Coordinate Systems and Geodetic Reference Frames

The International Earth Rotation and Reference Systems Service (IERS) is responsible for defining and maintaining the ITRS (International Terrestrial Reference System) and the ICRS (International Celestial Reference System).
ICRS/ICRF: The International Celestial Reference System is a quasi-inertial system with its origin at the solar system barycenter. Its orientation is based on the mean equator and equinox at epoch J2000.0.
ICRF3: The latest realization of the celestial frame, consisting of over 4500 coordinates for extragalactic radio sources (quasars) observed via Very Long Baseline Interferometry (VLBI).
ITRS/ITRF: The International Terrestrial Reference System is the conceptual geocentric system, while the ITRF (International Terrestrial Reference Frame) is the physical realization consisting of coordinates and velocities of a global station network.

Conventional Inertial ( $i_0$ ): A stable reference frame fixed relative to distant stars, unaffected by Earth's rotation, precession, or nutation.
Average Inertial ( $\bar{i}$ ): The system $i_0$ corrected for precession, described by parameters $\zeta_0$ , $\theta$ , and $z$ .
Instantaneous Inertial ( $i$ ): The system $\bar{i}$ corrected for nutation, described by parameters $\epsilon$ , $\Delta\epsilon$ , and $\Delta\psi$ .
Precession: A secular, circular movement of Earth's rotation axis caused by gravitational torques from the Sun and Moon. It has a period of approximately $26\,000$ years and a rate of roughly $50.3$ arcseconds per year.
Nutation: A periodic "nodding" motion superimposed on precession. The dominant component is the Bradley-nutation with a period of $18.6$ years.
Transformation: To rotate from the Average Inertial system to the Conventional Inertial system: $r_i = P r_{i0} = R_3(-z)R_2(\theta)R_3(-\zeta_0)r_{i0}$

GAST (Greenwich Actual Sidereal Time): The angle between the true vernal equinox and the Greenwich meridian. It is used to transform between the Momentary True Inertial System ( $i$ ) and the Momentary Earth-Fixed System ( $e$ ).
Calculation: $GAST = GMST + \text{equation of equinoxes}$ , where GMST is Greenwich Mean Sidereal Time.
This transformation is essential for relating celestial positions to Earth-based coordinates in real-time.

Momentary Terrestrial ( $e$ ): An instantaneous coordinate system that rotates with the Earth; it is non-inertial.
Conventional Terrestrial: Systems fixed to the Earth's surface, such as ITRF. Positions are often represented as geocentric Cartesian coordinates $(X, Y, Z)$ .
Global Geodetic: Systems based on a global datum like WGS84, which is used by GPS and aligned closely with ITRF.
ETRS89 (European Terrestrial Reference System 1989): A regional system fixed to the stable part of the Eurasian tectonic plate. It coincided with ITRS at epoch 1989.0.
EUREF89: The implementation of the European reference system in Norway, introduced in 1997 to replace older systems like NGO1948 and ED50.

Tectonic Motion: ITRF accounts for Earth's dynamics; some points in the frame move up to $10\,cm$ per year due to plate tectonics and isostasy.
Helmert Transformation: A 14-parameter transformation (7 parameters at a reference epoch and 7 rate-of-change components) is used to convert coordinates between different ITRF realizations (e.g., from ITRF2014 to ITRF2020).
Transformation Formula: $\begin{pmatrix} X \ Y \ Z \ \end{pmatrix}<em>{target} = \begin{pmatrix} X \ Y \ Z \ \end{pmatrix}</em>{source} + \begin{pmatrix} T_x \ T_y \ T_z \ \end{pmatrix} + \begin{pmatrix} D & -R_z & R_y \ R_z & D & -R_x \ -R_y & R_x & D \ \end{pmatrix} \begin{pmatrix} X \ Y \ Z \ \end{pmatrix}_{source}$
ITRF2020: The latest realization, including a plate motion model for 14 tectonic plates.