## 4 Verification and Accuracy Analysis of Single-Frequency Occultation Processing
### relTEC Correctness Verification
Figure 1: The expression of the reITEC change rate of the single-frequency pseudorange carrier phase combination and the dual-frequency carrier phase combination at different tangent heights, (**A**) is occultation event 1 and (**B**) is occultation event 2. Figure 2: Consistency comparison of the reITEC change rate calculated by single-frequency pseudo-range carrier phase combination and dual-frequency carrier phase combination, (**A**) is occultation event 1 and (**B**) is occultation event 2. The analysis in Figure 2 shows the consistency of the intensity of the linear relationship between the single-frequency and dual-frequency occultation processing. It shows that the two keep pace to a large extent in terms of their increasing or decreasing trends, even if they do not maintain a proportional relationship; that is, the characteristics of the position sequence in the occultation data processing are not considered. Therefore, the consistency of the relTEC results calculated by single-frequency and dual-frequency observations will be analyzed from the perspective of order. In statistics, [PERSON]'s correlation coefficient quantifies the correlation between independent variable X and dependent variable Y. It relies fully on the data sample ranking value. If Y tends to increase when X increases, [PERSON]'s correlation coefficient is positive, while if Y tends to decrease when X increases, [PERSON]'s correlation coefficient is negative. When X and Y are more and more close to a complete monotonic correlation, [PERSON]'s correlation coefficient will increase in absolute value, and when X and Y are completely monotonically correlated, the absolute value of [PERSON]'s correlation coefficient is 1. The fully monotonically increasing relationship means that for any two pairs of data (X\({}_{i}\), Y\({}_{i}\)) and (X\({}_{j}\), Y\({}_{j}\)), the results of X\({}_{i}\)-Y\({}_{i}\) and Y\({}_{i}\)-Y\({}_{j}\) always share the same plus or minus sign. [PERSON]'s correlation coefficient is calculated as  
\[\rho= \frac{\frac{1}{n}\sum_{i=1}^{n}\left(\left(\mathrm{R}(X_{i})-\overline{ \mathrm{R}(x)}\right)\cdot\left(\mathrm{R}(Y_{i})-\overline{\mathrm{R}(y)} \right)\right)}{\sqrt{\left(\frac{1}{n}\sum_{i=1}^{n}\left(\mathrm{R}(X_{i})- \overline{\mathrm{R}(x)}\right)^{2}\right)\cdot\left(\frac{1}{n}\sum_{i=1}^{n} \left(\mathrm{R}(Y_{i})-\overline{\mathrm{R}(y)}\right)^{2}\right)}} \tag{4}\]  
where \(\mathrm{R}(X_{i})\) represents the ith relTEC change rate sequence calculated by the occultation data at the BDS single-frequency received by FY3E; \(\mathrm{R}(Y_{i})\) represents the ith relTEC change rate sequence calculated by the occultation signal at the dual-frequency of BDS received by FY3E; and \(\overline{\mathrm{R}(x)}\) and \(\overline{\mathrm{R}(y)}\) represent the median of single-frequency and dual-frequency relTEC change rates, respectively. Figure 3 shows the distribution histogram of Spearman's correlation coefficient between single- and dual-frequency occultation processing of an occultation event on 4 January 2022. The dual-frequency carrier phase combination is used to calculate the relTEC change rate (as an independent variable X), and the single-frequency pseudorange-carrier phase combination is used to calculate the relTEC change rate (as a dependent variable Y), with an interval of 0.05. It is shown that the correlation coefficient between the rate of change of the dual-frequency solution and the rate of change of the single-frequency solution is greater than 0.95 in most (nearly 90%) of the occultation events of the BDS B1I signal. Moreover, the proportion of data with [PERSON]'s correlation coefficient greater than 0.8 is 96.8%, showing a strong correlation. The [PERSON]'s correlation coefficient of all data is greater than 0.7. Therefore, the relTEC change rate results of single- and dual-frequency observation have strong consistency related to order.