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Han et al. Satell Navig (2021) 2:18 Page 2 of 10

pollution between the satellite clock and the satellite orbit, Basic concepts of the space–time reference system
BDS-2 adopted a time synchronization technology which is Newton’s time and space reference system
completely diferent from that of GPS, GLObal NAvigation Under the framework of Newtonian mechanics, time and
Satellite System (GLONASS) and Galileo (Liu et al., 2009; space are independent of each other. Time passes evenly
Domnin et al., 2009). Two-Way Time and Frequency Trans- and space is uniform everywhere which satisfes Euclidean
fer (TWTFT) is used between the satellites and ground geometry. Under this assumption, the space–time reference
stations, including the master station and the time synchro- system can be simply understood as a Tree-Dimensional
nization/upload stations. Te results of the long-term oper- (3D) space frame and a clock carried by an observer. Te
ation of BDS-2 show that the time synchronization is very former is used as a space reference for direction and dis-
successful with the uncertainty of satellite-to-ground time tance, and the later as a time reference. A space frame is
comparison better than 1 ns, which efectively improves somewhat alike an electronic total station that can meas-
the orbit and time synchronization accuracy of the satellites ure distance and direction. Due to the fatness of space, the
(Liu et al., 2009; Han et al., 2013; Guo et al., 2018). For this measurement range of the space frame has no limits theo-
reason, the BDS-3 continues to use this time synchroniza- retically. Terefore, the space–time reference system can
tion technique, and further adds inter-satellite time com- be defned by a time scale and a 3D space reference system.
parison links to directly measure the distances and relative A time scale is composed of a starting point and the time
clock ofsets between the satellites (Liu et al., 2019; Chen unit, which has nothing to do with the choice of space posi-
et al. 2016). Te distinctive time synchronization technique tion and coordinates. A space reference system is defned by
is one of the important innovations of BeiDou Navigation its origin, orientation of axes and scale or unit of distance.
Satellite System (BDS). For convenience, it is usually required that the three axes
BDS is essentially a system of precision time measurement are orthogonal to each other and have the same coordinate
and time synchronization in a large-scale space near the unit. Tere is a special kind of space reference systems in
Earth. At present, with the daily frequency stability of the Newtonian mechanics, called inertial reference systems, in
BDS satellite atomic clock at the level of 1 × 10 −15 , the time which Newton’s law of inertia are satisfed. Te laws of phys-
synchronization uncertainty is required to reach 1 ns or ics have the same form of expression in all inertial reference
even higher, and the accuracy of the space–time measure- systems, which are called Galileo’s principle of relativity.
ment and observation model is required to reach the order Practically, a complete space reference system needs to
of picosecond (Han and Cai 2019). Te constellation of BDS defne not only a reference frame, but also some geometric
is composed of Geostationary Earth Orbit (GEO), Inclined parameters and related physical feld parameters, such as
Geo-Synchronous Orbit (IGSO) and Medium Earth Orbit the reference ellipsoid, the geocentric gravitational constant
(MEO) satellites, covering a space range of 40,000 km near and gravitational model in the terrestrial reference system,
the Earth (Tan, 2017). Although it is in the scope of the etc..
Earth’s gravitational feld, the infuence of the sun and moon
gravitational felds on the satellite orbit and satellite clock The concepts of relativistic space–time reference system
cannot be ignored (Petit and Wolf, 1994; Wolf and Petit, Compared with Newtonian mechanics, the concepts of
1995; Klioner, 2003). Under such a large scale of space and relativistic space–time reference system are much more
high-precision index requirements, the space–time theory complicated. First, the physical time and space are not
of Euclidean geometry and Newtonian mechanics is difcult absolutely independent. Tey are interrelated and cannot
to meet the needs of high-precision measurement modeling be completely separated. In fact, whether the objects in the
(the accuracy of Newtonian mechanics is equivalent to the universe are moving or static, whether the moving speed is
−8
level of 1 × 10 ). Relativity is the theoretical basis for pre- large or small, and whether the motion path is a straight or
cise orbit determination, time synchronization, and meas- a curve are all determined by the observer or the frame of
urement modeling of BDS (Han, 2002; Han and Cai, 2019). reference. Te space point of one observer may be a line in
As a typical realization of the basic theory of relativistic the eyes of another. What one sees is a straight line may be
space–time reference system, the defnition and realization a curve seen by another. Te spatial points, straight lines,
of BDS space–time reference system will be discussed in curves, even planes and curved surfaces in Euclidean geom-
depth below. At the same time, in view of the autonomous etry do not have complete objectivity. Te distance between
operation of navigation satellite constellation, the processing two diferent events in the universe is closely related to the
method of Earth Orientation Parameter (EOP) will also be observer or the frame of reference. For diferent spatial
briefy analyzed. points, the observer’s defnition of simultaneity is related to