## 4 Simulation Results  
### Synthetic Attenuation Field  
The synthetic cloud data used in this paper were produced from an idealized simulation of a mesoscale convective system using the WRF model. More details about the simulation can be found in Ref. [12]. This cumulus type cloud has very large vertical development, the cloud base is approximately 5 km in height, with the top around 10 km. One vertical slice of the cloud field that covers 80 km in width and 0-12.5 km in height was selected as the reference field. This slice contains a very high content of water, which mainly consists of supercooled cloud water. Some areas can produce specific attenuation up to 2.35 dB/km, while there are other areas with relatively thin clouds. Very few rainwater droplets are present in this slice. A \(320\times 50\) grid is used to define the field with each unit square being 250 m by 250 m, i.e., both the horizontal and vertical resolutions are 250 m.  
In the cloud LWC field generated by WRF, each unit square has a uniform cloud water mixing ratio given in g/g, which is defined as  
\[\beta=\omega_{w}/\omega_{a}, \tag{30}\]  
where \(\omega_{w}\) and \(\omega_{a}\) are the liquid water and dry air density in g/m\({}^{3}\), respectively. Note that other particles such as ice crystals and snow are not considered as they have minimal effects on the attenuation of microwave signals around 30 GHz [8].  
Dry air density is obtained by [23]  
\[\omega_{a}=11.612p\times\frac{300}{T_{c}}, \tag{31}\]  
in which \(T_{c}\) is the cloud liquid water temperature in Kelvin and \(p\) is the dry air pressure in kPa. We use the barometric law [24] to calculate \(p\):  
\[p(z)=P_{0}\times\left(\frac{T_{0}}{T_{0}+L_{b}\times z}\right)^{\frac{g_{0} \times M_{c}}{R_{0}\times L_{b}}}, \tag{32}\]  
where \(z\) is the altitude above sea level, \(P_{0}\) is static pressure at sea level (101.325 kPa), \(g_{0}\) is gravitational acceleration (9.80655 m/s\({}^{2}\)), \(M_{c}\) is molar mass of Earth's air (0.0289644 kg/mol) and \(R_{0}\) is universal gas constant (8.3144598 J/(mol\(\cdot\)K)). For each unit square in the cloud field, temperature \(T_{c}\) is calculated according to the altitude of the center of the square, assuming that the temperature lapse rate \(L_{b}\) is \(-0.0065\) K/m from the sea level temperature \(T_{0}\) of 288 K. Note that the constant lapse rate is an approximation for the cloudy atmosphere and will introduce an inaccuracy in the temperature profile. Nevertheless, it only serves as a means to translate the WRF-generated LWC field into the reference attenuation field, and using it does not affect the validation of the retrieval method per se.  
In order to generate the reference attenuation field, we use the equations provided by the International Telecommunications Union (ITU) [25], where the specific attenuation within a cloud is obtained by  
\[R(f,T_{c})=K_{l}(f,T_{c})\omega_{w}, \tag{33}\]  
where \(K_{l}\) is the cloud liquid water specific attenuation coefficient (details can be found on Page 2 in [25]) in (dB/km)/(g/m\({}^{3}\)), \(f\) is the signal frequency in GHz. With the temperature and pressure profile ready for each unit square, \(\omega_{w}\) can be calculated using Equations (30) and (31) and the specific attenuation \(R\) can then be calculated by Equation (33) with the signal frequency \(f=30\) GHz. The reference attenuation field is shown in Figure 3a.