# Research on Snow Load Characteristics on a Complex Long-Span Root Based on Snow-Wind Tunnel Tests
## 4 Analysis of Snow Distribution Characteristics  
### Analytical Method  
To deeply analyze the relationship between the snow distribution pattern and the shape characteristic of the roof, the empirical orthogonal function (EOF) analysis was adopted. This approach is generally used to analyze the structural features in data and extract major data feature quantities. Specifically, EOF analysis can decompose a variable field matrix **X** that changes with time into a time-independent spatial matrix **EOF** and a time matrix **PC**, as shown in Equation (9). In this study, the time-dependent field matrix was replaced by a field matrix that changed with the wind direction, and a wind direction related matrix could be obtained instead of the time matrix. The spatial matrix **EOF** generalizes the geographical distribution characteristics of the field; whereas, the wind direction matrix **PC** is composed of the linear combination of coefficients of the spatial points of the field. As the main information of the original field **X** is concentrated in the first several components of the spatial matrix **EOF**, the study of the original field **X** changing with wind direction can be converted into the investigation of the first components of the spatial matrix **EOF** and their wind direction matrix **PC**, respectively.  
\[\textbf{X}=\textbf{EOF}\times\textbf{PC} \tag{9}\]  
The specific method is illustrated in Equation (10)-(14). Firstly, a field matrix \(\textbf{X}_{m\alpha n}\) is obtained by collating the measured data, where the subscript \"\(m\)\" represents the number of spatial points under a certain inflow direction condition, namely 900 and the subscript \"\(n\)\" represents the number of inflow  
Figure 10: Unbalanced distributed shape coefficient on building roofs: (**a**) shape coefficient for umbrella-shaped roof indicated in the technical specification for cable structures; (**b**) shape coefficient for multi-span gable roof indicated in load code for design of building structures.  
cases, namely five. Based on the orthogonal decomposition theory, the spatial matrix **EOF** and its eigenvalue matrix \(\mathbf{\Lambda}\) can be derived, as shown in Equation (11)-(13), where **C** is the cross-product of the field matrix **X** and its transpose matrix \(\mathbf{\mathsf{X}}^{T}\). The \(k\)th column of the spatial matrix **EOF** represents the \(k\)th spatial mode \(\textbf{EOF}_{k}\). Finally, the wind direction matrix **PC** can be derived from the spatial matrix **EOF** and original field matrix **X**, as indicated in Equation (14).  
\[\textbf{X}_{m\times n}=\left[\begin{array}{cccc}x_{11}&x_{12}&\ldots&x_{13} \\ x_{21}&x_{22}&\ldots&x_{23}\\ \ldots&\ldots&\ldots&\ldots\\ x_{m1}&x_{m2}&\ldots&x_{mn}\end{array}\right] \tag{10}\]  
\[\textbf{C}_{m\times n}=\frac{1}{n}\textbf{X}\times\textbf{X}^{T} \tag{11}\]  
\[\textbf{C}_{m\times n}\times\textbf{EOF}_{m\times n}=\textbf{EOF}_{m\times n }\times\textbf{\Lambda}_{m\times n} \tag{12}\]  
\[\textbf{\Lambda}_{m\times m}=\left[\begin{array}{cccc}\lambda_{1}&0&\ldots &0\\ 0&\lambda_{2}&\ldots&0\\ \ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&\lambda_{m}\end{array}\right] \tag{13}\]  
\[\textbf{PC}_{m\times n}=\textbf{EOF}_{m\times n}^{T}\times\textbf{X}_{m\times n} \tag{14}\]