# Evaluation of the GPS Precise Orbit and Clock Corrections from MADOCA Real-Time Products
## 5 SISRE Computation  
The SISRE is a comprehensive measure of the fidelity of the MADOCA products, reflecting the combined effects of satellite orbit and clock errors projection in the line of sight. The SISRE's contributions and statistical characterization are at the same time of crucial significance for the development of advanced-receiver autonomous-integrity monitoring (ARAIM) algorithms. The instantaneous SISRE calculated by an arbitrary set of orbit errors including radial (R), along-track (A), and cross-track (C) varies with the orientation of the line-of-sight vector and the receiver locations [25]. As such, the concept of global averaging is adopted for SISRE calculation in this paper where \"the global average SISRE\" means the root-mean-square SISRE across the points within the satellite's coverage. The GPS Standard Positioning Service performance standard [26] provides two methods for computing the global average SISRE for a satellite at a specific instant epoch. One of them is Brute Force RMS by which the global average SISRE can be computed numerically from instantaneous SISREs, but the amount of calculation is very large; and the other is the Piecewise RMS that computes the global average SISRE by the satellite orbit errors and the total timing error. In this paper we use the latter method. Accordingly, the following two SISRE metrics are considered in our study, orbit-only contribution to SISRE can be described as:  
\[\mathrm{SISRE}(\mathrm{orb})=\sqrt{w_{R}^{2}*R^{2}+w_{A,C}^{2}*(A^{2}+C^{2})}, \tag{4}\]  
here, \(R\), \(A\), \(C\) denote, respectively, the radial orbit error, along-track orbit error, and cross-track orbit error while \(w_{R}\) and \(w_{A,C}\) are weight factors for statistical contribution of radial and along/cross-track errors to the line-of-sight ranging error which depend on the altitude of the GNSS satellite [19]. For GPS, \(w_{R}\) = 0.98 and \(w_{A,C}\) = 0.141. Analogous to (4), the formula for the global average rms SISRE that combines orbit and clock errors is:  
\[\mathrm{SISRE}=\sqrt{\left(w_{R}R-cT\right)^{2}+w_{A,C}^{2}*(A^{2}+C^{2})}, \tag{5}\]  
where, \(c\) and \(T\) are the speed of light and total timing error, respectively. It can be seen from the equation above that the radial orbit error and the clock error play a vital role in the calculation process of SISRE, and there is a significant negative correlation between them. The SISRE values obtained from Equation (5) can reasonably evaluate the average ranging error of the entire surface of the earth. It is noteworthy that the satellite orbit errors in radial, along-track, and cross-track directions are needed in the SISRE budget, and the performance of the satellite orbit in the radial direction plays a significant  
Figure 6: Seven-day average-STD clock errors of all available GPS satellites. role during positioning and navigation, yet the orbit coordinate values provided in these two products are represented in the ECEF frame. Before conducting the evaluation and calculating the SISRE values, therefore, we convert the ECEF frame to RAC frame. The relevant conversion model is assumed to be known and thus will not be discussed. Interested readers are recommended to refer to [PERSON] and [PERSON] [8], [PERSON] and [PERSON] [27] and [PERSON] and [PERSON] [28]. For the purpose of a comprehensive perspective, the orbit-error-only SISRE values and the SISRE values of all available satellites were calculated and the corresponding results are summarized in Figure 7. There are two key messages revealed from this figure. One is that the orbit-error-only SISRE values of all satellites are improved by about 2 to 3 cm after Helmert transformation. The other is that the clock errors contribute an imm moderate part of the SISRE budget of almost all GPS satellites. However, the SISRE values are still slightly improved after conducting Helmert transformation. Hence, in the subsequent long-term MADOCA real-time product evaluation, the statistics will be based on the orbit messages after Helmert transformation.