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International Journal of Navigation and Observation

Volume 2008 (2008), Article ID 175468, 7 pages

http://dx.doi.org/10.1155/2008/175468

## GPS Time and Frequency Transfer: PPP and Phase-Only Analysis

^{1}Royal Observatory of Belgium, 1180 Brussels, Belgium^{2}Instituto Nazionale di Ricerca Metrologica, 10135 Torino, Italy

Received 31 August 2007; Revised 15 February 2008; Accepted 8 May 2008

Academic Editor: Patrizia Tavella

Copyright © 2008 Pascale Defraigne et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To compute precise point positioning (PPP) and precise time transfer using GPS code and phase measurements, a new software named *Atomium* was developed by the Royal Observatory of Belgium. *Atomium* was also adapted to perform a phase-only analysis with the goal to obtain a continuous clock solution which is independent of the GPS codes. In this paper, the analysis strategy used in *Atomium* is described and the clock solutions obtained through the phase-only approach are compared to the results from the PPP mode. It is shown that the phase-only solution improves the stability of the time link for averaging times smaller than 7 days and that the phase-only solution is very sensitive to the station coordinates used. The method is, however, shown to perform better than the IGS clock solution in case of changes in the GPS receiver hardware delays which affects the code measurements.

#### 1. Introduction

GPS-based geodetic time and frequency transfer are based on a consistent modeling of both code and carrier phase measurements, and is presently widely recognized for its high precision of about 100 picoseconds at each epoch [1–5]. Its major limitation concerns the discontinuities at the day boundaries, which are caused by the noise of the code measurements. Indeed, the carrier phase measurements contain an unknown initial ambiguity (integer number of cycles) and can, therefore, only provide information on the clock evolution or frequency transfer. The absolute synchronization error between two clocks is obtained from the code measurements, so that its accuracy is limited by the code noise (a few nanoseconds). As GPS data are classically analyzed by daily batches, the absolute clock synchronization error is determined for each day from the code measurements gathered during that day. Due to the colored signature of the code noise, implying large and long-term temporal variations in all code measurements, the clock solutions show discontinuities at the day boundaries. The origin of this colored noise in the codes is presently not yet fully understood [6], but it reduces our ability to access the true clock signal and stresses the necessity to develop a rigorous approach for continuous geodetic time transfer. One solution was already proposed in [7]; it consists in performing the PPP analysis on a longer data batch, reducing consequently the number of day-boundary jumps. A second approach was proposed in [8], where two methods were tested: a clock handover, analyzing daily data batches with one common observation epoch between two consecutive days (midnight), or a stacking of the carrier phase ambiguities, through a stacking of normal equations for consecutive days.

In order to have at our disposal a straightforward
tool for the rapid computation of clock solutions in a PPP mode, the ROB
developed *Atomium. Atomium* performs a
least squares analysis of single station GPS code and carrier phase data to
provide for each observation epoch a clock receiver solution consisting of the
synchronization error between the receiver clock and the IGS time scale (IGST). Recently, *Atomium* has been adapted to provide also a single station phase-only
solution. Some preliminary results were presented in [9], and a more
comprehensive study of the approach and of its sensitivity to the input
parameters is presented here.

The basic observable in the phase-only solution is the difference between successive carrier phase observations from the same satellite. Using a least squares adjustment, this observable is used to obtain the evolution of the clock synchronization error. The advantage of the phase-only approach is that it provides a continuous solution across the day boundaries, but the drawback is that the absolute clock synchronization errors of course cannot be determined due to the presence of the carrier phase ambiguities. This means that the continuity of the clock solution is interrupted after each tracking interruption.

For the computation of TAI [10] or any other time scale, each period of continuous clock solutions can be combined with a calibrated time transfer method such as two way satellite time and frequency transfer (TWSTFT) which allows synchronizing remote clocks with an accuracy better than 100 picoseconds and has a reliable long-term stability. However, TWSTFT measurements are disturbed by a diurnal variation of 1–3 nanoseconds peak to peak and their resolution is poor (one point each 2 hours in the best case). On the other hand, the GPS phase-only analysis has no calibration, but a very high resolution (5 minutes) and good short-term frequency stability. Combining the two methods will take advantage of the strong points of each of them: the accuracy for TWSTFT and the stability for GPS carrier-phase [10].

The first part of this paper presents the analysis procedure
used by *Atomium* to produce station position and clocks using PPP, and
shows the clock solutions obtained for several stations in comparison with the
IGS clock solution (IGS combined solution [11]) as well as the solutions
obtained by other software packages (NRCan and Bernese 5.0). The second part
describes the theoretical background of the phase-only approach implemented in *Atomium,* compares the phase-only with the PPP results, and investigates the sensitivity
of the phase-only method to its input parameters.

#### 2. PPP Approach

Precise point positioning (PPP) consists
in determining the station position, receiver clock, and tropospheric zenith
path delay without using any measurements from other stations. This approach
requires the availability of precise satellite orbits and clocks provided by an
external source. The PPP procedure is fully described in [12]; we just recall
here the main principle and its implementation in *Atomium*.

The observation equations for carrier phases ( and ) and pseudoranges ( and )
are with the geometric distance receiver-satellite, the satellite clock error, the receiver clock error, the tropospheric delay,
the ionospheric delay, *λ* the
carrier wavelength, the phase
ambiguity,
the instrumental code delay,
and the noise.

*Atomium* is
based on the ionosphere-free combinations of and and of and ,
named and , respectively. The observations are used
at the 5-minute sampling rate. The satellite positions are obtained using a Neville
interpolation on 12 points of the IGS sp3 files in which the satellite
positions are given at a 15-minute sampling rate. The satellite clock
corrections are extracted from the IGS clock files
in which the sampling rate is 5 minutes. The station position is corrected for
its time variations due to degree 2 solid Earth tides as given in the IERS conventions
[13] and ocean loading effects, using the FES2004 model [14]. The respective elevation (no-azimuth) and nadir-de- pendent absolute
corrections for the receiver and satellite antenna phase center variations as
made available by the IGS (ftp://igscb.jpl.nasa.gov/pub/station/general/igs05.atx)
are applied, and the carrier phase measurements are corrected for phase windup
taking into account the satellite attitude. Periods of eclipse events are
eliminated. The instrumental code delays are considered as constant and not
included in the present version of the software.

The tropospheric delay is expressed
as the sum of the hydrostatic and wet delays, both being the product between a
given mapping function (mf) and the
zenith path delay (zpd). The
hydrostatic part is introduced using the Saastamoinen a priori model [15] with
the dry Niell mapping function [16] and the wet part uses the Niell wet mapping
function. The wet zpd is estimated
with a 2-hour sampling rate, with linear interpolation between the epochs of
consecutively estimated zpd’s. Using a least squares scheme, with a
weighting for the codes and carrier phases associated with the noise level of
each observation type and satellite elevation, *Atomium* provides an
estimation of(i)the receiver clock
delay, at each epoch (5-minutes sampling rate), with respect to IGST;(ii)the position for the whole day
(optional: the position can be fixed to a given value);(iii)the zpd,
with a sampling rate of 2 hours;(iv)the phase
ambiguities. Figure 1 presents the clock solution
obtained with *Atomium* in a PPP mode
for three IGS stations over a period of 15 days. Two of them (NRC1 and BRUS)
are equipped with an H-Maser, while the station PTBB is equipped with a cesium
clock. NRC1 and BRUS have been chosen for their, respectively, characteristic large
and small day boundary jumps. Figure 1 also shows the differences between the *Atomium* clock solution and the combined
IGS clock solution, and this for the period January-Febuary 2007. The differences between the *Atomium* clock results and the IGS clock
solution for all the stations analyzed (not all are shown here) show a constant
bias ranging between 0 and 200 picoseconds plus some small variations with an
rms between 50 and 100 picoseconds.

In order to validate the *Atomium* software, the clock results
obtained from the PPP analysis were also compared with the results obtained
from two other software capable of performing PPP: NRCan [12] and Bernese 5.0
[17]. The Bernese software does not directly interpolate the IGS orbits, as
done by the NRCan and *Atomium* software, but it transforms the SP3 positions to an inertial frame, performs a
numerical integration of the equations of motion and then converts the obtained
satellite positions back to the earth-fixed system.

Figure 2 shows the comparison between
the results obtained for IENG over one week with *Atomium,* NRCan, and Bernese. The
NRCan solution has been corrected for a constant bias of about 4 nanoseconds
with respect to the other solutions. This bias is caused by the fact that the
version of the NRCan software used in this work is using relative corrections
for the satellite and antenna phase center variations while the Bernese, *Atomium,* and the IGS use absolute corrections since November 2006. The NRCan bias
remains visible in the lower part of Figure 2 which displays the differences
between the different clock results and the IGS clock solution. We can see that
the differences have similar amplitudes with an rms ranging between 30 and 100 picoseconds
over the analyzed week, but note that the rms is affected both by the accuracy
of the solution during the different days, and its precision.

#### 3. Phase-Only Approach

##### 3.1. Description of the Method

As explained above, the use of PPP for time transfer applications is mainly limited by the presence of day-bo- undary jumps in the clock solution. These day-boundary jumps are station-dependent and can be large for some sta- tions, see Figure 1 for NRC1, or [18]. One solution to suppress the day-boundary jumps is to exploit the continuity of the phase measurements across midnight and to provide a continuous solution without using any pseudoranges.

The phase-only approach developed here is based on this principle and has the advantage of being simple and rapid to use. It consists of using the phase differences between two consecutive epochs (using a 5-minute sampling interval) as basic observable, to create the normal equations accordingly, and to solve for the clock differences between the consecutive epochs.

The first derivative of (1) is used to generate the phase derivatives of the ionosphere-free carrier phase measurements: so that where , , and are the differences between the satellite clock errors, the receiver clock synchronization errors, and the tropospheric delays, at times and .

This new quantity does not contain
any ambiguity as long as there is no tracking interruption. In the occurrence
of a cycle slip, the corresponding phase difference will be rejected as an
outlier. The phase-only normal equations are constructed and solved in a
similar way as for PPP. However, using as observations differences between
successive carrier phase measurements, the position can only be determined with
a limited precision similar to the classic triple difference phase processing.
For this reason, we choose in this stage to fix this parameter to a given value.
The sensitivity of the results to the station position (fixed or free) will be
discussed in Section 3.5. The phase-only version of *Atomium* therefore provides
(i)the receiver clock derivative at each epoch;(ii)the zpd, with a sampling rate of 2 hours.
The phase-only clock solution is then
obtained from integrating the receiver clock derivative at each epoch obtained
from the least squares estimation.

Similar to the strategy used in the PPP mode, the phase-only analysis is also done in daily batches. But, now the successive days are linked by computing the difference between the first observation of the day and the last observations of the previous day exactly as the differences between two successive observations within a day. This guarantees the continuity of the clock solution across the day boundaries. In case of an actual tracking interruption, the phase-only solution is interrupted and the integration must be restarted.

##### 3.2. Discontinuities in the IGS Products

While the phase-only analysis has the potential to provide a continuous solution, still some jumps with amplitudes up to 120 picoseconds can appear at the day boundaries. These jumps are caused by the nonperfect overlapping of the IGS orbit and clock files at midnight. The jumps are geographically correlated as their amplitudes depend on the visible satellites at midnight GPS time. This is illus- trated in Figure 3, where the stations in Europe (BRUS, IENG) show small jumps, while stations in North-America (USN3, USNO, NRC1, NISU) show a large jump (above 100 picoseconds).

The day-boundary jumps in the
phase-only solution are reduced in *Atomium* by eliminating the satellites with a discontinuity in the IGS orbit and clock
information using a 2-sigma threshold. This is, however, only possible when the receiver clock is an H-maser as for le ss stable clocks the day-boundary phase jumps are lost within the clock noise. The impact
of the jumps in that case is, however, partly reduced by a proper weighting of
the satellite orbits accounting for the orbit accuracy deduced from the 7-day
arc evaluations and given in the IGS reports for IGS final orbits.

##### 3.3. Comparison between the Phase-Only, the PPP, and the IGS Receiver Clock Solutions

The continuous clock solution obtained from the phase-only analysis was compared over one month with the IGS and PPP clock solutions. Figure 4 presents the three solutions for the IGS station NRC1 which is, during winter time, characterized by large day-boundary discontinuities. On a daily basis, the phase-only solution and the PPP solution are equivalent, but the day-boundary jumps are absent from this phase-only solution and the improvement brought by this new method is obvious.

Thanks to the elimination of the day-boundary jumps, our phase-only solution improves the stability of the time link for averaging times up to 7 days, as shown in Figure 5 for NRC1-IGST. This improvement confirms the results obtained in [8]. Of course, the importance of the improvement depends on the size of the day-boundary jumps in the PPP (or equivalently IGS) solution.

##### 3.4. Comparison with the NRCan Multiday Batch Analysis

We also compared our phase-only analysis with the continuous (or multiday) PPP solution obtained with the NRCan software [7]. This software is able to analyze a data batch of several days with sequential least squares, using both code and phase observables. The resulting multiday PPP solution does not suffer from day-boundary discontinuities and one single position can be estimated for the whole period processed.

Figure 6 shows the comparison between
the *Atomium* phase-only results and
the NRCan 30-day PPP solution for the stations NRC1, USN3, and PTBB. For NRC1
and USN3, the *Atomium* solution shows
a drift with respect to the NRCan and IGS solution; this drift is due to the
position fixed within the *Atomium* phase-only analysis; this problem will be discussed in more detail in Section 3.5.
For PTBB, a large jump occurs in the IGS solution between mjd 54144 and 54146; this
jump is caused by a jump in the code data at mjd 54145.4 and was later
confirmed as a change of the receiver hardware delay between 2006 and 2007 (G.
Petit, personal communication). There is no IGS clock solution for PTBB for the
day with the jump as the IGS analysis rejected the solution because of this
problem. As our phase-only approach only uses the carrier phases and neglects the code measurements, it is insensitive to
the code jump. However, in the multiday NRCan solution, the jump is distributed
over about 10 days inducing an artificial drift of the solution during these
days. Also the IGS solution is degraded in this case. This is confirmed by the
comparison with TWSTFT measurements between PTBB and another station (USN3 in
our case, see Figure 7): neither a discontinuity nor a frequency drift is
present in the TWSTFT solution and only the *Atomium* phase-only approach is in agreement with the results of this independent
technique. Note that in that case, the PTBB and USN3 coordinates have been
chosen in order to have no drift between the IGS solution and the phase-only
solution (see Section 3.5 for the sensitivity of the results to station
position).

##### 3.5. Sensitivity to the Station Position

All the results presented up to now have been obtained by fixing the station position in the analysis. As explained before, the station coordinates can be determined using phase differences between successive epochs, but with a limited precision due to the strong correlation between the estimated position and the clock solution. To quantify the sensitivity of the phase-only analysis to the station coordinates, a test was performed using the IGS station BRUS which is characterized by small day-boundary discontinuities. The phase-only solution has been computed over one month either by determining the station position for each day, or by fixing the station position to a given value. In the case of the fixed station position, two different positions separated by 3 mm in the Y-component are tested. The results are presented in Figure 8: the three curves deviate with respect to the IGS solution. Furthermore, we can observe a significant drift between the two solutions obtained with fixed positions, leading to a disagreement of 1.5 nanoseconds after only one month. The drift between the IGS solution and the phase-only solution is, therefore, most probably due to the imperfect station coordinates introduced or determined during the analysis and possible small modeling differences with respect to the IGS solution. The choice of the station coordinates with a subcentimeter precision is, therefore, clearly a crucial point in the phase-only analysis.

#### 4. Conclusions

A new tool for precise GPS time and
frequency transfer has been presented. It is called *Atomium* and is based
on a least-squares analysis of GPS measurements. A PPP solution can be obtained
using both codes and carrier phases, and a phase-only solution is proposed in
order to reduce the day-boundary jumps or the intraday jumps which are due to
the usage of the code measurements. The
accuracy of the results obtained in the PPP mode has been determined from the
comparison of these results with the IGS combined clock products. The bias
between the *Atomium* clock solutions
and the corresponding IGS clock products is station-dependent, while always remaining
below 200 picoseconds. In addition, some small variations, with an rms between
50 and 100 picoseconds, are present. They are similar to the variations
between two other software performing PPP (Bernese 5.0 and NRCan) and the IGS
clock solution.

A continuous frequency transfer method, based on a phase-only analysis, is also presented in this paper. As demonstrated, the advantage of the phase-only analysis is that it can be combined with a calibrated time transfer method as two way satellite time and frequency transfer (TWSTFT) taking advantage of the strong points of each of the methods: the accuracy for TWSTFT and the stability and high resolution for GPS carrier-phase [9].

Our continuous clock solutions are based on the least squares analysis of the differences between the consecutive GPS phase measurements. The analysis is performed in daily data batches, but the successive days are linked by computing the differences between the first observations of the day and the last observations of the previous day. The results show that a satellite-dependent weighting is necessary to assure that the phase-only solution does not show day-boundary discontinuities (up to 120 picoseconds) due to the nonperfect overlapping of the IGS clock files at midnight. We have demonstrated that the phase-only clock solution improves the stability of the time link for averaging times up to 7 days; the amplitude of the improvement depends of course on the amplitude of the day-boundary discontinuities in the PPP solution. The high sensitivity of the phase-only clock solution to the station coordinates fixed in the analysis was also demonstrated, indicating that a subcentimeter precision is needed for the position. Finally, the efficiency of the phase-only solution in case of changes in the GPS receiver hardware delays was shown; contrarily to the results obtained using both GPS code and phase observations, the phase-only solution is not affected by these hardware delay variations.

#### Acknowledgments

The authors would like to acknowledge the NRCan team for the availability of their software that they used for comparison of results, and the IGS for the availability of their products.

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