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 approximated by 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 form (1) above. For more details on SLR-based degree-1 coefficeints, see Cheng, Tapley and Ries (2010).

Another way to obtain an estimate of degree-1 time-variable gravity coefficients uses GRACE / GRACE-FO data with the output of a numerical ocean model. The original method was presented in Swenson, Chambers and Wahr (2008). Sun et al. (2016) extended the method to include and ocean mass distribution that is consistent with the so-called sea-level-equation solved for the self-gravitation patterns of sea level rise related to land water storage change (from hydorlogy, ice and snow etc.). Degree-1 coefficients computed based on Sun et al. (2016) are available here (Technical Note TN-13). These coefficients are expressed in the form (2) above. Please note that as of RL06, we now provide one TN-13 for each of JPL, CSR, and GFZ, representing degree-1 coefficients from the respective Level-2 standard spherical harmonic coefficeints.


Sun, Y., R. Riva, and P. Ditmar (2016), Optimizing estimates of annual variations and trends in geocenter motion and J2 from a combination of GRACE data and geophysical models, J. Geophys. Res. Solid Earth, 121, doi:10.1002/2016JB013073.

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 (2008), J. Estimating geocenter variations from a combination of GRACE and ocean model output J. Geophys. Res., 113, 8410

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