In surveying and civil engineering, auto level formula plays a crucial role in determining elevation differences between points on the ground. The auto level (also known as a dumpy level or automatic level) is a precise optical instrument used for leveling work, ensuring that structures like roads, bridges, and buildings are constructed on the correct gradient. Understanding how to use the auto level formula accurately ensures precise measurement, reducing errors in design and construction.
What Is an Auto Level?
An auto level is a surveying instrument used to establish or verify points in the same horizontal plane. It consists of a telescope mounted on a tripod that automatically maintains a horizontal line of sight. This instrument is used along with a leveling staff to measure height differences between various ground points.
The key feature of an auto level is its compensator mechanism, which automatically adjusts the line of sight to be horizontal even when the instrument is slightly tilted. This makes it faster and more accurate than manual leveling instruments.
Principle of Auto Level
The working principle of an auto level is based on the concept of a horizontal line of sight. The instrument creates a straight line parallel to the Earth’s surface at a specific height. When readings are taken on a leveling staff placed at different points, the height difference (also called rise or fall) between those points can be calculated using the auto level formula.
In simple terms:
Height of Instrument (HI) = Elevation of Benchmark (BM) + Backsight (BS)
Once the HI is known, the elevation (RL) of other points can be found using:
RL = HI – Foresight (FS)
These basic formulas form the foundation for all leveling calculations.
Main Components of an Auto Level
Before understanding the formulas, it’s essential to know the parts involved in an auto level setup:
- Telescope: For viewing the leveling staff.
- Objective Lens: Magnifies the staff readings.
- Leveling Screws: Used to adjust the instrument’s position.
- Circular Bubble: Ensures the instrument is roughly horizontal.
- Compensator: Automatically corrects small tilts.
- Tripod Stand: Provides stability.
- Leveling Staff: Graduated rod used for measuring readings.
Basic Auto Level Formulas
The auto level formula revolves around three essential readings taken at each setup:
- Backsight (BS): Reading on a point of known elevation (Benchmark).
- Intermediate Sight (IS): Reading on points between the benchmark and foresight.
- Foresight (FS): Reading on the final point before moving the instrument.
Let’s understand the formulas in detail.
1. Height of Instrument (HI) Method
This is the most commonly used method in leveling.
Formula:
HI = RL of BM + BS
RL of Next Point = HI – FS
Example:
| Point | BS (m) | IS (m) | FS (m) | HI (m) | RL (m) |
|---|---|---|---|---|---|
| BM | 1.500 | – | – | 101.500 | 100.000 |
| A | – | 1.200 | – | – | 100.300 |
| B | – | – | 1.800 | – | 99.700 |
Explanation:
- HI = RL (BM) + BS = 100.000 + 1.500 = 101.500
- RL (A) = HI – IS = 101.500 – 1.200 = 100.300
- RL (B) = HI – FS = 101.500 – 1.800 = 99.700
2. Rise and Fall Method
This method calculates the difference between consecutive readings.
Formulas:
- Rise = Previous Reading – Next Reading (if positive)
- Fall = Next Reading – Previous Reading (if positive)
- RL of Next Point = RL of Previous Point + Rise – Fall
Example:
| Station | Reading (m) | Rise (m) | Fall (m) | RL (m) |
|---|---|---|---|---|
| BM | 1.500 | – | – | 100.000 |
| A | 1.200 | 0.300 | – | 100.300 |
| B | 1.800 | – | 0.600 | 99.700 |
Here:
- Rise from BM to A = 1.500 – 1.200 = 0.300
- Fall from A to B = 1.200 – 1.800 = –0.600 (so fall = 0.600)
The rise and fall method provides a check on accuracy by ensuring the sum of all rises minus the sum of all falls equals the difference between the last and first reduced levels.
Key Terms in Auto Level Formula
1. Benchmark (BM)
A point of known elevation used as a reference for leveling operations.
2. Reduced Level (RL)
The vertical height or depth of a point relative to the benchmark.
3. Backsight (BS)
The first reading taken after setting up the instrument, usually on the benchmark.
4. Foresight (FS)
The final reading before shifting the instrument to a new position.
5. Intermediate Sight (IS)
Readings taken between BS and FS to determine levels at various points.
6. Height of Instrument (HI)
Elevation of the line of sight above the benchmark, calculated as RL + BS.
Steps to Calculate Using Auto Level Formula
- Set up the Auto Level: Mount the instrument on a tripod and level it using screws and the circular bubble.
- Take Backsight Reading (BS): On the benchmark or known RL point.
- Compute HI: Add BS to the known RL.
- Take Intermediate and Foresight Readings: On all points to be leveled.
- Calculate RLs: Subtract each IS or FS from HI.
- Check the Arithmetic: (Sum of BS – Sum of FS) = Last RL – First RL
This ensures the accuracy of your observations.
Practical Example of Auto Level Formula
Let’s take a real-world example to understand calculations.
| Point | BS (m) | IS (m) | FS (m) | HI (m) | RL (m) |
|---|---|---|---|---|---|
| BM | 1.250 | – | – | 101.250 | 100.000 |
| A | – | 1.500 | – | – | 99.750 |
| B | – | – | 2.000 | – | 99.250 |
| C | 1.800 | – | – | 101.050 | 99.250 |
| D | – | – | 1.400 | – | 99.650 |
Calculations:
- HI₁ = RL (BM) + BS = 100.000 + 1.250 = 101.250
RL (A) = 101.250 – 1.500 = 99.750
RL (B) = 101.250 – 2.000 = 99.250 - HI₂ = RL (B) + BS = 99.250 + 1.800 = 101.050
RL (D) = 101.050 – 1.400 = 99.650
This gives a complete level profile for the section surveyed.
Checks for Accuracy
Every leveling operation must satisfy the arithmetic check:
ΣBS – ΣFS = Last RL – First RL
If both sides of the equation are equal, the leveling is accurate.
Advantages of Using Auto Level Formula
- Provides accurate elevation data.
- Simple mathematical relationship makes calculations fast.
- Reduces human error with automatic compensation.
- Useful for road grading, building foundations, and drainage layout.
- Ensures precise slope and gradient determination.
Applications of Auto Leveling
- Road Construction: Setting longitudinal and cross-sections.
- Building Construction: Leveling foundation and floors.
- Canal and Drain Works: Maintaining uniform gradients.
- Topographic Surveys: Determining ground profiles.
- Bridge Construction: Setting pier and abutment levels.
Common Errors in Auto Leveling
- Instrument not leveled properly before readings.
- Staff held incorrectly (not vertical).
- Incorrect reading order (BS, IS, FS).
- Parallax error due to improper focusing.
- Recording mistakes in field notes.
Proper care, double-checking calculations, and maintaining the instrument ensure accuracy.
FAQs About Auto Level Formula
1. What is the basic formula for auto level?
The basic formula is:
Height of Instrument (HI) = RL + BS
Reduced Level (RL) = HI – FS
2. What is the difference between BS and FS?
BS is taken on a known elevation (benchmark), while FS is taken on the last point before changing the instrument position.
3. Which method is more accurate – HI or Rise and Fall?
The Rise and Fall method is more accurate because it allows cross-verification of each level difference.
4. What is the importance of the arithmetic check in leveling?
It ensures no major observation or recording errors have occurred.
5. What is the role of the compensator in an auto level?
The compensator automatically maintains the horizontal line of sight, improving accuracy and efficiency.
Conclusion
The auto level formula is the foundation of accurate leveling in civil engineering. By using the relationships between backsight, foresight, height of instrument, and reduced level, surveyors can precisely determine elevation differences. Whether constructing roads, bridges, or buildings, mastering this formula ensures that every structure stands on a perfectly leveled base — essential for safety, durability, and performance.
