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34

Journal of Global Positioning Systems (2021)

Vol. 17, No. 1: 34-47

DOI: 10.5081/jgps.13.1.34

BDS Real-time Satellite Clock Offsets Estimation

with Three Different Datum Constraints

Guanwen Huang1, Wei Xie1*, Wenju Fu2, Pingli Li3, Haohao Wang1 and Fan Yue1

1. College of Geology Engineering and Geomatics, Chang’an University, Xi’An 710054, China.

2. State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan

University, Wuhan 430079, China.

3. The 20th Research Institute of China Electronic, Technology Group Corporation, Xi’An 710068, China.

* Correspondence: chdxiewei@chd.edu.cn

Abstract A clock offset datum should be selected to

separate the satellite and receiver clock offset when

estimating the Global Navigation Satellite System

(GNSS) satellite clock offset. However, the applicable

conditions and performance of the estimated satellite

clock offset vary for different clock offset datums. In

this paper, the BeiDou Navigation Satellite System

(BDS) real-time satellite clock offset is estimated by

using the undifferenced (UD) model with three datum

constraints: receiver clock datum, satellite clock

datum, and zero-mean condition (ZMC) datum. The

constraint conditions of three clock offset datums are

discussed, the transformation relationship among the

three datums constraints is derived, and the

characteristics of three clock offset datums are

analyzed. One hundred stations were used to perform

the experiments, and the results show mean standard

deviation (STD) values of ±0.118, ±0.124, and ±0.101

ns for the receiver clock, satellite clock, and ZMC

datum, respectively. The mean clock offset model

precisions with the three datum are ±0.497, ±0.646,

and ±0.442 ns, respectively. The frequency stability

with ZMC datum results showed the best performance

when the integration time is less than 10,000 s. For

precise point positioning (PPP), the ZMC datum

results show better performance among the three

datum constraints. This study can provide a reference

for clock offset datum selection for GNSS satellite

clock offset estimation.

Keywords: Clock offset datum;·Receiver clock

datum;·Satellite clock datum;·ZMC datum;·Satellite

clock offset estimation; Satellite Clock performance

1 Introduction

The real-time satellite clock offset is one of the key

elements for real-time precise point positioning

(RT-PPP) and satellite clock performance monitoring

and evaluation. PPP technology has been

demonstrated as an effective tool that can be used in

areas such as GNSS meteorology [Lu, et al, 2015],

precise orbit determination of low earth orbit (LEO)

satellites [Bock, et al, 2002], time and frequency

transfer [Defraigne, et al, 2015], precision agriculture

[Guo, et al, 2018], earthquake and tsunami early

warning. Satellite clock performance monitoring is

significantly crucial for satellite clock offset

prediction, satellite clock offset estimation and

integrity monitoring of satellite [Xie, et al, 2019].

The development of Chinese BeiDou Navigation

Satellite System (BDS) followed the ‘three-step’

strategy. The first phase was the demonstration system

(BDS-1) established in 2003, in which three

geostationary orbits (GEO) satellites were launched.

The BDS regional navigation satellite system (BDS-2)

was established on 27 December 2012. A constellation

Editor in charge: Dr. Yang Gao

35

of 14 satellites, including five GEO satellites, five

inclined geosynchronous orbits (IGSO) satellites, and

four medium earth orbit (MEO) satellites were

launched [Yang, et al, 2019]. The BDS-3 was

completed on 31 July 2020. It contains 30 satellites,

which can provide global positioning, navigation, and

timing (PNT) services, and whose quality is based on

the performance of the real-time satellite clock.

The atomic clock equipped on a navigation

satellite is easily affected by the external environment

and its characteristics, it is difficult to use the

mathematical model for prediction. Therefore, the

real-time satellite clock offset should be estimated

using the ground observation stations, and the satellite

and receiver clock offset parameters are estimated

simultaneously when estimating the satellite clock

offsets. There is a linear dependency relationship

between the satellite and receiver clock offset,

resulting in rank deficiency in the observation

equations. Therefore, one clock offset datum should

be selected to eliminate the rank deficiency [Liu et al.,

2019]. The clock offset datum is the precondition that

ensures the stability and continuity of the clock offset

datum. Once the clock offset datum is missing or

interrupted, the clock offset cannot be estimated. In

previous studies on clock offset datum selection, Jiang

et al. [2019) estimated the satellite clock offsets by

using the ZMC datum to separate satellite and receiver

clock offsets, and Fu et al. [2018) selected a receiver

clock as a time reference when performing the

GPS/BDS satellite clock estimation. By employing

three different kinds of reference stations, internal

cesium; external cesium; and external hydrogen-maser,

as clock offset datum when conducting satellite clock

estimation[Kamil et al. 2019]. Furthermore, Chen et al.

[2018] applied one satellite clock as a clock offset

datum to estimate the GPS/BDS/Galileo satellite clock

offset. However, the transformation relationship and

characteristics of three clock offset datums have not

been discussed. Furthermore, the satellite clock

performance comparison among the estimated

real-time satellite clock offset estimations with three

different datum constraints has not been reported.

In this paper, we focus on BDS real-time satellite

clock offset estimation with three different datum

constraints including one receiver clock datum, one

satellite clock datum, and the ZMC datum. BDS

real-time satellite clock offset estimation with three

datum constraints was conducted, and the satellite

clock performance of the estimated clock offset was

compared by using three datum constraints. This study

is organized as follows: after this introduction, the

observation and functional model is introduced, and

the constraint condition and transformation

relationship of three clock offset datums are discussed.

Then, the characteristics of the three datum constraints

are analyzed. Next, the BDS real-time satellite clock

offset estimation experiments with three datum

constraints are conducted, and the clock performance

in terms of clock offset accuracy, clock offset model

precision, frequency stability, and PPP for three

different clock offset datums are compared and

analyzed. Finally, the conclusions are presented.

2 Methodology

In this section, the undifferenced (UD) observation

and functional model are introduced. Then, the

constraint conditions of the satellite clock, receiver

clock, and ZMC datum are provided. Finally, the

transformation relationship of the three datums

constraints is derived.

2.1 Observation model

The raw observation equation of the code and

carrier phase can be expressed as follows:

,

=+()+,+,

+

+,

(1)

,

=+()+,,

+

,

++,

(2)

wherein and represent the receiver and satellite,

respectively, refers to the different frequency bands,

is the geometric distance between one satellite and

receiver, is the speed of light in vacuum, and

and are the receiver and satellite clock offsets,

respectively, , and are the code hardware

delays on the frequency in meters of the receiver

and the satellite, respectively, , and are the

phase delays on the frequency in cycles of the

receiver and satellite, respectively, ,

is the

ionospheric delay on the first frequency, is the

ionospheric coefficient for different frequencies, and it

36

can be expressed as =/, is the

wavelength of the carrier phase in meters, ,

is the

carrier phase ambiguity in cycles, and are the

wet mapping function and zenith wet delay of the

tropospheric delay, respectively, and ,

and ,

are the measurement noise and multipath error for the

code and carrier phase, respectively.

In the data processing of the dual-frequency

observation model, the ionospheric-free (IF)

combination is selected to calculate based on the two

observations at two different frequency bands,

therefore, the first-order ionospheric delay is

eliminated. After the IF combination is applied, the

code hardware delays of the receiver and satellite can

be absorbed by the receiver and satellite clock offset,

respectively. The phase hardware delays of the

receiver and satellite can be absorbed by the

ambiguity parameter. Therefore, the observation

model can be reformulated as follows:

,

=+()++ (3)

,

=+()+

,

++ (4)

where and are the re-parameterized receiver

and satellite clock offset, respectively. The IF

combination of dual-frequency code hardware delay

can be absorbed by them, and they can be expressed

as =+,, =+

. Furthermore,

,

is the re-parameterized float IF phase ambiguity,

which absorbs the phase and code hardware delays of

the receiver and satellite and can be expressed as

,

=,

+,

+

, [Wang, et

al, 2019]. It should be noted that the phase wind-up,

earth rotation correction, relativistic effect, satellite

antenna phase center offsets (PCO), phase center

variations (PCV), solid tide and ocean tide, and pole

tide are carefully considered. When the satellite clock

offset estimation is conducted using the UD

observation model, the satellite orbit and station

coordinates are fixed. Therefore, the parameters,

including the satellite and receiver clock offset, the

wet delay of the tropospheric and the float ambiguities,

should be estimated.

2.2 Functional model

We assume that there are satellites observed by

stations, and pseudorange and carrier phase

observations in total at an epoch. According to

observation models (3) and (4), ,

and ,

on

the left side of the equation are moved to the right,

and the observation model can be transformed into a

functional model. The satellite and receiver clock

offsets are estimated as white noise, the troposphere is

regarded as a constant during a period of time, and the

ambiguity is also a constant for each arc if no cycle

slip occurs. Therefore, the satellite and receiver clock

offset parameters are placed in a matrix, and the

troposphere and ambiguity parameters are fed into

another matrix. The functional model based on the

observation model of all stations can be expressed as:

×=

×()

()×+

×()

()×

× (5)

where denotes the residuals vector, is the

coefficient matrix of receiver and satellite clock offset,

is the satellite and receiver clock offset parameter

vector. is the coefficient matrix of the troposphere

and ambiguity, is the troposphere and the

ambiguity parameter vector, and is the observation

vector of the code and carrier phases. Then, the

normal equation can be expressed as follows:

= (6)

where =

, =

and

=

.

It is noted that there is a linear dependency

between the receiver and the satellite clock offset,

resulting in rank deficiency, whose number is 1 in the

matrix, resulting in (6) cannot be determined.

Therefore, one clock offset datum should be selected

to eliminate the rank deficiency, and then the receiver

and satellite clock offset can be separated.

2.3 A satellite as clock offset datum

The clock offset value of one satellite is set as zero for

each epoch when the clock offset datum is one

satellite. Assuming that the clock offset of the th

satellite is zero, it can be expressed as follows:

= 0 (7)

where is the order of the clock offset datum

satellite among all of satellites. According to the

37

constraints condition, it can be expressed in the

following matrix form:

×()

()×= 0

×()

= [0,0, … ,0,0. . .0,1, 0

] (8)

According to the adjustment of indirect observations

with the constraints model, and combining (6) and (8),

the matrix can be reformulated as:

=+ (9)

Then, the rank deficiency of the matrix can be

eliminated, and the satellite clock offset can be

estimated with the satellite clock datum.

2.4 A receiver as clock offset datum

The clock offset value of one receiver is set to zero

when the clock offset datum is one receiver. Assuming

the th receiver clock offset is zero, it can be

expressed as follows:

= 0 (10)

where is the order of the clock offset datum

receiver among all of receivers. According to the

constraints condition, it can be expressed in the

following matrix form:

×()

()×= 0

×()

= [ 0,

0,0, … ,0,0,0 … . .1] (11)

where the element ‘1’ is the 1st row and (++

+)th column in the vector. According to the

adjustment of indirect observations with the

constraints model, combining (6) and (11), the matrix

can be reformulated as:

=+ (12)

Then, the rank deficiency of the matrix can be

excluded, and the satellite clock offset can be

estimated with the receiver clock datum.

2.5 ZMC as clock offset datum

The mean value of all satellite clock offsets is set as

zero when the clock offsets datum is ZMC, and it can

also be expressed as the sum value of all satellite

clock offsets being zero, which can be expressed as:

= 0 (13)

where is the number of satellites. According to the

constraints condition, it can be expressed in matrix

form:

×()

()×= 0

×()

= [

×, 0

×()]

×= [1,1, … 1]

(14)

where the elements of from the 1st column to the

th column are ‘1’. According to the adjustment of

indirect observations with the constraints model and

combining (6) and (14), the matrix can be

reformulated as

=+ (15)

Then, the rank deficiency of the matrix can be

excluded, and the satellite clock offset can be

estimated with the ZMC datum.

2.6 Transformation relationship analysis of three

datum restraints

It can be seen from (8), (11), and (14) when different

clock offset datum constraints are applied to satellite

clock offset estimation, their difference is the

constraint condition. After selecting the clock offset

datum, the estimated parameter matrix can be

expressed as

= (+)= (16)

= (+)= (17)

= (+)= (18)

where, , , and are estimated parameter

matrices based on the satellite clock offset datum,

receiver clock offset datum, and ZMC datum,

respectively. , , and are the cofactor

matrices for the satellite clock offset datum, receiver

clock offset datum, and ZMC datum, respectively.

According to (11), (16), and (17), the relationship

between and can be expressed as:

==(+)= (19)

Considering that the vector is rank deficiency, and

the matrix and the constraints condition (, ,

) are not linearly dependent, therefore =

== 0. Then, can be expressed as:

=()+()(20)

38

= (+)()=

() (21)

Therefore, the can be expressed as:

=(+)==

() (22)

Then, the relationship between and can be

expressed as:

= [()] (23)

Therefore, the satellite clock offset can be transformed

from the receiver clock datum to the satellite clock

datum by using the vector and the clock offset

with the receiver clock datum. Similarly, the

relationship between and , , and can

also be transformed.

3 Characteristics of three datums constraints

Three datum constraints can be theoretically

transformed. In fact, the clock offset datum noise,

frequency stability, and applicable conditions for

different clock offset datums are different. Therefore,

the characteristics of three clock offset datums are

analyzed in this section.

The receiver clock offset value is constrained to

zero for each epoch when the clock offset datum is

one receiver, and the remaining satellites and receiver

clock offset are estimated with respect to this clock

offset datum. The advantage is that it can ensure that

all satellite clock offsets have a value and this value is

not equal to zero, which is beneficial to all satellite

clock performance monitoring and all satellite clock

performance can be evaluated by using the satellite

clock offset. The disadvantage is that one receiver

clock offset value is set as zero, which can negatively

impact the performance evaluation of this receiver

clock. Furthermore, during real-time satellite clock

offset estimation, the observation data of the clock

offset datum receiver cannot be received by the user

owing to the network delay of real-time observation

data transmission, temporary modem failure

[EI-Mowafy, et al, 2017], or other factors. There is no

observation data in the clock offset datum receiver,

and the satellite clock offset cannot be estimated at

that time. To ensure that the satellite clock offset can

be estimated continuously, the clock offset datum

should be changed into another receiver. However, all

of satellite clock offset values would jump almost the

same amount at that time. If this clock offset is used

for frequency stability estimation and modeling of the

satellite clock offset, the performance of the satellite

clock cannot be reflected due to the clock offset jump,

which is not good for satellite clock performance

evaluation, and the modeling of the satellite clock

offset is also inaccurate. When each epoch of the

clock offset datum receiver has observation data, one

receiver clock can be used as a clock offset datum for

the clock offset estimation. Furthermore, when

regional network stations are used for real-time

satellite clock estimation, it is appropriate that the

receiver clock is selected as the datum.

When the clock offset datum is one satellite clock,

one satellite clock offset value is constrained to zero

for each epoch. All receiver clock offset values can be

reserved under these circumstances, which is

beneficial for the performance evaluation of the

receiver clock. The drawback is that one satellite

clock offset value is zero, which is unfavorable for all

satellite clock performance monitoring and evaluation.

When the clock offset is estimated, the receiver clock

offset needs to be reserved, and one satellite clock

offset value is zero does not affect its application, this

datum can be recommended.

When the real-time satellite clock offset is

estimated and the clock offset datum is ZMC, the

mean value of all satellite clock offsets is zero.

Therefore, all satellite and receiver clock offset values

are not zero, which is beneficial to all satellite and

receiver clock performance evaluations. Furthermore,

because all satellites and receiver clock offset values

are not equal to zero, the satellite and receiver clock

noise can be absorbed by itself. However, the satellite

signal cannot be tracked by any receiver on the ground

owing to the signal loss of lock or other failures, as

shown in Figure 1. It can be seen that C07 satellite

clock offset experienced data disruption at 13:44:30

because the C07 satellite could not be tracked by all

stations. Under this condition, the satellite clock offset

jump occurred, and the satellite clock offset data was

discontinuous, which is not suitable for all satellite

clock performance evaluations. A short-term clock

39

offset prediction [El-Mowafy et al. 2017, Zhao et al.

2020] of the C07 satellite clock may guarantee the

continuity of all satellite clock offsets. The ZMC

datum is suitable when the satellite and receiver clock

offset must be reserved. Furthermore, when the

number of estimated satellite clock offsets of all

epochs are the same, the ZMC datum is also suitable.

Fig.1 Time series of BDS satellite clock offset

4 Experiments and results

In this section, the data and processing strategy are

described first. Then, the frequency stability of the

clock offset datum is analyzed. Thereafter, the clock

performance of the three datum constraints is

evaluated in terms of clock offset accuracy, clock

offset model precision, frequency stability, and PPP.

4.1 Data and processing strategy

The BDS-2 was selected to conduct the experiment.

The simulated real-time satellite clock offset

estimation is applied. The 100 observation stations

from IGS Multi-GNSS Experiment (MGEX) network

[Montenbruck et al. 2017] evenly distributed all over

the world were used to estimate the BDS satellite

clock offset products. And the BDS signal can be

tracked by all of them. Figure 2 shows the geographic

distribution of the stations. The red dots indicate the

stations used to estimate clock offset, while the blue

triangles mean the stations applied for perform

RT-PPP. The experiment was simulated in real-time

mode, and the clock offset estimation was carried out

from 31 March to 6 April 2019 (DOY 090–096) by

using the rtclk (Real-Time Clock) software developed

by BeiDou Analysis and Service Center of Chang’an

University [Fu et al. 2019]. In the process of clock

offset estimation, observations with B1/B2 frequency

and 30 s sample rate were applied, the elevation mask

was set to 7°, and the GBM final orbit products were

used [Deng et al. 2014]. For quality control, when one

residual of observation was larger than 8.5 times the

median of normalized residuals, this residual with

respect to the observation data was deleted [Fu et al.

2018]. The details of the data process strategy for

clock offset estimation can be found in Table 1.

Fig. 2 Geographic distribution of stations. The red

points denote 100 stations are used for clock offset

estimation, the blue triangle denote 10 stations are

applied for PPP

To compare the satellite clock performance with

three different datum constraints, three schemes were

designed for comparisons, which are as follows:

Scheme 1: BDS real-time satellite clock estimation

using the TID1 receiver clock offset as datum. Scheme

2: BDS real-time satellite clock estimation using the

C08 satellite clock offset as datum. Scheme 3: BDS

real-time satellite clock estimation and the clock offset

datum imposed on ZMC datum of all satellite clock. It

is noted that for different constraints, the clock offset

processing of GEO/IGSO/MEO satellite are treated as

the same.

4.2 Clock offset datum stability analysis

The frequency stability is used to describe the random

fluctuation of the atomic clock output frequency

caused by noise. The overlapping Hadamard deviation

and overlapping Allan deviation were selected to

evaluate the frequency stability of the rubidium

atomic clock and the passive hydrogen maser (PHM),

respectively. Based on the clock offset data,

overlapping Hadamard deviation and overlapping

Allan deviation can be expressed as follows [Riley

10 11 12 13 14 15 16 17 18

-8

-4

0

4

8

C01 C02 C03 C04 C05

C06 C07 C08 C09 C10

C11 C12 C13 C14

Clock offset (10

-4

S)

Time (Hour)

40

2007]:

()=

()[3+ 3]

(24)

()=

()[2+]

(25)

where () indicates the frequency stability, is

the number of clock offset data, is the smooth

factor, and is the smooth time.

Table 1 Data process strategy for BDS real-time

clock estimation

Items Strategy

Observations Ionospheric-free (IF)

combination

Signal selection B1/B2

Sampling rate 30 s

Elevation mask 7

°

Weight 1 when

>30

°;

2sin(

) for

<30

°

Estimator Sequential least squares

Satellite orbit Fixed to the GBM final orbit

Satellite antenna phase

center and variation

ESA model [Dilssner et al.

2014]

Phase wind-up effect Corrected [Wu et al. 1993]

Relativistic effect Corrected

Station coordinate Fixed to the IGS weekly

solution

Station displacement Solid earth tide, pole tide, ocean

tide

Receiver antenna phase

center and variation igs14.atx

Satellite clock Estimated as white noise

Receiver clock Estimated as white noise

Tropospheric delay

Saastamoinen model and GMF

mapping functions, estimated

every one hour

Phase ambiguities Constants for each continuous

tracking arc

Clock offset datum TID1 receiver clock / C08

satellite clock / ZMC

Maciuk [2009] has demonstrated that the

frequency stability of the satellite clock using the

clock offset datum receiver equipped with a

hydrogen-maser clock is better than that of the

internal and cesium atomic clock when conducting

satellite clock estimation. There is a close correlation

between the reference clock and the frequency

stability of the estimated clock offset. Therefore, when

the satellite clock offset estimation is based on the

receiver clock offset datum, the receiver equipped

with a hydrogen-maser clock is usually selected to

ensure a high-precision satellite clock. It should be

noted that in this study, the TID1 station was equipped

with an external H-maser and a rubidium atomic clock

installed on the C08 satellite clock during the

experiment [Han et al. 2011]. The frequency stability

of the TID1 receiver and C08 satellite clock was

evaluated using IGS and GBM rapid products,

respectively. The mean sub-daily frequency stability

of the C08 satellite clock and TID1 receiver clocks

was evaluated using overlapping Hadamard deviation

and overlapping Allan deviation methods, respectively,

and it is presented in Figure 3. It was observed that the

frequency stability of TID1 receiver clock was

significantly better than that of C08 satellite clock,

and the frequency stability of the receiver clock was

about one order of magnitude higher than that of the

C08 satellite clock.

Fig. 3 The mean sub-daily frequency of the C08

satellite and TID1 receiver clock

4.3 Satellite clock offset accuracy

To evaluate the quality of the estimated satellite clock

101102103104105

10-15

10-14

10-13

10-12

10-11

Frequency stability

Averaging Interval (S)

C08

TID1

41

with three different clock offset datum, the clock

offset accuracy is compared to the GBM final clock

products because the satellite orbit is fixed to the

GBM when estimating BDS real-time satellite clock

offsets. The standard IGS clock offset evaluation

procedure is adopted, in which the STD and root mean

square (RMS) are applied to assess the clock quality.

The STD indicates the values of the processing quality

of the phase observations and shows the quality of the

clock solution, while the RMS values can reflect the

consistency between code and phase observations,

which shows the accuracy of code observations [Ge et

al. 2012; Liu et al. 2019].

The experiment was conducted from DOY 090 to

096, 2019; considering that the convergence time in

DOY 090 and the C07 satellite experienced the loss of

lock in DOY 095, the result can be impacted by the

ZMC datum [Odijk et al. 2016]. Therefore, the results

of DOY 090 and 095 have been excluded. The mean

STD and RMS of the BDS satellite clock with three

datum constraints are shown in figure 4. For the three

datum constraints, it can be seen that STD is better

than 0.2, 0.1, and 0.25 ns for GEO, IGSO, and MEO

satellite clock offset, respectively. In terms of the

RMS, the GEO, IGSO, and MEO satellite clock offset

are better than 1.0, 0.5, and 2.0 ns. The poor STD and

RMS of the MEO satellite clock offset can be

attributed to the fact that the observation stations

tracked by the BDS satellite are limited around the

world. Compared to the IGSO satellite, the relatively

poor STD and RMS of the GEO satellite clock is due

to the weak geometry strength of the GEO satellite

orbit compared to that of the IGSO; and geometry

observation conditions change slowly, resulting in the

decreased orbit accuracy of the GEO satellite

compared to the IGSO satellite, which then impacts

the clock offset estimation accuracy [Liu et al. 2017;

Zhao et al. 2013].

The mean STD and RMS for GEO, IGSO, and

MEO satellite clock with three datum constraints are

shown in Table 2. For the STD, the value between the

TID1 and ZMC datum shows almost the same value

for the GEO satellite clock, the difference is 0.002 ns

which can be neglected. However, the STD is the

worst for the GEO satellite clock when the clock

offset datum is the C08 satellite clock. For the IGSO

and MEO satellite clock, the STD with TID1 receiver

clock datum and C08 satellite clock datum is almost

identical, and the clock offset accuracy with the ZMC

datum is slightly better than the TID1 receiver and

C08 satellite clock. For the RMS, the ZMC datum

shows the best accuracy regardless of whether it is

GEO, IGSO, or MEO satellite, and the RMS of GEO,

IGSO, and MEO satellite clock offset is 0.449, 0.177,

and 0.787 ns, respectively. Compared to the TID1

receiver clock datum, the improvements in clock

accuracy are 27.11%, 45.54%, and 44.07% for the

GEO, IGSO, and MEO satellite clock, respectively.

The improvement is 30.50%, 10.61%, and 41.00% for

GEO, IGSO, and MEO satellite clock when compared

to the C08 satellite clock datum, respectively.

Therefore, it can be inferred that the consistency

between code and phase observations of the ZMC is

better than the TID1 receiver and C08 satellite clock

datum.

For real-time satellite clock estimation with three

datum constraints, the mean STD is 0.118, 0.124, and

0.101 ns for the TID1 receiver clock, C08 satellite

clock, and ZMC datum, respectively. In terms of RMS,

the mean RMS is 0.661, 0.646, and 0.405 ns with

TID1 receiver clock, C08 satellite clock datum, and

ZMC datum, respectively. The mean differences of

STD and RMS between the TID1 receiver and C08

satellite clock datum are 0.006 and 0.015 ns, which

shows that these two datum constraints have the same

comparable clock offset accuracy. However, the STD

and RMS of the ZMC show better performance. It has

been demonstrated that three datum constraints can be

theoretically transformed, and the clock offset

accuracy should be the same for three different clock

offset datum. However, when the clock offset datum is

the receiver or satellite clock, the clock offset datum is

set as zero [Liu et al. 2019; Loyer et al. 2012], the

clock offset datum noise can be absorbed into other

receiver or satellite clock, resulting in a decrease in

the accuracy of the satellite clock offset, and therefore,

there is a clock offset accuracy difference for the

different clock offset datum. For the ZMC datum, all

satellite and receiver clock offsets can be estimated,

and all clock offsets are not equal to zero, thus the

clock noise can be decreased and the clock accuracy

42

can be improved. Therefore, the STD and RMS with

ZMC are better than those of the TID1 receiver and

C08 satellite clock offset datum.

Fig.4 STD and RMS of each BDS satellites with

three datums constraints

Table 2 Clock offset accuracy of BDS GEO,

IGSO and MEO satellites with three

datums constraints

STD RMS

TID1 C08 ZMC TID1 C08 ZMC

GEO 0.109 0.122 0.111 0.616 0.646 0.449

IGSO 0.075 0.070 0.058 0.325 0.198 0.177

MEO 0.218 0.217 0.169 1.407 1.334 0.787

4.4 Satellite clock offset model precision

The satellite clock offset model precision is also called

fitting precision, which is the RMS of the fitting

residuals of the satellite clock offset model, and it can

reflect the noise level characteristics of the atomic

clock, which can directly determine the accuracy and

stability of real-time clock offset prediction and

estimation [Huang et al. 2013]. Therefore, the satellite

clock offset model precision is selected to evaluate the

clock quality. The rubidium atomic clocks are

equipped on the BDS-2 satellites [Han et al. 2011],

and the frequency drift is apparent. Therefore, the

quadratic polynomial model was employed to fit the

daily clock offset. The RMS is applied to express the

noise level, and it can be expressed as follows:

=+()+

()+ (26)

=

(27)

where is clock offsets; , and are the

phase, frequency, and frequency drift, respectively;

and are the observation and reference epochs,

respectively; is the fitting residuals, which is the

difference between the satellite clock offset value and

the satellite clock offset model value for each epoch

[Xie et al. 2019]. After is calculated, the clock

offset model precision can be obtained.

The mean satellite clock offset model precision

of each BDS satellite clock with three datum

constraints is shown in figure 5. For the GEO satellite

clock, the satellite clock offset model precision is

almost the same when the TID1 receiver clock and

ZMC are used as clock offset datum, while the

satellite clock offset model precision of C08 satellite

clock datum is significantly worse than that of the

TID1 receiver clock datum and ZMC datum. The

clock offset model precision of the TID1 receiver

clock datum shows the poorest performance among

the three datum constraints for the IGSO satellite

clock, and there is a small difference in clock offset

model precision between the ZMC and C08 satellite

clock datum. For the MEO satellite clock, the ZMC

datum shows the best clock offset model precision

among the three datum constraints. The values for the

C11, C12, and C14 satellite clocks are 0.357, 0.227,

and 0.244 ns, respectively. The clock offset model

precision with TID1 receiver clock datum is slightly

worse than that of the ZMC datum, the worst satellite

clock offset model precision is the C08 satellite clock

datum. Overall, the mean satellite clock offset model

precision is 0.497, 0.580, and 0.442 ns for the TID1

receiver clock, C08 satellite clock, and ZMC datum,

respectively, and this difference is small. Compared to

the TID1 receiver and C08 satellite clock datum, the

satellite clock offset model precision can be improved

by approximately 11.06% and 23.79% when the ZMC

datum is used. Therefore, when estimating BDS

real-time clock offset, the clock offset datum using the

ZMC method is more suitable for modeling the

satellite clock offset.

0.0

0.1

0.2

0.3

0.4

STD ( ns )

TID1 C08 ZMC

C01

C02

C03

C04

C05

C06

C07

C08

C09

C10

C13

C11

C12

C14

0

1

2

3

RMS ( ns )

BDS Satellite

43

Fig. 5 The mean fitting precision of each BDS

satellites with three datums constraints

4.5 Frequency stability

The mean sub-daily frequency stability of the satellite

clock for five days with the three datum constraints is

shown in Figure 6. The sub-daily frequency stability

shows almost the same variation trends among the

three datum constraints, which indicates that clock

frequency stability variation is not impacted by the

clock offset datum. Furthermore, the visible ‘bump’

has appeared for C11 satellite clock at an integration

times of 2,000 s, which can be attributed to the impact

of the special period existed in the C11 satellite clock

offset. The special period may be caused by the

hardware noise of its atomic clock [Huang et al. 2018,

Wang et al. 2016]. Furthermore, the frequency

stability decreases when the integration time exceeds

10,000 s for some GEO and IGSO satellite clocks,

such as C04, C07, and C10 satellite clocks, which can

also be attributed to the impact of periodic terms.

For integration times of 100 s, 1000 s, 10000 s,

and 20000 s, the frequency stability of the different

clock offset datums is shown in Figure 7. It can be

seen that when the averaging interval time is 100 s,

the frequency stability with the C08 satellite clock

datum exhibits the poorest performance. The satellite

clock frequency stability with the ZMC datum is

slightly better than that with the TID1 receiver clock

offset datum.

When the averaging interval is 1000 s, the mean

frequency stability is 1.15×10-13, 2.06×10-13, and

1.09×10-13 for GEO satellite with the TID1 receiver

clock datum, C08 satellite clock, and ZMC datum,

respectively. As for IGSO satellite clock, the mean

frequency stability is 1.79×10-13, 2.51×10-13, and

1.68×10-13 for TID1 receiver clock datum, C08

satellite clock datum, and ZMC datum, respectively.

For the MEO satellite clock, the mean frequency

stability with the TID1 receiver clock datum, C08

satellite clock datum, and ZMC datum is 1.51×10-13,

2.31×10-13, and 1.41×10-13, respectively. The GEO

satellite clock presents the best performance, and the

frequency stability with the ZMC datum shows the

best performance.

When the averaging interval is 10000 s, the mean

frequency stability for GEO, IGSO, and MEO satellite

clock is 5.62×10-14, 7.64×10-14, and 5.57×10-14 with

TID1 receiver clock datum, 6.68×10-14, 8.02×10-14,

and 6.17×10-14 for the C08 satellite clock datum,

4.31×10-14, 5.88×10-14, and 3.63×10-14 for the ZMC

datum, respectively. Compared to the TID1 receiver

clock and C08 satellite clock, the frequency stability

with the ZMC datum can be improved by 23.34%,

23.04%, 34.84% and 35.40%, 26.63%, and 41.13%

for GEO, IGSO, and MEO satellite clock, respectively.

Moreover, the frequency stability performance of the

MEO satellite clock is better than that of the GEO and

IGSO satellite clocks.

When the integration time is longer than 10000 s,

the frequency stability variation is more complex.

Previous studies have demonstrated that the periodic

terms exist in the BDS satellite clock offset [Wang et

al. 2016], and the frequency stability is impacted by

this periodic terms when the integration time is longer

than 10000 s. The degree of frequency stability of

each satellite clock affected by periodic terms is still

unknown, and it should be studied in the future. When

the averaging interval is less than 10000 s, the

frequency stability difference can be attributed to the

difference in clock datum frequency stability.

Fig. 6 Sub-daily frequency stability of each BDS

satellites with three datum constraints

C01

C02

C03

C04

C05

C06

C07

C08

C09

C10

C13

C11

C12

C14

0.0

0.5

1.0

1.5

MEO

IGSO

Fitting precision (ns)

BDS Satellite

GEO TID1 C08 ZMC

10

1

10

2

10

3

10

4

10

5

10

-15

10

-14

10

-13

10

-12

10

-11

10

1

10

2

10

3

10

4

10

5

10

1

10

2

10

3

10

4

10

5

Frequency stability

Averaging Interval (S)

C01 C02 C03 C04 C05 C06 C07

C08 C09 C10 C13 C11 C12 C14

TID1

Averaging Interval (S)

C08

Averaging Interval (S)

ZMC

44

Fig. 7 Frequency stability of each BDS satellite clock

4.6 PPP

One of the applications of real-time estimated satellite

clock offset is in RT-PPP. To analyze the difference in

satellite clock performance with three datum

constraints, the BDS kinematic PPP is conducted by

using three different clock offset products.

Furthermore, PPP using GBM final clock offset

products is also performed for comparison. For PPP,

10 stations are selected to conduct the PPP (figure 2).

Among them, four stations are involved in the clock

offset estimation, while another six are not. The

sample rate of observations is 30 s, the satellite orbit is

GBM final orbit, and the ambiguity is set as a float

solution. The observation data are from DOY 091 to

096, 2019, except for DOY 095, 2019. The RMS is

calculated after PPP convergence, and the mean RMS

for each station is shown in figure 6.

For most stations, the BDS kinematic PPP

accuracy in the east, north, and up components present

almost the same performance for the three datum

constraints, ranging from 7 to 14 cm, 4 to 11 cm, and

14 to 28 cm, respectively. Due to the poor accuracy of

the GEO satellite orbit, the accuracy of BDS

kinematic PPP is poorer than that of GPS [Zhou et al.

2020]. However, with the construction of BDS-3, the

PPP accuracy can be significantly improved in the

future [Jiao et al. 2019]. The mean PPP accuracy of

ten stations with the three datum constraints and

GBMs are presented in Table 3. The mean PPP

accuracies in the east, north, and up directions are

10.51, 6.67, and 21.51 cm with the TID1 receiver

clock datum, 11.97, 6.76, and 21.08 cm with C08

clock datum, and 10.82, 6.58, and 20.08 cm with

ZMC datum, respectively. ZMC datum shows better

positioning performance among the three datum

constraints. Considering that the STD of the clock

accuracy of the three clock offset datums is between

0.101 and 0.121 ns, the difference between the three

datum constraints and GBM is within the normal

range.

Fig. 8 RMS of BDS kinematic PPP in east, north and

up for each station

Table 3 Mean PPP accuracy with three datums and

GBM (units: cm)

TI

D1

C0

8

ZM

C

GB

M

GBM-T

ID1

GBM-

C08

GBM-Z

MC

Eas

t

10.

51

11.

97

10.

82

8.6

6

1.85 3.31 2.16

Nor

th

6.6

7

6.7

6

6.5

8

5.8

8

0.79 0.88 0.70

Up 21.

51

21.

08

20.

08

18.

29

3.22 2.79 1.79

0

4

8

12

Frequency stability / (×10

-13

)

TID1 C08 ZMC

GEO IGSO MEO

0

2

4

0.0

0.4

0.8

1.2

C01

C02

C03

C04

C05

C06

C07

C08

C09

C10

C13

C11

C12

C14

0.0

0.4

0.8

1.2

BDS Satellite

0

5

10

15

20

East (cm)

TID1 C08 ZMC GBM

0

5

10

15

North (cm)

ANMG

DARW

DGAR

HOB2

NTUS

PNGM

PTGG

SIN1

USUD

YAR3

0

10

20

30

40

Up (cm)

Station

45

5 Conclusions

This study presents the BDS real-time satellite clock

offset estimation with three datum constraints,

including one receiver clock, one satellite clock, and

ZMC. It proves that the three datum constraints can be

theoretically transformed. Furthermore, the

characteristics of the three datum constraints were

analyzed. The receiver clock is recommended as the

clock offset datum under the following conditions: the

clock offset datum of each epoch has observation data,

and the regional stations network is used to estimate

the real-time satellite clock offset. If all the receiver

clock offsets are reserved and one satellite clock offset

value being zero does not affect the application of the

satellite clock offset, one satellite can be used as a

clock offset datum. When the number of satellite

clocks of all epochs is the same and the stations are

globally distributed, the ZMC datum is suitable.

To compare the clock offset quality of the three

datum constraints, 100 stations from the MGEX

network are used to estimate the BDS real-time

satellite clock. The BDS satellite clock performance

with three datum constraints is evaluated in terms of

clock offset accuracy, fitting precision, frequency

stability, and PPP. For clock offset accuracy, the mean

STD is 0.118, 0.124, and 0.101 ns for the clock offset

datum for the TID1 receiver clock, C08 satellite clock,

and ZMC datum, respectively. The mean RMS is

0.661, 0.646, and 0.405 for the TID1 receiver clock

datum, C08 satellite clock datum, and ZMC,

respectively. The mean fitting precision of the TID1

receiver clock datum, C08 satellite clock datum, and

ZMC are 0.497, 0.580, and 0.442 ns, respectively. As

for the frequency stability, when the integration is less

than 10000 s, the clock offset datum with ZMC shows

better performance. In terms of PPP, the ZMC shows

better positioning performance among three datum

constraints. The reason for the numerical difference

among the three datum constraints can be attributed to

the difference in frequency stability for the satellite

and receiver clock. For the ZMC datum, all satellites

and receiver clock offsets can be estimated, and the

satellite and receiver clock noise can be absorbed.

Therefore, clock performance is better.

Acknowledgments

The IGS, IGS-MGEX, GBM are greatly

acknowledged for providing the Multi-GNSS tracking

data, sinex coordinates, erp, satellite orbit and clock

products. This work was funded by the Programs of

the National Natural Science Foundation of China

(41774025, 41731066, 41904038) , the National Key

R&D Program of China (2018YFC1505102), the

Special Fund for Technological Innovation Guidance

of Shaanxi Province (2018XNCGG05), the Special

Fund for Basic Scientific Research of Central

Colleges (grant no. CHD300102269305,

CHD300102268305, Chang’an University), and the

China Postdoctoral Science Foundation

(2019M662713), and the Grand Projects of the

Beidou-2 System (GFZX0301040308), the

Fundamental Research Funds for the Central

Universities, CHD(300102261713).

Data Availability

BDS precise ephemeris was retrieved from:

ftp://ftp.gfz-potsdam.de/pub/GNSS/products/mgnss.

BDS raw observations from MGEX stations were

retrieved from: ftp://cddis.gsfc.nasa.gov/pub/gnss/

data/daily/.

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Authors Guanwen Huang is currently a

professor of Chang’an University,

Xi’An, P. R. China. He received

a B.Eng, M.Sc., and Ph.D. in

Surveying Engineering from

Chang’An University, in 2005,

2008, and 2012. His research

activities include GNSS Precise

Point Positioning, real-time satellite clock model, and

their applications.

Wei Xie is currently a Ph,D.

candidate at the College of

Geology Engineering and

Geomatics, Chang’An University,

Xi’an, P.R. China. He completed

his B.Sc. at Hunan University of

Science and Technology in 2017.

His research is focused on GNSS

satellite clock offset estimation and satellite clock

performance evaluation. Wenju Fu received his Ph.D.

from Chang’An University,

Xi’An, P.R.China in 2018. He

is currently a postdoctoral

fellow at the State Key

Laboratory of Information

Engineering in Surveying,

Mapping and Remote Sensing (LIESMARS), Wuhan

University. His research interest includes GNSS

Precise Point Positioning, precise satellite clock, and

LEO applications.

Pingli Li is an engineer at the

20th Research Institute of China

Electronic, Technology Group

Corporation, Xi’An, P.R. China.

He received his Bachelor and

M.Sc. from Chang'An University

in 2015 and 2018. His research

focuses on GNSS precise

satellite clock offset estimation.

Haohao Wang is currently a

Master graduate student at the

College of Geology Engineering

and Geomatics, Chang’An

University, Xian, P.R. China.

His research focuses on GNSS

real-time satellite clock offset

estimation.

Fan Yue is currently a Master

graduate student at the College of

Geology Engineering and

Geomatics, Chang’An University,

Xi’an, P.R. China.