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GPS Solutions (2021) 25:4

https://doi.org/10.1007/s10291-020-01035-5

ORIGINAL ARTICLE

Two methods todetermine scale‑independent GPS PCOs

andGNSS‑based terrestrial scale: comparison andcross‑check

WenHuang1,2 · BenjaminMännel1· AndreasBrack1· HaraldSchuh1,2

Received: 30 March 2020 / Accepted: 18 September 2020

Abstract

The GPS satellite transmitter antenna phase center offsets (PCOs) can be estimated in a global adjustment by constraining

the ground station coordinates to the current International Terrestrial Reference Frame (ITRF). Therefore, the derived PCO

values rest on the terrestrial scale parameter of the frame. Consequently, the PCO values transfer this scale to any subsequent

GNSS solution. A method to derive scale-independent PCOs without introducing the terrestrial scale of the frame is the pre-

requisite to derive an independent GNSS scale factor that can contribute to the datum definition of the next ITRF realization.

By fixing the Galileo satellite transmitter antenna PCOs to the ground calibrated values from the released metadata, the GPS

satellite PCOs in the z-direction (z-PCO) and a GNSS-based terrestrial scale parameter can be determined in GPS + Galileo

processing. An alternative method is based on the gravitational constraint on low earth orbiters (LEOs) in the integrated

processing of GPS and LEOs. We determine the GPS z-PCO and the GNSS-based scale using both methods by including the

current constellation of Galileo and the three LEOs of the Swarm mission. For the first time, direct comparison and cross-

check of the two methods are performed. They provide mean GPS z-PCO corrections of

−186 ±25

mm and

−221 ±37

mm

with respect to the IGS values and

+1.55 ±0.22

ppb (parts per billion) and

+1.72 ±0.31

in the terrestrial scale with respect

to the IGS14 reference frame. The results of both methods agree with each other with only small differences. Due to the

larger number of Galileo observations, the Galileo-PCO-fixed method leads to more precise and stable results. In the joint

processing of GPS + Galileo + Swarm in which both methods are applied, the constraint on Galileo dominates the results.

We discuss and analyze how fixing either the Galileo transmitter antenna z-PCO or the Swarm receiver antenna z-PCO in

the combined GPS + Galileo + Swarm processing propagates to the respective freely estimated z-PCO of Swarm and Galileo.

Keywords GNSS· PCO· Galileo· Terrestrial scale· LEOs

Introduction

In October 2017, the European GNSS Agency (GSA)

released a comprehensive set of satellite metadata for the

Galileo FOC satellites. The available data set includes space-

craft properties, optical surface characteristics, the attitude

law, and the phase center positions of the transmitter antenna

with respect to the satellite centers of mass (PCOs) as well

as azimuth- and nadir-dependent phase variations (PVs).

Together with similar information released for the IOV sat-

ellites in December 2016, for the first time, this information

became available for a whole GNSS. Since then, several

studies have discussed resulting improvements in the geo-

detic analysis (Bury etal. 2019; Katsigianni etal. 2019;

Zajdel etal. 2019; Li etal. 2019). In particular, the informa-

tion about the PCOs and the PVs allows improved process-

ing and new investigations, which are discussed within this

study.

In order to realize the Terrestrial Reference System

(TRS), which is the basis for all geodetic measurements on

the earth, the geodetic datum has to be defined, i.e., origin,

orientation, and scale have to be specified. Theoretically,

GNSS can provide a terrestrial scale thanks to (1) centim-

eter-level accurate satellite orbits (Männel 2016) and (2)

the precision of the GNSS phase measurements (observa-

tion error less than 2mm). However, to link both orbit and

* Wen Huang

wen.huang@gfz-potsdam.de

1 Deutsches GeoForschungsZentrum GFZ,

14473Telegrafenberg,Potsdam, Germany

2 Institute ofGeodesy andGeoinformation Science,

Technische Universität Berlin, Strasse des 17. Juni 135,

10623Berlin, Germany

GPS Solutions (2021) 25:4

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observation, information is required about the transmitting

point (reference for the observation) with respect to the sat-

ellite center of mass (reference for the orbit). Obviously,

an unconsidered offset in the radial direction (i.e., in the

z-direction of the spacecraft body-fixed frame) will shift

the determined station heights and bias the eventually esti-

mated terrestrial scale parameter. Unfortunately, the posi-

tion of the transmitting point is usually not disclosed by

the GNSS providers. For some recently launched satellites,

ground calibrated PCOs are now provided, e.g., for Galileo,

BeiDou-3, QZSS, and GPS III. For most currently and for-

merly operational satellites, however, the PCOs and the PVs

have to be determined in global adjustments (Schmid etal.

2007). Over the past years, several PCO and PVs sets have

been estimated for the different constellations, for example,

by Steigenberger etal. (2016) and Schmid etal. (2016). Due

to the high correlation between station height, troposphere

delay, and the offsets of transmitting and tracking anten-

nas, accurate calibrations of the tracking antennas are a

prerequisite for estimating the transmitting antenna offsets.

The corresponding robot calibrations are provided in the

International GNSS Service (IGS) antenna exchange format

(ANTEX). Moreover, thanks to a recent effort by Geo + + ,

signal-specific (including Galileo frequencies) and multi-

GNSS calibrations are available for many receiver antennas

used within the IGS tracking network, for example, in the

ANTEX file for IGS repro3 igsR3_2057.atx provided by

Villiger (2019). In addition, the terrestrial scale had to be

fixed, for example, to the current ITRF solution, to avoid a

poorly conditioned normal equation system with less precise

estimates. Consequently, the derived transmitter offsets and

any further derived geodetic products are not independent

of this ITRF scale anymore (Haines etal. 2015).

By fixing the transmitter antenna patterns of Galileo

satellites to the ground calibrated values, a GNSS-based

terrestrial scale becomes achievable. However, with the

first operational Galileo satellites launched in 2012, a cor-

responding Galileo-only solution could cover only the most

recent years (i.e., from 2017 onward). To process a long-time

solution and determine the terrestrial scale back in time, the

PCOs, which are independent of the terrestrial scale and

derived by other techniques such as very long baseline inter-

ferometry (VLBI) and satellite laser ranging (SLR), are still

required for GPS and GLONASS. We present two different

approaches to re-adjust these offsets. It will cross-check and

compare both approaches and present the resulting terrestrial

scale values for an exemplary period (first half of 2019).

To improve readability, we will use the following naming

convention. PCOs describe the offset between the center of

mass and mean transmitting point onboard the spacecraft

and the offset between the antenna reference point and mean

transmitting center for receiving antennas. Deviation from

the mean transmitting or receiving point is described by

PVs, which are nadir and azimuth or elevation and azimuth

dependent, respectively. Transmitter phase centers are iden-

tified by the satellite system in a superscript (e.g.,

PCOGPS

Receiving antennas are indicated by subscripts (e.g.,

PCOLEO

). The estimated PCO differences in the z-direction

with respect to the a priori values are indicated by z-

ΔPCO

e.g., z-

ΔPCOGPS

and z-

ΔPCOLEO

. The manuscript is struc-

tured as follows. Following this introduction, the two PCO

determination approaches are introduced in more detail and

the validation and comparison scheme is explained. Subse-

quently, the processing period, the selection of ground sta-

tions, the quality check of the Swarm orbits, and the details

about the processing strategy are introduced. The results are

presented and discussed in the section afterward. The sum-

mary and our conclusion are given in the last section.

Methods forphase center offset estimation

This section describes two methods used to derive

PCOGPS

in the z-direction (z-

PCOGPS

) without fixing the terrestrial

scale. Also, the validation procedure used later and the defi-

nition of the assessed cases are described. Both approaches

rely on additional observations, either ground Galileo

observations or GPS observations onboard low earth orbit-

ers (LEOs). Figure1 presents the basic setup consisting

of ground stations, GPS and Galileo satellites, and LEOs.

The satellite orbits are constrained by celestial mechanics.

Ground-based and space-based observations connect the

Fig. 1 Schematic diagram of the two methods of determining scale-

independent GPS z-PCO

GPS Solutions (2021) 25:4

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antenna phase center of different transmitters and receiv-

ers. The estimated coordinates of the ground station network

have a scale factor with respect to the a priori terrestrial

frame, in our case IGS14 (Rebischung and Schmid 2016).

Method I: Galileo withcalibrated antenna offsets

The basic idea of this method is to separate z-

PCOGPS

and

terrestrial scale by adding Galileo (GAL) observations.

With the

PCOGAL

fixed to the calibrated values provided

by the GSA, a reliable scale-independent network solu-

tion is achieved. As GPS and Galileo are observed by the

same stations whose coordinates are now estimated scale

independently from the underlying reference frame, also

the z-

PCOGPS

can be estimated scale independently. This

method will fail if there is any systematic bias between inde-

pendently estimated station coordinates for GPS and Galileo.

Villiger etal. (2018, 2019) reported translational biases of

several mm when applying the L1 and L2 PVs of GPS to the

Galileo E1 and E5 signals. With the signal-specific antenna

corrections provided by Geo + +, this systematic discrep-

ancy should not occur anymore. This assumption was tested

by processing GPS and Galileo solutions independently

in the framework of the next IGS reprocessing campaign

(repro3). However, due to the different satellite PCOs used

(z-

PCOGPS

from igs14.atx and z-

PCOGAL

from GSA) a ter-

restrial scale bias of 1.16

0.27ppb (part per billion) was

observed in the GFZ submission (Männel etal. 2020). When

taking this terrestrial scale into account, GPS and Galileo-

based coordinates agree on the level of a few millimeters in

the height component. By fixing the antenna offsets of Gali-

leo to the calibrated values, z-

ΔPCOGPS

have been computed

by Villiger etal. (2020). He reported a system-wise change

−150

mm and

−221

mm for the z-

PCOGPS

by using robot

calibrations and chamber calibrations for ground stations,

respectively. The re-adjusted PCOs have been updated in the

IGS repro3 ANTEX file (igsR3_2077.atx) and will be used

in the IGS repro3 processing.

Method II: gravitational constraint

The orbits of the LEOs are scale independent as their radial

position is constrained by orbital dynamics (so-called gravi-

tational constraint). Therefore, the estimation of scale-inde-

pendent z-

PCOGPS

becomes possible. However, there are

three limitations. Firstly, there are not enough space-based

observations to solve for all z-

PCOGPS

, LEO orbits, GPS

orbits, etc. Therefore, ground- and space-based observa-

tions have to be combined. This approach is known as an

integrated or one-step approach and has been studied for the

past 15years. It was already used to determine z-

PCOGPS

Haines etal. (2015) and Männel (2016). To transfer the scale

constraint offered by the space-based observations requires

a fully consistent estimation of GNSS satellite orbits and

clocks which link ground- and space-based observations.

The second limitation is the availability and quality of the

space-based observations. And thirdly, an error in the a pri-

ori calibrated z-

PCOLEO

can significantly bias the derived

PCOGPS

Validation

In general, the validation of z-

PCOGPS

is challenging as the

phase center offsets cannot be observed by space geodetic

techniques. However, we can evaluate the different z-

PCOGPS

estimated by both methods by comparison and cross-check.

First of all, scale independence can be mathematically

assessed by comparing the correlation between scale and

phase center parameters. Using only ground-based GPS

observations results in a large correlation coefficient of the

two parameters (Schmid etal. 2007). Using both methods

with different observations and constraints (on

PCOGAL

PCOLEO

) allows, secondly, to assess the agreement between

the z-

PCOGPS

estimates. For this purpose, we designed six

cases that are listed in Table1. Different combinations of

included satellites, PCO constraints, and estimated satel-

lite z-PCO increments with respect to the a priori values

(

ΔPCO

) are selected for the different cases. Since our focus

is on the satellite z-PCO and the terrestrial scale, the satellite

PCOs in x- and y-directions are always kept fixed. In case 1,

as a reference case, we want to show the problem of a high

correlation between the terrestrial scale and the z-

ΔPCOGPS

In cases 2 and 3, z-

ΔPCOGPS

are estimated by fixing either

PCOLEO

PCOGAL

. Moreover, we combine GPS and

Table 1 Six cases for the estimation of the GPS z-PCO and GNSS-

based terrestrial scale deriving

The column named “Satellites” shows the satellites included in the

processing. “Fixed” means that the corresponding satellite PCOs are

fixed to the a priori values. The last column shows the estimated cor-

rections of satellite z-PCO in the processing. The name of each case

is based on the included satellites and the estimated z-

ΔPCO

Cases Satellites Fixed Estimated

1 G-G GPS - z-

ΔPCOGPS

2 GL-G GPS

LEOs

PCOLEO

ΔPCOGPS

3 GE-G GPS

GAL

PCOGAL

ΔPCOGPS

4 GEL-G GPS

GAL

LEOs

PCOGAL

PCOLEO

ΔPCOGPS

5 GEL-GL GPS

GAL

LEOs

PCOGAL

ΔPCOGPS

ΔPCOLEO

6 GEL-GE GPS

GAL

LEOs

PCOLEO

ΔPCOGPS

ΔPCOGAL

GPS Solutions (2021) 25:4

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Galileo ground-based observations and space-based GPS

observations in cases 4, 5, and 6. In case 4, both

PCOGAL

and

PCOLEO

are fixed to estimate the z-

ΔPCOGPS

. In cases 5

and 6, we determine z-

ΔPCOLEO

or z-

ΔPCOGAL

jointly with

ΔPCOGPS

while only fixing

PCOGAL

PCOLEO

, respec-

tively. These two cases allow the ultimate cross-check with

the known Galileo and LEO offsets. However, it is debatable

whether the gravitational constraint can be transferred from

the GPS space-based observations to the Galileo satellites

or reversely. (Unfortunately, space-based observations are

available only for GPS.) This question will be discussed in

the section of the results. To improve the readability, we

name the six cases based on the included satellites and the

estimated z-

ΔPCO

. For example, GEL-GE means that GPS

(G), Galileo (E), and Swarm (L) satellites are all included in

the processing and z-

ΔPCOGPS

(-G) and z-

ΔPCOGAL

(-E) are

estimated, while only the PCOs of Swarm satellites are fixed.

Processing period andground station

selection

We selected day 1 to 180 of 2019 as our processing period.

During this period, the Galileo constellation already had

24 satellites in operation. All selected ground stations are

tracking both GPS and Galileo, and the network is glob-

ally and evenly distributed. As a prerequisite for the ter-

restrial scale realization, the stations should have accurate

coordinates that are offered within the IGS products (i.e., in

IGS14 reference frame). There are 68 to 94 stations (only

a few days less than 75) that are selected for different days.

The majority of the stations for each daily processing have

Galileo-calibrated receiver antennas (Fig.2), and for the oth-

ers, the GPS L2 calibrations are applied for E5a. We used

only stations that provide observations in RINEX3 format to

the IGS data centers. The station number increases around

DOY (day of the year) 87 because more stations started to

offer RINEX3 observations from that day onward. Figure3

presents the selected 75 stations for the processing of DOY

1 as an example.

Swarm orbit quality

For the gravitational constraint strategy, we included the

three spacecraft of the Swarm mission, which is a mini-

satellite constellation mission to survey the geomagnetic

field (Friis-Christensen etal. 2008). The three satellites

(Swarm-A, B, and C) are operated in two different orbit

configurations. Swarm-A and Swarm-C are flying at a mean

altitude of 450km and in an 87.4° inclined orbital plane,

while Swarm-B has a higher inclination of 88° and a larger

mean altitude of 530km. To check the quality of the LEO

observation data and to verify our orbit determination, we

did a Swarm-only reduced-dynamic POD by using IGS

final orbit and 30-s clock products. The data sampling rate

is 30-s and the arc length is 24-h. The determined orbits

are validated by comparing with an external solution and

by SLR observation residual validation. The daily orbit is

compared with the official precise orbit products which are

offered by the European Space Agency (ESA, Olsen 2019)

in the along-track, cross-track, and radial directions. The

orbit RMS values averaged over 180days are presented in

Table2. In general, the orbits of the three Swarm satellites

are determined in similar accuracy with about 30-mm RMS

in the along-track direction, 14-mm RMS in the cross-track

direction, and 24-mm RMS in the radial direction. We used

all SLR observations of the high-quality Yarragadee sta-

tion in Australia during the 180days to validate the Swarm

orbits. The statistical results are also listed in Table4, and all

the epoch-wise solutions are shown in Fig.4. With the most

observations, the SLR residuals of Swarm-B have the largest

mean (4.2mm) and the smallest variation (

19.5mm). With

similar numbers of observations, the SLR residuals (with

variation) of Swarm-A and Swarm-C are 3.7

25.1mm and

Fig. 2 Number of stations selected for each day

Fig. 3 Distribution of the 75 stations selected for January 1, 2019.

Stations with Galileo antenna calibrations are marked in blue. Red

denotes the stations using phase center corrections of GPS for the

Galileo signals

GPS Solutions (2021) 25:4

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2.4

25.0mm. Compared with previous studies, the orbit

quality of our solution is at a comparable level.

Processing strategy

We use the software PANDA (Liu and Ge 2003) for the pro-

cessing. We performed the one-step method (Montenbruck

and Gill 2000) to estimate the orbits of GPS, Galileo, and

Swarm satellites, the PCOs in the z-direction of different

satellites, and the other parameters simultaneously. Detailed

information on the orbit modeling, processing configura-

tion, metadata, and estimated parameters is listed in Table3.

As the initial release of the antenna correction file for IGS

Repro3, igsR3_2057.atx includes IGS estimated

PCOGPS

(Schmid etal. 2016), the GSA calibrated

PCOGAL

and the

ground calibrated

PCOstation

for multi-GNSS. Depending on

the cases in Table1, the satellite PCOs in the z-direction are

fixed to their a priori values or estimated freely. To derive

scale-independent z-

PCOGPS

and a GNSS-based terrestrial

scale, the coordinates of the ground stations are constrained

only by applying no-net-rotation condition. The scale of the

ground-tracking network is not constrained in this study,

and the scale is derived by Helmert transformation between

the estimated coordinates of the ground network and the a

priori coordinates (i.e., IGS14). Montenbruck etal. (2018)

reported in-flight calibrated PVs for the Swarm satellites of

up to 25mm. As these in-flight calibration results might not

be independent of the scale provided by VLBI and SLR, we

decided not to apply them in this study.

Results

We analyze the results from three aspects. Firstly, consider-

ing the relationship that 130-mm error in GPS z-

PCOGPS

leads to one ppb terrestrial scale (Zhu etal. 2003), we dis-

cuss both the estimated z-

ΔPCOGPS

and the derived terres-

trial scale with respect to IGS14. The further comparisons

and the estimation quality analysis are based on the daily

estimates, the formal error of the estimates, and the cor-

relation coefficient of z-

ΔPCOGPS

and scale. The variation

of the estimated daily z-

ΔPCO

values, the formal error of

ΔPCO

, and the derived scale between the processed days

are shown by the empirical standard deviation (STD) of their

time series with respect to the mean. Both satellite-specific

results and the results averaged over satellites (system-wise)

are discussed in detail. Secondly, the z-

ΔPCO

estimated by

fixing only the

PCOGAL

in GEL-GL will be analyzed. The

impact of fixed

PCOGAL

on the estimation of z-

ΔPCOLEO

shown. At last, mainly based on GEL_GE, the z-

ΔPCOGAL

estimated by fixing only the

PCOLEO

is analyzed. The effect

of transferring the gravitational constraint directly to GPS

and indirectly to Galileo via GPS satellites and ground sta-

tions is discussed.

Estimated z‑

1PCOGPS

andterrestrial scale

In Fig.5, the satellite-specific z-

ΔPCOGPS

with respect to

the IGS values and averaged over the 180 processed days

are shown as blue bars. The vertical lines denote the empiri-

cal STD for each time series. The last bar in each plot pro-

vides the mean value over all satellites; correspondingly,

the empirical STD of the constellation-wise value is smaller

than that of the satellite-specific values. The formal errors of

ΔPCOGPS

and their empirical STD are presented as green

bars. Due to the evenly distributed ground network and satel-

lite constellation, the formal errors are quite similar within

one case. There is no obvious block-specific phenomenon

visible. Although the z-PCOs of GPS satellites in the same

Table 2 The validation of the orbits of Swarm satellites

The direction-specific RMS values of orbit differences compared to

the office products are averaged over 180days. The SRL validation

is based on the observations of Yarragadee station. The residuals are

averaged over epochs

Orbit RMS compared to official

products [mm]

SRL residuals [mm]

Along-track Cross-track Radial Epochs Mean/STD

Swarm-A 30.9 15.1 25.3 1781 3.7

25.1

Swarm-B 29.7 12.2 21 4083 4.2

19.5

Swarm-C 30.3 14.8 25.1 1650 2.4

25.0

Fig. 4 SLR observation residuals for the Swarm-A, B, and C POD

solution for all passes of the Yarragadee station in Australia for

180days. The gaps are caused by missing SLR observations

GPS Solutions (2021) 25:4

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block are similar, the z-PCOs corrections are similar for all

satellites in every case.

In the case G-G, the estimated z-

ΔPCOGPS

values are

smaller than 100mm, but with large empirical STD (100 to

130mm), formal error (about 46mm), and empirical STD of

formal error (about 22mm) among all cases. The reason for

this is the high correlation between the estimated z-

ΔPCOGPS

and the terrestrial scale. Slight changes in any inputs of the

estimation (e.g., the ground station network) lead to very

different solutions; therefore, the precision of the estimated

ΔPCOGPS

and the scale is low.

In the other five cases, the PCOs of either the Galileo

or the Swarm satellites or both are fixed. Consequently,

the precision of the z-PCO estimates is improved. In

general, the results of the five cases show collective

shifts of z-

PCOGPS

with respect to the IGS values, and

the satellite-specific values have a good agreement

among the five cases. Comparing the results based

on the gravitational constraint (GL-G) and on Gali-

leo (GE-G), the z-

ΔPCOGPS

values have differences of

about 30mm for all satellites. The empirical STD of

ΔPCOGPS

, the formal error of z-

ΔPCOGPS

, and the

empirical STD of the formal error of GL-G are 12mm,

5mm, and 3mm larger than those of GE-G, respectively.

That means the precision of the LEO-PCO-fixed case is

slightly lower than that of the Galileo-PCO-fixed cases.

It is explained by the stronger constraint transferred by

Table 3 Processing details (orbit modeling, processing configurations, metadata, and estimated parameters)

Orbit modeling

Atmosphere drag DTM94 (Berger etal. 1998) only for LEOs

Earth gravity field EIGEN-GRACE02S (Reigber etal. 2005); up to degree and order 150

Earth radiation analytical, box-wing models applied

N-body perturbation JPL DE405 (Standish 1998)

Relativistic corrections IERS 2010 conventions (Petit and Luzum 2010)

Solid earth and pole tide IERS 2010 conventions

Ocean tide FES2004 (Lyard etal. 2006)

Solar radiation pressure Reduced ECOM (Springer etal. 1999) for GPS; a priori box-wing model + reduced ECOM (Montenbruck etal.

2015) for Galileo; box-wing for LEOs

Configurations and metadata

Arc length 24h

Cut-off elevation 7° for ground stations and 3° for LEOs

Observations Zero-difference ionosphere-free phase and code measurements; 5-min sampling rate for both ground and

onboard observations

Weighting Satellites are equally weighted, and observations are weighted depending on elevation angles

Swarm attitude Antenna position and star-camera-based spacecraft attitude (quaternion data provided by operators)

Swarm macro-model Taken from Montenbruck etal. (2018)

Swarm receiver PCOs and PVs Satellite-specific ionosphere-free combined PCOs offered by ESA (Siemes 2019); chamber calibrated; PVs are

not applied

Station receiver PCOs and PVs igsR3_2057.atx

GPS PCOs and PVs igsR3_2057.atx

Galileo PCOs and PVs igsR3_2057.atx

Ambiguity fixing Only within ground stations

Parameters

Station coordinate no-net-rotation with respect to IGS14 which is aligned to ITRF2014 (Altamimi etal. 2016)

GPS and Galileo orbits Initial epoch state vector and five solar radiation pressure parameters (initial orbital elements are generated from

broadcast ephemeris)

LEOs orbits Initial epoch state vector; piece-wise empirical force (90-min interval) and atmosphere drag (four hours inter-

val) parameters (initial orbital elements are generated from official products)

GPS, Galileo, and LEOs PCOs satellite-specific daily solution of ionosphere-free combined PCOs in the z-direction; fixed to a priori values or

freely estimated depending on the case

Earth rotation Rotation pole coordinates and UT1 for 24-h intervals, piece-wise linear modeling

Tropospheric delay For each ground station; piece-wise constant zenith delays for 1-h intervals; piece-wise constant horizontal

gradients for 4-h intervals

Satellite and receiver clocks Epoch-wise; pre-eliminated

GPS Solutions (2021) 25:4

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many more observations from 24 Galileo satellites com-

pared to the three Swarm satellites, which is verified

later in this section.

In the GEL-G case, the PCOs of both Galileo satellites

and Swarm satellites are fixed, but the results are quite sim-

ilar to GE-G. Similar results are obtained in GEL-GL in

which only Galileo PCOs are fixed. This demonstrates that

the Galileo satellites are dominating the results due to the

larger number of observations. In GL-G and GEL-GE, only

the PCOs of Swarm satellites are fixed. However, the result

differences between GL-G and GEL-GE are larger than the

result differences between GE-G, GEL-G, and GEL-GL.

The z-

ΔPCOGPS

values in GEL-GE are collectively larger

than that of GL-G by about 10mm. The empirical STD of

ΔPCOGPS

, the formal error of z-

ΔPCOGPS

, and the empiri-

cal STD of formal error are all smaller in GEL-GE than

in GL-G. The differences between GL-G and GEL-GE are

caused by including Galileo.

The time series of daily system-wise (averaged over

satellites) z-

ΔPCOGPS

and the corresponding terrestrial

scale are shown in Fig.6 for G-G and in Fig.7 for the

other five cases. The corresponding mean values and the

standard deviations of all the time series are presented in

Table4. Comparing the upper (z-

ΔPCOGPS

) and the lower

(scale factor) plots, we can see the relationship between

the two parameters. The variation of the time series in G-G

is quite large (103mm empirical STD for z-

ΔPCOGPS

and

0.823ppb empirical STD for terrestrial scale). The solu-

tions of the Galileo-PCO-fixed solutions (GE-G, GEL-G,

and GEL-GL) are very similar. The time series of GL-G

and GEL-GE have larger variation and -20 to -40mm dif-

ferences in mean values of z-

ΔPCOGPS

than those of the

Galileo-PCO-fixed solutions. By including Galileo satel-

lites, GEL-GE is more stable and closer to the Galileo-

PCO-fixed solutions than GL-G.

The impact of the 20 additional stations after DOY 89

(Fig.2) on the estimates is not visible. Only a slight decrease

of the formal errors is observed in the analysis.

Fig. 5 Estimated GPS z-PCO differences with respect to IGS values

(blue) and their formal errors (green). The name of each case is based

on the included satellites and the estimated z-

ΔPCO

, for example,

GEL-GE means that GPS, Galileo, and Swarm satellites are included

and z-

ΔPCOGPS

and z-

ΔPCOGAL

are estimated. Each bar denotes the

solution averaged over 180 processing days. The vertical lines denote

the empirical standard deviation of the time series with respect to the

mean. The x-label presents the space vehicle number of the satellites,

and the satellites are sorted by block-wise

Fig. 6 Time series of the differences between the estimated GPS

z-PCO and IGS values averaged over satellites (upper) and the cor-

responding terrestrial scale with respect to IGS14 (lower) in the case

including GPS only without fixing the scale

GPS Solutions (2021) 25:4

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4 Page 8 of 13

The quality of the estimation in the different cases is

also reflected in the correlation coefficients of the estimated

ΔPCO

and the terrestrial scale. The corresponding cor-

relation coefficients averaged over satellites and days are

presented in Table4. Overall, the coefficients are very

stable with variations smaller than 0.01. G-G shows the

largest correlation coefficient of z-

ΔPCOGPS

and terres-

trial scale (0.87) which agrees to the analysis mentioned

above. The correlation coefficient can be reduced effec-

tively by introducing LEOs or by processing together with

Galileo and fixing

PCOGAL

. Derived by different num-

bers of observations, the Galileo-PCO-fixed case GE-G

is more effective than the LEOs-PCO-fixed case GL-G in

de-correlation (reduction of 0.74 versus 0.35). Due to the

stronger impact of Galileo on transferring the constraint

compared to Swarm, the correlation coefficient nearly does

not change after fixing

PCOLEO

additionally (GEL-G and

GEL-GL). The correlation coefficients in GEL-GL show

that the fixed Galileo PCOs can separate the derived ter-

restrial scale and the estimated z-

ΔPCO

for both GPS and

LEOs. In GEL-GE, with only three PCO-fixed LEOs, the

correlation between z-

ΔPCO

and terrestrial scale is identi-

cal for GPS and Galileo satellites (0.56) and is close to that

of GL-G (0.52).

To investigate the impact of the numbers of Galileo and

Swarm satellites on the estimation, GE-G is processed again

by only including three Galileo satellites (E101, E210, and

E212) in three different orbital planes (GE-G*). The statis-

tic of the solution for GE-G* is presented in Table4. With

fewer Galileo satellites, the results of GE-G* are different

from those of GE-G with a system-wise difference of 25mm

for the estimated z-

ΔPCOGPS

. This is caused by the weaker

geometry and fewer observations of the three Galileo satel-

lites compared to the full system. Without the advantage of

more satellites, the precision of GE-G* becomes lower than

that of the GL-G with 13-mm larger empirical STD of the

ΔPCOGPS

and 0.1 ppb larger empirical STD of the scale.

Moreover, the correlation coefficient between z-

ΔPCOGPS

and the terrestrial scale increases from 0.13 (GE-G) to 0.54

(GE-G*), which exceeds that of GL-G by 0.02. In a sum-

mary, due to the much faster geometry change, including

three Swarm satellites gives more precise z-

ΔPCOGPS

than

including three Galileo satellites.

Besides the internal comparison and cross-check between

the different cases, we also compared our results with other

studies. The system-wise z-

ΔPCOGPS

derived by GE-G is

Fig. 7 Time series of the estimated GPS z-PCO differences with

respect to IGS values averaged over satellites (upper) and the cor-

responding terrestrial scale with respect to IGS14 (lower) for the

five cases. The name of each case is based on the included satel-

lites and the estimated z-

ΔPCO

, for example, GEL-GE means that

GPS, Galileo, and Swarm satellites are included and z-

ΔPCOGPS

and

ΔPCOGAL

are estimated

Table 4 The estimated z-

ΔPCOGPS

averaged over satellites and pro-

cessed days and the scale factor with respect to IGS14 averaged over

the processed days. The empirical standard deviations of the time

series (STD) are also given. The correlation coefficients of the esti-

mated satellite PCOs in the z-direction and the terrestrial scale. The

values are averaged over satellites and processed days. Zero val-

ues are due to the fixing to the a priori values. The dash means not

included

Case z-

ΔPCOGPS

[mm] mean/

STD

Scale [ppb] mean/STD z-

ΔPCOGPS

vs. Scale z-

ΔPCOGAL

vs. Scale z-

ΔPCOLEO

vs. Scale

G-G − 33/103 0.31/0.82 0.87 – –

GL-G − 221/37 1.72/0.31 0.52 – 0

GE-G − 186/25 1.55/0.22 0.13 0 –

GEL-G − 188/23 1.56/0.22 0.12 0 0

GEL-GL − 184/24 1.54/0.23 0.13 0 0.09

GEL-GE − 201/33 1.66/0.29 0.56 0.56 0

GE-G* − 161/50 1.44/0.41 0.54 0 –

GPS Solutions (2021) 25:4

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Page 9 of 13 4

between the robot-calibration-based solution and the cham-

ber-calibration-based solution in Villiger etal. (2020). We

compared the estimated GNSS-based scale with the scale

determined by the VLBI and SLR. As reported by Altamimi

etal. (2016), the scale factors determined by VLBI and

SLR with respect to the ITRF 2014 are about + 0.77ppb

and -0.77ppb, respectively. The GNSS-based scales derived

by GL-G and GE-G cases are + 1.72ppb and + 1.55ppb,

respectively. Therefore, the GNSS-based scale derived by

both methods agree with each other well and agree with

the VLBI-based scale better than the SLR-based scale does.

After removing the systematic errors in SLR data by Luc-

eri etal. (2019), the scale derived by SLR is about + 1ppb

toward ITRF 2014 scale. Therefore, the scales determined

by GNSS in this study, by VLBI, and by SLR have an agree-

ment within differences smaller than 1ppb.

Estimated z‑

1PCOLEO

In the case GEL-GL, z-

ΔPCOGPS

and z-

ΔPCOLEO

are esti-

mated simultaneously by fixing the PCOs of all Galileo sat-

ellites to the GSA values. Figure8 shows the time series of

the estimated satellite-specific z-

ΔPCOLEO

. The mean values

and the empirical STD are

−2.2 ±2.5

mm,

−2.6 ±2.1

mm,

and

−1.1 ±2.4

mm for Swarm-A, B, and C satellites, respec-

tively. The plots of Swarm-A and Swarm-C are very similar,

but that of Swarm-B is slightly different from them. This

can be explained by their orbital configuration introduced

in Section “Swarm orbit quality.” During DOY 55 to 57, the

orbits of Swarm-B have a 10-mm larger RMS with respect

to the official products than the other days, which might be

caused by some unknown behavior of the spacecraft. This

is assumed to cause the large deviation of the estimated

ΔPCOLEO

on those days. It also affects all the time series

of z-

ΔPCOGPS

, z-

ΔPCOGAL

, and the scale derived by the

LEOs-PCO-fixed cases. Therefore, we concluded that orbit

modeling quality has a large impact on the estimation.

All of the three time series show a periodic behavior. The

periodicity might be related to the draconitic period, i.e.,

the time between two passages of the satellite through its

ascending node, of Swarm, Galileo, and GPS. The impact of

the periodicity can also be observed from the time series var-

iation of z-

ΔPCOGPS

and z-

ΔPCOGAL

estimated in GL-G and

GEL-GE in which only the PCOs of LEOs are fixed (Figs.7

and 12). From the magnitude of the estimated z-

ΔPCOLEO

the importance of the PCO accuracy of the LEOs can be

realized. The scale factor between GL-G and GEL-GL has

a 0.2 ppb (1.3mm at the equator) difference, while the esti-

mated z-

ΔPCOLEO

in GEL-GL is 1–2mm with respect to

the a priori values which are fixed in GL-G. Additionally,

we processed an update for GL-G (GL-G*) using artificially

modified Swarm PCOs by adding the estimated z-

ΔPCOLEO

to the values offered by ESA. The time series of z-

ΔPCOGPS

estimated in GL-G* is presented in Fig.9 (green). The

curve of the updated case is systematically shifted from the

curve of GL-G by 48mm. However, the z-

ΔPCOGPS

aver-

aged over satellite and processed days is

−173

mm which is

much closer to the Galileo-PCO-fixed solution in GEL-GL

(red) than that of GL-G (purple). This comparison shows the

importance of the accuracy of the LEO PCOs again.

Estimated z‑

1PCOGAL

In the case GEL-GE, z-

ΔPCOGAL

and z-

ΔPCOGPS

are esti-

mated simultaneously by fixing the PCOs of the three LEOs

to a priori values and without constraint on the terrestrial

scale. This allows us to discuss the estimated z-

ΔPCOGAL

with respect to the GSA values. Since the z-

ΔPCOGPS

and

the terrestrial scale derived in GEL-GE have small dif-

ferences with respect to the solutions derived by the Gal-

ileo-PCO-fixed cases (GE-G, GEL-G, and GEL-GL), the

estimated z-

ΔPCOGAL

in GEL-GE are small. The satellite-

specific z-

ΔPCOGAL

values are presented as blue bars in

Fig. 8 Time series of the estimated Swarm-A, B, and C z-PCO dif-

ferences with respect to the a priori values (ESA offered) in the case

including GPS, Galileo, and Swarm satellites and fixing only the

PCO of Galileo satellites

Fig. 9 Time series of the estimated GPS z-PCO differences with

respect to IGS values averaged over satellites in three cases. GL-G

includes GPS and Swarm satellites and the PCOs of Swarm satellites

are fixed. GEL-GL includes GPS, Galileo, and Swarm satellites, and

only the PCOs of Galileo satellites are fixed. GL-G* is an update pro-

cessing of GL-G by modifying the z-

ΔPCOLEO

artificially

GPS Solutions (2021) 25:4

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4 Page 10 of 13

Fig.10. The z-

ΔPCOGAL

averaged over satellites is only

−21

mm. The empirical STD values of the z-

ΔPCOGAL

time

series are about 80mm to 100mm for different Galileo sat-

ellites, and they are about 30mm to 50mm larger than that

of the GPS satellites shown in Fig.5. The reason is that the

gravitational constraint on LEOs is transferred only via the

GPS satellites and the ground stations to Galileo (Fig.1).

Since the selected ground network changes day by day dur-

ing the processed period, the impact on individual Galileo

satellites will change. However, the constraints on the LEO

orbits affect z-

ΔPCOGPS

directly by onboard GPS observa-

tions; therefore, the variations of z-

ΔPCOGPS

are smaller.

Evaluating the impact on the whole Galileo constellation,

the empirical STD of z-

ΔPCOGAL

averaged over all satellites

is only 10mm larger than that of GPS. The formal errors

of z-

ΔPCOGAL

and the empirical STD of the formal errors

are only about 6mm and 2mm larger than those of the esti-

mated z-

ΔPCOGPS

. In general, the gravitational constraint on

the LEOs acts on the estimation of z-

ΔPCOGAL

. We also see

some unexpected phenomena on satellite E102. We expected

a systematic change of the estimated z-

ΔPCOGAL

due to

the scale change, but the absolute value of the estimated

ΔPCOGAL

of E102 is much larger than all the other satel-

lites. For example, the z-

ΔPCOGAL

of E101 and E102 has a

−123

mm difference. However, the a priori (GSA) z-PCOs

of the two satellites have a

+87

mm difference. That means

our estimated z-PCO values of the two satellites are much

closer than their a priori values. This result agrees with the

study by Steigenberger etal. (2016).

For the four Galileo satellites E219, E220, E221, and

E222, which were launched in July 2018, we found larger

formal errors than for the other Galileo satellites. This is

likely caused by fewer ground-based observations that were

available during the first 43 processed days, as a part of the

ground stations was not offering data for them. The numbers

of observations are shown in Fig.11. Moreover, the observa-

tions of satellites E222 and E219 reach a similar number as

the other satellites in mid-2019. This is due to the limited

capability of some ground receivers which only observe sat-

ellites with PRN smaller than 32 (Mozo 2018).

We also plot the time series of system-wise z-

ΔPCOGAL

and z-

ΔPCOGPS

in Fig.12. As explained above, the variation

of the z-

ΔPCOGAL

time series is slightly larger (10mm) than

that of z-

ΔPCOGPS

. However, due to the fixed LEO PCOs

and the same ground stations, they agree with each other.

Summary andConclusions

Using two different methods based on (1) the Galileo sys-

tem with ground-calibrated antenna offsets and on (2) the

gravitational constraint on LEO orbits, we determined

the scale-independent GPS z-PCO and the corresponding

GNSS-based terrestrial scale. Applying the first method,

Fig. 10 The estimated Galileo z-PCO differences with respect to

igsR3_2057.atx (upper) and their formal errors (lower). Each bar

denotes the solution averaged over 180 processing days. The thin

errors bars denote the empirical standard deviation of the time series

Fig. 11 Number of ground-based observations that are used for the

processing of Galileo satellites E102 (as a reference), E219, E220,

E221, and E222 during the 180 processed days

Fig. 12 Time series of the estimated GPS and Galileo z-PCO differ-

ences with respect to igsR3_2057.atx averaged over satellites in the

case including GPS, Galileo, and Swarm satellites and only fixing the

PCOs of Swarm satellites

GPS Solutions (2021) 25:4

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Page 11 of 13 4

we found a

−186 ±25

mm z-PCO correction with respect

to the IGS values, and a

+1.55 ±0.22

ppb terrestrial scale

with respect to the IGS14. The results of the gravitational

constraint method are

−221 ±37

mm for the z-PCO and

+1.72 ±0.31

ppb for the terrestrial scale. The solutions

derived by the two independent methods with different

observations and metadata agree well with each other. The

Galileo-based solution agrees very well with the latest study

by Villiger etal. (2020). Moreover, these two solutions

also agree with the VLBI-based scale (+ 0.77ppb) better

than the SLR-based scale (-0.77) does. Compared with the

updated SLR-based scale without systematic errors (Luceri

etal. 2019), the scales determined by GNSS in this study,

by VLBI, and by SLR agree with each other with differences

smaller than 1ppb.

Since Galileo offers many more observations which

transfer the constraints than the Swarm constellation, Gali-

leo dominated the results of the case in which the PCOs

of both Galileo and LEOs are fixed. Based on the correla-

tion coefficient of z-

ΔPCOGPS

and scale, the formal error

of z-

ΔPCOGPS

and the empirical STD of the time series,

the precision and stability of the solution derived by the

Galileo-PCO-fixed method is higher than that derived by

the LEO-PCO-fixed method. This is mainly caused by the

different number of satellites and observations from Galileo

and Swarm. If Galileo is reduced from the full constellation

to only three satellites, the better geometry of the Swarm-

based solution leads to better results.

The joint estimation of z-

ΔPCOGPS

and z-

ΔPCOLEO

only fixing

PCOGAL

showed that the z-

ΔPCOGPS

and the

derived scale factor are very close to the solutions derived by

the case including GPS and Galileo and fixing the

PCOGAL

Consequently, the constraint from Galileo is very strong and

is nearly unaffected by including LEOs. The z-

ΔPCOLEO

precisely estimated at 1 to 2mm with respect to the values

offered by ESA. This shows the small difference between

the two methods again. Moreover, the accuracy of the

LEOs’ PCOs is very important for the gravitational con-

straint method. We realized some periodic variations in the

ΔPCOLEO

time series. This is also visible in the time series

of z-

ΔPCOGPS

, z-

ΔPCOGAL

and scale derived by applying

only the gravitational constraint and might be related to the

draconitic period of GPS and Swarm constellations. Based

on the unusual results of the Swarm-B satellite in three days,

the importance of orbit modeling quality is shown.

The z-

ΔPCOGAL

estimated by only fixing

PCOLEO

the GPS, Galileo, and Swarm joint processing differs on

average by

−21

mm from the GSA values. This difference

corresponds to 0.13ppb difference in the terrestrial scale.

The estimated z-

ΔPCOGPS

and scale have slight differences

from the results derived by the case, which includes only

GPS and LEOs. The precision and stability of z-

ΔPCOGAL

are both worse than those of the simultaneously estimated

ΔPCOGPS

. The gravitational constraint on the Swarm

orbits is partially transferred to the Galileo satellites. This

situation can be improved by including more LEOs and

moreover, by including Galileo space-based observations,

which might be available in the future.

A future study including a longer processing period (at

least three years) and more LEOs will be performed to study

the GNSS-based terrestrial scale in detail.

Acknowledgment Wen Huang is financially supported by the Chinese

Scholarship Council. The authors want to thank ESA, GSA, and IGS

for offering the data and products.

Funding Open Access funding enabled and organized by Projekt

DEAL.

Data Availability The data of the Swarm satellites are provided by

ESA (ftp://swarm -diss.eo.esa.int). GNSS ground observations and

related products are provided by IGS (https ://www.igs.org/produ cts).

The Galileo metadata is offered by GSA (https ://www.gsc-europ a.eu/

suppo rt-to-devel opers /galil eo-satel lite-metad ata).

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.

Wen Huang received his M.Sc.

degree in the GeoEngine Pro-

gram at the University of Stutt-

gart in 2016. He is a Ph.D. stu-

dent of Technische Universität

Berlin and works at the GFZ

German Research Centre for

Geosciences, Potsdam, where he

is working on precise orbit deter-

mination and the combination of

ground- and space-based GNSS

observations.

Benjamin Männel received his

Ph.D. in Geodesy from the Insti-

tute of Geodesy and Photogram-

metry at ETH Zurich in 2016. He

is leading the IGS Analysis

Center at the GFZ German

Research Centre for Geo-

sciences, Potsdam. Currently, his

main research interests are the

combination of ground- and

space-based GNSS observations

and the impact of surface loading

corrections on geodetic

products.

GPS Solutions (2021) 25:4

1 3

Page 13 of 13 4

Andreas Brack received his M.Sc.

degree in Electrical and Com-

puter Engineering from Techni-

cal University of Munich

(TUM), Munich, Germany, in

2012. In 2019, he completed his

Ph.D. on high-precision GNSS

applications at TUM. He is a

researcher at the GFZ German

Research Centre for Geo-

sciences, Potsdam, where he is

working on precise GNSS orbit

and clock determination, posi-

tioning, and atmospheric

sensing.

Harald Schuh is Director of

Department “Geodesy” at the

GFZ German Research Centre

for Geosciences, Potsdam, and

Professor of “Satellite Geodesy”

at Technische Universität Berlin.

He has engaged in space geo-

detic 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).