Why Do We Need This Correction?
In ordinary differential leveling, we assume that the line of sight (Line of Collimation) is perfectly horizontal and the earth is perfectly flat. However, for long sights (usually > 200 meters), the Curvature of the Earth and Atmospheric Refraction cause significant errors in the staff readings, leading to inaccurate elevation calculations.
1. Curvature Correction (Cc)
Because the earth is spherical, the ground curves downwards as you move away from the leveling instrument. However, the telescope looks straight ahead. This causes the horizontal line of sight to hit the leveling staff much higher than the actual true level line.
- Effect: The observed staff reading is always greater than the true reading.
- Sign: The correction is always Negative (-).
- Formula: Cc = 0.0785 × D2
2. Refraction Correction (Cr)
Light rays do not travel perfectly straight. As light reflects off the leveling staff and travels through the atmosphere to your telescope, it bends downwards due to varying air densities. This makes the staff appear slightly higher, effectively lowering your reading.
- Effect: It partially cancels out the error caused by curvature.
- Sign: The correction is always Positive (+).
- Magnitude: It is roughly 1/7th of the curvature correction.
- Formula: Cr = 0.0112 × D2
3. Combined Correction (C)
Since Curvature is a negative correction and Refraction is a positive correction, the net effect is always negative. Surveyors simply use the Combined Correction formula to find the total error in one step.
How to Eliminate This Error in the Field
If you do not want to calculate these mathematical corrections manually, you can completely eliminate the errors using a field technique called Reciprocal Leveling.
By setting up the Auto Level exactly midway between the Backsight (BS) staff and the Foresight (FS) staff, the curvature and refraction errors applied to both staffs will be identical. When you subtract the FS from the BS to find the elevation difference, the errors perfectly cancel each other out.
Solved Numerical Example
Solution:
1. Given D = 1.2 km
2. Formula: C = 0.0673 × D2
3. Calculation: 0.0673 × (1.2)2
4. 0.0673 × 1.44 = 0.0969 meters
Result: Subtract 0.0969 m (approx 97 mm) from the observed staff reading to get the true level.
Error Table for Quick Reference
| Distance (m) | Curvature (m) | Combined Error (m) |
|---|---|---|
| 100 m | 0.0008 | 0.00067 (Negligible) |
| 250 m | 0.0049 | 0.0042 (4 mm) |
| 500 m | 0.0196 | 0.0168 (17 mm) |
| 1000 m (1 km) | 0.0785 | 0.0673 (67 mm) |