Ellipsoidal Heights, Geoid Undulation, and Grid-to-Ground Factor on a 1.8 km Line
A GNSS static campaign established two control points 1.846 km apart inside a UTM zone. Post-processed ellipsoidal heights are hA = 152.340 m and hB = 148.910 m. The national hybrid geoid model gives undulations NA = -32.15 m and NB = -32.07 m. The UTM grid distance from the adjusted coordinates is 1,846.118 m; the mean grid scale factor along the line is 1.000405 and the mean ellipsoidal height is 150.6 m (R = 6,371,000 m). The drainage design that depends on these points requires orthometric height differences good to +/-0.02 m, and a contractor proposes to set out a 1.8 km pipeline using UTM grid bearings and grid distances taken directly from the coordinates.
1. Compute the orthometric heights of A and B and the orthometric height difference. Explain why it is not the same as the ellipsoidal height difference, and state the consequence for the drainage design.
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Orthometric heights (H = h - N):
- HA = 152.340 - (-32.15) = 184.490 m
- HB = 148.910 - (-32.07) = 180.980 m
- Orthometric difference HB - HA = -3.510 m
Ellipsoidal difference: 148.910 - 152.340 = -3.430 m.
The two differ by 0.080 m, which is exactly the change in geoid undulation along the line (-32.07 vs -32.15). The ellipsoidal difference measures separation between two ellipsoid-parallel surfaces and says nothing about gravity; water does not flow along it. Only the orthometric difference represents height above the geoid, which is what the hydraulic design uses.
Consequence: accepting the raw GNSS height difference would put the pipeline grade 80 mm out over 1.846 km - four times the stated +/-0.02 m tolerance. The geoid model is not an optional refinement here; it is mandatory, and its accuracy must also be justified (a 2-3 cm geoid error still consumes most of the tolerance).
2. Compute the combined scale factor and the ground distance between A and B. Quantify the error the contractor would introduce by setting out with grid distances.
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Elevation factor: R/(R + h) = 6,371,000 / (6,371,000 + 150.6) = 6,371,000/6,371,150.6 = 0.99997636.
Combined factor: k x EF = 1.000405 x 0.99997636 = 1.0003814.
Ground distance: 1,846.118 x 1.0003814 = 1,846.822 m.
Error if grid distance is used: 1,846.822 - 1,846.118 = 0.704 m too short over 1.846 km, i.e. about 381 ppm, or roughly 0.38 m per kilometre.
Judgement points to state explicitly:
- The distortion is one-signed and cumulative along the alignment, so it does not average out; a 1.8 km pipe laid to grid distances would finish 0.70 m from its designed endpoint.
- Bearings are also affected: the grid bearing differs from the geodetic azimuth by the meridian convergence, which is a rotation, not a scale change. Scale alone does not fix the setup.
- The correct field procedure is to convert design coordinates to ground by dividing by the combined factor (or work in a low-distortion ground projection) and to apply convergence to bearings.
3. The client asks whether the GNSS-derived orthometric heights can replace spirit-levelled bench marks for the drainage design. What do you advise?
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Not without verification, and the reasoning should be stated in error-budget terms:
- Geoid model error: national hybrid models are commonly good to 2-5 cm (1 sigma), and they may contain long-wavelength tilt that does not cancel between A and B. A 3 cm tilt over 1.8 km already exceeds the +/-0.02 m tolerance.
- GNSS height error: ellipsoidal heights from static GNSS are typically 2-3 times weaker than the horizontal solution, especially over short sessions.
- Combined: the RSS of geoid and GNSS error plausibly exceeds the design tolerance, whereas differential spirit levelling over 1.8 km is comfortably inside it (a few millimetres).
Recommendation: use the GNSS/geoid heights for reconnaissance and design screening, but tie the site to at least two spirit-levelled bench marks and carry a levelling loop through the critical structures. If GNSS heights are used for construction, state the method, the geoid model identity/version, the observation session lengths, and an estimated uncertainty in the report - and confirm a closed loop or a redundant check against a levelled bench mark before any grade is set.