolation is performed, the error is zero at the RUC network points, which are spaced 20 km apart in the horizontal direction and 25 hPa apart in the vertical direction. Thus, the horizontal spatial scale of the error is approximately 40 km. This is significantly smaller than the spatial correlation of wind errors and is therefore considered a pure random term. Additionally, the residuals of the weighted summation model are random, as the summation already accounts for the spatial and temporal correlations. Therefore, in Equation (13), \(e_{G}^{m}\) represents the pure random corrections for measurement calibration errors, nominal wind field determination errors, and model residuals. When constructing the above wind uncertainty model, it is assumed that the measurement errors, nominal wind field determination errors, and RUC system prediction errors are independent of each other. The following additional assumptions are also made:  
The wind uncertainty components along the three vertical directions are uncorrelated. Therefore, the corresponding 3 \(\times\) 3 correlation matrix is a diagonal matrix, and the components along the three directions can be analyzed separately. It should be noted that the correlation of different wind quantities is influenced by the choice of coordinate system. This patent adopts the wind directions along the three vertical axes. Compared to the horizontal components, the vertical wind component Wh is relatively small and is often neglected. However, the actual vertical wind may not be zero, and its presence could affect the accuracy of the predicted trajectory. In meteorological models, commonly used interpolation methods include linear interpolation, bilinear interpolation, spline interpolation, and inverse distance weighting (IDW), among others. For wind fields involving wind speed and direction, IDW is relatively common, but it is not well suited for capturing complex nonlinear variations. Therefore, a combination of spline interpolation and IDW is adopted. Inverse Distance Weighting (IDW) is an interpolation method based on Tobler's First Law of Geography. It uses the distance between the point to be estimated and the measured values as a weighting factor, and infers the attributes of the unknown point based on the known values.