Computing reduced levels by the height of instrument method

HI = RL of BM + BS. RL of any point = HI − staff reading.

Worked answer.

Step 1: Height of instrument (HI). The line of collimation is above the benchmark by the backsight reading:
HI = RL(BM) + BS = 100.000 + 1.525 = 101.525 m.

Step 2: RL of P (intermediate sight).
RL(P) = HI − IS = 101.525 − 0.875 = 100.650 m.

Step 3: RL of Q (foresight).
RL(Q) = HI − FS = 101.525 − 2.150 = 99.375 m.

Step 4: Arithmetic check (HI method). The standard check is
ΣBS − ΣFS = Last RL − First RL.

• ΣBS − ΣFS = 1.525 − 2.150 = −0.625 m.
• Last RL − First RL = 99.375 − 100.000 = −0.625 m.

They match, so the arithmetic is correct.

Note: the HI method's basic check does not test intermediate-sight calculations. The fuller check is Σ(each HI × number of IS and FS taken from it) − ΣIS − ΣFS = sum of all RLs except the first. Here: 101.525 × 2 − 0.875 − 2.150 = 200.025, and 100.650 + 99.375 = 200.025. This matches as well.

Interpretation. A smaller staff reading means the point is higher. P (0.875 m) is above the benchmark; Q (2.150 m) is below it.

HI method vs rise-and-fall method

• HI method: faster, fewer calculations, good for many intermediate sights, such as in profile levelling or contouring. Its weakness is the incomplete check on intermediate RLs.
• Rise-and-fall method: more calculations, but the check (ΣBS − ΣFS = ΣRise − ΣFall = Last RL − First RL) covers every reading, so it is preferred for precise work.

Common mistake. Adding the foresight to HI instead of subtracting it.

Likely follow-up. "What is a change point?" A point where both a foresight (from the old setup) and a backsight (from the new setup) are taken, allowing the instrument to be moved forward.