# Retrieval of Cloud Ice Water Path from FY-3F MWTS and MWHS
## 3 Description of Cloud Ice Water Algorithm  
Algorithms for retrieving IWP from satellite observation data have been extensively developed over the years [7; 8; 13], as illustrated in Equation (4):  
\[IWP=\mu D_{e}\rho_{i}(\Omega/\Omega_{N}) \tag{4}\]  
where \(\mu\) is the cosine of the zenith angle, \(D_{e}\) is the ice particle effective diameters, \(\rho_{i}\) is the ice particle bulk volume density, \(\Omega\) is the ice cloud scattering parameter, and \(\Omega_{N}\) is the normalized scattering parameter. [PERSON] and [PERSON] derived the relationship between the brightness temperature emanating from ice clouds \(T_{B}(z_{t},\mu)\) and that at the cloud base \(T_{B}(z_{b},\mu)\) using the two-stream approximation at microwave frequencies [7], as expressed in Equation (5):  
\[T_{B}(z_{t},\mu)=\frac{T_{B}(z_{b},\mu)}{1+\Omega(\mu)} \tag{5}\]  
where \(z_{t}\) and \(z_{b}\) are the heights of the top and base layers of clouds. Figure 2: Scatter plots of observed brightness temperatures from corresponding channels of MWTS-III and AMSU-A before and after applying the correction during 26–27 June 2024: panels (**a**,**c**) show scatter plots for the 23.8 GHz channel of MWTS-III before and after correction, while panels (**b**,**d**) present scatter plots for the 31.4 GHz channel of MWTS-III before and after correction. As shown in Equation (4), the IWP is proportional to \(\Omega\), while the relationship between \(\Omega_{N}\) and \(D_{e}\) is nonlinear and may depend on the specific ice particle size distribution and bulk density. Consequently, measurements at two different frequencies are required to determine both IWP and \(D_{e}\) for a given particle bulk density [7; 27]. In this study, a constant bulk density for ice particles of 920.0 kg/m\({}^{3}\) is assumed, making IWP uniquely related to \(\Omega\) and \(D_{e}\)[8]. The ratio of scattering parameters at two different frequencies, denoted as \(r\), is determined using Equation (4):  
\[r(D_{e})=\frac{\Omega_{89}}{\Omega_{166}}=\frac{\Omega_{N89}(x,m)}{\Omega_{N16 6}(x,m)} \tag{6}\]  
where \(x\) is the size parameter (\(x=\pi D/\lambda\)) and \(m\) is the complex refractive index. In 1992, [PERSON] established an empirical relationship between \(r\) and \(D_{e}\) using simulation data from a radiative transfer model and experimentally derived the relationship between \(\Omega_{N}\) and \(D_{e}\)[28]. This relationship is expressed in the Equations (7) and (8):  
\[D_{e}=a_{0}+a_{1}r+a_{2}r^{2}+a_{3}r^{3} \tag{7}\]  
\[\Omega_{N}=\exp\Bigl{\{}b_{0}+b_{1}\ln(D_{e})+b_{2}[\ln(D_{e})]^{2}\Bigr{\}} \tag{8}\]  
Among these, \(a_{i}\) (\(i=0,1,2,3\)) and \(b_{i}\) (\(i=0,1,2\)) are parameters dependent on the bulk density and size distribution of ice particles. In this case \(a_{0}=-0.24843\), \(a_{1}=3.86726\), \(a_{2}=-4.70782\), and \(a_{3}=4.67150\). Table 2 lists the parameters used in this study for calculating \(\Omega_{N}\). In summary, once \(T_{B}(z_{b},\mu)\) is determined, the IWP can be retrieved [8]. Over the ocean, \(T_{B}(z_{b},\mu)\) can be derived using the radiative transfer equation. For a non-scattering atmosphere, this equation is expressed as [29]:  
\[T_{b}=T_{s}[1-(1-\varepsilon)\gamma^{2}]-\Delta T(1-\gamma)[1+(1-\varepsilon) \gamma] \tag{9}\]  
Here, \(\gamma\) is atmospheric transmittance, \(\varepsilon\) is the surface emissivity, \(T_{s}\) is the surface temperature, and \(\Delta T\) is the temperature difference between the surface and the atmospheric mean temperature.