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Han et al. Satell Navig (2021) 2:18 Page 7 of 10

potentials of the space–time metric in GCRS are from Conceptually, two space-time reference systems are
the BCRS, the defnition of GCRS is conceptually derived mainly involved in BDS in the operation mode supported
from BCRS. by the master control station and monitor stations. One
is GCRS which is a unifed space-time reference for its
Geocentric terrestrial reference system precise orbit determination, time synchronization, and
Te defnition of GTRS is given by IAU2000 Resolution observation information processing, and the other is
B1.3. It is a space coordinate system whose origin is at the GTRS which is the reference of the Earth’s surface and
center of the Earth and makes diurnal apparent motion geostationary space and mainly used to express the posi-
with the Earth. In this reference system, the coordinates tions of the users and the ground stations.
of points on the solid surface of the Earth remain almost Te observation modeling of satellite precise orbit
unchanged except of the small changes caused by geo- determination and time synchronization in BDS are
physical efects. IUGG2007 Resolution 2 clarifed that all based on GCRS. Since the positions of satellites and
GTRS was a geocentric space–time coordinate system ground stations are separately expressed in two spa-
under the framework of the theory of relativity. Te coor- tial reference systems, GCRS and GTRS, the coordinate
dinate transformation between GTRS and GCRS is real- transformation between them is required. Te relation-
ized through a space rotation determined with EOP. ship can be written as follows:
Te International Terrestrial Reference System (ITRS)
maintained by IERS is an implementation of GTRS, [GCRS]=Q(t)R(t)W (t)[GTRS] (12)
which constitutes a set of prescriptions and conventions Here, Q(t), R(t), W (t) are respectively the matrices of
together with the modeling required to defne the ori- precession and nutation, the Earth rotation, and the polar
gin, scale, orientation, and time evolution. Te system is motion, and.
realized as the International Terrestrial Reference Frame
(ITRF) based upon the estimated coordinates and veloci- Q(t) ≡ R 3 (−E)R 2 (−d)R 3 (E+s) (13)
ties of a set of stations observed by VLBI, Lunar Laser
Ranging (LLR), GPS, Satellite Laser Ranging (SLR), and R(t) ≡ R 3 (ω E ) (14)
Doppler Orbitography and Radiopositioning Integrated
by Satellite (DORIS). According to IERS, ITRS meet the W (t) ≡ R 3 (−s )R 2 (x p )R 1 (y p ) (15)
′
following:
Where R , R , and R represent the rotation matrices
1
2
3
• Te origin is at the mass center of the Earth, includ- about the X, Y, and Z axes respectively. Te acquisition
ing oceans and atmosphere. and calculation methods of each parameter were given in
• Te unit of length (scale) is the meter (SI). IERS 2010 Technical Note (Petit and Luzum, 2010).
• Te orientation is initially given by the Bureau Inter- Te geocentric terrestrial reference system used in
national de I’Heure(BIH) orientation at 1984.0. BDS is called BeiDou Coordinate System (BDCS) and its
• Te time evolution of the orientation is ensured by defnition is consistent with ITRS (Wu, 2018). Te ori-
using a no-net-rotation condition with regard to hor- gin of the coordinates is at the mass center of the Earth,
izontal tectonic motions over the whole Earth. and the direction and scale of the coordinate axes are the
same as ITRF. Table 1 shows the reference ellipsoid and
Up to now, there are 12 ITRF realizations: ITRF89, the Earth’s gravitational feld constants defned in BDCS,
ITRF90, ITRF91, ITRF92, ITRF93, ITRF94, ITRF96, including the semi-major axis and the fatness of the ref-
ITRF97, ITRF2000, ITRF2005, ITRF2008, ITRF2014 erence ellipsoid, the geocentric gravitational constant
(https:// www. iers. org/). and the angular velocity of the Earth rotation.
Te latest implementation of BDCS is done by using
The space–time references of BDS more than 100 globally distributed ground stations as
A Global Navigation Satellite System (GNSS) involves a
large space–time range near the Earth. To achieve high- Table 1 The reference ellipsoid and geocentric gravitational
precision Positioning, Navigation, and Timing (PNT), the constant adopted in BDCS
unifed and high-precision space–time reference systems Semi-major axis a = 6,378,137.0 m
must be established. Te construction of observation Geocentric gravitational constant (Mass μ = 3.986,004,418 × 10 m /s 2
3
14
models for the precise satellite orbit determination, time of Earth’s atmosphere included)
synchronization, and other information process should Factor of ellipsoid fatness f = 1/298.257222101
be performed within a framework of the same space– Mean angular velocity of the earth ˙ Ω e = 7.292115 × 10 rad/s
−5
time benchmark. rotation