Rise and Fall Booking on a Long Section with a Misread Staff
A level run of 12 setups is being reduced by the rise-and-fall method to produce a longitudinal section for a 1.2 km road. The instrument is a dumpy level with a 4 mm per 30 m collimation error suspected but not yet tested. The staff is a 4 m telescopic pattern. At setup 7, the backsight to BM2 reads 1.842 m and the foresight to the next change point reads 0.463 m; the booker records the foresight as 0.863 m. The closing level on the starting benchmark differs from its known value by 0.412 m, and the arithmetic checks in the reduction sheet (sum of BS − sum of FS = last RL − first RL) appear to balance. The surveyor must find the error and decide whether the section is usable.
1. Show how the arithmetic check can balance even when a foresight has been misread, and explain what that tells you about the check.
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What the check actually verifies.
The rise-and-fall reduction rests on the identity:
ΣBS − ΣFS = last RL − first RL
This identity is a consequence of the booked numbers, not of the field observations. Whatever value is written in the FS column, the rise or fall computed from it, and the RL carried forward, are all derived from that same number. So the check will balance for any consistent set of entries, right or wrong.
Demonstration.
Suppose the true FS at setup 7 is 0.463 m but 0.863 m is booked. The booked FS is 0.400 m too large. That makes the fall 0.400 m too large (or the rise 0.400 m too small), so every subsequent RL is 0.400 m too low. The final RL is therefore 0.400 m low. But ΣFS is also 0.400 m too large, so ΣBS − ΣFS is 0.400 m too small - exactly matching the 0.400 m drop in the final RL. The check balances.
What the check catches and what it does not.
- It catches arithmetic slips: a wrong addition, a transposed digit in the RL column, a rise entered as a fall.
- It does not catch booking errors where the wrong number is used consistently downstream, nor field misreads, nor staff zero error, nor collimation error.
Implication. A balanced reduction sheet is necessary but not sufficient. It proves the arithmetic, not the survey. The surveyor must look elsewhere for the 0.412 m closing error.
2. Diagnose the likely source of the 0.412 m closing error and outline the checks that would confirm or eliminate each candidate.
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Candidate 1 - The misread foresight at setup 7.
A transposition of 0.463 to 0.863 is a 0.400 m error, which is very close to the 0.412 m closing error. This is the prime suspect. The magnitude match is strong evidence, but note the 0.012 m difference - that residual must be explained by the other candidates below.
Check: re-read the field book. A 4 m staff read to 0.463 m is a normal reading; 0.863 m is also normal, so the number itself gives no clue. Look for a booking pattern: was 0.863 written on a line where the previous FS ended in 63? Transposition errors often echo a nearby digit. The decisive check is to re-observe the two points at setup 7.
Candidate 2 - Collimation error.
A 4 mm per 30 m error accumulates with sight length. Over 12 setups, if the BS and FS lengths are unequal, the error does not cancel. The standard test is the two-peg test: set two pegs 30 m apart, take readings from the midpoint and from an extension, and compute the collimation error per unit length.
Check: if the error is 4 mm/30 m and the average imbalance between BS and FS is, say, 10 m over 12 setups, the accumulated error is 0.004 × (10/30) × 12 ≈ 0.016 m. That accounts for part of the 0.012 m residual, not the 0.400 m.
Candidate 3 - Staff zero error or a sticking telescopic section.
A telescopic staff that does not fully extend reads short by a fixed amount at every reading above the joint. This produces a constant offset that cancels between BS and FS only if both readings are on the same section. It does not cancel when one reading is below the joint and the other above.
Check: compare the staff against a steel tape over its full 4 m; check the joint for grit and wear; note whether the suspect readings straddle the joint.
Candidate 4 - Benchmark movement or a wrong BM value.
Check: re-level between the start and end benchmarks by a different route, or check against a second BM.
Ranking. The 0.400 m transposition at setup 7 explains 97% of the closing error and must be corrected first. Only after that correction should the residual 0.012 m be attributed to collimation and staff effects.
3. After correcting the misread, state what remains to be done before the section can be issued, and how the correction should be entered.
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Entering the correction.
Do not overwrite the original figure. Strike through 0.863 with a single line, write 0.463 beside it, and initial and date the change. Then re-reduce the sheet from setup 7 onward - every RL after that point shifts by +0.400 m. Re-run the arithmetic check; it should still balance, now against the corrected final RL.
What remains before issue.
- Recompute the closing error. After correction the misclosure should fall to about 0.012 m. Compare this with the permitted limit for the work - for a longitudinal section on a road, a common allowance is of the order of 12√k mm where k is the run in km, i.e. about 13 mm for 1.2 km. The residual is therefore marginal and must be explained, not simply accepted.
- Run the two-peg test. Establish the collimation error per 30 m. If it exceeds about 2-3 mm/30 m, either adjust the instrument or apply a computed correction to every reading proportional to sight length.
- Check the staff. Verify the telescopic joint and the zero of the staff against a tape.
- Balance the sights. For future runs, keep BS and FS lengths equal at each setup so collimation error cancels setup by setup rather than accumulating.
- Check the section against independent evidence. Spot levels on known features (culvert invert, existing road crown) will reveal any remaining systematic offset.
- Document. The report should state the raw misclosure, the identified cause, the correction applied, the residual misclosure, and the result of the two-peg test. A section issued with a silent 0.400 m shift in it is worse than no section at all, because the error is invisible to everyone downstream.