## 4 Development of an Iterative Algorithm for Retrieval of Optically Thick Cloud Properties  
Of the various ground based cloud retrieval techniques, the most extensively used and referenced approach for measurements from the MFRSR-MWR system is the Non-linear Least Squares (NLSQ) retrieval algorithm of [PERSON] and [PERSON] [12] hereafter referred as MIN, which is based on the parameterization of cloud droplet scattering properties at 415 nm on the effective radius and LWP using Mie theory. MIN uses an adjoint formulation of the radiative transfer [45] to maintain accuracy and improve its execution speed. In this approach, traditional Langley regression of the direct-normal irradiance taken on clear stable days are used to extrapolate the MFRSR's response to the TOA [37], and this calibration is applied to the total-horizontal irradiance. [PERSON] and [PERSON] [37] have indicated that the standard deviation of the traditional Langley analysis inferred extra-terrestrial response from a single retrieval at a difficult site to be approximately 5%. The MIN method is applied to measurements of atmospheric transmittance at 415 nm band because this band is completely insensitive to absorption by ozone and by permanent atmospheric gases, and the surface albedo is low and stable for natural surfaces (except for snow-covered surfaces). Moreover, for this wavelength, both the asymmetry parameter and the single scattering albedo (close to unity) show little dependence on droplet size. In another approach, [PERSON] _et al._[13] builds a parameterized regression based inversion of a plane parallel radiative transfer (RT) model hereafter referred as PAR based on the assumption of a  
Figure 4: Inter-comparison of column Integrated precipitable water (IPW) from MP-3000A, WVP-1500 (microwave radiometers), AERONET and Radiosondes (RAOBs) for **(a)** 10 September 2011, and **(b)** 11 September 2011. Red lines with which blue line is flanked represents the interval of confidence for the MP-3000A dataset based solely on temporal fluctuations of retrieval. fixed rural background aerosol mode for SGP ARM site. The parameterization based on regression fitting from the RT simulations is given by  
\[\tau_{e}=\frac{(1+r_{e})}{(1+0.4125\ r_{e})}\frac{1}{(1-r_{g})}\bigg{[}\left(P_{ 1}\mu_{0}+P_{2}\right)\frac{1}{T}+P_{3}-\frac{0.007}{T^{2}}\bigg{]} \tag{11}\]  
where \(\mathrm{P_{1}=-0.4330\times AOD+3.6659}\),  
\(\mathrm{P_{2}=-0.7686\times AOD+2.0895}\), and  
\(\mathrm{P_{3}=-0.1986\times AOD-5.7936}\)  
Here \(\mu_{0}\) is the cosine of the solar zenith angle, T is the atmospheric transmittance, \(\mathrm{r_{g}}\) is the surface albedo at 415 nm, \(\tau_{\mathrm{c}}\) and \(\tau_{\mathrm{c}}\) respectively are the visible COD and cloud droplet effective radii (\(\mathrm{R_{eff}}\)). The linear dependence of \(\mathrm{P_{1}}\), \(\mathrm{P_{2}}\) and \(\mathrm{P_{3}}\) with AOD agrees for AOD ranging from 0 to 0.2. Also, this method points out that AOD has an effect in the retrieval of cloud properties, but was unable to prove that introducing AOD improves the agreement between the retrieved properties and the true (and unknown) values of COD and \(\mathrm{R_{eff}}\). Of course, this approach can be extended to more regional realistic aerosol models for different locations as needed. In our analysis, we closely follow the MIN algorithm but instead of a NLSQ optimization, we develop an iterative inversion approach. This iteration approach forms the basis of our retrieval algorithm, hereafter referred to as \(\mathrm{RET}\). Initially, \(\mathrm{R_{eff}}\) is fixed at 8 \(\mathrm{\SIUnitSymbolMicro m}\) and the first estimation of COD from the Look-Up-Table (LUT) is obtained. Then \(\mathrm{Q_{415 nm}(R_{eff})}\) is estimated from Equation (9) assuming again \(\mathrm{R_{eff}=8\ \SIUnitSymbolMicro m}\).