Lecture 19 — Earth's Gravity Field & the Geoid

ESS 314 — Introduction to Geophysics
University of Washington · Spring 2026

Marine Denolle · Mon May 4, 2026 · JHN 111

By the end of this lecture, you should be able to…

  • [LO-19.1] Apply Newton's law on a spherical Earth: g=GME/RE 2g = G M_{E}/R_{E}^{\,2}.
  • [LO-19.2] Distinguish sphere ↔ ellipsoid ↔ geoid as three approximations to Earth's shape.
  • [LO-19.3] Apply the four gravity corrections — latitude, free-air, Bouguer, terrain.
  • [LO-19.4] Compute representative magnitudes (mGal) for realistic stations.

Course objectives addressed: LO-1, LO-2, LO-4.

A motivating observation

Two stations on the same line of longitude:

  • One at the foot of Mt. Rainier
  • One on Puget Sound, 30 km away

Both gravimeters read gg to ±10−8\pm 10^{-8} relative precision. They differ by tens of mGal. Why?

→ Three contributors: Earth's shape, station elevation, local geology.

Newton's law of universal gravitation

F=G m1m2r2,G=6.674×10−11 m3kg−1s−2F = G \, \frac{m_{1} m_{2}}{r^{2}}, \qquad G = 6.674 \times 10^{-11} \text{ m}^{3} \text{kg}^{-1} \text{s}^{-2}

Force on a unit test mass → gravitational acceleration:

g(r)=GMr 2g(r) = \frac{G M}{r^{\,2}}

The acceleration is the same for every test mass — a fact unknown until Galileo, central to general relativity.

Gravity is a field

Conservative force ⇒ potential UU exists with g=−∇U\mathbf{g} = -\nabla U.

U(r)=− G∫Vρ(r′)∣r−r′∣ dV′U(\mathbf{r}) = -\,G \int_{V} \frac{\rho(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\, dV'

  • Equipotentials are perpendicular to g\mathbf{g} everywhere.
  • A still water surface follows an equipotential — this is what "horizontal" means.
  • Plumb line points along g\mathbf{g} — this is what "vertical" means.

How gg varies with distance

Inverse-square law and free-air gradient

Left: g(r)=GME/r2g(r) = G M_{E}/r^{2}, log-log. Right: linear free-air gradient −0.3086 -0.3086\,mGal m⁻¹ near surface.

Numbers to carry

  • gg at sea level ≈9.81\approx 9.81 m s⁻² ≡\equiv 981 000 mGal.
  • gg at the summit of Everest ≈9.78\approx 9.78 m s⁻² — about 2.7 Gal lower.
  • gg at the ISS orbit (~408 km) ≈8.66\approx 8.66 m s⁻² — astronauts are not weightless, they are in free fall.
  • gg at geostationary orbit ≈0.22\approx 0.22 m s⁻².

The ISS is not in deep space. gg falls off slowly.

Three approximations to Earth's shape

Sphere, oblate spheroid, geoid

Each panel is the "next correction" on the previous one.

  • (a) Sphere (R=6371R = 6371 km): good to a few parts in 10310^{3}.
  • (b) Oblate spheroid (f=1/298.257f = 1/298.257): rotation flattens the poles by ~21 km.
  • (c) Geoid: real equipotential, lumpy at the ±100 m level globally.

Theoretical (normal) gravity at the ellipsoid

gn(φ)≈978 032.7 (1+5.30 ⁣× ⁣10−3sin⁡2φ−5.82 ⁣× ⁣10−6sin⁡22φ)  mGalg_{n}(\varphi) \approx 978\,032.7 \, \bigl( 1 + 5.30 \!\times\! 10^{-3} \sin^{2}\varphi - 5.82 \!\times\! 10^{-6} \sin^{2}2\varphi \bigr) \;\text{mGal}

  • Equator: gn≈978 033g_{n} \approx 978\,033 mGal
  • Poles: gn≈983 219g_{n} \approx 983\,219 mGal
  • Total equator-to-pole difference: ~5186 mGal, dominated by rotation.

Subtract this off: Δgraw=gobs−gn(φ)\Delta g_{\text{raw}} = g_{\text{obs}} - g_{n}(\varphi) — the latitude correction.

Measuring gg

Absolute gravimeters. Free-fall timed by laser interferometry. Accuracy ~5–10 µGal. Bulky, expensive.

Relative gravimeters. A test mass on a precisely calibrated "zero-length spring." Read differences between stations. Accuracy ~10 µGal. Field-portable.

The two methods are complementary: absolute instruments anchor the network; relative instruments fill it in.

Why we need 4 decimal places

A gravimeter measures gg to ~1 part in 10810^{8}.
Geological signals are 0.1–100 mGal.
Elevation effects can be 300+ mGal.

→ Without careful corrections, the geological signal is lost in the noise of station placement.

The next four slides take a single transect and walk through each correction.

The reduction chain

Topography, raw deviation, free-air & simple Bouguer, complete Bouguer

A buried 5-mGal target sits underneath a 300-mGal elevation effect. Each correction below removes a known, predictable contribution.

Free-air correction

A station at elevation hh is farther from Earth's centre, so gg is smaller.

dgdr=−2gr=− 0.3086 mGal m−1\frac{dg}{dr} = -\frac{2 g}{r} = -\,0.3086 \text{ mGal m}^{-1}

→ Add 0.3086 h0.3086\, h mGal back to the observation.

The free-air anomaly:

ΔgFA=gobs−gn(φ)+0.3086 h\Delta g_{FA} = g_{\text{obs}} - g_{n}(\varphi) + 0.3086\, h

Bouguer (slab) correction

There is rock between the station and the geoid, pulling the gravimeter down.

Approximate it as an infinite horizontal slab:

gslab=2πGρch≈0.0419 ⁣× ⁣10−3ρch mGalg_{\text{slab}} = 2\pi G \rho_{c} h \approx 0.0419\!\times\!10^{-3} \rho_{c} h \text{ mGal}

Subtract this from ΔgFA\Delta g_{FA}:

ΔgB=ΔgFA−0.0419 ⁣× ⁣10−3ρch\Delta g_{B} = \Delta g_{FA} - 0.0419\!\times\!10^{-3} \rho_{c} h

For "normal" crust (ρc=2.67\rho_{c}=2.67): combined elevation correction is ~0.197 mGal m⁻¹.

Terrain correction

The infinite-slab assumption fails in real terrain.

  • Mountains above the station pull up and laterally (away).
  • Valleys below the station leave a mass deficit.

→ Always add a positive correction. Today, computed numerically from a DEM.

The complete Bouguer anomaly:

ΔgCB=gobs−gn+FAcorr−Bcorr+Tcorr\Delta g_{CB} = g_{\text{obs}} - g_{n} + \text{FA}_{\text{corr}} - B_{\text{corr}} + T_{\text{corr}}

By construction, ΔgCB\Delta g_{CB} is due to lateral density variations near the survey — the geological target.

The reduction chain — what's left

After all four corrections, the residual reveals the buried body that motivated the survey.

In Lecture 20: how to read this residual to recover subsurface structure.

Worked example — alpine station at 45°N, 2000 m

Latitude term: gn(45∘)≈980 629g_{n}(45^{\circ}) \approx 980\,629 mGal.
Free-air: +0.3086×2000=+617+0.3086 \times 2000 = +617 mGal.
Bouguer (ρc=2.67\rho_{c}=2.67): −0.0419 ⁣× ⁣10−3×2.67×2000=−224-0.0419\!\times\!10^{-3}\times 2.67 \times 2000 = -224 mGal.
Terrain (rough alpine): ∼+10\sim +10 mGal.

Total elevation correction: ∼+400\sim +400 mGal. A 5-mGal target is two orders of magnitude smaller than the corrections.

→ Why precision and care in the reduction chain are non-negotiable.

Forward problem — what does the model predict?

Given:

  • Latitude φ\varphi
  • Elevation hh
  • Density model ρ(r)\rho(\mathbf{r}) for rock above the geoid

→ The reduction chain is a deterministic forward map.

The "inverse" problem of this lecture is mostly bookkeeping: extract the residual. The interesting inverse problem comes in Lecture 20.

Non-uniqueness — the punchline

Gravity is an integral of ρ\rho. Two density distributions can give the same surface gg.

Two specific ambiguities:

  • Depth ↔ density: deeper, denser body looks the same as shallower, lighter body.
  • Reduction density: choosing a wrong ρc\rho_{c} creates topography-correlated artifacts.

→ Gravity alone is rarely enough. Joint with seismic, magnetic, geological constraints.

Course connections

  • Backward: seismic tomography (L11–12) gives velocity → density via empirical scaling.
  • Forward: L20 — read residual anomalies for subsurface structure.
  • Forward: L21 — long-wavelength Bouguer signal = isostatic compensation.

Research horizon — gravity from space

GRACE-FO (NASA/GFZ, 2018–): two satellites tracked to fraction-of-µm separation. Measures time-varying gravity → ice loss, water storage, slow-slip events.

Open access: Tapley et al. (2019) Nature Climate Change, https://doi.org/10.1038/s41558-019-0456-2

Centimetre-level static geoid is now achievable. (Pail et al. 2023 review, open access.)

Societal relevance — the Earth is losing mass

Greenland ice loss measured by GRACE: ~270 Gt yr⁻¹, accelerating.

For the PNW:

  • The geoid itself moves at mm/decade — affects regional sea-level interpretation.
  • Aquifer-storage changes (Columbia Plateau) are reaching detection threshold for airborne gravity.

USGS SIR 2010-5101 (public domain) is the open-access entry point.

AI Literacy — AI as a tool

Two productive uses of ML in modern gravity:

  • Edge detection in gridded gravity-gradient data (CNNs).
  • Ensemble inversion: an ensemble of trained NNs produces a distribution of plausible models, capturing non-uniqueness explicitly.

ML cannot solve the underlying inverse-problem ambiguity (Green's third identity is mathematical, not algorithmic). But it can make the ambiguity visible by sampling the posterior.

Concept Check

  1. A free-air anomaly of +200+200 mGal at a 3-km alpine station; the simple Bouguer at the same station is −150-150 mGal. What does this tell you about local compensation?
  2. Sketch qualitatively the free-air and Bouguer anomalies you expect across a 4-km-deep ocean trench.
  3. Two stations at the same elevation and latitude on opposite sides of a vertical fault have Bouguer anomalies differing by 5 mGal. Can the data alone tell you the throw? What additional measurement would constrain it?