Global distortion of GPS networks associated ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, B07405, doi:10.1029/2006JB004675, 2007

Global distortion of GPS networks associated with satellite antenna model errors E. Cardellach,1,2 P. Elo´segui,1,3 and J. L. Davis1 Received 4 August 2006; revised 20 March 2007; accepted 16 April 2007; published 10 July 2007.

[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 yr1 level, which will impact high-precision crustal deformation studies. Citation: Cardellach, E., P. Elo´segui, and J. L. Davis (2007), Global distortion of GPS networks associated with satellite antenna model errors, J. Geophys. Res., 112, B07405, doi:10.1029/2006JB004675.

1. Introduction [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 appar1 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 Cientı´ficas – Institut d’Estudis Espacials de Catalunya, Barcelona, Spain.

Copyright 2007 by the American Geophysical Union. 0148-0227/07/2006JB004675$09.00

ent 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

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Table 1. Compilation of Values of the z Component of the Phase Center Offseta Block Type II/IIA, m

IIR-A/B, m

IIR-M, m

Reference

1.0230 2.3384 2.4582 0.9519 1.0220 2.3300

0.0000 1.3326 1.5535 0.6300 0.4000 1.1400/0.6690

– – – – – 0.6983

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 2005. Two additional IIR-M satellites (PRN 31 and 12) were launched in 2006. 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 1985. The last Block I satellite was deactivated in 1994. Other GPS configurations have included nine operational Block II (1989– 1990) and 19 IIA (1990 – 1997) satellites, 12 IIR-A and IIR-B satellites (1997 – 2004) 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., Elo´segui 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

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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.

2. Errors Induced by Phase Center Offsets of the GPS Satellite Antenna 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 TCM, 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.

[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, for Dz 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 cos ; T

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 Dr ¼ Dz

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rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 R2 1 2 þ 2 sin2 ’ Dz 0:94 þ 0:06 sin2 T T ð3Þ

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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 DRu ’ 34:2 26:3 sin min ; Dz

(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, DRu. The former is a constant whereas the second depends on the sine of the elevation angle, Dr ¼ DC þ DRu sin :

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ð5Þ

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).

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where DRu (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 = TCM + z and range for the erroneously assumed phase center position T + Dz. For a receiving antenna at position R, Dr is Dr ¼ jTCM þ z þ Dz Rj jTCM þ z Rj;

ð6Þ

where TCM 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 TCM, receiver and satellite clocks Cr and Cs, and the zenith atmospheric propagation delay t za. In its linearized form, the observation equation is ^ DR þ r ^ DTCM þ DCs þ DCr þ Dt za csc : Dr ¼ r

ð7Þ

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We parameterize the receiver position error in terms of its geocentric coordinates, and then transform them to errors in topocentric site coordinates (DRe, DRn, DRu). 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 3 ^ DTCM ¼ cos q wðt t Þ sin q Da þ sin q Dm : r 2

ð8Þ

[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 TCM 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 http://igscb.jpl.nasa.gov/overview/pubs.html). 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

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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 1996. 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 10° 10° 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. 3.1. 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 1996. (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

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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 (103 Dz ) Minimum Elevation Angle Parameter Mean vertical site positiona Network distortion amplitudeb Satellite clock Site clock Horizontal site position Radial orbital Atmospheric zenith delay

Symbol

0°

15°

hDRui

28 8