Reducing a Level Book: Rise and Fall vs Height of Instrument

Last updated 24 October 2026

Reduce the same level book both ways, run the arithmetic checks, compare the misclosure with the allowable and spread it over the set-ups.

The readings

A level run starts on benchmark BM-01 (reduced level, RL, 10.000 m), takes a level on a kerb at A and closes back on BM-01, with two change points (TP-1 and TP-2), so there are three set-ups of the level:

• BS (backsight): the first reading after setting up, on a point of known or just-found level.
• IS (intermediate sight): any reading in between, here the kerb at A.
• FS (foresight): the last reading before the level moves, on a change point or the closing benchmark.

Method 1: rise and fall

Compare each reading with the one before it from the same set-up. A smaller reading means the ground is higher: a rise. A bigger one is a fall. Each RL is the previous RL plus the rise or minus the fall.

For example, 1.425 at BM-01 then 1.836 at TP-1 is a fall of 0.411, so TP-1 is 10.000 − 0.411 = 9.589.

The arithmetic check: all three of these must be the same number.

• ΣBS − ΣFS = 4.047 − 4.057 = −0.010
• ΣRise − ΣFall = 1.523 − 1.533 = −0.010
• Last RL − first RL = 9.990 − 10.000 = −0.010

If they don't agree, there is a slip in the sums or the reduction, not in the field. Rise and fall checks every intermediate sight too, which is why it is preferred for work that will be checked.

Method 2: height of instrument (HI)

The height of the instrument's line of sight is the RL plus the backsight. Every other reading from that set-up is the HI minus the reading.

The same levels come out, with less writing, so HI is quick for setting out many levels from one set-up. Its simple check (ΣBS − ΣFS = last RL − first RL) does not test the intermediate sights. The full check does: Σ(each HI × the number of IS and FS taken from it) − ΣIS − ΣFS = the sum of every RL except the first. Here: 11.425 + 2 × 10.571 + 11.000 − 2.104 − 4.057 = 37.406, and 9.589 + 8.467 + 9.360 + 9.990 = 37.406.

Is the misclosure acceptable?

The run should have closed on 10.000 but came back at 9.990: a misclosure of −10 mm. The allowable is set by the specification, usually ±C√n mm for n set-ups or ±C√K mm for K kilometres of levelling. With C = 10 mm and three set-ups, the allowable is ±10√3 = ±17.3 mm, so the run is accepted.

Spreading the correction

The correction is +10 mm, shared equally between the three set-ups and added up as you go: +3.3 mm on everything from the first set-up's foresight, +6.7 mm on the second set-up's readings, +10.0 mm on the closing benchmark.

Doing it without the arithmetic

In Traverse & Level, type the start and closing benchmarks, then BS, IS and FS on one row per point (or paste them from Excel, or import a CSV). Choose Rise & Fall or HI and the app runs the arithmetic checks, compares the misclosure with your allowable, and shows the corrections and adjusted RLs with Accepted or Rejected. The levelling report prints with boxes for Surveyed, Checked and Approved by. The free traverse and level calculator does the same in the browser for a quick check.

More guides

• How to Adjust a Closed Traverse by the Bowditch Rule
• Traverse Misclosure and Precision: What 1 : 10,000 Means
• How to Put Survey Points from Excel into Google Earth
• Total Station CSV to KMZ with Site Photos
• How to Convert Easting and Northing to Latitude and Longitude
• How to Find Your UTM Zone

All 29 guides