# Research on Snow Load Characteristics on a Complex Long-Span Root Based on Snow-Wind Tunnel Tests
## 4 Analysis of Snow Distribution Characteristics
### Spatial Matrix Empirical Orthogonal Function (EOF)  
According to the EOF theory, each spatial mode \(EOF_{k}\) corresponds to an eigenvalue \(\lambda_{k}\). The contribution of \(EOF_{k}\) to the total variance is determined by its eigenvalue \(\lambda_{k}\); therefore, the significance of \(EOF_{k}\) can be defined by its eigenvalue. The larger the eigenvalue is, the more significant its influence is. Furthermore, if the eigenvalue error ranges (\(\lambda_{k}-\Delta\lambda\), \(\lambda_{k}+\Delta\lambda\)) between two modes overlap, the mode characteristics will be similar. Here, \(\Delta\lambda_{k}\) is determined according to Equation (15), where \(N^{*}\) represents the effective degrees of freedom of data. \[\Delta\lambda_{k}=\lambda_{k}\sqrt{\frac{2}{N^{*}}} \tag{15}\]  
Figure 11 illustrates the eigenvalues of the first few spatial modes. The first three spatial modes (\(EOF_{1}\), \(EOF_{2}\), and \(EOF_{3}\)) contribute the most to the total variance; therefore, only the characteristics of these three spatial modes are analyzed in the following. Compared with \(\lambda_{2}\) and \(\lambda_{3}\), the \(\lambda_{1}\) value for \(EOF_{1}\) leaps significantly. Evidently, \(EOF_{1}\) has an overwhelming influence on the snow distribution on the stadium roof. The eigenvalues for \(EOF_{2}\) and \(EOF_{3}\) are similar, but the significance of \(EOF_{2}\) is slightly more pronounced. Furthermore, a relatively large overlap exists in the error ranges between \(EOF_{2}\) and \(EOF_{3}\), indicating that their distribution patterns are similar. The distribution characteristic of \(EOF_{1}\) is illustrated in Figure 12. The values of \(EOF_{1}\) are all positive. The distribution pattern is indicated by the strip of local accumulations formed in the concave regions along the radial cables, similar to the unbalanced snowpack formed in the eave region between multi-span gable roofs (Figure 10b). In general, \(EOF_{1}\) further reflects the influence of the local roof shape on the snow load distribution on the building roof. The distribution characteristics of \(EOF_{2}\) and \(EOF_{3}\) are illustrated in Figure 13. The spatial characteristics of \(EOF_{2}\) and \(EOF_{3}\) are similar, as discussed previously; that is, the overall surface accumulations along the symmetry axes (\(0^{\circ}\) or \(90^{\circ}\) inflow direction) and skew surface accumulations between the axes (\(45^{\circ}\) inflow direction). These patterns are similar to the fan-shaped depositions on the umbrella-shaped roof (Figure 10a), reflecting the influence of the overall roof shape on the snow load distribution. Moreover, the magnitude of the shape coefficients for \(EOF_{2}\) and \(EOF_{3}\) are positive or negative in two perpendicular regions, taking on an opposite distribution pattern with a different inflow direction (the black or gray arrow in Figure 13). **PC** coefficients for \(EOF_{1}\), \(EOF_{2}\), and \(EOF_{3}\) are illustrated in Figure 14. Under the five inflow direction conditions, which range from 0\({}^{\circ}\) to 90\({}^{\circ}\), the proportion of the **PC** coefficients for \(EOF_{1}\) is the largest, indicating an overwhelming influence, as described earlier. The value fluctuates slightly depending on the wind direction and basically remains at 22.5. For \(EOF_{2}\) and \(EOF_{3}\), the **PC** coefficients vary sharply with the wind direction, that is, the **PC** coefficients are sensitive to the wind direction. Although the coefficients fluctuate dramatically, the curves of **PC** coefficients are symmetrical along the \(PC=0\) axis with varying inflow directions. Specifically, when the inflow originates from the 0\({}^{\circ}\) direction, the inflow direction is consistent with that of the negative state of \(EOF_{2}\) (gray inflow direction in Figure 13a). The overall snow distribution reflects the characteristics of \(EOF_{2}\), while the **PC** coefficient for \(EOF_{3}\) is close to zero.