Question 4 of 7: Map Projections — Elevation Factor and Laplace Correction

Question 4: Map Projections — Elevation Factor and Laplace Correction (20 marks — a 12, b 8)

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. Terrain distance/baseline and azimuth observations that must be reduced first to the reference ellipsoid before projection to a mapping plane.

Find. (a) the definition and use of the elevation factor (with a worked example); (b) the reason for the Complete Laplace Correction and its order of magnitude.

(a) Elevation factor. The elevation factor (sea-level or ellipsoid-reduction factor) is the multiplier that reduces a horizontal ground distance down to the ellipsoid. Geometrically it is the ratio of the ellipsoidal chord to the terrain distance, and for a mean Earth radius \(R\) and a station height \(H\) above the ellipsoid,

$$EF \;=\; \frac{R}{R+H}.$$

Approach. Apply \(EF\) to a representative ground line to show the reduction; the same factor multiplies GNSS baselines projected onto the ellipsoid.

1. Form the factor. For \(H=1500\ \text{m}\) and \(R=6371\ \text{km}\), $$EF=\frac{6\,371\,000}{6\,371\,000+1500}=\frac{6\,371\,000}{6\,372\,500}=\boxed{0.9997646}.$$
2. Reduce a ground line. A \(2000\ \text{m}\) horizontal ground distance becomes, on the ellipsoid, $$s_{\text{ell}}=2000\times0.9997646=1999.53\ \text{m},$$ a reduction of \(0.47\ \text{m}\) (\(\approx235\) ppm).

Its use: every horizontal ground distance (or GNSS baseline component) is multiplied by \(EF\) before projection, giving the corresponding length on the ellipsoid; the ellipsoidal length is later multiplied by the projection scale factor \(k\) to reach the grid, the two together forming the combined (grid) factor. Because \(H\) sits in the denominator, the higher the terrain the smaller the elevation factor — the reduction reaches hundreds of ppm in mountainous work and must never be neglected.

(b) Complete Laplace Correction — reason and magnitude. Reason: a total station or theodolite measures an astronomic azimuth — it is levelled to the local plumb line (the gravity vertical), not to the ellipsoidal normal. Because the two verticals differ by the deflection of the vertical, an observed astronomic azimuth must be corrected to a geodetic azimuth before it can be used on the ellipsoid. The Complete (full) Laplace Correction supplies that reduction:

$$\alpha \;=\; A \;-\; \eta\tan\varphi \;-\; \bigl(\xi\sin\alpha-\eta\cos\alpha\bigr)\cot z,$$

where \(A\) is the astronomic azimuth, \(\alpha\) the geodetic azimuth, \(\xi,\eta\) the deflection components and \(z\) the zenith distance of the target. The dominant term \(\eta\tan\varphi\) is the Laplace equation; the second (“complete”) term corrects for lines that are not horizontal and is needed for steep sights. Order of magnitude: deflection components are typically a few arc-seconds (up to \(\sim\!10''\) in rugged terrain); with \(\eta\sim5''\) at \(\varphi\approx49^\circ\), \(\eta\tan\varphi\approx5''\times1.15\approx6''\). So the correction is of order a few arc-seconds, up to \(\sim\!10''\) — small, but far larger than the sub-arc-second precision of modern geodetic azimuths, so it is mandatory in first-order work (a Laplace station).