## 2 Methodology
### Global Spectral Restoration and Feature Enhancement  
When the phenological changes in remote sensing scenes are quite evident, the spectral prior information of the fused image can only come from the coarse image \(C_{t_{1}}\) at the reference time. Most existing STF models neglect the preservation of spectral information from \(C_{t_{1}}\) during the fusion process. To improve the spectral consistency and detail restoration capability of the spatiotemporal fusion results, the GSRFE module was designed.  
The purpose of GSRFE is to use the spectral information from the coarse imagery at the predicted time to correct and enhance the obtained spatiotemporal fusion features. It combines Adaptive Instance Normalization (AdaIN) and spatial attention mechanisms to enhance and repair the input features. By adaptively adjusting the spectral distribution of the features, it improves the quality of spatiotemporal feature fusion. The design process of the GSRFE module is shown in Figure 4, which mainly includes the following steps:  
(1) Spectral information extraction and mapping: First, the coarse image feature \(L_{1}^{(i)}\) at the reference time is globally average-pooled to obtain a global feature vector. After passing through a series of fully connected layers, the scaling and shifting parameters \(\lambda^{(i)}\) and \(\beta^{(i)}\) are learned, which are used to adjust the spectral distribution of the fused feature map.  
(2) Spectral adaptive adjustment: First, the spatiotemporal fused feature \(F_{st}^{(i)}\) undergoes channel normalization to obtain the normalized feature \(F_{st}^{(i)^{\prime}}\). The learned parameters \(\lambda^{(i)}\) and \(\beta^{(i)}\) are then used to adjust the normalized feature.  
\[F_{st}^{(i)^{\prime\prime}}=\lambda^{(i)}\times F_{st}^{(i)^{\prime}}+\beta^{ (i)} \tag{16}\]  
(3) Spatial enhancement: To further correct the spectral inconsistencies caused by sensor system errors while retaining the detailed features from the spatiotemporal fusion information \(F_{st}^{(i)}\), the corrected feature \(F_{st}^{(i)^{\prime\prime}}\) is input into a spatial attention module. This module first applies average pooling and max pooling operations on the input features to obtain global information in the spatial dimension. These global features are then processed further through convolution operations, ultimately producing a spatial weight matrix \(W_{s}^{(i)}\). The process can be expressed as  
\[SA(x)=x\times\sigma\Big{(}\text{Conv}^{1}\Big{(}\text{LReLU}\Big{(}\text{Conv}^{ 3}(\text{cat}(\text{MP}(x),\text{AP}(x)))\Big{)}\Big{)}\Big{)} \tag{17}\]  
where \(\sigma(\cdot)\) denotes the Sigmoid activation function and \(\text{Conv}^{1}(\cdot)\) and \(\text{Conv}^{3}(\cdot)\) represent the operations of \(1\times 1\) convolution and \(3\times 3\) convolution, respectively. \(\text{MP}(\cdot)\) and \(\text{AP}(\cdot)\) refer to max pooling and average pooling, respectively.  
Finally, the features \(F_{st}^{(i)}\) and \(F_{st}^{(i)^{\prime\prime}}\) enhanced by the attention module are added together to further enhance the spatial information. These enhanced features are refined through two \(1\times 1\) convolutions and activation functions, ultimately producing the globally spectral-restored and feature-enhanced feature \(F_{st_{\text{corr}}}^{(i)}\).  
\[F_{st_{\text{corr}}}^{(i)}=\text{Conv}^{1}\Big{(}\text{LReLU}\Big{(}\text{ Conv}^{1}\Big{(}SA\Big{(}F_{st}^{(i)}\Big{)}+SA\Big{(}F_{st}^{(i)^{\prime\prime}} \Big{)}\Big{)}\Big{)}\Big{)} \tag{18}\]