Differential level circuit reduced from field notes

1The two equations that run the whole page

Differential levelling has exactly two arithmetic operations, applied alternately. A backsight is a rod reading taken on a point of known elevation, so adding it to that elevation gives the elevation of the line of sight itself. That value is the height of instrument, and it is not the height of the telescope above the ground or above the tripod plate; it is an elevation on the same datum as the benchmarks.

A foresight is a rod reading taken on a point whose elevation is wanted, so subtracting it from the height of instrument gives that elevation. Every line of a level page is one of those two operations, and the columns are laid out so that a backsight is always added and a foresight is always subtracted.

2Setups 1 and 2

Start on BM-1 at its published elevation and work the first two setups. The turning points are the only points that carry the elevation forward, which is why they are read to 0.001 ft and why they must be firm, well-defined objects that will not settle while the instrument is moved.

Notice that TP-2 comes out higher than TP-1 by 6.017 ft. That is a normal figure for a single setup on rolling ground and is worth a glance in the field: a turning point that jumps more than about 8 ft usually means the rod was read on the wrong foot mark.

3Setups 3 and 4

The same alternation carries the line to BM-2. Nothing in the arithmetic remembers where the line has been, so an error made at setup 2 propagates unchanged into every elevation after it. That is the reason the run has to be closed on a second benchmark or brought back to BM-1 before the elevations are trusted.

4The reduced page

Written out in the standard five columns, the page reads as follows. Blank cells are shown as a dash; a station has either a backsight or a foresight in a given row, and a turning point has both.

5The page check

Every backsight was added once and every foresight was subtracted once, so the net change in elevation from the first station to the last must equal the sum of the backsights less the sum of the foresights. Any addition or subtraction error inside the page destroys that identity, which is why the check is worked before the crew leaves the site.

The check is purely arithmetic. It proves the page was added correctly; it says nothing about whether the rod was plumb, whether the instrument was level, or whether a turning point settled between the foresight and the backsight taken on it.

6What still has to be checked in the field

The page check passes, so the arithmetic is sound and the elevations follow from the readings as recorded. The observations themselves are checked by closing the work: either run the line back from BM-2 to BM-1 and test the loop misclosure against a tolerance, or continue to a third benchmark of known elevation and compare.

Balancing the backsight and foresight lengths at each setup, as this crew did, removes the effect of a collimation error in the instrument and of curvature and refraction, because both errors are proportional to sight length and enter the two readings with opposite sign. Balanced sights are the cheapest error control available on a level line and cost nothing but a little pacing.