## 4 Verification and Accuracy Analysis of Single-Frequency Occultation Processing
### Reconstruction Excess Phase Doppler Correctness Verification  
In Section 2, after the ionospheric relTEC step is calculated by combining the pseudorange and carrier phase of frequency B\({}_{1}\), the quasi-carrier-phase observation B\({}_{3}^{*}\) of frequency band B\({}_{3}\) is then reconstructed with the combination of frequency B\({}_{1}\). In the double difference method for occultation processing, excess phase Doppler is a very important parameter [35]. The excess phase Doppler \(\Delta\)f of frequency B\({}_{1}\) is calculated by the excess phase delay:  
\[\Delta\mathrm{f}=-\frac{1}{\lambda}\frac{\mathrm{d}\widetilde{\mathrm{L}}}{ \mathrm{dt}} \tag{5}\]  
where \(\widetilde{\mathrm{L}}\) indicates the excess phase delay of the single-frequency carrier phase and the quasi-carrier-phase (or the excess phase delay of the dual-frequency double difference), and \(\lambda\) indicates the wavelength of frequency B\({}_{1}\). It is helpful to study the excess phase Doppler in the occultation process. Figure 4 shows the comparison of the excess phase Doppler calculated from the reconstructed quasi-carrier-phase observation B\({}_{3}^{*}\) and from actual dual-frequency observation, in which the horizontal axis is the excess phase Doppler, in Hz, and the vertical axis is the tangent point height, in km. Pink and blue dots represent the excess phase Doppler of the reconstructed B\({}_{3}^{*}\) and the excess phase Doppler of the actual observation B\({}_{3}\). Due to the coincidence of many data dots in Figure 4, it also shows the result of 100\(\times\) magnification at some positions. At the beginning of occultation, the tangent point of the ray passes through the ionosphere and the upper stratosphere, where the medium is relatively thin, the amplitude of the signal is relatively constant, and the excess phase Doppler is close to 0. As the ray drops in the atmosphere, the vertical gradient with stronger refractive index will cause the ray to become more curved, resulting in excess phase Doppler delay changes. The figure also shows that the overall trend of the excess phase Doppler of reconstructed B\({}_{3}^{*}\) and that of the actual observation B\({}_{3}\) is the same and highly consistent. No matter how many observations are conducted in an occultation event, the excess phase reconstructed by a single frequency is consistent with that of the real observation B\({}_{3}\). Next, the trend change between the reconstructed excess phase and the actual excess phase are compared, a first-order linear fitting of all the reconstructed excess phase Doppler of an occultation event is performed, and then the fitted function is compared with the function y = x. Figure 5 is the comparison of the excess phase Doppler function between BDS single-frequency observation reconstruction and BDS dual-frequency observation under two occultation events. The enlarged details are shown in the small box. The horizontal axis is the excess phase Doppler of real observation B\({}_{3}\), and the vertical axis is the excess phase Doppler of the reconstructed B\({}_{3}^{*}\) (the blue dots). At the same time, the fitting function of the single-frequency reconstructed B\({}_{3}^{*}\)s excess phase Doppler is represented by a blue curve, and the function y = x, which represents the 100% consistency of the two, is represented  
Figure 3: Spearman correlation coefficient distribution histogram of the reITEC change rate solved by dual-frequency carrier phase combination and single-frequency pseudorange carrier phase combination. by a red curve. As can be seen in Figure 5, when there is more data of an occultation event (Figure 5A), the fitting function of the single-frequency reconstructed B\({}_{3}^{*}\) excess phase Doppler is closer to the red curve. The two occultation events show that the blue data dots are distributed near y = x, and the red fitting curve is almost identical to y = x. The result of B\({}_{3}^{*}\) excess phase Doppler reconstruction is very promising.