## 3 Methods
### Virtual Image Generation
#### 3.2.2 Two-Stage CGAN Model  
In this study, a two-stage CGAN model (hereafter referred to as CGAN\({}_{\text{2S}}\)), including representation and generation stages, is presented to generate the virtual Sentinel-2 image (Figure 6). A representation stage for feature extraction is particularly designed to utilize all available information from Sentinel-1 and -2 images (Figure 7). The features extracted in the representation stage are fed into a CGAN-based generation stage to create the virtual Sentinel-2 image. Figure 6: Two-stage CGAN model applied in this study (S-2—Sentinel-2 optical image; S-1—Sentinel-1 SAR image). In the representation stage, various inputs, which are helpful in representing scattering characteristics in croplands, were first considered from the Sentinel-1 images. The dual-polarization-based radar vegetation index (RVI) was considered an additional SAR-based feature. The RVI was considered an indicator of the vegetation growth dynamics [52; 63; 64] and is defined as  
\[\text{RVI}=4\sigma_{VH}^{0}/(\sigma_{VV}^{0}+\sigma_{VH}^{0}), \tag{4}\]  
where \(\sigma_{VH}^{0}\) and \(\sigma_{VV}^{0}\) are the backscattering coefficients for VH and VV polarizations, respectively. Figure 8 shows the dual-polarization backscattering coefficients and RVI of the Sentinel-1 imagery on 29 July 2019. Three inputs, including two backscattering coefficients for VH and VV polarizations, and RVI, were utilized as SAR-based features. In addition to the above SAR-based features, multi-spectral bands in Table 1 were also used as inputs for feature extraction. The core part of Sentinel-2-based feature extraction is to extract optical features from all of the spectral information via a convolution operation. After concatenating five resampled red-edge and SWIR band images with VNIR band images, the convolution operation was applied to generate Sentinel-2-based features (see Figure 7). Consequently, the extracted features provide useful information to the discriminator of the CGAN model compared to that of each original spectral band. The Pix2 Pix architecture [34] shown in Figure 9 was adopted in this study because it can adequately reproduce spectral information [37; 38; 39; 47]. The Pix2 Pix model requires a paired set of SAR-based and optical-based features. The convolution layer in the representation stage is regarded as the input layer of the discriminator in the Pix2 Pix model. Figure 8: Sentinel-1 SAR dual-polarization images and radar vegetation index on 29 July 2019. Figure 7: Feature extraction procedures from Sentinel-1 and -2 images in the representation stage. A U-Net architecture with a specific encoder-decoder scheme [65] was utilized as the generator in this study (Figure 9). Virtual images are predicted from the generator through down-sampling for feature extraction and up-sampling for the reconstruction from the extracted features. Skip-connection was applied to mitigate the loss of detailed information during feature extraction. As shown in Figure 9, the down-sampling and up-sampling include several processing layers, such as convolution, batch normalization, and activation functions. Thus, many hyperparameters must be optimized. In this study, the optimal hyperparameters were determined through a trial-and-error approach. The structure and optimal hyperparameters are listed in Table 3. For feature enhancement and dimension reduction, we extracted the features by reducing their dimensions with the predefined kernel size of the patch and strides (4 and 2, respectively). Moreover, dropout was applied during the up-sampling instead of the input noise in Equation (1) [34]. The discriminator is a CNN-based model that discriminates the validity of the generated images from the generator. A PatchGAN, which tries to determine whether the generated images are real or fake in a predefined patch unit [34], not in an entire image, was adopted as the discriminator in this study.