Global distortion of GPS networks associated with satellite antenna model errors

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2006jb004675, 2007 Global distortion of GPS networks associated with satellite antenna model errors E. Cardellach, 1,2 P. Elósegui, 1,3 and J. L. Davis 1 Received 4 August 2006; revised 20 March 2007; accepted 16 April 2007; published 10 July [1] Recent studies of the GPS satellite phase center offsets (PCOs) suggest that these have been in error by 1 m. Previous studies had shown that PCO errors are absorbed mainly by parameters representing satellite clock and the radial components of site position. On the basis of the assumption that the radial errors are equal, PCO errors will therefore introduce an error in network scale. However, PCO errors also introduce distortions, or apparent deformations, within the network, primarily in the radial (vertical) component of site position that cannot be corrected via a Helmert transformation. Using numerical simulations to quantify the effects of PCO errors, we found that these PCO errors lead to a vertical network distortion of 6 12 mm per meter of PCO error. The network distortion depends on the minimum elevation angle used in the analysis of the GPS phase observables, becoming larger as the minimum elevation angle increases. The steady evolution of the GPS constellation as new satellites are launched, age, and are decommissioned, leads to the effects of PCO errors varying with time that introduce an apparent global-scale rate change. We demonstrate here that current estimates for PCO errors result in a geographically variable error in the vertical rate at the 1 2 mm yr 1 level, which will impact high-precision crustal deformation studies. Citation: Cardellach, E., P. Elósegui, and J. L. Davis (2007), Global distortion of GPS networks associated with satellite antenna model errors, J. Geophys. Res., 112,, doi: /2006jb Introduction 1 Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, USA. 2 Now at Institut d Estudis Espacials de Catalunya, Bellaterra, Spain. 3 Now at Institute for Space Sciences, Consejo Superior de Investigaciones Científicas Institut d Estudis Espacials de Catalunya, Barcelona, Spain. Copyright 2007 by the American Geophysical Union /07/2006JB004675$09.00 [2] Errors in the models of the GPS satellite antennas are a potential major source of error in high-precision GPS site positioning [e.g., Schmid and Rothacher, 2003; Ge et al., 2005]. The antenna models of the GPS satellites are typically described using two terms, the so-called phase center offset (PCO) and the set of phase center variations (PCV). The former accounts for the phase offset between a reference point in the satellite and a virtual point (the phase center) from which a hypothetical point source would be radiating spherical equiphase wavefronts [e.g., Balanis, 1982], and the latter accounts for the phase variations with elevation and azimuth angles that are required to correct for the nonsphericity of the real wavefronts. By far the greatest source of error between these two is the PCO, which, based on current estimates (e.g., G. Gendt and R. Schmid, IGS electronic mail messages 5149 and 5189, 2005), has been in error by 1 m or so. Because of the geometry of the GPS system, PCO errors most greatly affect estimates of the radial coordinate of site position. One of the main effects of these PCO errors has therefore been to introduce an apparent global-scale error into site coordinates [e.g., Zhu et al., 2003]. The scale error, and its first-order temporal variability, the scale rate, have thus naturally received the most attention in the last years [e.g., Steigenberger et al., 2006, and references therein]. However, PCO errors may introduce additional positioning errors such as geographically varying vertical position errors that cannot be accommodated by a simple global-scale parameter as well as horizontal errors. Studies of these errors, or about the effect of these errors on velocity estimates, are lacking, yet they might be important for geophysical applications where the highest accuracy is required. The aim of this paper is to quantify these errors using numerical simulations. [3] The GPS observable reflects the range between the electrical reference points for the GPS satellite transmitting and (user) receiving antennas. Ideally, the electrical reference point of an antenna is fixed with respect to a physical reference point near the antenna. For a GPS satellite, the physical reference point is (effectively) the satellite s center of mass, since the equations of motion governing the satellite orbit refer to that location. The electrical reference point is a point along the electrical path through the GPS transmitting antenna array from which the electrical path length to a set of (imaginary) receivers radially distributed around such point is constant. This point is also known as the phase center. To accurately relate the observed range to the location of the center of mass of the GPS satellite, the vector offset between the phase center and the satellite center of mass (i.e., phase center offset) must be well determined. In practice, no true electrical reference point 1of13

2 Table 1. Compilation of Values of the z Component of the Phase Center Offset a Block Type II/IIA, m IIR-A/B, m IIR-M, m Reference Kouba [1998] Schmid and Rothacher [2003] Schmid et al. [2005b] Ge et al. [2005] Ge et al. [2005] / G. Gendt and R. Schmid (IGS electronic mail messages 5149 and 5189, 2005) a See text for PCO sign convention. Entries on first and last row define the recent (pre-2007) and current (post-2006) IGS standards, respectively. The values in the last row are the average of the individual satellite PCO values in each block type. Actual values for individual satellites, as well as values for the x and y components of the PCO and their matching PCVs, can be found in the reference. Because of the correlation between PCO and PCVs, comparison between different sets of PCOs is strictly only possible with information on each corresponding PCV set. Matching PCV values for the PCOs in the first row are all zero. First Ge et al. row entry is for an unpublished JPL solution. The new Block IIR-M type is in operation since the launch of GPS satellite PRN 17 on 26 September Two additional IIR-M satellites (PRN 31 and 12) were launched in Information for this satellite block has not been used in this study. exists, since the antenna phase pattern is not spherical. In this case, a direction-dependent table or functional description of phase center variations is required. Furthermore, because PCVs are correlated with the PCO [Schmid and Rothacher, 2003; Schmid et al., 2005b], changes in the latter can be compensated by changes in the former, and vice versa. In this study, we use standard satellite PCO values (see Table 1), which typically result from minimizing the matching PCV values, and thus focus on PCO errors. [4] GPS consists of a nominal constellation of 24 operating satellites (with several spares) in nearly circular orbits, distributed in six orbital planes. Each satellite has a limited lifetime, and new satellites are continually launched to maintain a full constellation. There have been a number of design changes since the first (11) experimental GPS satellites, of the Block I design configuration, were launched between 1978 and The last Block I satellite was deactivated in Other GPS configurations have included nine operational Block II ( ) and 19 IIA ( ) satellites, 12 IIR-A and IIR-B satellites ( ) and three Block IIR-M satellites (2005 to present). The current GPS constellation therefore include satellites of Blocks II, IIA, IIR-A, IIR-B, and IIR-M. Each block of GPS satellites in principle possesses different PCOs and PCVs. However, there has been confusion regarding correct values for these. In fact, although the gain pattern of satellite antennas were measured in an outdoor range prior to launch of the satellites, PCOs were not measured but estimated from a theoretical analysis, and no PCVs were ever measured nor estimated (G. Mader and F. Czopek, private communication, 2007). As the GPS satellite constellation has evolved, the use of incorrect PCOs and PCVs has therefore introduced time-dependent errors in the estimated positions of ground-based GPS receivers, which we set to quantify in this paper. [5] It is not only extremely difficult to know, through measurement, modeling, or a combination, the instantaneous vector (PCO plus PCV) between the satellite center of mass and the electrical phase center [Mader and Czopek, 2002] but the center of mass might also change appreciably during the lifetime of the satellite as fuel is expended and structures (e.g., solar panels) age (M. Ziebart, personal communication, 2005). Electrical properties of antennas can be measured in anechoic chambers [e.g., Schupler et al., 1994], but the presence of electrically conducting structures associated with the satellite can potentially change the PCV significantly. (See, e.g., Elósegui et al. [1995], for a discussion of the effect of electrical scattering for a ground-based antenna.) [6] To deal with this problem, investigators have attempted to use the GPS observations to estimate satellite antenna characteristics as part of the multiparameter geodetic solutions. This procedure is mathematically ill-defined because several parameters involved in these solutions, such as the satellite and station antenna PCOs and PCVs are totally correlated, and the station vertical component and atmospheric parameters are highly correlated. A solution to this problem is possible only when independent information for some of those parameters becomes available. The GPS geodetic community has made great strides toward achieving this goal. Starting in the mid-1990s, antenna phase patterns for receiving antennas were estimated relative to a particular antenna, which was adopted as reference, using GPS data over short baselines [Mader, 1999]. Soon after, absolute ground antenna phase patterns were determined using a robotic system [Menge et al., 1998]. Using this information and fixing the coordinates of a set of reference sites to values provided by International Terrestrial Reference Frame (ITRF) models (determined, in turn, by other space geodetic techniques such as VLBI, SLR, and DORIS) it is possible to estimate satellite antenna phase patterns [Schmid and Rothacher, 2003; Ge et al., 2005; Schmid et al., 2005a]. [7] Several sets of satellite antenna phase patterns determined in this way exist. Table 1 shows PCO values for GPS satellites of four block designs. The standards adopted by the International GNSS Service (IGS) assume that all GPS satellites in a given block have identical properties, are point transmitters, and that there is a relative offset between block types. These (pre-2007) standards have been in use for over a decade to produce the IGS products, which form the basis for all subsequent geodetic and geophysical high-precision applications worldwide. However, recent PCO estimates can differ with respect to those standards, and among themselves, by more than 1 m, an indication of the difficulty of this approach. [8] Simulations have shown that PCO errors mostly affect estimates of clock errors, but that estimates of site position 2of13

3 This distortion, an artifact due to errors in the satellite antenna models, may be a source of misalignment found between the GPS-based reference frames and other frames used in geodesy and geophysics [Ray et al., 2004]. We use a simulation approach for determining the possible distortion in GPS position and velocity estimates due to PCO errors. We focus on the vertical component of site position because the effect on this component is larger than on the horizontal component. We begin by reviewing the satellite antenna geometry and describing the simulation method using a simplified model for the effects of PCO errors. We then use this method and a realistic model to estimate global distortion for ground-based GPS sites. Figure 1. Geometry of the phase center offset (PCO) error. (Not to scale.) The instantaneous range r is the distance between the phase center positions for the satellite T and the site R, both defined in a geocentric reference frame. The position of the satellite phase center is offset by z from the satellite center of mass T CM, whose trajectory in space is governed by orbital dynamics. The z component of the PCO (see text) points toward the center of the Earth, the nadir direction. A PCO error Dz induces a range error Dr, which depends on the nadir angle q, and has an effect on site position and other parameter estimates. are also affected by such errors. In particular, Zhu et al. [2003] found that 5% of the PCO error propagates into errors in the vertical component estimates. Hence meter level PCO errors (see Table 1) introduce vertical positioning errors at the 5-cm level or, equivalently, an apparent Earth s scale error equivalent to 7 parts per billion (ppb). This error has not remained constant in time due to changes in the GPS constellation, effectively introducing a fictitious scale variability [Ge et al., 2005]. In its continued mission and effort to improve the GPS accuracy, the IGS has adopted a revised set of PCOs (and matching PCVs) that takes into account the recent developments (G. Gendt and R. Schmid, IGS electronic mail messages 5149 and 5189, 2005). The PCV values of this latest set are the same for all satellites within each block design. The x and y axis PCO offsets are also specific for each block, but the z components (see below) are different for each individual satellite. The antenna array of the GPS satellites is composed of 12 elements arranged in two concentric rings. Intersatellite PCO (and PCV) differences could arise from relative phase differences among the elements of the antenna subassembly (T. Herring, personal communication, 2005). In addition, all PCOs and PCVs in the IGS model remain constant in time. [9] In this paper, we assess the effect of errors in PCO values on the apparent shape of the Earth as determined using a GPS network. We will refer to this error as network distortion. An important difference between network distortion and globally averaged errors is that the former cannot be corrected via a parameter transformation (i.e., rotation, translation, and scale), whereas the latter can. 2. Errors Induced by Phase Center Offsets of the GPS Satellite Antenna [10] To assess the effect of PCO errors on network distortion, we use a simplified model for the effects of PCO errors that helps us develop some intuition for the distortion effect and allows us quantify its order of magnitude. The model assumes that the error in the PCO is in the z component as measured in the satellite reference system. The z component of the satellite is the direction of the main lobe of the satellite antenna radiation pattern, and we assume that this direction is parallel to the vector from the center of mass of the satellite to the center of the (spherical) Earth (Figure 1), i.e., the satellite nadir direction. PCVs and signal propagation effects will be ignored in our simulations. The signal transmitted from the antenna phase center has originated at a location that differs from the assumed location by an offset Dz along the satellite z axis (Figure 1). (The sign convention adopted in this paper for the PCO error Dz is consistent with its definition by the IGS, i.e., Dz is positive for a modeled phase center offset that is closer to the center of the Earth than the true phase center offset.) The (ground-based) receiving antenna is located at an angle q with respect to the nadir direction. The error Dr in the calculation of the instantaneous range r = jt Rj between the satellite phase center position T and the receiving antenna R, fordz r, is Dr ¼ Dz cos q ðdz rþ: ð1þ [11] In the (topocentric) frame of the ground-based receiver, the GPS satellite is located at zenith angle z. From Figure 1, the nadir angle q and the zenith angle z are related by T sin q = R sin z, where T is the distance between the satellite antenna phase center and the center of the Earth, and R is the Earth s equatorial radius. The nadir angle can then be written as sin q ¼ R T cos ; ð2þ where is the elevation angle of the satellite in the topocentric frame. Combining equations (1) and (2), we can express the range error as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dr ¼ Dz 1 R2 T 2 þ R2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2 sin2 Dz 0:94 þ 0:06 sin 2 ð3þ 3of13

4 Figure 2. Phase center offset contribution to the range error versus satellite elevation angle after equation (3). A PCO error Dz = 1 m was used, which is commensurate with the most recent findings. uniformly in elevation angle between the minimum elevation angle min and a maximum elevation angle, which depends on the latitude of the site through the (55 ) inclination of the orbital planes of the GPS satellites. (The assumption of uniform distribution is simplistic but serves to illustrate the geometric effects of the error. The results remain unchanged if other ad hoc distributions such as doubling the number of observations at low with respect to high elevation angles are used.) The error is shown for three sites each representing a typical midlatitude, circumpolar, and polar location, respectively. As it might be expected from the form of equation (3), the clock parameter absorbs most of the PCO error, but it is sensitive to min. This estimated parameter varies almost linearly with min because it is (anti)correlated with the vertical position parameter. The sites at the three representative locations show a similar dependence with min. As an example, the sensitivity of the vertical error estimate of a typical midlatitude site to min is adequately described by the ad hoc relation (see Figure 2). The error has a large constant term and a small elevation angle-dependent term. It is this elevation angle dependence that causes the network distortion, as we will see below. [12] To examine the effect of PCO errors under this simplified model we also used a simplified error model during the simulated least squares analysis of the GPS phase observable. For this simulation, we will adjust only two relevant parameters of the GPS model, a clock error (or clock for simplicity), DC, and the vertical component of site position, DR u. The former is a constant whereas the second depends on the sine of the elevation angle, DR u Dz 34:2 26:3 sin min; ð5þ Dr ¼ DC þ DR u sin : ð4þ More parameters will be estimated when we develop a rigorous error model in section 3. [13] The clock term of equation (4) represents a simplified combination of time-dependent receiver and satellite clock errors, as well as ambiguity and instrumental offset terms [e.g., King et al., 1985]. In practice, GPS analysis software can eliminate the clock and instrumental parameters through the use of differencing [e.g., King et al., 1985] or, equivalently, difference operators [e.g., Bock et al., 1986; Schaffrin and Bock, 1988]. These methods will give identical parameter estimates, assuming the same data are used and the errors are propagated correctly. Our use of the simplified equation for the one-way phase enables us to perform our simulation without the bookkeeping required for the differencing techniques. [14] A unique ambiguity is associated with each unique satellite-site pair. These are freely estimated in the solution, and then fixed to integer values when possible. Different solutions may have different numbers of fixed ambiguities. Our approach ignores these details. Clearly, however, it is possible for some of the clock error to be partitioned into one or more ambiguity parameters in a real-life solution, and to degrade the capability to fix the ambiguities. [15] Figure 3 shows the error in the estimate of the clock and the vertical site position parameters of this simple model versus the minimum elevation angle of the observations. The observations are assumed to be distributed Figure 3. Error (symbols) in estimates of (top) clock parameter and (bottom) vertical component of site position versus minimum elevation angle for the PCO error shown in Figure 2. The elevation angles of the observations were uniformly distributed between the minimum elevation angle and a maximum elevation angle of (circles) 90, (open triangles) 70, and (solid triangles) 55, each representing a typical midlatitude, circumpolar, and polar site, respectively. The least squares fit (line) to the vertical position error of the (circles) midlatitude site is the ad hoc model of equation (5). 4of13

5 where DR u (in mm) is the error in the vertical component of site position, and Dz is in m. [16] Figure 3 shows that PCO errors can introduce vertical position errors at the level of tens of millimeters [Zhu et al., 2003; Ge et al., 2005]. As discussed above, these errors would be simply absorbed by a scale factor if the satellite geometry at all sites were identical, which is clearly not the case for GPS. For example, assuming identical minimum elevation angles at sites globally, already a rather unrealistic assumption, the lack of high-elevation observations at high-latitude sites can introduce vertical error differences with respect to lower latitude sites of 6 8 mm (Figure 3). The situation is further aggravated since minimum elevation angle differences among sites can amount to several degrees depending on local conditions. Thus differences in satellite geometry among sites could introduce vertical error differences at the several mm level that cannot be removed by a scale parameter, thus causing network distortion. Moreover, the distortion would be time dependent as the GPS constellation evolves since the PCO error may differ for different satellites. 3. Simulations [17] In the simplified study of the effects of PCO errors in section 2 we limited ourselves to a single satellite and site, and a geometry characterized by elevation angle only, and we focused on the errors in the estimated clock and vertical site position. The study we describe in this section includes a realistic GPS constellation and satellite geometry from multiple ground-receiving sites. It also includes additional parameters that better reflect the conditions of high-precision GPS applications. [18] We simulated the error Dr in the observed range due to a PCO error Dz using the vector formulation that leads to equation (1). This model is exact for a PCO error in the direction of the satellite z axis. We expressed the instantaneous range error as the difference between range for the actual satellite phase center position T = T CM + z and range for the erroneously assumed phase center position T + Dz. For a receiving antenna at position R, Dr is Dr ¼ jt CM þ z þ Dz Rj jt CM þ z Rj; ð6þ where T CM is the position of the satellite center of mass (Figure 1). We have focused on PCO errors along the satellite z axis because current estimates of PCO values for that component are an order of magnitude (or more) larger than for the other two components (see references in Table 1). These simulations further assume that the PCV values remain the same when PCOs are changed. [19] We used an expanded model with parameters representing adjustments to the three-dimensional positions of the receiving site R and the satellite center of mass T CM, receiver and satellite clocks C r and C s, and the zenith atmospheric propagation delay t a z. In its linearized form, the observation equation is Dr ¼ ^r DR þ ^r DT CM þ DC s þ DC r þ Dt z a csc : ð7þ We parameterize the receiver position error in terms of its geocentric coordinates, and then transform them to errors in topocentric site coordinates (DR e, DR n, DR u ). We use a simplified parameterization for the satellite orbital error, developed in Appendix A. The major term in this parameterization is an error Da in the satellite orbital radius, which, given the geometry, we might expect to be most greatly influenced. We also include a parameter representing the mean anomaly Dm at the reference epoch t, thereby enabling us to include the coupling between radial and along-track orbital position (Appendix A). From equation (A3), we have ^r DT CM ¼ cos q 3 2 w ð t t Þsin q Da þ sin q Dm : [20] When the error model Dr given by equation (6) is used in the left side of equation (7) then the least squares estimates of the parameters in the right side of equations (7) (8) represent the error in those parameters caused by PCO errors. [21] We performed inversions with several levels of complexity for the distribution of PCO errors within the GPS satellite constellation. We also used two different types of ground-based networks, a fictitious global grid and a realistic global network based on the current network used by the IGS. In all the simulations, the time-dependent positions for the center of mass T CM of each satellite were adopted from satellite orbit estimates available from standard GPS global solutions (e.g., J. Kouba, A guide to using International GPS Service (IGS) products, 2003, available at The simulated observations spanned 24 hours at a sampling rate of one set of measurements every 900 s. These observations, aside from the sampling rate, closely represent a standard data set in daily GPS geodetic solutions from which site velocities and other time-dependent parameters are later derived. (We generated observations at typically three times slower than standard sampling rates to avoid introducing potential interpolation errors in satellite orbital parameters. The sampling rate difference should have no impact on the conclusions derived from this study.) [22] Our simulation approach is conceptually similar to data analysis of standard GPS observations [e.g., Steigenberger et al., 2006, and references therein] where, in a multiparameter solution, we solve simultaneously at each site for the three components of position, a clock, and an atmospheric delay parameter, as well as a set of satellite orbital parameters (a clock parameter and two Keplerian elements per satellite) common to all sites. [23] We performed three types of simulations using this strategy whereby the complexity of satellite PCO errors is gradually increased to help better understand its effect on global vertical estimates: (1) simulations in which the PCO error is the same for all GPS satellites; (2) simulations in which the PCO error is the same for all but one or two GPS satellites; and (3) simulations in which the PCO error for each satellite is different and consistent with the latest findings [e.g., Ge et al., 2005]. For identification purposes, we label the three steps of this progressive ð8þ 5of13

6 Figure 4. Spatial variation of the error in the estimates of the vertical component of site position for a homogeneous PCO error Dz = 1 m for all satellites and a minimum elevation angle of 0. Overlayed (dots) are the 15-min satellite orbit projections on the Earth s surface of the GPS constellation of 22 July approach to the effect of PCO errors homogeneous, quasi-homogeneous, and realistic, respectively. [24] We used two selections of ground-based, globally distributed GPS sites to assess the PCO error effect on site velocity estimates throughout these simulations. One global network consisted of 614 sites spread across the Earth s surface on a regular grid, and the other a set of 90 stations from the IGS network used in the IGb00 reference frame solution [e.g., Ray et al., 2004]. Although the second type of network is more relevant from the geodetic and geophysics standpoint, the high density and homogeneity of the former enables a better understanding of the spatial variations of the induced global position and velocity errors Homogeneous PCO error [25] In these simulations, the PCO of all GPS satellites were in error by the same amount, i.e., Dz i = Dz, i =1,..., N, where N is the total number of satellites available. Figure 4 shows the resulting error in the estimate of the vertical component of site position due to a PCO error Dz = 1 m, which is commensurate with the PCO error of the most recent findings (e.g., G. Gendt and R. Schmid, IGS electronic mail messages 5149 and 5189, 2005). The GPS constellation used for this particular simulation corresponds to that of 22 July (The choice of date has no bearing on these results; other dates will be used below.) The minimum elevation angle in this simulation is 0, which from Figure 3 resulted in the smallest vertical position error. (Other elevation angles will be investigated below.) This simulation demonstrates that the vertical component is globally biased by 28 mm (both mean and median values), and that the spatial variations (i.e., network distortion) can amount to 8 mm amplitude peak to peak. (Hereafter, whenever we use the term network distortion, we refer to the peak-to-peak variation of the position error about the mean value over the Earth; see also definition in Table 2.) This bias was predicted by the simplified model presented in section 2 (see also Figure 3). The difference between the estimated bias values in Figure 3 and here reflects the higher accuracy of the parameterization in equation (7) with respect to the simplified dependence on only minimum elevation angle in equation (5). This bias could in principle be absorbed by a scale parameter in a seven-parameter transformation that is commonly used in GPS analysis to transform solutions between different reference frames [e.g., Bock, 1997]. For an Earth radius of 6371 km, a 28 mm vertical bias is equivalent to a scale factor error of 4.4 ppb. [26] The spatial variations represent an apparent distortion of the shape of the Earth that cannot be accommodated by a transformation. The vertical error in Figure 4 is largest in the midlatitude regions. The error then gradually diminishes toward the polar regions and, to a lesser extent, toward the equator. As predicted above, the spatially varying vertical error cannot be explained solely by the minimum elevation angle. There is a clear dependence of the error with the distribution of the elevation angles of the observations since sites in midlatitude locations (where the density of satellites at zenith is the greatest) are more affected than polar sites, for which high elevation angle observations are simply lacking. The network distortion shows a markedly zonal pattern that could be characterized by a degree two and order zero spherical harmonic. This pattern may indicate that PCO errors may be a source of error in some GPS deformation studies such as mass loading on the Earth s surface [e.g., Gross et al., 2004]. [27] Among the other parameters involved in the simulation, satellite clocks are most affected by the PCO error. In particular, we found that the satellite clocks absorb 97% of the PCO error, a result in good agreement with the simplified, intuitive model of equation (5) and with studies of other authors [e.g., Zhu et al., 2003]. Receiver clocks absorb 1% of the PCO error, the horizontal components of site position absorb less than 0.04%, and orbital parameters, and specially troposphere parameters, less than 0.01%, and thus are not affected significantly. A summary of these parameter error estimates is presented in Table 2. [28] We also found that the magnitude of the global vertical bias, for small variations in minimum elevation Table 2. Summary of Parameter Error Estimates Error ( 10 3 Dz ) Minimum Elevation Angle Parameter Symbol 0 15 Mean vertical site position a hdr u i Network distortion amplitude b pp dr u 8 10 Satellite clock DC s <974 <910 Site clock DC r <8 <10 Horizontal site position DR h <0.4 <0.2 Radial orbital Da <0.1 <0.1 Atmospheric zenith delay z Dt a <0.1 <5 a Site position error is defined to be DR = hdri + dr, where hdri is the mean error over the Earth. b Network distortion amplitude dr u pp is defined to be the peak-to-peak variation of dr about hdri over the Earth. 6of13

7 Figure 5. (a) Mean global vertical bias and (b) peak-topeak amplitude of the network distortion versus magnitude of the (circles) homogeneous PCO error for the GPS constellation of 22 July 1996 and minimum elevation angles between (light gray to black) 0 and 15, at5 steps. Within each star cluster, each of the nine stars is the (a) vertical bias and (b) network distortion for the realistic PCO error (see text below) of the GPS constellation of each 22 July date between 1996 and 2004 and (gray) 0 and (black) 15 minimum elevation angles. angle, varies linearly with the minimum elevation angle min of the observations, as expected from the results using the simplified model. Figure 5 shows, for the simulation of 22 July 1996, the global vertical bias and the amplitude of the network distortion for PCO errors (of magnitude) between 0.1 and 1.5 m and minimum elevation angles between 0 and 15. The linear fit to these errors can be expressed (in mm) as hdr u i ( min ) Dz, when min is in degrees and Dz in meters. This error model can be translated into a scale factor error (in ppb) as ( min ) Dz, with min and Dz in same units as above. These results are consistent with those reported by Zhu et al. [2003] and Ge et al. [2005] for minimum elevation angles between 10 and 15. [29] The implications of these simulations for accurate determinations of vertical components of site position are clear: meter level PCO errors can induce spatially varying vertical position errors of several millimeters (8 mm per meter of PCO error, from Figure 5 and Table 2) in addition to the apparent scale errors. Since PCO values in use for more than a decade now have been found to be in error by this order of magnitude (Table 1), these results suggest that PCO errors may be a significant source of network distortion. Moreover, since the GPS constellation, and hence the PCO error, has evolved, these simulations suggest that PCO error may be a significant source of temporal error, with potential impact on determinations of global vertical velocity fields. We explore this hypothesis in section Quasi-homogeneous PCO Error [30] As an intermediate step toward understanding the effects of PCO errors, we performed a simulation in which the PCO of all but a few GPS satellites is in error by the same amount. We assumed that Dz i = Dz = 1 m for all but one or two satellites, for which Dz = 1.1 m. This simulation is useful because recent estimates indicate that PCOs for different satellite blocks, and even for satellites within the same block, may differ by up to 0.6 m [Ge et al., 2005; G. Gendt and R. Schmid, IGS electronic mail messages 5149 and 5189, 2005]. [31] Figure 6 shows three examples of the difference in the estimate of the vertical component of site position between a simulation with a homogeneous PCO error (i.e., Figure 4) and a simulation for which the PCO error of one or two satellites are perturbed relative to the homogeneous PCO error. The PCO error of satellites PRN 9, 9 and 3, and 9 and 6 were perturbed in each of these simulations, respectively. These satellite combinations were purposely chosen to illustrate the effect of intersatellite PCO error variations. There exist multiple possible satellite combinations that could have served the same purpose. [32] Figure 6a shows that the vertical position error is largest for sites that lie on a wide fringe along the ground track of the satellite orbit whose PCO has been perturbed. The perturbed satellite passes directly overhead those sites at some epoch. At those epochs, the projection of the PCO error on the vertical component of site position is maximum and thus their error is also larger. The magnitude of the error is at the submillimeter level, partly because the perturbation introduced is small, and partly because the clock parameter of the perturbed satellite absorbs 98% of the PCO error, which is consistent with the simulation above. Figure 6b shows that when satellites have ground tracks that are similar the error adds constructively resulting in a pattern whose magnitude can be twice the magnitude of the individual errors. On the other hand, Figure 6c shows that if the ground tracks of the perturbed satellites do not overlap, the error grows where the tracks cross and becomes more diffuse where they do not cross. The crossing occurs at latitudes that approximately match the 55 inclination of the orbital planes of the GPS satellites. The higher density of ground track crossings of the GPS constellation at these latitudes than at other latitudes explains the spatial pattern of the network distortion that can be seen in Figure 4. [33] This set of simulations demonstrates that PCO irregularities can have an additional effect on vertical position error, and suggest that the network distortion is affected first by the larger average PCO error and then by satellite PCO differences Realistic PCO Errors [34] The PCOs of the GPS constellation are more complicated than the scenario reproduced in the previous simulations, and thus the effect of PCO errors on geodetic 7of13

8 were adopted by the IGS about a decade ago. (The IGS data processing centers have switched to the former values (G. Gendt and R. Schmid, IGS electronic mail messages 5149 and 5189, 2005) in late For identification purposes, the IGS standards are also known as absolute and relative antenna calibrations, respectively.) The mean value of the latest PCO error estimates is m, and has decreased by m between 1996 and (If one assumes that the latest set of PCO values are the true PCOs, the PCO error is then the negative of the value in Figure 7, for consistency with the definition of PCO error above and the IGS definition.) The standard deviation for the ensemble of satellites is 0.35 m and has also evolved with time, slightly increasing as the mix of satellite block types has also increased, especially in The two extreme values for the latest PCO error estimates, m (PRN 19, a Block II deactivated in 2001) and m (the PRN 19 replacement, a Block IIR launched in 2004), differ by more than 1 m. [35] To investigate the effect of realistic PCO errors, we simulated observations using the latest IGS PCO values (G. Gendt and R. Schmid, IGS electronic mail messages 5149 and 5189, 2005) and then used the pre-2007 standard IGS PCO values [Kouba, 1998] for the model in the least squares inversion. (Because simulations with and without the matching PCVs, which are zero and 12 mm for the pre-2007 and post-2006 sets, respectively, indicated that the effect was submillimeter, we set all PCVs to zero to ensure meaningful comparison between the two sets of PCOs.) For consistency, we used the orbital configuration of 22 July 1996 and a minimum elevation angle of 0. Figure 8 shows that the main features of the network distortion remain similar to those of prior simulations, indicating that the distortion is largely due to the average PCO error with a smaller contribution due to variation of individual satellite values. (We further examine this conclusion below.) The vertical component is globally biased by 38 mm in this simulation and the amplitude of the network distortion amounts to 11 mm in amplitude. The Figure 6. Spatial variation of the difference of the error in the estimates of the vertical component of site position between the homogeneous PCO error simulation (Figure 4) and a quasi-homogeneous simulation whereby an additional 0.1 m was added to the PCO error of GPS satellites (a) PRN 9, (b) PRN 9 and 3, and (c) PRN 9 and 6. Overlayed (dots) are the 15-min satellite orbit projections on the Earth s surface of these GPS satellites of 22 July parameter estimates is also expected to be more complicated. Figure 7 shows the time evolution of the most recent (post- 2006) mean PCO estimates and the intraconstellation variability relative to the (pre-2007) standard PCO values that Figure 7. Temporal variation of the (dots) mean PCO difference between the latest and the standard PCO estimates (i.e., latest minus standard) of all satellites in the GPS constellation and ±1 standard deviation. Statistics calculated at yearly intervals. Note that the PCO error Dz is the negative of this value (see text). 8of13

9 Figure 8. Radial error when PCO errors arise from differences between current and standard PCO values using a geometry defined by a ground network of (background color) 614 sites on a grid and overlayed (colored triangles) 90 IGb00 sites. The GPS constellation is that of 22 July 1996, and the minimum elevation angle is 0. magnitudes of these errors are both larger than in prior simulations, which, as predicted in Figure 5, results from the larger (in absolute value) mean PCO error of this simulation (i.e., hdzi = 1.37 m versus 1.0 m). Figure 8 also includes a second simulation in which only the 90 GPS sites from the IGS network used in the IGb00 realization of the ITRF were used. The vertical error difference between common sites present in both the regularly gridded network and the IGb00 network is less than 1 mm. We therefore focus on the results from the gridded network, rather than from IGb00, because the former enables a better understanding of global error patterns, which are then fully applicable to the latter. [36] To investigate further whether the similarity between the spatial pattern in Figures 4 and 8 is largely due to the average PCO error, we repeated these simulations, but using the GPS constellation of 22 July 2004 instead. We simulated two cases for this new epoch. In the first case, the PCO error was the same for all GPS satellites (i.e., a homogeneous PCO error) and equal to Dz = m, which is the mean value for that epoch (see Figure 7). In the second case, we used the realistic PCO errors for that epoch (see above). We found that the bias of the vertical component was 34 mm in both cases, and that the amplitude of the network distortion was 9 mm. These results are thus fully consistent with results from prior simulations (see Figure 5). Compared to the simulation with the quasi-homogeneous PCO error in Figure 6, the effect is significantly larger here because the perturbation is also larger, up to 0.6 m, and involved not only one or two satellites, but all 28 satellites that were available on this epoch. Figure 9 shows the vertical error difference between the homogenous and realistic cases and demonstrates that intersatellite PCO differences contribute only at 10 20% of the amplitude of the network distortion. Also consistent with prior simulations, Figure 9 illustrates how intersatellite PCO variations contribute to enhance the spatial pattern of the vertical error when satellite ground tracks overlap, as it happens to occur in this particular case with the set of satellites with smallest PCO values (black circles), and how PCO variations tend to cancel each other producing a more diffuse pattern when the ground tracks are more evenly spread, as is the case of the set of satellites with largest PCO values (red inverted triangles). [37] Standard GPS data analysis would rarely use observations down to a minimum elevation angle of 0 because several GPS systematic errors such as multipath tend to have a larger impact at low-elevation angles. As a result, minimum elevation angles between 7 and 15 are more typical in standard GPS data processing. However, Figure 5 shows that in the case of PCO errors, increasing the minimum elevation angle magnifies both the global vertical bias and the network distortion. To investigate this effect using realistic PCO errors, we simulated observations with a minimum elevation angle of 15 for the orbital configuration of 22 July Figure 10 shows that as predicted, the magnitude of the global vertical error has increased significantly, to a mean global value of 68 mm, and that the amplitude of the network distortion has also increased, to 17 mm. Whereas the degree-2 zonal spherical harmonic pattern continues to be present, a less pronounced pattern in longitude that looks like a degree-4 spherical harmonic has Figure 9. Spatial variation of the difference of the error in the estimates of the vertical component of site position between a homogeneous PCO error simulation with Dz = m for all satellites and the latest PCO error values. The same GPS constellation of 22 July 2004 and minimum elevation angle of 0 was used in both simulations. Overlayed are the 15-min satellite orbit projections on the Earth s surface of the four GPS satellites with (black dots) smallest PCO errors (PRN 19, 22, 18, and 11, whose PCO values are 0.668, 0.792, 0.911, and m, respectively) and (red inverted triangles) largest PCO errors (PRN 30, 27, 3, and 6, whose PCO values are 1.443, 1.449, 1.596, and m, respectively). 9of13

10 Figure 10. Radial error for the same simulation as for Figure 8, except here for a minimum elevation angle of 15. IGS standard four-character code of sites mentioned in the text is included for identification purposes. also emerged. This realistic simulation shows that spatial gradients in vertical position error of 5 8 mm can be seen across North America and Eurasia, and about twice that amount between some midlatitude and polar regions. The error pattern for IGb00 sites is consistent with this simulation. For example, the vertical position error between IGb00 sites in western (DRAO) and eastern (WES2) North America amounts to 5.1 mm, and between sites in southeast Australia (HOB2) and Antarctica (MCM4) to 10.4 mm. [38] Figure 11 shows the error in the estimates of the horizontal component of site position for the same simulation as in Figure 10. The magnitude of the horizontal error is less than 0.8 mm, with a mean value of 0.3 mm, and thus significantly smaller than the vertical error. Although the mean error is somewhat below the current typical precision of daily estimates of horizontal site position using GPS, horizontal errors at the 0.8 mm are commensurate with current GPS precision, especially if daily position estimates are averaged over several days, which is often the case. The most prominent feature of the horizontal error field is its markedly hemispherical, equatorially bearing pattern with two antipodal sinks at roughly 30 and 210 longitudes. A second noteworthy feature is a loose anticorrelation at midlatitude sites between the vertical and horizontal error fields. In particular, the horizontal error is generally small at the location of the four vertical error maxima in Figure 10 and large at the location of the minima. [39] Figure 7 shows that the average PCO error among all satellites has changed over time as satellites have become operational and/or have been decommissioned. Although the overall change (0.124 m between 1996 and 2004) is small compared to the average PCO error, our simulations suggest that the change can have significant implications on the temporal evolution of the network distortion, as already demonstrated for the scale rate parameter [Ge et al., 2005]. To investigate this possibility, we simulated the effect of a changing constellation using the orbital configuration of every date July 22 between 1996 and 2004, and a minimum elevation angle of 15. Figure 12 shows the temporal variation of the vertical bias and the network distortion. The vertical bias is largest in the early years ( ), when the average PCO error was also largest (see Figure 7), and then decreases as the PCO error also decreased, most significantly first in 2000 and later in The network distortion is mm throughout the period, and though its spatial pattern remains quite similar there are distinct differences between the two end epochs. These simulations thus demonstrate that small changes in the GPS constellation can change the vertical bias and the pattern of network distortion at the several millimeter level, thus potentially introducing spurious time-varying signals. [40] Indeed, a spurious vertical velocity results from the temporal variation of the vertical bias, whose value changed from 68 mm in 1996 to 60 mm in 2004, or about 1 mm yr 1. As indicated above, this global vertical velocity error can be accommodated by a scale rate parameter in a parameter transformation and thus will not be further discussed here. More importantly, a spurious signal also arises from the temporal variation of the network distortion, which is largely due to the change in the average PCO error. Unlike the former, this more subtle error cannot be corrected. Figure 13 shows the spatial distribution of this velocity error, which was calculated by first subtracting from each year snapshot of the vertical position error field in Figure 12 a globally averaged vertical rate (and offset) model, and then fitting a linear model to these residuals. [41] These simulations show that estimates of vertical velocity may be in error at the ±0.7 mm yr 1 level due to PCO errors, and that very distinct spatial patterns with large error gradients may exist. (The horizontal velocity error due to PCO errors is less than 0.1 mm yr 1 everywhere.) For example, the relative vertical velocity error between northern North America sites such as DRAO and SCH2, where postglacial rebound effects are of interest, is 1.1 mm yr 1. The error between southern coastal sites such as HOB2 and AUCK, which may be suited for sea level studies, is Figure 11. Horizontal component of site position error for the same simulation as for Figure of 13

11 0.7 mm yr 1. Relative vertical velocity errors across Eurasia, for example between ARTU toward the center of the plate and sites in western Europe could amount to 0.6 mm yr 1, and to YSSK in eastern Asia to 1.2 mm yr 1. [42] A number of studies have focused on seasonal site position variations to investigate the Earth s climate and water balance [e.g., Dong et al., 2002; van Dam et al., 2001; Davis et al., 2004; Lavallée et al., 2006]. To investigate the effect of PCO errors on estimates of seasonal vertical signals, we performed one simulation per month in 2003, with a minimum elevation angle of 15. Figure 14 shows the vertical error for a sample of five GPS sites. These simulations show that PCO errors could also introduce spurious, millimeter level seasonal signals in time series of vertical site position. The magnitude of the error is similar for simulations with minimum elevation angles down to 10. The monthly scatter mostly results from variations in the geometry of the GPS constellation throughout the year rather than variations of the PCO set. 4. Discussion and Conclusions [43] Our results confirm previous studies that an error in the satellite PCO is absorbed mainly in the scale of the (ground-based) network and in clock parameters. However, we have also demonstrated that this error introduces distortions, or apparent deformations, within the network, primarily in the vertical (or radial) component of site position. The PCO error leads to a vertical network distortion between 6 and 12 mm per meter of PCO error. On the basis of the latest PCO models for GPS satellites adopted by the IGS, PCO errors of amplitude slightly larger than 1m are reasonable assumptions until recently. The effects of PCO errors (scaling and distortion) depend on the minimum elevation angle used in the phase data analysis, becoming larger as the minimum elevation angle increases. Although each satellite may have a different PCO error, if the differ- Figure 12. Time variation of the radial error for every date July 22 between (top) 1996 and (bottom) The simulation of the first panel is identical to that of Figure 10. The simulations of the rest of panels are the same as for Figure 10, except for the epoch. Figure 13. Spatial variation of the error in the estimate of the vertical velocity due to temporal variations of PCO errors between 1996 and 2004 (see text). 11 of 13