Reciprocal levelling across a river: From A — BS at A = 1.490 m, FS at B = 2.195 m. From B — BS at B = 0.705 m, FS at A = 1.540 m. True difference in elevation A and B:
Correct Answer: C. 0.770 m
📚 Detailed Explanation: Reciprocal Levelling Across River — True Difference = 0.770 m
Why C (0.770 m) is correct: Reciprocal levelling averages the two apparent differences observed from opposite banks. Apparent diff from A-side: 2.195 − 1.490 = 0.705 m; from B-side: 1.540 − 0.705 = 0.835 m. True diff = (0.705 + 0.835)/2 = 0.770 m.
Setup near A (instrument at A, long sight to B):
BS on A = 1.490 m → HI_A = RL(A) + 1.490
FS on B = 2.195 m → RL(B) from A = HI_A – 2.195 = RL(A) + 1.490 – 2.195
Apparent diff (A→B) = 2.195 – 1.490 = 0.705 m [B appears lower than A by 0.705 m]
BS on A = 1.490 m → HI_A = RL(A) + 1.490
FS on B = 2.195 m → RL(B) from A = HI_A – 2.195 = RL(A) + 1.490 – 2.195
Apparent diff (A→B) = 2.195 – 1.490 = 0.705 m [B appears lower than A by 0.705 m]
Setup near B (instrument at B, long sight to A):
BS on B = 0.705 m → HI_B = RL(B) + 0.705
FS on A = 1.540 m → RL(A) from B = HI_B – 1.540 = RL(B) + 0.705 – 1.540
Apparent diff (B→A) = 1.540 – 0.705 = 0.835 m [A appears higher than B by 0.835 m]
True difference:
h = (0.705 + 0.835) / 2 = 1.540 / 2 = 0.770 m
A is 0.770 m higher than B.
Why the Two Apparent Differences Differ
| Source of Error | Effect from A setup | Effect from B setup | After Averaging |
|---|---|---|---|
| Collimation error | Long sight to B; error in one direction | Long sight to A; error in opposite direction | Cancels out |
| Earth curvature | Makes B appear lower (larger FS) | Makes A appear higher (larger FS) | Cancels out |
| Atmospheric refraction | Partial correction; still imparts systematic effect | Same systematic effect from other side | Cancels out |
- Apparent diff from A-side: 2.195 − 1.490 = 0.705 m.
- Apparent diff from B-side: 1.540 − 0.705 = 0.835 m.
- True difference = (0.705 + 0.835) / 2 = 0.770 m; A is higher than B.