Based on the real-time ambiguities of PPP estimated from a reference network, State Space Representation (SSR) data streams are adopted for real-time precise orbit clock correction. In this study, the UPD estimation strategy proposed by Ge et al. [

16] is applied. The single-difference between two satellites is adopted to eliminate the receiver terminal UPD. According to the consistency of UPD influence on various stations, the UPD is separated according to the data of multiple stations. Melbourne-Wübbena (WM) [

27,

28] combination observation values are adopted to calculate the float ambiguities of single-difference between satellites

$\nabla {N}_{r,WL}^{j,k}$, and multi-epoch is smoothed to obtain

$\nabla {\widehat{N}}_{r,WL}^{j,k}$ to weaken the influence of measured noise and multi-path error [

35]. The smoothed ambiguity and the corresponding noise are as follows:

where

${\sigma}_{i}^{2}$ is the error of the

$i$-th epoch. During the smoothing process, the quality of float ambiguities should be controlled to eliminate cycle slip or gross error properly. After the qualified floating wide-lane ambiguity is obtained, the relationship between the UPD and the ambiguity is as follows:

where

$\nabla {\widehat{n}}_{r,WL}^{j,k}$ is the most approximate integer of

$\nabla {\widehat{N}}_{r,WL}^{j,k}$ and can be calculated according to

$\nabla {\widehat{n}}_{r,WL}^{j,k}=\{\begin{array}{c}\mathrm{int}(\nabla {\widehat{N}}_{r,WL}^{j,k}+0.5),\nabla {\widehat{N}}_{r,WL}^{j,k}\ge 0\\ \mathrm{int}(\nabla {\widehat{N}}_{r,WL}^{j,k}-0.5),\nabla {\widehat{N}}_{r,WL}^{j,k}<0\end{array}$ (

$\mathrm{int}(\xb7)$ is the integer operator) The value range of

$\nabla {b}_{wl}^{j,k}$ is

$[-0.5,0.5]$, and considering that the UPD characteristic that

$\nabla {b}_{wl}^{j,k}\text{}=\text{}0.5$ is equivalent to

$\nabla {b}_{wl}^{j,k}\text{}=\text{}-0.5$, the sine and cosine function method [

24] irrelevant to the integer item should be adopted to estimate

$\nabla {b}_{wl}^{j,k}$:

where

$Nu{m}^{i,j}$ is the number of the stations participating in the calculation and

${P}_{s}$ is the weight of the decimal part of each measuring station; After

$\nabla {b}_{wl}^{j,k}$ is estimated ,

$\nabla {b}_{wl}^{j,k}$ is substituted into Equation (4) to obtain narrow-lane floating-point ambiguity:

where

$\nabla {\widehat{N}}_{r,IF}^{j,k}$ is the float ambiguity of the observation value of the ionosphere free combined model during PPP resolving process,

$\nabla {\widehat{N}}_{r,1}^{j,k}$ is the single-difference narrow-lane float ambiguity between two satellites,

$\nabla {\widehat{n}}_{r,1}^{j,k}$ is the most approximate integer of

$\nabla {\widehat{N}}_{r,1}^{j,k}$,

$\nabla {b}_{1}^{j,k}$ is the single-difference narrow-lane UPD, with the value range of

$[-0.5,0.5]$, and

$\nabla {b}_{1}^{j,k}$ is also estimated according to the sine and cosine function method:

where

$\nabla {\widehat{N}}_{r}^{j,k}$ is the float ambiguity,

$\nabla {\widehat{n}}_{r}^{j,k}$ is the most approximate integer of

$\nabla {\widehat{N}}_{r}^{j,k}$;

${\epsilon}_{r}^{j,k}$ is the residual of un-model atmosphere real-time precise orbit clock correction, and they are different from each other in different regions, so the

$\nabla {b}^{j,k}$ could absorb different characteristics of the regional stations is as follow:

The UPD estimated by the above method is relative to the reference satellite. For the unification of the UPD datum, the reference satellite UPD should be set to zero as the datum to convert the single-difference UPD $\nabla {b}_{wl}^{j,k}$ and $\nabla {b}_{1}^{j,k}$ into zero-difference UPD ${b}_{wl}^{j}$ and ${b}_{1}^{j}$. The zero-difference UPDs of all satellites have a parameter bias ${b}_{wl}^{k}$ and ${b}_{1}^{k}$, and such bias will be absorbed by the receiver clock correction at the user terminal during positioning. When the reference satellite selected for UPD estimation has been changed, the corresponding bias will be also changed, but it will not influence user positioning. Therefore, there is no need for UPD users to consider the inconsistency of the reference satellite during positioning.

A certain quantity of stations should be selected for UPD estimation, and if more stations are more uniformly distributed around the world, it is more favorable for UPD estimation, and the global UPD estimation model is the same as the regional UPD estimation model, but due to the adoption of single-difference between satellites in the UPD estimation model, globally distributed sites cannot have the same reference satellite, so it is necessary to consider the reference satellite recursive method in the global UPD estimation. Actually, it is difficult to obtain the real-time data of the global stations for UPD estimation, so only regional stations are needed for UPD estimation only serving for the designated regions, and the different distributions of the regional stations can cause different UPD results, thus influence the PPP-AR resolving effect of the reference station.