## Terrestrial Reference System

$$\global\def\fig#1{\scriptsize\textcolor{E83E8C}{fig.#1}}$$

### Definitions

A TRS is a spatial reference system co-rotating with the Earth in its diurnal motion in space. In such a system, positions of points anchored on the Earth solid surface have coordinates which undergo only small variations with time, due to geophysical effects (tectonic or tidal deformations). A TRF is a set of physical points with precisely determined coordinates in a specific coordinate system (cartesian, geographic, mapping...) attached to a Terrestrial Reference System. Such a TRF is said to be a realization of the TRS.

### Three main concepts are identified as follows:

#### 1. Ideal Terrestrial Reference Systems.

An ideal TRS is defined as a tri-dimensional reference frame (in the mathematical sense) close to the Earth and co-rotating with it. In the newtonian background, the geometry of the physical space considered as an euclidian affine space of dimension 3 provides a standard and rigorous model of such a system through the selection of an affine frame (O,E). O is a point of the space named origin. E is a vector base of the associated vector space.The currently adopted restrictions to E are to be orthogonal with same length for the base vectors. Moreover, one adopts a direct orientation. The common length of these vectors will express the scale of the TRS and the set of unit vectors collinear to the base its orientation:

$$\Large \lambda = \begin{Vmatrix}\overrightarrow{E_i}\end{Vmatrix}_{i=1,2,3} \newline~ \fig1$$

In the context of IERS, we consider the geocentric systems where the origin is close to the geocenter and the orientation is equatorial (Z axis is the direction of the pole). In this case, cartesian coordinates, geographical coordinates or plane (map) coordinates are currently used.
Under these hypothesis, the general transformation of the cartesian coordinates of any point close to the Earth from a TRS ($\fig1$) to a TRS ($\fig2$) will be given by a tri-dimensional similarity (T1,2 is a translation vector, λ1,2 a scale factor and R1,2 a rotation matrix):

$$\Large X^{(2)} = T_{1,2} + \lambda_{1,2}.R_{1,2}.X^{(1)} \newline~ \fig2$$

This concept can be generalized in the frame of a relativistic background model such as Einstein's General Relativity, using the spatial part of a local cartesian frame.

#### 2. Conventional Terrestrial Reference System (CTRS).

Such a system designates the set of all conventions, algorithms and constants which determine the estimation of coordinates of points in a specific ideal TRS.

#### 3. Conventional Terrestrial Reference Frame.

A CTRF is defined as a set of physical points with precisely determined coordinates in a specific coordinate system as a realization of an ideal Terrestrial Reference System. Two types of frames are currently distinguished (this is also valid for celestial references), namely dynamical or kinematical ones, depending whether a dynamical model is applied in the determination process of these coordinates, or not.