The spherical harmonic coefficients of degree 1 represent the distance between the center of mass of the Earth and its 'center of figure', which in practice is the center of a set of tracking stations on the surface of the Earth. Because of their physical meaning, time changes in degree 1 coefficients can be expressed in several equivalent forms

  1. As distances in mm along the Z (along the axis of rotation), X and Y axes;
  2. As (fully normalized) coefficients of the geopotential;
  3. As the changes in mass (per unit area) that would give rise to the geopotential coefficients, expressed either in kg/m^2 or cm of equivalent water thickness.

The relation between these forms can be found in Swenson et al (2008), equations 5 and 4.
One way to obtain these coefficients uses Satellite Laser Ranging to geodetic satellites. Coefficients obtained in that manner since 1992 are available here . These coefficients are expressed in the form (1) above. See Cheng, Tapley and Ries (2010).

Another way to obtain these coefficients combines GRACE data and the output of a numerical ocean model, and is applicable to land hydrology and cryospheric applications (Swenson, Chambers and Wahr, 2008). Coefficients computed in this manner are available here. These coefficients are expressed in the form (2) above.


Cheng, M.K., B.D. Tapley, J.C. Ries (2010) Geocenter Variations from Analysis of SLR data, IAG Commission 1 Symposium 2010, Reference Frames for Applications in Geosciences (REFAG2010), Marne-La-Vallee, France, 4-8 October 2010.

Swenson, S.; Chambers, D. & Wahr, J. Estimating geocenter variations from a combination of GRACE and ocean model output J. Geophys. Res., 2008, 113, 8410

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