Problem Statement
A level was set up at a point O and the distances to two staff stations A and B were 150 m and 250 m respectively. The observed staff readings on stations A and B were 2.725 m and 1.855 m. Find the correct difference of levels between stations A and B.
Solution
Calculate combined correction of curvature and refraction for the staff reading on A:
Combined correction = 0.0673 × d² = 0.0673 × (150/1000)² = 0.0673 × 0.0225 = 0.0015 m
Calculate combined correction of curvature and refraction for the staff reading on B:
Combined correction = 0.0673 × d² = 0.0673 × (250/1000)² = 0.0673 × 0.0625 = 0.0042 m
Calculate correct staff readings by subtracting the corrections:
Correct staff reading on station A = 2.7250 − 0.0015 = 2.7235 m
Correct staff reading on station B = 1.8550 − 0.0042 = 1.8508 m
Calculate the correct difference of level between stations A and B:
Correct difference of level = 2.7235 − 1.8508 = 0.8727 m
Therefore, the correct difference of level between stations A and B is 0.8727 m
Detailed Explanation
In surveying, when taking staff readings with a level, two corrections need to be applied to account for the Earth’s curvature and atmospheric refraction:
- Earth’s Curvature Correction: Due to the Earth’s curved surface, a horizontal line of sight from the level actually curves above the true horizontal, requiring a correction.
- Atmospheric Refraction Correction: Light rays bend slightly as they pass through the atmosphere, affecting the observed readings.
These two corrections are usually combined into a single formula:
Combined correction = 0.0673 × d² km
Where d is the distance in kilometers from the level to the staff
In this problem:
- For staff A, the distance is 150 m (0.15 km)
- For staff B, the distance is 250 m (0.25 km)
Since the correction is proportional to the square of the distance, the staff reading at B (which is farther away) requires a larger correction (0.0042 m) compared to staff A (0.0015 m).
When finding the difference in elevation between two points, we apply these corrections to the raw staff readings to get the true readings. The difference between these corrected readings gives us the actual elevation difference between points A and B.
In this case, the original readings were:
- Staff A: 2.725 m
- Staff B: 1.855 m
After applying corrections:
- Corrected Staff A: 2.7235 m
- Corrected Staff B: 1.8508 m
The positive difference (2.7235 – 1.8508 = 0.8727 m) indicates that point A is lower than point B by 0.8727 meters. This is because a higher staff reading corresponds to a lower ground elevation at that point.
