Understanding Reciprocal Levelling for True Difference in Level

This problem involves determining the true difference in level between two points, A and B, located on opposite sides of a river. This is a classic application of reciprocal levelling, a technique used to minimize or eliminate errors caused by factors such as collimation error and the combined effect of curvature and refraction.

The core principle of reciprocal levelling is to take readings on both points A and B from two different instrument setups: one near point A and the other near point B. By averaging the results obtained from these two setups, the systematic errors mentioned above, which tend to have the same magnitude but opposite effect when reading from opposite ends, cancel out.

Data from Levelling Setups

Let's list the staff readings obtained from the two setups:

Calculating Apparent Difference in Level

From each setup, we can calculate an apparent difference in level between A and B. Let's define the difference in level as the rise from A to B, i.e., \(RL_B - RL_A\). The apparent difference in level between two points from a setup is the difference between the reading on the back sight station and the reading on the foresight station.

• From Setup 1 (Instrument near A):

  Here, the reading on A (2.645 m) acts like a Back Sight, and the reading on B (2.30 m) acts like a Fore Sight, relative to the instrument's position near A. The apparent difference in level \(RL_B - RL_A\) is given by \( \text{Reading on A} - \text{Reading on B} \).

  \( \text{Apparent difference}_1 = 2.645 \text{ m} - 2.30 \text{ m} = +0.345 \text{ m} \)

  This positive value indicates that, from this setup's perspective, B appears to be 0.345 m higher than A (or A appears to be 0.345 m below B).

• From Setup 2 (Instrument near B):

  Here, the reading on B (1.665 m) acts like a Back Sight, and the reading on A (1.085 m) acts like a Fore Sight, relative to the instrument's position near B. The apparent difference in level \(RL_A - RL_B\) is given by \( \text{Reading on B} - \text{Reading on A} \).

  \( \text{Apparent difference in level (B to A)}_2 = 1.665 \text{ m} - 1.085 \text{ m} = +0.580 \text{ m} \)

  This positive value indicates that, from this setup's perspective, A appears to be 0.580 m below B (or B appears to be 0.580 m higher than A). So, the apparent difference in level from A to B is \( -0.580 \) m.

Formula for True Difference in Level (Reciprocal Levelling)

The true difference in level between A and B (let's calculate \(RL_A - RL_B\)) is the average of the apparent differences calculated from each setup, but we need to be careful with the sign convention.

Let \(r_{A1}\) and \(r_{B1}\) be readings on A and B from setup 1 (near A).

Let \(r_{A2}\) and \(r_{B2}\) be readings on A and B from setup 2 (near B).

The true difference in level \(H_{A \text{ to } B} = RL_A - RL_B\) is given by the formula:

\( H_{A \text{ to } B} = \frac{(r_{A1} - r_{B1}) + (r_{A2} - r_{B2})}{2} \)

Alternatively, if we want the true rise from A to B \(H_{B \text{ to } A} = RL_B - RL_A\), the formula is:

\( H_{B \text{ to } A} = \frac{(r_{B1} - r_{A1}) + (r_{B2} - r_{A2})}{2} \)

Let's use the formula for \(RL_A - RL_B\):

• From setup 1 (near A): Apparent \(RL_A - RL_B = r_{A1} - r_{B1} = 2.645 - 2.30 = +0.345\) m.
• From setup 2 (near B): Apparent \(RL_A - RL_B = r_{A2} - r_{B2} = 1.085 - 1.665 = -0.580\) m.

Now, substitute these values into the formula for the true difference \(RL_A - RL_B\):

\( H_{A \text{ to } B} = \frac{(+0.345) + (-0.580)}{2} \)

\( H_{A \text{ to } B} = \frac{0.345 - 0.580}{2} \)

\( H_{A \text{ to } B} = \frac{-0.235}{2} \)

\( H_{A \text{ to } B} = -0.1175 \text{ m} \)

This result \(RL_A - RL_B = -0.1175\) m means that the Reduced Level of A is 0.1175 m *lower* than the Reduced Level of B. In other words, A is 0.1175 m below B.

Let's re-check the formula used in some resources which might differ in sign convention or how the apparent difference is defined. A common formula for the true difference in level between A and B is half the difference between the apparent differences obtained from the two stations. Let the difference be \( (r_A - r_B) \). From setup 1 (near A): apparent diff \( d_1 = r_{A1} - r_{B1} = 2.645 - 2.30 = +0.345 \). From setup 2 (near B): apparent diff \( d_2 = r_{A2} - r_{B2} = 1.085 - 1.665 = -0.580 \). True difference \( = \frac{d_1 - d_2}{2} = \frac{+0.345 - (-0.580)}{2} = \frac{0.345 + 0.580}{2} = \frac{0.925}{2} = +0.4625 \). This positive result from this formula indicates that point A is higher than point B by 0.4625 m, or B is lower than A by 0.4625m. Let's verify this formula's derivation.

Using the earlier derivation: \(r_{A1} - r_{B1} = (H_B - H_A) + (e - c)D\) --- (1) (Apparent rise A to B from setup 1) \(r_{B2} - r_{A2} = (H_A - H_B) + (e - c)D\) --- (2) (Apparent rise B to A from setup 2) Let \(h = H_B - H_A\) (True rise A to B). \(r_{A1} - r_{B1} = h + (e - c)D\) \(r_{B2} - r_{A2} = -h + (e - c)D\) Subtracting the second from the first: \((r_{A1} - r_{B1}) - (r_{B2} - r_{A2}) = (h + (e - c)D) - (-h + (e - c)D) = h + (e - c)D + h - (e - c)D = 2h\). So, \(h = \frac{(r_{A1} - r_{B1}) - (r_{B2} - r_{A2})}{2}\). This \(h\) is the true rise from A to B (\(RL_B - RL_A\)).

• \(r_{A1} - r_{B1} = 2.645 - 2.30 = +0.345\)
• \(r_{A2} - r_{B2} = 1.085 - 1.665 = -0.580\)

True rise from A to B \( (RL_B - RL_A) = \frac{(+0.345) - (-0.580)}{2} = \frac{0.345 + 0.580}{2} = \frac{0.925}{2} = +0.4625 \) m.

A positive rise from A to B means B is higher than A. So, B is 0.4625 m above A, or A is 0.4625 m below B.

Conclusion on True Difference

The true difference in level between A and B, calculated using the reciprocal levelling formula, shows that B is 0.4625 m higher than A. This is equivalent to stating that A is 0.4625 m below B.

Revision Table: Key Concepts

Additional Information: Errors in Levelling

Several sources of error can affect levelling accuracy, especially over long sights like those encountered when levelling across a river. Reciprocal levelling is specifically designed to mitigate some of the most significant ones:

• Collimation Error: This occurs when the line of sight of the instrument is not truly horizontal. It causes readings to be consistently too high or too low. In reciprocal levelling, because the distance to the far point is the same from both setups, the collimation error \( e \cdot D \) has opposite effects on the difference calculation from each setup when defined consistently (e.g., reading on far - reading on near), thus cancelling out in the average.
• Curvature of the Earth: The earth is curved, so a horizontal line deviates from the level surface as distance increases. This effect makes staff readings appear too high. The correction for curvature is proportional to the square of the distance.
• Atmospheric Refraction: Light rays bend downwards when passing through the atmosphere, making objects appear higher than they are. This effect makes staff readings appear too low. Refraction partially counteracts the curvature effect.

The combined effect of curvature and refraction is typically \( (c) \), which causes the staff reading to be larger than it would be to a truly level surface at the same point. The magnitude of this combined error depends on the distance and atmospheric conditions. Reciprocal levelling cancels out this combined error \( c \cdot D \) as well because the distance is the same for the long sight in both setups.

The formula used: \( \text{True rise A to B} = \frac{(r_{A1} - r_{B1}) - (r_{A2} - r_{B2})}{2} \) effectively removes the combined \( (e-c)D \) term, yielding the true difference in elevation \( (H_B - H_A) \).