Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth

[1] Crustal motion can be described as a vector displacement field, which depends on both the physical deformation and the reference frame. Self-consistent descriptions of surface kinematics must account for the dynamic relationship between the Earth's surface and the frame origin at some defined center of the Earth, which is governed by the Earth's response to the degree-one spherical harmonic component of surface loads. Terrestrial reference frames are defined here as "isomorphic'' if the computed surface displacements functionally accord with load Love number theory. Isomorphic frames are shown to move relative to each other along the direction of the load's center of mass. The following frames are isomorphic: center of mass of the solid Earth, center of mass of the entire Earth system, no-net translation of the surface, no-net horizontal translation of the surface, and no-net vertical translation of the surface. The theory predicts different degree-one load Love numbers and geocenter motion for specific isomorphic frames. Under a change in center of mass of surface load in any isomorphic frame, the total surface displacement field consists not only of a geocenter translation in inertial space, but must also be accompanied by surface deformation. Therefore estimation of geocenter displacement should account for this deformation. Even very long baseline interferometry (VLBI) is sensitive to geocenter displacement, as the accompanying deformation changes baseline lengths. The choice of specific isomorphic frame can facilitate scientific interpretation; the theory presented here clarifies how coordinate displacements and horizontal versus vertical motion are critically tied to this choice.