Grid-to-Ground Mismatch on a Mountain Tunnel Alignment
A 1.2 km water tunnel is to be driven from two portals at elevations of 1,780 m and 1,820 m. The design alignment is a straight line defined in UTM zone 13N, NAD83(2011). GNSS control at both portals gives UTM coordinates with 8 mm horizontal repeatability. The UTM point scale factor at the site is 0.99960, and the elevation factor at 1,800 m is 0.99972. The contractor's total station and design drawings use ground distances; the CAD model is in UTM. The contractor sets two portal headings independently from the UTM coordinates and reports that the headings would miss by about 0.9 m if the tunnel were bored straight. The design azimuth is given as a grid azimuth, and the convergence angle at the site is about 0.7°. You are the surveyor of record and must decide what to correct before construction.
1. Compute the combined scale factor and the expected linear mismatch over 1,200 m. Explain whether the 0.9 m miss is explained by scale.
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The combined factor is the product of the point scale factor and the elevation factor: 0.99960 × 0.99972 = 0.99932. A ground distance of 1,200.00 m therefore maps to 1,200.00 × 0.99932 = 1,199.18 m on the grid, a difference of 0.82 m. The reported 0.9 m miss is consistent with mixing grid distances in CAD with ground distances in the field. The scale effect alone explains the order of magnitude. It does not explain a lateral heading error, because scale changes distance, not direction. The convergence angle is a separate effect: if a grid azimuth were used in a total station oriented to true north without correction, the lateral error over 1.2 km would be on the order of 1,200 × tan(0.7°) ≈ 14.7 m, far larger than 0.9 m. That means the contractor's 0.9 m is probably a distance mismatch, but the azimuth basis must still be checked before construction.
2. Explain the convergence issue and how you would convert between grid and ground azimuths.
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Grid north is not true north except on the central meridian. At this site the convergence angle is about 0.7°, so a grid azimuth differs from a true or ground azimuth by that amount. The sign depends on which side of the central meridian the project lies and on the projection definition; apply the projection's stated convention rather than guessing. To convert, compute the convergence at the point, then apply it with the correct sign to move between grid azimuth and true azimuth. If the project uses a local ground system, also apply the rotation between grid north and the local system's north. The practical check is to occupy both portals, measure a common reference direction, and compare the computed grid azimuth with the observed ground azimuth. A 0.7° error is 12 mm per metre, so it cannot be left as an office assumption.
3. Recommend the coordinate system and field procedure for construction.
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Use a project coordinate system whose combined factor is close to unity at the tunnel elevation, or define a local ground system with a documented transformation to UTM. Give the contractor a control table with both grid and ground coordinates and a stated combined factor. Set the total station on ground coordinates, convert design grid coordinates to ground using the combined factor and the convergence rotation, and verify by measuring between the two portal control points. Require a closed traverse or a reciprocal observation between portals before boring. Do not allow the CAD model and the field instrument to use different distance bases. Record the transformation parameters, the epoch, and the convergence sign in the survey report and on the control diagram.