Network Effect in Geocentre Motion

Geocentre motion is the motion of the centre of mass of the Earth system with respect to the geometric centre of figure of the solid Earth surface because of the continual deformation of the Earth by geophysical processes. This motion is important both in theory and in practice to understand and interpret various mass transport phenomena and their consequences, such as sea level rise, postseismic relaxation, polar ice melting, and glacial isostatic adjustment. Global reference frames for space geodetic point positioning are realised using measurements of the relative motion between satellites orbiting around the centre of mass on one hand and stations placed on the Earth's surface on the other. Therefore, reliable modelling of the geocentre motion is vital for the stability and the accuracy of these reference frames. In turn, the interpretation of many geodynamical quantities of current interest, such as the mean sea level, depends heavily on the quality of the adopted reference frame. Space geodetic measurement of the true geocentre motion, however, is difficult due to the discrete and therefore incomplete sampling of the Earth's surface by geodetic stations. In other words, there is a discrepancy between the centre of figure of the Earth surface and the centre of network of the stations, called the network effect, arising from the sampling bias of the geodetic network. In this work, we develop a method to estimate the magnitude of the network effect for a network of a given size \(N\). For a given crustal deformation model, we consider the Helmert parameters of transformation, that is, the parameters characterising a Euclidean similarity transformation, between the centre of figure frames before and after the deformation event. Our proposed estimate for the network effect, which we call the `expected bias', is the standard deviations of the changes in these parameters by the event as measured by a random network of the size \(N\). We show that, in accordance with probability theory, the expected bias scales as \(1/\sqrt{N}\), and we provide an explicit formula for this estimate in terms of the vector spherical harmonics expansion of the displacement field. We assess the effectiveness of the expected bias as an estimate of the network effect by simulating the displacement fields for two illustrative geodynamical processes: (instantaneous) coseismic deformation due to great earthquakes, and (time-dependent) elastic deformation due to surface water movements. We accordingly concentrate on the instantaneous changes and the secular drifts in the Helmert parameters for the two cases respectively. We found that, in both case studies, the network effect is often as large as the changes in the Helmert parameters themselves. Hence, current space geodetic networks are indeed inadequate for verifying the geocentre motion predictions by geophysical models accurately. Nevertheless, our simulations validate the expected bias to be a reasonable estimate of the network effect. Finally, we propose an alternative definition of the centre of network frame that assigns a weight proportional to the area a station represents to its measurements. We show that it can significantly reduce the network effect and improve the detection of geocentre motion in most cases