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BDS Real-time Satellite Clock Offsets Estimation with Three Different Datum Constraints

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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
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.
... Therefore, the batch-estimated receiver and satellite clocks are aligned as the clock relative to the reference clock. The reference clock strategies include selecting one satellite clock as zero, one receiver clock as zero and the mean value of satellite clocks as zero, which are equivalent to each other [36]. In order to ensure the stability of the satellite clock batch estimation, we select the zeromean condition as the reference clock [37]. ...
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