Uncertainty in the velocity between the mass ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, B10405, doi:10.1029/2012JB009196, 2012

Uncertainty in the velocity between the mass center and surface of Earth Donald F. Argus1 Received 2 February 2012; revised 31 July 2012; accepted 22 August 2012; published 24 October 2012.

[1] Using spectral analysis and data decimation, we estimate the uncertainty in the velocity between the cumulative mass center of Earth (CM) and geodetic sites on Earth’s surface. Knowing this velocity is crucial for evaluating space geodetic observations of continental uplift and subsidence in terms of postglacial rebound and sea level rise. We find SLR observations of satellite LAGEOS to constrain the X and Y components of the velocity of CM to 0.4 mm/yr and the Z component to 0.9 mm/yr. (95% confidence limits, X is in the direction of 0 N 0 E, Y of 0 N 90 E, and Z of 90 N.) The uncertainty in Z is high, so that the estimate includes the independent inference made jointly using site velocities, the rigid plate hypothesis, and models of postglacial rebound that the true velocity of CM has a Z component of 0.5–1.0 mm/yr relative to that in ITRF2008. Uncertainty in scale rate, an intermediate parameter in the determination of an ITRF, is 0.36 mm/yr for VLBI, 0.52 mm/yr for SLR, and 0.20 mm/yr for GPS. The scale of GPS depends on that of VLBI and SLR, but the low GPS uncertainty indicates that GPS results are, for the first time, unbiased by changing satellite Block types, evidently due to newly incorporated satellite phase center variations. GPS constrains the velocity of CM nearly as well as SLR, representing a technical advance given that a GPS satellite is not a sphere and responds strongly to solar radiation pressure. Citation: Argus, D. F. (2012), Uncertainty in the velocity between the mass center and surface of Earth, J. Geophys. Res., 117, B10405, doi:10.1029/2012JB009196.

1. Introduction [2] In this study we investigate the velocity between the mass center of Earth (CM) and the surface of solid Earth. CM is the mass center of solid Earth, the ice sheets, continental water, the oceans, and the atmosphere. Satellite LAGEOS, a dense sphere, rotates about CM along an orbit controlled by gravity. SLR laser pulses traveling between LAGEOS and sites on Earth’s surface allow the determination of site positions relative to CM. The velocity between the sites and CM can also be determined. (It is straightforward to understand that the positions of sites relative to CM can, on a given day, be determined. It is more difficult to conceptualize that the velocity between CM and sites moving relative to one another can also be determined, but this velocity can also be rigorously specified by fixing a site and three directions to other sites.) The mass center of Earth (CM) is the reference relative to which site velocities are usually estimated. In the universally adopted (ITRF) International Terrestrial Reference Frame, site velocities are specified relative to CM as observed by SLR. If the estimate of the velocity of CM were to change from one 1

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA. Corresponding author: D. F. Argus, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109-8099, USA. ([email protected]) This paper is not subject to U.S. copyright. Published in 2012 by the American Geophysical Union.

ITRF realization to the next, then all site velocities would change by the opposite amount (Figure 1). [3] The velocity between the mass center and surface of Earth is important for geoscience. First, CM relates closely to the mass center of solid Earth (CE), and CE is the reference relative to which predictions of site motions are determined in models of postglacial rebound and current ice loss [Farrell, 1972]. In postglacial rebound models [e.g., Peltier, 2004, 2007] the speed between CM and CE is negligible because ice loss occurs before 4 thousand years ago. Seven GPS sites in Germany (Dresden, Hohenbuenstorf, Karlsruhe, Kloppenheim, Potsdam, two at Wettzell, all with 14 yr of data) are estimated to be rising at ≈0.2 mm/yr in ITRF2008 [Altamimi et al., 2007]. If the velocity of CM in ITRF2008 were exact, then this observation would tightly constrain the current subsidence of the Fennoscandia peripheral bulge. [4] Second, the velocity of CM is important for understanding relative sea level rise. Eight tide gauges in the Netherlands record sea level rise at a mean rate of ≈1.5 mm/yr over the past 150 years [Woodworth and Player, 2003] (http:// www.psmsl.org/data/obtaining/). Four GPS sites in the Netherlands (Kootwijk, Westerbork, and two at Delft, all with 14 yr of data) are rising at ≈0.5 mm/yr in ITRF2008. If the value of CM in ITRF2008 were exact, then we could infer that there the ocean has risen at ≈2 mm/yr relative to CM. (Such calculations must be made worldwide; this example simply illustrates how the velocity of CM relates to sea level rise.) Distinguishing between land subsidence and sea rise is critical for assessing

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ARGUS: VELOCITY OF EARTH’S MASS CENTER

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uncertainty in the velocity of CM [Collilieux et al., 2009]. GPS has not been proven to tightly constrain the velocity of CM. A GPS satellite is not close in shape to a sphere and responds strongly to solar radiation pressure (Figure 2) [Bar-Sever and Kuang, 2004]. Rülke et al. [2008] are among the first to define a (TRF) terrestrial reference frame, including estimates of CM and scale, using just GPS. [9] Scale is an intermediate parameter in the determination of an ITRF. In truth relative site positions can be estimated

Figure 1. If the estimate of the velocity of Earth’s center were to change in the direction of the South pole, then the velocity of all sites would change by the same amount in the direction of the North pole. The amount by which the vertical (radial) and horizontal (lateral) components of site velocity change depends on location: at the North pole the up component of site velocity increases; along the equator the north component of site velocity increases. the effect of man’s management of water resources at many places (e.g., New Orleans [Dixon et al., 2006]). [5] Third, an accurate estimate of the velocity of CM is crucial for satellite altimetry [Morel and Willis, 2005]. Beckley et al. [2007, Figure 2] find, that when they substitute the velocity of CM in ITRF2005 [Altamimi et al., 2002] for that in ITRF2000 [Altamimi et al., 2007], estimated sea level rise increases by ≈1 mm/yr in the northern oceans and decreases by ≈1 mm/yr in the southern oceans. [6] Fourth, the velocity of CM relates to global plate motion. Argus et al. [2010, Figure 4] find that, if the plates rotate about CM, and if the velocity of CM were to be assumed known, then substituting ITRF2005 for ITRF2000 would significantly change estimates of relative plate velocity; and that accounting for uncertainty in the velocity of CM increases the (1–dimensional) uncertainty in relative plate velocity by a factor of roughly 2. [7] Doubts about how to treat the velocity of CM are evident in the literature. By adopting the velocities and uncertainties in an ITRF, the studies of Nocquet et al. [2005], Lidberg et al. [2007], and Sella et al. [2007] assume the velocity of CM to be known exactly. By solving for a translational velocity, the studies of Argus et al. [2010], Argus and Peltier [2010], Hill et al. [2010], and Wu et al. [2010] discard the SLR constraint on the velocity of CM. [8] The velocity of CM in ITRF2005 [Altamimi et al., 2007] differs from that in ITRF2000 [Altamimi et al., 2002] by 1.8 mm/yr, suggesting that the velocity of CM is not constrained tightly by SLR. But the velocity of CM in ITRF2008 [Altamimi et al., 2007] is nearly identical to that in ITRF2005, suggesting that these last two estimates are accurate. The sparse, uneven distribution of SLR sites contributes to

Figure 2. (top) SLR satellite LAGEOS is a sphere 0.6 m in diameter. (middle) The GPS Block II and IIA satellite has a body about 2 m across and solar panels with a wing span of 5 m. (bottom) The GPS Block IIR-M satellite has a body about 2 m across and wing span of 10 m.

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from any one space technique using either the speed of light or G*M, where G is the gravitational constant and M is Earth’s mass. In practice the size (scale) of the site–network polyhedron of a space technique is, in the determination of an ITRF, estimated each week. In ITRF2008 scale rate is defined to be the mean of VLBI and SLR. The quality of each technique can be assessed from the straightness of its time series of transformations of scale and CM to the ITRF. The accuracy of the resulting ITRF can be evaluated using the degree to which scale estimates agree among the four techniques. [10] This study has two purposes. First, by analyzing time series using spectral analysis and data decimation, we evaluate uncertainty in estimates of scale rate and the velocity of CM from the four techniques. Second, we present new GPS results from (JPL) Jet Propulsion Laboratory from a re– determination of satellite orbits and site positions using revised estimates of satellite phase center variations [Desai et al., 2011]. From JPL’s GPS results we find uncertainty in the velocity of CM to be ≈0.8 mm/yr and uncertainty in scale rate to be 0.2 mm/yr (95% confidence limits). Scale in GPS ultimately depends on scale in SLR and VLBI. The small uncertainty in GPS scale rate nevertheless indicates the GPS solution to be of high quality.

2. Data [11] We analyze thirteen series from four space techniques (Figure 3). The VLBI, SLR, and DORIS series for scale and the SLR and DORIS series for the velocity of CM are from the ITRF2008 analysis [Altamimi et al., 2011]. No such GPS series exists in the ITRF2008 analysis; the GPS series that we analyze was determined by JPL in Fall 2011 [Desai et al., 2011]. (VLBI does not provide an estimate of the velocity of CM.) [12] In the ITRF2008 analysis Altamimi et al. [2011] estimate, for each week, and for each space technique, a Helmert transformation. This Helmert transformation consists of 7 parameters: the 3 components of a rotation, the 3 components of a translation, and a scale accounting for expansion or contraction [Altamimi et al., 2011, equation 1] (Text S1 of the auxiliary material).1 Implicit in the analysis is that, for the satellite techniques, CM is at the origin of each weekly estimate. Also implicit is that, for all four space techniques, scale before the transformation is the value estimated from the space technique. Thus the SLR and DORIS series for the X, Y, and Z components of CM are the 3 components of the translation in the Helmert transformation. The VLBI, SLR, and DORIS series for scale also are from the Helmert transformation. [13] The GPS series for scale and CM are transformations from JPL’s reprocessed solution into ITRF2008. JPL’s revised GPS estimates of satellite orbits, clocks, and site positions from 1996 to 2011 [Desai et al., 2011] are determined using improved models of both satellite antenna phase center variations [Schmid et al., 2010, igs08.atx] and solar radiation pressure [Sibthorpe et al., 2010]. All estimates are in the IGS08 reference frame. The solid Earth and pole tide models follow the IERS [Petit and Luzum, 2010] standards. On each day the positions of about 24 GPS satellites and 80 sites are determined using the GMF and GPT troposphere models [Boehm et al., 2006, 2007] and GPSM10 solar radiation pressure model

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[Sibthorpe et al., 2010]. The estimates of satellite orbits, clocks, site positions, and Helmert transformations are available to all by anonymous ftp to sideshow.jpl.nasa.gov in directory / home/ftp/pub/JPL_GPS_Products_IGS08/ Final [Desai et al., 2011]. Time series of positions of 2500 sites are available at the same anonymous ftp site in the directory /home/ftp/pub/ usrs/mbh/ point [Heflin et al., 2011]. [14] There is no official GPS series for scale and CM in the ITRF2008 analysis of Altamimi et al. [2011]. Collilieux et al. [2011] examine series of Helmert transformations for four of the solutions that comprise the IGS (GPS) contribution to ITRF2008. In Figure S4 of the auxiliary material we analyze the series of transformations between this IGS solution (that contributes to ITRF2008) and the ITRF2005, not ITRF2008, reference frame. We also (in Figure S3 of the auxiliary material) perform data decimation and CATS spectral analysis on an SLR series from Center for Space Research provided to us by J. C. Ries (electronic communication, 2012). [15] The VLBI series represents a mean of seven analysis centers and comes from the International VLBI Service (IVS) [Schlüter and Behrend, 2007]. The SLR series is also a mean of seven analysis centers and comes from the International Laser Ranging Service (ILRS) [Pearlman et al., 2002]. The DORIS series are a mean of two analysis centers and comes from the International DORIS Services (IDS) [Willis et al., 2007]. JPL’s GPS observations ultimately are from the (IGS) International GNSS Service [Dow et al., 2009].

3. Method: Estimation of Rate Uncertainty [16] Space geodetic estimates of position are correlated in time [Langbein and Johnson, 1997; Mao et al., 1999; Williams et al., 2004]. Therefore rate uncertainties estimated from positions using linear propagation of errors are unrealistically small [Langbein and Johnson, 1997; Mao et al., 1999; Williams et al., 2004]. In this study we determine realistic rate uncertainties first by using the CATS spectral analysis algorithm [Williams, 2008], and second by using data decimation. 3.1. CATS Spectral Analysis [17] In the formulation of Williams [2008] there may be three kinds of error, each corresponding to a different value of spectral index in the power law Pðf Þ ¼ P0 =f a

where P is the power spectrum, P0 a constant, f the frequency, and a the spectral index. White noise (spectral index of 0), is a random error in which estimates are not correlated in time. Random walk (spectral index of 2) consists of a series of successive random steps. Flicker noise (spectral index of 1) is in between white noise and random walk. [18] The nature of the three kinds of error can be understood from the relationship between the (s) error in rate and the (T) time period of observation. [19] For white noise

1

Auxiliary materials are available in the HTML. doi:10.1029/ 2012JB009196.

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swh ¼ swh =T3=2

ð1Þ

[20] For flicker noise

h i1=2 sfl ¼ sfl =T 8= pf 1=2

ð2Þ

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Figure 3. Estimates of scale and the position of CM as a function of time from VLBI, SLR, GPS, and DORIS. Scale rate is computed along Earth’s radius. A rate, position, and sinusoid with period of 1 year is fit to each series. Estimates from VLBI before 1985 and from SLR before 1993 are omitted (pink dots); all other estimates are included (light blue dots). RMS– root mean square dispersion. Oscillation– given are the peak–to–peak amplitude of the sinusoid and the month of the maximum position. 4 of 15

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Figure 3. (continued) [21] For random walk srw ¼ srw =T

1=2

ð3Þ

[22] The study of Bos et al. [2008] is the basis of these relationships; we substitute into their equations 29, 30, and 31 the identities N = T*f and dt = 1/f. swh, sfl, and srw are values of the three kinds of noise output by the CATS algorithm; dt, the sampling period, is the inverse of f, the frequency of observations; T is the time period of observations. [23] For SLR, GPS, and DORIS, dt = 1/52 and sfl = 0.594 sfl / T. For VLBI, dt = 104 and sfl = 0.5 sfl / T. (There are 52 SLR data per year. Because we average GPS daily data into weekly means, there are 52 GPS data per year. There are about 104 VLBI data per year.) [24] Flicker noise has long been observed to be the dominant error source in geodetic time series [Mao et al., 1999; Williams et al., 2004]. Williams et al. [2004] find the mean spectral indices of 954 GPS series to be 0.9 (east), 1.0 (north), and 0.8 (up), showing flicker noise (spectral index of 1) to be the main error source. We [Argus and Gordon, 1996; Argus et al., 1999, 2005, 2010; Argus and Peltier, 2010] have for 15 years adopted the 1 / T form inherent in flicker noise; we have taken rate error to be s / T, where the systematic distance s is estimated empirically [Argus et al., 1999, 2010; Argus and Peltier, 2010] to be 4.5 mm in the horizontal and 10 mm in the vertical using the degree of consistency with plate motion and postglacial rebound, the degree of consistency between space techniques, and data decimation. [25] We invert the thirteen series using the CATS algorithm (maximum likelihood estimation); assume there to be white noise and flicker noise; and estimate an offset, a rate, and a sinusoid having a period of 1 year. We find white noise to amount to negligible rate error; flicker noise, the source of the rate error, is as in Table 1.

[26] If we were to not assume white and flicker noise, and instead solve for the spectral index, we would find spectral indices to be for scale 1.1 (VLBI), 1.2 (SLR), 1.1 (GPS); for CM from SLR 0.6 (X), 0.7 (Y), and 1.0 (Z); and for CM from GPS 0.9 (X), 0.9 (Y), and 1.3 (Z). The median spectral index is 1.0, consistent with there being white and flicker (1) noise as we assume. Moreover, because the physical processes in the estimation are identical for CM and scale as they are for site positions, rate uncertainties for the velocity of CM and scale rate estimated assuming white and flicker noise are realistic. 3.2. Data Decimation [27] We also evaluate rate errors by examining the variance of rate estimates determined by distinct (non–overlapping) time periods of data (Figures 4, 5, 6, and Figures S1–S5 of the auxiliary material). We first split a series into two, fit a rate to each of the two time periods, compute the sample standard deviation s:d: ¼

h

i1=2 S ðrai –ramean Þ2 =n

where n is the number of time periods, and plot the s.d. as a function of the length of the time period (Figure 7, open circle on right-hand side). We next split the series into three, then into four, next into five, and finally into eight; fit rates to the distinct time periods; compute the sample standard deviation for each experiment; and again plot the sample s.d. as a function of time period (Figure 7, open circles). [28] For samples of just a few values, the sample standard deviation tends to be less than the true standard deviation of the parent population, and the inference of the true sample standard deviation from the sample is quite uncertain. If by chance the values happen to be clustered together, then

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Table 1. Estimates of Uncertainty in Geodetic Scale Rate and Mass Center Velocitya Distance (mm) or Flicker Noise (mm/yr1/4) sc

X

Y

Z

Rate (mm/yr) sc

X

Y

Z

Method

VLBI 1985–2008 4.6 2.1 6

RMS dispersion CATS flicker noise Data decimation Mean

0.13 0.24 0.18

2.5 3.8 6

3.2 5.3 3

2.8 4.3 4.5

SLR 1993–2008 6.8 9.9 0.14 9 0.38 0.26

0.20 0.19 0.20

0.16 0.28 0.22

0.37 0.56 0.46


1.1 2.2 3

4.1 7.7 9

3.5 7.7 4.5

GPS 1996–2011 (JPL 2012) 5.7 12.3 0.08 9 0.12 0.10

0.29 0.56 0.42

0.29 0.28 0.28

0.46 0.56 0.51


3.5 14.1 6

5.7 11.6 7.5

6.5 13.9 7.5

DORIS 1993–2008 26.7 61.3 0.50 45 0.38 0.44

0.43 0.47 0.45

0.52 0.47 0.50

2.28 2.81 2.50


3.7 10.1 6

4.4 12.1 7.5

2.7 10.3 7.5

SLR 1985–2008 10.1 29.5 0.20 15 0.25 0.22

0.19 0.31 0.25

0.19 0.31 0.25

0.58 0.62 0.60


1.8 1.4 3

1.7 1.6 3

0.08 0.19 0.14

0.08 0.19 0.14

0.25 0.38 0.32


1.5 2.0 4

3.9 7.8 7.5

7.7 10.0 30

GPS 1994–2009 (IGS 2008) 6.4 14.0 0.07 9 0.50 0.28

0.29 0.47 0.38

0.37 1.87 1.12

0.52 0.56 0.54


1.1 3.7 3

4.1 7.7 9

3.5 7.9 4.5

GPS 1996–2011 (JPL 2010) 5.7 12.8 0.14 9 0.19 0.16

0.29 0.47 0.38

0.29 0.56 0.42

0.48 0.56 0.52


CSR SLR 1985–2008 3.8 5.0 6

a RMS dispersion– root mean square residual in mm of the fit of an offset, a rate, and a sinusoid with a period of 1 yr to the data (in Figure 1). CATS flicker noise– flicker noise (sfl in equation (2)) in mm/yr1/4 and inferred standard error (sfl) in rate in mm/yr. For SLR, GPS, and DORIS the (f) frequency in equation (2) is 52, yielding sfl = 0.594 sfl / T. For VLBI, f is roughly 104, yielding sfl = 0.500 sfl / T. Data decimation– systematic distance (s) in mm; and inferred standard error in rate (s = s/T, where T is the time period of observations) in mm/yr (from Figure 5). Standard errors in scale rate are given in this table; 95% confidence limits in scale rate follow the in the body of this article. The top four sets of results, 1 for each of 4 techniques, is the basis of the main text. The bottom three sets of results are for series in Figures S2, S3, S4, and S5 of the auxiliary material.

the sample standard deviation would be too small. Or if the values randomly obtained are more scattered than the overall population, the standard deviation would be too big. For samples of several values, the sample standard deviation tends to get closer to the true standard deviation, and the inference of the true standard deviation becomes more tightly constrained. Using Monte Carlo simulation of a Gaussian distribution (Table 2), we determine unbiased (Figure 7, light blue circles) inferences of the true standard deviation and their 95% confidence limits (Figure 7, error bars).

[29] We next fit by eye a curve having the form s = s / T to these points. We take uncertainty in rate for the whole series to be s divided by the total time period, as in Table 1. [30] Implicit in the data decimation estimate of the rate uncertainty is the assumption that the rate is constant. [31] The ratio of the data–decimation and CATS estimates of rate error ranges from 0.8 to 2.7 and has a median value of 1.2. The data–decimation error is larger than the CATS error for eight series and smaller for four series. Given that estimates of rate error are uncertain, this range of ratios is acceptable. Hereinafter we take the mean of the rate error from data decimation

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Figure 4. Decimating series of scale from VLBI, SLR, and GPS. In the second row the series is split in half; in the last row the series is split into 8 distinct time periods. Rates fit to each distinct time period are given in mm/yr; For each split the sample standard deviation (s.d) of the rate estimates is given. In Figure 7 this standard deviation is plotted on the vertical axis and the length of the time period is plotted on the horizontal axis. and that from CATS to be the estimate of the true error. Hereinafter 1–dimensional 95% confidence limits follow ‘’.

4. Results 4.1. Scale [32] We find estimates of the scale rate relative to ITRF2008 to cluster tightly: VLBI 0.05 mm/yr, SLR 0.12 mm/yr, and GPS 0.00 mm/yr (along Earth’s radius) (Figure 8a). The uncertainty in scale rate from VLBI, 0.36 mm/yr, is 30 per cent smaller than the uncertainty in scale rate from SLR, 0.52 mm/yr. This is mostly because scale rates fit to different time periods of data differ less for

VLBI than for SLR (Figure 4), and partly because the data time period is longer for VLBI, 24 yr, than for SLR, 16 yr. Thus, even though the (RMS) root mean square dispersion of positions is greater for VLBI than for SLR, the data decimation experiments indicate VLBI to be more certain than SLR. [33] Altamimi et al. [2011] find estimates of scale rate to differ between VLBI and SLR by a significant 0.31 0.13 mm/yr (along Earth’s radius; we convert their standard errors to 95% confidence limits). In contrast, we find estimates of scale rate from VLBI, SLR, and GPS to differ insignificantly by at most 0.17 0.63 mm/yr. Altamimi et al. [2011] estimate uncertainty in the velocity of CM from SLR to be 0.4 mm/yr. We estimate the uncertainty in

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Figure 5. Decimating series of the X, Y, and Z components of the velocity of CM from SLR. Otherwise identical to Figure 4. X and Y to be 0.4 mm/yr, identical to their uncertainty, but estimate the uncertainty in Z to be 0.9 mm/yr, twice their uncertainty. We conclude that Altamimi et al.’s [2011] estimates of uncertainty in scale rate and in the Z component of CM are unrealistically small. [34] The VLBI and GPS scale series have a peak–to–peak annual oscillation of 3 to 5 mm with a maximum in August to September. As we can identify no phenomenon causing Earth

to expand and contract by season, we suspect this oscillation to be due to error related to the uneven spatial distribution of sites between the northern and southern hemispheres. 4.2. Center of Mass [35] SLR constrains the X and Y components of the velocity of CM tightly, to 0.4 mm/yr, but constrains the Z component loosely, to 0.9 mm/yr.

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Figure 6. Decimating series of the X, Y, and Z components of the velocity of CM from GPS. Otherwise identical to Figure 4. [36] GPS constrains the velocity of CM to 0.8 mm/yr in X, 0.6 mm/yr in Y, and 1.0 mm/yr in Z. Given that the GPS satellites are not at all close to a sphere as is SLR’s LAGEOS, it is remarkable that GPS yields an estimate of the velocity of CM constrained close to as well as SLR does. [37] The Z component of CM has a peak to peak amplitude of 11 mm in the SLR series (Figure 5) but just 1.6 mm in the GPS series (Figure 6). In the loading model of

Collilieux et al. [2009], fluctuation of water, atmosphere, and snow causes an annual oscillation of CM of ≈5.5 mm, with a maximum position in February––Snow and ice accumulates in North America and Eurasia from December to March, causing CM to move toward North pole. The predicted oscillation is half that observed with SLR, but roughly 3 times larger than inferred from GPS. Ries [2011] suggests the model has insufficient snow and ice loading at

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Figure 7. Estimating uncertainty in scale rate and the X, Y, and Z components of the velocity of CM using the data decimation. Open circle– sample standard deviation as a function of the length of the time period the data are split into, from Figures 2, 3, 4, and Figure S1 of the auxiliary material. Light blue circle– estimate of the true standard deviation, correcting for the tendency for the true sample standard deviation to be less than that of the parent population using Table 2. Error bars– 95% confidence limits are also calculated from the sample standard deviation and from Table 2. An error budget (blue curve) is fit by eye to the points. In this error budget rate error as a function of time is taken to be s = s/T, where s is a systematic distance in mm and T is the time period in yr. The value of the systematic distance is given (in blue) in mm; the resulting standard error in rate of the whole series is given after the s. high latitudes in February; we suggest that half the SLR– observed oscillation is due to error. It is difficult to imagine how GPS could find an annual oscillation smaller than it truly is. We argue that the very small GPS–observed oscillation suggests that the true oscillation of water, snow, and ice between northern and southern hemisphere may be less than in the geophysical model. [38] There is an annual oscillation in Y having a peak–to– peak amplitude of 7 mm with a maximum in May in the SLR and in the GPS series, indicating that Earth’s water and atmosphere is truly moving seasonally. GRACE gravity observations show the Amazon river basin to have an annual mass oscillation of 2 1015 kg, with a maximum during the heavy rainfall in April; and southeast Asia to have an annual mass oscillation of 1 1015 kg, with a maximum during the monsoons in September (F. Landerer, electronic communication,

2012). This would generate a peak–to–peak oscillation between CM and CE of 3.1 mm, about half that observed. 4.3. DORIS [39] DORIS constrains scale rate more poorly than either VLBI or SLR. DORIS also poorly constrains the Z component of the velocity of CM (Table 1 and Figure S1 of the auxiliary material).

5. Interpretation 5.1. GPS Quality [40] GPS is providing an independent means by which to estimate the velocity of the center of mass of Earth (CM), but not of scale rate. The estimate of scale from GPS at a specific time depends on VLBI and SLR because antenna

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Table 2. Relationship Between Sample and True Standard Deviation in a Gaussian Distributiona Percentile N

2.5

15.9

50

84.1

97.5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.446 0.520 0.566 0.599 0.624 0.644 0.661 0.675 0.687 0.698 0.708 0.716 0.724 0.731 0.737 0.744 0.749

0.709 0.737 0.761 0.778 0.792 0.804 0.814 0.822 0.829 0.835 0.841 0.846 0.850 0.855 0.858 0.862 0.865

1.482 1.201 1.125 1.090 1.071 1.059 1.049 1.043 1.038 1.034 1.030 1.028 1.026 1.024 1.022 1.021 1.019

4.998 2.404 1.895 1.676 1.556 1.478 1.424 1.383 1.351 1.325 1.305 1.286 1.271 1.257 1.245 1.235 1.226

32.398 6.293 3.749 2.875 2.462 2.206 2.038 1.916 1.827 1.753 1.697 1.652 1.610 1.574 1.545 1.520 1.495

a

Given a sample standard deviation of 1 computed from a sample of N numbers, the 2.5, 15.9, (median) 50, 84.1, and 97.5 percentile values of the true standard deviation of the parent population in a Gaussian distribution. For example, given a sample standard deviation of 1 computed from two values, 2.5 per cent of the time the true standard deviation is greater 32.398, 2.5 percent of the time the true standard deviation is less than 0.446, and the median true standard deviation is 1.482. In Figure 5 the unbiased estimate of the true standard deviation (light blue circle) is inferred from the sample standard deviation (open circle) by multiplying by the (median) 50 percentile value for the appropriate sample size in this table; the 95% confidence limits are computed by multiplying by the 2.5 and 97.5 percentile values in this table.

phase variations of satellites (meaning phase center variations and antenna offsets) are, in GPS analysis, fixed to values [Schmid et al., 2007, 2010] determined assuming site positions in an ITRF, which are in turn determined using scale from VLBI and SLR. If the assumed antenna offsets and phase center variations were to not change with time, then the estimate of scale rate from GPS would be independent of VLBI and SLR. In truth GPS satellites with three different patterns of antenna phase variations have been swapped into and out of the GPS constellation: first there

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were just Block II and IIA satellites, next Block II–R(A) satellites were substituted, and then Block II–R(B) and IIR–M satellites were added. The patterns of phase center variations are determined assuming site positions in an ITRF; therefore the estimate of scale rate ultimately depends on VLBI and SLR. [41] We nevertheless infer, from the GPS scale series, that the quality of JPL’s GPS analysis is excellent, that is, that the estimates of satellite orbits, clocks, and site positions are robust over time. Two characteristics of the GPS series show this. First, we find a rate from GPS nearly equal to that from VLBI and SLR (Figures 3 and 8). Second, the series is straight, not crooked, as evident in the small rate uncertainty of 0.20 mm/yr that we find from data decimation and CATS spectral analysis. The scale rate uncertainty (0.20 mm/yr) for JPL’s latest solution [Desai et al., 2011] is a factor of 3 smaller than for JPL’s prior two solutions determined in 2010 (uncertainty 0.56 mm/yr) and in 2008 (also 0.56 mm/yr) (Figure S5 of the auxiliary material). This is due to better quality control [Desai et al., 2011], improved satellite antenna variations (igs08.atx file) [Schmid et al., 2010], and improved solar radiation pressure models [Sibthorpe et al., 2010]. 5.2. Independent Inferences of the Velocity of CM From Site Velocities [42] Over the past 20 years geodesists have analyzed time series for ‘geocenter’, that is, estimates of the evolution of CM relative to the center of network of geodetic sites (CN). If sites were distributed throughout the world, including on the ocean floor, than one might be able to determine the evolution of the mean position of the surface of solid Earth (CF) relative to CM. (In the literature CF is the mean position of the surface of solid Earth, not the true center of figure of solid Earth’s surface. Horizontal (lateral) motion changes CF but not the true center of figure). In practice, the geographic distribution of sites is insufficient to accurately estimate the velocity of CF. [43] Postglacial rebound models, however, provide a means by which to relate (the geometric) estimates of site velocity to (the dynamic) Earth’s mass center [Argus, 2007]. How site velocities constrain the velocity of Earth’s center is

Figure 8. (a) Estimates of scale rate and 95% confidence limits from (V) VLBI, (S) SLR, (G) GPS, and (D) DORIS. Error bars are from (solid) data decimation and (dashed) CATS spectral analysis. ‘R’ is the GPS scale rate of Rülke et al. [2008]. (b) Estimates of the velocity of CM from SLR and GPS and 95% confidence limits. Error ellipses are from data decimation (filled beige for SLR and green for GPS) and from CATS spectral analysis (dashed for SLR, dash-dotted for GPS). ‘CSR’ is the velocity of CM estimated using SLR by Center for Space Research [Ries, 2011]. (c) Estimates of the velocity of CE determined from horizontal site velocities assuming that, besides plate motion, sites on plate interiors not near the former ice sheets are moving negligibly laterally relative to CE, and 95% confidence limits. ‘GEODVEL2’ (blue-filled ellipse) is determined from the VLBI, SLR, and DORIS site velocities in Argus et al. [2010] and GPS site velocities that we estimate from the time series determined by M. B. Heflin and A. Moore (Jet Propulsion Laboratory, electronic communication, 2012) using JPL’s revised GPS satellite orbits [Desai et al., 2011]. The ‘ITRFVEL’ (pink-filled ellipse) is determined from the ITRF2008 site velocities. The velocity of CE determined from horizontal site velocities in ITRF2008 by Altamimi et al. [2012] is also plotted (though they interpret the determination to be CF, the mean velocity of Earth’s surface). (d) Estimates of the velocity of CE determined from vertical site rates assuming that sites on plate interiors move radially as predicted by postglacial rebound model ICE-6G VM5a T60 Rot (Peltier et al. manuscript in preparation, 2012), and overoptimistic 95% confidence limits, with the blue- and pink-filled ellipses denoting the same data sets as in Figure 8c. The yellow-filled ellipse is the velocity of CE determined from the same data as the blue using the same assumption except that the postglacial rebound model of Paulson et al. [2007, Geraldo A] is corrected for. (The model of Paulson et al. is based on ICE-5G and VM2–2 [Peltier, 2004]). The (unlabeled) pentagon is the estimate of Wu et al. [2011]. The nominal 95% confidence limits in any one estimate is overoptimistic in that uncertainty in the rebound model is not accounted for; the range of the estimates begins to describe the true 95% confidence limits. 11 of 15

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Figure 8 12 of 15

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straightforward [Argus, 2007]. The vertical (radial) components of site velocities constrain the velocity of Earth’s center. If the velocity of Earth’s center were wrong, then one side of Earth would appear to be rising (or falling), while the other side of Earth would appear to be falling (or rising). The horizontal (lateral) components of site velocities also constrain the velocity of Earth’s center. Changing the estimate of the velocity of Earth’s center changes the horizontal component of site velocities by different amounts in different places (Figure 1). If the estimate of the velocity of Earth’s center were wrong, then the plate interiors would appear to be deforming. Blewitt [2003] defines these two constraints to be the center of height (CH) and the center of lateral (CL) movement of Earth’s surface. [44] Postglacial rebound, Earth’s viscous response to unloading of the ice sheets over the past 25 kyr, violates the assumption implicit in the two determinations. Places in Canada, Fennoscandia, and Antarctica are rising as fast as 10 mm/yr; the margins of the ice sheets are furthermore moving away from the former ice sheets centers at ≈1 mm/yr [Johansson et al., 2002; Argus and Peltier, 2010, Argus et al., 2011]. In the inversion of vertical observations for the velocity of Earth’s center, we correct for the predictions of a postglacial rebound model. Because the predictions are relative to the center of mass of solid Earth (CE), we estimate the velocity of CE. Implicit in this vertical determination is the assumption that the plate interiors are moving vertically relative to CE as predicted by the postglacial rebound model. In the inversion of the horizontal observations for both the velocity of Earth’s center and the angular velocities of the plates, we omit velocities of sites beneath or along the margins of the late Pleistocene ice sheets. Implicit in this horizontal determination is the assumption that, besides plate motion, the parts of the plate interiors not near the former ice sheets are not moving horizontally relative to CE. (Places far from the former Laurentia ice center may be moving at about 1 mm/yr toward the ice center, suggesting that the horizontal determination of the velocity of CE may be tenuous.) [45] No phenomenon is known to sustain a significant velocity between CM and CE [Argus, 2007; Argus et al., 2010]. Earth’s viscous response to unloading of the ice sheets from 25 ka to 4 ka generates an insignificant speed between CM and CE. (W. R. Peltier (personal communication, 2010) calculates this speed to be, in the model of Peltier [2007], just 0.017 mm/yr.) This speed is miniscule because relative sea level at Pacific islands in the tropics was at a highstand of 2 m at 4 ka, requiring global ice sheet loss to have ended by 4 ka. [46] Current ice loss also could not sustain a significant speed between CM and CE. If current ice loss in Antarctica were 100 Gt/yr (enough to raise global sea level 0.28 mm/yr), then CM would move relative to CE toward North pole at 0.11 mm/yr [Argus, 2007, Appendix A; Metivier et al., 2010]. But 100 Gt/yr of current ice loss from Greenland, on the opposite side of Earth as Antarctica, would roughly cancel 100 Gt/yr of Antarctic ice loss. Although it is not known which continent is losing ice faster, estimates of current ice loss [Chen et al., 2009; Horwath and Dietrich, 2009; Peltier, 2009; van den Broeke et al., 2009; Jacob et al., 2012] inferred jointly from GRACE gravity observations and postglacial rebound models suggest current Antarctica and

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Greenland ice loss to differ by less than 100 Gt/yr, which would create a speed between CM and CE of less than 0.11 mm/yr. [47] We estimate the velocity of CE from two distinct data sets (Figures 8c and 8d). The first set is the ITRF2008 site velocities. The second set consists of the VLBI, SLR, and DORIS site velocities used in Argus et al. [2010] and GPS site velocities that we estimate from time series determined by JPL (M. B. Heflin and A. Moore, electronic communication, 2012; sideshow.jpl.nasa.gov/post/series.html). This latter set of series differs from that in Argus et al. [2011] in that they are based on the revised GPS satellite orbits [Desai et al., 2011]. In the discussion that follows and in Figures 8c and 8d we are not claiming to be estimating the velocity between CM and CE. Rather, we are arguing on geophysical grounds that the speed between CM and CE is insignificant, and that we are in Figures 8c and 8d comparing two independent sets of estimates of the velocity of Earth’s mass center, the first from SLR observations of LAGEOS, and the second inferred from site velocities using a plate model or a postglacial rebound model. [48] The velocity of CE estimated using JPL’s latest GPS horizontal observations using JPL’s site velocities is in X and Y is nearly equal to the velocity of CM from SLR, but in Z has a rate of +1.0 mm/yr (Figure 8c, blue-filled ellipse). Implicit in this horizontal determination is the assumption that, beside plate motion, the parts of the plate interiors not near the former ice sheets are not moving horizontally relative to CE. [49] The velocity of CE estimated using JPL’s latest GPS vertical observations is in X and Y is nearly equal to the velocity of CM from SLR, but in Z has a rate of +0.6 mm/yr to +0.8 mm/yr (Figure 8d). Implicit in this vertical determination is the assumption that the plate interiors are moving radially relative to CE as predicted by a postglacial rebound model, either that of (blue-filled ellipse) W. R. Peltier, D. F. Argus, and R. Drummond (A new model of global glacial isostatic adjustment based on the application of Global Positioning System observations of motion of Earth’s surface: ICE-6G (VM5a), manuscript in preparation, 2012) or that of (yellow-filled ellipse) Paulson et al. [2007]. [50] The velocity of CM from SLR is most uncertain along Z, the direction along which the determinations of the velocity of CE differ most from CM. In Z the 95% confidence limits in the horizontal and vertical determinations using JPL’s latest GPS data just overlap, and their intersection lies inside the 95% confidence limits of the velocity of CM from SLR. Perhaps the velocity of CM truly has a Z component of +0.8 mm/yr relative to that in ITRF2008. If this were true, then there would be 0.65 mm/yr (=0.8 sin(60 )) more uplift at high northern latitudes and 0.65 mm/yr more subsidence at high southern latitudes than in ITRF2008.

6. Conclusions [51] 1. Data decimation and CATS spectral analysis provides a means to estimate realistic uncertainty in scale rate and the velocity of CM, two quantities important for the definition of an ITRF. The realistic uncertainty in the velocity of CM is furthermore important for the study of postglacial rebound, current ice loss, sea level rise, and plate motion.

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[52] 2. We find the realistic uncertainty in the velocity between the mass center and surface of Earth to be 0.4 mm/yr in X and Y and 0.9 mm/yr in Z. Because the Z component of the velocity of CM is very uncertain, it must not be assumed to be known in space geodetic applications to Earth science. [53] 3. JPL’s latest GPS solution is of high quality, as evident in the small realistic uncertainty (0.2 mm/yr) in scale rate from data decimation and CATS spectral analysis. Differences in antenna phase center variations between satellites of different Block types do not bias GPS estimates of satellite orbits, site positions, and clocks. GPS is providing an independent estimate of the velocity of the center of mass of Earth (CM). [54] Acknowledgments. We are grateful to Z. Altamimi, X. Collilieux, and L. Metivier for determining the Helmert transformations from SLR, VLBI, and DORIS to ITRF2008; to R. Ferland for the IGS Helmert transformations from GPS to ITRF2008; to S. Desai, W. Bertiger, J. Gross, B. Haines, N. Harvey, C. Selle, A. Sibthorpe, and J. P. Weiss for the JPL GPS solution; to M. B. Heflin and A. Moore for JPL’s position-time series; to S. D. P. Williams and M. S. Bos for instruction on CATS spectral analysis; and to F. W. Landerer for GRACE estimates of seasonal mass fluctuations. We thank Erricos Pavlis and two anonymous reviewers for their suggestions and constructive criticism. D. Argus performed this research under contract by NASA at Jet Propulsion Laboratory, California Institute of Technology.

References Altamimi, Z., P. Sillard, and C. Boucher (2002), ITRF2000: A new release of the International Terrestrial Reference Frame for Earth science applications, J. Geophys. Res., 107(B10), 2214, doi:10.1029/2001JB000561. Altamimi, Z., X. Collilieux, J. Legrand, B. Garayt, and C. Boucher (2007), ITRF2005: A new release of International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters, J. Geophys. Res., 112, B09401, doi:10.1029/2007JB004949. Altamimi, Z., X. Collilieux, and L. Metivier (2011), ITRF2008: An improved solution of the international terrestrial reference frame, J. Geod., 85, 457–473. Altamimi, Z., L. Metivier, and X. Collilieux (2012), ITRF2008 plate motion model, J. Geophys. Res., 117, B07402, doi:10.1029/2011JB008930. Argus, D. F. (2007), Defining the translational velocity of the reference frame of Earth, Geophys. J. Int., 169, 830–838. Argus, D. F., and R. G. Gordon (1996), Tests of the rigid-plate hypothesis and bounds on intraplate deformation using geodetic data from very long baseline interferometry, J. Geophys. Res., 101, 13,555–13,572. Argus, D. F., and W. R. Peltier (2010), Constraining models of postglacial rebound using space geodesy: A detailed assessment of model ICE-5G (VM2) and its relatives, Geophys. J. Int., 181, 697–723, doi:10.1111/ j.1365-246X.2010.04562.x. Argus, D. F., W. R. Peltier, and M. M. Watkins (1999), Glacial isostatic adjustment observed using very long baseline interferometry and satellite laser ranging, J. Geophys. Res., 104, 29,077–29,083. Argus, D. F., M. B. Heflin, G. Peltzer, F. H. Webb, and F. Crampe (2005), Interseismic strain accumulation and anthropogenic motion in metropolitan Los Angeles, J. Geophys. Res., 110, B04401, doi:10.1029/ 2003JB002934. Argus, D. F., Gordon, R. G., M. B. Heflin, C. Ma, R. J. Eanes, P. Willis, W. R. Peltier, and S. E. Owen (2010), The angular velocities of the plates and the velocity of Earth’s center from space geodesy, Geophys. J. Int., 180, 913–960. Argus, D. F., G. Blewitt, W. R. Peltier, and C. Kreemer (2011), Rise of the Ellsworth mountains and parts of the East Antarctic peninsula observed with GPS, Geophys. Res. Lett., 38, L16303, doi:10.1029/2011GL048025. Bar-Sever, Y., and D. Kuang (2004), Emperically derived solar radiation pressure model for Global Positioning System Satellites, Interplanet. Progress Rep., 42–159, Jet Propul. Lab., Pasadena, Calif. [Available at http://ipnpr.jpl.nasa.gov/progress_report/42-159/159I.pdf.] Beckley, B. D., F. G. Lemoine, S. B. Luthcke, R. D. Ray, and N. P. Zelensky (2007), A reassessment of global and regional mean sea level trends from TOPEX and Jason-1 altimetry based on revised reference frame orbits, Geophys. Res. Lett., 34, L14608, doi:10.1029/2007GL030002. Blewitt, G. (2003), Self-consistency in reference frames, geocenter definition and surface loading of the solid Earth, J. Geophys. Res., 108(B2), 2103, doi:10.1029/2002JB002082.

B10405

Boehm, J., A. Niell, P. Tregoning, and H. Schuh (2006), Global Mapping Function (GMF): A new empirical mapping function based on numerical weather data, Geophys. Res. Lett., 33, L07304, doi:10.1029/2005GL025546. Boehm, J., P. J. Mendes Cerveira, H. Schuh, and P. Tregoning (2007), The impact of tropospheric mapping functions based on numerical weather models on the determination of geodetic parameters, in Dynamic Planet—Monitoring and Understanding a Dynamic Planet With Geodetic and Oceanographic Tools, Int. Assoc. Geod. Symp., vol. 130, edited by P. Tregoning and C. Rizos, Springer, pp. 837–843. doi:10.1007/978-3540-49350-1_118. Bos, M. S., R. M. S. Fernandes, S. D. P. Williams, and L. Bastos (2008), Fast error analysis of continuous GPS observations, J. Geod., 82, 157–166. Chen, J. L., C. R. Wilson, D. D. Blankenship, and B. D. Tapley (2009), Accelerated Antarctic ice loss from satellite gravity measurements, Nat. Geosci., 694, 859–862, doi:10.1038/ngeo694. Collilieux, X., Z. Altamimi, J. Ray, T. van Dam, and X. Wu (2009), Effect of the satellite laser ranging network distribution on geocenter motion estimation, J. Geophys. Res., 114, B04402, doi:10.1029/2008JB005727. Collilieux, X., L. Metivier, Z. Altamimi, T. van Dam, and J. Ray (2011), Quality assessment of GPS reprocessed terrestrial reference frame, GPS Solut., 15, 219–231, doi:10.1007/s10291-010-0184-6. Desai, S. D., W. Bertiger, B. Haines, N. Harvey, C. Sella, A. Sibthorpe, and J. P. Weiss (2011), Results from the reanalysis of global GPS data in the IGS08 reference frame, Abstract G53B-0904 presented at 2011 Fall Meeting, AGU, San Francisco, Calif., 5–9 Dec. Dixon, T., et al. (2006), Subsidence and flooding in New Orleans, Nature, 441, 587–588, doi:10.1038/441587a. Dow, J. M., R. E. Neilan, and C. Rizos (2009), The International GNSS Service in a changing landscape of Global Navigation Satellite Systems, J. Geod., 83, 191–198, doi:10.1007/s00190-008-0300-3. Farrell, W. E. (1972), Deformation of the Earth by surface loads, Rev. Geophys., 10, 761–797, doi:10.1029/RG010i003p00761. Heflin, M. B., A. W. Moore, and S. E. Owen (2011), Impact of ambiguity resolution and orbit reprocessing on the global reference frame, Abstract G53A-0879 presented at 2011 Fall Meeting, AGU, San Francisco, Calif., 5–9 Dec. Hill, E. M., J. L. Davis, M. E. Tamisiea, and M. Lidberg (2010), Combination of geodetic observations and models for glacial isostatic adjustment fields in Fennoscandia, J. Geophys. Res., 115, B07403, doi:10.1029/ 2009JB006967. Horwath, M., and R. Dietrich (2009), Signal and error in mass change inferences from GRACE: The case of Antarctica, Geophys. J. Int., 177, 849–864, doi:10.1111/j.1365-246X.2009.04139.x. Jacob, T., J. Wahr, W. T. Pfeffer, and S. Swenson (2012), Recent contributions of glaciers and ice caps to sea level rise, Nature, 482, 514–518, doi:10.1038/nature10847. Johansson, J. M., et al. (2002), Continuous GPS measurements of postglacial adjustment in Fennoscandia: 1. Geodetic results, J. Geophys. Res., 107(B8), 2157, doi:10.1029/2001JB000400. Langbein, J., and H. Johnson (1997), Correlated errors in geodetic time series: Implications for time-dependent deformation, J. Geophys. Res., 102(B1), 591–603, doi:10.1029/96JB02945. Lidberg, M., J. M. Johansson, H. G. Scherneck, and J. L. Davis (2007), An improved and extended GPS-derived 3D velocity field of the glacial isostatic adjustment (GIA) in Fennoscandia, J. Geod., 81, 213–230, doi:10.1007/s00190-006-0102-4. Mao, A., C. G. A. Harrison, and T. H. Dixon (1999), Noise in GPS coordinate time series, J. Geophys. Res., 104(B2), 2797–2816, doi:10.1029/ 1998JB900033. Metivier, L., M. Greff–Lefftz, and Z. Altamimi (2010), On secular geocenter motion: The impact of climate changes, Earth Planet. Sci. Lett., 296, 360–366, doi:10.2016/j.epsl.2010.05.021. Morel, L., and P. Willis (2005), Terrestrial reference frame effects on global sea level rise determination from TOPEX/Poseidon altimetric data, Adv. Space Res., 36, 358–368, doi:10.1016/j.asr.2005.05.113. Nocquet, J. M., E. Calais, and B. Parsons (2005), Geodetic constraints on glacial isostatic adjustment in Europe, Geophys. Res. Lett., 32, L06308, doi:10.1029/2004GL022174. Paulson, A., S. Zhong, and J. Wahr (2007), Inference of mantle viscosity from GRACE and relative sea level data, Geophys. J. Int., 171, 497–508, doi:10.1111/j.1365-246X.2007.03556.x. Pearlman, M. R., J. J. Degnan, and J. M. Bosworth (2002), The International Laser Ranging Service, Adv. Space Res., 30, 135–143, doi:10.1016/ S0273-1177(02)00277-6. Peltier, W. R. (2004), Global glacial isostasy and the surface of the ice-age earth: The ICE-5G (VM2) model and GRACE, Annu. Rev. Earth Planet. Sci., 32, 111–149, doi:10.1146/annurev.earth.32.082503.144359.

14 of 15

B10405


Peltier, W. R. (2007), History of Earth rotation, in Treatise on Geophysics, vol. 9, Evolution of the Earth, edited by G. Schubert, pp. 243–293, Elsevier, New York, doi:10.1016/B978-044452748-6/00148-6. Peltier, W. R. (2009), Closure of the budget of global sea level rise over the GRACE era: The importance and magnitudes of the required corrections for global glacial isostatic adjustment, Quat. Sci. Rev., 28, 1658–1674, doi:10.1016/j.quascirev.2009.04.004. Petit, G., and B. Luzum (Eds.) (2010), IERS Technical Note 36, 179 pp., Bundesamts für Kartogr. und Geod., Frankfurt, Germany. Ries, J. C. (2011), Seasonal geocenter motion from space geodesy and models, paper presented at GGOS Unified Analysis Workshop, Int. Earth Rotation and Ref. Syst. Serv., Zurich, Switzerland, 16 Sep. Rülke, A., R. Dietrich, M. Fritsche, M. Rothacher, and P. Steigenberger (2008), Realization of the Terrestrial Reference System by a reprocessed global GPS network, J. Geophys. Res., 113, B08403, doi:10.1029/ 2007JB005231. Schlüter, W., and D. Behrend (2007), The International VLBI Service for Geodesy and Astrometry (IVS): Current capabilities and future prospects, J. Geod., 81, 379–387, doi:10.1007/s00190-006-0131-z. Schmid, R., P. Steigenberger, G. Gendt, M. Ge, and M. Rothacther (2007), Generation of a consistent absolute phase center correction model for GPS receiver and satellite antennas, J. Geod., 81, 781–798, doi:10.1007/ s00190-007-0148-y. Schmid, R., X. Collilieux, F. Dilssner, R. Dach, and M. Schmitz (2010), Updated phase center corrections for satellite and receiver antennas, paper presented at IGS workshop, Int. GNSS Serv., Newcastle, U. K. Sella, G. F., S. Stein, T. H. Dixon, M. Craymer, T. S. James, S. Mazzotti, and R. K. Dokka (2007), Observations of glacial isostatic adjustment in

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“stable” North America with GPS, Geophys. Res. Lett., 34, L02306, doi:10.1029/2006GL027081. Sibthorpe, A., J. P. Weiss, N. Harvey, D. Kuang, and Y. Bar-Sever (2010), Empirical modeling of solar radiation pressure forces affecting GPS satellites, Abstract G54A-04 presented at 2010 Fall Meeting, AGU, San Francisco, Calif., 13–17 Dec. van den Broeke, M., J. Bamber, J. Ettema, E. Rignot, E. Schrama, W. Jan van de Berg, E. van Meijgaard, I. Velicogna, and B. Wouters (2009), Partioning recent Greenland mass loss, Science, 326, 984–986, doi:10.1126/ science.1178176. Williams, S. D. P. (2008), CATS: GPS coordinate time series analysis software, GPS Solut., 12, 147–153, doi:10.1007/s10291-007-0086-4. Williams, S. D. P., Y. Bock, P. Fang, P. Jamason, R. M. Nikolaidis, L. Prawirodirdjo, M. Miller, and D. J. Johnson (2004), Error analysis of continuous GPS position time series, J. Geophys. Res., 109, B03412, doi:10.1029/2003JB002741. Willis, P., L. Soudarin, C. Jayles, and L. Rolland (2007), DORIS applications for solid earth and atmospheric sciences, C. R. Geosci., 339, 949–959, doi:10.1016/j.crte.2007.09.015. Woodworth, P. L., and R. Player (2003), The Permanent Service for Mean Sea Level: An update to the 21st century, J. Coastal Res., 19, 287–295. Wu, X., M. B. Heflin, H. Schotman, B. L. A. Vermeersen, D. Dong, R. S. Gross, E. R. Ivins, A. W. Moore, and S. E. Owen (2010), Simultaneous estimation of global present-day water transport and glacial isostatic adjustment, Nat. Geosci., 3, 642–646, doi:10.1038/ngeo938. Wu, X., X. Collilieux, Z. Altamimi, B. L. A. Vermeersen, R. S. Gross, and I. Fukumori (2011), Accuracy of the International Terrestrial Reference Frame origin and Earth expansion, Geophys. Res. Lett., 38, L13304, doi:10.1029/2011GL047450.

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