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Unit 4 · Lesson 4-3
Terrestrial Frames and Earth Orientation
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Terrestrial Frames and Earth Orientation

Earth rotation, polar motion, ITRF, WGS 84, geodetic coordinates and the antenna’s point of view

Canberra, 2016
Geoscience Australia makes an unusual announcement: the entire country is going to move 1.8 metres north-north-east. On paper. Nothing physical is happening — or rather, nothing new. Australia sits on one of the fastest-moving tectonic plates on Earth, sliding north at about 7 cm a year. Its national coordinate system, GDA94, had been fixed to the continent's position in 1994. Every year since, the real continent had crept further from its own official coordinates, while GPS satellites — which work in an Earth-centred frame that does not ride with any plate — reported where things actually were. For a surveyor with a tape measure, 1.8 metres over two decades is invisible. For a self-driving tractor, a drone delivery, or a car using lane-level satellite navigation, it is the difference between the road and the ditch. In 2020 the new datum, GDA2020, took over. The ground you stand on is not a fixed frame. It rotates, its axis wanders, and its crust drifts. This lesson takes a satellite from the inertial frames of the last two lessons all the way down to a dish on that moving ground.
Watch first · presented by Shoaib Aziz
The numbers in this film are the labs' own: your Earth Rotation Angle matches SOFA to 1.7e-11°; the old sidereal time has drifted 1199″ from it by 2026; a 0.4 s error in UT1 moves the equator by 186 m; rotating the ISS velocity without ω × r gives 7.613 km/s instead of 7.233 (402 m/s wrong); geocentric and geodetic latitude differ by up to 11.5′ at 45°; and the station's look angles come out at azimuth 58°, elevation 43°, range 704 km.
Lesson 2 ended on a flaw: Greenwich sidereal time measures Earth's spin from the equinox, and the equinox slides. In 2000 the IAU rebuilt the chain around two new reference points on the equator:
CIP — Celestial Intermediate Pole. The axis whose motion is modelled by precession–nutation; Earth's rotation is defined about it. CIO — Celestial Intermediate Origin. A point on the CIP's equator chosen to have no motion along the equator as the pole moves — a "non-rotating origin". It moves with the pole, but never slides sideways. TIO — Terrestrial Intermediate Origin. The same idea on the Earth-fixed side, essentially Greenwich.
The angle from CIO to TIO is the Earth Rotation Angle. Because neither origin slides, it contains nothing but rotation — and it is exactly linear in UT1.
θERA=2π (0.7790572732640+1.00273781191135448 DU),DU=JDUT1−2451545.0\theta_{\mathrm{ERA}} = 2\pi\,\big(0.7790572732640 + 1.00273781191135448\,D_U\big), \qquad D_U = \mathrm{JD}_{\mathrm{UT1}} - 2451545.0

No T², no T³: a straight line. The 1.0027378… is the ratio of a solar day to a sidereal one. Compare the GMST polynomial of lesson 2, which hides precession inside the rotation angle.

The full IERS transformation from the celestial to the terrestrial frame is then three clean pieces, each with one physical job:
r⃗ITRS=W(t)⏟polar motion  R3(θERA)⏟Earth rotation  C(X,Y,s)⏟precession–nutation  r⃗GCRS\vec{r}_{\mathrm{ITRS}} = \underbrace{\mathbf{W}(t)}_{\text{polar motion}}\;\underbrace{\mathbf{R}_3(\theta_{\mathrm{ERA}})}_{\text{Earth rotation}}\;\underbrace{\mathbf{C}(X, Y, s)}_{\text{precession–nutation}}\;\vec{r}_{\mathrm{GCRS}}

GCRS →C→ CIRS →R₃(ERA)→ TIRS →W→ ITRS. X and Y are the CIP’s coordinates in the GCRS (they contain frame bias, precession and nutation together); s is a tiny angle that locates the CIO. IERS Conventions 2010, chapter 5.

💡Earth Orientation Parameters
Precession and nutation are computed from theory. The rest of Earth's orientation has to be measured, because it is driven by things like ocean currents, atmospheric winds and the liquid core. The IERS publishes these Earth Orientation Parameters (EOP) daily: UT1 − UTC — how far Earth's actual rotation is ahead of or behind atomic time. Always within 0.9 s while leap seconds exist. A 0.1 s error moves a point on the equator by 46 m. xₚ, yₚ — polar motion. The rotation axis wanders over the crust by a few tenths of an arcsecond — around 10 m on the ground. The main term is the Chandler wobble, a free oscillation of the whole Earth with a 433-day period, plus an annual term from seasonal mass shifts. dX, dY — small observed corrections to the modelled celestial pole. Bulletin A gives rapid values and predictions; Bulletin B and the C04 series give the final ones. Operational software downloads them every day. A frame conversion without EOP is accurate to roughly ten metres on Earth's surface — which is fine for a TV dish and fatal for laser ranging.
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Positions convert with a rotation matrix. Velocities do not, and this is the single most common bug in frame code. ITRS is turning at ω⊕\omega_\oplus relative to GCRS. A point that is fixed on the ground has zero velocity in ITRS, but moves at up to 465 m/s in GCRS. Rotating the vector is not enough: you also have to subtract the velocity the frame itself contributes. That is the transport theorem:
v⃗ITRS=M v⃗GCRS−ω⃗⊕×r⃗ITRS,ω⃗⊕=[0, 0, 7.2921159×10−5] rad/s\vec{v}_{\mathrm{ITRS}} = \mathbf{M}\,\vec{v}_{\mathrm{GCRS}} - \vec{\omega}_\oplus \times \vec{r}_{\mathrm{ITRS}}, \qquad \vec{\omega}_\oplus = [0,\,0,\,7.2921159\times10^{-5}]\ \mathrm{rad/s}

M is the full GCRS → ITRS matrix. For the ISS the correction is ~0.4 km/s. For a GEO satellite it cancels almost the entire 3.07 km/s, which is the point of GEO.

💡ITRS, ITRF, [WGS 84](https://en.wikipedia.org/wiki/World_Geodetic_System) — and why there is more than one
ITRS is the terrestrial reference system: origin at the centre of mass of the whole Earth including oceans and atmosphere; scale consistent with the SI metre; orientation continuous with the old BIH 1984 system; and a "no-net-rotation" condition so that, averaged over all plates, the frame does not spin relative to the crust. ITRF is its realisation, rebuilt every few years by combining four independent techniques — VLBI, satellite laser ranging, GNSS and DORIS — at a few hundred sites. Each station gets a position at a reference epoch and a velocity. ITRF2020 is the current one, and it now models seasonal motion and the slow relaxation of the crust after large earthquakes. WGS 84 is the US military realisation used by GPS. It has been re-aligned to the ITRF repeatedly; the January 2024 realisation, WGS 84 (G2296), agrees with ITRF2020 to within about 2 cm. For most purposes, WGS 84 is ITRF. For centimetre surveying, it is not. Regional datums (GDA2020 in Australia, ETRS89 in Europe, NAD83 in North America) are tied to one plate, so that coordinates on that plate barely change with time. That is the trade: a plate-fixed datum keeps your survey markers constant, while ITRF keeps the satellites honest.
🛰Measuring the sea to a millimetre
Global mean sea level is rising by roughly 3–4 mm per year, and the number comes from radar altimetry satellites (TOPEX/Poseidon, the Jason series, Sentinel-6 Michael Freilich) that measure the height of the ocean surface from orbit. Those heights are only as good as the satellite's orbit, and the orbit is only as good as the frame. An error of 1 mm/yr in where the ITRF puts Earth's centre of mass leaks almost directly into the sea-level trend. That is why the ITRF origin is maintained to a few millimetres, and its drift to under a millimetre per year: a climate record depends on the stability of a coordinate system.
Earth-fixed Cartesian coordinates are rarely what people want. They want latitude, longitude and height — and on an ellipsoid, latitude has two meanings.
Geocentric latitude ψ is the angle between the equator and the line to Earth's centre. Geodetic latitude φ is the angle between the equator and the normal to the ellipsoid — the local "up" a spirit level gives you (to within a few arcseconds of gravity anomaly). Maps, GPS receivers and launch-site coordinates all use geodetic latitude. The two differ by up to 11.5 arcminutes, which is 21 km on the ground.
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r⃗ECEF=[(N+h)cos⁡φcos⁡λ(N+h)cos⁡φsin⁡λ(N(1−e2)+h)sin⁡φ],N=a1−e2sin⁡2φ\vec{r}_{\mathrm{ECEF}} = \begin{bmatrix} (N + h)\cos\varphi\cos\lambda \\ (N + h)\cos\varphi\sin\lambda \\ \big(N(1-e^2) + h\big)\sin\varphi \end{bmatrix}, \qquad N = \frac{a}{\sqrt{1 - e^2\sin^2\varphi}}

WGS 84: a = 6378.137 km, f = 1/298.257223563, e² = f(2 − f). The inverse has no closed form worth remembering; iterate on φ (lab 3), which converges to 10⁻¹² rad in about four passes.

⚠Height above what?
A GNSS receiver natively computes ellipsoidal height h — height above the smooth mathematical WGS-84 ellipsoid. "Height above sea level" is height above the geoid, the lumpy surface of equal gravitational potential that the oceans would follow at rest. The two differ by the geoid undulation, which ranges from about −106 m (the Indian Ocean Geoid Low, just south of India and Sri Lanka) to +85 m (near New Guinea). A raw receiver on a beach in the middle of the Indian Ocean Geoid Low reports a height of about −100 m, and it is not broken. Every launch-site, ground-station and landing coordinate must say which height it means.
The last frame on the ground is the observer's own. Unit 3 used SEZ (south, east, zenith). Geodesy and most software use ENU (east, north, up); aviation and missiles use NED (north, east, down). They are the same frame with the axes relabelled — and a sign error between them is a classic. The ENU rotation follows directly from geodetic latitude and longitude:
ρ⃗ENU=[−sin⁡λcos⁡λ0−sin⁡φcos⁡λ−sin⁡φsin⁡λcos⁡φcos⁡φcos⁡λcos⁡φsin⁡λsin⁡φ](r⃗sat−r⃗site),az=atan2⁡(ρE,ρN),  el=arcsin⁡ρUρ\vec{\rho}_{\mathrm{ENU}} = \begin{bmatrix} -\sin\lambda & \cos\lambda & 0 \\ -\sin\varphi\cos\lambda & -\sin\varphi\sin\lambda & \cos\varphi \\ \cos\varphi\cos\lambda & \cos\varphi\sin\lambda & \sin\varphi \end{bmatrix}\big(\vec{r}_{\mathrm{sat}} - \vec{r}_{\mathrm{site}}\big), \qquad \mathrm{az} = \operatorname{atan2}(\rho_E, \rho_N),\ \ \mathrm{el} = \arcsin\frac{\rho_U}{\rho}

Both positions must be in the same Earth-fixed frame first. Use geodetic φ, not geocentric ψ — otherwise “up” tilts by up to 11.5′ and your antenna points beside the satellite.

💡The whole road, once
To point a dish at a satellite whose orbit you know in GCRS:
  1. Propagate the state to the pass time (Unit 5).
  2. Convert time: UTC → TT for the precession–nutation, UTC → UT1 (IERS) for the rotation angle.
  3. GCRS → ITRS: C(X, Y, s), then R₃(ERA), then W(xₚ, yₚ). Transport theorem for the velocity if you need Doppler.
  4. Station geodetic (φ, λ, h) → ITRS, using the station's ITRF position propagated to the date with its velocity.
  5. Difference, rotate into ENU, take azimuth and elevation.
Every one of those five lines is a place a real mission has lost a satellite.
↓But there is a problem
Every frame in this unit so far has been nailed to a planet: its centre, its spin axis, its crust. Now ride along with the satellite. Ahead of it, below it, out to the side — those are the directions that matter for a docking, a thruster firing, a collision warning. Operations teams have at least four different frames for "ahead, below and sideways", and they do not agree on which way is which.