PPP-RTK: from common-view to all-in-view GNSS networks

Abstract

Integer ambiguity resolution-enabled precise point positioning (PPP), otherwise known as PPP real-time kinematic (PPP-RTK), recovers the integer nature of ambiguities at a user receiver by delivering the satellite phase biases (SPBs) estimated from a global navigation satellite system (GNSS) network. Due to the rank-deficiency existing between the satellite and receiver phase biases and the ambiguities, the formulation of PPP-RTK model needs to choose a set of unknown parameters as the datum (or the S-basis). Despite the fact that there are non-unique datum choices, one prefers a PPP-RTK model where the estimable SPBs contain a minimum number of datum ambiguities. We will show that otherwise there will be discontinuities occurring in datum ambiguities that will lead to unfavorable jumps in the estimated SPBs and frequent ambiguity resolution (re-)initialization on the user side. For this to occur one normally restricts to a common-view (CV) network, where the satellites are commonly visible to all receivers involved, and constructs the PPP-RTK model by choosing the phase biases and the ambiguities, pertaining to one receiver, as the datum. In doing so the CV model is capable of estimating the SPBs with each bias containing only one datum ambiguity. In this contribution we extend the CV model to an all-in-view (AV) network case where the satellites tracked can differ across receivers, but at least one satellite is commonly visible; this is practical as the network size is normally consisting of baseline lengths of several hundreds of kilometers. Contrary to the CV model, in the AV model the phase biases and the ambiguities pertaining to one satellite is selected as the datum, such that, the number of datum ambiguities entering into the estimable SPBs is always at the minimum as the SPBs are formulated in a between-satellite single-differenced form. The benefits with AV model are that it relieves the stringent satellite visibility as required by the CV model and, at the same time, reduces to the best possible extent any jumps in the estimated SPBs as well as the necessary ambiguity resolution (re-)initialization on the user side. Experiments conducted using multi-GNSS data collected in both CV and AV networks verify that the AV model always outperforms the CV one, as measured by both the time-to-first-fix as well as the positioning accuracy when compared to very precise benchmark coordinates.

Appendix: Proof of Eqs. (18) and (19)

We consider a network where $n$ receivers ($r = 1, \ldots ,n$) track $m$ satellites ($s = 1, \ldots ,m$), and the receiver $r$ tracks $m_{r}$ satellites. Moreover, we assume that satellite 1 ($s = 1$) is commonly visible to all receivers. For brevity, we only consider the parameters on one frequency and omit the frequency index in the following. We write the vector of phase biases and ambiguities ${\overline{\mathbf{a}}}$ with its design matrix ${\overline{\mathbf{A}}}$ as

$$ \begin{aligned} {\overline{\mathbf{A}}} & = \left[ {\begin{array}{*{20}c} {\mathbf{B}} & {\mathbf{C}} & {\mathbf{D}} \\ \end{array} } \right] \\ {\overline{\mathbf{a}}} & = \left[ {\begin{array}{*{20}c} {\mathbf{Z}} & {{{\varvec{\Delta}}}_{r} } & {{{\varvec{\Delta}}}^{{s}} } \\ \end{array} } \right]^{T} \\ \end{aligned} $$

(A1)

where ${\mathbf{B}} = \mathrm{blkdiag}\left( {\lambda {\mathbf{I}}_{{m_{1} }} ,\lambda {\mathbf{I}}_{{m_{2} }} , \ldots ,\lambda {\mathbf{I}}_{{m_{n} }} } \right)$, ${\mathbf{C}} = \mathrm{blkdiag}\left( {{\mathbf{e}}_{{m_{1} }} ,{\mathbf{e}}_{{m_{2} }} , \ldots ,{\mathbf{e}}_{{m_{n} }} } \right)$ and ${\mathbf{D}} = \left( { - {\mathbf{d}}_{{m_{1} \times m}}^{T} , - {\mathbf{d}}_{{m_{2} \times m}}^{T} , \ldots , - {\mathbf{d}}_{{m_{n} \times m}}^{T} } \right)^{T}$ in which $\mathrm{blkdiag}\left( . \right)$ denotes a block diagonal matrix, ${\mathbf{I}}_{{m_{r} }}$ is an $m_{r}$ identity matrix, ${\mathbf{e}}_{{m_{r} }}$ is a $m_{r} \times 1$ vector of ones, and ${\mathbf{d}}_{{m_{r} \times m}}$ is the coefficient matrix of the satellite phase biases corresponding to the observations of receiver $r$. ${\mathbf{Z}} = \left[ {{\mathbf{z}}_{1} ,{\mathbf{z}}_{2} , \ldots ,{\mathbf{z}}_{n} } \right]$ in which ${\mathbf{z}}_{r} = [z_{r}^{1} ,z_{r}^{2} , \ldots ,z_{r}^{{m_{r} }} ]$ is the ambiguity vector corresponding to receiver $r$. ${{\varvec{\Delta}}}_{r} = \left[ {\overline{\delta }_{1} ,\overline{\delta }_{2} , \ldots ,\overline{\delta }_{n} } \right]$ and ${{\varvec{\Delta}}}^{{s}} = \left[ {\overline{\delta }^{1} ,\overline{\delta }^{2} , \ldots ,\overline{\delta }^{m} } \right]$ are the receiver phase biases and SPBs, respectively.

Since the column vectors of matrix ${\overline{\mathbf{A}}}$ is linearly dependent, the parameters of vector ${\overline{\mathbf{a}}}$ cannot be estimated unbiasedly. In what follows, we employ the S-system theory to identify and eliminate the rank deficiencies in a step-by-step manner. The linear dependence between the vector of receiver phase biases and SPBs is shown by

$$ \sum\limits_{r = 1}^{n} {{\mathbf{c}}_{r} + \sum\limits_{s = 1}^{m} {{\mathbf{d}}^{{s}} } } = 0 $$

(A2)

where ${\mathbf{c}}_{r}$ is the $r_{th}$ column of ${\mathbf{C}}$ and ${\mathbf{d}}^{{s}}$ is the $s_{th}$ column of ${\mathbf{D}}$. Equation (A2) indicates that one rank deficiency arises, for which we choose $\overline{\delta }^{1}$ as the datum to eliminate. Hence, we delete ${\mathbf{d}}^{1}$ in ${\overline{\mathbf{A}}}$ and rewrite the design matrix and parameter vector as

$$ \begin{aligned} {\hat{\mathbf{A}}} & = \left[ {\begin{array}{*{20}c} {\mathbf{B}} & {\mathbf{C}} & {{\hat{\mathbf{D}}}} \\ \end{array} } \right] \\ {\hat{\mathbf{a}}} & = \left[ {\begin{array}{*{20}c} {\mathbf{Z}} & {{\hat{\varvec{\Delta }}}_{r} } & {{\hat{\varvec{\Delta }}}^{{s}} } \end{array} } \right]^{T} \end{aligned} $$

(A3)

where ${\hat{\mathbf{D}}} = \left[ {{\mathbf{d}}^{2} ,{\mathbf{d}}^{3} , \ldots ,{\mathbf{d}}^{m} } \right]$, ${\hat{\varvec{\Delta }}}_{r} = \left[ {\hat{\hat{\delta }}_{1} ,\hat{\hat{\delta }}_{2} , \ldots ,\hat{\hat{\delta }}_{n} } \right]$, and ${\hat{\varvec{\Delta }}}^{{s}} = \left[ {\hat{\hat{\delta }}^{2} ,\hat{\hat{\delta }}^{3} , \ldots ,\hat{\hat{\delta }}^{m} } \right]$, in which

$$ \begin{aligned} \hat{\hat{\delta }}_{r} & = \overline{\delta }_{r} - \overline{\delta }^{1} \\ \hat{\hat{\delta }}^{{s}} & = \overline{\delta }^{{s}} - \overline{\delta }^{1} \quad \left( {s > 1} \right) \\ \end{aligned} $$

(A4)

are the redefined receiver phase bias and SPB.

The linear dependence between-receiver phase biases and ambiguities is indicated by

$$ \sum\limits_{k = 1}^{{m_{r} }} {{\mathbf{b}}_{{r_{k} }} } - \lambda {\mathbf{c}}_{r} = 0\quad \left( {r = 1, \ldots ,n} \right) $$

(A5)

where ${\mathbf{b}}_{{r_{k} }}$ is the $k_{th}$ column vector of ${\mathbf{B}}_{r}$ and ${\mathbf{B}}_{r}$ denotes the coefficient matrix of the ambiguities corresponding to receiver $r$, i.e., $\left[ {{\mathbf{B}}_{1} ,{\mathbf{B}}_{2} , \ldots ,{\mathbf{B}}_{n} } \right] = {\mathbf{B}}$. To eliminate this rank deficiency of size $n$, we choose $z_{r = 1, \ldots ,n}^{1}$ as the datum, which is guaranteed by the fact that all receivers track satellite 1. Hence, we delete the first column vector of ${\mathbf{B}}_{r = 1, \ldots ,n}$ and obtain

$$ \begin{aligned} {{\hat{\hat{A}}}} & = \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{B}}}} & {\mathbf{C}} & {{\hat{\mathbf{D}}}} \\ \end{array} } \right] \\ {{\hat{\hat{a}}}} & = \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{Z}}}} & {{{\tilde{\tilde{\varvec{\Delta }}}}}_{r} } & {{\hat{\varvec{\Delta }}}^{{s}} } \\ \end{array} } \right]^{T} \\ \end{aligned} $$

(A6)

where ${\hat{\mathbf{B}}} = \left[ {{\hat{\mathbf{B}}}_{1} ,{\hat{\mathbf{B}}}_{2} , \ldots ,{\hat{\mathbf{B}}}_{n} } \right]^{T}$, ${\hat{\mathbf{Z}}} = \left[ {\hat{z}_{1} ,\hat{z}_{2} , \ldots ,\hat{z}_{n} } \right]$, and ${{\tilde{\tilde{\varvec{\Delta }}}}}_{r} = \left[ {\tilde{\tilde{\delta }}_{1} ,\tilde{\tilde{\delta }}_{2} , \ldots ,\tilde{\tilde{\delta }}_{n} } \right]$ in which

$$ \begin{aligned} {\hat{\mathbf{B}}}_{r} & = [{\mathbf{b}}_{{r_{2} }} ,{\mathbf{b}}_{{r_{3} }} , \ldots ,{\mathbf{b}}_{{r_{{m_{r} }} }} ] \\ {\hat{\mathbf{z}}}_{r} & = [z_{r}^{2} - z_{r}^{1} ,z_{r}^{3} - z_{r}^{1} , \ldots ,z_{r}^{{m_{r} }} - z_{r}^{1} ] \\ \tilde{\tilde{\delta }}_{r} & = \overline{\delta }_{r} - \overline{\delta }^{1} + \lambda z_{r}^{1} \\ \end{aligned} $$

(A7)

where $\tilde{\tilde{\delta }}_{r}$ is the estimable receiver phase bias and is the same as that in Eq. (19).

Finally, the rank deficiency originating from SPBs and ambiguities is identified by

$$ \sum\limits_{k = 1}^{n} {{\mathbf{b}}_{{k_{s} }} } + \lambda {\mathbf{d}}^{{s}} = 0\quad \left( {s > 1} \right) $$

(A8)

in which the ${\mathbf{b}}_{{k_{s} }}$ is the column vector of ${\hat{\mathbf{B}}}_{k}$ corresponding to satellite $s$. To eliminate these $m - 1$ rank deficiencies, we choose a pivot receiver, denoted as receiver $x\left( s \right)$, for each satellite and choose the corresponding ambiguities $z_{x\left( s \right)}^{{s}}$ ($s > 1$) as the datum, yielding estimable parameters as

$$ \begin{aligned} \tilde{\tilde{\delta }}^{{s}} & = \overline{\delta }^{{s}} - \overline{\delta }^{1} - \lambda \left( {z_{x\left( s \right)}^{{s}} - z_{x\left( s \right)}^{1} } \right)\quad \left( {s > 1} \right) \\ \tilde{\tilde{z}}_{{r}}^{{s}} & = \left( {z_{{r}}^{{s}} - z_{x\left( s \right)}^{{s}} } \right) - \left( {z_{r}^{1} - z_{x\left( s \right)}^{1} } \right)\quad \ge \left( {r \ne x\left( s \right) \, for \, s, \, s > 1} \right) \\ \end{aligned} $$

(A9)

which are identical to the estimable SPBs and ambiguities in Eq. (19).

We remark that although the choice of ambiguity datum is not unique when addressing the rank deficiency between-receiver phase biases and ambiguities as shown by Eq. (A5), only the choice of $z_{r = 1, \ldots ,n}^{1}$ makes Eq. (A8) satisfied for arbitrary AV cases. Otherwise, Eq. (A8) may involve the column vector corresponding to the receiver phase biases, resulting in the estimable SPBs that may contain many ambiguities. This is the reason why the AV model requires at least one CV satellite.