Contours, an Overlong Chain and a Closing Error
A class carries out two exercises. Part A. On a 1:50,000 Survey of India sheet a student measures 4.2 cm between spot height 412 and spot height 388; both lie on the same road. She also measures the nala crossing that road as 1 mm wide on the sheet, and locates a temple from its conventional sign at the nala–road junction, which she intends to use as a control point. Part B. The same class measured a rectangular school playground with a 20 m chain, later found to be 20.05 m long against a standard tape. The recorded sides are 84.0 m and 56.0 m. A chain traverse run round the same plot closed with an error of 0.9 m on a perimeter of 280 m. Part C. The class must also map a small irregular pond and its silted margin. The teacher asks four things: the ground gradient, the true width of the nala, the correct area, and whether the traverse is acceptable.
1. Work out the gradient, the nala width and the reliability of the temple as a control point.
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Gradient
Map distance 4.2 cm at 1:50,000 → 4.2 × 50,000 = 2,10,000 cm = 2,100 m ground distance. Fall = 412 − 388 = 24 m. Gradient = 24 ÷ 2,100 = 0.01143, that is 1 in 87.5, or 1.14% (about 11.4 m per kilometre).
Note that map distance is horizontal distance, which is what a gradient requires - no slope correction is needed here. Note also that 1.14% is gentle; a road at that gradient would be regarded as easy, so if the student expected a steep hill, she has probably measured between the wrong two contours.
Nala width
1 mm × 50,000 = 50 m. A fifty-metre-wide stream is impossible for a nala of this order, and the reason is simple: at 1:50,000 a stream is drawn as a conventional single line, not to scale. The width cannot be read off the sheet at all. To get it, use a 1:10,000 or larger map, or measure it in the field. This is a scale-reading trap, not a measurement.
Temple as a control point
A temple is drawn as a point symbol whose position is generalised. At 1:50,000, 0.5 mm of plotting error is 25 m on the ground, and the symbol is placed where it can be seen on the sheet, not necessarily at the temple's true coordinates. It may be used as an identifiable reference in a description, but not as a survey control point. Control should come from a triangulation station, a GPS fix, or a benchmark.
2. Correct the area for the chain error and judge whether the closing error is acceptable.
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Chain correction
A chain of nominal 20 m that is actually 20.05 m is too long. Every recorded '20 m' was really 20.05 m, so recorded lengths are smaller than the truth and must be multiplied up.
Correction factor = 20.05 ÷ 20 = 1.0025.
- 84.0 × 1.0025 = 84.21 m
- 56.0 × 1.0025 = 56.14 m
- True area = 84.21 × 56.14 = 4,727.6 m²
- Recorded area = 84.0 × 56.0 = 4,704.0 m²
- Error = +23.6 m², i.e. 0.5% too small.
This matches the rule that for a small chain error of 1 in 400 in length, the area error is about twice that, 1 in 200. Half a percent is negligible for a school playground but would matter for a plot boundary or a compensation payment.
Closing error
Relative closing error = 0.9 ÷ 280 = 1 in 311.
For ordinary chain survey, the closing error should not exceed about 1 in 1,000 on reasonably level ground, and 1 in 2,000 where care has been taken. 1 in 311 is three times the permissible limit, so the traverse is not acceptable.
Before re-surveying, check the likely causes in order: the chain length itself (already known to be out by 1 in 400, which accounts for part of it); sloping ground measured along the slope but booked as horizontal; faulty ranging or an offset taken at the wrong point; and simple booking or arithmetic error. Correcting the chain length alone reduces the error to about 1 in 800, still short of the standard - so there is at least one further mistake to find.
3. Which survey method would you choose for the irregular pond and its silted margin, and why not the others?
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Choice: plane table, radiation and intersection, with the table set at one or two stations.
Reasoning. The pond has an irregular, curved boundary and a soft, silty margin that cannot be walked with a chain without disturbance, and the margin itself is the feature being mapped - a mud strip whose edge shifts. A plane table lets the surveyor plot each boundary point as it is sighted, so the curve can be picked up point by point and no angles or distances need to be booked and reduced afterwards. Silted margins are unsafe to walk; radiation from a firm station on the bank, with intersection for points that cannot be approached, keeps the surveyor off the mud.
Why not chain survey alone. A chain or chain-and-offset survey works best for a straight-sided, accessible area. Chaining across silt is both slow and a disturbance to the very feature being measured, and a curved boundary needs a dense offset grid, which multiplies error.
Why not a total station or theodolite as first choice. Both are more accurate and would be preferable if available, particularly for the margin, but they require a trained operator and a control framework the class does not have, and for a pond of a few hundred square metres the plane table's accuracy is sufficient.
Why not GPS alone. A hand-held GPS gives roughly 3–5 m accuracy - too coarse for a silt margin only a metre or two wide, and unusable under tree cover on the bank.
Practical point. Whatever the method, fix the pond's high-water line and the edge of vegetation as two separate lines; confusing the two is the commonest error in mapping a tank or pond, and it is exactly the error that produced the 24 m/yr figure in another exercise.