## 3 Observation Sufficiency to Distinguish the Convective Transition Relationship
### Separation Using IMERG Precipitation  
A bootstrapping procedure ([PERSON] et al., 2018) was used to resample this COSMIC-IMERG dataset (with replacement) to create many simulated samples with the same size as the original record. This process involves drawing random samples from the original dataset as a means of estimating confidence intervals. Using an average of the water vapor between the 200-850 hPa layer to define the free tropospheric water vapor, conditional average precipitation is determined over thousands of soundings for different temperature ranges. Figure 5 shows one such set of transition curves, representing these data analyzed over the global tropical (\(\pm\)20-degree) latitudes. The black, orange, blue, and red line colors represent the resampled data corresponding to four 1-degree temperature ranges (temperature bins centered at 264, 265, 266, and 267 K, respectively), averaged over the 200-850 hPa layer. Each thin line in each panel represents one realization of conditional-average precipitation rate (mm h\({}^{-1}\)) as a function of the integrated column water vapor (CWV, units of mm) within the 200-850 hPa layer, showing a nonlinear rise of precipitation with water vapor. This dependence shifts slightly for each temperature (but in a manner more complex than Clausius-Clapeyron, as outlined in Section 2). The number on top of each panel in Figure 5 indicates the total number (hereafter denoted by N) of observations used for each temperature range. For N = 5000, there is a hint of separation between these ranges, but one can immediately visualize that discrimination  
Figure 5: Precipitation rate as a function of 200–850 hPa integrated column water vapor (pickup curves) after (**left**) 50,000 observations, (**center**) 20,000 observations, and (**right**) 5000 observations taken from the COSMIC-IMERG dataset, subsetted into the global \(\pm\)20-degree latitude range. The black, orange, blue, and red colors refer to observations corresponding to four 1-degree temperature ranges (each centered at 264, 265, 266, and 267 K, respectively), averaged over the same 200–850 hPa layer. among neighboring ranges would have low confidence. For N = 50,000 (left panel), the four curves appear to be more clearly separated by the 1-degree ranges. To assess this more quantitatively, an area metric was defined, corresponding to the area between adjacent curves as depicted in Figure 6. A standard statistical method tests the null hypothesis that the area between curves is zero, using the bootstrap ensemble. For this exercise, the area was determined for precipitation rates between 0.3 and 0.5 mm h\({}^{-1}\) and includes the area between the four temperatures. Figure 7 shows the area between the curves as a function of N. The blue line represents the median, and the lower edge of the error-bar is the fifth percentile. The red vertical line at N = 16,000 is where the fifth percentile line crosses the zero-area line, which represents the minimum N needed to separate the four temperature ranges for this entire global tropical area at the 5% confidence level. In other words, when the black curve is above zero, there is over 95% confidence that the curves are distinct. For particular regions, within the tropical latitudes, the convective transition characteristics are not necessarily the same; different oceanic basins may require a larger (or smaller) value than N = 16,000 noted for the tropical latitudes as a whole. This is because distinguishing the curves depends on the number of samples in high water vapor, strongly precipitating part of the range (the low-precipitation samples are also required but are generally plentiful). Different choices of region can yield less (or more) frequent excursions into such conditions. As one example, Figure 8 shows the same set of transition curves for the West Pacific Ocean domain within \(\pm\)20-degree latitudes. In this region, the transition curves separate by four 1-degree temperature ranges for a smaller number of observations, between N = 2000 and 10,000.