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. Canada’s legacy vertical datum CGVD28 and its replacement CGVD2013 (adopted 2013), together with the geometric relationship among terrain, geoid and ellipsoid.
Find. (a) the conceptual difference and precision of each; (b) the height system each uses; (c) the reference surface of each; (d) one simple CGVD28→CGVD2013 transformation method.
(a) Fundamental difference and precision. CGVD28 is a levelling-based, benchmark-realized datum: it was established by decades of precise spirit levelling constrained to mean sea level observed at a few tide gauges, and it exists physically as the network of stamped bench marks. It degrades with distance from the tide gauges, carrying systematic distortions reaching several decimetres (up to about 1 m) across the country. CGVD2013 is a gravimetric, geoid-model-based datum: it is defined by a single equipotential surface of the Earth’s gravity field, realized through the CGG2013 geoid model and accessed by GNSS, and it is consistent nationwide at roughly the few-centimetre level with no build-up over distance. The conceptual shift is from “heights are what the bench marks say” to “heights come from GNSS plus a published geoid model.”
(b) Height system. CGVD28 is based on normal-orthometric heights: its levelled height differences were corrected with normal (latitude-dependent, ellipsoidal) gravity rather than observed gravity, because gravity values were not available along most of the levelling lines. CGVD2013 is based on orthometric heights \(H\) (height above the geoid along the plumb line, in the Helmert sense). The two also differ in how the height is obtained: CGVD28 propagates heights through the adjusted levelling network from the tide-gauge constraints; CGVD2013 obtains \(H = h - N\) from a GNSS ellipsoidal height \(h\) and the CGG2013 geoid undulation \(N\).
(c) Reference surface. CGVD28’s reference surface is mean sea level as determined at a handful of Canadian tide gauges around 1928, tied together by levelling — a surface that is not a true equipotential and is distorted by the levelling network. CGVD2013’s reference surface is a rigorously defined equipotential surface (geoid) fixed by the globally adopted geopotential value \(W_0 = 62\,636\,856.0\ \text{m}^2\,\text{s}^{-2}\), realized by the CGG2013 model.
(d) Simple transformation. The simplest practical method is to apply NRCan’s published height-transformation grid \(\delta H(\varphi,\lambda)=H_{\text{CGVD2013}}-H_{\text{CGVD28}}\): interpolate \(\delta H\) at the station’s latitude/longitude and add it to the CGVD28 height, \(H_{2013}=H_{28}+\delta H\). Equivalently, if a GNSS ellipsoidal height \(h\) is available, compute \(H_{2013}=h-N_{\text{CGG2013}}\) directly.
