## 3 Methodology
### Singular Value Decomposition Spectrum Analysis (SSA)  
In this study, due to the varying positions of missing values in the images on different dates, traditional interpolation methods may not be effectively applied. Therefore, we adopted Singular Spectrum Analysis (SSA) to address the missing values caused by orbital coverage in the FY-3C MWRI PMW data. SSA is an advanced time-series analysis method widely used in remote sensing data processing and analysis. The basic principle of SSA is to decompose the time series into a series of components, including trends, periodic components, and noise, in order to reveal the intrinsic structure and dynamic characteristics of the data. This method helps to improve data quality, fill missing values, and enhance the accuracy of data analysis by extracting the main components of the time series. The processing steps of SSA can be summarized in four key stages:  
1. **Embedding:** The time series is constructed into a trajectory matrix to facilitate the analysis of its intrinsic structure. For a given time series (\(x_{i}\)), this step constructs a trajectory matrix (\(X\)) by selecting a window length (\(L\)).
2. **Singular Value Decomposition (SVD):** SVD occurs on the trajectory matrix (\(X\)), decomposing it into the product of three matrices, as follows: \[X=U\Sigma V^{T}\] (3) where \(U\) and \(V\) are orthogonal matrices, and \(\sum\) is a diagonal matrix containing the singular values.
3. **Reconstruction:** Based on the magnitude of the singular values, select certain singular values and their corresponding singular vectors to reconstruct the time series, separating the main components of the time series. The formula is as follows: \[X^{\prime}=U_{k}\Sigma_{k}V_{k}^{T}\] (4) where \(X^{\prime}\) is the matrix reconstructed based on the selected singular value \(k\), \(U_{k}\), \(\sum_{k}\), and \(V_{k}^{T}\) are matrices composed of the selected singular vectors and singular values from the original singular value decomposition. By selecting different \(k\) values, various components of the time series (such as trends or periodic components) can be extracted.
4. **Diagonal Averaging:** perform diagonal averaging on the reconstructed trajectory matrix to convert it back into the time-series format.  
When processing global microwave images, we first treat the surface temperature data in each image as a time series and apply the SSA method to decompose and reconstruct these time series. Considering that the missing values in microwave images appear in long strips and their positions vary by date, we analyze the surface temperature data of each day as a separate time series. For each time series, we select an appropriate window length T to construct the trajectory matrix X. After applying Singular Value Decomposition (SVD) to the trajectory matrix, we classify the components into signal and noise parts based on the magnitude of the singular values, with the signal part typically corresponding to the larger singular values. We then reconstruct the signal components and use diagonal averaging to restore the time series. The reconstructed time series is used to fill in the missing values in the original data, as shown in Figures 3 and 4.