This data product provides weekly values of the spherical harmonic coefficients of the gravity field complete to degree and order 5, derived from satellite laser ranging up to five geodetic satellites, computed by Minkang Cheng, John Ries and Byron Tapley (2011).

This is an experimental product; the coefficients of degree 3-5 are of lower quality than those of degree 1 and 2 due to the nature of the tracking data. The data are provided for experienced users who wish to assess the quality of the product.

The coefficients of degree 1 and 2 have clear physical interpretations. The degree 1 coefficients represent the relative position of the geocenter (the center of mass of the Earth) relative to the center of figure (defined, in practice, by the position of laser tracking stations on the surface of the Earth). The degree 2, order 0 coefficient is described here. The degree 2, order 1 coefficients are related to the relative position of the principal figure axis of the Earth relative to the rotational axis. In the absence of external torques, those two axes would be very close [Wahr, 1987 and 1990]. However, as a consequence of excitations from tidal and other mass redistribution within the Earth system (dominated by seasonal variations), the two axes are not completely aligned, and the rotational axis precesses around the Earth’s principal figure axis (causing the ~435 day Chandler wobble).

The coefficients are given as variations relative to the mean gravity field GGM02C, which is provided in this file. All coefficients are normalized. The format for the GGM02C coefficients is (6X,2I3,2D21.14,2D11.4). The variations should be added to GGM02C to obtain the full coefficient.

Each week's information consists of 21 lines with the epoch and the 5x5 variations. The EPOCH card provides the Modified Julian Date of the first day of the week as well as the mid-point of the arc given in years. The format is (6X,F9.1,F15.6). The next 20 lines contain the degree, order, the delta C and S coefficients, followed by the C and S sigmas and the epoch in years. The epoch is provided again to make it easy to extract selected harmonics. The variations and their sigmas are scaled by 1.0e10. The format for the GEOEST card is (6X,2I3,4F12.5,F15.6).

The degree 1 terms are obtained by converting the estimated geocenter translation to gravity harmonics.

Note that no atmosphere-ocean de-aliasing product and no rate terms have been applied, so the signal variations are related to the full Earth System.

Data Access

The data can be downloaded here.

Acknowledgement and Citation

This research is sponsored by the NASA GRACE Science Team (NASA Grant NNX08AE99E) and the NASA MEaSURES program (JPL contract JPL1368074). When using these data, please acknowledge receiving the data from "", and cite:

Cheng, M., J. C. Ries, and B. D. Tapley (2011), Variations of the Earth's Figure Axis from Satellite Laser Ranging and GRACE, J. Geophys. Res., 116, B01409, 2011, DOI:10.1029/2010JB000850.


Wahr, J. M. (1987), The Earth’s C21 and S21 gravity coefficients and the rotation of core, Geophys. J. R. Astron. Soc., 88(1), 265–276.

Wahr, J. M. (1990), Correction and update to ‘The Earth’s C21 and S21 gravity coefficients and the rotation of core’, Geophys. J. Int., 101, 709–711, doi:10.1111/j.1365-246X.1990.tb05578.x.

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