Height Systems — CGVD28 vs CGVD2013: 18-Geom-A3 Geodesy and Positioning, December 2018, Q3

Question 3: Height Systems — CGVD28 vs CGVD2013 (20 marks — a 5, b 5, c 5, d 5)

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); precise geodetic levelling as the classical means of obtaining physical heights; and levelled height differences of up to 2 km (mountainous) versus ≤100 m (flat).

Find. (a) the conceptual difference and precision of each datum; (b) the height system each uses; (c) the fundamental measurements needed for orthometric/dynamic heights; (d) the order of magnitude of the orthometric correction in mountainous versus flat terrain.

Ellipsoidal height h, geoid undulation N and orthometric height H, with h = H + N. GNSS delivers h; the CGG2013 geoid supplies N; the orthometric height H (the CGVD2013 datum) is referred to the geoid’s equipotential surface.

(a) Fundamental difference and precision. CGVD28 is a levelling-based, benchmark-realized datum: established by decades of precise spirit levelling constrained to mean sea level at a few tide gauges, and existing physically as the network of stamped bench marks. It degrades with distance from the gauges, carrying systematic distortions of several decimetres (up to ~1 m) across the country. CGVD2013 is a gravimetric, geoid-model-based datum: defined by a single equipotential surface of the gravity field (fixed by $W_0=62\,636\,856.0\ \text{m}^2\text{s}^{-2}$), realized through the CGG2013 geoid and accessed by GNSS, and consistent nationwide at 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. The two datums use different height systems. CGVD28 heights are normal-orthometric: the levelled height differences were corrected with normal (theoretical ellipsoidal) gravity rather than observed gravity — the normal-orthometric correction — and propagated from the tide-gauge constraints, so they are not true orthometric heights. CGVD2013 heights are (Helmert) orthometric heights $H$ — height above the $W_0$ geoid along the curved plumb line — obtained as $H=h-N$ from a GNSS ellipsoidal height $h$ and the CGG2013 undulation $N$.

(c) Fundamental measurements for orthometric/dynamic heights. Precise geodetic levelling of these physical height systems requires two fundamental measurement types along the line: (1) levelled height differences $\delta n$ — the raw back-sight − fore-sight differences of precise spirit (or digital) levelling between consecutive bench marks; and (2) surface gravity $g$ observed (or interpolated from a gravity model) at those bench marks. Combining them gives the path-independent geopotential number $C=\int g\,dn$, from which orthometric height follows as $H=C/\bar g$ (with $\bar g$ the mean gravity along the plumb line) and dynamic height as $H^{\text{dyn}}=C/\gamma_{45}$ (a single reference gravity). Gravity is what makes the height physically meaningful and path-independent; levelling alone is not enough.

(d) Order of magnitude of the orthometric correction. The orthometric correction (OC) converts a levelled height difference into an orthometric height difference; it exists because level surfaces are not parallel, so its magnitude scales with the height difference and with the departure of gravity from normal, roughly $OC\sim(\Delta g/\gamma)\,\Delta H$.

Approach. Estimate $OC$ from a representative gravity anomaly $\Delta g$, normal gravity $\gamma\approx 981\,000$ mGal, and the two given height differences.

Mountainous case ($\Delta H\approx2$ km). In the Rockies characteristic anomalies are of order $\Delta g\sim100$ mGal, so $$OC_{\text{mtn}}\;\approx\;\frac{\Delta g}{\gamma}\,\Delta H\;=\;\frac{100}{981\,000}\times 2000\ \text{m}\;=\;\boxed{0.2\ \text{m}}\ \text{(a few decimetres)}.$$ This is large enough that it must be applied in precise mountain levelling.
Flat case ($\Delta H\le100$ m). With gentler relief $\Delta g\sim30$ mGal and a small height difference, $$OC_{\text{flat}}\;\approx\;\frac{30}{981\,000}\times 100\ \text{m}\;=\;\boxed{3\ \text{mm}}\ \text{(a few millimetres)},$$ i.e. essentially negligible for most work.

So the orthometric correction is of the order of a decimetre or two in the mountains and a few millimetres in flat terrain — roughly two orders of magnitude apart, driven by both the larger height difference and the larger gravity anomalies in rugged country.

Check

These are order-of-magnitude estimates using representative anomalies (Rockies $\sim$100 mGal, plains $\sim$30 mGal); the exact correction for a specific line must be computed from the actual observed gravity and levelling data along that line.

Quantity	Value / order of magnitude
Orthometric correction, mountainous line ($\Delta H\approx2$ km)	~0.2 m (decimetres)
Orthometric correction, flat line ($\Delta H\le100$ m)	~3 mm (millimetres)
Height system realized	CGVD28: normal-orthometric / CGVD2013: Helmert orthometric
CGVD28 precision / CGVD2013 precision	decimetres–1 m / few cm

Concept & Theory

Topic. Vertical datums, geopotential numbers, orthometric/dynamic heights, and the orthometric correction to levelling.

Key relations. $h=H+N$; geopotential number $C=\int g\,dn$; orthometric $H=C/\bar g$ (Helmert $\bar g=g+0.0424H$, $g$ in gal, $H$ in km); dynamic $H^{\text{dyn}}=C/\gamma_{45}$; order of magnitude $OC\sim(\Delta g/\gamma)\Delta H$; $W_0=62\,636\,856.0\ \text{m}^2\text{s}^{-2}$.

Why this works. Because equipotential surfaces converge, a purely geometric sum of levelled differences is path-dependent; weighting each $\delta n$ by gravity gives the path-independent geopotential number, and dividing by an appropriate gravity value yields a consistent height. The correction that turns raw levelling into orthometric height therefore grows with both the height climbed and the local gravity anomaly — hence large in mountains, tiny on the plains.

Common pitfall. Adding levelled height differences without the gravity-based orthometric correction over large elevation changes; and assuming CGVD28 is an equipotential surface — it is not, which is why it is distorted.

Source. Heiskanen & Moritz, Physical Geodesy, ch. 4 (heights, orthometric correction); NRCan Canadian Geodetic Survey, CGVD2013 / CGG2013 documentation.

Quantity	Value / order of magnitude
Orthometric correction, mountainous line (\(\Delta H\approx2\) km)	~0.2 m (decimetres)
Orthometric correction, flat line (\(\Delta H\le100\) m)	~3 mm (millimetres)
Height system realized	CGVD28: normal-orthometric / CGVD2013: Helmert orthometric
CGVD28 precision / CGVD2013 precision	decimetres–1 m / few cm

Question 3 of 7: Height Systems — CGVD28 vs CGVD2013

Question 3: Height Systems — CGVD28 vs CGVD2013 (20 marks — a 5, b 5, c 5, d 5)