7.5 Differential Leveling, Height of Instrument, and Closure

The Height-of-Instrument Method

Differential leveling carries an elevation from a benchmark of known height to new points using a level and a graduated rod. Two readings define each setup:

• A backsight (BS) is a rod reading taken on a point of known elevation. Because the rod reads up from the point, the line of sight is above it, so the backsight is added.
• A foresight (FS) is a rod reading on a point whose elevation you are finding, so it is subtracted.

The key intermediate value is the height of instrument (HI), the elevation of the level's line of sight:

• HI = known elevation + backsight
• New point elevation = HI - foresight

A turning point (TP) is a temporary point that receives a foresight from one setup and a backsight from the next, letting you leapfrog the instrument across long or steep ground while preserving the elevation chain.

Worked Level Run

Start at benchmark BM-1 at elevation 100.00 ft. The notes below carry the elevation to BM-2:

HI at BM-1 = 100.00 + 4.20 = 104.20. Elevation of TP-1 = 104.20 - 2.50 = 101.70, then its new HI = 101.70 + 5.15 = 106.85, and so on to BM-2 = 106.40 - 4.10 = 102.30 ft.

The Arithmetic Check

Every level run carries a built-in page check that catches arithmetic blunders:

sum of backsights - sum of foresights = final elevation - initial elevation

For the run above: 12.40 - 10.10 = +2.30 ft, and 102.30 - 100.00 = +2.30 ft. They agree, so the arithmetic is internally consistent. Note this check verifies the math only; it does not detect a misread rod.

Loop Closure and Distributing Error

When a run closes back on a benchmark of known elevation (or loops to its start), the difference between the computed and the published elevation is the misclosure. Acceptable misclosure is often expressed as a function of distance, for example C x sqrt(M), where M is the loop length in miles and C is an order-of-accuracy constant (a common third-order value is about 0.05 ft x sqrt(miles)).

To adjust, distribute the misclosure with opposite sign to the running elevations:

• Equal distribution: divide the misclosure by the number of setups and apply cumulatively.
• Distance weighting: apportion the correction in proportion to the distance leveled to each point.

Example. A loop returns to BM-1 reading 100.04 ft against a true 100.00 ft, a +0.04 ft misclosure over four setups. Apply -0.01 ft of cumulative correction per setup so the final reading matches 100.00 ft. Always close and adjust before using elevations for grades, contours, or construction stakeout.

Common FS Traps

• Adding a foresight or subtracting a backsight reverses the sign of the elevation change.
• Forgetting that a turning point is both a foresight and the next backsight breaks the chain.
• Reporting elevations before checking closure invites carrying an undetected error into design grades.

Balancing Sights to Cancel Systematic Error

The most important field habit in differential leveling is balancing the backsight and foresight distances at each setup. When the rod is the same distance behind and ahead of the instrument, the small errors from Earth curvature, atmospheric refraction, and instrument collimation affect both readings almost equally and therefore cancel in the difference (BS minus FS). Keeping sights short and balanced also limits the rod-reading error that grows with distance.

Profile and Construction Leveling

The same height-of-instrument arithmetic supports profile leveling for roadway and pipeline design and construction stakeout for grades. Two terms recur:

• A grade rod is the rod reading that would place the rod foot exactly at the design elevation: grade rod = HI - design elevation.
• Cut or fill is the difference between the design grade and the existing ground at a station; a foresight larger than the grade rod indicates the ground is low (fill needed).

Orders of Accuracy

Leveling is specified by order and class, which set the allowable loop misclosure. The misclosure limit takes the form C x sqrt(M), where M is the loop distance in miles. Tighter (lower-numbered) orders use a smaller constant: first-order work demands a far smaller C than third-order work. On the FS you are more likely to be given the constant and asked whether a run passes than to memorize each value, but you should recognize that the allowable error grows with the square root of distance, not linearly, because random errors accumulate that way.

Always compare your computed misclosure against the stated limit before accepting the elevations, and distribute any acceptable misclosure with opposite sign before the numbers reach a design drawing.