Change in Elevation Calculator (Back Sight & Fore Sight)

An essential tool for surveyors and civil engineers to determine elevation changes accurately.

What are Change in Elevation using Back Sight and Foresight Calculations?

The process of determining the change in elevation using back sight and foresight calculations is a fundamental surveying technique known as **differential leveling**. It is used to find the difference in height between two points. A **Back Sight (BS)** is a reading taken on a leveling staff held at a point of known elevation (a benchmark), while a **Fore Sight (FS)** is a reading on a staff at a point whose elevation needs to be determined. This method is crucial for civil engineers, surveyors, and construction professionals to ensure that structures are built at the correct heights and slopes.

The core principle involves establishing a horizontal line of sight with a leveling instrument (like a dumpy level or auto level). By subtracting the foresight reading from the backsight reading, we can calculate the vertical difference, or the ‘rise’ (an increase in elevation) or ‘fall’ (a decrease in elevation), between the two points.

Formulas for Elevation Calculation

There are two primary methods for calculating elevation changes: the Height of Instrument (HI) method and the Rise and Fall method. This calculator uses both to provide a comprehensive result.

1. Height of Instrument (HI) Method

The HI method involves first determining the elevation of the instrument’s line of sight. The formulas are:

• Height of Instrument (HI) = Known Elevation (RL) + Back Sight (BS)
• New Elevation (RL) = Height of Instrument (HI) – Fore Sight (FS)

2. Rise and Fall Method

The Rise and Fall method directly calculates the difference in height between points without first finding the HI. A positive result indicates a rise, and a negative result indicates a fall.

• Rise or Fall = Back Sight (BS) – Fore Sight (FS)
• New Elevation (RL) = Known Elevation (RL) + Rise (or – Fall)

Explanation of Variables in Elevation Calculations

Variable	Meaning	Unit	Typical Range
RL	Reduced Level (Elevation of a point relative to a datum)	Meters / Feet	Varies by location
BS	Back Sight (Reading on a point of known RL)	Meters / Feet	0.1 – 4.0
FS	Fore Sight (Reading on a point of unknown RL)	Meters / Feet	0.1 – 4.0
HI	Height of Instrument (Elevation of the instrument’s line of sight)	Meters / Feet	Varies by location

For more details on these calculations, you might find a guide on the rise and fall method helpful.

Practical Examples

Example 1: Metric Units (Meters)

A surveyor needs to find the elevation of a new point (TP1) from a benchmark (BM1) with a known elevation.

• Known Elevation (BM1): 150.000 m
• Back Sight (BS) reading on BM1: 2.855 m
• Fore Sight (FS) reading on TP1: 1.230 m

Calculation:

1. Height of Instrument (HI): 150.000 m + 2.855 m = 152.855 m
2. Elevation of TP1: 152.855 m – 1.230 m = 151.625 m
3. Change in Elevation (Rise): 2.855 m – 1.230 m = +1.625 m

Example 2: Imperial Units (Feet)

A construction crew is setting a foundation corner (P2) relative to a known point (P1).

• Known Elevation (P1): 512.50 ft
• Back Sight (BS) reading on P1: 4.72 ft
• Fore Sight (FS) reading on P2: 5.98 ft

Calculation:

1. Height of Instrument (HI): 512.50 ft + 4.72 ft = 517.22 ft
2. Elevation of P2: 517.22 ft – 5.98 ft = 511.24 ft
3. Change in Elevation (Fall): 4.72 ft – 5.98 ft = -1.26 ft

Understanding the basics of a leveling survey calculator can provide further insights into these examples.

How to Use This Change in Elevation Calculator

Follow these simple steps to determine the elevation of a new point:

1. Select Units: Choose between ‘Meters (m)’ and ‘Feet (ft)’ from the dropdown menu. Ensure all your input values use this selected unit.
2. Enter Known Elevation: Input the starting elevation of your benchmark (BM) or known point in the first field.
3. Enter Back Sight (BS): Input the reading you took from the leveling staff placed on the known point.
4. Enter Fore Sight (FS): Input the reading from the staff on the new point whose elevation you want to find.
5. Review Results: The calculator automatically updates the ‘New Point Elevation’, ‘Height of Instrument (HI)’, and the ‘Change in Elevation’ (as a Rise or Fall). The diagram also updates to provide a visual representation.

Key Factors That Affect Elevation Calculations

The accuracy of the change in elevation using back sight and foresight calculations is paramount. Several factors can introduce errors:

• Instrument Calibration: The leveling instrument must be properly calibrated. An uncalibrated instrument will have a line of sight that is not perfectly horizontal, leading to errors.
• Staff Condition: The leveling staff must be straight, have clear markings, and be held perfectly plumb (vertical). A slanted staff will give an incorrect reading.
• Ground Stability: The instrument and staff must be on firm, stable ground. Sinking of the tripod or the turning point between BS and FS readings will cause significant errors.
• Reading/Recording Errors: Human errors, such as misreading the staff, transposing numbers when recording, or incorrect calculations, are common. It’s crucial to double-check all readings and arithmetic.
• Atmospheric Conditions: Heat shimmer (refraction) near the ground can cause the line of sight to bend, affecting readings over long distances. It’s best to keep sight lengths relatively short and equal.
• Earth’s Curvature: For long sight distances (typically over 100m or 300ft), the curvature of the Earth becomes a factor, causing the level line to deviate from the line of sight. This is usually negligible for most construction and site work. To learn more, see our guide on the height of instrument method.

Frequently Asked Questions (FAQ)

What is a Back Sight (BS)?

A Back Sight is the first reading taken with a leveling instrument after it has been set up. It is taken on a point with a known elevation, such as a benchmark or a previously established turning point. It is sometimes called a “plus sight” because it is added to the known elevation to determine the Height of Instrument (HI).

What is a Fore Sight (FS)?

A Fore Sight is the last reading taken before moving the leveling instrument. It is taken on a point whose elevation is to be determined. It’s often called a “minus sight” because it is subtracted from the Height of Instrument (HI) to find the elevation of the new point.

What is a Turning Point (TP)?

A Turning Point is an intermediate point used in a leveling traverse to carry the elevation forward. It’s a temporary, stable point where a Fore Sight is taken from one instrument setup, and then a Back Sight is taken from the next instrument setup.

What is the difference between the HI method and the Rise and Fall method?

The HI method calculates the elevation of the instrument’s line of sight first, then subtracts foresight readings. The Rise and Fall method directly calculates the elevation difference between successive points (BS – FS). The Rise and Fall method provides an extra arithmetic check on intermediate sights, making it slightly more robust for long traverses. For more information, see backsight and foresight in surveying.

Why is it important to have equal BS and FS distances?

Keeping the distance from the instrument to the Back Sight point and the Fore Sight point roughly equal helps to cancel out systematic errors from instrument miscalibration and the effects of Earth’s curvature and atmospheric refraction.

What does a negative change in elevation mean?

A negative change in elevation signifies a “Fall”. This means the new point (where the foresight was taken) is lower in elevation than the starting point (where the backsight was taken).

Can I use this calculator for a leveling traverse with multiple setups?

This calculator is designed for a single instrument setup. For a traverse, you would calculate each segment sequentially. The ‘New Elevation’ from the first setup becomes the ‘Known Elevation’ for the second setup, and so on.

What is “Reduced Level” (RL)?

Reduced Level (RL) is another term for the elevation of a point. It refers to the vertical height of a point above or below a specified reference datum (like mean sea level or a local benchmark). Check out our reduced level calculation tool for more.