Absolute phase center corrections of satellite and receiver antennas
Results of the estimation of azimuth-dependent phase center variations (PCVs) of GPS satellite antennas using global GPS data are presented. Significant variations of up to ±3–4 mm that are demonstrated show excellent repeatability over eight years. The application of the azimuthal PCVs besides the nadir-dependent ones will lead to a further reduction in systematic antenna effects. In addition, the paper focuses on the benefit of a possible transition from relative to absolute PCVs. Apart from systematic changes in the global station coordinates, one can expect the GPS results to be less dependent on the elevation cut-off angle. This, together with the significant reduction of tropospheric zenith delay biases between GPS and VLBI, stands for an important step toward more consistency between different space geodetic techniques.
KeywordsSatellite antenna Phase center variations Tropospheric zenith delay Elevation cut-off
For several years, absolute receiver antenna phase center variations (PCVs) from two independent approaches (calibration in an anechoic chamber and field calibration with a robot) have been available that are in good agreement with each other as well as with the relative PCVs published by the IGS (International GPS Service; see e.g., Rothacher 2001). Nevertheless, the latter are still used, although it is well known that systematic errors are the consequence (e.g., Schmid et al. 2005). The problem of a terrestrial scale change of about 15 ppb with respect to ITRF2000 that initially prohibited the adoption of absolute PCVs could be solved, in the meantime, by the estimation of nadir-dependent satellite antenna PCVs (Schmid and Rothacher 2003). The latter are also called “absolute”, namely, in terms of compatibility to the absolute receiver antenna corrections and not in a real absolute sense. At the moment, several IGS analysis centers are working on the generation of best possible satellite antenna correction values in order to prepare for a transition from relative to absolute PCVs within the IGS.
Ge and G. Gendt (2005) showed that it is not sufficient to use block-specific antenna correction values, as there are significant differences in the phase center behavior between certain subgroups of the satellite blocks or even between individual satellites (see also Schmid et al. 2005). This holds particularly true for the phase center offsets, but possibly for the PCVs as well. First results of the reprocessing of a global GPS network performed at the Technical Universities of Munich and Dresden (Steigenberger et al. 2004) pointed at the necessity to use long periods of data for the estimation of phase center offsets and variations of individual satellites, as weeks with poor determinability alternate with periods of high accuracy for the estimated parameters (high accuracy for small angles between the orbit plane and the direction to the sun). Since these long-term results are not yet fully available, the present paper will not consider the PCV details of individual satellites.
Rather, the focus of this paper is on a more detailed description of the overall PCV model of the satellite antenna as well as on the benefit from the transition from relative to absolute receiver and satellite antenna PCVs, when estimating global GPS parameters. The first paragraph of the results section addresses the completion of the nadir-dependent satellite antenna PCV models by estimating antenna correction values for Block I satellites by means of data from 1994. The second paragraph is devoted to azimuth-dependent satellite antenna PCVs that may be expected due to the antenna assembly and the power supply (Czopek and Shollenberger 1993). The estimates are primarily analyzed in terms of repeatability and resolution. Recently, PCV maps including azimuthal variations were also derived from on-orbit data of Jason-1 and GRACE A and B by Haines et al. (2004).
Finally, the third paragraph of the results section deals with the comparison of global GPS solutions estimated using relative and absolute PCVs, respectively, for the receiver and the satellite antennas. At first, coordinate sets corresponding to different elevation cut-off angles are compared in order to confirm the suggestion that the use of absolute PCVs might reduce the dependence of coordinate results on the elevation cut-off angle (Schmid and Rothacher 2003). The fact that the mismodeling of an elevation-dependent effect can cause coordinate results depending on the selected elevation cut-off angle was stated by Elósegui et al. (1995), Hatanaka et al. (2001) and others. At the end of the paper, the influence of antenna PCVs on the biases between total tropospheric zenith delays derived from different space geodetic techniques is discussed. Schuh and Böhm (2003) who analyzed the GPS and VLBI (very long baseline interferometry) derived troposphere parameters for identical times at 11 co-located sites found out that all mean values of the total zenith delays derived by GPS are larger than those derived by VLBI. Haas et al. (2003) get similar results for the Onsala Space Observatory from an analysis of long time series of integrated precipitable water vapor (IPWV) from four different techniques (GPS, VLBI, water vapor radiometers (WVRs) and radiosondes). The reduction of systematic biases between GPS, VLBI and other space geodetic techniques such as satellite laser ranging (SLR) are of particular importance with regard to the combination of the different techniques and the establishment of an integrated global geodetic observing system (IGGOS) (Rummel et al. 2000, 2002). Any unmodeled systematic effect present within one single technique will complicate or even rule out such a combination.
Let us start, however, by describing the relation between “horizontal” satellite antenna offsets and azimuthal PCVs, the antenna assembly of the different satellite blocks, and the data sets used.
As a first step, we have to define the azimuth angle under which a specific station is seen from the satellite. We adopt the convention here that the y-axis corresponds to the nominal rotation axis of the solar panels, the z-axis points toward the Earth, and the x-axis completes the right-hand system (Hugentobler et al. 2001). In analogy to the receiver antenna where the azimuth counts clockwise from the north toward the east direction, the azimuth at the satellite antenna is also chosen to count clockwise for these studies, but now from the y-axis in the satellite-fixed coordinate system toward the x-axis when looking toward the z-axis.
Regarding the antenna assembly of the different satellite blocks (Block I, Block II/IIA, Block IIR), not all the facts of interest in connection with azimuthal PCVs are clearly known. According to Aparicio et al. (1995), each antenna comprises 12 helical elements, arranged in two concentric circles on the Earth-facing satellite panel. The inner circle is composed of four elements fed with 90% of the total power, and the outer circle contains the eight remaining elements. Even though the radii of the two circles vary from block to block, there is no indication of an unequal spacing of the single elements within one circle in any of the publications. Czopek and Shollenberger (1993) have shown that the field intensity is not completely symmetrical in the plane perpendicular to the z-axis, because there are only four elements in the center. The maximum gain is achieved in the two planes that intersect two elements of the center quad. The variation of the field intensity caused by the outer circle is minimized due to the increase in the number of elements, providing greater symmetry. As a result and due to the fact that the outer circle is only fed with 10% of the total power, one may expect to see mainly a fourfold pattern in the azimuthal PCVs. This is also clearly demonstrated in Fig. 3 of Haines et al. (2004).
The PCVs were modeled as 2-D piece-wise linear functions with user-defined resolution in both elevation and azimuth direction. The value in nadir direction was constrained to be identical for all different azimuth directions, and the sum of all PCV values was constrained to be zero. As it was not possible to estimate separate phase center corrections for L1 and L2, all results refer to the ionosphere-free linear combination LC (Schmid and Rothacher 2003).
The authors estimated 1-day GPS solutions using double-difference phase data from globally distributed IGS stations with the Bernese GPS Software (Hugentobler et al. 2001) for three different time spans: 1–9 January 1994, 14–19 July 2002 and 17–31 October 2002. The data set from 1994 was generated as part of the reprocessing of a global GPS network performed by the Technical Universities of Munich and Dresden (Steigenberger et al. 2004). Due to the fact that three Block I satellites (PRN03, 12, 13) were active in early 1994, these data allow to estimate phase center correction values for that specific satellite block, too. Moreover, the data are interesting in terms of long-term consistency of the estimated parameters. The second set (July 2002) corresponds to the data used in (Schmid and Rothacher 2003). During the third time span the VLBI campaign CONT02 (see below) took place, so that the tropospheric estimates from GPS could be compared to independent VLBI results at co-located sites.
CONT02, a 2-week campaign of continuous VLBI observations was initiated by the International VLBI Service for Geodesy and Astrometry (IVS). The data of the eight participating stations were analyzed at Deutsches Geodätisches Forschungsinstitut (DGFI) with OCCAM (Titov et al. 2001). As great care was taken to use identical models and the same parameterization for the common parameters in the GPS and the VLBI analysis, a comparison of the tropospheric parameters derived from the two independent techniques should be meaningful (D. Thaller et al., to be published). For further comparisons, a preliminary data set of one of the WVRs located at Onsala was available (Elgered and Haas 2003).
The satellite antenna PCVs were estimated together with all relevant global parameters, namely site coordinates, site-specific troposphere parameters, orbit parameters and Earth rotation parameters. For the receiver antennas, absolute phase center corrections from robot calibrations were applied (Wübbena et al. 2000). For further details see (Schmid and Rothacher 2003). Due to an averaging over approximately 150 stations available, at least regarding the two data sets from 2002, local effects (i.e., effects not common to all stations) such as multipath or troposphere variations are expected to have only minor influence on the results.
Nadir-dependent phase center variations
Moreover, the data from 1994 could be used to check the consistency of the Block II/IIA pattern at two time intervals more than 8 years apart. It has to be noted that differences of up to 3 mm show up when comparing the results from 1994 and 2002, ignoring the maximum nadir angle that is poorly determined. These discrepancies could arise from a scale drift of about 0.2 ppb/year that appears in the reprocessing results using absolute PCVs (Steigenberger et al. 2004). Another explanation could be the slow change in the composition (number of Block I, II/IIA, IIR satellites) of the space segment over time.
Azimuth-dependent phase center variations
As described above, the azimuthal PCVs are always overlaid by the effect of phase center offset errors in the xy-plane. In order to eliminate this dominating and disturbing effect, one offset correction (Δx, Δy) was applied to each 1-day solution. Whereas the daily offset corrections are subject to strong variations (mainly due to correlations with orbital parameters) and necessitate long time spans of data for a precise determination, the azimuthal PCVs are much more stable.
Impact on global GPS solutions
Besides demonstrating a new systematic effect, it is important to clarify the question whether and how the estimation of global GPS parameters is affected. In this context, the effect of a possible transition from relative to absolute PCVs on site coordinates (e.g., on the global scale) as well as on tropospheric parameters is analyzed here. One set of daily, shortly called “relative”, solutions was estimated using the official IGS PCV set igs_01.pcv that contains relative receiver antenna PCVs only. In contrast, the IGS test set pcv_abs_proposed11.tst containing absolute receiver and satellite antenna PCVs was introduced to compute the “absolute” solutions. The latter file only includes nadir-dependent satellite antenna PCVs, so that the azimuth dependence mentioned above is not taken into account here. For all the following investigations, the data set from the second half of October 2002 was used.
Bias of the tropospheric zenith delay (dry and wet part) due to a height difference of 10 m between two instruments for different station heights (Saastamoinen 1973; standard atmosphere)
Station height h (m)
Tropospheric bias Δδρtrop (mm) corresponding to Δh = 10 m
Bias in the estimated tropospheric zenith delays between GPS, VLBI and WVR (17–31 October 2002; height difference between the GPS antenna and the VLBI telescope removed)
GPS–VLBI (Mean of eight stations)
GPS–WVR (Onsala, wet part only)
VLBI–WVR (Onsala, wet part only)
5.4 mm ±4.3 mm
−1.8 mm ±2.7 mm
The tropospheric biases between GPS and VLBI resulting from the use of relative PCVs correspond well to the results of Schuh and Böhm (2003) who analyzed 11 co-located sites altogether. Their investigations also showed that the bias is always positive, the individual values ranging from +1.4 mm to +13.5 mm (Onsala: +4.8 mm). Haas et al. (2003) compared GPS and VLBI with data from WVRs and radiosondes, but for the Onsala Space Observatory only. The biases of the integrated precipitable water vapor (IPWV) they got from an analysis of long time series have to be multiplied by a factor of about 6 in order to get the biases of the wet zenith delay (Bouma 2002). From pair-wise comparisons they got a bias of about +2 mm and +3.5 mm between GPS and VLBI and between GPS and WVR, respectively. However a combination of the four techniques yielded biases of about +0.5 mm (GPS–VLBI) and +3 mm (GPS–WVR).
It has been shown that it is possible to estimate azimuth-dependent antenna PCVs of the GPS satellites from global GPS measurements. Although the results coincide very well with the expectations arising from the geometrical arrangement of the helical elements, it has to be annotated that still not enough is known about the satellite antennas. As the PCVs reach an order of magnitude of ±3–4 mm, they cannot be ignored in high-precision applications. The repeatability of the azimuthal PCVs between different satellites and from day to day as well as after several years is excellent. This leads to the conclusion that changes in the nadir-dependent PCVs over the years might have a different cause, e.g., variations in the global scale.
The investigations of the coordinate results obtained with relative respective absolute PCVs have shown that GPS antennas are a very critical error source. A transition from relative to absolute PCVs would cause jumps of 2–10 mm in all three components. The advantage would be, however, that the dependence of the coordinate results on the elevation cut-off angle could be significantly reduced. As the determination of relative PCVs is not possible down to the horizon, observations below an elevation angle of 10° should not be used without switching to absolute PCVs.
The comparison of tropospheric parameters from GPS, VLBI and WVR has pointed out significant biases that complicate any effort in combining the different space geodetic techniques. It could be demonstrated that these biases are reduced significantly when switching from relative to absolute PCVs: an important step toward more consistency between the techniques. An inter-technique combination will only be beneficial, if each single technique corrects for its specific systematic effects. Another issue in this context could be highlighted once more: the problem that calibration results have to be available for any combination of antenna and radome.
The authors thank Dr. V. Tesmer (DGFI) for the analysis of the VLBI data and Prof. Dr. G. Elgered (Onsala Space Observatory) and his group for making available the preliminary WVR data set.
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