# Accuracy Analysis of GNSS Hourly Ultra-Rapid Orbit and Clock Products from SHAO AC of iGMAS
## Appendix A The POD Processing by Combined Serial and Parallel Threads (CSPT)  
The undifferenced (UD) observation model for code pseudorange (P) and carrier phase \(\Phi\) observations can be showed by distance, as follows:  
\[\mathrm{P}^{\mathrm{s}}_{\mathrm{r,i}}=\rho^{\mathrm{s}}_{\mathrm{r}}-\Delta \mathrm{t}^{\mathrm{s}}+\Delta\mathrm{t}_{\mathrm{r}}+\mathrm{c.}(\mathrm{d} _{\mathrm{r,i}}-\mathrm{d}^{\mathrm{s}}_{\mathrm{i}})+\mathrm{m}^{\mathrm{s}}_ {\mathrm{r}}\mathrm{\cdot ZTD}_{\mathrm{r}}+\mathrm{I}^{\mathrm{s}}_{\mathrm{ r}}+\varepsilon^{\mathrm{s}}_{\mathrm{P,r,i}} \tag{1}\]  
\[\mathrm{\Phi}^{\mathrm{s}}_{\mathrm{r,i}}=\rho^{\mathrm{s}}_{\mathrm{r}}- \Delta\mathrm{t}^{\mathrm{s}}+\Delta\mathrm{t}_{\mathrm{r}}+\mathrm{\lambda _{i}}\cdot(\mathrm{b}_{\mathrm{r,i}}-\mathrm{b}^{\mathrm{s}}_{\mathrm{i}}+ \mathrm{N}^{\mathrm{s}}_{\mathrm{r,i}})+\mathrm{m}^{\mathrm{s}}_{\mathrm{r,i} }\mathrm{\cdot ZTD}_{\mathrm{r}}-\mathrm{I}^{\mathrm{s}}_{\mathrm{r}}+ \varepsilon^{\mathrm{s}}_{\mathrm{\varnothing,r,i}} \tag{2}\]  
where s, r, and i (i = 1, 2) are on half of satellites, receivers, and frequencies, respectively. \(\mathrm{\rho}^{\mathrm{s}}_{\mathrm{r}}\) is the geometric distance between satellite and receiver, \(\Delta\mathrm{t}^{\mathrm{s}}\) and \(\Delta\mathrm{t}_{\mathrm{r}}\) denote satellite and receiver clock bias, respectively. c is light velocity. \(\mathrm{d}_{\mathrm{r,i}}\) and \(\mathrm{d}^{\mathrm{s}}_{\mathrm{i}}\) refer to receiver and satellite code bias at frequency i. \(\mathrm{m}^{\mathrm{s}}_{\mathrm{r}}\) and \(\mathrm{ZTD}_{\mathrm{r}}\) represent the mapping function and zenith tropospheric delay at receiver position. \(\mathrm{I}^{\mathrm{s}}_{\mathrm{r}}\) denotes the ionospheric delay from receiver to satellite. \(\mathrm{\lambda}_{\mathrm{i}}\) is the wave length for frequency i. \(\mathrm{b}_{\mathrm{r,i}}\), and \(\mathrm{b}^{\mathrm{s}}_{\mathrm{i}}\) illuminate the uncalibrated phase delays (UPDDs) for receivers and satellites. \(\mathrm{N}^{\mathrm{s}}_{\mathrm{r,i}}\) is the integer ambiguity between receiver and satellite pairs at frequency i. \(\mathrm{\epsilon}^{\mathrm{s}}_{\mathrm{P,r,i}}\) and \(\mathrm{\epsilon}^{\mathrm{s}}_{\mathrm{\varnothing,r,i}}\) denote the sum of the noise and other un-modeled corrections. Other corrections, such as phase center offset (PCO), phase center variations (PCV), and ocean tide loading, etc., are corrected by models (introduced in Section 2). In order to eliminate the low order ionospheric delays, the ionosphere-free (IF), combined between double frequencies, are applied in this contribution. It should be noted that the time consumption would increase dramatically if four satellite navigation systems were combined to estimate orbit and clock parameters. Four threads were running at the same time to generate the orbit and clock products for GPS, GLONASS, Galileo, and BDS by distributed parallelism mode, as Formulas (3)-(6) shows. Formulas (4)-(6) also show GLONASS, Galileo, and BDS are coupled with GPS to improve the accuracy of the orbit and clock products for GLONASS, Galileo, and BDS, respectively. \[\left\{\begin{array}{l}\mathrm{P}_{\mathrm{r,IF}}^{\mathrm{S}}=\mu_{\mathrm{r }}^{\mathrm{S}}\cdot\left(\psi(\mathrm{t},\mathrm{  
files and strongly constrained. Moreover, a reference clock is fixed when estimated the satellite and receiver clock biases in order to remain the same time reference, and then the estimated satellite and receiver clock biases subtract another reference clock bias to eliminate the clock bias difference in these four threads. The reference clock should be connected with hydrogen atomic clock and showed better performance, such as ALGO, PTBB, WTZR, NRC1, PIE1 or ONSA. \[\left\{ \begin{array}{c}\Delta_{\rm HF}^{\rm S}=\Delta_{\rm HF}^{\rm S}-\Delta _{\rm tr,IF,ref}^{\rm S}\\ \Delta_{\rm tr,IF}^{\rm S}=\Delta_{\rm tr,IF}^{\rm S}-\Delta_{\rm tr,IF,ref}^{ \rm S}\end{array} \right.