Gravity Field and Geoid: 18-Geom-A3 Geodesy and Positioning, December 2018, Q2

Question 2: Gravity Field and Geoid (20 marks — a 4, b 4, c 4, d 4, e 4)

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 Earth’s gravity field described relative to a normal (ellipsoidal) field; the distinction between the full gravity field (gravitation + centrifugal) and the pure gravitational field; and the geoid as an equipotential reference surface.

Find. (a) the generic and refined-Bouguer gravity anomalies; (b) the accuracy of a modern geoid and what limits it; (c) why a geoid model is needed; (d) why gravity plumb lines curve and whether gravitational plumb lines do; (e) the deflection of the vertical and a survey-reduction use.

(a) Gravity anomaly — generic and refined Bouguer. In its most generic form the gravity anomaly is the difference between the Earth’s actual gravity and the normal (reference-ellipsoid) gravity at corresponding points:

$$\Delta g \;=\; g_P \;-\; \gamma_Q,$$

where $g_P$ is observed gravity at a point $P$ (classically reduced to the geoid) and $\gamma_Q$ is the normal gravity of the level ellipsoid at the corresponding point $Q$. The Bouguer anomaly builds on this by first applying the free-air (height) reduction and then removing the attraction of the topographic masses between the station and the geoid, modelled as an infinite slab of thickness $H$: $\Delta g_B=g_{\text{obs}}+0.3086\,H-0.0419\,\rho\,H-\gamma_0$ (mGal, $H$ in m, density $\rho$ in g/cm³). The refined (complete) Bouguer anomaly adds a terrain correction $\delta g_T$ that accounts for the departure of the real topography from that flat slab (the pull of hills above the station and the missing mass of valleys below it), so that

$$\Delta g_{B,\text{ref}} \;=\; g_{\text{obs}} \;+\; 0.3086\,H \;-\; 0.0419\,\rho\,H \;+\; \delta g_T \;-\; \gamma_0 .$$

The refined Bouguer anomaly therefore isolates, as cleanly as the slab-plus-terrain model allows, the effect of subsurface density — which is why it is the anomaly used in geophysical interpretation, whereas the free-air anomaly is used in Stokes’ geoid integral.

Gravity anomaly Δg = g (actual, at P) − γ (normal, at the corresponding ellipsoid point Q). The refined Bouguer anomaly additionally strips the free-air, Bouguer-slab and terrain-correction effects to isolate subsurface density.

(b) Accuracy of a modern geoid and its limits. A modern regional geoid — such as Canada’s CGG2013, built by the remove–compute–restore technique from satellite (GRACE/GOCE) plus dense terrestrial/airborne gravity — is typically accurate to the few-centimetre level (about 1–3 cm) in well-surveyed areas, degrading to the decimetre level where gravity coverage is sparse. Its accuracy is limited by: (i) incomplete/uneven gravity coverage (gaps and errors in historical data, sparse data over oceans, mountains and remote regions); (ii) datum and systematic errors in the assembled gravity databases; (iii) the limited short-wavelength resolution of satellite gravity, which must be filled by terrestrial data and a terrain-density model; (iv) terrain and crustal-density uncertainty in the topographic reduction; and (v) truncation of the Stokes integral to a finite cap. To improve future geoids we would need denser and more accurate (systematic airborne) gravimetry, better global models from new gravity-field satellite missions, and improved high-resolution terrain-density models.

(c) Why a geoid model is needed. The geoid is the reference surface for physical (orthometric) heights, whereas GNSS delivers only geometric ellipsoidal heights $h$. A geoid model supplies the undulation $N$ that links the two through $H = h - N$, so it is what lets a GNSS user obtain the orthometric heights required for engineering, drainage and levelling — and it is precisely how CGVD2013 is realized. More broadly the geoid is the physical, equipotential vertical datum to which heights refer (water flows from high $H$ to low $H$), and it is needed to reduce observations from the terrain to the ellipsoid and to convert between height systems.

(d) Why plumb lines curve. A plumb line is everywhere tangent to the local field vector and orthogonal to the equipotential surfaces. Gravity plumb lines are curved for two reasons (2 points): (1) the equipotential surfaces of the real field are not parallel — they converge and diverge because gravity varies with latitude and height — so a line kept perpendicular to them must bend; and (2) the field is irregular, being perturbed by lateral density variations and topography, which deflects the vertical from place to place. The same is true of the gravitational field (gravitation only, centrifugal force removed): its plumb lines are also curved (3 points), because gravitation still varies with position over a non-spherical, laterally heterogeneous Earth, so its equipotential surfaces are likewise non-parallel and irregular. Removing the centrifugal part changes the shape and orientation of the surfaces (and eliminates the smooth latitude-driven part of the curvature) but does not make them parallel; only a perfectly spherical, radially symmetric mass would give straight, radial plumb lines.

Deflection of the vertical θ: the angle between the true plumb line (along the local gravity vector) and the ellipsoidal normal. Its N–S and E–W components are ξ and η.

(e) Deflection of the vertical and its use. The geoid deflection of the vertical is the angle between the direction of the actual gravity vector (the plumb line / astronomic vertical) and the normal to the reference ellipsoid at a point. It is resolved into a north–south component $\xi$ (= astronomic minus geodetic latitude, $\Phi-\varphi$) and an east–west component $\eta$ ($=(\Lambda-\lambda)\cos\varphi$); typical values are a few arc-seconds, reaching $\sim\!10''$ in rugged terrain. Common application: in reducing terrestrial observations from the terrain to the ellipsoid we must relate astronomic directions (which a theodolite/total station references to the plumb line) to geodetic ones (referenced to the ellipsoidal normal). The deflection enters the reduction of an astronomic azimuth to a geodetic azimuth through the Laplace correction $A-\alpha=\eta\tan\varphi+(\xi\sin\alpha-\eta\cos\alpha)\cot z$, and it also corrects observed zenith/vertical angles; without it, precise azimuths and triangulation would carry an arc-second-level systematic error.

Concept & Theory

Topic. Gravity anomalies (generic, Bouguer, refined Bouguer), geoid accuracy, curved plumb lines, and the deflection of the vertical.

Key relations. Generic anomaly $\Delta g=g_P-\gamma_Q$; refined Bouguer $\Delta g_{B,\text{ref}}=g_{\text{obs}}+0.3086H-0.0419\rho H+\delta g_T-\gamma_0$; heights $h=H+N$; deflection components $\xi=\Phi-\varphi$, $\eta=(\Lambda-\lambda)\cos\varphi$; Laplace $A-\alpha=\eta\tan\varphi+\dots$.

Why this works. The disturbing potential $T$ (actual minus normal) makes the real equipotential surfaces non-parallel and irregular, so field lines (plumb lines) curve and tilt away from the ellipsoidal normal — that tilt is the deflection of the vertical, and its integral over the globe is what a geoid model captures. Removing centrifugal force (gravitation only) does not restore parallel surfaces, so those plumb lines curve too.

Common pitfall. Using a Bouguer anomaly in Stokes’ formula (it needs the free-air anomaly); and assuming gravitational plumb lines are straight — only a radially symmetric sphere would give that.

Source. Heiskanen & Moritz, Physical Geodesy, ch. 2–3 (anomalies, Bouguer/terrain, deflections); Torge & Müller, Geodesy, ch. 4–6; NRCan CGG2013 documentation.

Question 2 of 7: Gravity Field and Geoid

Question 2: Gravity Field and Geoid (20 marks — a 4, b 4, c 4, d 4, e 4)