GNSS antenna array-aided CORS ambiguity resolution

Abstract

Array-aided precise point positioning is a measurement concept that uses GNSS data, from multiple antennas in an array of known geometry, to realize improved GNSS parameter estimation proposed by Teunissen (IEEE Trans Signal Process 60:2870–2881, 2012). In this contribution, the benefits of array-aided CORS ambiguity resolution are explored. The mathematical model is formulated to show how the platform-array data can be reduced and how the variance matrix of the between-platform ambiguities can profit from the increased precision of the reduced platform data. The ambiguity resolution performance will be demonstrated for varying scenarios using simulation. We consider single-, dual- and triple-frequency scenarios of geometry-based and geometry-free models for different number of antennas and different standard deviations of the ionosphere-weighted constraints. The performances of both full and partial ambiguity resolution (PAR) are presented for these different scenarios. As the study shows, when full advantage is taken of the array antennas, both full and partial ambiguity resolution can be significantly improved, in some important cases even enabling instantaneous ambiguity resolution. PAR widelaning and its suboptimal character are hereby also illustrated.

Appendices

Appendix

Assuming all matrices and vectors involved have appropriate dimensions, the following properties of the Kronecker product $\otimes$ and vectorization operator vec (Rao 1973):

$$\begin{aligned}&(AB)\otimes (CD)=(A\otimes C)(B\otimes D) \end{aligned}$$

(23)

$$\begin{aligned}&\mathtt{vec }{(ABC)}=(C^T\otimes A)\mathtt{vec }(B) \end{aligned}$$

(24)

and the projector identity (Teunissen 2003)

$$\begin{aligned} QD_r(D_r^TQD_r)^{-1}D_r^T = I_r - e_r(e_r^TQ^{-1}e_r)^{-1}e_r^TQ^{-1} \end{aligned}$$

(25)

with $D_r^Te_r=0$, will be frequently applied in the derivations.

Appendix A: Derivation of some formulae in Sect. 2

To derive (5), we apply the error propagation law to

$$\begin{aligned} \left[ \begin{array}{l} y_1\\ \mathtt{vec }(\tilde{Y}) \end{array}\right] = \mathtt{vec }(YR_r)= (R_r^T\otimes I_{2fs}) \mathtt{vec }(Y) \end{aligned}$$

(26)

where $R_{r}=[c_{1}, D_{r}]$. This gives

$$\begin{aligned} \mathsf{D }\left[ \begin{array}{l} y_1\\ \mathtt{vec }(\tilde{Y}) \end{array}\right]&= (R_r^T\otimes I_{2fs}) \mathsf{D }(\mathtt{vec }(Y)) (R_r\otimes I_{2fs}) \nonumber \\&= (R_r^T\otimes I_{2fs}) (Q_r\otimes Q) (R_r\otimes I_{2fs}) \nonumber \\&= \left[ \begin{array}{l@{\quad }l} c_1^TQ_rc_1 &{} c_1^TQ_rD_r \\ D_r^TQ_rc_1 &{} D_r^TQ_rD_r \end{array}\right] \otimes Q \end{aligned}$$

(27)

To derive the first equation of (8), we apply the invertible transformation (6) to (3). It follows:

$$\begin{aligned}&\Big [\big (1,~-c_1^TQ_rD_r(D_r^TQ_rD_r)^{-1}\big )\otimes I_{2fs}\Big ] \left[ \begin{array}{l} y_1\\ \mathtt{vec }(\tilde{Y}) \end{array}\right] \nonumber \\&= y_1 - (c_1^TQ_rD_r(D_r^TQ_rD_r)^{-1}\otimes I_{2fs})\mathtt{vec }(\tilde{Y}) \nonumber \\&= y_1 - \mathtt{vec }(\tilde{Y}(D_r^TQ_rD_r)^{-1}D_r^TQ_rc_1) \nonumber \\&= y_1 - YD_r(D_r^TQ_rD_r)^{-1}D_r^TQ_rc_1 \nonumber \\&= y_1 - Y \big [I_r - Q_r^{-1}e_r(e_r^TQ_r^{-1}e_r)^{-1}e_r^T \big ]c_1 \nonumber \\&= YQ_r^{-1}e_r(e_r^TQ_r^{-1}e_r)^{-1} \end{aligned}$$

(28)

where the identity (25) was applied. One can easily work out the variance matrix (8) using this identity.

Appendix B: Derivation of variance matrix (16)

For the geometry-free model, we replace $g$ with $I_s$ in (13) and further use the differencing matrix $D_{2f}^T\otimes I_s$ to eliminate troposphere design matrix:

$$\begin{aligned}{}[D_{2f}^T\Gamma \otimes I_s]~~\mathrm{and}~~D_{2f}^T\tilde{Q}D_{2f}\otimes W^{-1} \end{aligned}$$

(29)

with $\tilde{Q}=\frac{1}{r}Q_{f}+\sigma _{\iota }^{2}\nu \nu ^{T}$. This gives the normal matrix of $k$ epochs for ambiguities as

$$\begin{aligned} \big ( \Gamma ^TD_{2f}(D_{2f}^T\tilde{Q}D_{2f})^{-1}D_{2f}^T\Gamma \big )\otimes (k\overline{W}) = \tilde{N} \otimes (k\overline{W})\nonumber \\ \end{aligned}$$

(30)

We now concentrate on the first part $\tilde{N}$ only and use the identity (25) to rewrite it as

$$\begin{aligned} \tilde{N} = \Gamma ^T \tilde{Q}^{-1}\Gamma - \Gamma ^T\tilde{Q}^{-1}e_{2f}(e_{2f}^T\tilde{Q}^{-1}e_{2f})^{-1}e_{2f}^T\tilde{Q}^{-1}\Gamma \nonumber \\ \end{aligned}$$

(31)

Using matrix inversion lemma gives:

$$\begin{aligned} \tilde{N}^{-1} = (\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1} + c_{\hat{\tau }_\mathrm{free}}^2qq^T \end{aligned}$$

(32)

with

$$\begin{aligned} c_{\hat{\tau }_\mathrm{free}}^{-2}&= e_{2f}^T\tilde{Q}^{-1}e_{2f} - e_{2f}^T\tilde{Q}^{-1}(\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}\Gamma ^T\tilde{Q}^{-1}e_{2f} \end{aligned}$$

(33)

$$\begin{aligned} q&= (\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}\Gamma ^T\tilde{Q}^{-1}e_{2f} \end{aligned}$$

(34)

We first work out $c_{\hat{\tau }_\mathrm{free}}^{-2}$. Using the analogous projector identity

$$\begin{aligned} I_{2f} - \Gamma (\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}\Gamma ^T\tilde{Q}^{-1} = \tilde{Q}\Gamma _{\bot }(\Gamma _{\bot }^T\tilde{Q}\Gamma _{\bot })^{-1}\Gamma _{\bot }^T\nonumber \\ \end{aligned}$$

(35)

with $\Gamma _{\bot }=[0,~I_f]^T$, we get

$$\begin{aligned} c_{\hat{\tau }_\mathrm{free}}^{-2}&= e_{2f}^T\Gamma _{\bot }(\Gamma _{\bot }^T\tilde{Q}\Gamma _{\bot })^{-1}\Gamma _{\bot }^Te_{2f}\\&= e_f^T\left( \frac{1}{r}Q_p+\sigma _{\iota }^2\mu \mu ^T\right) ^{-1}e_f \end{aligned}$$

where use is made of $\tilde{Q}=\frac{1}{r}Q_f+\sigma _{\iota }^2\nu \nu ^T$.

Now we work out the expression for $q$. Premultiplying the matrix identity (35) with $(\Gamma ^T\Gamma )^{-1}\Gamma ^T$ gives

$$\begin{aligned}&(\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}\Gamma ^T\tilde{Q}^{-1} \nonumber \\&\quad = (\Gamma ^T\Gamma )^{-1}\Gamma ^T \big (I_{2f}-\tilde{Q}\Gamma _{\bot }(\Gamma _{\bot }^T\tilde{Q}\Gamma _{\bot })^{-1}\Gamma _{\bot }^T \big ) \nonumber \\&\quad = \!\Lambda ^{-1}[I_f,0]\left( I_{2f}-\tilde{Q} \left[ \begin{array}{l} 0\\ I_f \end{array}\right] \left( \frac{1}{r}Q_p+\sigma _{\iota }^2\mu \mu ^T\right) ^{-1}[0,I_f] \right) \nonumber \\ \end{aligned}$$

(36)

Hence

$$\begin{aligned} q&= (\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}\Gamma ^T\tilde{Q}^{-1}e_{2f} \nonumber \\&= \Lambda ^{-1} \left( e_f + \sigma _{\iota }^2\mu \mu ^T\left( \frac{1}{r}Q_p+\sigma _{\iota }^2\mu \mu ^T\right) ^{-1}e_f \right) \end{aligned}$$

(37)

It is not difficult to verify that

$$\begin{aligned} I_f + \sigma _{\iota }^2\mu \mu ^T\left( \frac{1}{r}Q_p+\sigma _{\iota }^2\mu \mu ^T\right) ^{-1} = I_f + \alpha \mu \mu ^TQ_p^{-1} \end{aligned}$$

with $\alpha =[(r\sigma _{\iota }^2)^{-1} + \mu ^TQ_p^{-1}\mu ]^{-1}$. Hence, for $q$ we find

$$\begin{aligned} q = \Lambda ^{-1} (I_f + \alpha \mu \mu ^TQ_p^{-1}) e_f \end{aligned}$$

(38)

It is rather easy to prove:

$$\begin{aligned} (\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}\otimes \frac{1}{k}\overline{W}^{-1}=Q_{\hat{a}\hat{a}}^\mathrm{(fixed)} \end{aligned}$$

(39)

Finally, combining (30), (32) and (39) yields (16).

Appendix C: Derivation of variance matrix (17)

To specify the time variation of troposphere design matrix $g$ and elevation-dependent weight matrix, we assign the epoch index $t$ to $g$ and $W$. The normal matrix of $k$ epochs reads

$$\begin{aligned} \left[ \begin{array}{l@{\quad }l} (\Gamma ^T\tilde{Q}^{-1}\Gamma )\otimes k\overline{W} &{} (\Gamma ^T\tilde{Q}^{-1}e_{2f})\otimes k\overline{W}\bar{g}\\ (e_{2f}^T\tilde{Q}^{-1}\Gamma )\otimes k\bar{g}_t^T\overline{W} &{} c_{\hat{\tau }|a}^{-2}\sum _{t=1}^k g_t^TW_tg_t \end{array} \right] \end{aligned}$$

(40)

with $\overline{W}=\frac{1}{k}\sum _{t=1}^kW_t$, $\bar{g} =\left( \sum _{t=1}^kW_t\right) ^{-1}\sum _{t=1}^k W_tg_t$, $\sum _{t=1}^k W_tg_t= k\overline{W}\bar{g}$ and

$$\begin{aligned} c_{\hat{\tau }|a}^{-2}=e_{2f}^T\tilde{Q}^{-1}e_{2f}= e_{2f}^T \left( \frac{1}{r}Q_{f}+\sigma _{\iota }^{2}\nu \nu ^{T}\right) ^{-1}e_{2f} \end{aligned}$$

(41)

Reducing the ZTD parameter, the normal matrix of ambiguities over $k$ epochs is

$$\begin{aligned} \left( Q_{\hat{a}\hat{a}}^\mathrm{(fixed)}\right) ^{-1}-\frac{(\Gamma ^T\tilde{Q}^{-1}e_{2f}\otimes \overline{W}\bar{g}) (e_{2f}^T\tilde{Q}^{-1}\Gamma \otimes \bar{g}^T\overline{W})}{k^{-2}c_{\hat{\tau }|a}^{-2}\sum _{t=1}^k g_t^TW_tg_t}\nonumber \\ \end{aligned}$$

(42)

Using the matrix inversion lemma, we obtain the variance matrix (43) of ambiguities.

$$\begin{aligned} Q_{\hat{a}\hat{a}} = Q_{\hat{a}\hat{a}}^\mathrm{(fixed)}+ \frac{k^2c_{\hat{\tau }|a}^2 Q_{\hat{a}\hat{a}}^\mathrm{(fixed)}(\Gamma ^T\tilde{Q}^{-1}e_{2f}e_{2f}^T\tilde{Q}^{-1}\Gamma \otimes \overline{W}\bar{g}\bar{g}^T\overline{W}) Q_{\hat{a}\hat{a}}^\mathrm{(fixed)}}{\sum _{t=1}^k g_t^TW_tg_t- (e_{2f}^T\tilde{Q}^{-1}\Gamma \otimes \bar{g}^T\overline{W}) Q_{\hat{a}\hat{a}}^\mathrm{(fixed)} (\Gamma ^T\tilde{Q}^{-1}e_{2f}\otimes \overline{W}\bar{g})k^2c_{\hat{\tau }|a}^2} = Q_{\hat{a}\hat{a}}^\mathrm{(fixed)} + \frac{U}{v} \end{aligned}$$

(43)

Let us now focus on the fraction $U/v$ of the second term only. We first simplify its numerator $U$. Substituting (39) into it yields:

$$\begin{aligned} U&= c_{\hat{\tau }|a}^2 \underbrace{(\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}\Gamma ^T\tilde{Q}^{-1}e_{2f}}_q \nonumber \\&\times \underbrace{e_{2f}^T\tilde{Q}^{-1}\Gamma (\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1}}_{q^T} \otimes \bar{g}\bar{g}^T \nonumber \\&= c_{\hat{\tau }|a}^2 qq^T\otimes \bar{g}\bar{g}^T \end{aligned}$$

(44)

Substituting (39) into the denominator $v$ of fraction yields:

$$\begin{aligned} v&= \sum _{t=1}^k g_t^TW_tg_t - \frac{k}{c_{\hat{\tau }|a}^{-2}} e_{2f}^T\tilde{Q}^{-1}\Gamma (\Gamma ^T\tilde{Q}^{-1}\Gamma )^{-1} \nonumber \\&\times \Gamma ^T\tilde{Q}^{-1}e_{2f}\bar{g}^T\overline{W}\bar{g} \end{aligned}$$

(45)

$$\begin{aligned}&= \sum _{t=1}^k g_t^TW_tg_t - \frac{k}{c_{\hat{\tau }|a}^{-2}}\left( c_{\hat{\tau }|a}^{-2} - c_{\hat{\tau }_\mathrm{free}}^{-2}\right) \bar{g}^T\overline{W}\bar{g} \end{aligned}$$

(46)

where use is made of (33) and (41). Further substituting the identity

$$\begin{aligned} \sum _{t=1}^k(g_t^TW_tg_t)=\sum _{t=1}^k(g_t-\bar{g})^TW_t(g_t-\bar{g})+ k\bar{g}^T\overline{W}\bar{g} \end{aligned}$$

(47)

into (46) gives

$$\begin{aligned} v = \sum _{t=1}^k(g_t-\bar{g})^TW_t(g_t-\bar{g})+ k\frac{c_{\hat{\tau }|a}^2}{c_{\hat{\tau }_\mathrm{free}}^2}\bar{g}^T\overline{W}\bar{g} \end{aligned}$$

(48)

Therefore,

$$\begin{aligned} \frac{U}{v} = \frac{1}{k} c_{\hat{\tau }}^2qq^T\otimes P_{\bar{g}}\overline{W}^{-1} \end{aligned}$$

(49)

where

$$\begin{aligned} c_{\hat{\tau }}^2&= c_{\hat{\tau }_\mathrm{free}}^2 \left[ 1+\frac{c_{\hat{\tau }_\mathrm{free}}^2}{c_{\hat{\tau }|a}^2} \frac{\sum _{t=1}^k(g_t-\bar{g})^TW_t(g_t-\bar{g})}{k\bar{g}^T\overline{W}\bar{g}}\right] ^{-1} \\ P_{\bar{g}}&= \bar{g}(\bar{g}^T\overline{W}\bar{g})^{-1}\bar{g}^T\overline{W} \end{aligned}$$

Finally, inserting (49) into (43) yields (17).