18-Geom-A3 Geodesy and Positioning · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, December 2017 — 3 hours, closed book (approved Casio/Sharp calculators only). SEVEN numbered questions; six constitute a complete paper and each is of equal value (20 marks); the per-part marking scheme printed on page 4 of the paper is reproduced in each answer. Most answers are required in essay format. All seven questions are solved below for completeness.
Reference texts: Vaníček & Krakiwsky, Geodesy: The Concepts (2nd ed., North-Holland); Hofmann-Wellenhof, Lichtenegger & Wasle, GNSS — Global Navigation Satellite Systems (Springer, 2008); Torge & Müller, Geodesy (4th ed., de Gruyter); Heiskanen & Moritz, Physical Geodesy (Freeman); Snyder, Map Projections — A Working Manual (USGS PP 1395); Natural Resources Canada geodetic references for NAD83(CSRS), CGVD2013 and the CGG2013 geoid model. Canadian datums/regulators throughout (NRCan, Canadian Geodetic Survey).
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, and the determination of the geoid by Stokes’ integral from gravity data.
Find. (a) the generic and free-air gravity anomalies; (b) the two observation types feeding Stokes’ formula and their use; (c) the accuracy of a modern geoid and what limits it; (d) why a geoid model is needed in geodetic computations.
(a) Gravity anomaly — generic and free-air. In its most generic form the gravity anomaly is the difference between the Earth’s actual gravity and the normal (reference-ellipsoid) gravity evaluated at corresponding points:
$$\Delta g \;=\; g_P \;-\; \gamma_Q,$$
where \(g_P\) is the observed gravity at a point \(P\) (classically on the geoid) and \(\gamma_Q\) is the normal gravity produced by the level ellipsoid at the corresponding point \(Q\) (on the ellipsoid, or on the telluroid in the modern Molodensky sense). The free-air gravity anomaly specialises this: the surface-measured gravity \(g_{\text{obs}}\) is reduced down to the geoid using only the free-air (height) correction — the vertical gradient of normal gravity, \(0.3086\ \text{mGal/m}\) — with no account of the masses between the point and the geoid, and normal gravity is then subtracted:
$$\Delta g_{\text{FA}} \;=\; g_{\text{obs}} \;+\; 0.3086\,H \;-\; \gamma_0 ,$$
with \(H\) the orthometric height in metres and \(\gamma_0\) the normal gravity on the ellipsoid at the station’s latitude. Because it omits the Bouguer (mass) term, the free-air anomaly is the anomaly used directly in Stokes’ formula for geoid computation.
(b) The two data types for Stokes’ formula. Stokes’ formula recovers the geoid undulation \(N\) (spherical approximation) as
$$N \;=\; \frac{R}{4\pi\gamma}\iint_{\sigma} S(\psi)\,\Delta g\; d\sigma ,$$
a global integral of the gravity anomalies weighted by Stokes’ kernel \(S(\psi)\). Its practical evaluation requires two fundamental observation types. (1) Gravity observations — measured values of gravity \(g\) (from terrestrial, marine, airborne and, for the long wavelengths, satellite gravimetry), which after reduction become the free-air anomalies \(\Delta g\) that form the integrand; ideally they must cover the entire globe because Stokes’ kernel does not vanish with distance. (2) The positions and heights of the gravity stations — the geodetic latitude/longitude and elevation of each measurement; these are needed both to reduce the raw gravity to an anomaly (the free-air correction and the latitude-dependent normal gravity \(\gamma_0\)) and to compute the spherical distance \(\psi\) between the computation point and each data element that enters the kernel \(S(\psi)\). In short: gravity supplies the values to be integrated, positions/heights supply the reduction and the geometry of the integration.
(c) Accuracy of a modern geoid and its limits. A modern regional geoid model — 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. The accuracy is limited by: (i) incomplete and uneven gravity coverage — gaps and errors in historical terrestrial data, and sparse data over oceans, mountains and remote regions; (ii) datum and systematic errors in the assembled gravity databases; (iii) the limited spatial resolution of satellite gravity for the short wavelengths, which must be supplied by terrestrial data and a digital 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 gravity (systematic airborne gravimetry), better global models from new gravity-field satellite missions, and improved high-resolution terrain-density models.
(d) 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 a modern geoid 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 the various height systems.
