Distance Measurement, Levelling and Errors
Principles of surveying, chain and tape corrections, levelling with height of instrument and rise-and-fall methods, curvature and refraction, reciprocal levelling, and theory of errors.
Surveying fixes the relative positions of points. The two basic measurements are distances and heights.
Principles
- Work from the whole to the part (control first, details later).
- Fix each new point by at least two independent measurements.
Tape corrections
| Correction | Formula | Sign |
|---|---|---|
| Temperature | α L (T − T₀) | + if hotter |
| Pull (tension) | (P − P₀) L/(A E) | + if more pull |
| Sag | w² L³/(24 P²) | Always − |
| Slope | h²/(2L) (approximately) | Always − |
| Standardisation | L(l′ − l)/l | Depends on tape |
Normal tension is the pull at which the pull and sag corrections cancel.
Wrong tape length: true length = measured length × (actual tape length/nominal tape length). Areas use the square of that ratio.
Levelling
- Back sight (BS): first reading after setting up, on a point of known level. Fore sight (FS): last reading before moving. Intermediate sights in between.
- Height of instrument (HI) method: HI = RL + BS; RL of a point = HI − its staff reading. Check: ΣBS − ΣFS = last RL − first RL.
- Rise and fall method: compares consecutive readings; it checks intermediate sights too (ΣBS − ΣFS = ΣRise − ΣFall = last RL − first RL).
Curvature and refraction
- Curvature correction c_c = 0.0785 d² m (d in km), negative.
- Refraction correction = 0.0112 d² (one-seventh of curvature), positive.
- Combined correction = −0.0673 d² m.
- Distance to the visible horizon: d = 3.855 √h km (h in m), with refraction.
- Reciprocal levelling eliminates curvature, refraction and collimation errors: true difference = average of the two apparent differences.
Theory of errors
- Systematic errors follow a rule and can be corrected; random errors follow the normal distribution.
- Most probable value of equally weighted observations = their mean.
- Probable error of a single observation: 0.6745 √(Σv²/(n − 1)); of the mean: that divided by √n.
- Weights are inversely proportional to variance; for levelling lines, inversely proportional to length.
Worked example
A 30 m tape is found to be 30.02 m long. A line measured with it is 600 m.
- True length = 600 × 30.02/30 = 600.4 m.
Practice
What is the combined curvature and refraction correction (m) for a sight of 2 km? (Give 4 decimal places.)
Show answer and explanation
Answer: -0.2692−0.0673 × 4 = −0.2692 m.
A 20 m tape is actually 19.95 m long. A distance measured with it is 500 m. What is the true distance in m?
Show answer and explanation
Answer: 498.75True = 500 × 19.95/20 = 498.75 m (a short tape over-measures).
The RL of a benchmark is 100.000 m. The back sight on it is 1.250 m and the fore sight on point A is 2.150 m. What is the RL of A?
Show answer and explanation
Answer: 99.1HI = 101.250 m; RL of A = 101.250 − 2.150 = 99.100 m.
A 30 m tape weighing 0.6 kg is held at a pull of 6 kg (w = 0.02 kg/m). What is the sag correction in mm? (Give 1 decimal place.)
Show answer and explanation
Answer: 12.5c = w²L³/(24P²) = 0.0004 × 27,000/864 = 0.0125 m = 12.5 mm (negative).