## 3 Ground-Based Instrumentation and Data Processing
Here, I\({}_{0}\) is directly related to the calibration constant, C = exp(-c), where c is the external calibration coefficient making the calibration coefficient of the MFRSR a critical parameter. The resultant atmospheric transmittances are therefore calculated under cloudy conditions as the ratio of calibrated diffuse irradiance to the TOA irradiance value. However, in order to obtain simultaneously both the COD and R\({}_{\mathrm{eff}}\) of the prevailing thick clouds, we require an additional measurement which can give the total liquid water content in the cloud layer, termed as liquid water path, in the zenith direction which is the main function of the MWR. In particular, both column integrated precipitable water vapor (PWV) and cloud liquid water path (LWP) are defined as,  
\[PWV=\frac{1}{\rho_{wt}}\int\rho_{v}(z)dz \tag{3}\] \[\mathrm{and}\;\;LWP=\int w(z)dz \tag{4}\]  
where \(\rho_{wt}\) is the density of liquid water (\(\sim\)10\({}^{6}\) g\(\cdot\)m\({}^{-3}\)), \(\rho_{v}(z)\) is the vertical distribution of water vapor density (expressed in g\(\cdot\)m\({}^{-3}\)) and _w(z)_ is the vertical distribution of cloud liquid water content (expressed in g\(\cdot\)m\({}^{-3}\)). Here we express PWV in cm and LWP in g\(\cdot\)m\({}^{-2}\). Using the above relations, the cloud droplet effective radius (R\({}_{\mathrm{eff}}\)), an important property governing the cloud radiative transfer [28], is defined as the ratio of third to second moments of the size distribution of the cloud droplet number concentration. For a spatially homogeneous cloud,  
\[R_{\mathit{eff}}=\frac{\int N(r)r^{3}dr}{\int N(r)r^{2}dr} \tag{5}\]  
where N(r) is the size distribution of the cloud droplets and r (= R\({}_{\mathrm{eff}}\)) is the cloud droplet radius. In case of vertically inhomogeneous clouds,  
\[R_{\mathit{eff}}=\frac{\iint N(r,z)r^{3}drdz}{\iint N(r,z)r^{2}drdz} \tag{6}\]  
The double integrals are taken over droplet radius and over the depth of the cloud. Evaluating the definition of R\({}_{\mathrm{eff}}\) in terms of measured LWP and \(\tau_{c}\) (or COD),  
\[LWP=\frac{4\pi}{3}\rho_{wt}\iint N(r,z)r^{3}drdz \tag{7}\] \[\tau_{c}=\iint\pi^{-2}Q_{e}(r)N(r,z)drdz \tag{8}\]where Q\({}_{e}\)(r) is the extinction efficiency for a cloud droplet of radius (r). For \(\lambda\) = 415 nm and for modified gamma size distribution with width \(p\) = 7 [44], the weighted extinction efficiency can be approximated by,  
\[Q_{415 nm}\left(r\right)=2.00196\,+0.36411\,r^{-0.70043} \tag{9}\]  
which is only a slight correction to \(Q_{415 nm}\left(r\right)\approx 2\), yielding the simple and often used result  
\[R_{eff}\approx\frac{3}{2}\frac{LWP}{\rho_{wt}\tau_{c}} \tag{10}\]  
The profiling Microwave Radiometer (MWR) at CCNY contains two radio frequency (RF) subsystems sharing the same antenna and antenna pointing system in the same cabinet. The water vapor profiling subsystem utilizes sky brightness temperature observations at selected frequencies between 22 and 30 GHz (21 channels). The temperature profiling subsystem utilizes sky brightness temperature observations at selected frequencies between 51 and 59 GHz (14 channels). The profiler nominally views in zenith direction and makes use of atmospheric radiation measurements in the RF domain and provides high resolution vertical profiles of temperature, humidity, water vapor and low resolution profiles of liquid density at 58 altitude levels, starting with 50 m steps from the surface up to 500 m, then 100 m steps to 2 km, and 250 m steps from 2 to 10 km. In addition, surface relative humidity, temperature and barometric pressure measured by the radiometer are used in the determination of profiles. An internally mounted and vertically pointed Infrared Thermometer (IRT) indicates the presence of cloud and measures cloud base temperature, if clouds are present. Knowing cloud base temperature yields the vapor density at cloud base (at saturation) and when combined with the retrieved temperature profile, yields cloud base altitude.