## 4 Calibration and Characterization
### Calibration Examples  
In this subsection, the results of one example of each of the calibration procedures mentioned in the previous section are shown. Measurements have been performed in clear sky conditions, after the sunset, at a ground temperature \(T_{gr}\) of \(9^{\circ}\)C, and with the receiving chain stabilized in temperature. The first example corresponds to the calibration of the C-band V-polarization channel. In this case, six calibration points are measured, two hot ones pointing to the microwave absorber, two cold ones pointing to the sky at zenith, and two matched load measurements. The measurements have been taken using the following parameters: the acquisition band \(B_{r}\) is 90 MHz, the RBW is 100 kHz, the \(N_{\rm SWP}\) is 1001, the SWT is 9 ms, and the \(N_{\rm AVG}\) is 1000. Using these measurements, the \(a\) and \(b\) parameters are obtained by solving (14) and (15) using a linear regression method. Since the back-end stage provides power spectrum measurements, \(a\) and \(b\) parameters may be obtained either by averaging the whole spectrum or for particular sub-bands. If the power spectrum is averaged over the whole acquisition bandwidth, the values obtained are \(a=0.21\) nW/K, and \(b=106.74\) nW, with a \(R^{2}\) parameter (i.e., coefficient of determination) equal to 0.9971. The results of the averaged hot-cold calibration for the C-band V-polarization are shown in Figure 10a. On the other hand, Figure 10b shows the \(a\) and \(b\) values for sub-bands equal to the RBW. In this case, it can be appreciated that the gain \(a\) fluctuates around its average value, and the bias \(b\) slightly increases with frequency. Moreover, the band-limiting effect of the filters generates outlier \(a\) values (minimum and maximum peaks) at the band edges. The second example corresponds to the calibration of the X-band H-polarization channel. In this case, thirteen calibration points are measured, two hot ones pointing to the microwave absorber, nine cold ones pointing to the sky at different zenith angles, and two matched load measurements. The measurements have been taken using the following parameters: the acquisition band \(B_{r}\) is 100 MHz, the RBW is 100 kHz, the \(N_{\rm SWP}\) is 1001, the SWT is 9 ms, and the \(N_{\rm AVG}\) is 1000. The cold calibration points have been obtained by changing the elevation angle of the radiometer from the \(0^{\circ}\) (zenith) to \(70^{\circ}\), in steps of \(10^{\circ}\), and one extra measurement has been acquired at the zenith position. This corresponds to values from one to approximately three equivalent atmospheres. Figure 11 shows the results of the tipping curve equation defined in (22) for the X-band H-polarization channel, as a function of the equivalent number of atmospheres (secant of the elevation angle). With these values and using a robust (outlier values were discarded) non-linear regression method, the total atmosphere attenuation at zenith direction \(L_{atm}\) has been estimated to be 0.9701, with a \(R^{2}\) parameter equal to 0.9453. Outlier values measured at lower zenith angles are induced by the small (but enough) internal temperature differences that occur when changing the position of the radiometer, despite internal fans are installed to minimize this effect. Once estimated the \(L_{atm}\), the hot-cold calibration can be performed using several cold points to determine the unknown parameters. The results of the hot-cold calibration for the X-band averaged in the whole bandwidth are shown in Figure 11. The \(a\) and \(b\) parameters are obtained as in the previous case. The obtained values are \(a\) equal to 0.63 nW/K, and \(b\) equal to 89.30 nW, with a \(R^{2}\) parameter equal to 0.9987. Note that in the tipping curves regression the estimation of the \(a=0.65\) nW/K was also very similar. Figure 10: _Cont._  
Figure 11: Hot-cold calibration of the X-band H-polarization channel using tipping curves. Subfigure (**a**) shows the power tipping curves as a function of the equivalent number of atmospheres.