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GPS Solutions (2023) 27:12

https://doi.org/10.1007/s10291-022-01341-0

RESEARCH

Two‑epoch centimeter‑level PPP‑RTK withoutexternal atmospheric

corrections using best integer‑equivariant estimation

AndreasBrack1· BenjaminMännel1· HaraldSchuh1,2

Received: 14 July 2022 / Accepted: 27 September 2022

Abstract

Ambiguity resolution enabled precise point positioning (PPP-AR or PPP-RTK) without atmospheric corrections requires the

user to estimate tropospheric and ionospheric delay parameters. The presence of the unconstrained ionosphere parameters

impedes fast and reliable ambiguity resolution, so a time-to-first-fix of around 30min for GPS-only solutions is generally

reported, which can, to some extent, be reduced when combining multiple GNSS. In this contribution, we investigate the

capabilities of almost instantaneous PPP-RTK, using only a few observation epochs at a sampling interval of 30s, with

the ionosphere-float model. The considered key elements are (a) the MSE-optimal best integer-equivariant estimator, (b) a

combination of dual-frequency GPS, Galileo, BDS, and QZSS, (c) an area with good visibility of BDS and QZSS, and (d)

a proper weighting of the PPP-RTK corrections. We provide a formal and simulation-based analysis of kinematic and static

PPP-RTK with perfect, i.e., deterministic, clock and bias corrections as well as corrections computed from only a single

reference station. The results indicate that, on average, one can expect centimeter-level positioning results with just slightly

more than two epochs already with single-station corrections. This is confirmed with real four-system GNSS data, for which

the availability of two-epoch centimeter-level horizontal positioning results is 99.7% during an exemplary day.

Keywords Multi-GNSS· Precise point positioning (PPP)· Integer ambiguity resolution· Best integer-equivariant

estimation· PPP-AR· PPP-RTK

Introduction

For precise point positioning (PPP), a global navigation

satellite systems (GNSS) user requires precise satellite

orbit and clock products. Static PPP is capable of reach-

ing a centimeter-level positioning accuracy but only with a

convergence time of several hours (Zumberge etal. 1997;

Kouba and Héroux 2001; Bisnath and Gao 2008). When

also provided with satellite phase biases, a single-receiver

user can recover the integer property of the carrier phase

ambiguities and reduce the long convergence times through

ambiguity resolution (AR), known as PPP-RTK or PPP-AR

(Wübbena etal. 2005; Laurichesse etal. 2009; Mervart etal.

2008). Further strategies and methods for PPP-RTK have,

for instance, been formulated and demonstrated in Collins

(2008), Ge etal. (2008), Bertiger etal. (2010), Teunissen

etal. (2010), Zhang etal. (2011), or Geng etal. (2012).

An overview and comparison are provided in Teunissen and

Khodabandeh (2015), and the interoperability of different

PPP-RTK corrections is shown in Banville etal. (2020) for

products of the International GNSS Service (IGS).

Successful AR instantly leads to centimeter-level PPP-

RTK results, but fast and reliable AR still remains a chal-

lenge, mainly due to the presence of ionospheric delays. A

time-to-first-fix (TTFF) of about 30min with 1Hz GPS

data is reported in Geng etal. (2011), and a very similar

value in Zhang etal. (2019) with 30-s data. Similar results of

34/22min with 30-s GPS data for kinematic/static PPP-RTK

are obtained in Li and Zhang (2014), which are reduced to

20/16min when integrating GLONASS data. Geng and Shi

(2017) reported convergence times of 25 and 6min with

GPS and GPS + GLONASS using partial AR, and Li etal.

(2018) showed that the convergence time of static PPP-RTK

can be further reduced to around 9min when also including

BDS and Galileo data. In Li etal. (2020), Galileo PPP-RTK

* Andreas Brack

brac[email protected]

1 GFZ German Research Centre forGeosciences,

Telegrafenberg, 14473Potsdam, Germany

2 Chair ofSatellite Geodesy, Technische Universität Berlin,

Str. Des 17. Juni 135, 10623Berlin, Germany

GPS Solutions (2023) 27:12

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with two to five frequencies is analyzed, with resulting con-

vergence times between 22 and 15min with 30-s data. In a

simulation study, Psychas etal. (2020) show that sub-deci-

meter horizontal positioning errors can be expected within

25min for GPS and 6.5 and 4.5min for dual and triple fre-

quency GPS + Galileo + BeiDou with partial AR and deter-

ministic PPP-RTK corrections. Different criteria used in the

above studies when defining the TTFF or the convergence

time make it difficult to compare those numbers, but a gen-

eral trend to faster solutions when combining GNSS constel-

lations or using advanced AR strategies is clearly visible.

Faster centimeter-level positioning solutions can be

obtained when also ionospheric corrections are provided

to the user. However, as demonstrated in Psychas etal.

(2018), these have to be at the level of a few centimeters

to noticeably accelerate reliable AR. This requirement is

not met by current global ionosphere solutions, such as the

total electron content maps provided by the IGS, which have

an uncertainty of several decimeters for GNSS frequencies

(Brack etal. 2021b). Instantaneous PPP-RTK results using

(interpolated) ionospheric corrections from one or more

nearby reference stations are reported in Teunissen etal.

(2010), Banville etal. (2014), and Psychas etal. (2022),

however, at the cost of requiring rather dense local or

regional GNSS networks.

An alternative to fixing the ambiguities to integers is to

use the best integer-equivariant (BIE) ambiguity estimator

(Teunissen 2003). The resulting position estimates are MSE

optimal, and a closed-form expression for normally distrib-

uted data is provided in Teunissen (2003). An extension of

the BIE principle for elliptically contoured distributions

is presented in Teunissen (2020), and a sequential scalar

approximation is proposed in Brack etal. (2014). An evalu-

ation of the BIE estimator based on simulations is given

in Verhagen and Teunissen (2005), and its performance for

multi-GNSS single-baseline RTK positioning is analyzed in

Odolinski and Teunissen (2020).

In Brack etal. (2021a), it is demonstrated that when

combining all available GNSS with a partial AR approach,

one can reach centimeter-level single-baseline RTK results

within 3.3 epochs of 30-s data when ionospheric delays have

to be estimated. A similar positioning performance should

also be feasible with PPP-RTK without atmospheric correc-

tions, which is the topic of this contribution. We will make

use of (a) the MSE-optimal BIE estimator, (b) a combina-

tion of dual-frequency GPS, Galileo, BDS2/3, and QZSS,

a proper weighting of the PPP-RTK corrections. Simulated

GNSS data in the area of Perth, Australia, will be used to

show that with corrections from only a single reference sta-

tion just slightly more than two epochs are required on aver-

age to reach centimeter-level static or kinematic PPP-RTK

results, whereas using deterministic corrections often only

one epoch is sufficient. This result will be confirmed with

real GNSS data, where centimeter-level horizontal position

estimates are obtained after two epochs with an availability

of 99.7%, thus demonstrating that almost instantaneous PPP-

RTK is feasible with the current GNSS constellations, even

without any atmospheric corrections.

PPP‑RTK observation model

The single-system undifferenced, uncombined code and car-

rier phase observations

r,f

and

𝜑s

r,f

between receiver r and

satellite s on frequency f can be modeled as

with

⋅

]

being the expectation operator,

𝜌s

the geometric

range between satellite s and receiver r,

dtr

and

dts

the

receiver and satellite clock offsets,

𝜏r

the residual zenith

tropospheric delay (ZTD) at receiver r with the mapping

function

the first-order slant ionospheric delay on the

first frequency with the coefficient

𝜇

𝜆

depending on the

wavelengths

𝜆f

dr,f

and

the receiver and satellite code

biases,

𝛿r,f

and

𝛿s

the respective phase biases, and

r,f

the

integer phase ambiguity.

When processing the data of a GNSS receiver or network,

not all parameters in (1) can be unbiasedly estimated due to

rank deficiencies in the underlying system model. Estima-

ble combinations of the parameters resulting in a full-rank

model can be determined using S-system theory (Baarda

1973; Teunissen 1985) by constraining a minimum set of

parameters.

We assume that precise orbital positions are available. We

further assume that for the reference stations providing the

PPP-RTK corrections, the coordinates are a priori known,

so that the ranges

𝜌s

can be removed from the model. A set

of estimable parameters together with their definitions is

given in Table1, where the first receiver and satellite are

chosen as pivot (cf. Odijk etal. 2016). In the case of a local

network, the tropospheric mapping functions are (almost)

identical so that the ZTDs can only be estimated relative to

the pivot receiver, whose tropospheric slant delays are then

included in the satellite clock corrections. This also holds if

only one reference receiver is used. The PPP-RTK correc-

tions provided to the user contain the estimates of the satel-

lite clocks

d

, the satellite phase biases



𝛿

, and the satellite

code biases



for

f>2

On the user side, the observed-minus-computed obser-

vation equations follow from (1) by replacing the ranges

𝜌s

with

gs,T

Δx

, where

is the satellite-to-receiver unit vector,

(1)

E[ps

r,f

]=𝜌s

+dt

−dts+ms

𝜏

+𝜇

r,f

−ds

[𝜑s

r,f]=𝜌s

r+dtr−dts+ms

r𝜏r−𝜇fis

r+𝜆f

(

𝛿r,f−𝛿s

,f+as

r,f

)

GPS Solutions (2023) 27:12

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and

Δxr

is the unknown increment of the user position from

the initial value. The PPP-RTK corrections are applied to

the user observations, and the corresponding parameters are

removed. Other than that, the user parameters are the same

as defined in Table1 for

r≠1

For the multi-GNSS model, we assume that all param-

eters except the user position

Δxr

and the ZTDs

𝜏r

𝜏1r

are set up per constellation. This implies that the biases are

assumed constellation specific. A calibration of inter-system

biases on overlapping frequencies that would enable the use

of a common pivot satellite is not considered (Odijk and

Teunissen 2013; Paziewski and Wielgosz 2015). With the

satellite phase biases applied, the user can recover the inte-

ger estimable double-difference ambiguities with the pivot

receiver and use AR techniques.

Solving thePPP‑RTK user positioning model

Let the integer ambiguity parameters of the PPP-RTK user

model be collected in the vector

a∈ℤn

and all real-valued

parameters, including the position increment

Δxr

, in

b∈ℝp

We consider three estimators in this contribution, which are

briefly introduced in the following.

The first one is the ambiguity float estimator



, for which

the integer property of the ambiguities is disregarded. For

short observation time spans, its precision is driven by the

code data. The high precision of the carrier phases only

starts to have an impact when multiple epochs with time-

constant ambiguities are combined.

The second estimator is the ambiguity fixed estimator



for which all ambiguity parameters are fixed to integers. The

integer least-squares (ILS) ambiguity estimator is given by

(2)



=arg min

z∈

ℤn

‖



a−z

‖2

Q

with



the float ambiguity solution and

Q

its covariance

matrix. It is optimal in the sense of maximizing the probabil-

ity of correct ambiguity estimates (Teunissen 1999) and is

efficiently implemented in the LAMBDA method (Teunissen

1995). If the ambiguities are resolved correctly, the position-

ing precision is immediately driven by the carrier phases, but

wrong ambiguity estimates can lead to large positioning errors,

so one will usually prefer the float solution



if the ambiguity

success rate is too low.

The third estimator is the BIE estimator

(Teunissen 2003).

For normally distributed data, the BIE ambiguity estimates are

a weighted sum of integers

and

is the conditional least-squares estimator assuming

the ambiguities given by

. For computational reasons,

the infinite set

ℤn

in (3) has to be replaced by a finite set,

which is chosen as the set of integers within an ellipsoidal

region around



. Its radius is defined in the metric of

Q

and is derived from a central Chi-squared distribution with

degrees of freedom and a significance level of

𝛼=10−5

see Teunissen (2005). The BIE solution

is MSE-optimal

and unbiased within the class of integer-equivariant esti-

mators, which also contains the float and fixed solutions



and



(Teunissen 2003). As the variances of the BIE esti-

mates

are equal or smaller than those of the float and any

admissible fixed or partially fixed solution, they can serve

as a benchmark for analyzing the best possible performance

of a GNSS model. This is done in the following in order

to evaluate the limits of the convergence time of the PPP-

RTK model that can be achieved with the current GNSS

constellations.

(3)

=�

z∈ℤn

exp

�

−1

2‖

a−z‖2

Q

�

∑

u∈ℤnexp

�

−1

‖



a−u

‖

Q

�

Table 1 Estimable parameter

combinations with single-

system undifferenced and

uncombined code and carrier

phase observations; the receiver

and satellite with index

are

chosen as pivot;

(

⋅

)1r=(

⋅

)r−(

⋅

;

(

⋅)GF =

𝜇2−𝜇1

[−(⋅),1 +(⋅),2

]

;

(

⋅)IF =

𝜇2−𝜇1[

𝜇2(⋅),1 −𝜇1(⋅),2

]

;

the entries marked as

(

⋅

)∗

are

only relevant in a local network

Parameters Definition Condition

Receiver clocks

d

dt1r+d1r,IF

r≠1

Satellite clocks

d

s+ds

−dt1−d1,

−

(

𝜏1

)∗

ZTDs

𝜏 r

𝜏r

(

𝜏1

r)∗

(r≠1)∗

Ionospheric delays



r+dr,GF −ds

Rec. code biases



f−

IF −𝜇f

r≠1, f>2

Sat. code biases



,f−ds

IF −𝜇

GF −d

1,f

1,IF

+𝜇

f>2

Rec. phase biases



𝛿r

𝛿

1r,f−(d1r,IF −𝜇fd1r,GF)∕𝜆f+a

r≠1

Sat. phase biases



𝛿s

,f−(ds

IF −𝜇

GF −d

1,IF

+𝜇

1,GF

)∕𝜆

−𝛿

1,f

−as

1,f

Integer ambiguities

as

r,f

−a

r≠1, s≠1

GPS Solutions (2023) 27:12

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Experimental setup

The multi-GNSS PPP-RTK positioning capabilities of com-

bined GPS (G), Galileo (E), BDS 2 + 3 (C), and QZSS (J)

are analyzed with the ambiguity float, ILS and BIE solu-

tions. The station PERT (Trimble Alloy) in Perth, Australia,

acts as the user receiver, and the corrections are provided

by the stations CUT0, CUTA, CUTB, and CUTC (all Trim-

ble NetR9, distance to PERT 22.4km) forming a local

receiver array or by the individual station NNOR (Septen-

trio PolaRx5TR, distance to PERT 88.5km). The number

of visible satellites with an elevation angle greater than 10°

during April 1, 2022, is shown in Fig.1.

The observations are weighted according to exponen-

tial noise amplification factors as defined in Euler and

Goad (1991). The zenith-referenced standard deviations

of the considered dual-frequency code observations are

estimated from a different day of double-differenced code

data from the local CUT* receiver array in a least-squares

sense (Teunissen and Amiri-Simkooei 2008) and given in

Table2. For the tropospheric modeling, we use the global

mapping function (Boehm etal. 2006) and the a priori cor-

rections from the blind MOPS model (MOPS, 1999). The

GFZ precise multi-GNSS orbit products are applied (Deng

etal. 2017).

The PPP-RTK strategy used in this contribution is imple-

mented as follows: The above-mentioned PPP-RTK correc-

tions are computed on an epoch-by-epoch basis from either

a single reference receiver (CUT0 or NNOR) or from the

local CUT* receiver array. AR techniques are not applied

on the network side. In the simulations, also perfect, i.e.,

deterministic, corrections are considered. The user station

PERT then applies these corrections together with their

uncertainty as given by their covariance matrix. Neglect-

ing the uncertainty of the corrections can strongly degrade

the position performance (Psychas etal. 2022). No delay is

assumed between the generation of the PPP-RTK correc-

tions and the user positioning. The float solution is computed

with a recursive least-squares implementation, and the ILS

and BIE ambiguity solutions



and

are computed anew

in each epoch. We consider both static and kinematic posi-

tioning, where for the latter, the coordinates are assumed to

be completely unlinked in time. The user ambiguities and

receiver phase biases are assumed time-constant, and the

ZTD is modeled as a random walk with a process noise of

mm∕

√h

. All other parameters including the ionospheric

delays are assumed unlinked in time, so that the results do

not depend on the ionospheric activity.

Formal andsimulation analysis

The average ambiguity float positioning precision against the

number of epochs is shown in Fig.2 for the east component,

where the estimation is started every 10min during April 1,

2022. We can clearly see the benefit of combining multiple

systems, so that, for instance, the kinematic precision with

single-station corrections reaches 30cm after around 15

epochs for GPS, 6 epochs for GPS + Galileo, and 4 epochs

when also including BDS and QZSS. The static counterparts

0612 18 24

Time [h]

Number of satellites

Comb.

Fig. 1 Number of visible GNSS satellites at the station PERT during

April 1, 2022

Table 2 Considered signals with the estimated zenith-referenced

standard deviations of the code observations in [cm]. The zenith-ref-

erenced carrier phase standard deviations are assumed as 2mm

GPS Galileo BDS QZSS

Signal L1/L2 E1/E5a B1/B3 L1/L2

St. dev. [cm] 36/19 23/19 40/18 42/21

0.2

0.4

0.6

Prec. east [m]

Kinematic positioning

5101

Epochs [30 sec]

0.2

0.4

0.6

Prec. east [m]

Static positioning

G+E

G+E+C+J

Fig. 2 Average ambiguity float kinematic and static PPP-RTK posi-

tioning precision of the east component with single-epoch, single-

station corrections (solid lines) and with deterministic corrections

(dashed lines)

GPS Solutions (2023) 27:12

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converge slightly faster. Khodabandeh and Teunissen (2015)

demonstrate that PPP-RTK with single-station corrections

is equivalent to single-baseline RTK positioning. Applying

the theoretically assumed perfect corrections instead of the

single-station corrections translates to having no measure-

ment noise at the reference receiver for single-baseline RTK

positioning. Assuming a similar noise level at both receiv-

ers, the uncertainty of the PPP-RTK user observations with

the corrections applied is then essentially reduced to half,

so that the same precision values are obtained faster. When

extending the single reference station to an array or network

of receivers, the user positioning precision will be between

these two shown extremes. In all presented cases, however,

a centimeter-level precision within at most a few epochs can-

not be expected.

In comparison, the ambiguity fixed precision values in

the first epoch are already between 9mm for the GPS-only

case with single-station corrections and 3mm for the four-

system case with perfect corrections. These values are the

conditional standard deviations assuming the ambiguities

known and do not reflect the uncertainty of the ambiguity

estimators, but rather the highest precision that can possibly

be reached with any ambiguity estimator. A formal measure

of the strength of the GNSS model for ambiguity resolution

is the ambiguity dilution of precision (ADOP, Teunissen

1997). The average ADOP values of the above PPP-RTK

examples are shown in Fig.3. Again, we see an improve-

ment when combining systems or increasing the quality of

the corrections. Odijk and Teunissen (2008) found that an

ADOP ≤0.12

generally allows for reliable ILS ambiguity

resolution with a failure rate of less than 0.1%. This seems

promising for the four-system cases with perfect corrections,

but when we look at the actual average TTFF with ILS and

a failure rate constraint of 0.1% in Table3 (the integer boot-

strapping failure rate as a tight upper bound is used, see

Verhagen etal. 2013), we see that on average still around

15 epochs are required in this case, which is far from the

envisioned almost instantaneous solutions. The increase of

the convergence time of the four system solution compared

to GPS + Galileo is attributed to more frequently rising sat-

ellites. As the BIE estimator does not fix the ambiguities

to integers, the concept of a success rate does not apply. In

the following, it is investigated how this increased model

strength as measured by the ADOP translates into high posi-

tioning precision with the BIE estimator.

The simulated horizontal positioning errors of two kine-

matic GPS + Galileo PPP-RTK examples with single-station,

single-epoch corrections with five and seven observation

epochs and 10,000 samples are shown in Fig.4. We can see

the basic properties of the involved estimators: The float

solution (gray) is at the several decimeter level, correctly

fixed ILS solutions (green) are at the sub-centimeter level,

whereas incorrectly fixed ILS solutions (red) can have large

errors (no fixing criterion in form of an acceptance test of

the ILS ambiguity solution is applied). These are avoided to

0.1

0.2

0.3

0.4

ADOP [cycle]

Kinematic positioning

5101

Epochs [30 sec]

0.1

0.2

0.3

0.4

ADOP [cycle]

Static positioning

G+E

G+E+C+J

Fig. 3 Average ADOP values for kinematic and static PPP-RTK posi-

tioning with single-epoch, single-station corrections (solid lines) and

with deterministic corrections (dashed lines)

Table 3 Average TTFF in epochs for kinematic and static PPP-RTK

with corrections from a single station or perfect corrections. The fix-

ing criterion is an integer bootstrapping failure rate of 0.1% or lower.

One epoch corresponds to 30s

Single-station corr Perfect corr

kinematic static kinematic static

G 47.8 35.6 31.2 23.9

G + E 27.2 26.5 15.2 14.9

G + E + C + J 34.6 34.5 15.5 15.5

-1 01

East error [m]

-1

North error [m]

5 epochs

-1

East error [m]

-1

7 epochs

Fig. 4 Simulated horizontal positioning errors for kinematic

GPS + Galileo PPP-RTK examples with single-epoch, single-station

corrections. The float solution is shown in gray, the ILS solution in

green/red for correct/incorrect ambiguity estimates, and the BIE solu-

tion in blue

GPS Solutions (2023) 27:12

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12 Page 6 of 11

some extent with the BIE solutions (blue), which are more

concentrated around the true position. The ILS success rates

of the two examples are 69.6% and 81.4%, and the 3D RMS

errors of the float/ILS/BIE solutions are 88.4/25.8/22.2cm

and 57.8/4.5/4.0cm, showing the RMS error optimality of

the BIE estimator. An exemplary cumulative distribution

of the 3D positioning error for the same setup with three

observation epochs is shown in Fig.5; see also Verhagen and

Teunissen (2005), Brack (2019), or Odolinski and Teunis-

sen (2020) for similar results. While the BIE solution has

the smallest RMS error, the ILS solution has a much higher

probability of very small positioning errors but is also more

likely to result in large errors. Interestingly, the ILS success

rate of this example is 26.6%, implying that correct ambigu-

ity estimates



are not necessarily required for centimeter-

level errors, which occur at about 50%. Finally, the 3D RMS

positioning errors of the above example relative to the float

solution against the number of epochs are shown in Fig.6;

see also Brack (2019) or Odolinski and Teunissen (2020).

The fixed solution is worse than the float solution in the first

epochs due to the very low ILS success rate and eventually,

with increasing success rate, converges to the dashed line,

for which the ambiguities are assumed known. The BIE esti-

mator always has the minimum RMS error. Its results are

close to the float solution for very low ambiguity precision

in the first epochs and close to the fixed solution for high

ambiguity precision, which was proven in Teunissen (2003).

The capabilities of the three estimators for kinematic

and static PPP-RTK are analyzed by means of the average

simulated RMS positioning error, shown for the east com-

ponent in Fig.7, and by means of the average probability

of obtaining an absolute positioning error of less than 3cm

for the horizontal components and 15cm for the vertical

component in Fig.8. Again, solid and dashed lines represent

the setup with single-station and deterministic corrections,

respectively. The float results confirm the formal analysis in

Fig.2, i.e., fast centimeter-level results cannot be expected.

The BIE results always lead to the smallest RMS positioning

errors. For strong positioning models in which centimeter-

level results are actually possible, the RMS errors with ILS

are very close to the BIE results. Almost instantaneous cen-

timeter-level results are only obtained with the four-system

combination, where the RMS error of the BIE estimator with

012345

3D positioning error [m]

0.5

Cumulative distributio

Float

ILS

BIE

Fig. 5 Simulated cumulative distribution of 3D positioning errors for

a kinematic GPS + Galileo PPP-RTK example with single-epoch, sin-

gle-station corrections and three observation epochs

246810 12 14

Epochs [30 sec]

0.2

0.4

0.6

0.8

Ratio of 3D RMS pos. err.

0.2

0.4

0.6

0.8

ILS success rate

Float

ILS

BIE

Fig. 6 Ratios of simulated 3D RMS positioning errors relative to the

float solution. The dashed line marks the theoretically minimal values

assuming the ambiguities known. The ILS success rate is shown in

black

Fig.7 Average simulated RMS

positioning error of the east

component for kinematic and

static PPP-RTK with single-

epoch, single-station corrections

(solid lines) and with determin-

istic corrections (dashed lines)

0.2

0.4

RMS east pos. error [m]

Kinematic, G Kinematic, G+E+C+J

Float

ILS

BIE

5101520

Epochs [30 sec]

0.2

0.4

Static, G

Kinematic, G+E

5101520

Epochs [30 sec]

Static, G+E

5101

520

Epochs [30 sec]

Static, G+E+C+J

GPS Solutions (2023) 27:12

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Page 7 of 11 12

single-station corrections is at the sub-decimeter level using

two epochs and the sub-centimeter level using three epochs.

With deterministic corrections, about one fewer epoch is

required. This result is confirmed by the complementary

availability values in Fig.8, showing a very high availabil-

ity of precise positioning solutions with two to three epochs

with single-station corrections and one to two epochs with

perfect corrections. This figure also demonstrates again that

the ILS solution can have a higher probability of very pre-

cise solutions than the BIE solution, cf. Figure5. That is,

although the RMS errors of the BIE and ILS estimators are

very similar, their characteristics are still quite different.

Real‑data analysis

Judging from the simulation results in the previous section,

we can expect centimeter-level four system PPP-RTK results

in generally not more than three epochs, quite often with

one or two epochs, depending on the quality of the correc-

tions. This is now verified with real GNSS data. We focus

on the kinematic case, but a big benefit of the static case can

anyway not be expected when only a few epochs are used.

Positioning solutions are initialized at every 30s observa-

tion epoch.

With real data, it is obviously not possible to generate

perfect PPP-RTK corrections that can be assumed determin-

istic. To demonstrate the improvements in the positioning

capabilities with corrections of higher quality, we compare

the results using single-station corrections from CUT0 and

corrections from the four-station array CUT0/A/B/C. The

same set of satellites is available in both cases. The result-

ing empirical RMS positioning errors of the three estimators

are given in Table4 for different numbers of epochs. The

optimality of the BIE estimator is also confirmed with the

real data, where for five and ten epochs, the fixed solution is

essentially equally good in terms of the RMS error. Already

by using a local array to compute the corrections instead of

a single station, the PPP-RTK performance can be consider-

ably improved, with an absolute improvement of around one

meter for the vertical components in the first epoch and a

relative improvement of the BIE RMS errors between 7.5%

and 77.3%. The obtained empirical RMS positioning errors

cannot keep up with the simulations, which promised sub-

centimeter values already for three epochs. The reason is

that the NetR9 receivers at the CUT* stations do not track

Fig. 8 Average simulated avail-

ability of precise kinematic and

static PPP-RTK results with

single-epoch, single-station

corrections (solid lines) and

with deterministic correc-

tions (dashed lines). Precise is

defined as an error of less than

3cm for the horizontal compo-

nents and 15cm for the vertical

component

0.5

Kinematic, G

Availability of precise position sol.

Kinematic, G+E+C+J

Float

ILS

BIE

Kinematic, G+E

5101520

Epochs [30 sec]

0.5

Static, G

5101520

Epochs [30 sec]

Static, G+E

5101

520

Epochs [30 sec]

Static, G+E+C+J

Table 4 Empirical GPS + Galileo + BDS + QZSS east/north/up RMS

positioning errors of the station PERT in [cm] with corrections from

CUT0 and the array formed by CUT0/A/B/C. The lowest values are

marked in bold. BDS satellites with a PRN larger than 30 are not

included (see text)

GPS Solutions (2023) 27:12

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12 Page 8 of 11

BDS satellites with a PRN larger than 30, so the number of

visible satellites is clearly reduced.

In a second example, we therefore make use of the station

NNOR tracking all BDS satellites to compute the PPP-RTK

corrections. The empirical RMS positioning errors are given

in Table5. The horizontal RMS errors using the optimal BIE

estimator are at the decimeter level with one observation

epoch, at the centimeter level with two observation epochs,

and at the sub-centimeter level with three or more epochs,

confirming that two-epoch centimeter-level PPP-RTK with-

out atmospheric corrections is possible.

The positioning errors of the east, north, and up com-

ponent with one and two observation epochs are shown in

Fig.9. The corresponding empirical rates of a positioning

error of less than 3cm for the horizontal components and

15cm for the up component are given in Table6, where for

the values in the parenthesis only the horizontal components

are considered. With one epoch, the horizontal positioning

error of the ILS (red) and BIE (blue) solutions is less than

3cm in 97.6% and 87.6% of the cases. With two epochs,

only six and nine of the 2,879 epochs exceed these limits,

and with three epochs, this limit is always met with ILS

and exceeded twice with the BIE estimator. The results in

Table6 confirm the simulation results in the previous sec-

tion: While the BIE solutions are RMS optimal, see Tables4

and 5, ILS often leads to a higher availability of very small

positioning errors.

Conclusions

The user performance of PPP-RTK with the current GNSS

constellations and without atmospheric corrections was

analyzed. In the literature, often convergence times of sev-

eral tens of minutes are reported for this model in order to

reach centimeter-level positioning results. This contribution

focused on the feasibility of almost instantaneous precise

solutions. The main conclusions can be summarized as

follows:

The BIE estimator provides MSE optimal positioning

results within the class of integer-equivariant estimators

and is in this sense superior to the ambiguity float and any

admissible ambiguity fixed solution. Its results can therefore

be seen as benchmark results to evaluate the highest achiev-

able performance of a given GNSS model. To this end, a

simulation analysis of kinematic and static PPP-RTK with

perfect corrections and single-station corrections was con-

ducted. It showed that a high availability of almost instanta-

neous centimeter-level results with one to three observation

epochs can only be expected when combining all available

Table 5 Empirical GPS + Galileo + BDS + QZSS east/north/up RMS positioning errors of the station PERT in [cm] with corrections from the

station NNOR. The lowest values are marked in bold

1 epoch 2 epochs 3 epochs 4 epochs

Float 59.2/62.0/386.8 50.3/48.0/172.1 45.6/38.6/115.2 42.2/31.4/87.3

NNOR ILS 14.6/14.5/97.7 3.7/1.8/14.2 0.4/0.6/6.3 0.4/0.6/6.1

BIE 11.2/11.4/83.6 2.2/1.4/8.5 0.6/0.6/5.8 0.4/0.6/5.8

0612 18 24

-2

East error [m]

1 epoch

0612 18 24

-2

North error [m]

0612 18 24

Time [h]

-20

Up error [m]

0612 18 24

2 epochs

0612 18 24

Time [h]

-0.05

0.05

-0.05

0.05

-0.5

0.5

Fig. 9 East, north, and up GPS + Galileo + BDS + QZSS positioning

errors of the station PERT with single-station PPP-RTK corrections

from the station NNOR. The float solution is shown in gray, the ILS

solution in red, and the BIE solution in blue

Table 6 Empirical rates for GPS + Galileo + BDS + QZSS positioning

errors of the station PERT with corrections from the station NNOR

of less than 3/3/15cm for the east/north/up components in [%]. The

values in parenthesis result from only the horizontal components. The

highest values are marked in bold

1 epoch 2 epochs 3 epochs

Float 0.0 (0.2) 0.0 (0.2) 0.0 (0.5)

ILS 91.6 (97.6) 95.7 (99.8) 96.7 (100)

BIE 83.2 (87.6) 97.5 (99.7) 97.8 (99.9)

GPS Solutions (2023) 27:12

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Page 9 of 11 12

systems, namely GPS, Galileo, BDS, and QZSS, but is gen-

erally possible.

This result was confirmed with real data recorded at

the station PERT, with corrections provided by the sta-

tion CUT0, a local array of four receivers around CUT0,

or the station NNOR. Although perfect PPP-RTK correc-

tions can, of course, not be provided, the results showed how

the performance of the PPP-RTK user model could already

be improved when using a local array of receivers instead

of a single receiver, thereby improving the empirical BIE

RMS position errors by 7.5% to 77.3%. Using PPP-RTK

corrections from the station NNOR, horizontal positioning

errors of less than 3cm are obtained with the BIE estimator

in 87.6% of the cases with one observation epoch, and in

99.7% of the cases with two observation epochs. Two-epoch

centimeter-level horizontal PPP-RTK results without atmos-

pheric corrections are, therefore, indeed feasible. While this

precision cannot be reached for the vertical component, its

empirical two-epoch RMS error using the BIE estimator was

reduced to 8.5cm as compared to 14.2cm with ILS.

It has to be stressed, however, that with the current con-

stellations such short observation spans require a good sat-

ellite visibility. For instance, BDS satellites with a PRN

larger than 30 are not tracked by the CUT* stations. As a

consequence of removing these satellites, centimeter-level

RMS positioning errors with two epochs were no longer pos-

sible, see Table4. In view of the developments of the GNSS

constellations in the previous years, however, the number

of available GNSS satellites can be expected to increase

further in future so that similar results will also be possible

on a global scale. Further improvements toward instantane-

ous results can be expected when observations on additional

frequencies are included. Psychas etal. (2021) concluded

that the frequency separation is more important than the

number of frequencies in terms of the ambiguity resolution

performance, so that, for instance, adding E6 observations

to a Galileo E1/E5a solution is more beneficial than adding

both E5b and E5. As the computational burden of the BIE

estimator increases with the ambiguity dimension, this result

can prove very beneficial for extending the presented study

to multi-GNSS solutions with three or more frequencies.

Author contributions The study was designed, and the data analysis

was performed by AB. The manuscript was written by AB. All authors

contributed to the discussion about the content and provided comments

on the manuscript.

Funding Open Access funding enabled and organized by Projekt

DEAL.

Data availability RINEX data of CUT0/A/B/C provided by the GNSS-

SPAN Group at Curtin University, Perth, Australia, are available at

http:// saegn ss2. curtin. edu/ ldc/; RINEX data of the stations PERT and

NNOR provided by the IGS (Johnston etal. 2017) are available at

https:// cddis. nasa. go v/ ar chi ve/ gnss/ data/; GFZ multi-GNSS orbit prod-

ucts (Deng etal. 2017) are available at ftp:// ftp. gfz- potsd am. de/ GNSS/

produ cts/ mgex/. This support is gratefully acknowledged.

Declarations

Conflict of interest The authors have no relevant financial or non-fi-

nancial interests to disclose.

Open Access This article is licensed under a Creative Commons Attri-

bution 4.0 International License, which permits use, sharing, adapta-

tion, distribution and reproduction in any medium or format, as long

as you give appropriate credit to the original author(s) and the source,

provide a link to the Creative Commons licence, and indicate if changes

were made. The images or other third party material in this article are

included in the article's Creative Commons licence, unless indicated

otherwise in a credit line to the material. If material is not included in

the article's Creative Commons licence and your intended use is not

permitted by statutory regulation or exceeds the permitted use, you will

need to obtain permission directly from the copyright holder. To view a

copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.

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Publisher's Note Springer Nature remains neutral with regard to

jurisdictional claims in published maps and institutional affiliations.

Andreas Brack received his Ph.D. degree in electrical and computer

engineering on the topic of high precision GNSS from Technical Uni-

versity of Munich (TUM), Germany, in 2019. He is a researcher at the

Geodesy Department of the GFZ German Research Centre for Geo-

sciences, Potsdam, Germany, where he is working on precise GNSS

orbit and clock determination, positioning, and atmospheric sensing.

Benjamin Männel received his Ph.D. in Geodesy from the Institute of

Geodesy and Photogrammetry at ETH Zurich, Switzerland, in 2016. He

is leading the IGS Analysis Center at the GFZ German Research Centre

for Geosciences, Potsdam, Germany. Currently, his main research inter-

ests are the combination of ground and space-based GNSS observations

and the impact of surface loading corrections on geodetic products.

Harald Schuh is Director of the Geodesy Department at the GFZ Ger-

man Research Centre for Geosciences, Potsdam, Germany, and profes-

sor for Satellite Geodesy at Technische Universität Berlin, Germany.

He has engaged in space geodetic research for more than 40years. He

was Chair of the IVS (2007–2013), President of the IAU Commission

19 “Rotation of the Earth” (2009–2012), and President of the IAG

(2015–2019).