The Swarm mission of the European Space Agency (ESA) offers excellent opportunities to study the ionosphere and to provide temporal gravity field information for the gap between the Gravity Recovery and Climate Experiment (GRACE) and its follow-on mission (GRACE-FO). In order to contribute to these studies, at the Institut für Erdmessung (IfE) Hannover, a software based on precise point positioning (PPP) batch least-squares adjustment is developed for kinematic orbit determination. In this paper, the main achievements are presented.

The approach for the detection and repair of cycle slips caused by
ionospheric scintillation is introduced, which is based on the
Melbourne–Wübbena and ionosphere-free linear combination. The results show
that around 95 % of cycle slips can be repaired and the majority of the
cycle slips occur on

The Swarm mission was launched on 22 November 2013 and is the first
constellation of satellites of the European Space Agency (ESA) to study the dynamics of the Earth's magnetic
field and its interaction with the Earth system

A first performance evaluation of the RUAG 8-channel GPS receiver was carried
out by

In this contribution, the developments for kinematic POD of Swarm satellites
at the Institut für Erdmessung (IfE) are reported. This approach is based on the precise point positioning
(PPP) technique

Since the GPS data quality is decisive for the obtainable orbit accuracy, we
first analyse the in-flight performance of the Swarm onboard GPS
receivers in Sect. 2, e.g. the tracking performance and observation noise, especially
under the influence of ionospheric scintillation. A mandatory step in the
preprocessing is detecting and, if possible, repairing the cycle slips in
carrier phase observations in order to reduce the number of ambiguities and
therefore strengthen the orbit. Due to the large phase noise caused by
ionospheric scintillations, cycle slips in Swarm receivers are difficult to
detect at ionospheric-active areas. An approach based on the
Melbourne–Wübbena and ionosphere-free linear combination is introduced for
this task. In Sect.

Update of Swarm carrier loop bandwidth

Being a purely geometrical approach without using any dynamic models on LEO
satellites, the quality of the kinematic orbits relies completely on the GPS
observations as well as on the introduced GPS orbit and clock products, and
it is sensitive to measurement errors and the observation geometry

The Swarm onboard dual-frequency GPS receivers are developed by RUAG Space

The daily average number of tracked GPS satellites was increased from 7.6 to
7.7 after the first update of receiver bandwidth

The first reason is the reacquisition time of the Swarm receiver. When a GPS
satellite sets, the receiver cannot immediately track another GPS satellite
in view and needs around 84 s before the first receiver update (red) and
55 s after the update to acquire another satellite (blue), shown in
Fig.

Another reason for tracking less than 7 GPS satellites is the loss of the lock of
signal. There are also two reasons for the loss of the GPS signal, and one is due to
the weak signal strength for the GPS satellites at low elevations. However,
the loss of lock on GPS satellites at high elevations occurs mainly at polar
and equatorial areas under the influence of ionospheric scintillations, which
is evidenced by the change of the Total Electron Content (TEC). The occurrence of
loss of lock in September 2015 is shown in Fig.

Although the amount of the loss of lock on GPS satellites at higher elevations was decreased from 37 to 11 after the update, the receiver becomes unstable for the signals from lower elevations. The amount of the loss of lock increased almost fourfold, from 58 to 256. The reacquisition time in this situation was decreased from around 17 to 11 s after the update.

Reduction of reacquisition time for Swarm GPS receivers

As a dual-frequency receiver, the Swarm GPS receiver provides code
observations

Due to a software issue in the RINEX converter

The quality of the kinematic orbit determination depends mainly on the
available dual-frequency carrier phase observations. To analyse the phase
noise, the second-order differences (

Carrier phase noise for

The resulting

Figures

The

Strong ionospheric scintillations cause not only large noise but also cycle slips in the carrier phase observations. Therefore, a proper cycle-slip detection is necessary to set up correctly the ambiguities. Since the typical GPS satellite visibility for LEOs is mostly 40 min, the necessity of estimating additional ambiguities for even shorter segments will further decrease the geometric strength of the positioning.

A classical method for cycle-slip detection is to use the TurboEdit method

However, the code observations before DoY 102, 2016 contain large noise due
to the RINEX converter issue

Thus, a forward and backward moving window averaging (FBMWA) method is
applied to reduce the influence of large noise

Illustration of
the proposed cycle-slip detection approach based on Melbourne–Wübbena
combination, PRN23 of Swarm A on DoY 333, 2015:

Since with the Melbourne–Wübbena linear combination only the difference of
cycle slips on

The geometric distances are computed with the Medium Precise Orbit (MEO)
provided by ESA in the Swarm Level 1b product and removed from the right-hand
side of Eq. (

After removing the geometric distances term, the differences of receiver-clock error and ambiguity on

After removing the geometric distances and receiver-clock error, only the
ambiguity term remains. If there is no cycle slip, the

Illustration of the proposed cycle-slip detection approach based on
an ionosphere-free linear combination from PRN23 of Swarm A on DoY 333, 2015. The following are shown:

Together with the equation from Melbourne–Wübbena linear combination, the
cycle slip on

The statistics of the repaired cycle slips for Swarm satellites during 1
year from DoY 245, 2015 to DoY 244, 2016 are listed in Table

Statistics of the cycle slips for Swarm A, B and C during 1 year from DoY 245, 2015 to DoY 244, 2016

Summary of the measurement and error models used for Swarm kinematic orbit determination.

Global distribution of cycle slips on Swarm A (red), B (green) and C
(blue)

In order to see the impact of update of bandwidth of the tracking loop, the
cycle slips in September 2015 are additionally shown in Fig.

As a purely geometric approach, the kinematic orbit is sensitive to the GPS
measurement errors. As a consequence, it is important to screen the outliers.
The epoch-differenced ionosphere-free linear combination

In this section, the approach for kinematic orbit determination at IfE will be introduced first. Then 1-year kinematic Swarm orbits are validated with the reduced-dynamic orbits and the SLR residuals. The fully populated variance–covariance matrices of the kinematic orbits obtained from the adjustment are shown and discussed.

A MATLAB-based PPP software was developed at IfE Hannover using the
least-squares adjustment method. The ionosphere-free linear combinations of P
code and phase are computed and introduced as observables in the
least-squares adjustment model to eliminate the first-order ionospheric
effect. The GPS orbits and satellite clock errors are computed based on the
CODE final GPS orbits and 5 s clock products provided by the Center for
Orbit Determination in Europe (CODE)

Outlier detection and anomalous GPS satellite for Swarm A on DoY
271, 2015, here PRN26 (purple) and PRN32 (green);

Position residuals for Swarm A (red), B (green) and C (blue) with respect to
ESA reduced-dynamic orbits for DoY 333, 2015. The time series are offset by
10 cm for Swarm A and C. The legend indicates the rms (

The 1 Hz data is processed over 30 h, with 3 h overlap in the previous and following day. In this way, the removal of some GPS observations can be avoided due to the short arc at the day boundary. This increases the stability of the ambiguities at the day boundary. Because the observation time saved in the RINEX file is synchronized to the receiver internal clock, which has a large clock drift, a time offset is added to synchronize the internal clock with GPS time everyday at the day boundary. This causes a clock jump in the code observations and a clock jump as well as a phase jump in the carrier phase observations. In order to get a continuous arc at the day boundary, these phase jumps need to be fixed. The clock jump is first determined from the average of each epoch-differenced code observations (removing the a priori geometric distances) at the day boundary. Then, the phase jump is the average of each epoch-differenced phase observations (removing the a priori geometric distances) minus the clock jump.

The phase noise is weakly elevation-dependent (see Fig.

Differences of kinematic orbits with respect to ESA reduced-dynamic orbits of
Swarm C of different institutes from DoY 1 to 7, 2016. The time series are
intentionally shifted by 15, 5,

rms and mean of position residuals of IfE and ESA kinematic orbits with respect to ESA reduced-dynamic orbits for Swarm A, B and C during 1 year (DoY 245, 2015 to DoY 244, 2016). The values for ESA kinematic orbits are given in brackets.

The three unknown position coordinates

One-year orbits from 2 September 2015 to 31 August 2016 are computed. Here
only exemplary but typically results are shown. The position residuals for
Swarm A, B and C on DoY 333, 2015 with respect to the reduced-dynamic orbits provided
by ESA in the Swarm Level 2 product are shown in Fig.

The RMSEs in along-track, cross-track and radial directions are around
1.5, 1 and 2 cm, respectively. The
noise caused by the ionospheric scintillations shown in Fig.

The RMSEs and average of position residuals for Swarm A, B and C of IfE
and ESA kinematic orbits from DoY 245, 2015 to DoY 244, 2016 are listed in
Table

Spectral analysis of Swarm C orbits from and

Daily RMSEs of the ionosphere-free carrier phase residuals and the daily average STEC derived from Swarm C over all GPS satellites for 1 year of data from DoY 245, 2015 to DoY 244, 2016.

SLR residuals over 1 year from DoY 245, 2015 to 244, 2016, for

Correlation matrix for 30 h time span in

The computed kinematic orbits of Swarm C are also compared with the kinematic
orbits from ESA

In order to analyse the remaining deviations, a spectral analysis is
performed for Swarm C for March 2016 in Fig.

An additional possibility to assess the orbit quality is to analyse the
residuals of GPS observations

The comparison with reduced-dynamic orbits and the residuals of GPS
observations is an internal validation of the orbits. Independent satellite
laser ranging (SLR) measurements provide another external validation of the
orbits. The GPS-derived kinematic orbits are first interpolated with a
polynomial of degree 5 to the observation time of SLR measurements. Then, the
computed ranges between the ground stations of tracking networks and
the GPS-derived kinematic orbits are compared with the observed laser ranges. The
coordinates of the ground stations are given in ITRF2014

Correlation matrix for 30 h time span in

The least-squares estimation provides not only the coordinates but also the
variance–covariance information, which is important for gravity field
recovery

Figure

The correlation structures of the

In this contribution, we present the approach for the determination of the IfE
Swarm kinematic orbit with PPP. Since the GPS data quality is a prerequisite
for the high quality of kinematic orbit, a thorough analysis was first performed. Although only 8 channels are available, the requirements for an
accurate positioning are fulfilled. If the reacquisition time can be
shortened, the tracking performance can be further improved. Carrier phase
observations at both frequencies are strongly disturbed by ionospheric
scintillations, where noise larger than 0.2 m degrades the detection of
cycle slips. The approach is proposed, which allows for a successful cycle-slip
repairing of around 95 %. It is worth mentioning that almost all the
cycle slips occur on

Precise kinematic orbits at IfE are generated with the PPP method. The differences between our kinematic orbits and the ESA reduced-dynamic orbits are at 1.5, 1.5 and 2.5 cm level in along-track, cross-track and radial directions, respectively. The daily rms values of ionosphere-free carrier phase residuals are at 4 mm level, showing strong internal consistency. The spectral analysis of the orbits highlights the once-per-revolution period and higher harmonics as well as the dominant noise processes. A comparison with orbits from other institutes shows very good agreement with the rms level. A comparison with SLR underlines the accuracy of the kinematic orbits of 3–4 cm. Due to the ambiguities in the carrier phase, the coordinates are correlated. The mathematical temporal correlation between two successive epochs is larger than 0.6 for the 30 s data rate. The correlation decreases with increased sampling frequency. For a 1 s data rate, the correlation between two successive epochs is below 0.1.

The Swarm kinematic orbits from AIUB can be found at

The Swarm kinematic orbits from IfG TU Graz can be found at

The Swarm kinematic orbits from ESA can be found at

Swarm Level 1b and Level 2 products can be found

The CODE orbits and clock products can be downloaded from:

The Swarm kinematic orbits from IfE Hannover are still under evaluation and will be accessible for the public soon.

LR developed the code, designed the experiments and prepared the manuscript. SS contributed to the interpretation of the results and correction of the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Dynamics and interaction of processes in the Earth and its space environment: the perspective from low Earth orbiting satellites and beyond”. It is not associated with a conference.

This project is part of the Consistent Ocean Mass Time Series from LEO Potential Field Missions (CONTIM) and funded by the Deutsche Forschungsgemeinschaft (DFG) under the SPP1788 Dynamic Earth which is gratefully acknowledged. We would like to thank ESA for providing Swarm data. The Swarm kinematic orbits have been made available by ESA, AIUB and IfG, TU Graz. GPS orbits and clocks have been obtained from the Center for Orbit Determination in Europe (CODE). The SLR residuals are kindly provided by Anno Löcher of Institute of Geodesy and Geoinformation, University of Bonn. TU Delft is also very appreciated for providing the PCV maps. The support of all these institutions is gratefully acknowledged.The publication of this article was funded by the open-access fund of Leibniz Universität Hannover.The topical editor, Monika Korte, thanks two anonymous referees for help in evaluating this paper.