# Portable L-Band Radiometer (PoLRa): Design and Characterization
## 3 Radiometer Characterization
### Uncertainty Characterization  
With uncorrelated variables, the systematic uncertainty of calibrated noise temperature at the switch-port reference plane can be expressed using the variance formula as [30]:  
\[\Delta T_{\text{in}}^{p}\ =\ \sqrt{\left(\Delta T_{\text{RS}}\frac{\partial T_{ \text{in}}^{p}}{\partial T_{\text{RS}}}\right)^{2}+\left(\Delta T_{\text{ACS}} \frac{\partial T_{\text{in}}^{p}}{\partial T_{\text{ACS}}}\right)^{2}+\left( \Delta u_{\text{RS}}\frac{\partial T_{\text{in}}^{p}}{\partial u_{\text{RS}}} \right)^{2}+\left(\Delta u_{\text{ACS}}\frac{\partial T_{\text{in}}^{p}}{ \partial u_{\text{ACS}}}\right)^{2}+\left(\Delta u^{p}\frac{\partial T_{\text{ in}}^{p}}{\partial u^{p}}\right)^{2}}\,, \tag{12}\]  
where the \(\Delta\) prefix refers to the uncertainty associated with the proceeding variable. The systematic uncertainties \(\Delta u_{\text{RS}}\), \(\Delta u_{\text{ACS}}\), \(\Delta u^{p}\) of the measured voltages \(u_{\text{RS}}\), \(u_{\text{ACS}}\), \(u^{p}\) are \(<\)0.01 mV. When converted to temperature units by multiplication by the gain \(G\) (\(\sim\)5 K/mV), these are much smaller than the uncertainties \(\Delta T_{\text{RS}}\ =\ \Delta T_{\text{ACS}}\simeq\) 1K of measured physical temperatures \(T_{\text{RS}}\), \(T_{\text{ACS}}\). Thus Equation (12) can be simplified to:  
\[\Delta T_{\text{in}}^{p}\ \equiv\ \sqrt{\left(\Delta T_{\text{RS}}\frac{ \partial T_{\text{in}}^{p}}{\partial T_{\text{RS}}}\right)^{2}+\left(\Delta T _{\text{ACS}}\frac{\partial T_{\text{in}}^{p}}{\partial T_{\text{ACS}}} \right)^{2}}\,, \tag{13}\]  
with:  
\[\frac{\partial T_{\text{in}}^{p}}{\partial T_{\text{RS}}}\ =\ \frac{u^{p}-u_{ \text{RS}}}{u_{\text{RS}}-u_{\text{ACS}}}+1, \tag{14}\]  
and:  
\[\frac{\partial T_{\text{in}}^{p}}{\partial T_{\text{ACS}}}\ =\ \frac{u_{ \text{RS}}-u^{p}}{u_{\text{RS}}-u_{\text{ACS}}}, \tag{15}\]  
where the partial derivatives are calculated from the substitution of Equations (8) and (9) into (10). Given the specified temperature sensor uncertainty \(\Delta T_{\text{RS}}\ =\ \Delta T_{\text{ACS}}\simeq\) 1K, and the ACS RMSE \(\Delta T_{\text{ACS}}\) = 0.66 K from Section 3.1, the systematic uncertainty \(\Delta T_{\text{in}}^{p}\) of PoLRa noise temperature measurements \(T_{\text{in}}^{p}\) at the input ports \(p\ =\ \left\{\text{H},\text{V}\right\}\) can be calculated from Equation (13). We calculate \(\Delta T_{\text{in}}^{p}\) for a range of \(u^{p}\), covering the range 50 K \(\leq T_{\text{in}}^{p}\leq\) 350 K expected for measurements of terrestrial scenes. The total uncertainty \(\Delta T_{\text{in,tot}}^{p}\) of measured noise temperatures at the radiometer ports \(p\ =\ \left\{\text{H},\text{V}\right\}\) is then calculated as the root sum square of the systematic and statistical contributions:  
\[\Delta T_{\text{in,tot}}^{p}\ =\ \sqrt{\Delta T_{\text{in}}^{p}\,{}^{2}+\text{NEAT} ^{2}}\,. \tag{16}\]  
The systematic uncertainty \(\Delta T_{\text{in}}^{p}\) and total uncertainty \(\Delta T_{\text{in,tot}}^{p}\) are plotted for two different integration times in Figure 8. The uncertainty reaches a minimum when the measured noise temperature is roughly in-between the two calibration references (the RS and the ACS), and increases when the measured noise temperature requires extrapolation beyond the calibration reference points. Additional uncertainty sources, such as non-linearity, mismatch, and isolation [31], have been neglected in this analysis, as they are considered small compared to the uncertainty associated with the temperature sensors. Linearity estimates are provided with the detector, and mismatch between components and switch ports were all measured below \(-\)20 dB. The above uncertainty analysis has only considered internal uncertainty sources impacting the noise temperatures \(T_{\text{in}}^{p}\) measured at the input ports \(p\ =\ \left\{\text{H},\text{V}\right\}\) of the switch. When the antenna is viewing a natural footprint at the ground, additional uncertainty sources will arise, including potential radio-frequency interference (RFI).