Question 4 of 7: Map Projections — Elevation, Scale and Grid Factors; MTM

Question 4: Map Projections — Elevation, Scale and Grid Factors; MTM (20 marks — a 8, b 4, c 5, d 3)

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 observations that must be reduced first to the ellipsoid and then to a transverse-Mercator grid (UTM, and the Canadian MTM/3TM).

Find. (a) definition and use of the elevation factor; (b) definition and use of the scale factor; (c) the grid (combined) factor; (d) what MTM/3TM is and two conceptual differences from UTM.

(a) Elevation factor. The elevation factor (also called the 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,

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

Its use: every horizontal ground distance is multiplied by \(EF\) before projection, giving the corresponding distance on the ellipsoid. As a worked example, for \(H = 1000\ \text{m}\) and \(R = 6371\ \text{km}\), \(EF = 6\,371\,000/6\,372\,000 = \boxed{0.9998431}\); a 1000 m ground line becomes \(1000 \times 0.9998431 = 999.843\ \text{m}\) on the ellipsoid. Because \(H\) sits in the denominator, the higher the terrain the smaller the elevation factor — the reduction can reach hundreds of ppm in mountainous work and must never be neglected.

(b) Map projection scale factor. The scale factor \(k\) is the ratio of a distance on the projection (grid) to the corresponding distance on the ellipsoid at a point: \(k=\dfrac{\text{grid distance}}{\text{ellipsoidal distance}}\). Because a curved surface cannot be flattened without distortion, \(k\) is point-dependent: on a transverse-Mercator projection it is a minimum on the central meridian (UTM \(k_0=0.9996\)) and grows toward the zone edges (\(k>1\)). Its use: after a distance has been reduced to the ellipsoid, multiply by \(k\) to carry it up to the grid; equivalently a grid distance is divided by \(k\) to return to the ellipsoid.

(c) Grid factor (combined scale factor). The grid factor, or combined scale factor, is the single multiplier that converts a horizontal ground distance directly to its grid distance. It is the product of the two preceding reductions — the elevation factor (ground→ellipsoid) and the scale factor (ellipsoid→grid):

$$\text{Grid factor } CF \;=\; EF \times k \;=\; \frac{R}{R+H}\,k.$$

For the example above with \(EF = 0.9998431\) and a point scale factor \(k = 0.99970\), \(CF = 0.9998431 \times 0.99970 = \boxed{0.9995431}\), so a 1000 m ground line plots as \(999.543\ \text{m}\) on the grid. In practice every field distance is multiplied by \(CF\) before entering grid (plane) coordinate computations, and grid distances are divided by \(CF\) to recover ground distances.

(d) MTM (3TM). MTM — the Modified (or 3-degree) Transverse Mercator — is the Canadian provincial grid built on the same transverse-Mercator projection as UTM but with 3°-wide zones (hence “3TM”). Two conceptual differences from UTM: (1) Zone width and scale distortion — MTM zones are half as wide (3° vs 6°), so the maximum scale distortion is much smaller, which is why MTM is preferred for high-accuracy provincial cadastral and engineering surveys. (2) Central-meridian scale factor — MTM uses \(k_0 = 0.9999\) on its central meridian, whereas UTM uses \(k_0 = 0.9996\); MTM also uses provincially defined central meridians and false-easting/zone conventions rather than the global UTM scheme.