Curvature and refraction corrections in levelling
Why long sights read too high, the 0.0785 d² and 0.0112 d² corrections, and the distance to the visible horizon.
A level's line of sight is a straight line, but the level surface it should follow curves with the Earth. Over long sights the difference matters.
Curvature correction
The horizontal line of sight rises above the level surface, so the staff reading is too large. The correction is negative:
Cc = − d²2 R = − 0.0785 d²
with d in km and Cc in metres, taking Earth's radius R = 6370 km.
Refraction correction
Light bends downward through the atmosphere, curving the line of sight back towards the Earth. This reduces the error. Refraction is taken as one-seventh of curvature and is positive:
Cr = + 17 × 0.0785 d² = + 0.0112 d²
Combined correction
C = − 0.0673 d²
Worked example
A staff reading of 2.865 m is taken at 1.5 km.
C = − 0.0673 × 1.5² = − 0.151 m
Correct staff reading = 2.865 − 0.151 = 2.714 m.
Distance to the visible horizon
Turn the formula around: from a height h (m) above the sea, the horizon is at
d = √( h0.0673 ) = 3.855 √h km
A lighthouse keeper 25 m up sees about 19 km.
How to avoid it
Keep back sight and fore sight distances equal. The same correction then applies to both readings and cancels in the difference. This is the main reason levels are set up midway between staff stations, and why reciprocal levelling is used across rivers where that isn't possible.
Quick check
The combined correction for curvature and refraction over 1 km is about: