In the height of instrument (HI) method of levelling, you first find the height of the line of sight above the datum, then subtract each staff reading from it. HI = RL of the bench mark + back sight, and RL of any point = HI minus the intermediate sight or fore sight on that point. When the level is moved, a new HI is worked out from the RL of the change point plus the new back sight. The basic check is sum of back sights minus sum of fore sights = last RL minus first RL, and a second check, explained below, also covers every intermediate sight.
The HI method (also called the height of collimation or line of collimation method) is the quickest way to reduce a level book with many intermediate sights, such as a road centreline profile, a canal or drain section, or a grid of spot levels on a house plot. It is also the method site engineers use when setting out levels, because it turns a design RL straight into the staff reading the chainman must see. This guide pairs with our rise and fall method worked example, so you can compare the two on real numbers.
Practise: levelling questions appear in the basic civil engineering MCQ set (NEC chapter on surveying) and in the Lok Sewa civil engineering MCQs.
Terms used in the HI method
- Bench mark (BM): a fixed point of known reduced level (RL). On most building sites in Nepal this is a temporary bench mark (TBM) set on a gate pillar, a road kerb or a peg in concrete, with an assumed RL such as 100.000 m.
- Back sight (BS): the first staff reading after setting up the level, taken on a point of known RL.
- Intermediate sight (IS): any reading between the BS and the FS from the same set-up.
- Fore sight (FS): the last reading before the level is moved.
- Change point (CP): a firm point that gets an FS from one set-up and a BS from the next.
- Height of instrument (HI): the RL of the line of sight (line of collimation) for one set-up. It stays the same for every reading from that set-up, as long as the instrument is not disturbed.
Keep one rule in mind: a staff reading is the vertical distance from the line of sight down to the point. A bigger reading means a lower point.
HI method step by step
- Draw the level book with columns: Station, BS, IS, FS, HI, RL, Remarks. Write each reading on its own row in the order it was taken. At a change point, put the FS and the new BS on the same row.
- First row: HI = RL of the BM + BS. Write it in the HI column.
- For each IS or FS from the same set-up: RL = HI minus the reading.
- At a change point, first find its RL from the FS (RL = old HI minus FS). Then work out the new HI = that RL + the new BS.
- Carry on with the new HI until the next change point or the last reading.
- Add up the BS, IS and FS columns and apply both arithmetic checks.
- If you closed on a second bench mark, find the misclosure and compare it with the allowed error before using the levels.
The arithmetic checks
Check 1: back sights and fore sights
Sum of BS minus sum of FS = last RL minus first RL. This is the check given in most lecture notes, for example the King Saud University levelling computation notes. It proves the HI values and the RLs of change points and the last point are worked out correctly. It does not touch the intermediate sights at all, so a wrong subtraction on a profile point would pass this check.
Check 2: covers the intermediate sights
Every RL from a set-up equals that set-up's HI minus its reading. Adding them all up gives the full check:
Sum of all RLs except the first = sum of (each HI x the number of IS and FS taken from it) minus sum of IS minus sum of FS.
Count only the IS and FS readings for each HI, not the BS. If this check agrees, every RL in the book, including each intermediate point, is arithmetically right. Notes such as the Salahaddin University levelling lecture warn that with only check 1, errors in intermediate RLs can go undetected. Check 2 closes that gap, and it takes two minutes with a calculator.
Worked example 1: road centreline profile
A rural road is being surveyed for a longitudinal section, with staff readings on the centreline every 20 m from chainage 0+000 to 0+120. The work starts on BM1 with an RL of 1335.250 m (an assumed value for this example; on a real project use the RL given in the survey report or the nearest known bench mark) and closes on BM2, whose RL is known to be 1330.672 m. The level was set up three times.
| Station | BS | IS | FS | HI (m) | RL (m) | Remarks |
|---|---|---|---|---|---|---|
| BM1 | 1.425 | 1336.675 | 1335.250 | Known RL | ||
| 0+000 | 1.870 | 1334.805 | ||||
| 0+020 | 2.315 | 1334.360 | ||||
| 0+040 | 2.960 | 1333.715 | ||||
| CP1 | 0.845 | 3.105 | 1334.415 | 1333.570 | Change point | |
| 0+060 | 1.240 | 1333.175 | ||||
| 0+080 | 1.785 | 1332.630 | ||||
| 0+100 | 2.430 | 1331.985 | ||||
| CP2 | 1.120 | 2.890 | 1332.645 | 1331.525 | Change point | |
| 0+120 | 1.655 | 1330.990 | ||||
| BM2 | 1.980 | 1330.665 | Known 1330.672 | |||
| Sum | 3.390 | 14.255 | 7.975 |
How the first rows are worked out:
- HI 1 = 1335.250 + 1.425 = 1336.675 m.
- 0+000: 1336.675 minus 1.870 = 1334.805 m. 0+020: 1336.675 minus 2.315 = 1334.360 m.
- CP1: 1336.675 minus 3.105 = 1333.570 m. The level is moved, and HI 2 = 1333.570 + 0.845 = 1334.415 m.
- CP2: 1334.415 minus 2.890 = 1331.525 m. HI 3 = 1331.525 + 1.120 = 1332.645 m.
- BM2: 1332.645 minus 1.980 = 1330.665 m.
Checking the example
Check 1: sum BS minus sum FS = 3.390 minus 7.975 = minus 4.585. Last RL minus first RL = 1330.665 minus 1335.250 = minus 4.585. It agrees.
Check 2: HI 1 has 4 readings after the BS (three IS and the FS on CP1), HI 2 has 4, and HI 3 has 2.
- Sum of HI x count = 1336.675 x 4 + 1334.415 x 4 + 1332.645 x 2 = 13,349.650
- Minus sum IS (14.255) and sum FS (7.975) = 13,327.420
- Sum of all RLs except BM1 = 13,327.420. It agrees, so every intermediate RL is correct too.
Closing error and correction
The observed RL of BM2 is 1330.665 m against the known 1330.672 m, so the closing error is minus 7 mm (the line reads 7 mm low). Textbooks commonly give the allowed error as C x the square root of K mm, with K the route length in km and C about 100 for rough, 25 for ordinary, 12 for accurate and 6 for precise levelling. For a route of 0.24 km, the ordinary limit is 25 x 0.49 = 12.2 mm and the accurate limit is 5.9 mm. The 7 mm error is acceptable for ordinary road profile work. Your contract or department may set its own limit, so check it.
Spread the 7 mm equally over the three set-ups: add about 2 mm to RLs from set-up 1, 5 mm to set-up 2 and 7 mm to set-up 3 (rounded to whole millimetres).
| Station | Observed RL (m) | Correction (mm) | Corrected RL (m) | Gradient to next point |
|---|---|---|---|---|
| 0+000 | 1334.805 | +2 | 1334.807 | -2.23% (fall 0.445 in 20 m) |
| 0+020 | 1334.360 | +2 | 1334.362 | -3.22% |
| 0+040 | 1333.715 | +2 | 1333.717 | -2.70% |
| 0+060 | 1333.175 | +5 | 1333.180 | -2.72% |
| 0+080 | 1332.630 | +5 | 1332.635 | -3.23% |
| 0+100 | 1331.985 | +5 | 1331.990 | -4.97% |
| 0+120 | 1330.990 | +7 | 1330.997 | end |
The gradients use the uncorrected level differences (the correction changes them by at most 3 mm in 20 m). Over 120 m the road falls 3.81 m, an average grade of about 3.2%. The steeper 5% length between 0+100 and 0+120 is the kind of thing a profile is meant to show before the designer fixes the formation levels. You can turn any level difference and distance into percent, degrees or 1 in n with the slope calculator.
Worked example 2: setting out levels on a house site
On building sites the HI method is used in reverse. You know the design RL and want the staff reading that marks it. The formula is simply: required staff reading = HI minus design RL.
Suppose a TBM on the gate pillar is given an assumed RL of 100.000 m and the auto level reads 1.385 on it. HI = 101.385 m.
| What is being set | Design RL (m) | Staff reading to mark (m) |
|---|---|---|
| Top of finished plinth | 100.600 | 101.385 - 100.600 = 0.785 |
| Bottom of footing PCC | 98.500 | 2.885 |
| Drain invert at start | 99.700 | 1.685 |
| Drain invert 12 m away at 1 in 100 | 99.580 | 1.805 |
The chainman holds the staff against a peg or the side of a column and moves it up or down until the instrument reads the required value; the foot of the staff is then at the design level. For the excavation, the staff stands on the trench bottom: if it reads more than 2.885 the pit is too deep, if less, dig more.
Inverted staff readings. For a lintel or beam soffit above the line of sight, hold the staff upside down with its foot against the soffit. Book the reading as negative. RL = HI minus (minus reading) = HI + reading. An inverted reading of 0.715 from the same set-up gives a soffit RL of 101.385 + 0.715 = 102.100 m. Write "inverted" in the remarks column so nobody subtracts it by habit.
HI method vs rise and fall method
| Point | Height of instrument | Rise and fall |
|---|---|---|
| Working | RL = HI minus reading | Compare each reading with the one before |
| Subtractions | One per reading | One per reading plus the rise/fall step |
| Basic check | Sum BS - sum FS = last RL - first RL | Sum BS - sum FS = sum rise - sum fall = last RL - first RL |
| Intermediate sights | Only covered if you also do check 2 | Covered by the normal check |
| Best for | Profiles, cross-sections, spot levels, setting out | Fly levelling, bench mark transfer, exam answers unless the question says HI |
Both methods use the same field readings and give the same RLs. The difference is only in the arithmetic. If an exam question says "reduce by the HI method and apply the usual checks", show both checks; many examiners give marks for check 2.
Common mistakes seen on Nepali sites and in exams
- Using the old HI after a change point. Once the level moves, every following reading must be subtracted from the new HI. This is the most common exam error.
- Writing the FS and new BS of a change point on different rows. It breaks the count for check 2 and makes the HI column confusing.
- Counting the BS in check 2. The multiplier for each HI is the number of IS and FS readings only.
- Leaving out an IS from the sum. Check 2 then fails by the size of the missing reading, which is actually a quick way to find it.
- Weak change points. A CP on loose soil, a brick placed on mud or a grass verge sinks between the FS and the BS, and every level after it is wrong by the same amount. Use a kerb, a rock, a peg driven flush or a steel change plate.
- Bumping the tripod. On busy sites a labourer brushing the tripod changes the HI mid set-up. Re-read the BS before you leave the set-up; if it differs by more than a millimetre or two, redo the set-up.
- Staff not vertical. A tilted staff always reads too high. Ask the staff holder to rock it slowly towards and away from the instrument and book the smallest reading.
- Losing the TBM. Many house sites in the Kathmandu valley mark the TBM in chalk on a gate pillar that is later plastered over. Paint it, photograph it and set a second TBM on something that will not be disturbed.
- Very long sights. Curvature and refraction together lower the reading by about 0.0673 x d squared metres, with d in km. At 100 m that is only 0.7 mm, but it grows fast. Keep sights balanced and under about 60 to 80 m on site work.
Where the HI method fits in Nepal
Road, irrigation and water supply surveys in Nepal often run from a bench mark of known value to a long chain of intermediate sights on chainages and cross-section points, which is the case the HI method suits. Larger projects tie their levels to control points of the national network maintained by the Survey Department, while house plots usually use an assumed TBM. Whichever you use, write its description and RL at the top of the level book so the next person can find it.
If you also need horizontal control for the same survey, see our guide to whole circle and reduced bearings. For quick reduction in the field, enter your readings in the levelling calculator, which works by both rise and fall and HI and shows the arithmetic check. More free tools are on the engineering tools page.
Frequently asked questions
What is the formula for the height of instrument?
HI = RL of the point where the back sight is taken + the back sight reading. For the first set-up that point is the bench mark; after a move it is the change point.
What is the arithmetic check in the HI method?
Sum of BS minus sum of FS = last RL minus first RL. To also check intermediate sights: sum of RLs except the first = sum of (each HI x number of IS and FS from it) minus sum of IS minus sum of FS.
Why is the HI method faster?
Each RL needs only one subtraction from a value that stays the same for the whole set-up, so a profile with twenty intermediate sights is quick to reduce.
How is an inverted staff reading handled?
Book it as a negative number. RL = HI + the reading, so the point is above the line of sight.
Is the HI method less accurate than rise and fall?
No. The field accuracy is the same because the readings are the same. Only the standard arithmetic check is weaker, and check 2 fixes that.
Test yourself with more surveying and levelling questions on the engineering MCQ practice page.
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