# Portable L-Band Radiometer (PoLRa): Design and Characterization
## 3 Radiometer Characterization
### Active Cold Source Characterization
We use this along with the assumed linearity between ACS noise and physical temperature to formulate a cost function (\(CF\)) to minimize and obtain the losses \(L^{\rm H}\) and \(L^{\rm V}\) by least squares fit:  
\[CF~{}=~{}\sum\
olimits_{p~{}=~{}\{\rm H,V\}}\sum\
olimits_{i~{}=~{}1}^{n} \left[T_{\rm ACS}^{p,i}-T_{\rm ACS,mod}^{p}\right]^{2}+\sum\
olimits_{i~{}=~{} 1}^{n}\left[T_{\rm ACS}^{{\rm H},i}-T_{\rm ACS}^{{\rm V},i}\right]^{2}, \tag{7}\]  
where \(T_{\rm ACS}^{{\rm H},i}\) and \(T_{\rm ACS}^{{\rm V},i}\) are the ACS noise temperatures derived from Equation (5), and using voltages \(u_{\rm ACS}\) available from sky measurements \(i~{}=~{}\{1,2,\ldots,n\}\). The first term in the \(CF\) enforces the linearity of ACS noise with its physical temperature, and the second term enforces \(T_{\rm ACS}^{\rm H}~{}=~{}T_{\rm ACS}^{\rm V}\). The \(CF\) is minimized using a numerical global minimum finder to obtain optimal \(L^{\rm H}\) and \(L^{\rm V}\). For an ideal measurement system, the resulting linear fit parameters \(m^{p}\) and \(b^{p}\) used in Equation (6) would be identical for \(p~{}=~{}\{\rm H,V\}\), but this is not the case in practice. To obtain the optimal polarization-independent linear temperature dependence of the ACS, the linear fit parameters\(m\,=\,\left\langle m^{\rm H},m^{\rm V}\right\rangle\) and \(b\,=\,\left\langle b^{\rm H},b^{\rm V}\right\rangle\) can be averaged over the two polarizations, which is equivalent to a linear fit of all \(T_{\rm ACS}^{p,i}\) values versus \(\overline{T}_{\rm ACS}\). Figure 3a shows the setup for these sky measurements at the Davos-Laret Remote Sensing Field Laboratory [28]. The antenna was oriented towards the south with approximately 70\({}^{\circ}\) altitude angle. Sky measurements were performed at 5-min intervals over an approximately 11-h period between 7 and 8 May 2020. Evening to nighttime (17:00-06:00 local time) measurements were used in order to maximize the range of physical temperatures while also avoiding solar intrusion into the antenna. Potential galactic noise intrusion was also investigated using a night sky calculator, and estimated from our equatorial coordinates to be less than 1 K [25], with the worst case occurring at the beginning of the measurement period. Figure 5 shows the physical temperatures and the measured detector voltages. The evening cooling period provided a \(\sim\)25 K temperature variation. Note that the detector on the PoLRa is an inverse slope detector, so lower voltages correspond to higher absolute power levels. Figure 6 provides the calibrated cold load brightness temperatures \(T_{\rm ACS}^{p,i}(\overline{T}_{\rm ACS})\) versus the ACS physical temperature \(\overline{T}_{\rm ACS}\) along with the linear fit line over both polarizations \(T_{\rm ACS,mod}\) and the 95% confidence interval of this fit line. Table 1 shows the parameter values resulting from the cost function (_CF_) minimization procedure. Figure 5: (**a**) Measured physical temperatures, and (**b**) raw voltages from detector measured during sky measurement versus time of day. Figure 6: Measured physical temperature of the active cold source (ACS) \(\overline{T}_{\rm ACS}\) versus calibrated ACS noise temperature \(T_{\rm ACS}^{p,i}\), and linear fit \(T_{\rm ACS,\_mod}\), for sky-measurement-based characterization of the ACS. The dotted line shows the 95% confidence interval (CI) of the linear model. The color bar indicates the local time at which each measurement was taken between 7 and 8 May 2020.