---
title: "Reciprocal levelling; RL of closed traverse stations"
subject: "Surveying I"
chapter: 5
key: "semester_3/surveying_i:reciprocal-leveling-closed-traverse-bcdefgb"
source: https://www.bctengineer.com/q/semester_3/surveying_i/reciprocal-leveling-closed-traverse-bcdefgb
site: BCT Engineering
---

# Reciprocal levelling; RL of closed traverse stations

Surveying I · Chapter 5: Leveling

- Asked in: 2080 Bhadra, 6 marks
- Video: [reciprocal levelling numerical | survey](https://www.youtube.com/watch?v=2u5QBEY1c1I) — CE (simplified engineering)
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/reciprocal-leveling-closed-traverse-bcdefgb
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-5

## Question

To find out the RL of each station of closed traverse BCDEFGB, the true RL of B station needs to be found out. Station B is near the river bank and the known RL of A, which lies opposite to the river bank, was 1200 m. For that purpose, reciprocal levelling was done and the following observations were carried out between two points A and B, 1200 m apart. If the RL of A is 1200 m, find the true RL of B.

| Instrument near at | Staff at A (m) | Staff at B (m) |
|---|---|---|
| A | 1.379 | 2.918 |
| B | 1.210 | 2.739 |

Find the correct RL of each station.

| Line | Length (m) | Station | Calculated RL (m) |
|---|---|---|---|
| BC | 50.00 | B | 1198.406 |
| CD | 54.00 | C | 1201.500 |
| DE | 64.00 | D | 1201.600 |
| EF | 60.00 | E | 1201.900 |
| FG | 58.00 | F | 1201.700 |
| GB | 40.00 | G | 1200.800 |

## Short answer

True RL of B is 1198.466 m. The traverse RLs were computed from an assumed RL of B = 1198.406 m, so add 0.060 m to each: C 1201.560, D 1201.660, E 1201.960, F 1201.760, G 1200.860 m.

## Answer

Reciprocal levelling is used to carry the RL across the river from A to B, because a level set midway would be impossible. The error in the far reading (collimation, curvature and refraction) is the same in both sets, so it cancels in the mean.

**Notation:** level near A: staff on A (near) $a_1 = 1.379$, staff on B (far) $b_1 = 2.918$. Level near B: staff on A (far) $a_2 = 1.210$, staff on B (near) $b_2 = 2.739$. Rise of B above A is BS $-$ FS.

#### Step 1: Apparent differences of level

$$
H_1 = a_1 - b_1 = 1.379 - 2.918 = -1.539\ \text{m}
$$

$$
H_2 = a_2 - b_2 = 1.210 - 2.739 = -1.529\ \text{m}
$$

#### Step 2: True difference of level and error

$$
H = \frac{H_1 + H_2}{2} = \frac{-1.539 + (-1.529)}{2} = -1.534\ \text{m}
$$

$$
e = \frac{H_2 - H_1}{2} = \frac{-1.529 - (-1.539)}{2} = +0.005\ \text{m} \quad (\text{error in the far reading over 1200 m})
$$

B is lower than A by 1.534 m.

#### Step 3: True RL of B

$$
\text{RL of B} = 1200.000 - 1.534 = 1198.466\ \text{m}
$$

#### Step 4: Correct RLs of the stations of the traverse BCDEFGB

The traverse RLs were calculated from an assumed RL of B = 1198.406 m. The true RL of B is 1198.466 m, so every station is corrected by

$$
1198.466 - 1198.406 = +0.060\ \text{m}
$$

| Station | Calculated RL (m) | Correction (m) | Correct RL (m) |
|---|---|---|---|
| B | 1198.406 | +0.060 | 1198.466 |
| C | 1201.500 | +0.060 | 1201.560 |
| D | 1201.600 | +0.060 | 1201.660 |
| E | 1201.900 | +0.060 | 1201.960 |
| F | 1201.700 | +0.060 | 1201.760 |
| G | 1200.800 | +0.060 | 1200.860 |

The traverse is closed on B, so the lengths of the lines (50, 54, 64, 60, 58, 40 m) are not needed for this constant shift. If the loop levelling returns to B with a misclosure, it is distributed in proportion to the length of the lines and added to these values.

**Answer: true RL of B = 1198.466 m; each calculated RL is increased by 0.060 m, giving C = 1201.560, D = 1201.660, E = 1201.960, F = 1201.760, G = 1200.860 m.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Reciprocal levelling is used because:
   A. The level cannot be midway across an obstacle like a river
   B. The staff is too short
   C. The ground is level
   D. BM is unknown
   Answer: A. Readings are taken from both banks. (Key point: Purpose)

2. True difference of level is:
   A. Their sum
   B. The mean of the two apparent differences
   C. The first apparent difference
   D. The difference of the two
   Answer: B. (-1.539 - 1.529)/2 = -1.534 m. (Key point: Mean difference)

3. Why does the mean of the two apparent differences give the true difference?
   A. The errors add up to double the true value
   B. The staff is read twice so personal errors vanish
   C. The far-reading error is the same in both sets but enters the differences with opposite sign, so it cancels in the mean
   D. Both sets have no error
   Answer: C. Collimation, curvature and refraction act on the far reading in each set; they appear with opposite sign in H1 and H2 and cancel when averaged. (Key point: Why the mean works)

4. The true RL of B is:
   A. 1201.534 m
   B. 1198.406 m
   C. 1198.534 m
   D. 1198.466 m
   Answer: D. 1200 - 1.534 = 1198.466. (Key point: True RL of B)

## Related in Surveying I

- [Reciprocal levelling: RL of A, B at 1160.18 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-rl-b-1160-18) — Surveying I, chapter 5, asked 2×
- [Reciprocal levelling: RL of A, B at 1260.18 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-rl-b-1260-18) — Surveying I, chapter 5, asked 2×
- [Reciprocal levelling across river, 250 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-river-250m-eyepiece-height) — Surveying I, chapter 5
- [Reciprocal levelling AB 250 m, RL 1450.500](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-ab-250m-1-252) — Surveying I, chapter 5
- [Reciprocal levelling: why and formula](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-derivation) — Surveying I, chapter 5, asked 3×
- [True length: tape found 2.5 cm too long](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-825-plus-750-m) — Surveying I, chapter 2

## Related in other subjects

- [Traverse field procedure between two known stations](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atraverse-field-procedure-between-known-stations) — Surveying II, chapter 1
- [Closed traverse ABCDA missing data, C(1200, 2000)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-traverse-abcda-c-1200-2000) — Surveying II, chapter 1
- [Building near river bank; stop construction?](https://www.bctengineer.com/question.md?key=semester_8%2Fengineering_professional_practice%3Abuilding-on-river-bank-flood) — Engineering Professional Practice, chapter 7
- [RL of hill station P from stations R and Q](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-station-signal-r-q-p) — Surveying II, chapter 3