## 4 Warm Core Structure Retrieval Experiment  
### Global Scene-Dependent Atmospheric Retrieval Testbed (GSDART)  
In this paper, the retrieval experiment of the warm core structure is based on GSDART, which was developed by [PERSON] and [PERSON] based on MiRS [21]. The central component of GSDART is 1 DVAR. Figure 8 illustrates the main modules of GSDART and the retrieval process for the warm core structure. Figure 7: Striping index of individual channels of the MWHS (**a**) and MWTS (**b**) on FY-3F (black) and FY-3E (grey). The 1 DVAR retrieval algorithm leverages microwave observation data in conjunction with other auxiliary data to simultaneously retrieve multiple physical parameters. Its strength lies in emphasizing the physical condition constraints on the retrieved parameters and ensuring consistency in the physical relationships among the retrieved physical parameters. The theoretical foundation of 1 DVAR is rooted in Bayesian principles, which assume that both the observation errors and background errors follow Gaussian distributions. With ARMS serving as the observation operator, and given the atmospheric background field, background field covariance matrix, observation operator error matrix, observation information, etc., the following cost function is minimized:  
\[\mathrm{J}(\mathrm{X})=[-\frac{1}{2}(\mathrm{X}-\mathrm{X}_{0})^{\mathrm{T}} \mathrm{B}^{-1}(\mathrm{X}-\mathrm{X}_{0})]+[-\frac{1}{2}(\mathrm{Y}_{\mathrm{ m}}-\mathrm{Y}(\mathrm{X}))^{\mathrm{T}}\mathrm{E}^{-1}(\mathrm{Y}_{\mathrm{m}}- \mathrm{Y}(\mathrm{X}))] \tag{8}\]  
where \(\mathrm{X}_{0}\) and \(\mathrm{B}\) represent the mean value and the covariance matrix of the atmospheric state variables, respectively. \(\mathrm{Y}_{\mathrm{m}}\) denotes the observed data, while \(\mathrm{Y}(\mathrm{X})\) stands for the simulated observations calculated from the atmospheric state \(\mathrm{X}\) through the observation operator (ARMS). E represents the error matrix of the observation data and the observation operator. Minimizing the cost function \(\mathrm{J}(\mathrm{X})\) is equivalent to solving the Equation (9):  
\[\frac{\partial\mathrm{J}(\mathrm{X})}{\partial\mathrm{X}}=0 \tag{9}\]  
To solve the equation above, we need to assume that the observation operator is locally linear. That is, \(\mathrm{Y}(\mathrm{X})=\mathrm{Y}(\mathrm{X}_{0})+\mathrm{K}(\mathrm{X}- \mathrm{X}_{0})=\mathrm{Y}(\mathrm{X}_{0})+\mathrm{K}\Delta\mathrm{X}\). Given this assumption, Equation (9) can be transformed into the following:  
\[\Delta\mathrm{X}_{\mathrm{n}+1}=[\mathrm{B}\mathrm{K}_{\mathrm{n}}^{\mathrm{T} }(\mathrm{K}_{\mathrm{n}}\mathrm{B}\mathrm{K}_{\mathrm{n}}^{\mathrm{T}}+ \mathrm{E})^{-1}][\mathrm{Y}_{\mathrm{m}}-\mathrm{Y}(\mathrm{X}_{\mathrm{n}}) +\mathrm{K}_{\mathrm{n}}\Delta\mathrm{X}_{\mathrm{n}}] \tag{10}\]  
where n represents the iteration number. \(\mathrm{K}\) represents the Jacobian matrix of the observation operator \(\mathrm{Y}\) with respect to the atmospheric state variables \(\mathrm{X}\). The iterative process of 1 DVAR begins by utilizing a given background field \(\mathrm{X}_{\mathrm{b}}\) and an initial guess field \(\mathrm{X}_{0}\) (which we define as \(\mathrm{X}_{\mathrm{b}}=\mathrm{X}_{0}\), so the initial \(\Delta\mathrm{X}_{0}=0\)). It then simulates the initial background brightness temperature \(\mathrm{Y}(\mathrm{X}_{0})\) through the observation operator. If the error at this point already satisfies the convergence criteria, no further iteration is performed. Otherwise, the iterative loop is entered. Within this process, the convergence condition is determined using a \(\mathrm{X}^{2}\) test:  
Figure 8: Global scene-dependent atmospheric retrieval testbed (GSDART) used for retrievals of FY-3F warm core structure. \[\mathrm{X}^{2}=\frac{1}{\mathrm{N}}[\mathrm{Y_{m}}-\mathrm{Y(X_{n})}]^{\mathrm{T}} \mathrm{E}^{-1}[\mathrm{Y_{m}}-\mathrm{Y(X_{n})}] \tag{11}\]  
where N represents the number of channels. We consider the convergence condition to be achieved when \(\mathrm{X}^{2}\leq 1.0\).