## 3 Observation Sufficiency to Distinguish the Convective Transition Relationship
### Separation Precipitation Conditions Using Polarimetric Differential Phase  
At the same time that the COSMIC-IMERG dataset was being analyzed, the simulated value of \(\Delta\phi\) was computed. This would be the measurable value provided by each PRO constellation satellite to identify the presence of heavy precipitation within the moisture profile. The calculation of the simulated \(\Delta\phi\) follows the same procedure described in [21]. At the main GNSS frequency near 1.4 GHz, the specific propagation differential phase shift \(k_{DP}\) (units of deg km\({}^{-1}\)) scales nearly linearly with rain rate \(R\)[21] (Equation (1)). Referring \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Region**} & \multirow{2}{*}{**Temp (K)**} & \multicolumn{4}{c}{**Min Number of Temperatures Discriminated**} \\ \cline{3-5}  & & 263 & & \\  & & 263.5 & & \\  & & 264 & & \\  & & 264.5 & & \\  & & 265 & & \\  & & 266 & & \\  & & 266.5 & & \\  & & 267 & & \\ \hline \multirow{5}{*}{\begin{tabular}{c} West \\ Pacific Ocean \\ (1 K) \\ \end{tabular} } & 262 & & & \\  & 263 & & & \\  & 264 & & & \\  & & 265 & & \\  & & 266 & & \\  & & 267 & & \\ \hline \multirow{5}{*}{\begin{tabular}{c} East \\ Pacific Ocean (1 K) \\ \end{tabular} } & 261 & & & \\  & 262 & & & \\  & 263 & & & \\  & 264 & & & \\  & 265 & & & \\  & 267 & & & \\ \hline \multirow{5}{*}{\
\begin{tabular}{c} Indian Ocean \\ (1 K) \\ \end{tabular} } & 262 & & & \\  & 263 & & & \\  & 264 & & & \\  & 265 & & & \\  & 266 & & & \\  & 267 & & & \\ \hline \multirow{5}{*}{
\begin{tabular}{c} Atlantic Ocean (1 K) \\ \end{tabular} } & 262 & & & \\  & 263 & & & \\ \cline{1-1}  & 264 & & & \\ \cline{1-1}  & 265 & & & \\ \cline{1-1}  & 266 & & & \\ \cline{1-1} \cline{2-1}  & **N/A** & **N = 5000** & **N = 10,000** & **N = 20,000** & **N = 30,000** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Minimum number of joint moisture-precipitation observations (N) needed to separate the transition curves by 0.5 K (global tropics) and 1 K (West Pacific, East Pacific, Indian and Atlantic oceans). The color shading corresponds to N rounded up to the values in the legend (bottom row). The number of rows in the temperature range shading is one less than the number of temperatures listed, since each row represents the number of observations needed to distinguish the temperature indicated from the temperature above. In some regions, insufficient data was available for lowest and highest temperature ranges (e.g., no data below 262 K and above 266 K in the Atlantic Ocean). The gray shading indicates that no analysis was done for this number of temperatures. back to Figure 1, \(k_{DP}\) was computed for each IMERG grid box intercepted by the lowest 200-km ray path, and \(\Delta\phi\) represents its (200-km) path integrated value,  
\[\Delta\phi = \sum_{i\,=\,1}^{N}k_{DP,i}d_{i} \tag{1}\]  
\[R = \frac{1}{N}\sum_{i\,=\,1}^{N}r_{i} \tag{2}\]  
where the subscript \(i\) refers to each 0.1-degree IMERG grid box, and N is the number of IMERG grid boxes intercepted by the 200-km path (depicted in Figure 1). While \(\Delta\varphi\) cannot provide a unique estimate of the ray path-averaged precipitation (the same \(\Delta\varphi\) could occur from different precipitation conditions along the ray path), it can be related to the probability of a precipitation rate being exceeded [14]. Obviously, this is a simplification (short straight-line ray paths, no vertical variation) of the actual ray paths. The presence of heavy precipitation will be determined from each PRO whenever the measured \(\Delta\phi\) exceeds a threshold value. From the representative dataset, Table 2 shows the fraction of the observations that exceed two different \(\Delta\phi\) thresholds, for the full global tropics and four sub-basins within these same latitudes. For this study, a threshold of \(\Delta\phi\) > 2 mm was used as a threshold indicating heavy precipitation. On average, this \(\Delta\phi\) threshold corresponds to a path-average rain rate of 2 mm h\({}^{-1}\)[38].