The rise and fall method of levelling finds the reduced level (RL) of each point by comparing every staff reading with the one just before it. If the new reading is smaller, the ground has risen by the difference; if it is larger, the ground has fallen. Add each rise to, or subtract each fall from, the previous RL. The work is checked with one equation that covers every reading, including intermediate sights: sum of back sights minus sum of fore sights = sum of rises minus sum of falls = last RL minus first RL.
Rise and fall is the method most IOE, PU, PoU and KU survey camps ask students to use for fly levelling and profile levelling, and it is a regular numerical question in semester exams, the NEC licence exam and Lok Sewa engineering papers. On site it is used for road and canal profiles, building plinth levels and drainage falls. This guide walks through the terms, a full worked example, the arithmetic check, closing error and how it compares with the height of instrument (HI) method.
Terms you need first
- Bench mark (BM): a point of known RL. It can be a government survey bench mark or a temporary bench mark (TBM) you set on a fixed object on site.
- Back sight (BS): the first reading taken after the level is set up, usually on a point of known RL.
- Fore sight (FS): the last reading before the level is moved.
- Intermediate sight (IS): any reading between the BS and FS from the same setup.
- Change point (CP): a firm point where an FS is taken from one setup and a BS from the next. It carries the levels forward.
The rule to remember: a staff reading is the distance from the line of sight down to the point. A smaller reading means the point is higher.
Rise and fall method step by step
- Write each reading in its own row, in the order it was taken, in the BS, IS or FS column. At a change point, the FS and the next BS go in the same row.
- From the second row on, subtract the current reading (IS or FS) from the previous reading of the same setup (the BS or the IS just above).
- A positive answer is a rise. A negative answer is a fall. Write it in the matching column.
- RL of the point = previous RL + rise, or previous RL minus fall.
- After a change point, compare the next reading with the BS taken at that change point, not with its FS.
- Add up the BS, FS, rise and fall columns and apply the arithmetic check.
Worked example with arithmetic check
A level is set up and a BS of 1.585 m is taken on a bench mark of RL 100.000 m. Intermediate sights of 1.920 m and 2.355 m are taken on points A and B, and an FS of 2.680 m on change point C. The level is moved, a BS of 0.845 m is taken on C, an IS of 1.350 m on D and a final FS of 0.960 m on E.
| Point | BS | IS | FS | Rise | Fall | RL (m) |
|---|---|---|---|---|---|---|
| BM | 1.585 | 100.000 | ||||
| A | 1.920 | 0.335 | 99.665 | |||
| B | 2.355 | 0.435 | 99.230 | |||
| C (CP) | 0.845 | 2.680 | 0.325 | 98.905 | ||
| D | 1.350 | 0.505 | 98.400 | |||
| E | 0.960 | 0.390 | 98.790 | |||
| Sum | 2.430 | 3.640 | 0.390 | 1.600 |
How the rows are worked out:
- A: 1.585 minus 1.920 = minus 0.335, a fall. RL = 100.000 minus 0.335 = 99.665.
- B: 1.920 minus 2.355 = minus 0.435, a fall. RL = 99.230.
- C: 2.355 minus 2.680 = minus 0.325, a fall. RL = 98.905.
- D: compare with the new BS at C. 0.845 minus 1.350 = minus 0.505, a fall. RL = 98.400.
- E: 1.350 minus 0.960 = 0.390, a rise. RL = 98.790.
Arithmetic check:
- Sum of BS minus sum of FS = 2.430 minus 3.640 = minus 1.210
- Sum of rises minus sum of falls = 0.390 minus 1.600 = minus 1.210
- Last RL minus first RL = 98.790 minus 100.000 = minus 1.210
All three agree, so the booking and arithmetic are correct. Note that this check only proves the calculation. It does not prove the field readings were right. For that you need to close on a known level. You can enter the same readings in the levelling calculator to confirm each RL.
Closing error and how to distribute it
Suppose point E is a second bench mark with a known RL of 98.800 m and the route is 0.8 km long. The observed RL is 98.790, so the closing error is 98.790 minus 98.800 = minus 0.010 m, or 10 mm too low.
Surveying textbooks commonly give the permissible closing error as C x the square root of K, where K is the distance in km and C in mm is about 100 for rough levelling, 25 for ordinary levelling, 12 for accurate levelling and 6 for precise levelling. For 0.8 km these limits work out to about 89.4, 22.4, 10.7 and 5.4 mm. The 10 mm error is within the ordinary and accurate limits. Your course, department or contract may set its own limit, so check the specification.
The usual way to distribute the error is equally per setup. With two setups, add 5 mm to every RL found from setup 1 (A, B, C) and 10 mm to every RL from setup 2 (D, E). The corrected RL of E then becomes 98.800, matching the bench mark.
Rise and fall vs height of instrument method
| Point | Rise and fall method | Height of instrument method |
|---|---|---|
| Working | Compare each reading with the one before | HI = RL + BS; RL = HI minus reading |
| Check | Covers every reading, including IS | Sum BS minus sum FS = last RL minus first RL; does not check IS |
| Speed | Slower, more subtraction | Faster with many IS, as in profile and cross-section work |
| Best for | Fly levelling, bench mark transfer, exams | Road and canal sections, contouring |
In the example above, HI for setup 1 = 100.000 + 1.585 = 101.585, and HI for setup 2 = 98.905 + 0.845 = 99.750. Both methods give the same RLs. Where you then need gradients between points, the slope calculator converts a level difference and distance into percent, degrees or 1 in n.
Common mistakes in exams and in the field
- Comparing the reading after a change point with the FS at that point instead of the new BS.
- Writing the FS and BS of a change point on different rows.
- Leaving out the IS readings when adding up rises and falls.
- Choosing a change point on loose soil or grass, where the staff can sink between the two readings.
- Not holding the staff vertical. A tilted staff always reads too high.
For more practice on levelling and other survey topics, try the questions in the engineering MCQ practice tool.
Frequently asked questions
What is the arithmetic check in the rise and fall method?
Sum of BS minus sum of FS = sum of rises minus sum of falls = last RL minus first RL. All three must give the same value.
Why is the rise and fall method more accurate than the HI method?
The field readings are the same, so neither is more accurate in the field. Rise and fall is safer for calculation because its check also covers intermediate sights.
Is a smaller staff reading a rise or a fall?
A smaller reading than the previous one means the point is higher, so it is a rise.
Which method is better for profile levelling?
The HI method is quicker when there are many intermediate sights, but many surveyors still check the final booking with rise and fall.
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