TY - JOUR
AU - Zheng,, Fu
AB - SUMMARY Global positioning system (GPS) position time-series generated using inconsistent satellite products should be aligned to a secular Terrestrial Reference Frame by Helmert transformation. However, unmodelled non-linear variations in station positions can alias into transformation parameters. Based on 17 yr of position time-series of 112 stations produced by precise point positioning (PPP), we investigated the impact of network configuration and scale factor on long-term time-series processing. Relative to the uniform network, the uneven network can introduce a discrepancy of 0.7–1.1 mm, 21.3–27.5 μas and 1.3 mm in terms of root mean square (RMS) for the translation, rotation and scale factor (if estimated), respectively, no matter whether the scale factor is estimated. The RMS of vertical annual amplitude differences caused by such network effect reaches 0.5–0.6 mm. Whether estimating the scale factor mostly affects the Z-translation and vertical annual amplitude, leading to a difference of 1.3 mm when the uneven network is used. Meanwhile, the annual amplitude differences caused by the scale factor present different geographic location dependences over the north, east and up components. The seasonal signals derived from the transformation using the uniform network and without estimating scale factor have better consistency with surface mass loadings with more than 41 per cent of the vertical annual variations explained. Simulation studies show that 40–50 per cent of the annual signals in the scale factor can be explained by the aliasing of surface mass loadings. Another finding is that GPS draconitic errors in station positions can also alias into transformation parameters, while different transformation strategies have limited influence on identifying the draconitic errors. We suggest that the uniform network should be used and the scale factor should not be estimated in Helmert transformation. It is also suggested to perform frame alignment on PPP time-series, even though the used satellite products belong to a consistent reference frame, as the origin of PPP positions inherited from satellite orbits and clocks is not so stable during a long period. With Helmert transformation, the seasonal variations would better agree with surface mass loadings, and noise level of time-series is reduced. Loading of the Earth, Reference systems, Satellite geodesy, Time-series analysis 1 INTRODUCTION Over the past decades, Global Positioning System (GPS) position time-series have presented tremendous advances in geodetic and geophysical investigations, such as the establishment of Terrestrial Reference Frames (Altamimi et al. 2011, 2016), measuring plate motions (Larson et al. 1997; Altamimi et al. 2012; Avsar et al. 2017), as well as the analysis of surface mass redistribution (Kusche & Schrama 2005; Gu et al. 2017). GPS time-series are affected by unmodelled periodical signals, especially the routinely observed seasonal variations. Dong et al. (2002) demonstrated that about 40 per cent of the vertical GPS annual displacements can be explained by seasonal mass redistributions, and the remaining might be related to thermoelastic deformations and other systematic model errors. Another spurious signal discovered in recent years is the systematic errors at harmonics of the GPS draconitic year (Ray et al. 2007; Rebischung et al. 2016). The GPS draconitic errors are suggested to greatly relate to long-period GPS satellite orbit modelling deficiencies (Rodriguez-Solano et al. 2011, 2014; Griffiths & Ray 2012), and station-specific errors, such as imperfections in ocean and atmosphere tidal models, might also contribute some (Tregoning & Watson 2009; Abraha et al. 2017). When performing GPS time-series analysis, these unmodelled signals must be taken into careful consideration, otherwise they may bias the determination of velocity and its uncertainties and corrupt noise analysis (Blewitt & Lavallee 2002; Bogusz & Klos 2015). With the continuous expansion of global GPS networks, it is preferable to generate long-term GPS time-series by precise point positioning (PPP, Zumberge et al. 1997) because of its high computational efficiency. Before using PPP time-series to investigate geophysical signals, it is usually essential to transform PPP solutions to a secular International Terrestrial Reference Frame (ITRF, http://itrf.ign.fr/ITRF_solutions/index.php) or related IGS reference frame (http://acc.igs.org/reference_frame.html) by Helmert transformation (Heflin et al. 1992; Zumberge et al. 1997). For example, one may require the station positions in IGb08 frame (Rebischung et al. 2012), while the operational satellite products delivered by the International GNSS Service (IGS; Dow et al. 2009) are based on IGS14 frame (Rebischung & Schmid 2016). Besides, the full history of IGS products are not really self-consistent due to the continuous evolution of IGS frame. The position time-series estimated using such products contain internal inconsistency, which can cause inappropriate geophysical interpretations. Even though the second reprocessed products (Griffiths 2019) are in the unified IGb08 frame from 1994-01-02 to 2015-02-14, spanning nearly 21 yr, the follow-on final products in this frame extend only to 2017-01-28 due to the implementation of IGS14. This frame transition means that when estimating point positions over the full GPS time spans, the satellite products still are not consistent. Additionally, reprocessing campaigns conducted every few years are too time-consuming and take too much efforts, so the reprocessed products cannot timely solve the inconsistency of reference frame. Given these facts, the frame alignment of fiducial-incomplete or fiducial-inconsistent PPP solutions becomes even necessary. By combining PPP technique and Helmet transformation, long-term GPS time-series of massive networks can be efficiently produced in a consistent reference frame. For example, GPS time-series of thousands of stations are routinely processed by the Jet Propulsion Laboratory (JPL, https://sideshow.jpl.nasa.gov/post/series.html) and Nevada Geodetic Laboratory (NGL, http://geodesy.unr.edu/PlugNPlayPortal.php). The JPL determines GPS orbits and clocks in a no-net-rotation (NNR) frame and computes the transformation parameters from the NNR frame to IGS14 frame with roughly eighty global stations. The PPP solutions including over two thousand stations are generated using the determined satellite products, and then are aligned to IGS14 using the corresponding transformation parameters. Note that the alignment of GPS quasi-instantaneous network to ITRF/IGS frame requires the two frames to be geometrically similar. The ITRF/IGS frame is realized by a set of global points with coordinates at the reference epoch and linear velocities, which does not consider short-term variations, such as seasonal or subdaily variations (Dong et al. 2003). However, the estimated GPS positions inherently contain these unmodelled non-linear variations. In this case, improper frame transformation can introduce significant aliasing errors to transformation parameters (Tregoning & van Dam 2005; Collilieux et al. 2011), which can further affect the analysis of transformed position time-series. In recent years, several different literatures have examined the influence of different Helmert transformation strategies. Tregoning & van Dam (2005) found that the estimated scale factor can absorb part of station displacements from the simulated atmospheric pressure loading, introducing scale errors of 0.3 ppb and height errors of 4 mm. They also emphasized that such influence become even worse in the case of the uneven regional network. Collilieux et al. (2007) showed that the scale factors estimated from the uneven and uniform networks differ by an annual term of 1 mm. The similar network effect on scale factor was also confirmed by Collilieux et al. (2009) using the Satellite Laser Ranging (SLR) network. Collilieux et al. (2011) suggested that the conventional transformation using the largest set of stations as reference stations is not optimal whether the scale factor is estimated or not. More recently, Wei et al. (2016) theoretically explained the propagation mechanism of surface loading deformations on transformation parameters, and demonstrated by data simulation that the loading deformations and uneven distribution of GPS network together can explain about 30 per cent of the annual variations of GPS scale factor. It is noted that most of the previous studies are based on simulated experiments. Chen et al. (2018) demonstrated that the transformation parameters can result in a scatter of 0.1 mm for seasonal signals using real GNSS data. However, the seasonal terms have already been estimated in the stacked secular frame, which is different from the case we discussed here. In this study, real GPS time-series estimated by PPP with inconsistent satellite products are aligned to IGS14 frame with and without scale factor estimated using different network configurations. Results from different transformation solutions are analysed and compared, aiming at providing some practical suggestions about frame transformation of long-term GPS time-series, especially on how to avoid the impact of improper transformation strategies on the geophysical signals in position time-series as much as possible. We also would like to answer the following question: for long-term PPP processing, if the used satellite orbits and clocks have been generated in a consistent reference frame, whether it is still necessary to align daily solutions to a secular reference frame by using Helmert transformation. The paper is organized as follows. Section 2 introduces method to generate GPS position time-series and Helmert transformation strategies. The transformation parameters estimated with different transformation strategies are analysed and compared in Section 3. Section 4 evaluates the impact of Helmert transformation strategies on GPS seasonal and draconitic signals. In Section 5, we assess the level of agreement between GPS seasonal signals and surface mass loading deformations. The frame transformation of the nominally consistent GPS time-series is performed and analysed in Section 6. Finally, Section 7 concludes the main results and give some significant suggestions. 2 POSITION TIME-SERIES AND HELMERT TRANSFORMATION 2.1 Method to calculate GPS time-series In this study, a set of 112 global stations were selected from the IGS14 frame. The station distribution is illustrated in Fig. 1. The observations of these stations from 2002-01-01 to 2019-10-31 were processed in daily PPP mode using PANDA software (Liu & Ge 2003). The station positions were loosely constrained and estimated with other auxiliary parameters by fixing satellite orbits and clocks, and Earth Rotation Parameters (ERPs). The used orbits and clocks, including the second reprocessed and subsequent final products, are from two IGS Analysis Centers (ACs), the European Space Agency (ESA, ftp://ftp.igs.org/pub/center/analysis/esa.acn) and Massachusetts Institute of Technology (MIT, ftp://ftp.igs.org/pub/center/analysis/mit.acn). Note that these products are in IGb08 frame until 2017-01-28, while in IGS14 frame from 2017-01-29 to 2019-10-31. The ERPs are from the International Earth Rotation and Reference Systems Service (IERS) 08/14 C04 products. Figure 1. Open in new tabDownload slide Distribution of stations involved in real GPS time-series. The full-network consists of all 112 stations (black and red dots), while the subnetwork consists of 50 core stations (red dots). Figure 1. Open in new tabDownload slide Distribution of stations involved in real GPS time-series. The full-network consists of all 112 stations (black and red dots), while the subnetwork consists of 50 core stations (red dots). The GPS ionosphere-free observations were used for PPP processing. The phase centre offsets (PCOs) and phase centre variations (PCVs) of satellite and receiver antennas were corrected with igs08.atx/igs14.atx. The tropospheric delay was corrected using the GMF/GPT2 model (Boehm et al. 2006; Lagler et al. 2013) with the remaining estimated as piecewise constant. The first order ionospheric delay was eliminated using ionosphere-free observations and the higher was corrected. Solid Earth tide, ocean tide and pole tide were corrected according to IERS Conversions 2010 (Petit & Luzum 2010). No atmospheric pressure loading was applied. To obtain more reliable station positions, double-differenced ambiguity resolution was applied by combining the zero-differenced ambiguities from simultaneously observed stations of PPP solutions with the method proposed by Ge et al. (2005). 2.2 Helmert transformation strategies Due to the usage of inconsistent satellite products in PPP solutions, the resulting station positions are based on different reference frames (IGb08 and IGS14) during the periods from 2002-01-01 to 2017-01-28 and from 2017-01-29 to 2019-10-31. The inconsistent PPP positions were uniformly aligned to IGS14 by Helmert transformation. For comparison purpose, seven-parameter transformation (estimating three translations, three rotations, one scale factor) and six-parameter transformation (estimating three translations, three rotations) were parallelly performed. Two GPS networks that distinguish by station number and distribution were used to examine the network effect. The first network, called full-network hereafter, uses all 112 stations (black and red dots in Fig. 1) to estimate transformation parameters. This full-network is unevenly distributed with 70 per cent of the stations densely located in the Northern Hemisphere, especially in Europe and North America. Being a subset of the former, the second is a uniform subnetwork, in which 50 core stations (red dots in Fig. 1) are used in the estimation of transformation parameters. These core stations were carefully selected to increase the robustness of frame alignment, among which 43 stations have at least 15 yr of data and the remaining seven stations have longer than 10 yr of data. In this paper, the station used to compute transformation parameters is denoted as ‘transformation station’, and the corresponding network of such stations is called ‘transformation network’. According to the option of scale factor and the transformation network adopted in frame transformation, four different transformation solutions are listed in Table 1. The ‘7helm’ and ‘6helm’ are used to denote seven-parameter and six-parameter transformations, respectively. The ‘full’ and ‘sub’ represents the transformations using the uneven full-network and uniform subnetwork, respectively. Table 1. Solutions with different transformation strategies. Solution . Helmert parameter . Network . 7helm_full Translation, rotation, scale Uneven full-network with all 112 stations 7helm_sub Translation, rotation, scale Uniform subnetwork with 50 core stations 6helm_full Translation, rotation Uneven full-network with all 112 stations 6helm_sub Translation, rotation Uniform subnetwork with 50 core stations Solution . Helmert parameter . Network . 7helm_full Translation, rotation, scale Uneven full-network with all 112 stations 7helm_sub Translation, rotation, scale Uniform subnetwork with 50 core stations 6helm_full Translation, rotation Uneven full-network with all 112 stations 6helm_sub Translation, rotation Uniform subnetwork with 50 core stations Open in new tab Table 1. Solutions with different transformation strategies. Solution . Helmert parameter . Network . 7helm_full Translation, rotation, scale Uneven full-network with all 112 stations 7helm_sub Translation, rotation, scale Uniform subnetwork with 50 core stations 6helm_full Translation, rotation Uneven full-network with all 112 stations 6helm_sub Translation, rotation Uniform subnetwork with 50 core stations Solution . Helmert parameter . Network . 7helm_full Translation, rotation, scale Uneven full-network with all 112 stations 7helm_sub Translation, rotation, scale Uniform subnetwork with 50 core stations 6helm_full Translation, rotation Uneven full-network with all 112 stations 6helm_sub Translation, rotation Uniform subnetwork with 50 core stations Open in new tab 2.3 Simulation experiment Though this contribution aims at the frame transformation of real GPS time-series, a simulation experiment was also performed here for a thorough understanding of the aliasing effect of seasonal signals on transformation parameters. The simulated position time-series for all 112 stations are generated using their reference coordinates and velocities from IGS IGb08.snx file plus the surface mass loading deformations, covering the period from 2002-01-01 to 2012-08-20. The surface mass loading models at stations are provided by the École et Observatoire des Sciences de la Terre (EOST) Loading Service (http://loading.u-strasbg.fr/displacements.php) in the centre of figure of solid earth (CF) frame, including the contribution of atmospheric loading (.atmib extension), hydrology loading (.gldas extension) and non-tidal ocean loading (.ecco extension). For comparison purpose, another similar simulated time-series without considering the seasonal surface loadings were also prepared. The two simulated time-series, which are in IGb08 frame theoretically, were transformed to IGS14 frame by using the same transformation strategies as described in Section 2.2. For clarity, these two simulated position time-series with and without the surface mass loading deformations are denoted as SOL-L and SOL-NL in the following sections, respectively. 3 ANALYSIS OF HELMERT TRANSFORMATION PARAMETERS In this section, the transformation parameters estimated with different transformation strategies are evaluated, and the results from the real GPS time-series are compared with that from the simulated time-series. 3.1 Analysis using real GPS time-series 3.1.1 Analysis of estimated transformation parameters Figs 2 and 3 show the transformation parameters derived with ESA and MIT time-series, respectively. The scale factor in millimeters has been multiplied by 6.4 (1 ppb = 6.4 mm on the Earth surface). When the reference frame of satellite orbits switches from IGb08 to IGS14, a clear offset is observed in the transformation parameter time-series, especially for the three translations and X-rotation. This reveals that the inconsistent frames of satellite products used in PPP solutions will result in irreconcilable datum inconsistency within GPS time-series, which should be carefully considered. Figure 2. Open in new tabDownload slide Transformation parameters derived with ESA time-series. The vertical dashed–dotted line denotes the frame change of satellite orbits from IGb08 to IGS14. Figure 2. Open in new tabDownload slide Transformation parameters derived with ESA time-series. The vertical dashed–dotted line denotes the frame change of satellite orbits from IGb08 to IGS14. Figure 3. Open in new tabDownload slide Transformation parameters derived with MIT time-series. The vertical dashed–dotted line denotes the frame change of satellite orbits from IGb08 to IGS14. Figure 3. Open in new tabDownload slide Transformation parameters derived with MIT time-series. The vertical dashed–dotted line denotes the frame change of satellite orbits from IGb08 to IGS14. It is visible in Figs 2 and 3 that the transformation parameters from IGb08 to IGS14 (before 2017-01-29) are very noteworthy. In particular, the Z-translation and scale can reach up to about 10 mm when considering the influence of seasonal variations. As the IGb08 and IGS14 frames are derived from ITRF2008 (Altamimi et al. 2011) and ITRF2014 (Altamimi et al. 2016), respectively, the frame differences between ITRF2008 and ITRF2014 should take the inescapable and main responsibility (http://itrf.ensg.ign.fr/doc_ITRF/Transfo-ITRF2014_ITRFs.txt). A positive rate for Z-translation and a negative rate for scale factor can be clearly observed, which are due to the datum discrepancies between ITRF2008 and ITRF2014. As shown in Figs 2 and 3, before 2017-01-29 (transition to IGS14 for satellite products), the MIT translations and rotations are less fluctuant than the ESA parameters. Especially, the ESA Y- and Z-rotation have visible trends, which cannot be attributed to the frame discrepancies between ITRF2008 and ITRF2014. For IGS AC processing, NNR constraints are applied to the selected stations to align GPS orbits to IGS reference frame. Therefore, it may have some deficiencies of NNR constraints for ESA orbits. The piecewise linear variations of Z-translation for both ESA and MIT cannot be explained by the frame discrepancies between ITRF2008 and ITRF2014 neither. Moreover, the Z-translation pattern is also different among ESA and MIT solutions, although the identical data processing strategies and network configuration were adopted in PPP and frame transformation. It is demonstrated that the origin of the reference frame determined by GPS technique alone is not reliable enough due to serious collinearity issues of GPS parameters (Rebischung et al. 2013) and is particularly sensitive to data processing strategies especially satellite orbit modelling deficiencies (Collilieux et al. 2010; Meindl et al. 2013). Therefore, all of these indicate that the reference frame of PPP positions inherited from the consistent satellite orbits are still not so stable. Further discussions will be provided in Sections 3.2 and 6.3. The seasonal variations of transformation parameters are induced by the aliasing of seasonal variations in station position time-series, when we align PPP time-series to linear IGS14. This mechanism has been mentioned in Section 1 and will be further discussed in Section 3.2 using simulation studies. Note that MIT translations present more significant seasonal oscillations. 3.1.2 Impact of transformation network used in frame alignment To investigate the impact of transformation network and scale factor in frame alignment, Fig. 4 illustrates the transformation parameter differences between ESA different solutions. Comparing the transformation parameters from the full-network and subnetwork transformations, the network effect induces comparable differences for translation and rotation parameters, regardless of seven-parameter or six-parameter transformations (Figs 4a and b). The RMS of translation, rotation and scale factor (when estimated) differences are 0.7–1.1 mm, 22.1–27.5 μas (100 µas ≈ 3.1 mm of equatorial rotation) and 1.3 mm, respectively. Clear seasonal signals, which are firstly observed in transformation parameter time-series (Figs 2 and 3), are also visible in their differenced time-series (Figs 4a and b), especially for the translation and scale factor. Due to the linear characteristic of IGS14 frame, seasonal variations (mainly surface mass loading deformations) in instantaneous station positions can easily alias into the estimated transformation parameters (Tregoning & van Dam 2005; Collilieux et al. 2011). As demonstrated by eqs (9) and (13) in Wei et al. (2016), the uneven transformation network can worsen the effect of loading aliasing errors on transformation parameters. In this study, the full-network and subnetwork significantly differ in station number and distribution, so the common seasonal surface loadings contained in transformation stations are also diverse. These common loading differences are consequently absorbed into transformation parameters. What's more, the influence of uneven transformation network and seasonal surface loadings on the rotation is much smaller than that on the translation and scale factor in theory (Wei et al. 2016). This is why the rotation presents less significant seasonal oscillations. Figure 4. Open in new tabDownload slide Transformation parameter differences between ESA different Helmert transformations. Figure 4. Open in new tabDownload slide Transformation parameter differences between ESA different Helmert transformations. To gain a detailed insight into the seasonal variations of transformation parameters, the Lomb–Scargle periodogram (Scargle 1982) was computed and then smoothed using a boxcar smoother (e.g. Ray et al. 2007). Taking the typical scale factor as an example, as shown in Fig. 5, an obvious reduction of annual peak can be found when switching the full-network transformation to subnetwork transformation. It indicates that the homogeneous distribution of transformation stations can mitigate the aliasing errors of seasonal signals into the scale factor to some extent. Figure 5. Open in new tabDownload slide Lomb–Scargle periodogram of the detrend scale factor from ESA seven-parameter transformations. Figure 5. Open in new tabDownload slide Lomb–Scargle periodogram of the detrend scale factor from ESA seven-parameter transformations. Following the annual and semi-annual signals, sharp peaks centred on the GPS draconitic harmonics are clearly observed in Fig. 5, which are originally found in GPS-derived geodetic products, such as satellite orbits and station positions (Ray et al. 2007; Rodriguez-Solano et al. 2014). A more extensive analysis reveals that the GPS draconitic harmonics also exist in translation and rotation parameters as well as in MIT transformation parameters (e.g. Fig. A1). It indicates that GPS draconitic errors in position time-series can get absorbed erroneously into the transformation parameters in a similar manner of seasonal variations. In addition, an unexpected signal at the period of about 120 days was detected in translation and scale parameters by Arnold et al. (2014) and Wei et al. (2016). Arnold et al. (2014), inferred that this unknown oscillation is likely to be induced by radiation pressure models. Considering the findings of this study, we suggest that this 120-day signal should be the third GPS draconitic harmonic (about 117 d). 3.1.3 Impact of scale factor used in frame alignment Figs 4(c) and (d) illustrates the impact of estimating scale factor on translation and rotation parameters. When the full-network is used, estimating the scale factor or not causes significant seasonal variations for the Z-translation, as can be noted in Fig. 4(c3). Meanwhile, a striking linear trend is also visible in the Z-translation, which may disturb the determination of station velocity. However, for the transformations with the subnetwork, the scale factor does not cause appreciable seasonal differences in any transformation parameters, as shown in Fig. 4(d). This is consistent with the simulation results in Wei et al. (2016). When exchanging the uneven full-network by the uniform subnetwork, the RMS of Z-translation differences induced by the scale factor is reduced from 1.27 mm to 0.57 mm. This reveals that the uniform transformation network is beneficial to moderate the discrepancy of Z-translation. No significant rotation differences are induced by the scale factor with a general RMS of 4.8–11.5 μas, demonstrating that the impact of scale factor on the rotation parameters is not sensitive to transformation networks. Overall, the differences of transformation parameters caused by the network effect generally stay at the level of about 1 mm RMS. In contrast, the parameter differences induced by the scale factor are relatively small, which is usually below 0.6 mm RMS for translations (except Z-translation differences between 7hem_full and 6helm_sub) and below 11.5 μas RMS for rotations. In this light, the network configuration indeed has a more significant impact than the scale factor in frame alignment. Although the estimated transformation parameters from ESA and MIT solutions present irreconcilable characteristics (Figs 2 and 3), however, for the two ACs, the differenced transformation parameters between different transformations exhibit almost the same variation patterns, just as illustrated in Figs 4 (ESA) and A2 (MIT). This result is of great importance, which actually demonstrates that the variation behaviours of transformation parameters between different transformation strategies are not dependent on the satellite products used in PPP, but result from the network configuration and the choice of scale factor in frame transformation. 3.2 Analysis using simulated position time-series Fig. 6 illustrates the transformation parameter time-series from the two simulated solutions, SOL-L and SOL-NL. It is visible that the transformation parameters from the SOL-L solution present significant seasonal variations, especially when the uneven full-network is used. Regarding the SOL-NL solution, no obvious seasonal signals are observed. This result allows us to conclude that the seasonal signals in the simulated transformation parameters result from the aliasing of surface mass loading deformations. Figure 6. Open in new tabDownload slide Transformation parameters derived from the simulated position time-series. Figure 6. Open in new tabDownload slide Transformation parameters derived from the simulated position time-series. Note that the surface mass loading deformations referred to here are not equal to the seasonal variations in real GPS time-series, but only the main part of the latter. Nearly half of the vertical GPS annual signals, and less than 20 per cent of the horizontal GPS annual signals can be explained by surface mass loadings. As shown by Figs 2, 3 and 6, the seasonal amplitudes of translation and scale parameters from the SON-L solution are generally smaller than that from the ESA and MIT solutions. To analyse the effect of surface mass loadings on seasonal signals in real transformation parameters, we estimated and compared the annual signals of translation and scale parameters from the ESA, MIT and SOL-L solutions over the common period from 2002-01-01 to 2012-08-20. The reduction ratio of annual amplitudes is defined as the following equation (Xu et al. 2017) $$\begin{eqnarray*} &&\hspace{-6pt} {\rm reduction\;ratio} \nonumber \\ && \quad = \frac{{{A_{\rm GPS}} - \sqrt {A_{\rm GPS}^2 + A_{\rm Loading}^2 - 2{A_{\rm GPS}}{A_{\rm Loading}}\cos \Delta \varphi } }}{{{A_{\rm GPS}}}}, \end{eqnarray*}$$(1) where |${A_{\rm GPS}}$| and |${A_{\rm Loading}}$| denote the annual amplitudes of transformation parameters derived from ESA/MIT time-series and SON-L time-series, respectively; |$\Delta \varphi $| represents the annual phase differences of transformation parameters between ESA/MIT time-series and SON-L time-series. Eq. (1) can indicate how much of the annual amplitudes in ESA/MIT transformation parameters results from the aliasing of surface mass loadings. As shown in Table 2, about 52 per cent of the annual signals in the scale factor are contributed by the aliasing of surface mass loadings when the uneven full-network is used. Meanwhile, in the case of the full-network, the contribution ratio of surface mass loading effect to the annual amplitudes of the Z-translation is more than 61 and 32 per cent for the ESA and MIT solutions, respectively. Regarding the transformations with the uniform subnetwork, the aliasing of surface mass loadings can explain about 43–50 per cent of the annual signals for the scale factor, about 42–46 per cent for the ESA Z-translation, and about 20 per cent for the MIT Z-translation. These results indicate that the aliasing of surface mass loadings is reduced by using well-distributed transformation network. Table 2. Contribution ratio (in per cent) of the surface mass loadings to the annual signals in ESA and MIT translation and scale parameters (covering the common period from 2002-01-01 to 2012-08-20). Solutions . Time-series . X-translation . Y-translation . Z-translation . Scale . 7helm_full ESA 9.3 −181.2 70.9 52.5 MIT 56.5 6.1 32.3 51.6 7helm_sub ESA −3.5 −33.5 46.4 50.3 MIT −0.9 8.4 20.8 42.9 6helm_full ESA 39.3 −163.0 60.9 – MIT 60.4 7.1 35.1 – 6helm_sub ESA −11.5 −12.6 42.3 – MIT 4.2 7.8 20.1 – Solutions . Time-series . X-translation . Y-translation . Z-translation . Scale . 7helm_full ESA 9.3 −181.2 70.9 52.5 MIT 56.5 6.1 32.3 51.6 7helm_sub ESA −3.5 −33.5 46.4 50.3 MIT −0.9 8.4 20.8 42.9 6helm_full ESA 39.3 −163.0 60.9 – MIT 60.4 7.1 35.1 – 6helm_sub ESA −11.5 −12.6 42.3 – MIT 4.2 7.8 20.1 – Open in new tab Table 2. Contribution ratio (in per cent) of the surface mass loadings to the annual signals in ESA and MIT translation and scale parameters (covering the common period from 2002-01-01 to 2012-08-20). Solutions . Time-series . X-translation . Y-translation . Z-translation . Scale . 7helm_full ESA 9.3 −181.2 70.9 52.5 MIT 56.5 6.1 32.3 51.6 7helm_sub ESA −3.5 −33.5 46.4 50.3 MIT −0.9 8.4 20.8 42.9 6helm_full ESA 39.3 −163.0 60.9 – MIT 60.4 7.1 35.1 – 6helm_sub ESA −11.5 −12.6 42.3 – MIT 4.2 7.8 20.1 – Solutions . Time-series . X-translation . Y-translation . Z-translation . Scale . 7helm_full ESA 9.3 −181.2 70.9 52.5 MIT 56.5 6.1 32.3 51.6 7helm_sub ESA −3.5 −33.5 46.4 50.3 MIT −0.9 8.4 20.8 42.9 6helm_full ESA 39.3 −163.0 60.9 – MIT 60.4 7.1 35.1 – 6helm_sub ESA −11.5 −12.6 42.3 – MIT 4.2 7.8 20.1 – Open in new tab Table 3 lists the mean values of translation parameters from ESA, MIT, SOL-L and SOL-NL solutions over the common period from 2002-01-01 to 2012-08-20. Two simulated solutions SOL-L and SOL-NL obtain almost the same mean translations. However, the translations of ESA and MIT solutions present significant discrepancies with each other for the X and Z components. When comparing the results of SOL-L/SOL-NL and ESA/MIT solutions, a largest translation offset of 1.8 mm can be found for the X and Z components. This result can further illustrate that the frame origin inherited from GPS orbits is not stable. Nevertheless, the MIT translations are more consistent with that of simulation experiments. It indicates that the datum of MIT products is more stable than that of ESA, which can also be concluded from more volatility of ESA translations than that of MIT translations mentioned in Section 3.1.1. Table 3. Mean values (in mm) of translation parameters from ESA, MIT solutions and simulation experiment over the common period from 2002-01-01 to 2012-08-20. Solution . 7helm_full . 7helm_sub . 6helm_full . 6helm_sub . . X . Y . Z . X . Y . Z . X . Y . Z . X . Y . Z . ESA −0.8 −2.0 −5.2 −1.4 −2.1 −4.7 −0.4 −2.1 −3.9 −1.3 −1.9 −4.2 MIT −1.4 −2.3 −3.6 −1.9 −2.3 −3.3 −1.0 −2.4 −2.1 −1.8 −2.1 −2.7 SOL-L −2.3 −1.9 −3.4 −2.5 −1.9 −3.3 −2.2 −1.9 −2.9 −2.4 −1.8 −3.1 SOL-NL −2.3 −1.8 −3.5 −2.4 −1.9 −3.4 −2.1 −1.8 −3.0 −2.2 −1.8 −3.2 Solution . 7helm_full . 7helm_sub . 6helm_full . 6helm_sub . . X . Y . Z . X . Y . Z . X . Y . Z . X . Y . Z . ESA −0.8 −2.0 −5.2 −1.4 −2.1 −4.7 −0.4 −2.1 −3.9 −1.3 −1.9 −4.2 MIT −1.4 −2.3 −3.6 −1.9 −2.3 −3.3 −1.0 −2.4 −2.1 −1.8 −2.1 −2.7 SOL-L −2.3 −1.9 −3.4 −2.5 −1.9 −3.3 −2.2 −1.9 −2.9 −2.4 −1.8 −3.1 SOL-NL −2.3 −1.8 −3.5 −2.4 −1.9 −3.4 −2.1 −1.8 −3.0 −2.2 −1.8 −3.2 Open in new tab Table 3. Mean values (in mm) of translation parameters from ESA, MIT solutions and simulation experiment over the common period from 2002-01-01 to 2012-08-20. Solution . 7helm_full . 7helm_sub . 6helm_full . 6helm_sub . . X . Y . Z . X . Y . Z . X . Y . Z . X . Y . Z . ESA −0.8 −2.0 −5.2 −1.4 −2.1 −4.7 −0.4 −2.1 −3.9 −1.3 −1.9 −4.2 MIT −1.4 −2.3 −3.6 −1.9 −2.3 −3.3 −1.0 −2.4 −2.1 −1.8 −2.1 −2.7 SOL-L −2.3 −1.9 −3.4 −2.5 −1.9 −3.3 −2.2 −1.9 −2.9 −2.4 −1.8 −3.1 SOL-NL −2.3 −1.8 −3.5 −2.4 −1.9 −3.4 −2.1 −1.8 −3.0 −2.2 −1.8 −3.2 Solution . 7helm_full . 7helm_sub . 6helm_full . 6helm_sub . . X . Y . Z . X . Y . Z . X . Y . Z . X . Y . Z . ESA −0.8 −2.0 −5.2 −1.4 −2.1 −4.7 −0.4 −2.1 −3.9 −1.3 −1.9 −4.2 MIT −1.4 −2.3 −3.6 −1.9 −2.3 −3.3 −1.0 −2.4 −2.1 −1.8 −2.1 −2.7 SOL-L −2.3 −1.9 −3.4 −2.5 −1.9 −3.3 −2.2 −1.9 −2.9 −2.4 −1.8 −3.1 SOL-NL −2.3 −1.8 −3.5 −2.4 −1.9 −3.4 −2.1 −1.8 −3.0 −2.2 −1.8 −3.2 Open in new tab Presented in Fig. 7 is the transformation parameter differences between different transformations for SOL-L time-series. It is noted that the differenced transformation parameters from the simulated time-series show quite similar variation patterns to that from the real GPS time-series. Together with Fig. 6, it is safely concluded that the uneven full-network really makes the aliasing of seasonal loading deformations more significant. Thus, the differenced transformation parameters also present obvious seasonal variations (Figs 7a and b). When the transformation network is not uniformly distributed, estimating the scale factor mainly affects the Z-translation, which is consistent with the case when real GPS time-series is adopted. These simulation results can support the conclusions about the influence of network effect and scale factor on transformation parameters in Section 3.1. Figure 7. Open in new tabDownload slide Transformation parameter differences between different Helmert transformations for the simulated time-series with surface mass loading deformations. Figure 7. Open in new tabDownload slide Transformation parameter differences between different Helmert transformations for the simulated time-series with surface mass loading deformations. 4 IMPACT OF HELMERT TRANSFORMATION STRATEGIES ON PERIODIC SIGNALS Real GPS time-series after frame alignment were analysed (outliers removed, offsets corrected, detrended, and seasonal variations estimated) by the Quasi-Observation Combination Analysis (QOCA) software (Dong et al. 2002). GPS annual and draconitic signals were derived from the resulting position residual time-series to investigate the impact of Helmert transformation strategies on periodic signals. 4.1 Impact on GPS annual signal Table 4 lists the RMS of annual amplitude and phase differences between different transformation solutions. At the first glance, the annual signal differences derived from ESA and MIT solutions stay at the quite comparable level, and present similar variation tendency among different comparisons. This result further indicates that GPS annual signal differences and the transformation parameter differences between different Helmert transformation solutions are much more induced by network configuration and scale factor, rather than by the specific orbits used in PPP. Although the following analysis will take more of the ESA results as examples, it is also suitable for MIT solutions in most cases. Table 4. RMS of annual amplitude and phase differences between different transformation solutions. Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . ESA 7helm_full versus 7helm_sub 0.21 0.34 0.65 17.01 39.11 21.71 6helm_full versus 6helm_sub 0.43 0.40 0.56 23.02 41.13 12.01 7helm_full versus 6helm_full 0.41 0.10 1.26 18.89 8.82 41.67 7helm_sub versus 6helm_sub 0.09 0.02 0.99 3.79 4.13 26.35 MIT 7helm_full versus 7helm_sub 0.18 0.25 0.53 13.39 36.32 21.15 6helm_full versus 6helm_sub 0.40 0.31 0.53 14.67 39.11 13.32 7helm_full versus 6helm_full 0.42 0.09 1.22 15.00 9.13 39.85 7helm_sub versus 6helm_sub 0.10 0.03 1.05 6.37 3.97 27.81 Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . ESA 7helm_full versus 7helm_sub 0.21 0.34 0.65 17.01 39.11 21.71 6helm_full versus 6helm_sub 0.43 0.40 0.56 23.02 41.13 12.01 7helm_full versus 6helm_full 0.41 0.10 1.26 18.89 8.82 41.67 7helm_sub versus 6helm_sub 0.09 0.02 0.99 3.79 4.13 26.35 MIT 7helm_full versus 7helm_sub 0.18 0.25 0.53 13.39 36.32 21.15 6helm_full versus 6helm_sub 0.40 0.31 0.53 14.67 39.11 13.32 7helm_full versus 6helm_full 0.42 0.09 1.22 15.00 9.13 39.85 7helm_sub versus 6helm_sub 0.10 0.03 1.05 6.37 3.97 27.81 Open in new tab Table 4. RMS of annual amplitude and phase differences between different transformation solutions. Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . ESA 7helm_full versus 7helm_sub 0.21 0.34 0.65 17.01 39.11 21.71 6helm_full versus 6helm_sub 0.43 0.40 0.56 23.02 41.13 12.01 7helm_full versus 6helm_full 0.41 0.10 1.26 18.89 8.82 41.67 7helm_sub versus 6helm_sub 0.09 0.02 0.99 3.79 4.13 26.35 MIT 7helm_full versus 7helm_sub 0.18 0.25 0.53 13.39 36.32 21.15 6helm_full versus 6helm_sub 0.40 0.31 0.53 14.67 39.11 13.32 7helm_full versus 6helm_full 0.42 0.09 1.22 15.00 9.13 39.85 7helm_sub versus 6helm_sub 0.10 0.03 1.05 6.37 3.97 27.81 Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . ESA 7helm_full versus 7helm_sub 0.21 0.34 0.65 17.01 39.11 21.71 6helm_full versus 6helm_sub 0.43 0.40 0.56 23.02 41.13 12.01 7helm_full versus 6helm_full 0.41 0.10 1.26 18.89 8.82 41.67 7helm_sub versus 6helm_sub 0.09 0.02 0.99 3.79 4.13 26.35 MIT 7helm_full versus 7helm_sub 0.18 0.25 0.53 13.39 36.32 21.15 6helm_full versus 6helm_sub 0.40 0.31 0.53 14.67 39.11 13.32 7helm_full versus 6helm_full 0.42 0.09 1.22 15.00 9.13 39.85 7helm_sub versus 6helm_sub 0.10 0.03 1.05 6.37 3.97 27.81 Open in new tab 4.1.1 Impact of transformation network used in frame alignment Generally speaking, the annual signal differences between the transformed position time-series directly result from the differences of the corresponding transformation parameters, and are related to the contribution of transformation parameters in geocentric coordinate system to different topocentric components. As mentioned before, compared to the impact of scale factor, the network configuration has more significant impact on translation and rotation parameters, so the amplitude and phase differences of GPS annual signal caused by transformation network are roughly larger than that caused by scale factor for the north and east components. As listed in Table 4, the annual amplitude differences induced by network effect stay at the level of 0.2–0.4 mm RMS in the north and east components, and at the level of 0.5–0.6 mm RMS in the up component. As shown in Figs 4(a) and (b), A2(a) and (b), the translation differences caused by the network effect are enlarged in the six-parameter transformations with respect to that from the seven-parameter transformations, especially for the X- and Z-translation. According to the transformation between geocentric and topocentric coordinate systems [refer to eq. (B1) in Appendix B], the X-translation contributes to all north, east and up components, while the Z-translation only affects the north and up components. Due to the increased X- and Z-translation differences in the comparison 6helm_full versus 6helm_sub, the RMS of annual amplitude differences is obviously increased by about 0.2 mm in the north component, and is slightly increased by less than 0.1 mm in the east component with respect to the results of the comparison 7helm_full versus 7helm_sub, as shown in Table 4. However, the vertical annual amplitude differences induced by the network effect are slightly reduced for the ESA solutions and nearly does not change for the MIT solutions, in spite of the increased X- and Z-translation differences. This should be due to the fact that the scale factor, which mainly affects the up component, does not work in the comparison 6helm_full versus 6helm_sub. 4.1.2 Impact of scale factor used in frame alignment As shown in Figs 4(d) and A2(d), the parameter discrepancy induced by scale factor are less than 0.6 mm RMS for translations and less than 11.5 μas RMS for rotations when the uniform subnetwork is used. With such small parameter differences, the impact of scale factor on the annual signals of the north and east components can be negligible, as estimating scale factor mainly affect the up component. Regarding the comparison 7helm_sub versus 6helm_sub, the RMS of annual amplitude differences is only 0.1 and 0.03 mm for the north and east components, respectively, but reach about 1 mm for the up component. When the uneven full-network is used, the Z-translation differences can be significantly scattered by scale factor, presenting obvious seasonal variations and linear trend (see Fig. 4c3). As the Z-translation only affects the north and up components [refer to eq. (B1) in Appendix B], the RMS of annual amplitude differences of the north and up components increases by 0.2–0.3 mm. These results indicate that the impact of scale factor on the annual signals is closely related to network configuration. As illustrated in Fig. 8, the annual amplitude differences in the North component are obviously latitude-dependent, whose absolute values decrease as the latitude rises. The magnitude of annual amplitudes derived from the 7helm_full solution is universally smaller than that from the 6helm_full solution. Although the amplitude differences in the east component are quite negligible, an obvious distribution characteristic along the longitude direction is also found. Regarding the up component, estimating the scale factor leads to an overestimation of annual amplitudes near the equator, while leads to an underestimation at high latitudes, especially in the Northern Hemisphere. According to the transformation matrix from geocentric to topocentric coordinate system [refer to eq. (B1) in Appendix B], the north and up components are closely related to the geodetic latitude and longitude, and the latitude plays a more important role, while the east component is only dependent on the longitude. This may partially explain the latitude-dependence of the north and up components and the longitude-dependence of the east component. Figure 8. Open in new tabDownload slide Amplitude differences of the annual signal between ESA different Helmert transformations. Figure 8. Open in new tabDownload slide Amplitude differences of the annual signal between ESA different Helmert transformations. In short, the annual amplitude differences caused by network configuration stay at the level of 0.2–0.6 mm RMS, and the network effect is influenced by the choice of scale factor. The impact of scale factor on the vertical annual amplitude can reach up to 1.3 mm RMS, and can be mitigated by using the uniform transformation network. It is also found that the annual amplitude differences induced by the scale factor are latitude-dependent for the north and up components, while they are longitude-dependent for the east component. 4.2 Impact on GPS draconitic harmonics The power spectra of GPS position residual time-series with the seasonal terms removed were computed and stacked over all stations (see Figs A3 and A4). It is found that GPS draconitic signals prevail at least until the ninth harmonic for all components. For a detailed analysis, Fig. 9 illustrates the interpolated numerical values of the power spectra at the first nine draconitic harmonics for ESA solutions. As GPS seasonal variations and draconitic harmonics are affected by the identical transformation parameters, their variation features among different transformations are also similar. the east component shows negligible spectral differences due to its immunization to the Z-translation and scale factor. Considering the north component, the two transformations using the uniform subnetwork show consistent draconitic harmonics at least over the first seven harmonics. Regarding the transformations 7helm_full and 6helm_full, their draconitic peaks are close at most of the harmonics, but slightly divergent at the third, fifth and seventh, which may be due to the fact that the impact of the scale factor is amplified by uneven network. Regarding the up component, the transformations adopting the same transformation network present similar draconitic peaks at the first three harmonics, indicating that the network effect is dominant, but the impact of scale factor gradually prevails at higher frequencies. Figure 9. Open in new tabDownload slide Numerical values of the power spectra at the first nine draconitic harmonics for ESA solutions. Figure 9. Open in new tabDownload slide Numerical values of the power spectra at the first nine draconitic harmonics for ESA solutions. It is visible that the draconitic signals derived from the seven-parameter transformations are generally weaker than that from the six-parameter transformations. Compared with ESA 7helm_full solution, the first nine draconitic harmonics of ESA 6helm_sub solution get strengthened by 10, −2 and 9 per cent in the north, east and up components, respectively. Similarly, this value is 11, −3 and 6 per cent for MIT solutions. It is concluded that estimating the scale factor and using the uneven transformation network can lead to certain underestimation of GPS draconitic errors for the north and up components. 5 COMPARISON OF SEASONAL SIGNALS WITH SURFACE MASS LOADINGS GPS annual variations derived with different transformation strategies are compared to surface mass loading deformations. The used surface mass loading deformations at stations is as same as that used in the simulation experiment. To quantify the consistency of GPS annual variations with the surface mass loadings, the reduction ratio of GPS annual amplitudes after surface mass loading correction is computed according to eq. (1), and the results are listed in Table 5. Note that several extreme outliers smaller than −100 per cent have been excluded from the statistical results. Table 5. Reduction ratio (in per cent of GPS annual signal after surface mass loading correction. Solution . ESA . MIT . . North . East . Up . North . East . Up . 7helm_full 7.2 11.2 21.8 10.3 11.4 21.4 7helm_sub 8.0 16.5 38.5 6.0 13.7 32.4 6helm_full 7.3 9.2 41.4 7.6 8.9 32.6 6helm_sub 7.8 17.3 44.8 5.7 13.7 41.1 Solution . ESA . MIT . . North . East . Up . North . East . Up . 7helm_full 7.2 11.2 21.8 10.3 11.4 21.4 7helm_sub 8.0 16.5 38.5 6.0 13.7 32.4 6helm_full 7.3 9.2 41.4 7.6 8.9 32.6 6helm_sub 7.8 17.3 44.8 5.7 13.7 41.1 Open in new tab Table 5. Reduction ratio (in per cent of GPS annual signal after surface mass loading correction. Solution . ESA . MIT . . North . East . Up . North . East . Up . 7helm_full 7.2 11.2 21.8 10.3 11.4 21.4 7helm_sub 8.0 16.5 38.5 6.0 13.7 32.4 6helm_full 7.3 9.2 41.4 7.6 8.9 32.6 6helm_sub 7.8 17.3 44.8 5.7 13.7 41.1 Solution . ESA . MIT . . North . East . Up . North . East . Up . 7helm_full 7.2 11.2 21.8 10.3 11.4 21.4 7helm_sub 8.0 16.5 38.5 6.0 13.7 32.4 6helm_full 7.3 9.2 41.4 7.6 8.9 32.6 6helm_sub 7.8 17.3 44.8 5.7 13.7 41.1 Open in new tab Comparing results from the seven-parameter and six-parameter transformations that use the identical transformation network, the contribution of surface mass loadings to vertical GPS annual signal is significantly underestimated by 6–20 per cent in the seven-parameter transformations. Similarly, relative to the transformations using the uniform subnetwork, the contribution of surface mass loadings to vertical GPS annual signal is also reduced by 3–17 per cent in the transformations with the uneven full-network. These results indicate that estimating scale factor and using uneven transformation network will significantly reduce the consistency of GPS annual variations and surface loading model. At worst, only 21 per cent of the vertical annual variations are explained by the surface mass loadings for the 7helm_full solutions. It is noted in Fig. 10(a) that most of the stations that locates in the Europe and the North America abnormally acquire negative reduction ratios. In contrast, more than 41 per cent of the vertical annual variations are explained in the 6helm_sub solutions. The level of agreement between GPS annual variations and surface mass loadings is encouragingly improved. The statistic result indicates that at least 93 per cent of the stations acquire positive reduction ratios for the up component. Especially, those stations located in the Europe obtain significant improvement, as shown in Fig. 10(d). Figure 10. Open in new tabDownload slide Reduction ratio of vertical GPS annual variations after surface mass loading correction for ESA solutions. Figure 10. Open in new tabDownload slide Reduction ratio of vertical GPS annual variations after surface mass loading correction for ESA solutions. As a typical example, Fig. 11 illustrates the vertical position residuals of station DGAR (7.27°S, 72.37°E) derived from the surface mass loading deformations and GPS time-series (only 7helm_full and 6helm_sub solutions are shown for clarity). The top is the position residual time-series, and the bottom is the fitted annual signal using sinusoidal function. At the first glance, three residual time-series present similar seasonal oscillations. Concerning the fitted annual signals, remarkable periodic differences are visible between 7helm_full and 6helm_sub solutions, which reveals that GPS annual signal is really affected by transformation strategies. Figure 11. Open in new tabDownload slide Vertical position residuals of DGAR (7.27°S, 72.37°E) derived from the surface mass loadings, ESA 7helm_full and 6helm_sub solutions. Figure 11. Open in new tabDownload slide Vertical position residuals of DGAR (7.27°S, 72.37°E) derived from the surface mass loadings, ESA 7helm_full and 6helm_sub solutions. It is demonstrated that the GPS annual signal derived from the six-parameter transformations are more consistent with the results of previous studies (Dong et al. 2002; Tsai 2011; Xu et al. 2017). They proved that about 40 per cent of the vertical annual displacements are contributed by surface mass loadings over the global area. When the scale factor is estimated in frame transformation, the GPS annual variations in the up component explained by surface mass loading are obviously weakened. If the transformation network (especially when the scale factor is estimated) is not carefully chosen, the contribution of surface mass loadings to GPS annual signal is also degraded. 6 FRAME TRANSFORMATION ON THE CONSISTENT TIME-SERIES In this section, we want to answer the question: if PPP processing used satellite orbits and clocks generated in a consistent reference frame, whether it is still necessary to align daily positions to a secular reference frame by Helmert transformation. The GPS time-series used here are part of the ESA original and transformed time-series, which span from 2002-01-01 to 2017-01-28 and are estimated using consistent IGb08 products. The symbol ‘no_helm’ will be used below to denote the original time-series without frame transformation. 6.1 Assessment of station position residuals Fig. 12 shows the Weighted RMS (WRMS) of ESA position residuals (with the annual and semi-annual variations removed) for the no_helm and transformed solutions. Compared to the no_helm solution, all transformed solutions achieve some reduction of WRMS by 0.2–0.3 mm in the horizontal components and by 0.1–0.2 mm in the up component, meaning that the frame alignment can reduce the noise level of GPS time-series to a certain extent. It should be mentioned that this reduction level of WRMS has excluded the influence of seasonal variations. Once taking the seasonal variations into consideration, larger WRMS differences would be achieved by frame transformation. Figure 12. Open in new tabDownload slide WRMS of station position residuals for ESA no_helm and transformed solutions. X-axis is for no_helm solutions, and Y-axis is for 7helm_full, 6helm_full, 7helm_sub, 6helm_sub solutions from top to bottom. Figure 12. Open in new tabDownload slide WRMS of station position residuals for ESA no_helm and transformed solutions. X-axis is for no_helm solutions, and Y-axis is for 7helm_full, 6helm_full, 7helm_sub, 6helm_sub solutions from top to bottom. 6.2 Analysis of GPS annual signals Table 6 shows the RMS of annual amplitude and phase differences between ESA no_helm and its transformed solutions over the period from 2002-01-01 to 2017-01-28. The annual signals from the no_helm and 6helm_sub solutions show the best level of agreement among all comparison cases. The RMS of vertical annual signal differences between the no_helm and 6helm_sub solutions is only 0.31 mm for the amplitude and 5.16° for the phase. This consistency of vertical annual signal is even better than that between different transformation solutions in Table 4. However, if the no_helm solution is adopted, the horizontal annual signals are more significantly affected than the vertical ones. The RMS of horizontal annual signal differences between the no_helm and 6helm_sub solutions is 0.3–0.5 mm for the amplitude and up to 30° for the phase. Table 6. RMS of annual amplitude and phase differences between ESA no_helm and transformed solutions (from 2002-01-01 to 2017-01-28). Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . no_helm versus 7helm_full 0.39 0.22 1.67 20.18 23.45 44.41 no_helm versus 7helm_sub 0.44 0.30 1.14 26.23 33.60 27.50 no_helm versus 6helm_full 0.77 0.23 0.76 29.29 24.40 14.28 no_helm versus 6helm_sub 0.51 0.29 0.31 28.24 33.16 5.16 Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . no_helm versus 7helm_full 0.39 0.22 1.67 20.18 23.45 44.41 no_helm versus 7helm_sub 0.44 0.30 1.14 26.23 33.60 27.50 no_helm versus 6helm_full 0.77 0.23 0.76 29.29 24.40 14.28 no_helm versus 6helm_sub 0.51 0.29 0.31 28.24 33.16 5.16 Open in new tab Table 6. RMS of annual amplitude and phase differences between ESA no_helm and transformed solutions (from 2002-01-01 to 2017-01-28). Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . no_helm versus 7helm_full 0.39 0.22 1.67 20.18 23.45 44.41 no_helm versus 7helm_sub 0.44 0.30 1.14 26.23 33.60 27.50 no_helm versus 6helm_full 0.77 0.23 0.76 29.29 24.40 14.28 no_helm versus 6helm_sub 0.51 0.29 0.31 28.24 33.16 5.16 Comparison . Amplitude (mm) . Phase (°) . . North . East . Up . North . East . Up . no_helm versus 7helm_full 0.39 0.22 1.67 20.18 23.45 44.41 no_helm versus 7helm_sub 0.44 0.30 1.14 26.23 33.60 27.50 no_helm versus 6helm_full 0.77 0.23 0.76 29.29 24.40 14.28 no_helm versus 6helm_sub 0.51 0.29 0.31 28.24 33.16 5.16 Open in new tab Table 7 lists the reduction ratio of GPS annual signal after surface loading correction for the ESA no_helm and transformed solutions during the period from 2002-01-01 to 2017-01-28. With a few expectations, GPS annual signal derived from the 6helm_sub solution still achieves the best agreement with the surface mass loadings among all transformations. For the no_helm solution, about 7.9, 14.0 and 40.6 per cent of the annual signal is explained by the surface mass loadings for the North, East and Up components, respectively. This ratio is comparable to the results of the 6helm_sub solution, but the performance of 6helm_sub solution is still slightly better. Table 7. Reduction ratio (in  per cent) of GPS annual signal after surface loading correction (from 2002-01-01 to 2017-01-28). Solution . North . East . Up . 7helm_full 6.5 12.4 23.3 7helm_sub 6.9 17.9 38.5 6helm_full 5.8 10.7 42.2 6helm_sub 6.5 18.6 43.2 no_helm 7.9 14.0 40.6 Solution . North . East . Up . 7helm_full 6.5 12.4 23.3 7helm_sub 6.9 17.9 38.5 6helm_full 5.8 10.7 42.2 6helm_sub 6.5 18.6 43.2 no_helm 7.9 14.0 40.6 Open in new tab Table 7. Reduction ratio (in  per cent) of GPS annual signal after surface loading correction (from 2002-01-01 to 2017-01-28). Solution . North . East . Up . 7helm_full 6.5 12.4 23.3 7helm_sub 6.9 17.9 38.5 6helm_full 5.8 10.7 42.2 6helm_sub 6.5 18.6 43.2 no_helm 7.9 14.0 40.6 Solution . North . East . Up . 7helm_full 6.5 12.4 23.3 7helm_sub 6.9 17.9 38.5 6helm_full 5.8 10.7 42.2 6helm_sub 6.5 18.6 43.2 no_helm 7.9 14.0 40.6 Open in new tab 6.3 Discussions on the origin of PPP positions As we know, the reference frame of PPP positions theoretically inherits the frame of the used satellite orbits and clocks. Here we give some notes on the origin of IGS final satellite products, which always confuses some PPP users. According to satellite dynamics, GPS satellites physically rotate around the centre of mass of the total Earth system (CM). During the routine data processing of IGS ACs, station positions are estimated simultaneously with satellite orbits and clocks usually by adding NNR constraints (Griffiths & Choi 2016). So, satellite orbits and clocks from this processing theoretically refer to the apparent CM frame. When generating the orbit files in the SP3 format, the centre of mass correction (CMC) for tidal loadings is conventionally applied in the transformation of GPS orbits from the natural inertial frame to the crust-fixed frame (Petit & Luzum 2010). For most AC products (including ESA and MIT products used in this study), CMC has been applied for the ocean tidal loading. However, these AC final and reprocessed orbits are still referenced to the apparent CM apart from the applied ocean tidal loading. Furthermore, the AC final satellite clocks are supposed to be adjusted again with the station coordinates fixed to ITRF before submission to IGS, which means that the final clocks follow the centre of network (CN) convention (http://acc.igs.org/sp3-comments.html). Consequently, the PPP positions estimated using AC final orbit and clock products are based on the CN frame theoretically. When the geodetic network sampling the Earth surface is well distributed and sufficiently extended, the CN can approximate the CF (Collilieux et al. 2010). The origin of IGS final products are the combined origin of AC final products. To testify this declaration, we conducted a comparison of GPS annual signal with the surface mass loading in CM frame (not shown here). It is frustrating that the reduction ratio of annual signal for the no_helm solution is only 33 per cent in the Up component, and is extremely −38 and −57 per cent for the north and east components, respectively. Obviously, the GPS annual signal has much better level of agreement with the CF-based surface mass loadings rather than CM-based loadings. In summary, Helmert transformation is suggested even though the PPP time-series have been generated using the satellite products in a consistent frame. The transformed time-series present slightly better agreement of annual signal with the surface mass loadings, and have smaller RMS of position residuals partly due to that some gross errors are found and deleted during Helmert transformation processing. Note that GPS satellite orbits are not reliable enough to realize the true instantaneous CM, just as mentioned before. The scattered translation parameters (especially the Z-translation) can also demonstrate this statement (see Figs 2, 3 and Table 3). The long-term stability of satellite products may also be affected by the imperfect constraints during the alignment to IGS reference frame. This is another important reason why we suggest to perform Helmert transformation even though the PPP time-series have been generated using consistent satellite products. 7 DISCUSSIONS AND CONCLUSIONS PPP technique has been frequently used to generate long-term GPS time-series for huge geodetic networks nowadays due to its high efficiency and precision. To overcome the frame inconsistency of GPS time-series, an essential task is to align PPP positions to a secular ITRF/IGS frame by Helmert transformation. However, the estimated transformation parameters can absorb part of the non-linear variations in station positions, which can disturb the subsequent time-series analysis. The purpose of this contribution is to investigate the impact of different transformation strategies on long-term GPS time-series processing, especially in the case of the inconsistent satellite products used in PPP. A total of 17 yr of data collected from 112 global GPS stations were processed by PPP using ESA and MIT satellite products. The resulting PPP solutions were aligned to IGS14 using uneven and relatively uniform networks as well as with and without scale factor estimated. It is found that the estimated transformation parameters contain significant seasonal variations. The simulation experiment demonstrates that 40–50 per cent of the annual signals in the scale factor are contributed by the aliasing of surface mass loadings. Meanwhile, about 40–70 per cent (ESA solutions) and 20–30 per cent (MIT solutions) of the annual variations in Z-translation can be explained by the aliasing of loadings. Relative to the uniform network, the uneven network induces the transformation parameter discrepancies of 0.7–1.1 mm, 21.3–27.5 μas and 1.3 mm RMS for the translation, rotation and scale factor (if estimated), respectively, regardless of whether the scale factor is estimated. Such network effect in the seven-parameter transformations can lead to a discrepancy of 0.5–0.6 mm RMS for vertical GPS annual amplitudes. When not estimating the scale factor, the RMS of annual amplitude differences caused by the network effect is slightly reduced for the up component, while it increases from 0.2 to 0.4 mm for the north component mainly due to the enlarged translation differences of X and Z components, and a slight increase of annual amplitude differences is also found for the east component. The Z-translation and vertical seasonal variations are most sensitive to whether the scale factor is involved in frame transformation, especially in the case of the uneven network. When exchanging the uneven full-network by the uniform subnetwork, the RMS of Z-translation differences caused by the scale factor is significantly reduced by 0.7 mm, and the RMS of annual amplitude differences is reduced by about 0.3, 0.1 and 0.3 mm for the north, east and up components, respectively. Regarding the annual amplitude differences between the seven-parameter and six-parameter transformations, there are obvious geographic location dependences in the three topocentric components, which is consistent with the transformational relation between the geocentric and topocentric coordinate systems. Compared to the transformation with the scale factor estimated, not estimating the scale factor can significantly improve the consistency of the derived GPS annual signals and surface loading deformations. When adopting the six-parameter transformation and using the uniform network, more than 41 per cent of the vertical annual variations are explained by surface loading deformations on the global scale. Similar to GPS seasonal signals, GPS draconitic harmonics can also be absorbed into the estimated transformation parameters. However, different Helmert transformation strategies have very small impact on identifying the draconitic errors in terms of its power magnitude, only up to 6–9 per cent in the up component. Comparing the original and transformed GPS time-series generated using consistent satellite products, the transformed time-series have lower noise level with smaller WRMS of station position residuals. Meanwhile, the seasonal signals derived from the six-parameter transformation using the uniform subnetwork have better consistency with the surface mass loadings. According to these results, we strongly suggest to use the uniform network when performing frame transformation. Although abundant IGS stations are located in the Europe and North America areas and their data quality is very good, never choose excess stations over there to guarantee the uniformity of observation network. What's more, the observation network is evolving in distribution and size over time in practical data processing, and the inclusion of scale factor in frame transformation can reduce the resulting negative impacts. Whereas, the estimated scale factor will also result in the aliasing of seasonal variations of position time-series, which can greatly disturb the seasonal signals of the transformed position time-series. On balance, we suggest not to estimate the scale factor when processing long-term GPS time-series. Furthermore, even though the consistent satellite products are used in PPP, we also propose an additional Helmert transformation before performing long-term time-series analysis. It is because the origin of PPP positions inherited from satellite orbits and clocks is not so stable during a long period. The proposed transformation strategies not only play a vital role in unifying the reference frame, but also can reduce the noise level of time-series. ACKNOWLEDGEMENTS This study is sponsored by the National Natural Science Foundation of China (41931075, 41774007, 41804024) and National Key Research and Development Program of China (2016YFB0501802). Thanks go to IGS for GPS observations and precise products provision. Shiwei Guo processed the data and wrote the manuscript. Chuang Shi and Na Wei designed the scheme of the paper and helped to write the paper. Na Wei and Min Li provided significant comments and interpretations. Lei Fan, Cheng Wang and Fu Zheng discussed the results and helped to plot the figures. REFERENCES Abraha K.E. , Teferle F.N., Hunegnaw A., Dach R., 2017 . 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Precise point positioning for the efficient and robust analysis of GPS data from large networks , J. geophys. Res. , 102 ( B3 ), 5005 – 5017 .. 10.1029/96JB03860 Google Scholar Crossref Search ADS WorldCat Crossref APPENDIX A: FIGURES REFERENCED IN THE PAPER Figure A1. Open in new tabDownload slide Lomb–Scargle periodogram of the detrend scale factor from MIT seven-parameter transformations. Figure A1. Open in new tabDownload slide Lomb–Scargle periodogram of the detrend scale factor from MIT seven-parameter transformations. Figure A2. Open in new tabDownload slide Transformation parameter differences between MIT different Helmert transformations. Figure A2. Open in new tabDownload slide Transformation parameter differences between MIT different Helmert transformations. Figure A3. Open in new tabDownload slide Power spectra of ESA position residual time-series. Annual and semi-annual terms have been removed. The vertical grey dashed–dotted lines indicate the first nine draconitic harmonics. Figure A3. Open in new tabDownload slide Power spectra of ESA position residual time-series. Annual and semi-annual terms have been removed. The vertical grey dashed–dotted lines indicate the first nine draconitic harmonics. Figure A4. Open in new tabDownload slide Power spectra of MIT position residual time-series. Annual and semi-annual terms have been removed. The vertical grey dashed–dotted lines indicate the first nine draconitic harmonics. Figure A4. Open in new tabDownload slide Power spectra of MIT position residual time-series. Annual and semi-annual terms have been removed. The vertical grey dashed–dotted lines indicate the first nine draconitic harmonics. APPENDIX B: TRANSFORMATION FROM THE GEOCENTRIC TO TOPOCENTRIC COORDINATE SYSTEM Suppose that the geocentric position of a given station at reference epoch t0 is |${P_0}( {{x_0},{y_0},{z_0}} )$|⁠, and its instantaneous position at epoch t1 is |${P_1}( {{x_1},{y_1},{z_1}} )$|⁠. The topocentric coordinate of |${P_1}$| relative to the geocentric reference point is expressed as $$\begin{eqnarray*} \left( {\begin{array}{@{}*{1}{c}@{}} {North}\\ {East}\\ {Up} \end{array}} \right) &=& \left( {\begin{array}{@{}*{3}{c}@{}} { - \sin B\cos L} & \quad { - \sin B\sin L} & \quad {\cos B}\\ { - \sin L} & \quad {\cos L} & \quad 0\\ {\cos B\cos L} & \quad {\cos B\sin L} & \quad {\sin B} \end{array}} \right) \nonumber \\ && \times \, \left( {\begin{array}{@{}*{1}{c}@{}} {{x_1} - {x_0}}\\ {{y_1} - {y_0}}\\ {{z_1} - {z_0}} \end{array}} \right) \end{eqnarray*}$$(B1) In which, (North, East, Up) are the topocentric coordinate components; B and L are the geodetic latitude and longitude of the reference point. © The Author(s) 2020. Published by Oxford University Press on behalf of The Royal Astronomical Society. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Helmert transformation strategies in analysis of GPS position time-series
JF - Geophysical Journal International
DO - 10.1093/gji/ggaa371
DA - 2020-11-01
UR - https://www.deepdyve.com/lp/oxford-university-press/helmert-transformation-strategies-in-analysis-of-gps-position-time-EBZXYZOe0O
SP - 973
EP - 992
VL - 223
IS - 2
DP - DeepDyve
ER -