Application of Clebsch-Gordan coefficients and isomorphic frame transformations to invert Earth’s changing geometrical shape for continental hydrological loading and sea level’s passive response

Abstract

The hydrological cycle induces mass exchange between the oceans and continents, and redistribution of water within the continents. In turn, relative sea level passively responds to the change in shape of both the geoid and the ocean bottom. The time variation in these two surfaces must self-consistently relate to the total mass redistribution. Here we show formally how, if given a time series of station coordinates from a global network of geodetic stations, we can invert for the separate contributions to the total mass redistribution due to water on the continents and water in the oceans. If we characterize the deformation of the solid Earth in terms of a truncated vector spherical harmonic expansion, the solution to this problem can be formulated in terms of Clebsch-Gordan coefficients, which are familiar for the calculation of branching ratios in elementary particle physics. As part of a rigorous solution to this problem, we formally address the definition of reference frame by applying the concept of an “isomorphic frame transformation,” which conserves the formal theoretical relationship between surface deformation and load distribution by a simultaneous translation of origin and change of degree-one load Love numbers.