Height of Instrument Calculation in Leveling

The height of instrument (HI) in surveying is the elevation of the line of sight of the leveling instrument above the adopted datum. It is a crucial value for determining the reduced levels (RL) of other points.

The fundamental formula for calculating the Height of Instrument (HI) from a Benchmark (BM) is:

\text{HI} = \text{RL}_{\text{BM}} + \text{Back Sight (BS)}

where $\text{RL}_{\text{BM}}$ is the Reduced Level of the Benchmark and BS is the back sight reading taken on the staff held vertically on the Benchmark.

Understanding Back Sight with Inverted Staff

Normally, a back sight (BS) is taken on a staff held vertically with its base on a point of known elevation (like a BM). The reading represents the vertical distance from the point to the line of sight.

When an inverted staff is used, it is typically held against a point that is above the line of sight, such as the bottom of a lintel or beam. The reading on the inverted staff measures the vertical distance from the line of sight upwards to the point where the staff is held.

In the context of calculating the Height of Instrument using the standard formula $\text{HI} = \text{RL}_{\text{point}} + \text{Reading}_{\text{at point}}$, if the reading is taken on an inverted staff, this reading is considered negative when substituted into the formula if the point is above the line of sight. Conversely, if we rearrange the formula to find the RL of a point above the line of sight, we use $\text{RL}_{\text{point above}} = \text{HI} + \text{Reading}_{\text{on inverted staff}}$.

In this question, the back sight (BS) reading of 1.005 m is taken on an inverted staff. This reading is used along with the RL of the Benchmark (BM) to find the Height of Instrument (HI).

The phrasing suggests that the BS is being related to the BM to establish the HI. The most logical interpretation in standard leveling practice when using an inverted staff as a "back sight" from a known RL (like a BM) is that the inverted staff reading is effectively a negative back sight in the standard formula when applied to the BM.

Step-by-Step Calculation

Given:

• Reduced Level of Benchmark ($\text{RL}_{\text{BM}}$) = 354.445 m
• Back Sight (BS) reading on inverted staff = 1.005 m

When a BS is taken on an inverted staff from a known point like a BM, the line of sight is below the point where the staff is held. The standard formula $\text{HI} = \text{RL}_{\text{BM}} + \text{BS}$ is used, but the BS reading from the inverted staff is treated as negative because it represents a measurement upwards from the line of sight to the point.

Therefore, the Height of Instrument is calculated as:

\text{HI} = \text{RL}_{\text{BM}} + (-\text{BS reading on inverted staff})

\text{HI} = \text{RL}_{\text{BM}} - \text{BS reading on inverted staff}

Substitute the given values into the formula:

\text{HI} = 354.445 \text{ m} - 1.005 \text{ m}

\text{HI} = 353.440 \text{ m}

Result

The Height of Instrument (HI) is 353.440 m.

This calculation shows that the line of sight of the instrument is 1.005 m below the Benchmark, resulting in a Height of Instrument of 353.440 m above the datum.