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Review and principles of PPP-RTK methods

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Abstract

PPP-RTK is integer ambiguity resolution-enabled precise point positioning. In this contribution, we present the principles of PPP-RTK, together with a review of different mechanizations that have been proposed in the literature. By application of \(\mathcal {S}\)-system theory, the estimable parameters of the different methods are identified and compared. Their interpretation is essential for gaining a proper insight into PPP-RTK in general, and into the role of the PPP-RTK corrections in particular. We show that PPP-RTK is a relative technique for which the ‘single-receiver user’ integer ambiguities are in fact double-differenced ambiguities. We determine the transformational links between the different methods and their PPP-RTK corrections, thereby showing how different PPP-RTK methods can be mixed between network and users. We also present and discuss four different estimators of the PPP-RTK corrections. It is shown how they apply to the different PPP-RTK models, as well as why some of the proposed estimation methods cannot be accepted as PPP-RTK proper. We determine analytical expressions for the variance matrices of the ambiguity-fixed and ambiguity-float PPP-RTK corrections. This gives important insight into their precision, as well as allows us to discuss which parts of the PPP-RTK correction variance matrix are essential for the user and which are not.

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Acknowledgments

This work has benefitted from the many fruitful PPP-RTK discussions we had with our colleagues from the Curtin GNSS Research Centre. The first author is the recipient of an Australian Research Council (ARC) Federation Fellowship (project number FF0883188). This work has been done in the context of the Positioning Program Project 1.01 “New carrier phase processing strategies for achieving precise and reliable multi-satellite, multi-frequency GNSS/RNSS positioning in Australia” of the Cooperative Research Centre for Spatial Information (CRC-SI). All this support is gratefully acknowledged.

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Appendix

Appendix

Proof of Lemma 2

As the phase observations in (74) are reserved for the ambiguities \(\tilde{a}_r\), the float solutions of \(d\tilde{t}_r\) and \(\tilde{\iota }_r\) are, respectively, determined as the IF and GF combinations of the code observations only, that is

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle d\hat{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}} = -[\mu _\mathrm{\scriptscriptstyle IF}^T\otimes D_m^T]p_r,\\ &{}\!\!\!\displaystyle \hat{\tilde{\iota }}_{r,\scriptscriptstyle \mathrm{GF}} = +[\mu _\mathrm{\scriptscriptstyle GF}^T\otimes D_m^T] p_r \end{array} \end{aligned}$$
(114)

which gives the first expression of (76). The second expression (76) follows by substituting the preceding equations into

$$\begin{aligned}&\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}} = [\Lambda ^{-1}\otimes D_m^T]\nonumber \\&\quad \times \left[ \phi _r+[e\otimes I_{m-1}]d\hat{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}} +[\mu \otimes I_{m-1}]\hat{\tilde{\iota }}_{r,\scriptscriptstyle \mathrm{GF}}\right] , \end{aligned}$$
(115)

together with the identity \(e\mu _\mathrm{\scriptscriptstyle IF}^T=I_2-\mu \mu _\mathrm{\scriptscriptstyle GF}^T\).

An application of the (co)variance propagation law to (76) gives (77). \(\square \)

Proof of Lemma 3

Upon resolving the DD ambiguities \(\check{\tilde{z}}_r\) (\(r=1,\ldots ,n\)), with \(\check{\tilde{z}}_1=0\), \(2(m-1)(n-1)\) redundant model’s misclosures contribute to the estimation procedure, namely

$$\begin{aligned} t_{\tilde{z}_r} = \hat{\tilde{z}}_r - \check{\tilde{z}}_r,\quad r=2,\ldots ,n \end{aligned}$$
(116)

According to the least-squares conditional adjustment, the unbiased estimators \(\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}\) and \(d\hat{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}}\) are corrected by the above misclosures to provide the best linear unbiased estimators (BLUEs) \(\check{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}\) and \(d\check{\tilde{t}}_{r,\scriptscriptstyle \mathrm{GF}}\) (Teunissen 2000). Adding the corrections, the BLUEs must remain unbiased and get uncorrelated with the underlying misclosures, see Teunissen and Khodabandeh (2013, p. 463). The unique corrections must, therefore, fulfill two conditions: (1) they must be zero-mean and (2) their covariances with the misclosures must be identical to those between the float estimators and the misclosures with a negative sign. Proposing the following corrections

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle \epsilon _{\tilde{a}_r} = \check{\tilde{z}}_r-\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}+\frac{1}{n}\sum _{j=1}^n (\hat{\tilde{a}}_{j,\scriptscriptstyle \mathrm{GF}}-\check{\tilde{z}}_j)\\ &{}\!\!\!\epsilon _{d\tilde{t}_r} = -Q_{d\hat{\tilde{t}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{{\scriptscriptstyle \mathrm{GF}}-1} (\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}-\check{\tilde{a}}_{r,\scriptscriptstyle GF}) \end{array} \end{aligned}$$
(117)

their zero-mean property follows, respectively, from

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle \mathsf {E}(\hat{\tilde{a}}_{j,\scriptscriptstyle \mathrm{GF}}) = \check{\tilde{z}}_j-\tilde{\delta },\quad j=1,\ldots ,n\\ &{}\!\!\!\displaystyle \mathsf {E}(\hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}) = \mathsf {E}(\check{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}), \end{array} \end{aligned}$$
(118)

while the second-property follows, respectively, from

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle Q_{\epsilon _{\tilde{a}_r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} = -Q_{\hat{\tilde{a}}_{r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} \\ &{}\!\!\!\displaystyle Q_{\epsilon _{d\tilde{t}_r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} = -Q_{d\hat{\tilde{t}}_{r},t_{\tilde{z}_r}}^{\scriptscriptstyle \mathrm{GF}} \end{array} \end{aligned}$$
(119)

Accordingly, the fixed solutions are obtained as

$$\begin{aligned} \begin{array}{ll} &{}\!\!\!\displaystyle \check{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}} = \hat{\tilde{a}}_{r,\scriptscriptstyle \mathrm{GF}}+\epsilon _{\tilde{a}_r}\\ &{}\!\!\!\displaystyle d\check{\tilde{t}}_{r} = d\hat{\tilde{t}}_{r}+\epsilon _{d\tilde{t}_r} \end{array} \end{aligned}$$
(120)

which proves (78). Applying the (co)variance propagation law to (78) gives (79). \(\square \)

Proof of Table 3

We first prove the geometry-free results where again use is made of the GNSS misclosure concept (Khodabandeh and Teunissen 2014). In the \(k\)-epoch case, as the ambiguities are assumed constant in time, any (co)variance matrix \(Q_{\hat{x}\hat{y}}^{\scriptscriptstyle GF}\) is corrected according to the least-squares conditional adjustment as

$$\begin{aligned} Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k] = Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}-[\frac{k-1}{k}]Q_{\hat{x}\hat{\tilde{a}}_r}^{\scriptscriptstyle GF} Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{{\scriptscriptstyle \mathrm{GF}}-1} Q_{\hat{\tilde{a}}_r \hat{y}}^{\scriptscriptstyle \mathrm{GF}} \end{aligned}$$
(121)

Setting \(\hat{y}=\hat{\tilde{a}}_r\), the above equation is specialized to

$$\begin{aligned} Q_{\hat{x}\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k] = \frac{1}{k}Q_{\hat{x}\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}} \end{aligned}$$
(122)

This gives the expressions of \(Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]\) and \(Q_{d\hat{\tilde{t}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]\) by setting \(\hat{x}=\hat{\tilde{a}}_{r,{\scriptscriptstyle \mathrm{GF}}}\) and \(\hat{x}=d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}\), respectively. The expression of \(Q_{d\hat{\tilde{t}}_rd\hat{\tilde{t}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]\) follows by setting \(\hat{x}=\hat{y}=d\hat{\tilde{t}}_r\) in (121), together with the identity

$$\begin{aligned} Q_{d\hat{\tilde{t}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}} Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{{\scriptscriptstyle \mathrm{GF}}-1} Q_{\hat{\tilde{a}}_r d\hat{\tilde{t}}_r}^{\scriptscriptstyle \mathrm{GF}}= Q_{d\hat{\tilde{t}}_rd\hat{\tilde{t}}_r}^{\scriptscriptstyle \mathrm{GF}}-c_{\check{\rho }}^2 C_s \end{aligned}$$
(123)

We now prove the geometry-based results. To do so, we first formulate the \([k(m-1)-\nu ](n-1)\) redundant misclosures brought by the geometry-based model (cf. Table 2). The geometry parametrization \(\Delta \tilde{x}_r\) gives \((m-1-\nu )(n-1)\) misclosures as

$$\begin{aligned} t_{\hat{g}_r}\!=\! (D_m^TG) ^{\bot T} [\frac{1}{k}\sum \limits _{i=1}^k (d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}(i)-d\hat{\tilde{t}}_{1,{\scriptscriptstyle \mathrm{GF}}}(i))],\; r\!=\!2,\ldots ,n\nonumber \\ \end{aligned}$$
(124)

while the time-invariance assumption on \(\Delta \tilde{x}_r\) gives \((k-1)(m-1)(n-1)\) misclosures as

$$\begin{aligned} t_{i,r}= [d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}(i)-d\hat{\tilde{t}}_{1,{\scriptscriptstyle \mathrm{GF}}}(i)]-[d\hat{\tilde{t}}_{r,{\scriptscriptstyle \mathrm{GF}}}(1)-d\hat{\tilde{t}}_{1,{\scriptscriptstyle \mathrm{GF}}}(1)]\nonumber \\ \end{aligned}$$
(125)

for \(r=2,\ldots ,n\) and \(i=2,\ldots ,k\).

Since the two misclosure vectors \(t_{\hat{g}}=[t_{\hat{g}_2}^T,\ldots ,t_{\hat{g}_n}^T]^T\) and \(t=[t_{2,2}^T,\ldots ,t_{2,n}^T,\ldots ,t_{k,n}^T]^T\) are uncorrelated, any (co)variance matrix \(Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k]\) is corrected to \(Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GB}}[k]\) as

$$\begin{aligned} Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GB}}[k]&= Q_{\hat{x}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k] -Q_{\hat{x}t_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k]\nonumber \\&-Q_{\hat{x}t}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{tt}^{-1} Q_{t\hat{y}}^{\scriptscriptstyle \mathrm{GF}}[k] \end{aligned}$$
(126)

Accordingly, the expressions of \(Q_{\hat{\tilde{a}}_r\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GB}}[k]\), \(Q_{d\hat{\tilde{t}}\hat{\tilde{a}}_1}^{\scriptscriptstyle \mathrm{GB}}[k]\) and \(Q_{d\hat{\tilde{t}}\hat{\tilde{a}}_1}^{\scriptscriptstyle \mathrm{GB}}[k]\) follow through the identities

$$\begin{aligned} Q_{\hat{\tilde{a}}_rt_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}\hat{\tilde{a}}_r}^{\scriptscriptstyle \mathrm{GF}}[k]&= \frac{1}{k}\frac{n-1}{n}c_{\hat{\rho }}^2 \Lambda ^{-1}e_\mu e_\mu ^T \Lambda ^{-1} \otimes \tilde{C}_s\nonumber \\ Q_{d\hat{\tilde{t}}t_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}\hat{\tilde{a}}_1}^{\scriptscriptstyle \mathrm{GF}}[k]&= \frac{1}{k}\frac{n-1}{n}c_{\hat{\rho }}^2 e_\mu ^T \Lambda ^{-1} \otimes \tilde{C}_s\nonumber \\ Q_{d\hat{\tilde{t}}t_{\hat{g}}}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{t_{\hat{g}}t_{\hat{g}}}^{-1} Q_{t_{\hat{g}}d\hat{\tilde{t}}}^{\scriptscriptstyle \mathrm{GF}}[k]&= \frac{1}{k}\frac{n-1}{n}c_{\hat{\rho }}^2 \otimes \tilde{C}_s \end{aligned}$$
(127)

as well as

$$\begin{aligned} Q_{d\hat{\tilde{t}}t}^{\scriptscriptstyle \mathrm{GF}}[k]Q_{tt}^{-1} Q_{td\hat{\tilde{t}}}^{\scriptscriptstyle \mathrm{GF}}[k] = \frac{k-1}{k}\frac{n-1}{n}c_{\check{\rho }}^2 \otimes C_s \end{aligned}$$
(128)

with \(Q_{\hat{\tilde{a}}_rt}^{\scriptscriptstyle \mathrm{GF}}[k]=0\). \(\square \)

Proof of Tables 4 and 5

The proof goes along the same lines as the proof of Table 3, so it will not be presented here. \(\square \)

The exact of value of \(\gamma \) The exact value of \(\gamma \), introduced in Lemma 5, can be stated as

$$\begin{aligned} \gamma =\frac{(f_1^3-f_2^3)^2}{f_1^2f_2^2(f_1+f_2)^2+\epsilon \,\eta } \end{aligned}$$
(129)

with

$$\begin{aligned} \eta = \left[ \frac{f_1+f_2}{f_1-f_2}\right] ^2 (f_1^2+f_2^2)(f_1^4+f_2^4) \end{aligned}$$
(130)

The approximation, given in Lemma 5, follows by neglecting \(\epsilon \,\eta \), compared to the first term in the denominator of (129). \(\square \)

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Teunissen, P.J.G., Khodabandeh, A. Review and principles of PPP-RTK methods. J Geod 89, 217–240 (2015). https://doi.org/10.1007/s00190-014-0771-3

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