Precise orbit determination based on raw GPS measurements
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Abstract
Precise orbit determination is an essential part of the most scientific satellite missions. Highly accurate knowledge of the satellite position is used to geolocate measurements of the onboard sensors. For applications in the field of gravity field research, the position itself can be used as observation. In this context, kinematic orbits of low earth orbiters (LEO) are widely used, because they do not include a priori information about the gravity field. The limiting factor for the achievable accuracy of the gravity field through LEO positions is the orbit accuracy. We make use of raw global positioning system (GPS) observations to estimate the kinematic satellite positions. The method is based on the principles of precise point positioning. Systematic influences are reduced by modeling and correcting for all known error sources. Remaining effects such as the ionospheric influence on the signal propagation are either unknown or not known to a sufficient level of accuracy. These effects are modeled as unknown parameters in the estimation process. The redundancy in the adjustment is reduced; however, an improvement in orbit accuracy leads to a better gravity field estimation. This paper describes our orbit determination approach and its mathematical background. Some examples of real data applications highlight the feasibility of the orbit determination method based on raw GPS measurements. Its suitability for gravity field estimation is presented in a second step.
Keywords
Precise orbit determination Low earth orbiter Kinematic orbit Raw GPS observations Satellitetosatellite tracking highlow Time variable gravity1 Introduction
Kinematic orbit positions often serve as observations for gravity field estimation. Hence, their accuracy directly affects the quality of the gravity field estimates. We present a new method for kinematic orbit determination based on raw GPS measurements. The method can be seen as a variation of precise point positioning (PPP) (Witchayangkoon 2000). In contrast to currently used approaches, we directly use raw global positioning system (GPS) observations, as they are observed. No linear combinations or observation differences are used. Code and phase measurements on both frequencies are directly incorporated in a least squares adjustment. To achieve highest accuracies, a careful treatment of all errors affecting GPS observations is absolutely necessary. A major aspect of PPP is the knowledge of precise transmitter position and clock information. Besides this, errors like antenna center variations (ACV), relativistic effects, phase windup, and the ionosphere have to be taken into account. If these effects are handled properly, a position accuracy in the range of a few centimeters is achievable.
Several satellite missions to study Earth’s gravity field have been launched in the past two decades. These missions, starting with CHAMP (challenging minisatellite payload) (Reigber et al. 2001), followed by GRACE (gravity recovery and climate experiment) (Tapley et al. 2004) and GOCE (gravity field and steadystate ocean circulation explorer) (Drinkwater et al. 2007), have been very successful and have provided new insights into Earth’s gravity field. These three missions rely on diverse measurement techniques, all of them being equipped with a GPS receiver. The high accuracy kinematic orbits can be used as observations to determine Earth’s gravity field. This method is known as satellitetosatellite tracking in highlow mode (SSThl). It was the observation principle used for the CHAMP mission and served as a supplement to other measurements in case of GOCE and GRACE. A major advantage is that the principle can be transferred to any other low earth orbiter (LEO) equipped with a GNSS (global navigation satellite system) receiver.
In recent years, the scientific community has focused on the time variable component of the gravity field. GRACE is able to observe variations in the gravity field due to its highly accurate microwave ranging system. These results are of high interest to different communities like hydrology, glaciology, and geology (Cazenave and Chen 2010) and form the basis for research concerning climate change, sea level change, deglaciation and is, therefore, also of socioeconomical interest. Until recently, it was not possible to observe these variations in the gravity field through any other measurement technique, except for signals at large spatial scales. Facing a possible gap between the GRACE mission and its successor GRACE followon (Flechtner et al. 2014), an additional observation method becomes very important. First successful application of SSThl to derive mass variations was done by Prange (2010). Recent investigations (Weigelt et al. 2013; Baur 2013; Zehentner and MayerGürr 2014) indicate that it is possible to derive at least variations at medium scales (800–1000 km). We will use static and time variable gravity field results based on our kinematic orbits as a tool to validate our orbit determination method. For gravity field recovery, we incorporate the short arc integral approach introduced by MayerGürr (2006). For a detailed description, we recommend the reader to take a look at MayerGürr et al. (2006) or MayerGürr et al. (2010).
This paper is structured into five major sections. After the introduction, Sect. 2 describes the methodology used for precise kinematic orbit determination; in Sect. 3, the input for some real data applications and the work flow to compute the kinematic orbits is explained; Sect. 4 shows the validation of the produced kinematic orbits by means of orbit comparisons and gravity field estimates; finally, Sect. 5 summarizes the main points and draws some conclusions.
2 Method
Current approaches for POD (precise orbit determination) rely on the ionospherefree linear combination or on observation differences of GNSS signals. Stateoftheart approaches are, for example, given by Bock et al. (2011), van den IJssel et al. (2015) or Prange (2010). They all rely on observation combinations, which are utilized to mitigate systematic errors like the ionospheric influence. Our approach is founded on the basic rule that all observations shall be used as they are observed, including code and phase observations on all frequencies. This means no forming of linear combinations or differences, like single or double differences. A main drawback of linear combinations is the fact that the measurement noise is increased (Dach et al. 2007). If the original observations are used, the noise level is unchanged, but in contrast to linear combinations systematic errors are not eliminated or reduced. Thus, all systematic influences must be known beforehand, or at least estimated as parameters in the least squares adjustment. This approach was first described in Zehentner and MayerGürr (2014).
2.1 Basic observation equation
Based on the basic observation equations, with extensions described in the subsequent paragraphs, the design matrix for the leastsquares adjustment is set up. Observation equations are set up for each measurement individually. Due to the introduction of precise GPS orbits and clocks, the presented method can be seen as a variation of the PPP approach. The equation system is set up for a certain time span, e.g., one day, and solved iteratively by incorporating a variance component estimation to adjust the observation weights and account for outliers.
2.2 Additional parameters

antenna type

carrier frequency

observation type

signal direction (azimuth, elevation)

surrounding (multipath effects).
For transmitters, a parametrization with spherical harmonics is not applicable, because the view angle of a GNSS transmitter is restricted to approximately 14\(^\circ \)–17\(^\circ \), depending on the involved receivers. For a receiver on the ground, a view angle of 14\(^\circ \) is sufficient; in case of a LEO, the view angle is related to the altitude of the satellite. We use radial basis functions on a regular grid to parametrize transmitter ACVs. The principles of radial basis functions can, for example, be found in Eicker (2008). For the necessary regular distribution of the basis functions over the surface of the transmitter antenna, the triangle vertex algorithm as given by Kenner (1976) and Eicker (2008) was chosen.
The representations in Eqs. (8) and (9) depend on the carrier frequency f, the speed of light c, the peak electron density \(N_{\mathrm{max}}\), a shape parameter \(\eta \), and the magnetic field vector \(\mathbf{B_0}\). Fritsche et al. (2005) provide a value for the shape parameter (0.66) and an approximation formula to obtain \(N_\mathrm{max}\). Not shown in Eqs. (8) and (9) is a part of the thirdorder term we omit, as its magnitude is far below one millimeter as shown by Brunner and Gu (1991). To obtain the magnetic field vector, we make use of the International Geomagnetic Reference Field (IGRF) (Finlay et al. 2010) provided by the International Association of Geomagnetism and Aeronomy. It is given in a spherical harmonics expansion up to degree and order 13 at 5year intervals with linear interpolation between two consecutive models.
2.3 Extended observation equation
2.4 Observation weighting
The estimated a posteriori variance is compared to the given a priori value. If there is a significant difference, the observation weight is changed according to the empirical variance. The decision on the significance of a difference is made by application of a modified Huber Mestimator (Koch 2004). This method implicitly realizes an outlier detection and removal algorithm. The weight of huge outliers is continuously reduced after each iteration. Practically, such an outlier ends up with a weight close to zero and, therefore, does not contribute to the estimation of the parameters. The only requirement is that the redundancy at each epoch is high enough to reliably detect outliers. Modern spaceborne GPS receivers are tracking up to 12 satellites simultaneously and due to the fully populated GPS transmitter constellation the average number of tracked satellites is in the range of 8 to 10.
3 Real data applications
3.1 Input data
We investigate the performance of our method for precise kinematic orbit determination by applying it to real observation data from GRACE and GOCE. The GRACE time series starts in January 2003 and ends in November 2014, whereas for GOCE we processed the whole lifespan of the satellite from November 2009 to October 2013. Precise orbit and clock information is taken from the Center for Orbit Determination in Europe (CODE) (Dach et al. 2009). Precise orbit positions are given at 15min intervals. Clock corrections are provided with different sampling rates. If available highrate clock corrections with sampling rates of 5 s were used (Bock et al. 2009). For periods in which no 5 s sampled corrections were available, 30 s precise clocks were utilized. In addition to precise ephemerides, we used differential code biases also provided by CODE.
Used observations are the code and phase measurements on the original two carrier frequencies of GPS. For GRACE, six observation types are available: three code (C/A, P1 and P2) and three phase (L1C, L1P and L2P) measurements. The receiver of GOCE provides the same three code observations but only L1C and L2P phase observations. In both cases, we used the P1, P2, L1C and L2P observations. To fully exploit the observation data, we used the highest available sampling rate, which is 10 s for GRACE and 1 s for GOCE. For a priori observation weights, we introduced uniform values, except for the final orbit computation where we introduced azimuth and elevationdependent observation weights. The only preprocessing step was to find cycle slips, which was done by finding jumps in the Melbourne–Wübbena combination. In case of a detected jump, a new ambiguity was set up for this GPS satellite. All computed solutions are float solutions.
3.2 Orbit processing
The first step was to estimate ACVs for all three receivers. This was done individually for each satellite by setting up the daily normal equations and eliminating all epochdependent parameters and the phase ambiguities. Then, normal equations of a longer time span were combined and solved to get the ACV parameters. To guarantee a mean value of 0 for the ACVs, the parameters were regularized separately for each observation type. The weights of each daily solution and the regularization matrix were determined by variance component estimation. For each receiver, we estimated a common correction for L1 and L2 phase observations, and two individual parameter sets for the P1 and P2 code observations. For GRACE, a maximum degree of 50 for the spherical harmonics expansion was sufficient. The ACV estimation for GOCE was done up to maximum degree and order 60. For example, maximum spherical harmonic degree 60 corresponds to a halfwavelength resolution of 3\(^\circ \). For each GRACE satellite, we had to estimate two sets of ACV parameters, as we observed that the ACVs differ depending on whether the occultation antenna is switched on or off. The same effect was already observed by Montenbruck and Kroes (2003) with CHAMP data. To avoid mapping of transmitter ACVs into the estimated receiver ACVs, nadir angledependent values provided by the IGS were introduced in this first orbit computation.
The second step was to estimate ACVs for all transmitters active during the used time span. The previously estimated receiver ACVs were now introduced as known corrections. One set of ACV parameters was set up for each individual transmitter, without introducing any a priori values, e.g., the IGS values. The separation of transmitters was based on the satellite vehicle number (SVN), based on the assumption that the ACVs are constant for a certain SVN, even if the transmitted pseudorandom noise code (PRN) is changed. Similar to our approach for the receivers, we estimated a common correction for L1 and L2 phase observations and individual corrections for P1 and P2 code observations. Triangle vertex level for localization of the radial basis functions was 20 and restricted to a nadir angle of 17\(^\circ \). For details on the triangle vertex distribution, see Eicker (2008). This results in 101 parameters per observation type and transmitter.
The estimated ACVs were introduced as corrections to the observations for the third orbit computation. The observation residuals of this solutions were then used to generate accuracy maps for each observation type, as described in Sect. 2.4. Together with the ACVs, the accuracy maps were used in the fourth and final step to compute the final kinematic orbit, along with the covariance matrix for each epoch.
4 Results
4.1 Orbit comparisons
One problem in orbit validation is that there are no absolute reference values available for the satellite position. One possibility is to compare with a dynamic or reduceddynamic orbit. These products incorporate different force models, like an a priori gravity field model. They are far smoother than kinematic orbits, but they heavily depend on the choice of the background models which can introduce additional errors. Therefore, they cannot serve as an absolute reference, but they can help to characterize the noise level of the kinematic orbit.
We investigated the differences with respect to a reduceddynamic orbit. For the GRACE satellites, the L1B data product GNV1B provided by the Jet Propulsion Laboratory (JPL) (Case et al. 2010) was chosen. For GOCE, we used a reduceddynamic orbit solution produced by the Astronomical Institute at the University of Bern (AIUB) (Bock et al. 2011), which is provided in the GOCE level 2 SST_PSO (Gruber et al. 2008) product.
Figures 5, 6, 7, and 8 clearly show that the noise level for the radial component is higher. This is an inherent property of GPS. Due to the observation geometry, the vertical component is worse determined than the horizontal components. This is already known from groundbased applications, where a rule of thumb states that the vertical component is worse by a factor of 1.5 to 2 (HoffmannWellenhof et al. 2008). For GRACE, a degradation of the accuracy occurs for the time frame 2007 until the beginning of 2008 (Figs. 5, 6). This is due to the fact that for this time span only 30 s sampled precise GPS clocks are available, which were interpolated to the 10 s observation sampling. In general, it can be seen that the rms values for all three components are stable and do not show large time dependency, for example, related to the 11year solar cycle.
Mean daily rms values for along, cross and radial components of all three satellite missions over the entire time frame
rms (mm)  Along  Cross  Radial 

GRACE A  2.4  1.8  5.2 
GRACE B  2.4  1.7  5.3 
GOCE (ITSG)  3.4  3.0  7.3 
GOCE (AIUB)  3.7  3.2  8.0 
4.2 Gravity field results
Data and parameters used in the gravity field estimation
Data/parameter  GRACE A & B  GOCE 

Kinematic orbit  ITSG  ITSG 
Epoch covariance information  Used  Used 
Empirical covariance function  Used  Used 
Accelerometer data  Used  Used 
Minimum arc length  15 min  15 min 
Maximum arc length  45 min  45 min 
Data sampling  10 s  1 s 
Tide system  Tidefree  Tidefree 
atm./oc. dealiasing  AOD1B  AOD1B 
Maximum D/O  60  60 
Accelerometer bias  Linear/arc  Linear/arc 
A priori model  GOCO03S  GOCO03S 
All monthly solutions were produced independently for each satellite. This provides the opportunity to combine different solutions on the basis of normal equations. Finally, we produced a time series of GOCE only solutions (ITSGGO) and of GRACE only solutions (ITSGGR). For validation purposes, GRACE solutions from CSR (CSRRL05) (Bettadpur 2012) are used. We also computed GOCE monthly solutions based on the kinematic orbits produced at AIUB in the frame of the highlevel processing facility (HPF) (Visser et al. 2006). These solutions are similar to those included in the GOCE timewise model release 4, except for the maximum degree and order, as they were produced inhouse applying the same approach. From here on, these solutions are denoted as AIUBGO.
The validation of the monthly gravity field models can be also done by comparing them with monthly GRACE solutions derived from highly accurate microwave measurements. A direct comparison is not feasible, due to the high noise level of the SSThl solutions in contrast to the GRACE solutions. As an alternative approach, we derived gravity variations for selected regions and compared them to values derived from a GRACE time series. In particular, we studied the two most prominent regions where variations can be expected: the Amazon River basin, which exhibits the highest annual variations due to the changing water content; and Greenland, which also exhibits annual variation, but more importantly features the biggest trend in terms of mass change. This trend is due to the massive ice loss caused by the global climate change (Velicogna 2009; Cazenave and Chen 2010).
Figures 10 and 11 clearly demonstrate the ability to derive the annual signal in the Amazon River basin and the trend in Greenland from our kinematic orbits. At this stage, it must be emphasized that each monthly gravity field solution is a standalone solution. This means we did not apply a Kalmanfilter approach or any postprocessing of coefficients, in contrast to other recent publications, for example, by Weigelt et al. (2013) or Baur (2013).
4.3 Other satellite missions
As mentioned in the introduction, the principle of SSThl can be transferred to any satellite mission carrying a geodetic GPS receiver. This allowed us to compute kinematic orbits and gravity fields for several satellites which are not originally dedicated to the study of Earth’s gravity field. We used data from the satellite missions Swarm (FriisChristensen et al. 2006), TerraSARX (Werninghaus et al. 2007), TandemX (Krieger et al. 2007), MetOpA, MetOpB (Edwards et al. 2006) and Cosmic (Rocken et al. 2000). For each satellite, the previously described work flow was used to generate kinematic orbits. This includes estimation of transmitter and receiver ACVs as well as generation of individual accuracy maps for each satellite. Due to high computational burden, the estimation of transmitter ACVs was skipped for MetOpA, MetOpB and Cosmic. Instead transmitter ACVs derived from GRACE data were used.
Figure 13 shows that in principle all missions can be used to estimate Earth’s gravity field. The worst results are achieved with Cosmic data, even though the shown solution is a combination of all five active COSMIC satellites. This is mainly due to the fact that the quality of the GPS observations is very low compared to other missions. Additionally, the attitude determination of the Cosmic satellites is rather poor, with an accuracy of approximately 2\(^\circ \) (Hwang et al. 2008) in each axis, and the orbits of the satellites feature a high altitude of 800 km in combination with a low inclination of 72\(^\circ \).
The results obtained with the satellites MetOpA and B are also clearly inferior to solutions based on, for example, TerraSARX or TandemX. The reasons can be found in the high orbital altitude of approximately 800 km and a slightly worse positioning accuracy. The degradation in positioning accuracy can be attributed to the fact that the receivers of MetOpA and MetOpB are restricted to a maximum number of 8 tracked GPS satellites.
We achieved promising results with the two radar satellites TerraSARX and TandemX. Especially for the low degrees (2–10), the accuracy is comparable to the results obtained with GRACE or GOCE. The gravity field solutions obtained with Swarm data are slightly inferior compared to TerraSARX or TandemX. Only by combining data of all three Swarm satellites, a similar accuracy can be achieved, even though two of the satellites are orbiting at a lower altitude of 430 km. This indicates a slightly worse positioning accuracy for the Swarm satellites, possibly also due to the fact that the Swarm GPS receivers are only tracking a maximum of 8 GPS satellites simultaneously.
5 Conclusions
Based on the presented results, we conclude that the presented approach for precise orbit determination is applicable. The main advantage of our method is the fact that observations are directly used as they are observed by the receiver. This preserves the original measurement accuracy and gives the possibility to fully exploit the contained information of each individual observation type. The inclusion of new observables (new L5 signal) is straight forward, because the observation equations are set up individually and no combinations are used. The approach also includes a realistic observation weighting scheme as well as a sophisticated handling of different error sources. Precise knowledge of these errors, for example receiver and transmitter ACVs or precise transmitter orbits and clocks, can be seen as the Achilles’ heel of our approach. Furthermore, ionospheric modeling shall be reviewed and maybe adopted for LEO positioning. Validation shows that the produced orbit solutions feature the same or better accuracies as compared to existing methods. Gravity field estimates based on our kinematic orbits are slightly better than solutions based on kinematic orbits published by other research institutions. An important aspect is that the orbits show the potential to assess time variations in Earth’s gravity field. We also showed that the principle of SSThl can be transferred to any satellite mission equipped with a geodetic GNSS receiver.
With further work towards optimizing the orbit computation and a detailed analysis of remaining errors, SSThl can be a great opportunity to fill a possible gap between GRACE and its successor GRACE followon. Apart from being a gap filler, we demonstrated that SSThl can be seen as an additional viable method to quantify changes in Earth’s gravity field and thus can contribute to a better understanding of processes in the system Earth. All orbits presented here are freely available at: ftp://ftp.tugraz.at/outgoing/ITSG/tvgogo/orbits.
Notes
Acknowledgments
The presented work was funded by the Austrian Research Promotion Agency (FFG) in the frame of the Austrian Space Applications Programme Phase 9 (Prjnr.: 840126). We would like to thank the German Space Operations Center of the German Aerospace Center (DLR) for providing GRACE data, the Information System and Data Center at the German Research Center for Geosciences (GFZ) for providing TerraSARX and TandemX data, the European Space Agency for providing GOCE and Swarm data, and the COSMIC Data Analysis and Archive Center for providing COSMIC, MetOpA and MetOpB data.
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