Problem Statement
Reciprocal levels were taken with a dumpy level and following observations were recorded:
| Instrument Position | Staff Reading (m) | |
|---|---|---|
| At Station A | At Station B | |
| Near Station A | 1.225 | 1.375 |
| Near Station B | 0.850 | 0.500 |
R.L. of station A is known to be 626.155. Calculate the R.L. of station B. Also, calculate the error in line of collimation and state clearly whether it is inclined upwards or downwards.
Step-by-Step Solution
Key Information & Setup
- Staff reading at A (instrument near A) = 1.225 m
- Staff reading at B (instrument near A) = 1.375 m
- Staff reading at A (instrument near B) = 0.850 m
- Staff reading at B (instrument near B) = 0.500 m
- R.L. of station A = 626.155 m
- Goal: Find R.L. of station B and error in line of collimation
Step 1: Calculate Apparent Difference in Level with Instrument near Station A
The apparent difference in level between stations can be calculated by finding the difference in staff readings:
Apparent difference in level = Staff reading at B – Staff reading at A
Apparent difference in level = 1.375 – 1.225 = 0.150 m
Since the result is positive, station A appears higher than station B in this observation.
Apparent difference in level (instrument near A) = 0.150 m (A being higher)
Step 2: Calculate Apparent Difference in Level with Instrument near Station B
Similarly, we calculate the apparent difference when the instrument is near station B:
Apparent difference in level = Staff reading at B – Staff reading at A
Apparent difference in level = 0.500 – 0.850 = -0.350 m
Since the result is negative, station A appears higher than station B in this observation as well.
Apparent difference in level (instrument near B) = -0.350 m (A being higher)
Step 3: Calculate True Difference in Level between Stations
In reciprocal leveling, the true difference in level is the average of the apparent differences:
True difference in level = (Apparent difference 1 + Apparent difference 2) ÷ 2
True difference in level = (0.150 + (-0.350)) ÷ 2
True difference in level = -0.100 m
The negative sign indicates that station A is actually lower than station B by 0.100 m.
True difference in level = -0.100 m (A being lower than B)
Step 4: Calculate the R.L. of Station B
Using the known R.L. of station A and the true difference in level:
R.L. of station B = R.L. of station A + true difference in level
R.L. of station B = 626.155 + (-0.100)
R.L. of station B = 626.255 m
Note: Since the true difference is negative (A is lower than B), we actually add 0.100 m to the R.L. of station A.
Step 5: Calculate the Error in Line of Collimation
To calculate the error in line of collimation, we first determine what the correct reading on station A should have been when the instrument was at B:
Correct reading on station A = Reading on station B + true difference in level
Correct reading on station A = 0.500 + 0.100 = 0.600 m
The observed reading on station A with instrument at B was 0.850 m.
Error of collimation = Observed reading – Correct reading
Error of collimation = 0.850 – 0.600 = 0.250 m
Since the observed reading is greater than the correct reading, the line of collimation is inclined upwards.
Error in line of collimation = 0.250 m upwards
Final Results
The R.L. of station B is 626.255 m.
The error in line of collimation is 0.250 m upwards.
Explanation of Reciprocal Leveling
Purpose and Importance of Reciprocal Leveling:
- Eliminating Instrumental Errors: Reciprocal leveling eliminates errors due to imperfect adjustment of the level, particularly when stations are separated by obstacles like rivers or valleys.
- Accuracy in Difficult Terrain: It provides high precision for significant elevation differences when direct leveling would introduce substantial errors.
- Error Detection: By taking readings from both sides, systematic errors in the instrument become detectable and quantifiable.
- Field Verification: The method allows surveyors to verify the condition of their equipment in the field.
Line of Collimation Error Interpretation:
If observed reading > correct reading → Line of collimation inclined upwards
If observed reading < correct reading → Line of collimation inclined downwards
Reciprocal leveling is particularly valuable in surveying operations where high precision is required across difficult terrain features or when the condition of surveying equipment must be verified in the field.
