25.4 : Influence of Earth's Curvature and Atmospheric Refraction on Leveling

During leveling, the Earth's curvature and atmospheric refraction introduce deviations in the line of sight from a true horizontal reference. When the line of sight is leveled, it remains perpendicular to the plumb line only at a single point. Beyond this, it deviates due to the Earth’s curvature, represented by the correction C. For a sight distance D, the deviation can be derived using the relationship:

This relationship shows that the deviation increases quadratically with distance. Over a horizontal distance of 1 kilometer, it is approximately 0.0785 meters.

In addition to curvature, atmospheric refraction bends the line of sight downward, further deviating it from the true horizontal. The bending effect is approximately one-seventh of the Earth's curvature deviation, amounting to about 0.0112 meters over 1 kilometer. This refraction arises from variations in the refractive index of air with altitude, causing light rays to curve slightly. 

The combined influence of Earth's curvature and atmospheric refraction is approximately 0.0675 meters over 1 kilometer. As with curvature alone, this combined effect increases with the square of the distance, which requires careful adjustments during leveling operations to maintain accuracy. Correcting for these deviations is critical for high-precision applications, such as determining benchmark elevations and establishing horizontal planes over long distances.