## 2 Materials and Methods
### Statistical Metrics  
NUCAPS has been validated in research literature and as an ongoing effort in operations [4; 5; 26]. Ref. [26] provides a thorough discussion on the metrics used and their philosophical rationale, which we employ in this study for comparability and briefly summarize below. For temperature, the root-mean-square error (RMSE) for each pressure level is given by:  
\[\Delta T_{L,j}=\hat{T}_{L,j}-T_{L,j} \tag{2}\]  
\[RMSE\big{(}\Delta T_{L,j}\big{)}=\sqrt{\frac{1}{n_{j}}\sum_{j=1}^{n_{j}}\big{(} \Delta T_{L,j}\big{)}^{2}} \tag{3}\]  
where \(\hat{T}\) is the retrieved temperature in Kelvin, \(T\) is the dropsonde temperature in Kelvin, \(L\) is the level index, \(j\) is the field of regard index, and \(n\) is the total number of matched profiles. The temperature bias (\(\mu\)) takes the form:  
\[\mu\big{(}\Delta T_{L,j}\big{)}=\frac{1}{n_{j}}\sum_{j=1}^{n_{j}}\Delta T_{L,j} \tag{4}\]  
The bias is useful for assessing the accuracy of the retrieval with the observed value. The standard deviation (\(\sigma\)) is a combination of the bias and the RMSE:  
\[\sigma\big{(}\Delta T_{L,j}\big{)}=\sqrt{RMSE\big{(}\Delta T_{L,j}\big{)}^{2 }-\mu\big{(}\Delta T_{L,j}\big{)}^{2}} \tag{5}\]  
Water vapor is usually examined as a percent difference from the dropsonde value instead of using its absolute magnitude. This is because water vapor is retrieved as a layer abundance (also called column density) and can vary on exponential scales with height. The percent difference for each level \(L\) is expressed as:  
\[\Delta q_{L}=\frac{\hat{q}_{L,j}-q_{L,j}}{q_{L,j}} \tag{6}\]  
where \(\hat{q}\) is the retrieved water vapor column density converted to mixing ratio (g water/kg air) and q is the dropsonde water vapor (converted from relative humidity to column density). The water vapor RMSE and bias are then expressed as:  
\[RMSE\big{(}\Delta q_{L,j}\big{)}=\sqrt{\frac{\sum_{j=1}^{n_{j}}W^{\prime}{}_ {L,j}\big{(}\Delta q_{L,j}\big{)}^{2}}{\sum_{j=1}^{n_{j}}W^{\prime}{}_{L,j}} }\ \times\ 100\% \tag{7}\]  
\[\mu\big{(}\Delta q_{L,j}\big{)}=\frac{\sum_{j=1}^{n_{j}}W^{\prime}{}_{L,j} \big{(}\Delta q_{L,j}\big{)}}{\sum_{j=1}^{n_{j}}W^{\prime}{}_{L,j}}\ \times\ 100\% \tag{8}\]  
where \(W\) is the weighting factor. Weighting factors are used to account for large differences in the observed and retrieved water vapor values in a very dry atmosphere [5]. We use \(W^{\prime}{}_{L,j}=q_{L,j}{}^{2}\) for the RMSE calculation and \(W^{\prime\prime}{}_{L,j}=q_{L,j}\) for the bias calculation for consistency with past studies. It follows that the standard deviation for \(\Delta q\) at level \(L\) is then:  
\[\sigma(\Delta q_{L})=\sqrt{RMSE\big{(}\Delta q_{L,j}\big{)}^{2}-\mu\big{(} \Delta q_{L,j}\big{)}^{2}} \tag{9}\]  
Dropsondes measure environmental parameters at 2 Hz (pressure, temperature, and humidity) and 4 Hz (wind speed and direction), leading to observations on thousands of pressure levels, which vary for each profile. The dropsonde water vapor data were natively measured as relative humidity, which we converted to mixing ratio. These dropsonde temperature and water vapor mixing ratios profiles were linearly interpolated to theNUCAPS 100-level fixed pressure levels. The NUCAPS water vapor fields are all reported on pressure layers (see Section 2.1). To ensure all data were on the same vertical grid, the NUCAPS water vapor values were converted from pressure layers to pressure levels for this study. Since the NUCAPS layer pressure is the midpoint between the two NUCAPS pressure levels, water vapor was also converted from layer \(L^{\prime}\) to level \(L\) using a simple two-point mean:  
\[q_{L,j}=\frac{q_{L^{\prime},j}+q_{L^{\prime}-1,j}}{2} \tag{10}\]  
We chose to evaluate NUCAPS variables on the 100-level fixed pressure grid because that is how they are delivered to forecasters and research users.