Question 1 of 8: Coordinate Systems, Reference Frames and Datums

Question 1: Coordinate Systems, Reference Frames and Datums (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. The distinction between a spatial reference system (the theoretical definition of origin, orientation and scale) and its reference frame (the physical realization through station coordinates and velocities), applied to the ITRF and to the Canadian NAD83(CSRS) realization.

Find. (a) the meaning of “kinematical” vs “dynamical” coordinate-system definitions; (b) the ITRF coordinate system under the kinematical approach; (c) the space-geodetic techniques that define and give access to the ITRF; (d) the similarities and differences between NAD83(CSRS+epoch) and ITRF, with the order of magnitude of any offset.

(a) “Kinematical” versus “dynamical” definition. A dynamical definition ties the coordinate system to the Earth’s dynamics — its gravity field and rotation. The classical geodetic system fixed its origin at the centre of mass, its Z-axis along the mean rotation axis, and its scale through physical constants, so the axes were defined by, and drifted with, the dynamical figure of the Earth. A kinematical definition instead fixes the frame geometrically through the coordinates and, crucially, the velocities of a set of reference stations, with the orientation constrained by a convention (a “no-net-rotation” condition) rather than by the physics of Earth rotation. “Kinematical” therefore means the frame is defined by the motion of points (positions changing linearly in time), decoupled from any specific dynamical model; the origin is still geocentric but the orientation evolves by convention, not by tracking the physical spin axis instantaneously.

(b) The ITRF coordinate system. Under the kinematical approach the ITRF is a right-handed, geocentric, equatorial Cartesian system \((X,Y,Z)\): the origin is the Earth’s centre of mass (including oceans and atmosphere); the Z-axis is directed toward the IERS Reference Pole (the conventional mean pole); the X-axis lies in the equatorial plane toward the IERS Reference Meridian (near Greenwich); the Y-axis completes the right-handed set. Its scale is metric (SI metre, consistent with the geocentric relativistic frame). What makes it kinematical is that each defining station is given both a position at a reference epoch \(t_0\) and a linear velocity, and the time evolution of the orientation is fixed by a no-net-rotation condition with respect to a global plate model. A point’s coordinates are thus \(\mathbf{X}(t)=\mathbf{X}(t_0)+\dot{\mathbf{X}}\,(t-t_0)\).

(c) Techniques that define and access the ITRF. The ITRF is a combination of four independent space-geodetic techniques, each contributing what it measures best: VLBI (Very Long Baseline Interferometry) — fixes the orientation and scale and the tie to the celestial frame (the only technique that senses the inertial frame directly); SLR (Satellite Laser Ranging) — defines the origin (geocentre) and scale most robustly; GNSS (GPS/GLONASS, via the IGS) — densifies the frame with thousands of stations and provides the everyday access; and DORIS — a French Doppler system giving dense, uniform global coverage (especially over oceans). The IERS combines the four solutions, using co-location sites where several instruments share local ties, to produce a single set of coordinates and velocities. A user “accesses” the ITRF in practice through the IGS GNSS network and its precise orbit/clock products.

(d) NAD83(CSRS+epoch) versus ITRF. Similarities: both are three-dimensional, Earth-centred, right-handed Cartesian reference frames realized from space geodesy (ITRF is geocentric; the NAD83 origin was only intended to be geocentric and sits about 2 m from the true geocentre); both are kinematical (coordinates carry velocities and a stated epoch); and NAD83(CSRS) is defined through a published 14-parameter (Helmert + rates) transformation from a specific ITRF solution, so their internal precision is comparable at the centimetre level. Differences: NAD83(CSRS) is attached to the stable interior of the North American plate, so a mark on that plate appears almost stationary in it, whereas in the ITRF the same mark drifts at the full plate velocity (≈ 1–2 cm/yr); and the two frames have slightly different origin and orientation, so coordinates for one mark differ by roughly 1–2 m in position (the historic NAD83 non-geocentricity of about 2 m). In short: same concept and precision, but a metre-level datum offset plus a growing epoch-dependent difference driven by plate motion.