Levelling with a Dumpy Level: Rise and Fall Method Worked Example
Levelling finds the height of points relative to a reference, and it is needed for road profiles, building plinth levels, drain gradients and contour maps. In the surveying lab and on site, most Indian students start with the dumpy level and a levelling staff. This article explains the terms, the temporary adjustments of the instrument, and how to book and reduce readings using the rise and fall method, with a complete worked example and arithmetic checks.
Key Terms
| Term | Meaning |
|---|---|
| Benchmark (BM) | A point of known reduced level, such as a GTS benchmark or a temporary benchmark on a permanent structure |
| Reduced level (RL) | Height of a point above the chosen datum |
| Back sight (BS) | First staff reading after setting up the instrument, taken on a point of known RL |
| Fore sight (FS) | Last reading before shifting the instrument |
| Intermediate sight (IS) | Any reading between BS and FS from the same setup |
| Change point (CP) | A point where both FS and BS are taken, used to move the instrument forward |
| Height of instrument (HI) | RL of the line of collimation = RL of BM + BS |
The staff reading is larger when the ground is lower, because the staff extends further down from the horizontal line of sight. This is the single idea behind both reduction methods.
Temporary Adjustments of a Dumpy Level
- Setting up: open the tripod legs, place the instrument on top and fix it roughly level by adjusting the legs, with the telescope at a comfortable height.
- Levelling up: turn the telescope parallel to two foot screws and centre the bubble by turning them in opposite directions. Turn the telescope through 90° and centre the bubble using the third foot screw. Repeat until the bubble stays central in all directions.
- Eliminating parallax: focus the eyepiece on the cross hairs against a bright background, then focus the objective on the staff until there is no apparent movement between the image and the cross hairs when you move your eye.
Rise and Fall Method
In this method, each reading is compared with the previous reading taken from the same instrument setup. If the current reading is smaller, the point is higher, so there is a rise. If it is larger, there is a fall.
Difference = previous reading - current reading positive -> Rise negative -> Fall RL of current point = RL of previous point + Rise (or - Fall) Arithmetic check: ΣBS - ΣFS = ΣRise - ΣFall = Last RL - First RL
Unlike the height of instrument method, the rise and fall method checks every intermediate sight, so errors in intermediate readings are caught by the arithmetic check. It is slightly longer but more reliable for profile levelling.
Worked Example
A level is set up near a benchmark of RL 100.000 m. Readings in order: 1.585 (BS on BM), 1.950, 2.300, 2.755 (FS on change point CP1). The instrument is shifted and readings 1.120 (BS on CP1), 0.865 and 1.405 (FS on the last point) are taken.
| Station | BS | IS | FS | Rise | Fall | RL (m) | Remarks |
|---|---|---|---|---|---|---|---|
| 1 | 1.585 | 100.000 | BM | ||||
| 2 | 1.950 | 0.365 | 99.635 | ||||
| 3 | 2.300 | 0.350 | 99.285 | ||||
| 4 | 1.120 | 2.755 | 0.455 | 98.830 | CP1 | ||
| 5 | 0.865 | 0.255 | 99.085 | ||||
| 6 | 1.405 | 0.540 | 98.545 | End point | |||
| Sum | 2.705 | 4.160 | 0.255 | 1.710 |
Step by step: Stn 2: 1.585 - 1.950 = -0.365 -> fall 0.365 -> 100.000 - 0.365 = 99.635 Stn 3: 1.950 - 2.300 = -0.350 -> fall 0.350 -> 99.285 Stn 4: 2.300 - 2.755 = -0.455 -> fall 0.455 -> 98.830 (instrument shifted; next comparison starts from BS 1.120 at CP1) Stn 5: 1.120 - 0.865 = +0.255 -> rise 0.255 -> 99.085 Stn 6: 0.865 - 1.405 = -0.540 -> fall 0.540 -> 98.545 Checks: ΣBS - ΣFS = 2.705 - 4.160 = -1.455 ΣRise - ΣFall = 0.255 - 1.710 = -1.455 Last RL - First RL = 98.545 - 100.000 = -1.455 OK
All three values agree, so the arithmetic is correct. Note that the arithmetic check only proves the booking and calculation are right. It does not detect a wrong staff reading in the field. To check field work, close the levelling back on the starting benchmark or on another benchmark of known RL.
Closing Error and Its Distribution
Suppose a closed loop that should return to RL 100.000 instead gives 99.988. The closing error is -0.012 m. A common way to distribute it is in proportion to the number of setups or the distance from the start. With four setups, a correction of +0.003 m is added cumulatively at each change point: +0.003 at the first, +0.006 at the second and so on. Permissible closing error depends on the type of work and the length of the route, and is set by the project specification.
Common Mistakes
- Holding the staff tilted; always rock it slowly towards and away from the instrument and take the minimum reading.
- Disturbing the instrument between back sight and fore sight.
- Moving the change point plate before the new back sight is taken.
- Comparing the first reading after a shift with the old FS instead of the new BS.
- Writing an IS reading in the FS column, which breaks the arithmetic check.
Summary
A dumpy level gives a horizontal line of sight after careful levelling and parallax removal. In the rise and fall method, each reading is compared with the previous one from the same setup, and the difference gives a rise or fall. The arithmetic check ΣBS - ΣFS = ΣRise - ΣFall = Last RL - First RL confirms the calculation, while closing on a known benchmark confirms the field work.