Tacheometric surveying is one of the methods of angular surveying in which a tachometer is used to determine both the horizontal and vertical distance between two points. Tacheometric Surveying eliminates chain surveying as it becomes tedious to work with chain or tape in rough grounds, deep ravines, swampy areas, stretches of water bodies etc. It is also a less time-consuming method as compared to chain surveying. Hence the demand for Tacheometric Surveying among construction industries has peaked in recent years. The accuracy of this surveying technique ranges from 1/1000 to 1/10000.

This comprehensive article will cover all the details about Tacheometric Surveying including its principle, instruments involved, methods, errors and advantages of this technique. This article will definitely help in boosting up your preparation for SSC JE CE, RRB JE and also for GATE CE.

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Tacheometric Surveying

Tacheometric surveying is an angular surveying technique that employs optical instruments to measure horizontal and vertical distances between two points. It eliminates the need for traditional chain and tape methods and relies solely on instrumental observations to determine the distance between the instrument and staff stations. This method involves the use of specialized instruments such as the transit theodolite and stadia rod.

What is Tacheometric Surveying?

Tacheometric surveying, also known as tacheometry, is an angular surveying method used to determine both horizontal and vertical distances between two points. It is a rapid surveying technique that relies solely on instrumental observations to determine the horizontal distance from the instrument to the staff stations. Tacheometry eliminates the need for chains, tapes, or separate leveling instruments. This method utilizes specialized instruments such as the transit theodolite and stadia rod. Unlike traditional surveying methods that require linear measurements using chains or tapes, tacheometry offers a faster and less tiring approach. It is commonly employed in various applications, including the preparation of topographic maps, surveying in challenging terrains, reconnaissance surveys for highways and railways, and the establishment of secondary control points.

Tacheometry is defined as an optical method for determining horizontal and vertical distances as well as horizontal and vertical angles. Other names for tacheometry are Tachemetry or Telemetry. Tacheometry uses two important instruments: Tacheometer and Stadia Rod. 

Tacheometer

It is a transit theodolite fitted with a stadia diaphragm. The stadia consist of two stadia hairs at equal distances, one above and the other below the horizontal hair of the cross-hairs. 

Fig.1. Stadia Diaphragm

Characteristics of a Tacheometer

• Value of multiplying constant, k=100
• Value of additive constant, C=0
• Telescope should be fitted with an anallactic lens which is a type of convex lens.
• Magnifying power of the eye-piece should be kept high to make the staff graduations clearly visible even from a large distance.
• The aperture of the objective is kept roughly between 35 mm to 45 mm to make the image sharper. 

Stadia Rod

The ordinary levelling staff in Tacheometric Surveying having 5 mm graduations can be used only for short distances, and hence the requirement for a better alternative arises. A Stadia rod is a special type of levelling staff which can be sighted clearly from long distances as well. The length of the stadia rod varies from 3 to 5 m, and the width range is from 50 to 150 mm. The graduations in stadia rod are made bold and simple, which facilitates the reading even from a large distance. The stadia rod can be placed vertically or normal to the line of sight.

Fig.2. Stadia rod

Principles of Tacheometric Surveying

The principle of Tacheometric surveying is based on the property of an isosceles triangle, i.e. the ratio of the distance of the base from the apex and the length of the base is always constant.

The horizontal distance is determined using the mathematical formula : 

\( \mathrm{D}=\mathrm{k} \times \mathrm{S} \times(\cos \theta)^{2}+\mathrm{C} \times \cos \theta\)

Where,

• S = Staff intercept = difference between top and bottom reading
• k = Multiplying constant 
• C = Additive constant
• 𝜭 = Vertical angle measured by Tacheometer

And, the vertical distance is calculated as : 

\(\mathrm{V}=\mathrm{k} \times \mathrm{S} \times \sin \theta \times \cos \theta+\mathrm{C} \times \sin \theta\)

• It should be noted that if an Anallactic lens is used, then k=100 and C=0.

Principles of Tacheometric Surveying

The principle of Tacheometric Surveying is based on the optical geometry of isosceles triangles. In this method, the horizontal distance and vertical height between two points are calculated using the staff intercept, vertical angle, and instrument constants. This eliminates the need for measuring tapes or chains and offers a quick and efficient way to survey uneven or inaccessible terrain. Below are some commonly used parameters and their typical values in Tacheometric calculations:

Methods of Tacheometric Surveying

There are four methods of determining distances using optical means 

Stadia Method of Tacheometry

This is the most popular method of Tacheometric Surveying. The horizontal distance between the staff station and the instrument station can be determined and the elevation of the staff station with respect to the line of sight can be calculated with a single observation from the instrument station. 

Fig.3. Stadia Method

The horizontal distance D between the instrument station A and staff station B and the difference in their elevations can be determined using the staff intercept s and the vertical angle 𝜭.

Stadia method is further classified into two types : 

• Fixed hair system: In this system, the vertical distance between the upper and lower stadia hair is fixed, and this fixed distance is called the stadia interval (i). The Stadia interval is not changed during the measurement. When the tacheometer is targeted at staff, then staff intercept (s) is obtained from the difference of readings of upper hair and lower hair. Fixed hair system is most commonly used in practice.

Fig.4. Fixed hair stadia method

• Movable hair system: In this stadia method of tacheometry, the distance between the upper hair and lower hair (i.e. the stadia interval, i) is varied by moving the stadia hairs vertically by the micrometre screws. Here, the staff intercept (s) is fixed. The staff used has two vanes called targets which are fixed at a constant spacing (generally 2 or 3m). While taking the reading, the stadia interval is varied and is measured corresponding to the staff intercept. It is quite difficult to measure the stadia interval accurately, and thus the movable hair method is seldom used. 
•  It should be noted that unless otherwise specified, the stadia system refers to fixed hair system

Fig.5. Movable hair stadia method

Comparison: Fixed Hair vs Movable Hair Stadia Method

In the Stadia Method of Tacheometric Surveying, there are two systems – Fixed Hair and Movable Hair – based on how the stadia interval is handled. Each has its own features, usage conditions, and accuracy levels. The key differences between these two methods are highlighted below:

Tangential Method of Tacheometry

Stadia hairs are not required in the tangential method of Tacheometry. This method can even be used when the telescope is not provided with a diaphragm. The staff has two targets at a fixed distance (s) apart. 

Fig. 6. Tangential method

The vertical angles \(\theta_1\) and \(\theta_2\) are measured for the two targets. These vertical angles and the staff intercept is required to calculate the horizontal distance D as well as the difference in elevations. 

Range Finding

This method is used to determine the horizontal distance and direction of a line without going to the far end of the line. The instrument used is called a rangefinder. A fixed base is used to compute the ranges. The instruments and methods used are based on the measurement of either the base angles or the angle of parallax. By this method, only horizontal distance can be measured, but with the help of a level, the vertical distances can also be measured. 

Subtense Bar System 

Subtense bar is a bar of fixed length. It has two targets at a fixed distance (s) apart. The horizontal angle subtended between the instrument station and the two targets is measured. There is less chance of error due to refraction because the line of sight does not change its path. 

Fig.7. Subtense Bar Method

Errors in Tacheometric Surveying

Errors in Tacheometric Surveying occur due to humans, instruments and natural causes. 

Personnel Error

• Error due to faulty centering, levelling and bisection: If the instrument is not levelled and centred properly, this type of error may creep in.
• Error due to non-verticality of staff: Sometimes the staff is not held perfectly vertically because of which some measurement faults occur. A plumb bob is required in this case to ensure verticality.
• Error due to wrong reading of staff: It may sometimes happen that the reading taken from the staff intercept is not properly recorded and because of this, the whole process becomes erroneous. To avoid this error, it is recommended to read the staff readings for all three stadia wires and to check where the difference of readings of upper and middle stadia is equal to the difference of readings of middle and lower stadia.
• Error due to faulty reading of the vertical angle: As we know, the vertical angle is also measured in Tacheometric Surveying. Hence, it is advisable to take that readings properly as well.

Instrument Error

• Errors due to faulty instrument constants: With continuous use of Tacheometer, the multiplying and additive constants may get changed and because of this, error occurs.
• Error due to imperfect adjustment of instrument: Some adjustments in the instrument are made in the manufacturing process and they cause an error in the Tacheometry. 
• Error due to stadia rod: When there is a fault in graduations and standardisation, it causes an error.

Error due to Natural Causes

• Error due to wind: Strong winds make it difficult to hold the staff in the desired position. 
• Error due to Sun: Due to excessive heat, sometimes there occurs unequal expansion of the instrument and error may creep in. 
• Atmospheric refraction: Atmospheric refraction tends to change the path of line of sight and error occurs. It is advisable to apply refraction correction after the readings have been recorded. 

Errors in Tacheometric Surveying

Various types of errors can occur during Tacheometric Surveying due to human mistakes, faulty instruments, or environmental conditions. Understanding these errors and taking proper precautions is essential for ensuring accuracy in distance and elevation measurements. Here is a summary of the major error types, their causes, and preventive measures:

Advantages of Tacheometric Surveying

The advantages of Tacheometric Surveying are as follows: 

• It is a very quick method.
• No chains or tapes are required to measure the horizontal distance.
• It is an economical method.
• It can be used for any terrain, including hilly areas, rough grounds, marshy and swampy lands, as well as stretches of rivers.

Limitations of Tacheometric Surveying

Some of its limitations are as follows. 

• The man holding the staff must reach the point whose elevation must be determined.
• The staff station must be clearly visible from the instrument station.

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