A study on multi-GNSS phase-only positioning

The global navigation satellite systems (GNSS) carrier phase measurements form the basis of high-precision satellite positioning. These measurements are often accompanied by their code counterparts to enable one to compute single-epoch ambiguity-resolved positioning solutions. To avoid unwanted code modelling errors, such as code multipath, one may opt for a phase-only solution and take recourse to carrier phase measurements of two successive epochs. In this paper we study the ambiguity resolution performance of a dual-epoch phase-only model, upon which the unknown positioning parameters are assumed to be completely unlinked in time. With the aid of closed-form analytical results, it is investigated how ambiguity resolution performs when dealing with high-rate phase data. It is thereby shown that multi-GNSS integration makes near real-time centimetre-level phase-only positioning possible. Our analytical analysis is supported by means of numerical results.


Introduction
Global navigation satellite systems (GNSS) have been widely used as an effective means of delivering ubiquitous positioning and navigation services. While the pseudo-range (code) measurements form the basis of GNSS standard positioning, it is the provision of their carrier phase counterparts that leads to ultra precise positioning solutions, see e.g. Blewitt (1989), Remondi (1989), Teunissen (1995), Tiberius and de Jonge (1995), Jonkman et al (2000), Hauschild et al (2008), Gunther and Henkel (2012), and Odolinski et al (2015). This is because the GNSS carrier phase measurements are, approximately, two * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. orders of magnitude more precise than the code measurements. The starring role of carrier phase measurements in GNSS positioning is particularly pronounced when their unknown ambiguities are successfully resolved to their integer values using methods of integer ambiguity resolution (Teunissen 1995, Han 1997, Hassibi and Boyd 1998, Xu et al 2012, Khodabandeh and Teunissen 2018. Once the ambiguities are resolved, the carrier phase measurements will act as ultra precise code measurements, thus making fast and precise positioning and navigation possible. Integer ambiguity resolution is therefore considered to be the key to fast and precise GNSS parameter estimation. Next to GNSS, integer ambiguity resolution has also found a widespread usage in other interferometric techniques, such as very long baseline interferometry (Hobiger et al 2009), interferometric synthetic aperture radar (Kampes and Hanssen 2004), or underwater acoustic carrier phase positioning (Viegas and Cunha 2007).
In real-time positioning applications where only a single epoch of GNSS data is utilised, carrier phase measurements have to be accompanied by code measurements to make the underlying model solvable for the to-be-resolved phase ambiguities. However, if data of multiple epochs are considered, one can exploit the change in the satellite geometry and deliver phase-only solutions (Teunissen 1997a). The main advantage of phase-only positioning over its phase-and-code version is that unwanted code modelling errors, such as code multipath, can be avoided (Remondi andBrown 2000, Codol andMonin 2011). In (ibid), ambiguity resolution was not considered and phase-only models were only used to compute positioning solutions. As a consequence, a rather long observation timespan (∼1 h) is required to obtain centimetre-level solutions. To shorten the stated time-span, Teunissen et al (1997) introduced the principle of phase-only ambiguity resolution. Recent contributions have demonstrated that relatively fast phase-only ambiguity-resolved positioning (∼10-30 s) is indeed feasible (Huisman et al 2010, Wang et al 2018. To realise fast phase-only positioning, Wang et al (2018) studied the ambiguity-resolution performance of GPS triplefrequency for short baselines. It was shown that millimetrelevel positioning can be achieved by using only two successive epochs of phase data. In the aforementioned contributions, phase-only positioning is based on a rather stringent assumption in that the baseline parameters are assumed to behave unchanged in time. It is therefore the goal of the present contribution to address whether such assumption is really needed, particularly, for a multi-GNSS setup where GNSS receivers track a large number of satellites, collecting large amounts of phase data on multiple frequencies. To this end, we study the ambiguity resolution performance of a dual-epoch phase-only model upon which the unknown baselines are treated as fully unlinked-in-time parameters. It is investigated how ambiguity resolution performs when dealing with high-rate multi-GNSS data, e.g. with 5 Hz or 1 Hz sampling-rates.
The remainder of this paper is organised as follows. Section 2 is devoted to the GNSS short-baseline model on which our analysis is based. Closed-form expressions for the variance matrices of the model's parameter solutions are derived and the prominent role of multi-GNSS integration in achieving successful ambiguity-resolution is highlighted. In section 3, we study the phase-only ambiguity resolution performance using the concept of ambiguity dilution of precision (ADOP) (Teunissen 1997a), thereby quantifying the impact the measurement sampling-rate has on the model's ambiguity resolution strength. In section 4, real-world multi-GNSS data-sets, of different sampling-rates, are analysed to provide numerical insights into the positioning performance of the proposed dual-epoch phase-only model. Finally, concluding remarks are provided in section 5.
We make use of the following notation. The n-dimensional space of real numbers is denoted by R n , while the n-dimensional space of integers is denoted by Z n . The identity matrix and vector of ones are denoted by I and e, respectively. E(·) denotes the mathematical expectation operator. The capital Q is reserved for (co)variance matrices, with Q xy being the n × p covariance matrix of two random vectors x ∈ R n and y ∈ R p . Thus Q xx indicates the variance matrix of x. The transpose of a matrix is shown by the superscript T, i.e. (·) T . To express closed-form analytical results in a compact manner, the Kronecker matrix product ⊗ is employed (Henderson et al 1983).

The dual-epoch kinematic phase-only model
In this section we discuss the GNSS short-baseline model on which our analysis is based. The precision of the parameter solutions, obtained by such model, is quantified and the prominent role taken by the multi-GNSS integration in increasing the model's ambiguity success-rate is pointed out.

Double-differenced observation equations
Let ∆ϕ i ∈ R f(m−1) denote the vector of double-differenced (DD) observed-minus-computed carrier phase measurements that are collected by two GNSS receivers tracking m common satellites, on f frequencies, at epoch i (i = 1, 2). The distance between the two receivers is assumed short enough so that the DD atmospheric delays can be neglected. With these in mind, the corresponding linearised observation equations read (Teunissen 1997b, Khodabandeh and Teunissen 2018) where the unknown DD ambiguities a ∈ Z f(m−1) are linked to the measurements through the f × f diagonal matrix Λ = diag(λ 1 , . . . , λ f ), with λ j being the wavelength of the phase data on frequency j (j = 1, …, f ). The 3-vector ∆b i contains the unknown increments of the three-dimensional baseline b i at epoch i. The corresponding (m − 1)× 3 coefficient matrix is indicated by G i = D T m A i and is assumed fullrank, where the m × 3 matrix A i contains the receiver-satellite direction vectors. The m × (m − 1) matrix D m forms betweensatellite differences. The orbital corrections are assumed applied to the observed-minus-computed measurements. In the above equations and in the following, the epoch subscript i is used to emphasise the time dependency of the quantities. In the case of the DD ambiguities a however, the subscript i is omitted. This is because the phase ambiguities are constant in time if no cycle slips occur. It is also important to remark that code data are used to compute the approximate values needed for linearising the observation equations.
Why two epochs: If we stick to only one single epoch of phase data, the system of equations (1) would then be underdetermined as there are f (m − 1) equations in f (m − 1)+3 unknowns, i.e. f (m − 1) ambiguities plus three baseline components. The rank-defect is 3. Assuming that the augmented matrix [G 1 , G 2 ] is of full column rank, at least two epochs of phase data are required to have the model (1) solvable for the unknowns a and ∆b i (i = 1, 2). Under such assumption, the solvability condition of (1) is given by m ≥ 7. (2) Thus phase data of a minimum number of seven satellites is needed. Although phase data of more than two epochs can also be considered, here we confine our study to a two-epoch scenario as it has the potential to realise near real-time phaseonly positioning when dealing with high-rate GNSS data, e.g.
with 5 Hz or 1 Hz sampling-rates. It should be remarked, for a single-GNSS setup and under unfavourable measurement conditions like urban canyons, that the solvability condition (2) may not always hold (Montenbruck et al 2017).
Why kinematic: In the literature, the phase-only model (1) has been studied under the assumption that the baseline parameters are static over time, i.e. b 1 = b 2 , see e.g. (Tiberius and de Jonge 1995, Huisman et al 2010, Wang et al 2018. In the present contribution, we relax this stringent assumption and treat the baselines as fully unlinked-in-time parameters. In other words, no restriction is placed on the temporal behaviour of the two baseline vectors b i (i = 1, 2). The strength of such 'kinematic' model is less than that of its 'static' counterpart in the sense that it has three more unknowns (i.e. the baseline vector of the second epoch). Our goal is therefore to verify whether or not the stated weakness of the kinematic model (1) is of practical relevance for a multi-GNSS landscape where positioning users can track a large number of satellites (∼20), collecting large amounts of phase data on multiple frequencies.

Precision of the baseline solutions
In order to evaluate the variance matrices of the parameter solutions obtained by (1), the stochastic model of the DD phase measurements ∆ϕ i (i = 1, 2) is assumed given as (Khodabandeh and Teunissen 2015a) where W i = (D T m CD m ) −1 , with the m × m diagonal co-factor matrix C = diag(w −1 1 , . . . ,w −1 m ) whose positive elements w s (s = 1, …, m) are the satellite elevation-dependent weights. The zenith-referenced standard deviation of the undifferenced phase measurements is denoted by σ ϕ . The factor '2' in (3) indicates that the variance of the phase measurements is doubled by forming between-receiver differences. According to the second expression of (3), time correlation between the phase measurements ∆ϕ i (i = 1, 2) is assumed to be absent.
We first present the variance matrix of the parameters of interest, i.e. the baseline parameters, for which we distinguish the following two cases: • Ambiguity-float: the case that the DD ambiguities are unknown; • Ambiguity-fixed: the case that the DD ambiguities are successfully resolved as integers.
Lemma 1 (Lemma.). (Phase-only baseline precision) Letb i andb i , respectively, be the least-squares ambiguity-float and -fixed solutions of the baseline parameters b i as given in (1). Their respective variance matrices can be given as

The least-squares inverse of G i is given by G
The proof is given in the appendix.
From the expressions (4), one can infer the precision of the baseline solutions obtained by the phase-only model (1). Let us first consider the second expression in which the DD ambiguities are assumed to be fixed to their integers. The variance matrix Qb ibi is driven by the precision of the phase data σ ϕ , the number of frequencies f and the receiver-satellite geometry through the co-factor matrix (G T i W i G i ) −1 . The latter is identical to the co-factor of the code-based baseline positioning (Teunissen 1997a), meaning that Qb ibi is just a downscaled version of its code-based counterpart. Considering that the precision of code-based short-baseline positioning is at the metre-level and that the phase data are almost two orders of magnitude more precise than the code data, the precision of the phase-only ambiguity-fixed solutionsb i (i = 1, 2) is at the millimetre to centimetre level. Achieving such high-precision solutions relies on the assumption that one successfully maps the unknown ambiguities a to their correct integers, thus requiring an application of ambiguity resolution (Teunissen 1995).
Before applying ambiguity resolution, one should, however, first investigate whether it is worthwhile to do so as the phase-only ambiguity-float solutionsb i may already inherit the high-precision of their ambiguity-fixed versionsb i . In the first expression of (4), the ambiguity-float variance matrix Qb ibi is written as the sum of its ambiguity-fixed counterpart Qb ibi and the positive-definite matrix (2σ 2 ϕ /f)G + i Qρρ G +T i . The latter, in its turn, is driven by the co-factor matrix Qρρ. The inverse of the co-factor matrix Qρρ is formed as a 'weighted sum' of the projectors P ⊥ Gi (i = 1, 2). This implies, when G 2 ≈ G 1 , that the inverse-matrix Q −1 ρρ is near rank defect (near singular). This is because the projectors are singular matrices (Teunissen et al 2005). In such cases, the entries of Qρρ are rather large, indicating that the ambiguity-float solutionsb i are poorly estimable. Unfortunately, the approximation G 2 ≈ G 1 holds true for highrate GNSS data as the receiver-satellite geometries change rather slowly in time. This is due to the high-altitude orbits of the GNSS satellites. The higher the sampling rate, the closer the approximation G 2 ≈ G 1 , thereby the poorer the precision of the solutionsb i becomes.
To address to what extent the precision of the ambiguityfloat baseline solutions is poorer than their fixed versions, we use the concept of gain numbers (Teunissen 1997a). The gain numbers tell us how many times the variance of a function of the baseline solutionb i gets smaller after ambiguityfixing. Since different functions of the baseline can have distinct responses to ambiguity resolution, we employ the average precision gainγ defined as follows  (6), together with the corresponding number of satellites (bottom) for a GPS+Galileo data-set, cf. figure 3. The black solid lines correspond to the elevation weights ws = 1, while the grey dashed lines correspond to the elevation-dependent weights given in (18). in which | · | denotes the determinant of a matrix. The following lemma shows the dependency ofγ on the sampling-rate of the phase measurements.
Lemma 2 (Lemma.). (Average precision gain) Let τ be the measurement sampling period, i.e. the reciprocal of the sampling rate. Also let the m × 3 matrixȦ contain the timederivatives of the entries of (1/2) for small values of τ , the average precision gain of the phaseonly baseline solutions can be approximated bȳ , where the eigenvalues α k and β k are, respectively, the roots of the characteristic equations Proof. The proof is given in appendix.
Thus the average precision againγ is almost inversely proportional to the squared sampling period τ 2 and the positive scalars α and δ. To get some insight into (6), we present timeseries of √ α and √ δ in figure 1. While the scalar √ α is of the order of 10 −4 Hz, the scalar √ δ varies between 0.6 to 1. This implies, for small values of τ , that in which τ is in seconds and the symbol ∝ means 'is of the order of'. The square-root gain √γ indicates how many times the standard deviation of the ambiguity-float baseline solutions is roughly larger than its fixed version. Accordingly, when 1 Hz phase measurements are considered (τ = 1 s), the precision of the phase-only ambiguity-float solutionsb i is expected to be about tens of metres, i.e. 10 4 times millimetres. By increasing the sampling period to 10 s (i.e. τ = 10 s), the stated precision improves to several metres which is still way poorer than the sub-centimetre precision of the phase-only fixed solutionsb i .

Role of the multi-GNSS integration
That the phase-only model (1) delivers poorly estimable ambiguity-float baseline solutions may make one inclined to conclude that the model is not applicable to fast precise positioning. This would indeed be the case if successful ambiguity resolution is not realised. The decision whether or not ambiguity resolution is successful is determined by the probability of correct integer estimation, the so-called ambiguity successrate (Teunissen 1999). The maximum possible success-rate, i.e. the integer least-squares (ILS) success-rate, is governed by the ambiguity variance matrix Qââ. Assuming normallydistributed phase measurements, the ILS success-rate is given by the multivariate integral (ibid) in which the ILS pull-in region S 0 is characterised as with the notation || · || 2 Qââ = (·) T Q −1 aâ (·). According to (9), one only needs the ambiguity variance matrix Qââ as input to evaluate the ILS success-rate. We therefore present a closed-form expression of Qââ for the phase-only model (1).
Lemma 3 (Lemma.). (Phase-only ambiguity precision) Letâ be the least-squares solution of the ambiguity parameters a as given in (1). Then, the ambiguity variance matrix can be given as with W = (1/2) ∑ 2 i=1 W i and the projectors P e = (1/f )ee T and P ⊥ e = I − P e . Proof. The proof is given in appendix.
As with the float baseline variance matrices (4), the ambiguity variance matrix Qââ, in (11), also contains the near rank defect co-factor matrix Qρρ. Does this indicate that successful ambiguity resolution is not feasible? To answer  When a minimum number of satellites (m = 7) is considered, the single-frequency success-rates are almost zero, while the dual-frequency success-rates can be larger than 99% only for the low-rate scenario of τ = 30 s. Upon increasing the number of tracked satellites to m = 20 however, the dual-frequency success-rates can become almost 100% for even high-rate scenarios of τ = 0.2 and 1 s. This highlights the important role played by the number of satellites, and therefore by the multi-GNSS integration, in delivering successful ambiguityresolved solutions using the phase-only model (1). This notion will be made precise in the following section.

ADOP-analysis
So far we have learned that an increase in the number of tracked satellites can lead to almost 100% ambiguity successrate for the dual-epoch phase-only model (1). Interestingly, this even holds true for high-rate phase data. In this section we further study the ambiguity resolution strength of (1) and provide explicit links between the model's ambiguity resolution performance and the measurement sampling rate. To this end, we make use of the ADOP. The ADOP is defined as (Teunissen 1997a) From the ADOP, one can infer an upper bound for the bootstrapped ambiguity success-rate (Teunissen 2000). The smaller the ADOP, the higher the upper bound of the success-rate becomes. For ADOP smaller than 0.14 cycles, the stated upper bound remains always higher than 99%.

Single-epoch RTK ADOP as reference
For the sake of comparison, we take the phase-and-code model of single-epoch real-time kinematic (RTK) as reference to analyse the ADOP of the phase-only short-baseline model (1). The corresponding observation equations follow from (1) by sticking to the first epoch i = 1 and including the DD observedminus-computed code measurements ∆p 1 ∈ R f(m−1) , that is Compare (1) with (13). With the single-epoch model (13), an ambiguity-float baseline solution can be obtained through the second set of the equations, i.e. the code observation equations.
The phase-and-code RTK model (13) has been widely used in the literature and its ambiguity resolution performance has been studied for several multi-GNSS scenarios (Odolinski et al 2015, Brack 2017, Paziewski et al 2018. It is therefore important to address how much our phase-only ADOP differs from its rather well-studied RTK counterpart. Let the positive scalar ε be the phase-to-code variance ratio. The variance matrix of the DD code measurements ∆p 1 follows then by dividing the entries of Q ϕ1ϕ1 , in (3), by ε. The ADOP of the RTK model (13) can be expressed as (Odijk and Teunissen 2008) ) 1 2(m−1) . To provide insight into (14), let us first make some approximation. The term √ 2 w o is bounded from above by 2 for equal elevationdependent weights w s = 1 (s = 1, …, m). The precision of current GNSS phase data is also about 1% of their wavelength. Thus (σ ϕ /λ) ≈ 0.01. Since the code data are almost two orders of magnitude less precise than the phase data (i.e. ε ≈ 10 −4 ), the term (1 + 1/ε) is about 10 4 . We therefore arrive at the following approximation When seven satellites (m = 7) are tracked, the dual-frequency RTK ADOP is about 0.02 × 100 1 4 ≈ 0.06 cycles which is way smaller than 0.14 cycles. This shows, in the absence of code multipath, that single-GNSS, single-epoch ambiguity resolution is possible with the dual-frequency RTK model (13). By switching to the single-frequency case however, the RTK ADOP is about 0.02 × 100 1 2 ≈ 0.20 cycles, when m = 7. The single-frequency ADOP reduces to about 0.06 cycles if the number of satellites increases to m = 13. That is why one has to use multi-GNSS data to achieve successful single-epoch ambiguity resolution with the single-frequency RTK model (Odolinski et al 2015).

Dual-epoch phase-only ADOP compared
We now compare the ADOP corresponding to the phase-only model (1) with that of the RTK model (13). The following two phase-only cases are considered: • Kinematic: the case that the unknown baseline vectors b i (i = 1, 2) are assumed fully unlinked in time.

ADOP KIN
Proof. The proof is given in appendix.
As the first expression of (16) shows, the ADOP and the baseline's average precision gain (5) are closely intertwined. The smaller the average precision gainγ, the smaller the ADOP becomes (Teunissen 1997a). Also note the presence of the factor 1/ √ 2 in the first two expressions. The factor indicates that the use of two epochs of phase data ∆ϕ i (i = 1, 2), in the phase-only model (1), can decrease the corresponding ADOP. We now focus our attention on the second expression, i.e. the ADOP of the static phase-only model. With α k being of the order of 10 −8 Hz 2 (figure 1), the second expression can be further approximated for small values of τ as .
With the approximation √ε /α ≈ 100, the above ADOP-ratio is about 10 for the single-frequency case, with a minimum number of seven satellites, when τ = 1 s. This ratio reduces to about 2.7 for the dual-frequency case and can be further decreased to only 1.1 when the number of satellite increases to m = 20.
The last expression of (16) enables us to quantify the extent to which the ADOP-performance of the phase-only model (1) improves by switching from kinematic to static mode. Recall, from figure 1, that the scalar √ δ varies between 0.6 and 1. Thus the last expression approaches its maximum value 1, the larger the numbers of frequencies f and satellites m. Figure 2 shows ADOPs corresponding to models (1) and (13) as functions of the number of satellites m for different sampling periods τ . For both the single-and dual-frequency cases, the gap between the ADOPs of the phase-only kinematic and static models gets smaller when more than eight satellites are tracked. The phaseonly ADOPs approach their single-epoch RTK versions, the larger the number of satellites. For the low-rate cases (i.e. 10 and 30 s cases), the dual-frequency phase-only ADOPs gets even smaller than their RTK versions when over 15 satellites are tracked. As with the success-rate results in table 1, our ADOP analysis highlights the capability of the dual-epoch phase-only model (1) for delivering high-precision ambiguityresolved positioning solutions.

Results and discussions
To provide numerical insights into the positioning performance of the dual-epoch phase-only model (1), GPSplus-Galileo data-sets of a short-baseline, located in Perth, Australia, are analysed. The baseline configuration is shown in figure 3. The underlying carrier signals considered are on L1(E1) and L5(E5a) frequencies. Further information about the GNSS data-sets are provided in table 2. As shown in the table, we consider two data-sets of different sampling rates, one with the sampling period of τ = 10 s and the other with τ = 1 s. In both cases, the zenith-referenced standarddeviation of the undifferenced phase measurements is set to be σ ϕ = 2 mm (cf. equation (3)). To form the satellite elevationdependent weights w s in (3), a sinusoidal elevation-weighting strategy is applied as where θ s (s = 1, …, m) are the elevation angles of the satellites commonly tracked by the two receivers.
In figure 4 we show the positioning errors corresponding to the 10 s data-set. The ambiguity-float, correctly-fixed and wrongly-fixed baseline solutions are depicted by grey, green and red dots, respectively. When looking from left to right of the figure, it can be observed how an increase in the number of frequencies and satellites reduces the number of wrongly-fixed solutions. It should be remarked, for the dualfrequency Galileo case (left), that the solvability condition (2) does not always hold. This is because, in contrast to GPS, the Galileo constellation is not yet fully deployed. We therefore only consider the time-period in which m ≥ 7. Integrating GPS data, but in a single-frequency mode (middle), the empirical success-rate increases from 69.3% to 92.8%. When the multi-GNSS phase data of the second frequency L5 (E5a) are also used (right), the stated success rate is over 99.9%.
Let us now consider the phase-only positioning performance for the 1 Hz data-set. The corresponding results are shown in figure 5. As predicted by the formal results in table 1, we observe a considerable drop in the ambiguity success-rate upon decreasing the sampling period from τ = 10 s to τ = 1 s. Only 14.7% of the dual-frequency Galileo solutions (left) are correctly fixed. This number increases to 48.5% for the singlefrequency GPS-Galileo case. In case of the dual-frequency multi-GNSS phase data however, all the solutions are correctly fixed, leading to cm-level positioning solutions. This is expected as the corresponding ADOP is well below 0.14 cycles.
To conclude this section, we verify the impact the choice of the sampling period τ has on the baseline precision gain. A comparison between the two figures 4 and 5 (right-panels) reveals that the dispersion of the ambiguity-float solutions gets considerably larger by decreasing the sampling period from τ = 10 s to τ = 1 s. The dispersion of the correctly-fixed solution remains almost unchanged. Using the positioning errors in the East, North and Up directions, we estimate the float and fixed baseline variance matrices Qb ibi and Qb ibi for about 500 samples to numerically evaluate the average precision gainγ in (5). The corresponding square-root values are presented in figure 6. As predicted by the formal result (8), the square-root gain √γ is of the orders of 10 4 and 10 3 for the 10 s (τ = 10 s) and 1 Hz (τ = 1 s) sampling rates, respectively. As illustrated in the bottom-panel of the figure, the corresponding ratio of the two stated square-root gains fluctuates around 10 which is identical to the ratio of their underlying sampling-rates (i.e. 1 Hz over 0.1 Hz).

Concluding remarks and outlook
In this contribution we studied the positioning performance of the dual-epoch phase-only model (1). Closed-form expressions for the variance matrices of the corresponding parameter solutions were derived and the extent to which the baseline precision can be improved by successful ambiguity solution was quantified. In case of 1 Hz GNSS phase measurements, the ambiguity-float baseline precision is about tens of metres. The corresponding standard deviation is inversely proportional to the measurements' sampling rate (cf. (8)). Thus by decreasing the sampling rate to 0.1 Hz (10 s), the stated precision improves to several metres which is still way poorer than the sub-centimetre precision obtained by its fixed counterpart.
In the proposed phase-only model, the unknown baselines of the two involved epochs are treated as fully unlinked-intime parameters. Such 'kinematic' model is therefore weaker than that of its 'static' counterpart in which the baseline is assumed to behave constant in time. It was, both analytically and numerically, demonstrated that such model's weakness is not of practical relevance for a multi-GNSS landscape (cf figure 2). A modest increase in the number of tracked satellites, that can be brought by a multi-GNSS integration, makes the ambiguity-resolution performance of the kinematic phaseonly model almost the same as that of its static version. With the use of multi-GNSS high-rate phase data of two successive epochs, one can therefore deliver near real-time cm-level ambiguity-resolved positioning solutions (figure 5).
With the proven positioning capability of model (1), it is foreseen that the applicability of phase-only positioning will be extended to observable carrier phase signals that are broadcast by low earth orbit satellite communication systems (Khalife et al 2020).
In this study, attention was focused on the precision of the phase-only model's parameter solutions. A study on the testing procedures with which one can ensure the model's integrity against potential modelling errors, such as phase cycle slips Teunissen 2015b, Zaminpardaz andTeunissen 2019), is the topic of future works.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: http://saegnss2. curtin.edu.au/ldc/.
where u 1 = [1, 0] T and u 2 = [0, 1] T . The ambiguity variance matrix (11) follows from the inversion of the reduced normal matrix as (W 1 + W 2 ) = 2W. The fixed baseline variance matrix in (4) is directly obtained by the inversion of N bb , as the ambiguities (and therefore both N aa and N ab ) are absent for the fixed case.
Finally, the float baseline variance matrix in (4) is obtained from the following matrix-inversion identity 2), the inverse of the float and fixed baseline variance matrices follow as . and its transpose gives with 2G = (G 1 + G 2 ) and G 12 = G 2 − G 1 . Substitution of ∑ 2 i=1 G T i WG i = 2G T WG + (1/2)G T 12 WG 12 , G 12 ≈ τĠ and G T 2 WG 2 − G T 1 WG 1 ≈ τ (Ġ T WG + G T WĠ) into (A.8), together with an application of the determinant factorization rule (Koch 1999), gives with X = G T WG + (τ /2) 2ĠT WĠ and Y = (Ġ T WG + G T WĠ). From the definition of the generalized eigenvalues (Teunissen 1997a), as given in (7), follows that . (A.10) In case of GNSS, the entries of X −1 Y are of the order of 10 −4 , meaning that the last term in (A.9) is almost equal to 1. For small values of τ , we have |Qbb|/|Qbb| ≈ |Qb ibi |/|Qb ibi |. Thus the first expression of (6) follows by substituting (A.10) into (A.9), while the second expression follows by neglecting the presence of quantities compared to the terms (1/α k ) (k = 1, 2, 3). (4). The proof follows from the determinant rule (Odijk and Teunissen 2008):

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