Geocenter Variations

Variations in the Earth’s geocenter reflect the largest scale variability of mass within the Earth system, and are essential inclusions for the complete recovery of surface mass change from time-variable gravity. The Earth’s geocenter is the translation between the Earth’s center of mass (CM) and center of figure (CF) reference frames. Measurements of time-variable gravity from GRACE and GRACE Follow-On (GRACE-FO) are set in an instantaneous center of mass (CM) reference frame. For most science applications of time-variable gravity, a coordinate system with an origin coinciding with the Earth’s center of figure (CF) is required. Geocenter variations are represented by the degree one spherical harmonic terms.

Important

The exclusion of degree one terms can have a significant impact on estimates of ocean mass, ice sheet mass change, and terrestrial hydrology due to far-field signals leaking into each regional estimate [71].

calc_degree_one.py calculates coefficients of degree one by combining GRACE/GRACE-FO spherical harmonic products with estimates of ocean bottom pressure (OBP) following Sutterley and Velicogna [60], Swenson et al. [62]. The method assumes that the change in global surface mass density, \(\Delta\sigma(\theta,\phi)\), can be separated into individual land and ocean components using a land-function \(\vartheta(\theta,\phi)\) [62].

(4)\[\begin{split} \Delta\sigma(\theta,\phi) &= \Delta\sigma_{land}(\theta,\phi) + \Delta\sigma_{ocean}(\theta,\phi)\\ \Delta\sigma_{ocean}(\theta,\phi) &= \vartheta(\theta,\phi)~\Delta\sigma(\theta,\phi)\end{split}\]

The oceanic components of the change in degree one spherical harmonics (\(\Delta C^{ocean}_{10}\), \(\Delta C^{ocean}_{11}\), and \(\Delta S^{ocean}_{11}\)) can then be calculated from the changes in ocean mass, \(\Delta\sigma_{ocean}(\theta,\phi)\) [62, 73]. If the oceanic contributions to degree one variability (\(\Delta C^{ocean}_{10}\), \(\Delta C^{ocean}_{11}\), and \(\Delta S^{ocean}_{11}\)) can be estimated from an ocean model, then the unknown complete degree one terms (\(\Delta C_{10}\), \(\Delta C_{11}\), and \(\Delta S_{11}\)) can be calculated from the residual between the oceanic degree one terms and the measured mass change over the ocean calculated using all other degrees of the global spherical harmonics from GRACE/GRACE-FO [60, 62].

The calc_degree_one.py program will output geocenter files in ascii format for each GRACE/GRACE-FO month following Sutterley and Velicogna [60]. Uncertainties in geocenter due to a combination of error sources can be estimated using the monte_carlo_degree_one.py program.

Load Love Numbers

The degree one Love number of gravitational potential \(k_1\) is defined so that the degree one terms describe the offset between the center of mass (CM) of the combined surface mass and deformed solid Earth, and the center of figure (CF) of the deformed solid Earth surface [5, 69]. For the CF coordinate system, this means

(5)\[ k_1 = -(h_1 + 2\ell_1)/3\]

where \(h_1\) and \(\ell_1\) are the degree one vertical and horizontal displacement Love numbers.

Geocenter and Degree One

Fully-normalized degree one variations can be converted to cartesian geocenter variations using the following relation:

(6)\[\begin{split} \Delta X &= a\sqrt{3}~\Delta C_{11} \\ \Delta Y &= a\sqrt{3}~\Delta S_{11} \\ \Delta Z &= a\sqrt{3}~\Delta C_{10}\end{split}\]

The geocenter class has utilities for converting between spherical harmonics and geocenter variation along with readers for different geocenter datasets.