18-Geom-A3 Geodesy and Positioning · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, May 2015 — 3 hours, closed book (approved Casio/Sharp calculators only). EIGHT numbered questions; six constitute a complete paper and each is of equal value. Most answers are required in essay format — clarity and organization are marked. All eight questions are solved below for completeness.
Reference texts: Vaníček & Krakiwsky, Geodesy: The Concepts (2nd ed., North-Holland); Torge & Müller, Geodesy (4th ed., de Gruyter); Hofmann-Wellenhof, Lichtenegger & Wasle, GNSS — Global Navigation Satellite Systems (Springer, 2008); Snyder, Map Projections — A Working Manual (USGS PP 1395); Heiskanen & Moritz, Physical Geodesy (Freeman, 1967); Natural Resources Canada / Canadian Geodetic Survey references for NAD83(CSRS), CGVD2013 and CGG2013. Canadian datums and regulators throughout (NRCan; Ontario CORS network).
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 geopotential number and the physical height systems (dynamic, orthometric) built from it, together with Canada’s change of vertical datum from CGVD28 to CGVD2013.
Find. (a) definition and physical meaning of the geopotential number; (b) whether/how it is observable; (c) definitions of dynamic and orthometric height; (d) their conceptual difference; (e) whether orthometric height is exactly computable; (f) the fundamental difference between CGVD28 and CGVD2013.
(a) Geopotential number. The geopotential number \(C\) of a point \(P\) is the difference in gravity potential \(W\) between the geoid (\(W_0\)) and the point (\(W_P\)): \(C = W_0 - W_P = \int_0^P g\,\mathrm{d}n\), where \(g\) is gravity along the plumb line and \(\mathrm{d}n\) is the levelled increment. It is a potential difference (units \(\text{m}^2\,\text{s}^{-2}\), or the geopotential unit g.p.u. \(=10\,\text{m}^2\,\text{s}^{-2}\)), not a length. Physically it expresses the work per unit mass needed to raise a body from the geoid to the point against gravity; because it is a potential difference it is unique to the point and independent of the levelling path taken to reach it.
(b) Determining \(C\) by observation. Yes. \(C\) is obtained by combining spirit levelling with gravity measurements: along a levelling line we measure the raw height increments \(\delta n\) and, at each set-up, the gravity \(g\) (with a gravimeter). The geopotential number then accumulates as the sum \(C=\sum \bar{g}\,\delta n\) of gravity times levelled increment from a benchmark on the geoid to the point. Because \(g\,\delta n\) is an exact potential increment, the result is path-independent (loop misclosures theoretically vanish), which is exactly why \(C\) — not the raw levelled height — is the rigorous observable of vertical geodesy.
(c) Dynamic and orthometric heights. The dynamic height is the geopotential number scaled by a single constant reference gravity: \(H^{\text{dyn}} = C/\gamma_0\), where \(\gamma_0\) is normal gravity at a chosen standard latitude (usually 45°). The orthometric height is the geometric distance from the geoid to the point measured along the (curved) plumb line, obtained as \(H = C/\bar{g}\), where \(\bar{g}\) is the mean actual gravity along the plumb line between the geoid and the point (in the Helmert realization, \(\bar g \approx g + 0.0424\,H\) with \(g\) in gal and \(H\) in km, i.e. 0.0424 mGal per metre of height).
(d) Conceptual difference. The dynamic height divides \(C\) by a constant gravity, so points on the same equipotential surface receive exactly the same dynamic height — it is a “levelled”, water-consistent height but is not a geometric distance (it carries a correction, not a metric length). The orthometric height divides \(C\) by the true mean gravity along the plumb line, so it is a real geometric height above the geoid; but because \(\bar g\) varies from place to place, two points on the same equipotential surface can have slightly different orthometric heights. In short: dynamic height is geometrically fictitious but physically consistent; orthometric height is geometrically real but not perfectly equipotential-consistent.
(e) Accuracy of orthometric height. Not exactly. The orthometric height requires the mean actual gravity along the plumb line inside the topography, which cannot be measured directly — it depends on the unknown density distribution of the rock between the geoid and the surface. It is therefore always approximated (e.g. by the Helmert Poincaré–Prey reduction assuming a standard crustal density of 2670 kg m−3). The residual density uncertainty makes the orthometric height inexact at the millimetre-to-few-centimetre level in rugged, high terrain, whereas the geopotential number and the dynamic height (which need only surface gravity) are rigorously determinable.
(f) CGVD28 vs CGVD2013. The fundamental difference is how the reference surface is defined and realized: CGVD28 is a levelling-and-tide-gauge datum (mean sea level at a few tide gauges, propagated across the country by spirit levelling and materialized as bench marks), whereas CGVD2013 is a gravimetric geoid-based datum (a single defined equipotential surface, \(W_0=62\,636\,856.0\ \text{m}^2\text{s}^{-2}\), realized by the CGG2013 geoid model and accessed by GNSS as \(H=h-N\)). CGVD28 heights are also normal-orthometric (levelled differences corrected with normal gravity only), whereas CGVD2013 heights are Helmert orthometric heights referred to the geoid. The shift is from “heights are what the bench marks say” to “heights come from GNSS plus a published geoid model.”
