With recent improvements in space geodesy, the Earth's center of mass (CM) and center of figure (CF) are no longer indistinguishable. The current origin of the International Terrestrial Reference Frame (ITRF) is defined as the CM, which shows measured seasonal variations of several millimeters to 1 cm with respect to true CM. As scientists study Earth's dynamic deformations on seasonal and shorter timescales and begin to compare observed geocenter motion with predictions from geophysical models, the reference frame origin presents significant error due to missing the geocenter motion. This paper discusses the nature of the origin of the ITRF and explores the sensitivity of GPS measurements to geocenter motion. We find that since the values of nonlinear geocenter motion are not included in the positions of the ITRF sites, the behavior of the current ITRF origin reflects CM on secular timescale but reflects CF on seasonal and short timescales. The nature of the ITRF origin depends on both the adopted kinematic model and unmodeled network motion. The realized ITRF origin should be defined by a new nomenclature to reflect its nature accurately. By the new nomenclature, the origin will maintain its current long-term stability, while improving its stability on seasonal timescales to the submillimeter level. With the degree-1 deformation approach, GPS measurements are able to provide potentially valuable information on geocenter variations on seasonal and short timescales.