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18-Geom-A3 Geodesy and Positioning · December 2014

Question 3 of 7: Height Systems — CGVD28 vs CGVD2013

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Exams, December 2014 — 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). 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); Snyder, Map Projections — A Working Manual (USGS PP 1395); Natural Resources Canada geodetic references for NAD83(CSRS) and CGVD2013. Canadian datums/regulators throughout (NRCan, Canadian Geodetic Survey).

Question 3: Height Systems — CGVD28 vs CGVD2013 (20 marks)

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 the terrain, geoid and ellipsoid.

Find. (a) the conceptual difference; (b) the height system each uses; (c) the reference surface of each; (d) their precision; (e) one simple CGVD28→CGVD2013 transformation method.

EllipsoidGeoid (N above ellipsoid)TerrainhNHh = N + H
Relationship among the surfaces: ellipsoidal height h, geoid undulation N and orthometric height H, with h = H + N. GNSS delivers h; a geoid model supplies N; the orthometric height H is referred to the vertical datum’s equipotential (geoid) surface.

(a) Fundamental difference. 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 small number of tide gauges, and it exists physically as the network of stamped bench marks. 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 geoid model CGG2013 and accessed by GNSS. The shift is from “heights are what the bench marks say” to “heights come from GNSS plus a published geoid model,” giving a consistent, nationwide, easily maintained datum.

(b) Height system. They are based on different height systems. CGVD28 heights are normal-orthometric: the levelled height differences were corrected with normal (theoretical) gravity rather than observed gravity, so they are only an approximation to true orthometric heights. CGVD2013 heights are (Helmert) orthometric heights \(H\) — height above the geoid measured along the plumb line, consistent with the real gravity field. They also differ in how the height is obtained: CGVD28 propagates heights through levelled height differences from the tide-gauge origin; 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 five/six 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) with the globally adopted geopotential value \(W_0 = 62\,636\,856.0\ \text{m}^2\text{s}^{-2}\), realized by the CGG2013 model.

(d) Precision. CGVD28 degrades with distance from the tide gauges: it carries systematic distortions reaching several decimetres (up to about 1 m) across the country because of accumulated levelling error and the non-equipotential origin. CGVD2013, being gravimetric and GNSS-accessible, is consistent nationwide at roughly the few-centimetre level (geoid model accuracy of a few cm), with no build-up over distance.

(e) Simple transformation. Apply the published height-difference between the two datums at the point of interest. The simplest practical method is to use NRCan’s transformation grid \(\delta H(\varphi,\lambda) = H_{\text{CGVD2013}} - H_{\text{CGVD28}}\) (distributed with GPS·H / TRX): 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.