--- title: "Reciprocal levelling: why and formula" subject: "Surveying I" chapter: 5 key: "semester_3/surveying_i:reciprocal-leveling-derivation" source: https://www.bctengineer.com/q/semester_3/surveying_i/reciprocal-leveling-derivation site: BCT Engineering --- # Reciprocal levelling: why and formula Surveying I · Chapter 5: Leveling - Asked in: 2073 Shrawan, 6 marks; 2069 Chaitra, 6 marks; 2075 Asoj, 4 marks (asked 3 times) - Video: [Reciprocal Levelling|Curvature, Refraction and Collimation Correction](https://www.youtube.com/watch?v=UYZKNd7XKH0) — Master Civil Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/reciprocal-leveling-derivation - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-5 ## Question When and why is reciprocal levelling used? Derive the formula for reciprocal levelling (suggest the best method to transfer RLs from one bank of a river to the other, and the errors removed by it). ## Short answer Used when instrument cannot be set midway (e.g., across a river). True difference H = [(a1 - b1) + (a2 - b2)]/2, where a1,b1 and a2,b2 are readings from near A and near B. Errors due to collimation, curvature and refraction cancel. Best method: take multiple rapid sets from both banks and use mean. ## Answer **When and why used:** reciprocal levelling is used when the difference of level between two points that are far apart and separated by an obstacle (a wide river, lake, valley) has to be found and the instrument cannot be set midway so that the backsight and foresight are equal. A single setting would include the errors of collimation, curvature and refraction, which are proportional to the sight distance. By taking readings from both banks, these errors cancel. **Derivation:** let A and B be the two points, level 1 near A and level 2 near B. Let $e$ be the combined error in the far staff reading (error due to collimation + curvature $-$ refraction), which has the same value and the same sign for both far sights because the distance is the same. The error in the near sights is negligible. Level near A: staff readings $a_1$ (near, on A) and $b_1$ (far, on B). Level near B: $a_2$ (far, on A) and $b_2$ (near, on B). First set: $$ H_1 = a_1 - (b_1 - e) = (a_1 - b_1) + e $$ Second set: $$ H_2 = (a_2 - e) - b_2 = (a_2 - b_2) - e $$ The true difference H is the same in both, so $H_1 = H_2$: $$ (a_1 - b_1) + e = (a_2 - b_2) - e \;\Rightarrow\; e = \frac{(a_2 - b_2) - (a_1 - b_1)}{2} $$ Adding $H_1$ and $H_2$ gives: $$ H = \frac{(a_1 - b_1) + (a_2 - b_2)}{2} $$ So the **true difference of level** is the mean of the two apparent differences. $H$ is the rise of B above A (BS $-$ FS is a rise), so RL of B $=$ RL of A $+ H$; a negative $H$ means B is lower than A. **Best method:** the instrument is set near each bank in turn and readings are taken on the two staffs with the same instrument, the same staff, and the same observer, in as short a time as possible, and the mean taken of several sets. The errors removed are: 1. Error due to **collimation** (line of sight not horizontal), 2. Error due to **curvature** of the earth, and 3. Error due to **refraction** of the atmosphere, assuming that it is the same for the two sets. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why does reciprocal levelling cancel the errors? A. Because the far sight is the same length in both sets, so the error e is equal and has opposite effect on the two differences B. Because the near sights have the larger error C. Because the instrument is levelled twice D. Because the staff is inverted Answer: A. The error e in the far reading is the same in magnitude in both sets, adding to one difference and subtracting from the other. (Key point: Reason for use) 2. In the answer's derivation, what is H1 with the level near A? A. H1 = (a1 − b1) + e B. H1 = (a1 − b1) − e C. H1 = (a1 + b1) − e D. H1 = a1 − b1 only Answer: A. The far reading b1 contains error e, so H1 = a1 − (b1 − e) = (a1 − b1) + e. (Key point: Derivation of set 1) 3. Adding H1 and H2 gives which result? A. H = (a1 − b1) − (a2 − b2) B. H = [(a1 − b1) + (a2 − b2)]/2 C. H = e D. H = (a1 + a2)/2 Answer: B. Adding eliminates e, giving H as the mean of the two apparent differences. (Key point: Final formula) 4. How is RL of B obtained from H? A. RL of B = RL of A + H, and negative H means B is lower B. RL of B = RL of A − H always C. RL of B = H D. RL of B = RL of A + e Answer: A. H is BS − FS, the rise of B above A, so RL of B = RL of A + H. (Key point: Reading the sign of H) ## Related in Surveying I - [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: 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; RL of closed traverse stations](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-closed-traverse-bcdefgb) — Surveying I, chapter 5 - [Reciprocal levelling and its significance](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-significance) — Surveying I, chapter 5, asked 3× - [Principles of levelling; reciprocal levelling](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aprinciples-leveling-and-reciprocal) — Surveying I, chapter 5, asked 2× ## Related in other subjects - [Reciprocal trigonometrical levelling formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Areciprocal-trig-leveling-formula) — Surveying II, chapter 3 - [Euler's formula: one end fixed, other hinged](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-one-fixed-one-hinged) — Strength of Materials, chapter 8, asked 2× - [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 - [Dry and wet river intakes; unstable bank](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Awet-dry-intake-unstable-bank) — Water Supply Engineering, chapter 5