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Types of Errors and Corrections in Linear Surveying

Types of Errors and Corrections in Linear Surveying

In linear surveying, particularly when using a chain or tape, various factors cause discrepancies in measurements. These differences are known as 'Errors'. To maintain the accuracy of a survey, it is essential to understand and correct these errors.

रेखीय सर्वेक्षण (Linear Survey) में त्रुटियों के प्रकार और सुधार

1. Cumulative Errors

These are also known as systematic biases in some contexts, where the error continues to increase in one direction (either always positive or always negative).

Cause: Carelessness of the surveyor or equipment defects (e.g., the chain being longer or shorter than the standard length).
Characteristic: As the length of the survey increases, this error also accumulates. This is why it is called 'Cumulative'.
Remedy: Frequently checking the equipment against a standard tape and taking readings with extreme care.

2. Systematic Errors

These errors follow a specific mathematical rule or pattern.

Cause: Variations in temperature, tension (pull) on the tape, or tape sag.
Remedy: These can be corrected using specific Mathematical Formulas.

3. Accidental or Random Errors

Small errors that remain even after all systematic corrections are applied are called accidental errors.

Nature: They can be positive or negative and often tend to cancel each other out over a long series of measurements.
Analysis: These are analyzed using the Theory of Probability.

Statistical Analysis of Errors

Surveying observations generally follow a Normal Distribution or Gaussian Distribution.

Most Probable Value (MPV)

Since the 'True Value' of any quantity is impossible to determine, we consider the Arithmetic Mean of several observations as the Most Probable Value.

The difference between an observed value ( $x$ ) and the mean ( $\mu$ ) is called the Residual ( $v$ ):

v = x - \mu

Standard Deviation ( $\sigma$ )

Also known as the RMS (Root Mean Square) error, it indicates the spread of observations around the mean.

\sigma = \sqrt{\frac{\sum v^2}{n}}

(Where $n$ = number of observations)

Standard Error of the Mean

This indicates the precision of the calculated mean:

E_{mean} = \frac{\sigma}{\sqrt{n}}

Corrections in Linear Measurement

For NTS Study readers, remembering these formulas is vital for accuracy:

Correction for Absolute Length ( $C_a$ ):
$C_a = \frac{L \cdot c}{l}$
Temperature Correction ( $C_t$ ):
$C_t = \alpha(T_m - T_o)L$
Sag Correction ( $C_s$ ): (This is always negative)
$C_s = \frac{W^2 L}{24 P^2}$

Numerical Example

Question: A line was measured with a 30m steel tape and found to be 900m. During the survey, the following conditions were noted:

The tape was 0.05m longer than its standard length.
The field temperature was 35°C, while the tape was calibrated at 20°C.
Coefficient of thermal expansion for steel ( $\alpha$ ) = $12 \times 10^{-6} / ^\circ C$ .
Calculate the True Length.

Solution:

Step 1: Correction for Absolute Length ( $C_a$ )

Since the tape is longer than the standard, the correction is Positive.

Measured Length ( $L$ ) = 900 m
Actual Tape Length ( $l'$ ) = $30 + 0.05 = 30.05 \text{ m}$
Standard Tape Length ( $l$ ) = 30 m
$\text{Corrected Length} = \left( \frac{l'}{l} \right) \times L$
$\text{Corrected Length} = \left( \frac{30.05}{30} \right) \times 900 = 901.50 \text{ meters}$

Step 2: Temperature Correction ( $C_t$ )

Since $T_m (35^\circ C) > T_o (20^\circ C)$ , the tape has expanded, making the correction Positive.

$C_t = \alpha(T_m - T_o)L$
$C_t = 0.000012 \times (35 - 20) \times 900$
$C_t = 0.000012 \times 15 \times 900 = \mathbf{0.162 \text{ meters}}$

Step 3: Final True Length

\text{Total Length} = \text{Length from Step 1} + C_t

\text{Total Length} = 901.50 + 0.162 = \mathbf{901.662 \text{ meters}}

Conclusion: Without corrections, the reading was 900m, but the scientifically true length is 901.662m. In civil engineering, this difference of 1.662m could cause a major construction error, which is why these corrections are mandatory.

NTS Study Pro-Tip: Always remember:
If the tape is Long $\rightarrow$ Correction is (+).
If the tape is Short $\rightarrow$ Correction is (-).

🏗️ Surveying: Complete Study Guide & Index

📔 Part 1: Fundamentals of Surveying

Surveying: A Bird's Eye View – Meaning and significance of land surveying.
Fundamental Principles – Classification and types of surveying.
Primary Division – Understanding Plane vs. Geodetic Surveying.
Representative Fraction (RF) – Utilization of scales and reduction factors.

📏 Part 2: Linear Measurement & Chain Survey

Chain Surveying – Basic procedures and workflow.
Errors & Adjustments in Chaining – Deficiencies in measurement and their remedies.
Distance Measurement Methods – Detailed discussion on linear surveying tools.
Tape Corrections – Adjustments for Sag, Temperature, and Pull.

🧭 Part 3: Angular & Instrumental Survey

Compass Surveying – Magnetic bearing survey and its applications.
Plane Table Surveying – Equipment used and graphical methods.
Theodolite Surveying – Horizontal and vertical angle measurement.
Total Station – Components, features, and modern digital use.

🏔️ Part 4: Levelling & Elevation

Need for Levelling – Why vertical measurement is vital in civil engineering.
Key Concepts: RL & Datum – Definitions of Reduced Level, Datum, and Benchmarks.
Operating Levelling Instruments – Handling Auto Level and Tilting Level.

🛰️ Part 5: Modern Technologies

Remote Sensing – Information on INSAT and IRS Series satellites.
GIS & LIS Systems – Geographic data management and functionality.
Laser Scanning – Advanced application and control.
Geoid & Ellipsoid – Understanding the mathematical shape of the Earth.

📝 Part 6: Practice & Quizzes (MCQs)

Surveying Quiz 1 (01-25) – GPS, Remote Sensing, and Photogrammetry.
Surveying Quiz 2 (26-50) – Ranging, EDM, and Tacheometry.
Surveying Quiz 3 (51-75) – Contouring and HI Method Levelling.
Surveying Quiz 4 (76-100) – Transition Curves and Bowditch Rule.
Surveying Quiz 5 (101-125) – Plane table and Compass
Surveying Quiz 6 (125-150) – Theodolite Surveying and Levelling
Surveying Quiz 7 (151-175) – Tacheometry, Curves, Modern Surveying Instruments (EDM/GPS)
Surveying Quiz 8 (175-200) – Area & Volume Calculation, Minor Instruments

📚 Quick Revision Resources

Surveying IS Codes with Latest Revision Years

1. General Surveying & Instruments

IS 1491:1959 – Specification for Prismatic Compass (Liquid and Non-liquid).
IS 1963:1981 – Specification for Bubbles for Surveying Instruments.
IS 2988:1995 – Glossary of Terms Relating to Surveying Instruments.
IS 1634:1992 – Code of Practice for Design and Construction of Storage for Surveying Instruments.

2. Chain and Tape Surveying

IS 1492:1970 – Specification for Metric Surveying Chains.
IS 1269 (Part 1):1997 – Material and Construction of Steel Tapes.
IS 1269 (Part 2):1997 – Woven Metallic and Glass Fibre Tapes.
IS 1659:2006 – Specification for Invar Tapes for High Precision Measurement.

3. Theodolite and Tacheometry

IS 8002:1976 – Specification for Surveying Chain Vertical Vernier Theodolite.
IS 8330:1976 – Specification for Tilting Levels (Optical).
IS 8636:1977 – Specification for Tacheometers.

4. Leveling and Contouring

IS 9128:1992 – Specification for Tilting Levels.
IS 9573:1980 – Specification for Automatic Levels.
IS 1779:1961 – Specification for 4-metre Leveling Staff (Folding Type).

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