How to Use This Calculator
Using this reduced level calculator is straightforward. Just follow these steps with your field data:
- Enter Benchmark Elevation: Input the known Reduced Level of your Benchmark (BM) into the first field. This is your starting point.
- Input Backsight (BS): Enter the staff reading you took on the benchmark immediately after setting up your instrument.
- Input Foresight (FS): Enter the staff reading taken on the new point (Point A) whose elevation you want to find.
- Click Calculate: Hit the blue "Calculate RL" button. The tool will instantly compute the new Reduced Level.
- View Steps: Click "Show Steps" to see exactly how the formula was applied (HI = BM + BS, then RL = HI - FS).
Complete Guide to Reduced Level (RL) and Surveying Methods
Surveying is the backbone of civil engineering, ensuring that structures like roads, bridges, and skyscrapers are built on level ground and at the correct elevations. At the heart of surveying lies the concept of Leveling—the process of determining the relative heights of different points on the earth's surface. Whether you are a student, a site engineer, or a land surveyor, understanding how to calculate the Reduced Level (RL) is a critical skill.
This comprehensive guide will walk you through the definitions, the mathematics behind the Height of Instrument (HI) method, field procedures, and how to avoid common errors. By the end of this article, you will have a deep understanding of leveling computations.
1. What is Reduced Level (RL)?
Reduced Level (RL) refers to the vertical distance of a specific point above or below a standard reference surface, known as a datum. In most government and major infrastructure projects, the datum used is the Mean Sea Level (MSL).
- If a point has an RL of 100.000m, it means the point is exactly 100 meters above the Mean Sea Level.
- If a point has an RL of -5.000m, it is 5 meters below the Mean Sea Level (common in underwater surveying or deep mining).
Calculating RL is essential for determining the "Cut" (removing soil) or "Fill" (adding soil) required to bring the ground to the desired construction level (Formation Level).
2. Essential Leveling Terminology
Before performing calculations, it is vital to understand the vocabulary used in a surveyor's field book:
| Term | Detailed Definition |
|---|---|
| Datum Surface | An imaginary level surface (like Mean Sea Level) from which vertical distances are measured. In local surveys, an arbitrary datum (e.g., assuming a specific step is RL 100.00) is often used. |
| Benchmark (BM) | A fixed reference point of known elevation. Permanent BM: Established by government agencies (e.g., GTS Benchmarks). Temporary BM (TBM): A fixed point (like a fire hydrant) established at the end of a day's work to resume surveying later. |
| Backsight (BS) | The first reading taken after the leveling instrument (Dumpy Level or Auto Level) is set up and leveled. It is always taken on a point of known elevation (usually a Benchmark or a Change Point). It is also called a "Plus Sight" (+S) because it is added to the RL to find the instrument height. |
| Foresight (FS) | The last reading taken from an instrument setup before the instrument is moved. It is taken on a point whose elevation needs to be determined. It is also called a "Minus Sight" (-S) because it is subtracted from the HI. |
| Intermediate Sight (IS) | Any staff reading taken on a point between the Backsight and Foresight on the same instrument setup. These are used to determine the RL of points that are not change points. |
| Change Point (CP) / Turning Point (TP) | A point where both a Foresight and a Backsight are taken. This occurs when the surveyor needs to move the instrument to a new location. The FS is taken to establish the point's RL, and then a BS is taken from the new position to establish the new Height of Instrument. |
3. The Height of Instrument (HI) Method Explained
The Height of Instrument (HI) method, also known as the Collimation Method, is one of the two primary methods for reducing levels (the other being the Rise and Fall method). The HI method is generally preferred in construction setting-out and profile leveling because it is faster and involves less calculation when there are many Intermediate Sights.
The Mathematical Logic
The calculation follows a simple two-step cycle for every instrument setup:
Step 1: Find the Instrument Height
When you look back at a benchmark, you are measuring how high your telescope is above that benchmark.
HI = Reduced Level of BM + Backsight (BS)
Step 2: Find the New Level
When you look forward at a new point (staff station), you subtract that reading from your instrument height to see how high the ground is.
RL = HI - Staff Reading (FS or IS)
Arithmetic Check for HI Method
In any survey, errors can creep in. To verify your math, you should perform an arithmetic check at the bottom of your level book page. For the HI method, the check is:
∑ BS - ∑ FS = Last RL - First RL
(The sum of all Backsights minus the sum of all Foresights must equal the difference between the Last Reduced Level and the First Reduced Level). Note that this check validates the Reduced Levels of the Change Points but does not check the Intermediate Sights. This is the main disadvantage of the HI method compared to the Rise and Fall method.
4. Step-by-Step Field Procedure for Leveling
Knowing the formula is one thing, but gathering the data requires precision in the field. Here is the standard operating procedure for running a level loop:
Phase 1: Setup
- Tripod Setup: Extend the tripod legs to a comfortable height. Spread them widely and firmly press the shoes into the ground to ensure stability. The tripod head should be relatively level by eye.
- Mounting: Screw the Dumpy Level or Auto Level securely onto the tripod head.
- Leveling Up: Use the three foot-screws. Turn two screws simultaneously in opposite directions to bring the bubble to the center line. Then, turn the third screw to center the bubble completely in the circular vial. Rotate the telescope 180° to check if the bubble stays centered.
- Focusing: Aim the telescope at a bright background and rotate the eyepiece to make the crosshairs crisp (eliminating parallax). Then, focus on the leveling staff.
Phase 2: Taking Readings
- Staff Man: The assistant holds the leveling staff vertically on the Benchmark.
- Backsight: The surveyor takes the reading on the Benchmark. This is the BS.
- Intermediate Points: The staff man moves to various points required for the survey (e.g., every 10 meters along a road centerline). The surveyor takes readings. These are IS.
- Change Point: When the staff is too far away, or the terrain changes drastically, a Change Point is selected. A reading is taken (FS).
- Moving: The instrument is moved to a new position. The staff man must not move the staff from the Change Point during this time.
- New Backsight: From the new position, a reading is taken on the staff at the Change Point (BS), establishing a new HI.
5. Comparison: HI Method vs. Rise and Fall Method
Students often ask which method is better. The choice depends on the type of survey.
| Feature | Height of Instrument (HI) Method | Rise and Fall Method |
|---|---|---|
| Speed | Faster. Requires fewer calculations per point. | Slower. Requires calculating the difference for every single point. |
| Accuracy Check | Checks RL of Change Points only. No check for Intermediate Sights. | Checks arithmetic for all points (BS, FS, and IS). |
| Best Use Case | Profile leveling, plotting contours, and setting out construction levels where many readings are taken from one setup. | Fly leveling, establishing Benchmarks, and high-precision surveys where arithmetic accuracy is paramount. |
| Visualization | Easier to visualize the "plane of sight". | Easier to visualize the terrain slope (rising or falling). |
6. Common Sources of Error in Leveling
Even with a perfect calculator, bad data leads to bad results. Watch out for these errors:
- Collimation Error: If the line of sight is not perfectly horizontal when the bubble is centered, all readings will be inclined. This is checked using the "Two-Peg Test".
- Parallax: If the crosshairs are not focused correctly relative to the image, the reading will shift as you move your eye.
- Staff Verticality: If the staff man holds the staff at an angle, the reading will be too high. The staff should always be rocked back and forth; the surveyor records the lowest reading.
- Earth's Curvature & Refraction: Over long distances, the earth curves away from the straight line of sight, and light bends due to air density. To minimize this, keep BS and FS distances roughly equal (balancing sights).
7. Practical Applications of Reduced Levels
Why do we go through all this trouble? RL data is used for:
- Highway Engineering: Determining the longitudinal section of a road to balance cut and fill volumes.
- Drainage & Sewerage: Ensuring pipes are laid at a specific gradient (slope) so gravity can move the water.
- Contour Maps: Creating topographic maps that show the lay of the land for architects.
- Building Foundations: Ensuring the floor slab is perfectly level and at the height specified by the architect relative to the road.
Frequently Asked Questions (FAQ)
Why do we add Backsight (BS) to Reduced Level?
Because the Backsight is a reading taken looking "back" at a known height. By adding the staff reading to the known ground height, you are calculating the absolute elevation of the telescope's crosshair (the Height of Instrument).
Can a Reduced Level be negative?
Yes. If a point is below the reference datum (like Mean Sea Level), the Reduced Level will be negative. This is common in dredging operations, areas like the Dead Sea, or deep mines.
What is the difference between Geodetic and Plane Surveying?
In Plane Surveying, we assume the earth is flat. This is acceptable for small areas (less than 250 sq km). In Geodetic Surveying, the curvature of the earth is taken into account for high precision over large areas.
What is a 'fly leveling'?
Fly leveling is a type of differential leveling done to determine the elevation of a distant point. It involves a series of Backsights and Foresights with no Intermediate Sights, "flying" from the start to the end point quickly.
How do I correct a closing error?
If you start at a Benchmark and finish at a known Benchmark, but your calculated RL is slightly different, that is a closing error. If the error is within permissible limits, it is distributed evenly (or cumulatively) among the intermediate stations.