What is levelling formula?

Levelling is a fundamental surveying technique used to determine the relative heights or elevations of various points on the earth's surface. The core "levelling formulas" primarily involve calculating the Height of Instrument (H.I.), determining the Reduced Levels (R.L.) of points, and performing a crucial arithmetic check to ensure accuracy.

Core Levelling Concepts and Terminology

Before diving into the formulas, understanding key terms is essential:

• Reduced Level (R.L.): This is the vertical distance of a point above or below a specific reference plane, known as a datum. Often, the mean sea level is used as a datum.
• Back Sight (B.S.): The very first staff reading taken from a levelling instrument setup on a point with a known R.L. It's usually a positive reading added to the R.L. to find the H.I.
• Fore Sight (F.S.): The last staff reading taken from a particular instrument setup, typically on a Change Point (C.P.) or the final point of the survey. It's subtracted from the H.I. to get the R.L.
• Intermediate Sight (I.S.): Any staff reading taken between a Back Sight and a Fore Sight from the same instrument setup. Like F.S., it's subtracted from the H.I. to determine the R.L.
• Height of Instrument (H.I.): The elevation of the line of sight of the levelling instrument above the chosen datum.

The Height of Instrument (H.I.) Method Formulas

The Height of Instrument method is one of the most widely used techniques for reducing levels in surveying. It involves systematically determining the H.I. for each instrument setup and then using this H.I. to calculate the R.L. of all observed points.

1. Calculating the Height of Instrument (H.I.)

The H.I. for any instrument setup is determined by adding the known Reduced Level (R.L.) of the point where the Back Sight (B.S.) reading was taken, to that B.S. reading. This applies to each section of the levelling run where the instrument's position, and thus its H.I., might change.

• Formula: H.I. = R.L. of Previous Point + B.S.

2. Determining New Reduced Levels (R.L.)

Once the H.I. is established for a particular instrument setup, the Reduced Level of any other point visible from that setup can be calculated by subtracting the staff reading (either an Intermediate Sight or a Fore Sight) taken at that point from the H.I.

• Formula: New R.L. = H.I. – Staff Reading (I.S. or F.S.)

The Essential Levelling Arithmetic Check Formula

To ensure accuracy and detect any computational errors in the H.I. method, a vital arithmetic check is performed at the completion of a levelling run. This check compares the sums of the Back Sights and Fore Sights with the difference between the final and initial Reduced Levels. If the results match, it confirms the mathematical correctness of the calculations.

• Formula: Σ B.S. – Σ F.S. = Last R.L. – First R.L.

  • Σ B.S. represents the sum of all Back Sight readings.
  • Σ F.S. represents the sum of all Fore Sight readings.
  • Last R.L. is the Reduced Level of the very last surveyed point.
  • First R.L. is the Reduced Level of the starting point (often a Benchmark).

If both sides of this equation yield the same value, the arithmetic reductions are considered correct.

Practical Application Example: Applying Levelling Formulas

Let's illustrate these formulas with a simple levelling survey from point A (a Benchmark) to point E.

Station	B.S. (m)	I.S. (m)	F.S. (m)	H.I. (m)	R.L. (m)	Remarks
A	1.500			101.500	100.000	B.M.
B		0.800		101.500	100.700
C (CP1)	2.100		0.500	103.100	101.000	C.P.
D		1.200		103.100	101.900
E			2.000		101.100	Final Pt
Sums	3.600		2.500

Step-by-Step Calculations:

1. Station A (Benchmark): The R.L. is given as 100.000 m.
2. Calculate H.I. at A: H.I. = R.L. of A + B.S. at A = 100.000 + 1.500 = 101.500 m.
3. Calculate R.L. at B: R.L. = H.I. - I.S. at B = 101.500 - 0.800 = 100.700 m.
4. Calculate R.L. at C (Change Point): R.L. = H.I. - F.S. at C = 101.500 - 0.500 = 101.000 m.
5. Calculate New H.I. at C (after instrument moved): H.I. = R.L. of C + B.S. at C = 101.000 + 2.100 = 103.100 m.
6. Calculate R.L. at D: R.L. = H.I. - I.S. at D = 103.100 - 1.200 = 101.900 m.
7. Calculate R.L. at E (Final Point): R.L. = H.I. - F.S. at E = 103.100 - 2.000 = 101.100 m.

Applying the Arithmetic Check:

Using the formula Σ B.S. – Σ F.S. = Last R.L. – First R.L.:

• Left Side:

  • Σ B.S. = 1.500 + 2.100 = 3.600 m
  • Σ F.S. = 0.500 + 2.000 = 2.500 m
  • Σ B.S. – Σ F.S. = 3.600 – 2.500 = 1.100 m

• Right Side:

  • Last R.L. (at E) = 101.100 m
  • First R.L. (at A) = 100.000 m
  • Last R.L. – First R.L. = 101.100 – 100.000 = 1.100 m

Since 1.100 m = 1.100 m, the arithmetic calculations are verified as correct.

Importance of Accurate Levelling

Accurate levelling is critical for a vast range of civil engineering and construction applications, including:

• Designing and constructing roads, railways, and bridges.
• Establishing gradients for drainage systems and pipelines.
• Setting out building foundations and structural elements.
• Calculating earthwork volumes for cut and fill operations.
• Creating topographic maps and elevation models.

The formulas for calculating the Height of Instrument, determining Reduced Levels, and especially the arithmetic check, form the backbone of accurate levelling operations. Mastering these ensures the reliability of vertical control in surveying projects.