Estimating network effect in geocenter motion: Theory

Geophysical models and their interpretations of several processes of interest, such as sea level rise, postseismic relaxation, and glacial isostatic adjustment, are intertwined with the need to realize the International Terrestrial Reference Frame. However, this realization needs to take into account the geocenter motion, that is, the motion of the center of figure of the Earth surface, due to, for example, deformation of the surface by earthquakes or hydrological loading effects. Usually, there is also a discrepancy, known as the network effect, between the theoretically convenient center of figure and the physically accessible center of network frames, because of unavoidable factors such as uneven station distribution, lack of stations in the oceans, disparity in the coverage between the two hemispheres, and the existence of tectonically deforming zones. Here we develop a method to estimate the magnitude of the network effect, that is, the error introduced by the incomplete sampling of the Earth surface, in measuring the geocenter motion, for a network of space geodetic stations of a fixed size N. For this purpose, we use, as our proposed estimate, the standard deviations of the changes in Helmert parameters measured by a random network of the same size N. We show that our estimate scales as (Formula presented.) and give an explicit formula for it in terms of the vector spherical harmonics expansion of the displacement field. In a complementary paper we apply this formalism to coseismic displacements and elastic deformations due to surface water movements.