is the coefficient of degree 2, order 0 of the non-dimensional spherical harmonic

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Domenic Bruce
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1 Materials and Methods J is the coefficient of degree, order 0 of the non-dimensional spherical harmonic representation of the mass distribution of the Earth system. It is directly related to the diagonal elements of the inertia tensor of the Earth by J I = xx + I yy Ma I zz where the z-axis is orientated along the rotation axis, M is the total mass, and a the mean radius. If we consider a fluid layer at the Earth surface, the change of J associated with the fluid mass distribution is given by J 31 = 4πρ ( + k ) p( θ, λ) P ( cosθ ) Earth ag 0 Surface sinθ dθ dλ where k is the load Love number that accounts for the elastic response of the solid Earth ( k = ), ρ Earth is the mean Earth density, g 0 is the mean gravitational acceleration at the surface, p(θ, λ) is the atmospheric surface or oceanic bottom pressure (S1), θ is the colatitude, λ is the longitude and P is the Legendre polynomial of degree (S). The effect of a mass load on J is thus obtained by convolution with a weighting function which has a maximum at the equator, symmetric minima at the poles and passes through zero at latitudes ±35.3º, so that addition of mass equatorward (poleward) of these latitudes produces an increase (decrease) in J. High latitude regions have a particularly large impact on J, with mass variations at the poles having twice the effect of mass variations at the equator. 1

2 Satellite laser ranging (SLR) analyses (S3-S11) permit the accurate determination of the long wavelength gravitational field and its temporal changes; two decades ago, the pioneering work of Yoder et al. (S3) first demonstrated via SLR that the Earth s oblateness was changing, using the results of LAGEOS together with Earth rotation results from Lunar Laser Ranging (see also S4). More recently solutions have utilized multiple satellites enabling more robust solutions of the low degree harmonics (S9-S13). ECCO model products are available at Analyses presented in this study employ the ECCO- model simulation and Kalman filter assimilation products. A prototype of the ECCO- assimilation system is described in S14. The XBT data were provided by D.W. Behringer (NCEP, personal communication, 00). Text More details on the changes in oceanic bottom pressure and J forcing are given in Figs. S1 and S. These results are presented as the difference between averages computed over the two-year periods and , in order to illustrate the dramatic changes during the year 1998 (which was not included in the composites). Pronounced drops in oceanic mass content (as measured by ocean bottom pressure) occurred near latitude 60º in both the Southern and Northern Hemispheres, with changes in the SH dominating; compensating mass changes were distributed across the tropics and subtropics of both hemispheres, reaching latitudes of about 45º (Fig. S1, TOP). Since the J weighting function reverses sign at latitude 35.3º in both hemispheres, this pronounced

3 shift in the meridional mass distribution produces a large net change in oceanic J, with positive contributions at most latitudes dominated by the SH (Fig. S1, BOTTOM). Fig. S shows a more detailed view of the regional oceanic mass contributions to changes in J. The largest contribution was 41% from the Southern Ocean (south of 40º S); the Pacific Ocean (between 40º S and 40º N) contributed 30% and the Indian Ocean (north of 40º S) contributed 18%. The remaining 11% of the oceanic J change came from the Northern Pacific (north of 40º N) and Atlantic Oceans. The Arctic Ocean is not included in the model. Details of the post-1998 sub-polar glacial melting scenarios used in the calculations are given in Table S1. As shown in the main text the results are quite robust with respect to the post-1998 melting, with all three scenarios accounting for nearly all of the non-linear behavior remaining in the J series after subtraction of the modeled ocean effects. 3

4 Figure S1. TOP: Change in average oceanic bottom pressure between and , integrated zonally around the globe. Units are 10 9 Nm -1. BOTTOM: As above, but scaled to give the change in oceanic J. Units are m -1. 4

5 Figure S. The contribution of the average bottom pressure change between and to the change in oceanic J. Units are 10-4 m -. Table 5

6 Year / Scenario Obs. in year: ICE LO ICE ICE HI Table S1. Sub-polar glacial melting rates for the three post-1998 scenarios. The melting rates for 001 correspond to the observed values (S15) for the three pre-scenario years shown in the last column. Units are km 3 yr -1. Notes 6

7 S1. To compute the J contribution from ground water, the pressure is replaced by mass per unit area multiplied by g 0. S. B.F. Chao et al., J. Geophys. Res. 9, 9415 (1987). S3. C. F. Yoder et al., Nature (1983). S4. D.P. Rubincam, J. Geophys. Res. 89, 1077 (1984). S5. J.X. Mitrovica, W. R. Peltier, J. Geophs. Res. 98, 4509 (1993). S6. D. Han, J. Wahr, Geophys. J. Int. 10, 87 (1995). S7. W.R. Peltier, Rev. Geophys. 36, 603 (1998). S8. T.S. James, E.R. Ivins, J. Geophys. Res. 10, 605 (1997). S9. C.M. Cox, B.F. Chao, Science 97, 831 (00). S10. M.K. Cheng, C.K. Shum, B.D. Tapley, J. Geophys. Res. 10, 377 (1997). S11. R. S. Nerem et al., Geophys. Res. Lett. 7, 1783 (000). S1. B. F. Chao, A.Y. Au, J. Geophys. Res. 96, 6577 (1991). S13. A. Cazenave, F. Mercier, F. Bouille, J. M. Lemoine, Earth Planet Science Lett. 171, 549 (1999). S14. I. Fukumori, R. Raghunath, L. Fu, Y. Chao, J. Geophys. Res. 104, 5647 (1999). S

Non-isotropic filtering of GRACE temporal gravity for geophysical signal enhancement

Geophys. J. Int. (2005) 163, 18 25 doi: 10.1111/j.1365-246X.2005.02756.x GJI Geodesy, potential field and applied geophysics Non-isotropic filtering of GRACE temporal gravity for geophysical signal enhancement