# Risk Assessment of Landfalling Tropical Cyclones in China Based on Hazard Risk Theory
## 2 Materials and Methods
### Methodology
#### 2.3.3 ArcGIS Spatial Analysis  
As tropical cyclones have distinctive spatial characteristics, employing Geographic Information Systems (ArcGIS) for the management and analysis of spatial data is highly effective. This method utilizes spatial interpolation techniques such as Inverse Distance Weighting (IDW) and ANUSPLIN to expand point data from meteorological stations into areal representations. Primarily through raster overlay, multiple raster data layers are combined using specific mathematical operations to create a new raster data system. Additionally, overlaying visual information and vector layers enhances the precision of risk assessments and facilitates data visualization.  
1. Inverse Distance Weighting  
The Inverse Distance Weighting (IDW) method operates on the premise that the influence of each data point diminishes with increasing distance from the interpolation point. This approach is particularly effective in capturing local variations when the data points are densely and uniformly distributed [28]. IDW is favored for its straightforwardness and operational ease, accounting for both distance and direction in its calculations. Nonetheless, its performance can be significantly affected by the characteristics of the dataset itself.  
The mathematical formulation for the Inverse Distance Weighted (IDW) method is presented in Equation (1) [29]:  
\[\hat{Z}_{0}=\sum_{i=0}^{1}(Z_{2}W_{i}) \tag{1}\]  
where \(\hat{Z}_{0}\) is the estimated value at the point (\(x_{0}\), \(y_{0}\)), \(W_{i}\) signifies the ratio of weights between the interpolation point and the various data points, and \(n\) represents the total number of interpolation points.  
The computation of the weight coefficient \(W_{i}\) is crucial to the IDW methodology, typically defined by Equation (2).  
\[W_{i}=\frac{f\big{(}d_{ej}\big{)}}{\sum_{j=1}^{n}f\big{(}d_{ej}\big{)}} \tag{2}\]  
where \(n\) denotes the quantity of known data points. The function \(f\big{(}d_{ej}\big{)}\) articulates the established distance \(d_{ej}\) between these data points and the interpolation points. It employs a widely adopted formula for weight calculation, referred to as Equation (3).  
\[f\big{(}d_{ej}\big{)}=\frac{1}{d_{ej}^{b}} \tag{3}\]  
where \(b\) acts as a fixed constant; when this constant is set to 1 or 2, the method is converted into inverse distance weighting interpolation or inverse distance squared weighting interpolation, respectively.  
2. ANUSPLIN Spatial Interpolation  
ANUSPLIN employs the theory of planar smoothing splines and integrates various influencing factors as covariates to facilitate the analysis and interpolation of multivariate data [28]. This model has been extensively validated for its ability to maintain a balance between the accuracy and smoothness of the interpolation surface. The formula for the theoretical model is specified in Equation (4) [30].  
\[G_{i}=f(x_{i})+c^{T}y_{i}+h_{i} \tag{4}\]  
where \(G_{i}\) denotes the dependent variable for spatial data points, \(y_{i}\) serves as the independent covariate, \(c\) refers to the coefficient across \(n\) dimensions, and \(h_{i}\) is the random error associated with the independent variables, which is expected to equal zero.  
ANUSPLIN spatial interpolation characterizes the variability of variables across a region. This method does not inherently convey spatial resolution; rather, it primarily entails the substitution of numerical data and influencing factors into a polynomial equation for resolution.