from GPS observations omitting Chatham island and Guadalupe island, which is the angular velocity that they prefer. (In GEODVEL Chatham island and Guadalupe island are on the Pacific Plate.) The GEODVEL angular velocity differs insignificantly by 0.009° Myr⁻¹ from the one that Plattner et al. (2007) estimate using Chatham island, but differs significantly ($\chi^2 = 32.3, p = 4.5 \times 10^{-7}$) by 0.022° Myr⁻¹ from the one that Plattner et al. (2007) estimate using Guadalupe island. Plattner et al. (2007) assume the velocity of Earth's centre to be that in ITRF2000.

*Geology.* The GEODVEL North America–Pacific angular velocity differs significantly ($\chi^2 = 14.7, p = 0.0021$) from that of NUVEL-1A by 0.044° Myr⁻¹. The NUVEL-1A North America–Pacific angular velocity is likely biased because it comes from the biased circum-Pacific and Gulf of California data and the plate circuit going through an assumed single Africa Plate.

The GEODVEL angular velocity differs significantly ($\chi^2 = 61.0, p = 3.6 \times 10^{-13}$) by 0.094° Myr⁻¹ from the North America–Pacific angular velocity in our updated geological model. This anomaly 2A (3.2 Ma) angular velocity (4.0°N, 67.2°W, 0.792° Myr⁻¹) is determined from the plate circuit North America–Nubia–Antarctica–Pacific. The GEODVEL angular velocity also differs significantly ($\chi^2 = 39.3, p = 1.5 \times 10^{-8}$) from the one (50.8°N, 64.4°W, 0.694° Myr⁻¹) that we infer from the anomaly five rotation in a plate circuit: North America–Nubia anomaly 5y (9.8 Ma, Müller et al. 1999), Nubia–Antarctica anomaly 5o (11.0 Ma, Royer et al. 2006) and Antarctica–Pacific anomaly 5o (11.04 Ma, Cande & Stock 2004).

Along the spreading centre in the Gulf of California at 23.5°N, 108.5°W, GEODVEL indicates that the Pacific–North America Plate velocity is $52.4 \pm 0.9$ mm yr⁻¹ towards N54.5°W $\pm 1.0°$. This azimuth parallels the loosely constrained direction of Pacific–North America Plate motion inferred from transform azimuths and earthquake slip vectors in the Gulf. The rate is 4 mm yr⁻¹ greater than the mean spreading rate since 0.8 Ma (anomaly 1o) (DeMets 1995). Spreading rates corrected for outward displacement (DeMets & Wilson 2008) are: 48.5 mm yr⁻¹ (1o, 0.78 Ma), 48.2 (Jaramillo, 1.03 Ma), 46.0 (2, 1.86 Ma), 44.4 (2Ay, 2.58 Ma) and 44.5 mm yr⁻¹ (2Ao, 3.58 Ma). The difference in rate from the Pacific–North America rate indicated by GEODVEL is presumably because seafloor spreading in the Gulf of California records motion relative to the North America Plate of southern Baja California, which Plattner et al. (2007) find to be moving southeast relative to the Pacific Plate at 4–6 mm yr⁻¹.

# 7 DISCUSSION

## 7.1 Optimal method for estimating plate motions from geodesy

The velocity of Earth's centre is important for estimating accurate relative plate velocities. The approach used herein, applying the assumption of (lateral) plate rigidity to determine the velocity of Earth's centre, provides the best and most robust approach for estimating that velocity. Most differences between GEODVEL and prior sets of angular velocities are due to differences in the velocity of Earth's centre. The velocity of CE in GEODVEL is consistent with ITRF2000 and the velocity of CM in CSR00L01, but inconsistent with ITRF2005 and ITRF1997.

Also critical for estimating relative plate angular velocities of some plate pairs is the choice of what sites to include as lying on the stable interior of the plate without being affected by glacial

unloading or rebound or by plate boundary zone deformation. This is most important at five places: Algonquin Park, Onsala, Tromso, Metsahovi and Kiruna. All five have significant horizontal velocities due to glacial isostatic adjustment and also have velocities that are well constrained.

The difference between GEODVEL and other recent geodetic sets of plate angular velocities is small ($\gtrsim 0.02°$ Myr⁻¹), however, relative to the difference between GEODVEL and our geological set of angular velocities, as is discussed below.

## 7.2 Plate stability and rigidity

With our carefully chosen set of sites on the plate interiors and with the velocity of Earth's centre adopted herein, we find no evidence for measurable lateral deformation of the stable plate interiors not near late Pleistocene ice sheets. The size of the weighted room-mean square horizontal residual velocities is merely 0.52 and $0.57$ mm yr⁻¹ for the two plates (respectively North America and Eurasia) with the best geodetic data. This suggests that the displacement rate of any one site relative to the stable continental plate interior cannot be more than $\approx 1$ or $2$ mm yr⁻¹ for the sites with the best data and may be much less. The plates that are well populated with data but with the largest residuals are the Pacific Plate and the Nubia Plate with weighted root-mean square residuals respectively of 0.99 and 1.08 mm yr⁻¹, about twice as large as for North America and Eurasia. It is unclear whether residuals are larger for these two plates merely because the uncertainties in the velocities are larger or because the Pacific and Nubia plates deform more rapidly than the North America and Eurasia plates.

Argus & Gordon (1996) found that five North America sites and two Eurasia sites had upper bounds (at a 95 per cent confidence level) on their motion relative to their respective stable plate interior of $2$ mm yr⁻¹ or less. No site had an upper bound of $1$ mm yr⁻¹ or less. In GEODVEL we find that 15 North America places, 32 Eurasia places, nine Australia places, six Pacific places, five Antarctica places, three South America places, and two Nubia places have upper bound velocities of $2$ mm yr⁻¹ or less. Four North America places, eight Eurasia places, one Australia place and one South America place have upper bounds of $1$ mm yr⁻¹ or less.

Thus, the results presented here imply a slightly smaller upper bound on average deformation rates of stable continental plate interiors than the upper bound of $10^{-17}$ s⁻¹ (or, equivalently, $3 \times 10^{-4}$ Myr⁻¹) found by Gordon (1998) from the results of Argus & Gordon (1996).

## 7.3 Differences between GEODVEL and geological angular velocities

GEODVEL and our geological 2A model differ significantly for all but two of the 36 plate pairs formed between the nine largest plates (Antarctica, Australia, Eurasia, Nubia, North America, Pacific, Nazca, South America and Somalia). The median vector difference is 0.063° Myr⁻¹, which is up to $\approx 7$ mm yr⁻¹ at Earth's surface (Table 9b, lower left-hand side).

Among the 17 plate pairs sharing a boundary, the largest vector differences are 0.147° Myr⁻¹ (Nazca–South America), 0.102° Myr⁻¹ (Nazca–Antarctica), 0.094 Myr⁻¹ (Pacific–North America), 0.085° Myr⁻¹ (Nazca–Pacific) and 0.084° Myr⁻¹ (Antarctica–South America). Three of the five plate pairs include the Nazca Plate, reflecting its decelerating motion (Norabuena et al. 1998).
The largest well-constrained change in pole position for a plate pair with a common boundary is that for Nubia–Eurasia, with the GEODVEL pole of rotation lying 19.4° southwest of the geological pole. The two angular velocities differ by a vector of length 0.022° Myr⁻¹.

A large (15° to 20° ± 7°) clockwise change in direction of motion is indicated for Somalia–Antarctica motion.

The median angular difference between GEODVEL and geological relative angular velocities for the nine plate pairs sharing a mid-ocean ridge plate boundary is 0.044° Myr⁻¹. Eight spreading centres have slowed down while only two (Australia–Antarctica by 0–2 mm yr⁻¹ and Pacific–Antarctica by 1–3 mm yr⁻¹) have sped up. The largest decreases in spreading rate (by 4–10 mm yr⁻¹) involve the Nazca Plate (Nazca–Pacific spreading has slowed down by 4–8 mm yr⁻¹ while Nazca–Antarctica spreading has slowed down by 9–10 mm yr⁻¹). Antarctica–South America spreading has slowed down by 5–9 mm yr⁻¹. Other spreading rate decreases are more modest and range from 1 to 4 mm yr⁻¹. The inferred slowing for all plate pairs would be less than we found here by ≈1 mm yr⁻¹, however, if we had corrected for the outward displacement of reversal boundaries documented by DeMets & Wilson (2008).

This median angular velocity difference for spreading centres is about half of 0.084° Myr⁻¹, which is the median angular velocity difference for the five plate pairs sharing a circum-Pacific subducting plate boundary, which range from 0.032° Myr⁻¹ (Pacific–Australia) to 0.147° Myr⁻¹ (Nazca–South America). This suggests that systematic errors may accumulate in the geological plate circuit (Eurasia)–North America–Nubia–Antarctica–Pacific–(Nazca), perhaps partly due to horizontal thermal contraction of oceanic lithosphere (Kumar & Gordon 2009).

## 8 CONCLUSIONS

(1) Estimates of plate angular velocity depend on the estimate of the velocity of Earth's centre. A change in the velocity of Earth's centre of 1 mm yr⁻¹ typically results in a change in a plate angular velocity of 0.012° Myr⁻¹.

(2) Ice sheet loss, fluctuations of the ocean and atmosphere, and other phenomena generate a velocity between CM and CE of less than a few tenths of millimetres per year (Argus (2007); also see Appendix A). Given that the velocity of CM differs by 1.8 mm yr⁻¹ between ITRF2005 and ITRF2000, the velocity of CM estimated by SLR is not constrained well enough to reliably estimate plate angular velocities. The velocity of CE estimated from the velocities of places on Earth's surface is probably closer to the true velocity of CM than the velocity estimated using SLR.

(3) In a stable plate interior reference frame based on CE, places beneath and along the margins of the former ice sheets move horizontally away from their ice centres at up to 1.5 mm yr⁻¹ in viscous response to unloading of the ice sheets while places not near the late Pleistocene ice sheets move horizontally at less than ≈0.5 mm yr⁻¹ (weighted rms velocity).

(4) In GEODVEL we simultaneously estimate the velocity of CE and the angular velocity of the plates assuming that places not near the late Pleistocene ice sheets are moving with their respective plates relative to CE. The uncertainties in plate angular velocities incorporate the uncertainty in the velocity of CE.

(5) GEODVEL differs from other geodetic estimates of plate angular velocity partly because the velocity of Earth's centre differs

between studies, partly because we assign places to plates differently than others, and partly because the input data differ.

(a) GEODVEL differs substantially from REVEL (Sella *et al.* 2002) (by 0.028° Myr⁻¹ median vector difference) mainly because the velocity of Earth's centre differs greatly between the two studies.

(b) GEODVEL differs moderately from Kogan & Steblov (2008) (by 0.019° Myr⁻¹), partly because they assign to plates places moving horizontally in viscous response to unloading of the late Pleistocene ice sheets, and partly because the input data differ.

(c) GEODVEL differs moderately from Altamimi *et al.* (2007) (by 0.015° Myr⁻¹), partly because they assume the velocity of Earth's centre to be that of CM in ITRF2005, partly because they assign to plates places moving horizontally in viscous response to unloading of the late Pleistocene ice sheets, and partly because the input data differ.

(d) GEODVEL differs moderately from ITRFVEL (by 0.014° Myr⁻¹) entirely because the input data differ.

(6) Geodetic relative plate angular velocities and geological relative plate angular velocities averaged over the past 3.16 Myr differ significantly and by an amount that is roughly twice the difference between different geodetic estimates. The vector difference between a GEODVEL angular velocity and the corresponding geological angular velocity ranges from 0.019 to 0.147° Myr⁻¹ with a median of 0.063° Myr⁻¹. Many more changes in angular velocity indicate a slowing down of relative plate motion than indicate a speeding up of relative plate motion. The Nazca Plate, in particular, has noticeably slowed down in its eastward motion relative to its neighbours. The motion between eight of 10 plate pairs with a spreading plate boundary has slowed.

## ACKNOWLEDGMENTS

We are grateful to Saskia Goes, Mikhail Kogan, and an anonymous reviewer for their careful reviews. DFA performed research at Jet Propulsion Laboratory, under contract with the National Aeronautics and Space Administration (NASA). RGG performed research partly supported by National Science Foundation grants OCE-0453219 and OCE-0928961. PW performed research sponsored by the Centre National d'Etudes Spatiales (CNES); this paper is Institut de Physique du Globe de Paris contribution IPGP-2576. We created the colour topographic maps in Fig. 5 using program 'gclr' (written by R.W. Simpson, U.S. Geological Survey, Menlo Park, CA).

## REFERENCES

Altamimi, Z., Sillard, P. & Boucher, C., 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., Collilieux, X., Legrand, J., Garayt, B. & Boucher, C., 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**, B004949, doi:10.1029/2007JB004949.

Argus, D.F., 1996. Postglacial uplift and subsidence of earth's surface using VLBI geodesy: on establishing vertical reference, *Geophys. Res. Lett.*, **23**, 973–976.

Argus, D.F., 2007. Defining the translational velocity of the reference frame of Earth, *Geophys. J. Int.*, **169**, 830–838, doi:10.1111/j.1365-246X.2007.03344.x.

Argus, D.F. & Gordon, R.G., 1990. Comparison of Pacific–North America plate motion determined from very long baseline interferometry with that
determined from magnetic anomalies, transform faults, and earthquake slip vectors, *J. geophys. Res.*, **95**, 17315–17324.

Argus, D.F. & Gordon, R.G., 1991. No-net-rotation model of current plate velocities incorporating plate motion model NUVEL-1, *Geophys. Res. Lett.*, **18**, 2039–2042.

Argus, D.F. & Gordon, R.G., 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. & Gordon, R.G., 2001. Present tectonic motion across the Coast ranges and San Andreas fault system in central California, *Geol. Soc. Am. Bull.*, **113**, 1580–1592.

Argus, D.F. & Lyzenga, G.A., 1994. Site velocities before and after the Loma Prieta and Gulf of Alaska earthquakes estimated from VLBI, *Geophys. Res. Lett.*, **21**, 333–336.

Argus, D.F., Peltier, W.R. & Watkins, M.M., 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., Heflin, M.B., Peltzer, G., Webb, F.H. & Crampe, F., 2005. In-terseismic strain accumulation and anthropogenic motion in metropolitan Los Angeles, *J. geophys. Res.*, **101**, B04401, doi:10.1029/2003JB002934.

Beavan, J., Tregoning, P., Bevis, M., Kato, T. & Meertens, C., 2002. Motion and rigidity of the Pacific Plate and implications for plate boundary deformation, *J. geophys. Res.*, **107**(B10), 2261, doi:10.1029/2001JB000282.

Bevington, P., 1969. *Data Reduction and Error Analysis for the Physical Sciences*, McGraw-Hill Book Company, New York.

Bird, P., 2003. An updated digital model of plate boundaries, *Geochem. Geophys. Geosys.*, **4**, doi:10.1029/2001GC000252.

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.

Blewitt, G., Lavalee, D., Clarke, P. & Nurutdinov, K., 2001. A new global mode of Earth deformation: seasonal cycle detected, *Science*, **294**, 2342–2345.

Boucher, C., Altamimi, Z. & Sillard, P., 1998. *The 1997 International Terrestrial Reference Frame (ITRF97)*, International Earth Rotation and Reference Systems Service (IERS) Technical Note 27, Frankfurt-am-Main, Germany.

Boucher, C., Altamimi, Z., Sillard, P. & Feissel-Vernier, M., 2004. *The ITRF2000, International Earth Rotation and Reference Systems Service (IERS) Technical Note 31*, Frankfurt-am-Main, Germany.

Bouin, M. & Vigny, C., 2000. New constraints on Antarctic plate motion and deformation from GPS data, *J. geophys. Res.*, **105**, 28 279–28 293, doi:10.1111/j.1365-246X.2009.04087.x.

Briole, P., Willis, P., Dubois, J. & Charade, O., 2009. Potential volcanological applications of the DORIS system: a geodetic study of Socorro island (Mexico), *Geophys. J. Int.*, **176**, 581–590, doi: 10.1111/j.1365-246X.2009.04087.x.

Calais, E., DeMets, C. & Nocquet, J.M., 2003. Evidence for a post-3.16-Ma change in Nubia Eurasia North America plate motions?, *Earth planet. Sci. Lett.*, **216**, 81–92.

Calais, E., Harbinger, C., Hartnady, C. & Nocquet, J.M., 2006. Kinematics of the East African Rift from GPS and earthquake slip vector data, *Geol. Soc. Special Publ.*, **259**, 9–22, doi:10.1144/GSL.SP.206.259.01.03.

Calmant, S., Pelletier, B., Lebellegard, P., Bevis, M., Taylor, F.W. & Phillips, D.A., 2003. New insights on the tectonics along the New Hebrides subduction zone based on GPS results, *J. geophys. Res.*, **108**(B6), 2319, doi:10.1029/2001JB000644.

Cande, S.C. & Kent, D.V., 1992. A new geomagnetic polarity time scale for the late Cretaceous and Cenozoic, *J. geophys. Res.*, **97**, 13 917–13 951.

Cande, S.C. & Kent, D.V., 1995. Revised calibration of the geomagnetic polarity time scale for the late Cretaceous and Cenozoic, *J. geophys. Res.*, **100**, 6093–6095.

Cande, S.C. & Stock, J.M., 2004. Pacific-Antarctic-Australia motion and the formation of the Macquarie Plate, *Geophys. J. Int.*, **157**, 399–414, doi:10.1111/j.1365-246X.2004.02224.x.

Chase, C.G., 1978. Plate kinematics: the Americas, East Africa, and the rest of the world, *Earth planet. Sci. Lett.*, **337**, 355–368.

Chu, D. & Gordon, R.G., 1999. Evidence for motion between Nubia and Somalia along the southwest Indian ridge, *Nature*, **398**, 64–67.

Clark, T.A., Ma, C., Sauber, J.M., Ryan, J.W., Gordon, D., Shaffer, D.B., Caprette, D.S. & Vandenberg, N.R., 1990. Geodetic measurement of deformation in the Loma Prieta, California, earthquake with very long baseline interferometry, *Geophys. Res. Lett.*, **18**, 1215–1218.

Darby, D.J. & Meertens, C.M., 1995. Terrestrial and GPS measurements of deformation across the Taupo back-arc and Hikurangi fore-arc regions in New Zealand, *J. geophys. Res.*, **100**, 8221–8232.

DeMets, C., 1995. A reappraisal of seafloor spreading lineations in the Gulf of California: implications for the transfer of Baja California to the Pacific plate and estimates of Pacific-North America motion, *Geophys. Res. Lett.*, **24**, 3545–3548.

DeMets, C. & Dixon, T.H., 1999. New kinematic models for Pacific-North America motion from 3 Ma to present: I: evidence for steady motion and biases in the NUVEL-1A model, *Geophys. Res. Lett.*, **26**, 1921–1924.

DeMets, C. & Wilson, D.S., 2008. Toward a minimum change model for recent plate motions: calibrating seafloor spreading rates for outward displacement, *Geophys. J. Int.*, **174**, 825–841, doi:10.1111/j.1365-246X.2008.03836.x.

DeMets, C., Gordon, R.G., Argus, D.F. & Stein, S., 1990. Current plate motions, *Geophys. J. Int.*, **101**, 425–478.

DeMets, C., Gordon, R.G., Argus, D.F. & Stein, S., 1994. Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions, *Geophys. Res. Lett.*, **21**, 2191–2194.

Dietrich, R., Rulke, A. & Scheinert, M., 2005. Present-day vertical crustal deformation in West Greenland from repeated GPS observations, *Geo-phys. J. Int.*, **163**, 865–874, doi:10.1111/j.1365-246X.2005.02766.x.

Dong, D., Yunck, T. & Heflin, M., 2002. Origin of International Terrestrial Reference Frame, *J. geophys. Res.*, **108**(B4), 2200, doi:10.1029/2002JB002035.

Dow, J.M., Neilan, R.E. & Rizos, C., 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.

Engdahl, E.R, Van Der Hilst, R. & Buland, R., 1998. Global teleseismic earthquake relocation with improved travel times and procedures for depth determination, *Bull. seism. Soc. Am.*, **88**, 722–743.

Estabrook, C.H., Stone, D.B. & Davies, J.N., 1988. Seismotectonics of northern Alaska, *J. geophys. Res.*, **93**, 12 026–12 040.

Fagard, H., 2006. Twenty years of evolution for the DORIS permanent network: from its initial deployment to its renovation, *J. Geod.*, **80**, 429–456, doi:10.1007/s00190-006-0084-2.

Geirsson, H. et al., 2006. Current plate movements across the Mid-Atlantic ridge determined from 5 years of continuous GPS measurements in Iceland, *J. geophys. Res.*, **111**, B09407, doi:10.1029/2005JB003717.

Gordon, R.G., 1998. The plate tectonic approximation: plate nonrigidity, diffuse plate boundaries, and global reconstructions, *Ann. Rev. Earth planet. Sci.*, **26**, 615–642, doi:10.1146/annurev earth.26.1.615.

Gordon, R.G., DeMets, C. & Argus, D.F., 1990. Kinematic constraints on distributed lithospheric deformation in the equatorial Indian Ocean from present motion between the Australian and Indian plates, *Tectonics*, **9**, 409–422.

Gordon, R.G., Royer, J.-Y. & Argus, D.F., 2008. Space geodetic test of kinematic models for the Indo-Australian composite plate, *Geology*, **36**, 827–830, doi:10.1130/G25089A.1.

Greff-Leffitz, M., 2000. Secular variation of the geocenter, *J. geophys. Res.*, **105**, 25 685–25 692.

Gross, R.S. & Vondrak, J., 1999. Astrometric and space-geodetic observations of polar wander, *Geophys. Res. Lett.*, **26**, 2085–2088.

Hagedoorn, J.M. & Wolf, D., 2003. Pleistocene and Recent deglaciation in Svalbard: implications for tide-gauge, GPS and VLBI measurements, *J. Geodyn.*, **35**, 415–423.

Hammond, W.C. & Thatcher, W., 2004. Contemporary tectonic deformation of the Basin and Range province, western United States: 10 years of observation with the Global Positioning System, *J. geophys. Res.*, **109**, B08403, doi:10.1029/2003JB002746.

Heki, K., 1996. Horizontal and vertical crustal movements from three-dimensional very long baseline interferometry kinematic reference frame:
implication for the reversal time scale revision, *J. geophys. Res.*, **101**, 3187–3198.

Hilgen, F.J., 1991. Extension of the astronomically calibrated (polarity) time scale to the Miocene/Pliocene boundary, *Earth planet. Sci. Lett.*, **107**, 349–368.

Horner-Johnson, B.C., Gordon, R.G., Cowles, S.M. & Argus, D.F., 2005. The angular velocity of Nubia relative to Somalia and the location of the Nubia-Somalia-Antarctic triple junction, *Geophys. J. Int.*, **162**, 221–238, doi:10.1111/j.1365-246X.2005.02608.x.

Horner-Johnson, B.C., Gordon, R.G. & Argus, D.F., 2007. Plate kinematic evidence for the existence of a distinct plate between the Nubia and Somalian plates along the Southwest Indian Ridge, *J. geophys. Res.*, **112**, B05418, doi:10.1029/2006JB004519.

Hurst, K.J. et al., 2000. The coseismic signature of the 1999 Hector Mine earthquake, *Geophys. Res. Lett.*, **27**, 2733–2736.

Jackson, J. & McKenzie, D., 1988. The relationship between plate motions and seismic moment tensors, and the rates of active deformation in the Mediterranean and Middle East, *Geophys. J.*, **93**, 45–73.

Jestin, F., Huchon, P. & Gaulier, J.M., 1994. The Somalia plate and the east African rift system—present-day kinematics, *Geophys. J. Int.*, **116**, 637–654.

Johansson, J.M. et al., 2002. Continuous GPS measurements of post-glacial adjustment in Fennoscandia: 1. Geodetic results, *J. geophys. Res.*, **107**(B8), 2157, doi:10.1029/2001JB000400.

Khan, S.A., Wahr, J., Leuliette, E., van Dam, T., Larson, K.M. & Francis, O., 2008. Geodetic measurement of postglacial adjustments in Greenland, *J. geophys. Res.*, **113**(B2), B02402, doi:10.1029/2007JB004956.

Kogan, M.K. & Steblov, G.M., 2008. Current global kinematics from GPS (1995–2007) with the plate-consistent reference frame, 2008, *J. geophys. Res.*, **113**, B04416, doi:10.1029/2007JB005353.

Kohler, J., James, T.D., Murray, T., Nuth, C., Brandt, O., Barrand, N.E., Aas, H.F. & Luckman, A., 2007. Acceleration in thinning rate on western Svalbard glaciers, *Geophys. Res. Lett.*, **34**, L18502, doi:10.1029/2007GL030681.

Kreemer, C., Holt, W.E. & Haines, A.J., 2003. An integrated global model of present-day plate motions and plate boundary deformation, *Geophys. J. Int.*, **154**, 8–34.

Kreemer, C., Lavallee, D.A., Blewitt, G. & Holt, W.E., 2006. On the stability of a geodetic no-net-rotation reference frame and its implication for the International Terrestrial Reference Frame, *Geophys. Res. Lett.*, **33**, L17306, doi:10.1029/2006GL027058.

Kumar, R.R. & Gordon, R.G., 2009. Horizontal thermal contraction of oceanic lithosphere: the ultimate limit to the rigid plate approximation, *Geophys. Res.*, **114**, B01403, doi:10.1029/2007JB005473.

Langbein, J. & Johnson, H., 1997. Correlated errors in geodetic time series: implications for time-dependent deformation, *J. geophys. Res.*, **102**, 591–603.

Lemaux, J., Gordon, R.G. & Royer, J.-Y., 2002. The location of the Nubia–Somalia boundary along the Southwest Indian Ridge, *Geology*, **30**, 339–342.

Lourens, L., Hilgen, F.J., Laskar, J., Shackleton, N.J. & Wilson, D., 2004. The Neogene Period, in *A Geologic Time Scale 2004*, pp. 409–440, eds Gradstein, F., Ogg, J. & Smith, A., Cambridge University Press, London.

Mahmoud, S., Reilinger, R., McClusky, S., Vernant, P. & Tealab, A., 2005. GPS evidence for northward motion of the Sinai Block: implications for E. Mediterranean tectonics, *Earth planet. Sci. Lett.*, **238**, 217–224.

Mantovani, E., Viti, M., Babbucci, D. & Albarello, D., 2007. Nubia-Eurasia kinematics: an alternative interpretation from Mediterranean and North Atlantic evidence, *Ann. Geophys.*, **50**, 341–366.

Mao, A.L., Harrison, C.G. A. & Dixon, T.H., 1999. Noise in GPS coordinate time series, *J. geophys. Res.*, **104**, 2797–2816.

McKenzie, D. & Jackson, J., 1983. The relationship between strain rates, crustal thickening, paleomagnetism, finite strain, and fault movements within a deforming zone, *Earth planet. Sci. Lett.*, **65**, 182–202.

Meghraoui, M., Delouis, B., Ferry, M., Giardini, D., Huggenberger, P., Spotke, I. & Granet, M., 2001. Active normal faulting in the upper Rhine

graben and paleoseismic identification of the 1356 Basel earthquake, *Science*, **293**, 2070–2073.

Merkouriev, S. & DeMets, C., 2008. A high-resolution model for Eurasia-North America plate kinematics since 20 Ma, *Geophys. J. Int.*, **173**, 1064–1083.

Minster, J.B. & Jordan, T.H., 1978. Present-day plate motions, *J. geophys. Res.*, **383**, 5331–5354.

Müller, R.D., Royer, J.-Y., Cande, S.C., Roest, W.R. & Maschenkov, S., 1999. New constraints on the Late Cretaceous/Tertiary plate tectonic evolution of the Carribean, in *Caribbean Basins*, Vol. 4: Sedimentary Basins of the World, pp. 33–59, ed. Mann, P., Elsevier Science, Amsterdam.

Norabuena, E., Leffler-Griffin, L., Mao, A.L., Dixon, T., Stein, S., Sacks, I.S., Ocola, L. & Ellis, M., 1998. Space geodetic observations of Nazca-South America convergence across the central Andes, *Science*, **279**, 358–362.

Pagli, C., Sigmundsson, F., Lund, B., Sturkell, E., Geirsson, H., Einarsson, P., Arnadottir, T. & Hreinsdottir, S., 2007. Glacio-isostatic deformation around Vatnajokull ice cap, Iceland, induced by recent climate warming: GPS observations and finite element modeling, *J. geophys. Res.*, **112**, B08405, doi:10.1029/2006JB004421.

Patriat, P., Sloan, H. & Sauter, D., 2008, From slow to ultraslow: a previously undetected event at the Southwest Indian Ridge at ca. 24 Ma, *Geology*, **36**, 207–210.

Pearlman, M.R., Degnan, J.J. & Bosworth, J.M., 2002. The International Laser Ranging Service, *Adv. Space Res.*, **30**, 135–143.

Peltier, W.R., 1994. Ice age paleotopography, *Science*, **265**, 195–201.

Peltier, W.R., 1996. Mantle viscosity and ice-age ice sheet topography, *Science*, **273**, 1359–1364.

Peltier, W.R., 2004. Global glacial isostasy and the surface of the ice-age earth: the ICE-5G (VM2) model and GRACE, *Ann. Rev. Earth planet. Sci.*, **32**, 111–149.

Peltier, W.R. & Drummond, R., 2008. Rheological stratification of the lithosphere: a direct inference based upon the geodynamically observed pattern of glacial isostatic adjustment of the North American continent, *Geophys. Res. Lett.*, **35**, L16314, doi:10.1029/2008GL034586.

Petit, C. & Deverchere, J., 2006. Structure and evolution of the Baikal rift: a synthesis, *Geochem. Geophys. Geosyst.*, **7**, Q11016, doi:10.1029/2006GC01265.

Plattner, C., Malservisi, R., Dixon, T.H., LaFemina, P., Sella, G.F., Fletcher, J. & Suarez-Vidal, F., 2007. New constraints on relative motion between the Pacific Plate and Baja California microplate (Mexico) from GPS measurements, *Geophys. J. Int.*, **170**, 1373–1380, doi:10.1111/j.1365-246X.2007.03494.x.

Prawirodirdjo, L. & Bock, Y., 2004. Instantaneous global plate motion model from 12 years of continuous GPS observations, *J. geophys. Res.*, **109**, B08405, doi:10.1029/2003JB002944.

Roest, W.R. & Collette, B.J., 1986. The Fifteen Twenty Fracture Zone and the North American-South American plate boundary, *J. Geol. Soc.*, **43**, 833–843.

Roest, W.R. & Srivastava, S.P., 1991. Kinematics of the plate boundaries between Eurasia, Iberia, and Africa in the North-Atlantic from the late Cretaceous to the Present, *Geology*, **19**, 613–616.

Royer, J.-Y. & Gordon, R.G., 1997. The motion and boundary between the Capricorn and Australian plates, *Science*, **277**, 268–1274, doi:10.1126/ science.277.5530.1268.

Royer, J.-Y., Gordon, R.G. & Horner-Johnson, B.C., 2006. Motion of Nubia relative to Antarctica since 11 Ma: implications for Nubia-Somalia, Pacific-North America & India-Eurasia motion, *Geology*, **34**, 501–504.

Sato, T., Okuno, J., Hinderer, J., MacMillan, D.S., Plag, H.-P., Francis, O., Falk, R. & Fukuda, Y., 2006. A geophysical interpretation of the secular displacement and gravity rates observed at Ny-Alesund, Svalbard in the Arctic—effects of post-glacial rebound and present-day ice melting, *Geophys. J. Int.*, **165**, 729–743.

Sauber, J.M., Clark, T.A., Bell, L.J., Liswoski, M., Ma, C. & Caprette, D.S., 1993. Geodetic measurement of static displacement associated with the 1987–1988 Gulf of Alaska earthquakes, in *Contributions of Space Geodesy to Geodynamics: Crustal Dynamics*, Vol. 3, pp. 113–120, eds Smith, D.E. & Turcotte, D.L., America Geophysical Union.
Schlüter, W. & Behrend, D., 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.

Sella, G.F., Dixon, T.H. & Mao, A.L., 2002. REVEL: a model for Recent plate velocities from space geodesy, *J. geophys. Res.*, **107**(B4), 2081, doi:10.1029/2000JB000033.

Sella, G.F., Stein, S., Dixon, T.H., Craymer, M., James, T.S., Mazzotti, S. & Dokka, R.K., 2007. Observations of glacial isostatic adjustment in “stable” North America with GPS, *Geophys. Res. Lett.*, **34**, L02306, doi:10.1029/2006GL027081.

Smalley, R. et al., 2002. Geodetic determination of relative plate motion and crustal deformation across the Scotia-South America plate boundary in eastern Tierra del Fuego, *Geochem. Geophys., Geosyst.*, **4**, 1070, doi:1029/2002GC000446.

Soudarin, L. & Crétaux, J.F., 2006. A model of present-day tectonic plate motions from 12 years of DORIS measurements, *J. Geod.*, **80**, 609–624, doi:10.1007/s00190-006-0090-4.

Stamps, D.S., Calais, E., Saria, E., Hartnady, C., Nocquet, J.-M., Ebinger, C.J. & Fernandes, R.M., 2008. A kinematic model for the East African Rift, *Geophys. Res. Lett.*, **35**, L05304, doi:10.1029/2007GL032781.

Steblov, G.M., Kogan, M.K., King, R.W., Scholz, C.H., Burgmann, R. & Frolov, D.I., 2003. Imprint of the North American plate in Siberia revealed by GPS, *Geophys. Res. Lett.*, **30**, 1924, doi:10.1029/2003GL017805.

Stephenson, E.R. & Morrison, L.V., 1995. Long term fluctuations in the Earth's rotation: 700 B.C. to A.D. 1990, *Phil. Trans. R. Soc. Lond., Ser. A*, **351**, 165–202.

Tebbens, S.F. & Cande, S.C., 1997. Southeast Pacific tectonic evolution from early Oligocene to present, *J. geophys. Res.*, **102**, 12 061–12 084.

Thelin, G.P. & Pike, R.J., 1991. Landforms of the conterminous United States—a digital shaded-relief portrayal, *U.S. Geol. Surv. Map, I-2206*.

Titov, O. & Tregoning, P., 2005. Effect of post-seismic deformation on earth orientation parameter estimates from VLBI: a case study at Gilcreek, Alaska, *J. Geod.*, **79**, 196–202, doi:10.1007/s00190-005-0459-9.

Vicente, R.O. & Yumi, S., 1969. Co-ordinates of the pole (1899–1968) returned to the conventional international origin, *Publ. Int. Latit. Obs. Mizusawa*, **7**, 41–50.

Vogt, P.R. & Jung, W.Y., 2004. The Terceira Rift as hyper-slow, hotspot-dominated oblique spreading axis: a comparison with other slow-spreading plate boundaries, *Earth planet. Sci. Lett.*, **218**, 77–90.

Wahr, J., van Dam, T., Larson, K.M. & Francis, O., 2001. GPS measurement of vertical crustal motion in Greenland, *J. geophys. Res.*, **106**(D24), 33 755–33 759.

Wells, D.L. & Coppersmith, K.J., 1994. New emperical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement, *Bull. seism. Soc. Am.*, **84**, 974–1002.

Wenzel, M. & Schröter, J., 2007. The global ocean mass budget in 1993–2003 estimated from sea level change, *J. phys. Oceanogr.*, **37**, 203–213, doi:10.1175/JPO3007.1.

Williams, S.D.P., Bock, Y., Fang, P., Jamason, P., Nikolaidis, R.M., Prawirodirdjo, L., Miller, M. & Johnson, D.J., 2004. Error analysis of continuous GPS time series, *J. geophys. Res.*, **109**, B03412, doi:10.1029/2003JB002741.

Willis, P. & Ries, J.C., 2005. Defining a DORIS core network for Jason-1 precise orbit determination based on ITRF2000, methods and realization, *J. Geod.*, **79**, 370–378, doi:10.1007/s00190-005-0475-9.

Willis, P., Bar-Sever, Y.E. & Tavernier, G., 2005a. DORIS as a potential part of a global geodetic observing system, *J. Geodyn.*, **40**, 494–501, doi:j.jog.2005.06.011.

Willis, P., Boucher, C., Fagard, H. & Altamimi, Z., 2005b. Geodetic applications of the DORIS system at the French ‘Institut Géographique National’, *Compt. Rend. Geosci.*, **337**, 653–662, doi:10.1016/j.crte.2005.03.002.

Willis, P., Soudarin, L., Jayles, C. & Rolland, L., 2007. DORIS applications for solid earth and atmospheric sciences, *Compt. Rend. Geosci.*, **339**, 949–959, doi:10.1016/j.crte.2007.09.015.

Zumberge, J.F., Heflin, M.B., Jefferson, D.C., Watkins, M.M. & Webb, F.H., 1997. Precise point positioning for the efficient and robust analysis of GPS data from large networks, *J. geophys. Res.*, **102**, 5005–5017.

## APPENDIX A: ESTIMATES OF THE VELOCITY BETWEEN CM AND CE

In the postglacial rebound model of Peltier (2004), CM is moving relative to CE at 0.017 mm yr⁻¹ (W. R. Peltier, electronic communication 2008). This speed is less than the rough upper bound of 0.034 mm yr⁻¹ computed by Argus (2007, Appendix F).

In the ocean circulation model of Wenzel & Schröter (2007), the mass centre of the oceans moved relative to CE 9.8 m towards (36°N 15°E) the centre of the Mediterranean sea from 1993 to 2004. But, because the mass of the oceans is 5000 times less than the mass of Earth, CM moved relative to CE just 2.0 mm over the 11 yr, giving a rate of 0.18 mm yr⁻¹.

If Antarctica were losing ice fast enough to raise global sea level 1 mm yr⁻¹, then CM would be moving relative to CE towards the North pole at 0.35 mm yr⁻¹ (Argus 2007, Appendix E).

Greff-Lefftz (2000) maintains that the viscous response to unloading of the late Pleistocene ice sheets can cause the mean position of Earth's surface (CF) to be moving relative to CM at 0.2–0.5 mm yr⁻¹, suggesting that CE may be moving relative to CM at about this speed. In the postglacial rebound models of Peltier (1994, 1996, 2004), however, CF is moving relative to CE, respectively, at 0.02, 0.03 and 0.15 mm yr⁻¹, suggesting that the estimates of Greff-Lefftz (2000) are too high.

## APPENDIX B: ANALYSIS OF OBSERVATIONS FROM THE FOUR SPACE TECHNIQUES

### B.1 GPS

The GPS velocity model consists of estimates of the velocities of 662 sites (128 in Category Rigid, 39 in Category GIA and 495 in Category Boundary). A GPS site is a monument with a distinct DOMES number.

We determine this GPS velocity model from the estimates of position each day between 1991 January and 2007 August that scientists at Jet Propulsion Laboratory determine using the point positioning method of Zumberge et al. (1997) and the GIPSY data analysis algorithms. JPL first estimates the positions of 24 satellites as a function of time in a day using observations between the satellites and 50–100 ground sites. JPL next estimates, assuming these satellite orbits to be exact, the positions of up to roughly 900 ground sites. M. B. Heflin (at JPL) next transforms these position estimates by a translation and rotation into a reference frame in which the sites have a constant velocity except for offsets that he specifies. This is JPL's 'legacy series' (http://sideshow.jpl.nasa.gov/mbh/series.html), which is the laboratory's best complete position time-series as we prepare this study.

We next fit these estimates of position on each day with a position (three parameters, one for each of the east, north and up components), a velocity (three parameters, one for each component), offsets where and when needed (three parameters for each offset) and a sinusoid having a period of 1 yr (four parameters, an amplitude for each component and a phase identical for all three components). Requiring the phase to be identical for the three components means that, neglecting the velocity, the site moves each year back and forth along a line segment.

We next quote statistics for the 167 sites in Categories Rigid and GIA. We omit GPS estimates of position during 18 time intervals (16 because of failing antennas, two because of postseismic transients).
We estimate 83 offsets (57 at the times of antenna substitutions, five at the times of great earthquakes and 31 at times of neither an antenna substitution nor an earthquake). We thus estimate on average about one-half offset per site.

Four offsets (at iisc, ban2, mald and hyde) are due to the 2004 December 26 M 9.2 Sumatra earthquake. One offset (at hob2) is due to the 2004 December 24 M 8.1 Macquarie island earthquake. At iisc and ban2 we furthermore omitted 12 months of GPS estimates after the Sumatran earthquake because a postseismic transient may bias them. (We estimate an earthquake offset earthquake if the standard elastic dislocation model predicts a site moved more than 2 mm.)

A GPS antenna sometimes begins to fail, causing the GPS estimate of position to be wrong by tens of mm; scientists next substitute a good antenna for the bad one, and the GPS estimate of position becomes correct. In 16 such instances we delete estimates of position during time intervals one to 13 months long. At madr we delete 42 months of data because the antenna splitter failed. At hers we delete 41 months of data because the antenna failed.

Scientists often substitute one GPS antenna for another. We compute the height of the new phase centre to be

$$ (L_1 \times f_1^2 - L_2 \times f_2^2) / (f_1^2 - f_2^2), \quad (B1) $$

where $L_1$ and $L_2$ are heights (in m) of the phase centres of GPS's two radio frequencies, $f_1 = 154 \times f_0$ and $f_2 = 120 \times f_0$, where $f_0 = 10.23$ million cycles s⁻¹. We use this computed height to correct the GPS series. The GPS position nevertheless in many cases appears to be offset, in most cases in the vertical, but in some cases in the horizontal.

We choose whether to adopt an estimated offset mainly from the size of the estimated offset. We also consider whether there is a record of the antenna substitution in the log file and how much the fit improves if an estimated offset is used. Generally, we estimate an offset if the estimate of the offset exceeds about 10 mm in the vertical or 5 mm in the horizontal. (These threshold values of 5 and 10 mm are 25 per cent greater than the median root mean square dispersion of 4 mm for the east and north components and 8 mm for the up component). In close cases we estimate an offset if there is a record of an antenna substitution, but do not if there is not. In most cases estimating the offset reduces the root mean square dispersion (in mm) of the position estimates about the fit by more than 5 per cent.

We estimate offsets at the times of 57 of 175 antenna substitutions, roughly one-third of them.

## B.2 SLR

Richard J. Eanes (Center for Space Research, University of Texas at Austin) determined SLR velocity model CSR00L01 from 24 yr of observations beginning 1976 May and ending in 2000 March. From this model we use estimates of the velocities of 46 sites (19 in Category Rigid, one in Category GIA and 26 in Category Boundary), all of which meet the criteria that a site have observations over at least 3 yr and that the standard error in the two horizontal components of velocity be less than 3 mm yr⁻¹.

The velocity of each SLR site is estimated from observations at between one and seven lasers having observations over at least 2 yr. At Greenbelt observations at seven phase centres contribute to the velocity of the site; the effective time interval of observations at Greenbelt is 25.5 yr, which is the root sum square of 19.0 yr (at grf105), 12.9 yr (grf130), 7.3 yr (grf918), 5.4 yr (grf920), 4.3 yr (grf102), 3.7 yr (grf103) and 3.2 yr (grf101). At Wettzell observations at three phase centres contribute; the effective time interval of

observations at Wettzell is 19.6 yr, which is the root sum square of 12.6 yr (at wetzel), 11.8 yr (wet597) and 9.2 yr (wetz12). At each of six sites (Macdonald, Orroral, Arequipa, Matera, Potsdam and Kootwijk) two phase centres contribute; for example, the effective time interval of observations at Macdonald Observatory is 14.7 yr, which is the root sum square of 12.1 yr (at mcdon4) and 8.3 yr (at mcdon5). Lasers at a site are less than 1000 m apart (except at orroral, where lasers are 1900 m apart).

The uncertainties in velocity estimates in CSR00L01 suggest the quality of the data to vary greatly from site to site, especially in the vertical. Of the 20 sites in Category Rigid and Category GIA, six are constrained in the vertical very tightly (standard error of 0.2–0.35 mm yr⁻¹), three tightly (0.4–0.6 mm yr⁻¹), two less tightly (0.8–1.1 mm yr⁻¹) and 11 loosely (3.3–12.3 mm yr⁻¹). The 11 sites constrained tightly in the vertical are, in order of increasing error: Yaragadee, Graz, Greenbelt, Greenwich, Maui, Macdonald, Grasse, Wettzell, Orroral, Zimmerwald and Potsdam.

In the realistic error budget that we formulate by adding systematic error inversely proportional to the effective time interval of observation, the uncertainties vary less from site to site than in CSR00L01. For example, the smallest vertical standard errors increase from 0.2 to 1.0 mm yr⁻¹.

## B.3 VLBI

The VLBI velocity model, GSFC 2004b, consists of estimates of the velocities of 87 VLBI sites (26 in Category Rigid, 6 in GIA and 55 in Boundary). A VLBI site consists of one to three radio telescopes less than 1000 m apart. Chopo Ma (Goddard Space Flight Center) determined the velocity estimates from 25 yr of VLBI observations beginning in 1979 August and ending in 2004 March. Ma estimated offsets at sourdouh, whthorse and yakataga during the 1987 M 7.6 Gulf of Alaska earthquakes (Sauber et al. 1993; Argus & Lyzenga 1994); at fortords and presidio during the 1989 M 7.1 Loma Prieta (California) earthquake (Clark et al. 1990); at dss15 and mojave12 during the 1992 M 7.3 Landers (California) earthquake; and at gilcreek during the 2002 M 7.9 Denali (Alaska) earthquake (Titov & Tregoning 2005). Ma also estimated offsets due to antenna adjustments at eflsberg on 1996 October 1, at dss65 on 1997 April 15 and at ggao7108 on 2003 January 1.

## B.4 DORIS

The DORIS velocity model consists of estimates of the velocities of 60 sites (33 in Category Rigid, 5 in Category GIA and 22 in Category Boundary). A DORIS site consists of one to three monuments less than 1000 m apart (except for reya and noum, where beacons are, respectively, 4 and 9 km apart).

We determine this DORIS velocity model from the estimates of position each week between 1993 January and 2006 January that Pascal Willis (Institut Geographique National) determine using the methods of Willis et al. (2005b, 2007) and the GIPSY data analysis algorithms.

We impose measured velocity ties between 45 monuments handed to us by H. Fagard (Fagard 2006). The ties range in length from 7 mm to 9 km; the median length is 4 m. We impose the 45 ties having standard errors of 5 mm or less; we do not impose the 5 ties having a standard error of 10 mm or more. If we were to estimate the offsets corresponding to each of the 45 ties that we imposed, we would find the median component offset to be just 10 mm and the greatest component offset to be 52 mm.
We fit the estimates of position as a function of time in a manner identical to that for GPS, that is, with a position, a velocity, offsets where and when needed, and a sinusoid having a period of 1 yr.

We estimate a total of 11 offsets (Willis & Ries 2005). Five of them correspond to the imprecise ties: syob-sypb (Syowa), otta-ottb (Ottawa), sana-saob (Santiago), hbka-hbla (Hartebeesthoek) and gola-goma (Goldstone).

We estimate six offsets not at times of beacon substitutions:

(1) We estimate an offset of unknown cause of cola (Columbo), on the India Plate, in November 1994 to be 76 mm towards S45°E. The November 14 M 7.1 Mindoro (Philippines) earthquake and the November 15 M 6.5 earthquake in the Java Sea were too far from Columbo to permanently move the monument, but perhaps one of the two earthquakes shook the antenna and knocked it off a bit.

(2) We estimate an offset of unknown cause of soda (Socorro island), on the Pacific Plate, to be 72 mm towards S56°E. We estimate the velocity of Socorro island omitting observations from 1993 January to 1995 January, when the site moved about 0.2 m north-northwest towards the deflating volcano at the centre of the island (Briole *et al.* 2009).

(3) We estimate the movement of tria (Trista de Cunha), on the Nubia Plate, during four mb 4.7-4.8 earthquakes on 2004 July 29 to be 67 mm towards S16°W. We would not expect such small earthquakes to cause such a large offset (given a total seismic moment of $1.5 \times 10^{17}$ N m, the scaling relationship of Wells & Coppersmith (1994) gives slip of 86 mm across a fault plane of area 30 km²).

(4) We estimate the movement of adea (Dumont d'Urville), on the Antarctica Plate during the March 25 M 8.1 Balleny earthquake to be 23 mm towards N48°E, consistent with the prediction (30 mm, N49°E) of the dislocation model of Bouin & Vigny (2000).

(5) We estimate the movement of gola (Goldstone, California) during the 1999 October 16 M 7.1 Hector Mine earthquake to be 17 mm towards S77°W, in rough agreement with the prediction (8 mm, S78°W) of the dislocation model of Hurst *et al.* (2000).

(6) We estimate the movement of saka (Sakhalin island) during the 2003 September 25 M 8.3 Hokkaido earthquake to be 15 mm towards S28°W.

To avoid a post-seismic transient, we omit observations at area (Arequipa, Peru) after the 2001 June 23 M 8.4 Arequipa earthquake.

To avoid a postseismic transient, we omit observations at faia (Fairbanks, Alaska) after the 2002 November 3 M 7.9 Denali earthquake.

We omit observations from amsa and amsb (Amsterdam island) between 1993 and 1999 because the antenna was tilting; we use only observations from amtb after January 2000.

## APPENDIX C: ERROR BUDGET

Uncertainties in site velocities estimated from the dispersion of position estimates about a constant velocity are roughly 10 times smaller than the true uncertainties in site velocity because position estimates in successive months are highly correlated (Argus & Gordon 1996; Langbein & Johnson 1997). Mao *et al.* (1999) and Williams *et al.* (2004) determine realistic estimates of site velocity uncertainty by explicitly treating correlations of positions estimates in successive months (that is, by assuming coloured, as opposed to white, noise). In this study, we formulate a realistic error budget otherwise following the method of Argus & Gordon (1996). We take the true standard error in a component of site velocity to be the root sum square of a random error and a systematic error. We

compute the random error for GPS and DORIS from the dispersion of position estimates about a constant velocity and a sinusoid having a period of 1 yr, and for SLR and VLBI from the dispersion of position estimates about a constant velocity. We compute the systematic error to be a distance (as we specify next) in millimetres divided by the effective time of observation in years. The effective time for a site with an offset in position due to an antenna substitution or an earthquake is the root sum square of the length of the time intervals before and the length of the time interval after the offset. For each of the four techniques we determine the vertical distance to be the value that is just large enough to make the estimates of vertical site rate consistent between techniques; and the horizontal distance to be the value that is just large enough to make the estimates of horizontal site velocity consistent between techniques and with the parts of the plate interiors that are neither beneath nor along the margins of the late Pleistocene ice sheets being rigid; in the GEODVEL inversion we make the eight distances large enough to make the normalized sample standard deviations of the eight data subsets one (see for example table 4 of Argus *et al.* 1999). We list the eight distances in Table 1.

If we discard the requirement that sites on plates move as part of a rigid plate, the horizontal misfit (i.e. the normalized sample standard deviation) shrinks by 11 per cent (GPS –19 per cent, VLBI –5 per cent, SLR +4 per cent and DORIS –15 per cent). Imposing the constraint that the vertical site rates be consistent with the postglacial rebound model of Peltier (1996) would increase the vertical misfit (the normalized sample standard deviation) by 31 per cent (GPS +47 per cent, VLBI +2 per cent, SLR –21 per cent and DORIS –13 per cent).

To formulate the error budget for the ITRFVEL inversion, we multiply the horizontal components of the ITRF2005 site velocities by a factor of 2, then add in quadrature an error of 0.5 mm yr⁻¹; and we multiply the vertical components of the ITRF2005 site velocities by a factor of 5, then add in quadrature an error of 1.25 mm yr⁻¹. Inverting the ITRF2005 site velocities with this error budget results in a horizontal normalized sample standard deviation of 1.028 and a vertical normalized sample standard deviation of 1.302.

## APPENDIX D: MODELS OF POSTGLACIAL REBOUND

The postglacial rebound models of Peltier (1994, 1996, 2004) are fit mainly to geological observations of relative sea level over the past 20 kyr. The models account for ice sheets transforming into ocean water, as well as the gravitational effect of change of the solid Earth on the oceans. The models of Peltier (1996) and Peltier (2004) are also fit to two geophysical observables, the wander of Earth's spin axis since 1900 (Vicente & Yumi 1969; Gross & Vondrak 1999) and the increase in the spin rate after deducting the effect of tidal friction (Stephenson & Morrison 1995). The models have the following deglaciation histories and mantle viscosity profiles: Peltier (1994), ICE 4G, VM1; Peltier (1996), ICE 4G, VM2 and Peltier (2004), ICE 5G, VM2. The model of Peltier (2004) accounts for the effect of rotational feedback due to the wander of the spin axis, but the models of Peltier (1994, 1996) do not. In the model of Peltier (2004) rotational feedback generates a degree 2 order 1 pattern having maximum uplift and subsidence of 1.8 mm yr⁻¹ and maximum horizontal motion of 1.4 mm yr⁻¹.

The model of Peltier (1994) fits the geodetic horizontal site velocities well, but the models of Peltier (1996, 2004) fit them poorly. Sites along the margins of the late Pleistocene Laurentide ice sheets
are observed to be moving horizontally away from the ice centres at about 1 mm yr⁻¹, at about the speed predicted by the model of Peltier (1994), but at a speed 1–2 mm yr⁻¹ slower than predicted by the models of Peltier (1996, 2004). Sites on the interior of the North America Plate south of the Laurentide ice sheet are observed to be nearly stationary relative to one another, consistent with the model of Peltier (1994) but inconsistent with the models of Peltier (1996, 2004).

The models of Peltier (1994) and Peltier (1996) fit the geodetic vertical site rates better than does the model of Peltier (2004). Sites in eastern North America are observed to be moving vertically within about 1 mm yr⁻¹ of the uplift rates predicted by the models of Peltier (1994) and Peltier (1996), but at an uplift rate about 2 mm yr⁻¹ faster than that predicted by the model of Peltier (2004).

We know of no models of postglacial rebound available to all other than those of W. R. Peltier.

As we prepare this study, Peltier & Drummond (2008) are finding that they can fit both the horizontal and vertical observations by adding to the model of Peltier (2004) a thin layer of high viscosity beneath the elastic lithosphere.

## APPENDIX E: EARTH'S SCALE

Scientists estimate the scale of Earth (distances between Earth's centre and sites on Earth's surface) from VLBI using the speed of light and the time delay between radio telescopes. Scientists also estimate Earth's scale (distances between Earth's centre, sites on Earth's surface and satellites) from the three satellite techniques (SLR, GPS and DORIS) using observations of time delay, the speed of light and (GM) the product of G, the universal gravitational constant and *M*, the mass of Earth, oceans and atmosphere. The distance between phase centre and satellite mass centre is also important when estimating scale from the three satellite techniques.

The means by which we define Earth's scale differs between ITRFVEL and GEODVEL. In ITRF2005 Altamimi *et al.* (2007) assume Earth's scale to be that determined by VLBI; they estimate the scales and scale rates transforming GPS, SLR and DORIS position estimates into VLBI position estimates. For example, Altamimi *et al.* (2007) transform SLR positions into ITRF2005 using a scale rate

factor of 1 plus 0.08 parts per billion per year (which is 0.5 mm yr⁻¹ along Earth's radius, and 1.0 mm yr⁻¹ along Earth's diameter.)

In GEODVEL the scale for GPS, SLR and VLBI is that determined by the technique; the scale for DORIS is that minimizing differences with ITRF2000. (P. Willis estimates that a scale factor of 1 plus -0.10 parts per billion per year transforms DORIS position estimates in ITRF2000 into ITRF2005, roughly equal to the -0.08 parts per billion per year that transforms ITRF2000 into ITRF2005.)

M. B. Heflin and JPL scientists find, that to transform GPS estimates of position into ITRF2005, they must multiply by a scale factor of 1 plus 0.05 parts per billion per year (0.3 mm yr⁻¹ along Earth's radius, 0.6 mm yr⁻¹ along Earth's diameter). This 0.05 parts per billion per year is several times less than the 0.24 parts per billion per year that the JPL scientists find transforms GPS estimates into ITRF2000 (Altamimi *et al.* 2002).

To evaluate the degree to which velocity estimates depend on the scale factor, we compare GPS velocity estimates determined applying the scale factor of 1 plus 0.05 parts per billion with GPS velocity estimates determined not applying a scale factor. The horizontal velocity components differ negligibly between the two sets of positions (for the 167 sites in Categories Rigid and GIA), with the root mean square difference in the east and north components of the 167 GPS velocities being, respectively, 0.013 and 0.012 mm yr⁻¹. The vertical velocity components, however, differ more. The median uplift is greater by 0.66 mm yr⁻¹ in the set of velocities determined applying the scale, which is greater than the uplift at 0.32 mm yr⁻¹ that we would predict by multiplying 0.05 parts per billion by Earth's radius.

These vertical velocity differences alter GEODVEL only insomfar as enforcing vertical components of ties alter the translational and rotational velocities between techniques. If, when determining GEODVEL, we were to substitute a GPS velocity model determined applying the scale factor of 1 plus 0.05 parts per billion to transform into ITRF2005, we would find the plate angular velocities to be identical to 0.001° Myr⁻¹ with those in GEODVEL.

In sum, estimates of horizontal velocities depend negligibly on Earth's scale; estimates of vertical velocity depend significantly on Earth's scale. Not transforming by the scale factor when determining the GPS velocity model means that the GPS velocity model going into GEODVEL is independent of an ITRF.