A comparison of two PPP-RTK models: S-basis choice, network product precision, and user positioning performance

Abstract

PPP-RTK combines the advantages of both precise point positioning (PPP) and real-time kinematic (RTK) techniques. While constructing PPP-RTK models based on undifferenced and uncombined observations offers apparent benefits, these observation equations suffer from a rank deficiency issue. To address this problem, the Singularity-system (S-system) theory can be utilized. This theory imposes constraints on a minimal subset of parameters, known as the S-basis, by setting them to arbitrary values, typically zeros. Despite the existence of multiple options for the S-basis, prevailing research conventionally selects the parameters of one receiver—the pivot—as the S-basis. In this study, we depart from this practice by selecting the mean of receiver-related parameters as the S-basis. This departure prompts an exploration into how the S-basis choices influence PPP-RTK outcomes regarding network product precision and user positioning performance. Our comparative analysis of the mean receiver (MR) and pivot receiver (PR) models unveils distinctions in the combined product precision. These products include satellite clocks, satellite phase biases, and ionospheric delays (excluding tropospheric delays). The distinction emerges because the estimable satellite clocks in the PR model incorporate atmospheric delays specific to the pivot receiver, in contrast to the MR model, which integrates mean atmospheric delays from all receivers. Despite the distinction in the analytical form of combined product and its precision, both model results in similar positioning performance. This is because variations in product precision levels caused by selecting different atmospheric parameters as the S-basis can be nullified by the parameterized atmospheric delays on the user side. With the inclusion of tropospheric delays, the PR and MR models also demonstrate similar performance and yield more accurate user positioning when located near the pivot receiver compared to positions farther from the pivot receiver when employing ambiguity-float network products. This dependence on the pivot receiver stems from both models selecting the pivot receiver ambiguities as the S-basis, while opting for mean ambiguities across all receivers negates the integer nature of ambiguities. Our conclusion underscores that identical positioning outcomes in PR and MR PPP-RTK models rely on both models selecting the same ambiguities as the S-basis. This highlights the potential variability in PPP-RTK performance when different ambiguity parameters are selected as the S-basis, particularly in the absence of network integer ambiguity resolution.

Appendix: Proof of (27)

According to the definitions in Table 2, the sum of estimable satellite clocks and satellite phase biases of the MR model can be written as

$$\begin{aligned} d\tilde{\tilde{t}}^{s} + \tilde{\tilde{\delta }}_{,j}^{s} & = \left( {dt^{s} + d_{,IF}^{s} } \right) - \left( {d\overline{t}_{r} + \overline{d}_{r,IF} } \right) - m_{1}^{s} \overline{\tau }_{r} \\ & \quad + \left( {\delta_{,j}^{s} - d_{,IF}^{s} + \mu_{j} d_{,GF}^{s} } \right) - \left( {\overline{\delta }_{r,j} - \overline{d}_{r,IF} + \overline{d}_{r,GF} } \right) \\ & \quad - \lambda_{j} z_{1,j}^{s} - \lambda_{j} \overline{z}_{1r,j}^{1} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \left( {\delta_{,j}^{s} + \mu_{j} d_{,GF}^{s} } \right) \\ & \quad - \left( {\overline{\delta }_{r,j} + \overline{d}_{r,GF} } \right) - \lambda_{j} z_{1,j}^{s} - \lambda_{j} \overline{z}_{1r,j}^{1} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \left( {\delta_{,j}^{s} + \mu_{j} d_{,GF}^{s} } \right) \\ & \quad - \left( {\overline{\delta }_{r,j} + \overline{d}_{r,GF} } \right) - \lambda_{j} z_{1,j}^{s} - \lambda_{j} \overline{z}_{1r,j}^{1} + \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} - \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \left( {\delta_{,j}^{s} + \mu_{j} d_{,GF}^{s} } \right) \\ & \quad - \left( {\overline{\delta }_{r,j} + \overline{d}_{r,GF} } \right) - \lambda_{j} z_{1,j}^{s} - \lambda_{j} \overline{z}_{1r,j}^{1} \\ & \quad + \mu_{j} \left( {\overline{l}_{r}^{s} + \overline{d}_{r,GF} - d_{,GF}^{s} } \right) - \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \mu_{j} \overline{l}_{r}^{s} + \delta_{,j}^{s} \\ & \quad - \overline{\delta }_{r,j} - \lambda_{j} z_{1,j}^{s} - \lambda_{j} \overline{z}_{1r,j}^{1} - \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \mu_{j} \overline{l}_{r}^{s} + \delta_{,j}^{s} \\ & \quad - \overline{\delta }_{r,j} - \lambda_{j} \overline{z}_{r,j}^{s} + \lambda_{j} \overline{z}_{r,j}^{s} - \lambda_{j} z_{1,j}^{s} - \lambda_{j} \overline{z}_{1r,j}^{1} - \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \mu_{j} \overline{l}_{r}^{s} + \delta_{,j}^{s} \\ & \quad - \overline{\delta }_{r,j} - \lambda_{j} \overline{z}_{r,j}^{s} + \lambda_{j} \overline{z}_{1r,j}^{s} - \lambda_{j} \overline{z}_{1r,j}^{1} - \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \mu_{j} \overline{l}_{r}^{s} + \delta_{,j}^{s} \\ & \quad - \overline{\delta }_{r,j} - \lambda_{j} \overline{z}_{r,j}^{s} + \lambda_{j} z_{1r,j}^{1s} - \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} \\ & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \mu_{j} \overline{l}_{r}^{s} + \delta_{,j}^{s} \\ & \quad - \overline{\delta }_{r,j} - \lambda_{j} \overline{z}_{r,j}^{s} + \overline{\tilde{z}}_{r,j}^{s} - \mu_{j} \overline{\tilde{\tilde{l}}}_{r}^{s} \\ \end{aligned}$$

(A1)

where the items with an overline represent the mean of these receiver-related parameters, e.g., $d\overline{t}_{r} = \frac{1}{n}\sum\limits_{r = 1}^{n} {d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{t}_{r} }$.

Considering that

$$\begin{aligned} E\left[ { - \overline{\phi }_{r,j}^{s} } \right] & = dt^{s} - d\overline{t}_{r} - m_{1}^{s} \overline{\tau }_{r} + \mu_{j} \overline{l}_{r}^{s} \\ & \quad + \delta_{,j}^{s} - \overline{\delta }_{r,j} - \lambda_{j} \overline{z}_{r,j}^{s} \\ \end{aligned}$$

(A2)

we obtain

$$\begin{aligned} & d\hat{\tilde{\tilde{t}}}^{s} + \hat{\tilde{\tilde{\delta }}}_{,j}^{s} - m_{u}^{s} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tilde{\tilde{\tau }}}_{u} + \mu_{j} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tilde{\tilde{l}}}_{u}^{s} = - \overline{\hat{\phi }}_{,j}^{s} + \overline{\hat{\tilde{\tilde{z}}}}_{r,j}^{s} \\ & \quad + \mu_{j} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tilde{\tilde{l}}}_{u}^{s} - \overline{\hat{\tilde{\tilde{l}}}}_{r}^{s} } \right) - m_{u}^{s} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tilde{\tilde{\tau }}}_{u} \\ \end{aligned}$$

(A3)

from which (27) follows.