schematic mark

Geometric levelling · teaching model

Benchmarks of the great levelling

The difference between two staff readings carries height from one benchmark to the next. Three stations lead from A through B and C back to A — and there the calculation should return the height it started from. Change a reading and see how far the loop opens.

Loop calculation: three stations and the return to A

A100.000m

starting benchmark — height given in the exercise

Station 1 · A → B

+0.550m

Δh = w − p = 1.425 − 0.875

Backsight w₁ [m]

Foresight p₁ [m]

B100.550m

matches the reference

Station 2 · B → C

−0.530m

Δh = w − p = 1.120 − 1.650

Backsight w₂ [m]

Foresight p₂ [m]

C100.020m

matches the reference

Station 3 · C → A

−0.020m

Δh = w − p = 0.950 − 0.970

Backsight w₃ [m]

Foresight p₃ [m]

A′100.000m

the same benchmark A, height from the calculation after three stations

Return to A · gap f_h

0mm

A′ = A: the calculation returned to the height of benchmark A.

Previous result: none — nothing has been changed yet.

Loop balance

Δh₁+0.550m

Δh₂−0.530m

Δh₃−0.020m

Σ0.000m

Σw − Σp = 3.495 − 3.495 m

In a reading field the ↑ ↓ arrows change the value by 1 mm and the −5 and +5 buttons by 5 mm. Staff range in the exercise: 0.100–3.500 m. An empty reading is not zero — the calculation then pauses.

Exercise reference: we assume that B = H₀ + 0.550 m and C = H₀ + 0.020 m are true, so that you can see which point the measurement has shifted. In the field you do not know the truth — you have only the readings and the balance. The length of the gap on the left edge is schematic and linear up to 100 mm: a gap of 100 mm or more takes up 80% of the side length (the rest is shared between the two short segments of the mark).

What the closure says

Closure 0 mm — the calculation agrees.

The sum of the backsights equals the sum of the foresights, and points B and C agree with the exercise reference. This is agreement of the balance, not proof of a flawless measurement: mistakes of opposite signs also give zero (the “Errors cancel out” case).

One scale, 0–100 mm

Exercise threshold T5 mm

Now |f_h|0 mm

Previously |f_h|none

0100 mm or more (scale truncated)

Reading scenarios

Without JavaScript the scenarios and fields do not work. Five ready-made cases (reference loop, different sight-line height, small scatter, gross error, errors cancel out) and their results are described in the section What closure does not prove.

Reading scenarios

Height of benchmark A (H₀) Changes the stated heights; does not change the differences or f_h.

Exercise threshold T [mm] The exercise’s own threshold (0–100 mm), not a standard. Changes the assessment and the threshold bar.

One station

Two staffs, one horizontal sight line, one difference

The level stands between the points and defines a horizontal line of sight. The backsight reading w measures the distance from the starting point to that line, the foresight reading p from the end point. The difference w − p is the height difference: a positive value means that the foresight point lies higher [2]. The height of the sight line drops out of the result: raise it by 100 mm and both readings grow by the same amount (the “Different sight-line height” scenario).

The red dashed line is the level’s horizontal sight line. The bracket style matches the label and the marker in the list below: thin solid — w, dashed — p, thick — Δh; the letters A and B stand under the feet of the staffs. In a narrow layout the labels w, p, Δh are replaced by the list below the drawing. Original diagram, not to horizontal scale; the vertical has a constant scale (40 drawing units = 1 m) for all stations. Exercise data, invented.

Station 1 · A → B

Backsight staff, at point A: w = 1.425 m
Foresight staff, at point B: p = 0.875 m
Height difference Δh = w − p: +0.550 m — B lies 0.550 m higher than A

Limits of closure

Closure is a balance, not an accounting of the points

The loop returns to A and says only how large the sum of the mistakes is. It does not say at which station they arose, and it does not rule out mistakes of opposite signs. That is why checks independent of the balance are added in the field, for example a second reading on the second graduation of the staff or a measurement in the opposite direction [2] [3]. Five cases from the exercise, calculated for T = 5 mm and any H₀:

Reference loopf_h = 0 mm · B 0 · C 0
The calculation matches the reference. Still only a balance.
Different sight-line height, stn 1f_h = 0 mm · B 0 · C 0
Different readings, same difference. A different reading on the staff is not an error in itself.
Small scatterf_h = +3 mm · B 0 · C 0
Within the exercise threshold, but the loop does not show that station 3 is to blame.
Gross errorf_h = +100 mm · B 0 · C +100
Above the threshold. The balance says “somewhere”, the exercise reference shows “from C upwards”.
Errors cancel outf_h = 0 mm · B −100 · C 0
A perfect closure, yet point B is 10 cm lower. This very case is the reason not to read zero as proof.

Text log (second layer, ready to copy)

LEVELLING LOG · TEACHING MODEL · loop A → B → C → A
H(A) = 100.000 m · exercise threshold T = 5 mm
stn 1  A→B   w = 1.425 m   p = 0.875 m   Δh = +0.550 m   H(B) = 100.550 m
stn 2  B→C   w = 1.120 m   p = 1.650 m   Δh = −0.530 m   H(C) = 100.020 m
stn 3  C→A   w = 0.950 m   p = 0.970 m   Δh = −0.020 m   H(A′) = 100.000 m
Σw = 3.495 m · Σp = 3.495 m · f_h = Σw − Σp = 0 mm
Invented data; conventional layout; this is not a measurement or surveying instructions.

Scope and uncertainties

What this page assumes and what it does not check

Invented data. The readings, the elevations of A, B, C and the arrangement of heights are a conventional exercise; they do not come from any real line of benchmarks or from control network data. Points A–C are anonymous.
Sign and units to be confirmed. We assume f_h = Σw − Σp = H(A′) − H(A) and height difference Δh = w − p. The sign of the differences, the units and the interpretation of the misclosure should still be confirmed by a surveyor; that has not been done here.
Threshold T is yours. It is an exercise parameter, not a standard or a technical instruction. The page does not model adjustment or the choice of method.
Omitted effects. Sight lengths, refraction, collimation of the level and other systematic errors are not modelled — only six readings and their differences.
Does not replace a survey. A phone’s GPS does not give millimetre-level control here, and this page is not surveying instructions. Do not touch or disturb height marks; view them from publicly accessible places.
Sources not verified in this version. The links were carried over from version V2; they were not opened or re-checked while V3 was built.
Original drawings. The benchmark mark and the station diagram are illustrations, not a photograph or a map.

For the reviewer: what changed relative to V2

The “Whole loop / Stn 1–3” cards were replaced: the loop is visible all the time, and the detail of a station has its own section with a station selector.
Six scenarios were cut to five: the two counterexamples of compensation (backsight and foresight readings) were reduced to one, because they teach the same thing.
Saving in the browser’s storage was removed; nothing is stored, and reset restores the starting data.
Copying the log was moved to an expandable second layer.
The benchmark photograph from Wikimedia Commons is no longer embedded (the page works offline); it remains as a link in the sources.
Retained: the formula Δh = w − p, the sums Σw and Σp, the choice of H₀, the custom threshold T, all scenario values and their closures.

Sources

Where to check the rules

GUGiK · Produkty projektu GBDOT [GBDOT project products]Vertical control network — definition and use. Link from V2; not checked in V3.
NOAA / NGS · Geodetic Leveling (NOS NGS 3)Backsight minus foresight; the levelling line and sources of error (chapter 3 according to V2). Link from V2; not checked in V3.
ZPE · Pomiary sytuacyjne i wysokościowe [Planimetric and height surveys]Polish-language educational material on geometric levelling. Link from V2; not checked in V3.
Geoportal · Podstawowe osnowy, PRPOG [Basic control networks, PRPOG]Scope of the register and access to point data; the proper starting point for verifying a specific benchmark. Link from V2; not checked in V3.
Wikimedia Commons · Benchmark on the Old Town Hall in ToruńPhotograph from 2008, author: Pko, licence CC BY-SA 4.0 according to the V2 description. Link only — the photograph is not embedded in this page; it documents the mark, not its current condition or height.