The combined correction for curvature and refraction in levelling is expressed as:
Correct Answer: B. 0.0673 D²
📚 Detailed Explanation: Combined Curvature and Refraction Formula = 0.0673 D²
Why B (Cc = 0.0673D²) is correct: The combined correction for both curvature and refraction (in metres, D in km) is 0.0673D². This is derived by subtracting the refraction correction (0.0112D²) from the curvature correction (0.0785D²).
Curvature correction: Cc = 0.0785 × D² (corrects for Earth curving away)
Refraction correction: Cr = 0.0112 × D² (partial compensation; opposite sign to Cc)
Refraction correction: Cr = 0.0112 × D² (partial compensation; opposite sign to Cc)
Combined correction:
C = Cc – Cr = 0.0785D² – 0.0112D² = 0.0673 × D²
where D = distance in km, C = correction in metres.
Sign convention:
– Curvature makes staff reading appear too large (positive; must subtract)
– Refraction bends line of sight downward, reducing the apparent reading increase
– Net combined effect still makes reading appear too large → subtract 0.0673D² from observed reading
Formula Comparison Table
| Correction | Formula (D in km) | At D = 1 km | At D = 2 km |
|---|---|---|---|
| Curvature (Cc) | 0.0785 D² | 0.0785 m | 0.314 m |
| Refraction (Cr) | 0.0112 D² | 0.0112 m | 0.0448 m |
| Combined (C) | 0.0673 D² | 0.0673 m | 0.2692 m |
- Combined correction C = 0.0785D² − 0.0112D² = 0.0673D².
- Refraction reduces (partially corrects) curvature by 1/7: 0.0112 = 0.0785/7.
- Net correction is still negative (subtract from observed reading to get true reading).