## 2 Single-Frequency Occultation Data Processing Method Based on BDS  
The inversion algorithm based on BDS single-frequency radio occultation data means that when there is only observation of frequency band B\({}_{1}\), the pseudorange and carrier phase are used to calculate the ionospheric delay, and further reconstruct the inversion of occultation products (atmospheric refractive index) in the frequency band B\({}_{3}^{*}\). In this paper, the BDS observation data used to retrieve atmospheric occultation products is the B\({}_{1}\) carrier-phase measurement at 50 Hz sampling rate and B1I pseudorange measurement (C\({}_{1}\)) at the 1-50 Hz sampling rate. Due to the lack of data for the second frequency, the error caused by the ionospheric effect cannot be eliminated by difference combination. However, when only single-frequency measurement is used, the linear combination of B\({}_{1}\) and C\({}_{1}\) data can be used to construct B\({}_{3}^{*}\), but the result will be affected by the noise of the observation value C\({}_{1}\) itself [25; 26; 27]. This paper adopts a special filtering method to solve this problem. During data processing, with the help of a low-pass filter, the B\({}_{1}-\)C\({}_{1}\) data are filtered. At the same time, the regularization method [28] that minimizes the second derivative is used to interpolate to 50 Hz and repair the data gap so as to reduce the noise impact from the observations. The filter can be described as matrix F, given by the following formula:  
\[\mathrm{F}=(\mathrm{\widetilde{I}}+\gamma\mathrm{S}^{\mathrm{T}}\mathrm{S})^{ -1} \tag{1}\]  
where \(\mathrm{\widetilde{I}}\) is an incomplete identity matrix, which means the only diagonal element corresponding to the position where both B\({}_{1}\) and C\({}_{1}\) data are available is 1. Since in this paper the interpolation needs to be 50 Hz, the size of \(\mathrm{\widetilde{I}}\) is about 500 \(\times\) 500. Although there are many elements in the matrix \(\mathrm{\widetilde{I}}\), only about 10-500 elements on the diagonal are 1, and the other elements are all 0. The specific value of \(\mathrm{\widetilde{I}}\) is determined by the sampling rate of the carrier phase and the pseudorange. Matrix S is a second-order derivative finite-difference operator [29], S\({}^{\mathrm{T}}\) is its transposition, and \(\gamma\) is a regularization parameter that determines the smoothing strength. Here, \(\gamma=10^{6}\), corresponding to a low-pass filter with a cut-off frequency of \(\sim\)0.05 Hz, the cut-off frequency is selected to effectively eliminate the uncertain high-frequency noise in C\({}_{1}\) data from the signal BDS B1I. Therefore, the result of the filtered B\({}_{1}\) minus C\({}_{1}\) is obtained by the following formula:  
\[\mathrm{\overline{B_{1}\mathrm{-C_{1}}}}=\mathrm{F}(\mathrm{B_{1}\mathrm{-C_{ 1}}}) \tag{2}\]  
where \((\mathrm{B_{1}\mathrm{-C_{1}}})\) is a vector with about 500 elements, and the missing data point of B\({}_{1}\) or C\({}_{1}\) is 0. The expression \(\mathrm{k}(\mathrm{\overline{B_{1}\mathrm{-C_{1}}}})\) is the relative total electron content of the ionosphere, reITEC, in which the coefficient k is the conversion coefficient between the ionospheric delay and the total electron content. The carrier-phase measurement of B\({}_{3}^{*}\) is based on the carrier-phase measurement of B\({}_{1}\) and the result of filtered B\({}_{1}\mathrm{-C_{1}}\). The formula is as follows:  
\[\mathrm{B_{3}^{*}}=\mathrm{B_{1}\mathrm{-0.5(1-\frac{f_{1}^{2}}{f_{3}^{2}})( \mathrm{\overline{B_{1}\mathrm{-C_{1}}}})}} \tag{3}\]  
where f\({}_{1}\)= 1561.098 MHz is the observed signal frequency, and f\({}_{3}\) is the frequency of the observed signal to be constructed. In order to verify and analyze the reconstructed B\({}_{3}^{*}\) and single-frequency occultation processing method, this paper proposes constructing the B3I signal with a frequency of 1268.52 MHz.