Deviation of the vertical
As the horizontal axis of the theodolite is perpendicular to the vertical (geoid) and not the normal (ellipsoid), a correction similar to plate bubble error is applied to the directions, i.e.
where ζ is as in equation and β is the vertical angle. It may be ignored for small values of β.
Distance reduction to the ellipsoid
Reduction of distance to the ellipsoid:
The measured distance is reduced to its horizontal component by applying all the corrections appropriate to the method of measurement .
It is then reduced to the ellipsoid by reducing to MSL (geoid), A1B1, and from MSL to the ellipsoid, A2B2, in one computation.
where L = AB, the mean horizontal distance at ground level
H = mean height above MSL
N = height of the geoid above the ellipsoid
Rα = the radius of curvature of the ellipsoid in the direction α of the line
As already stated, values of N may not be available and hence the geoidal distance may have to be ignored. It should be remembered that if N = 6 m, a scale error of 1 ppm will occur if N is ignored. In the UK, the maximum value for N is about 4.5 m, resulting in a scale error of only 0.7 ppm, and may therefore be
ignored for scale purposes. Obviously it cannot be ignored in heighting.