6.3 The Nigeria Transverse Mercator Projection and Grid Belts

6.3 The Nigeria Transverse Mercator Projection and Grid Belts

Parameter source: the EPSG Geodetic Parameter Registry entries for Minna / Nigeria West Belt (EPSG:26391), Minna / Nigeria Mid Belt (EPSG:26392) and Minna / Nigeria East Belt (EPSG:26393), and the underlying projection definitions (EPSG:18151, 18152, 18153).

Getting these numbers wrong is not an academic failure. A wrong central scale factor puts every reduced distance out by a fixed proportion; a wrong false easting puts the whole parcel hundreds of kilometres from where it belongs.

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1. The belt parameters

Four features of this table are examinable and are the ones most often stated wrongly.

k0 = 0.99975. Not 0.9996 (that is UTM) and not 0.9999. A grid distance on NTM is shorter than the corresponding ellipsoidal distance at the central meridian by 0.00025, which is 25 cm per kilometre.

Latitude of natural origin = 4 deg N. NTM northings are measured from a parallel at 4 degrees north, not from the Equator. This is why NTM and UTM northings for the same point differ by roughly 442 km even before scale is considered.

False eastings are irregular numbers, and deliberately so. 230,738.26, 670,553.98 and 1,110,369.70 differ from each other by exactly 439,815.72 m in each step. The values were chosen so that eastings run continuously across the country from one belt to the next rather than resetting at each belt boundary - a design decision that makes NTM eastings nationally unique but makes it easy to forget which belt a coordinate belongs to, since the number alone will not always tell you.

The belts are 4 degrees wide, centred on their meridians: West spans about 2 deg 30' E to 6 deg 30' E, Mid 6 deg 30' E to 10 deg 30' E, and East 10 deg 30' E to 14 deg 30' E. Belt membership is decided by longitude, not by State. Lagos (about 3.4 deg E) is West Belt. Abuja (about 7.5 deg E) is Mid Belt. Port Harcourt (about 7.0 deg E) is Mid Belt, not East. Calabar (about 8.3 deg E) is Mid Belt. Enugu (about 7.5 deg E) is Mid Belt. Maiduguri (about 13.2 deg E) is East Belt. If a candidate is unsure, the rule is arithmetic: compare the longitude with 6 deg 30' E and 10 deg 30' E.

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2. Why a secant projection at all

The Transverse Mercator projection is conformal: it preserves angles locally, so shapes of small figures are true and a bearing measured on the ground converts to the grid by a simple correction. What it cannot preserve is scale.

If the cylinder were tangent at the central meridian, scale would be exactly 1 there and would grow steadily outward, reaching its maximum at the belt edge. By shrinking the whole projection slightly - setting k0 slightly below unity - the cylinder becomes secant, cutting the ellipsoid along two lines either side of the central meridian. Scale is then slightly too small between those lines, exactly right on them, and slightly too large outside them. The maximum distortion across the belt is roughly halved.

The lines of exact scale sit approximately where

$x \approx R\sqrt{2(1-k_0)}$

For NTM, with $k_0 = 0.99975$ and $R \approx 6{,}371{,}000$ m:

$x \approx 6{,}371{,}000 \times \sqrt{0.0005} \approx 6{,}371{,}000 \times 0.02236 \approx 142\text{ km}$

For UTM, with $k_0 = 0.9996$: $x \approx 6{,}371{,}000 \times \sqrt{0.0008} \approx 180$ km. The narrower NTM belt (4 degrees against UTM's 6) is matched by a scale factor closer to unity - which is the whole point of having a national grid rather than using the global one.

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3. Point scale factor

Away from the central meridian the point scale factor grows approximately as

$k \approx k_0\left[1 + \frac{{E'}^2}{2R^2k_0^2}\right]$

where $E' = E - E_0$ is the distance east or west of the central meridian.

Worked example. A station lies 100 km east of the Mid Belt central meridian.

$\frac{{E'}^2}{2R^2k_0^2} = \frac{(100{,}000)^2}{2 \times (6{,}371{,}000)^2 \times (0.99975)^2} = \frac{1.000\times10^{10}}{8.114\times10^{13}} \approx 1.232\times10^{-4}$

$k \approx 0.99975 \times (1 + 0.0001232) \approx 0.99987$

So at 100 km out the grid still shortens distances, but by about 13 cm per kilometre rather than 25 cm. At about 142 km, $k$ passes through 1.00000. At the belt edge, roughly 220 km out at these latitudes, $k$ has risen above unity and the grid lengthens distances.

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4. Reducing a ground distance to grid

Two corrections are needed, and candidates routinely apply only one.

Step 1 - reduce to the ellipsoid. A distance measured at mean height $H_m$ above the ellipsoid is longer than its projection onto the ellipsoid:

$S_{ellipsoid} = S_{ground} \times \frac{R}{R + H_m}$

Step 2 - project onto the grid. Apply the line scale factor $k_{line}$, taken as the mean of the point scale factors at the ends (or a three-point Simpson value on a long line):

$S_{grid} = S_{ellipsoid} \times k_{line}$

Combining:

$S_{grid} = S_{ground} \times \underbrace{\frac{R}{R + H_m} \times k_{line}}_{k_{combined}}$

Worked example. A 2,000.000 m line is measured at a mean height of 500 m, 100 km east of the Mid Belt central meridian.

• Height factor: $\dfrac{6{,}371{,}000}{6{,}371{,}500} = 0.99992153$
• Grid factor: $k_{line} \approx 0.99987$
• Combined: $0.99992153 \times 0.99987 \approx 0.99979$
• $S_{grid} = 2000.000 \times 0.99979 = 1999.58$ m

The correction is 42 cm on 2 km - about 1:4,800. Since regulation 3 of the Survey Regulations sets the standard of survey at an accuracy equivalent to a surround traverse closing at 1:3,000, a survey that omits the combined factor is on the edge of failing the standard for that reason alone. On a longer traverse at higher elevation it fails outright.

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5. Convergence of meridians

Grid north coincides with true north only along the central meridian. Elsewhere they differ by the convergence $\gamma$, approximated for a point at longitude $\lambda$ and latitude $\phi$ by

$\gamma \approx (\lambda - \lambda_0)\sin\phi$

East of the central meridian, grid north lies west of true north, and the convergence is conventionally positive; west of it, the sign reverses.

To convert an astronomically observed azimuth to a grid bearing:

$\alpha_{grid} = A_{true} - \gamma$

(strictly, less the small "t - T" arc-to-chord correction as well, which is negligible on the line lengths of ordinary cadastral work).

Worked example. A station at longitude 10 deg 30' E, latitude 10 deg 00' N, in the Mid Belt ($\lambda_0 = 8$ deg 30' E):

$\gamma \approx (10.5 - 8.5) \times \sin 10^\circ = 2^\circ \times 0.17365 = 0.3473^\circ \approx 20' 50''$

Twenty minutes of arc is roughly 6 m across a 1 km line - far beyond the regulation 5(B) requirement that three azimuth observations agree within 30 seconds. Convergence is not a refinement; it is a first-order correction.

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6. NTM against UTM

The last row is the trap. Belt conversion and datum transformation are two different operations. A GNSS receiver typically outputs WGS 84 geographic coordinates or WGS 84 UTM. Converting those to NTM requires:

1. a datum transformation from WGS 84 to Minna - a three-dimensional operation with its own parameters and its own accuracy (see section 6.7); then
2. a projection change from geographic coordinates on Clarke 1880 to the appropriate NTM belt.

Doing only the second - re-projecting WGS 84 coordinates straight into an NTM belt without transforming the datum - produces coordinates that look plausible and are wrong by the magnitude of the datum shift, typically of the order of a hundred metres or more. It is the single most common serious coordinate error in Nigerian practice, and the reason regulation 31(d) requires a description of the origin of coordinates on the face of every plan.

Note also that EPSG defines Minna / UTM zone 31N (26331) and Minna / UTM zone 32N (26332) - UTM grids computed on the Minna datum. A coordinate labelled only "UTM zone 32" is ambiguous until you know whether it is on Minna or on WGS 84.