## 3 Results
### Estimating the Vertical Resolution of Temperature and Water Vapor Retrievals  
The averaging kernel matrix (AKM) of the scene shows the degree to which the IR + MW retrieval updated the first guess measurements. AKM values range from 0 to 1, where smaller values indicate that the final retrieval is primarily from the first guess, while the \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Pressure** \\ **Layers (mb)** \\ \end{tabular} } & \multicolumn{6}{c}{**Clear to Partly Cloudy**} & \multicolumn{6}{c}{**Cloudy Scenes**} \\ \cline{2-13}  & \multicolumn{3}{c}{**IR + MW**} & \multicolumn{3}{c}{**MW-Only**} & \multicolumn{3}{c}{**IR + MW**} & \multicolumn{3}{c}{**MW-Only**} \\ \cline{2-13}  & **RMSE** & \(\mu\) & \(\sigma\) & **RMSE** & \(\mu\) & \(\sigma\) & **RMSE** & \(\mu\) & \(\sigma\) & **RMSE** & \(\mu\) & \(\sigma\) \\ \hline
300–0 & 1.21 & \(-\)0.34 & 0.20 & 1.37 & \(-\)0.81 & 0.20 & 4.59 & 0.78 & 0.90 & 1.62 & \(-\)1.08 & 0.22 \\ \hline
600–300 & 0.83 & \(-\)0.001 & 0.79 & 0.86 & 0.02 & 0.84 & 3.96 & \(-\)0.90 & 3.80 & 1.33 & 0.50 & 1.16 \\ \hline
1000–600 & 1.26 & 0.10 & 1.01 & 1.53 & 0.58 & 1.04 & 4.61 & 1.52 & 3.40 & 4.13 & 3.00 & 2.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average temperature (K) RMSE, bias (\(\mu\)), and standard deviation (\(\sigma\)) for broad pressure layers. Calculations were made using Equations (2)–(5). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Pressure** \\ **Layers (mb)** \\ \end{tabular}  } & \multicolumn{6}{c}{**Clear to Partly Cloudy**} & \multicolumn{6}{c}{**Cloudy Scenes**} \\ \cline{2-13}  & \multicolumn{3}{c}{**IR + MW**} & \multicolumn{3}{c}{**MW-Only**} & \multicolumn{3}{c}{**IR + MW**} & \multicolumn{3}{c}{**MW-Only**} \\ \cline{2-13}  & **RMSE** & \(\mu\) & \(\sigma\) & **RMSE** & \(\mu\) & \(\sigma\) & **RMSE** & \(\mu\) & \(\sigma\) & **RMSE** & \(\mu\) & \(\sigma\) \\ \hline
300–0 & 49.63 & \(-\)19.41 & 7.19 & 56.96 & \(-\)36.42 & 6.84 & 57.26 & \(-\)40.87 & 6.07 & 41.50 & \(-\)24.69 & 4.98 \\ \hline
600–300 & 46.35 & \(-\)22.07 & 39.72 & 46.00 & \(-\)2.45 & 44.52 & 65.83 & \(-\)17.93 & 59.25 & 43.74 & 21.10 & 37.24 \\ \hline
1000–600 & 26.05 & \(-\)9.51 & 19.24 & 22.17 &latter indicates that the first guess is replaced by the retrieved value. The typical values fall in the middle of this range. AKMs are calculated using the approach of [7]; these were developed for AIRSv5 [23] and inherited by NUCAPS [33]. Figure 5 shows the AKM of the NUCAPS coarse pressure levels for profiles where the IR + MW retrieval succeeded. Smaller lines show the AKMs of the individual profiles, while the thick colored lines show the average, and the thick horizontal lines show the standard deviation. The AKM values of retrievals collocated with the flight dropsondes range from 0 to 0.2 for temperature (Figure 5a) and between 0.25 and 5.0 for water vapor (Figure 5b), but the latter have greater variability from scene to scene. Note that the profile count of successful retrievals (N = 112) is inconsistent with the analysis in the previous section (N = 130) because the AKMs were generated offline using the science code, which differs from the NUCAPS-EDR. Furthermore, input data were downloaded using the direct broadcast network (Section 2.4) to lower data latency. Direct broadcast downlinks seven CrIS fields of regard instead of nine at the scan edge, which may impact cloud clearing. The vertical resolution of the retrieval can be estimated from the local maxima (or peaks) in the AKMs. The peaks are indicated on the plots using horizontal dashed lines, which are darker where more profiles have the same peak location on the coarse pressure levels. We estimated the peaks by searching for values where the averaging kernel for a given pressure level was larger than its neighboring points. More complex methods, such as full width at half-maximum (FWHM), can also be employed when AKMs are heavily skewed.