Problem Statement
A dumpy level was set up with its eye-piece vertically over a peg C. The height from the top of peg C to the centre of eyepiece was measured and found to be 1.578 m. The reading on the staff held on peg D was 1.008 m. The level was then moved and set up likewise at peg D. The height of eyepiece above D was 1.258 m and the reading on the staff held on peg C was 1.812 m. Find the R.L. of the point D if the R.L. of point C is 163.378 m.
| Instrument Position | Measurements (m) | |
|---|---|---|
| Height of Eyepiece | Staff Reading | |
| At Peg C | 1.578 (above C) | 1.008 (on D) |
| At Peg D | 1.258 (above D) | 1.812 (on C) |
Step-by-Step Solution
Key Information & Setup
- Height of eyepiece above peg C = 1.578 m
- Staff reading on peg D (instrument at C) = 1.008 m
- Height of eyepiece above peg D = 1.258 m
- Staff reading on peg C (instrument at D) = 1.812 m
- R.L. of point C = 163.378 m
- Goal: Find the R.L. of point D
Step 1: Calculate Apparent Difference in Elevation with Instrument at C
When the instrument is set up at peg C, the apparent difference in elevation is:
Apparent difference in elevation = Height of eyepiece above C – Staff reading on D
Apparent difference in elevation = 1.578 – 1.008 = 0.570 m
Since the result is positive, peg D appears higher than peg C in this observation.
Apparent difference in elevation (instrument at C) = 0.570 m (D being higher)
Step 2: Calculate Apparent Difference in Elevation with Instrument at D
Similarly, when the instrument is set up at peg D, the apparent difference in elevation is:
Apparent difference in elevation = Staff reading on C – Height of eyepiece above D
Apparent difference in elevation = 1.812 – 1.258 = 0.554 m
Again, since the result is positive, peg D appears higher than peg C in this observation as well.
Apparent difference in elevation (instrument at D) = 0.554 m (D being higher)
Step 3: Calculate True Difference in Elevation between Pegs
In reciprocal leveling, the true difference in elevation is the mean of the apparent differences:
True difference in elevation = (Apparent difference 1 + Apparent difference 2) ÷ 2
True difference in elevation = (0.570 + 0.554) ÷ 2
True difference in elevation = 1.124 ÷ 2 = 0.562 m
Since both apparent differences indicate that D is higher than C, the true difference confirms that point D is higher than point C by 0.562 m.
True difference in elevation = 0.562 m (D being higher than C)
Step 4: Calculate the R.L. of Point D
Using the known R.L. of point C and the true difference in elevation:
R.L. of point D = R.L. of point C + True difference in elevation
R.L. of point D = 163.378 + 0.562
R.L. of point D = 163.940 m
Final Result
Explanation of Dumpy Level Reciprocal Leveling
Special Considerations for Peg-to-Peg Reciprocal Leveling:
- Direct Instrument Setup: In this method, the instrument is set up directly over survey pegs rather than at intermediate points.
- Height of Eyepiece: The height of the eyepiece above the peg must be carefully measured and included in calculations.
- Error Elimination: This method effectively eliminates errors due to instrument adjustment, refraction, and earth’s curvature.
- Accuracy: When properly executed, this method can achieve high precision even over considerable distances.
Formula Application:
When instrument is at point P:
Apparent difference = Height of eyepiece above P – Staff reading on Q
When instrument is at point Q:
Apparent difference = Staff reading on P – Height of eyepiece above Q
This specialized form of reciprocal leveling is particularly valuable in precise surveying operations where direct transfer of elevations between specific points is required with minimal intermediate setups.
