Theodolite

The theodolite is the most accurate instrument used mainly for measuring horizontal and vertical angles. It can also be used for locating points on a line, prolonging survey lines, finding the differences in elevations, setting out grades, ranging curves, etc.

Depending upon the facilities provided for the reading of observations, the theodolites may be classified as simple vernier theodolite, micrometer theodolite, optical (glass arc) theodolite and electronic theodolite.

A modern theodolite is compact, light in weight, simple in design and can be used rough. All the movable parts and scales are fully enclosed and virtually dust and moisture proof. Its lower graduated circle defines the size of a theodolite. For example, a 20 cm theodolite means the diameter of the graduated circle of the lower plate is 20 cm. The size of the theodolites varies from 8 to 25 cm.

Theodolites may be classified into transit and non-transit theodolites. A theodolite is said to be a transit one when its telescope can be revolved through 180° in vertical plane about its horizontal axis, thus directing the telescope in exactly opposite directions.

A theodolite is said to be a non-transit one when its telescope cannot be revolved through 180° in a vertical plane about its horizontal axis. Such theodolites are obsolete nowadays. Examples are the Y-theodolite and everest theodolite (Figure 27.15).

 

 

Figure 27.15 Theodolite

 

 

Figure 27.16 Theodolite

 

An instrument used for measuring horizontal and vertical angles accurately is known as theodolite (Figure 27.16).

Classification of theodolites

Theodolites are primarily classified as

1. Transit theodolite
2. Non-transit theodolite

1. Transit theodolite: The theodolite whose telescope can be transited is called a transit theodolite. A transit telescope can be revolved through a complete revolution about its horizontal axis in a vertical plane.
2. Non-transit theodolite: The theodolite whose telescope cannot be transited is called a non-transit theodolite. A non-transit telescope cannot be revolved through a complete revolution about its horizontal axis in vertical plane.

Non-transit theodolites are inferior as compared to transit theodolites. These have become almost outdated nowadays.

The theodolites are classified as follows

1. Vernier theodolite: In this type of theodolites, verniers are provided for reading horizontal and vertical graduated circles.
2. Glass arc theodolite: In this type of theodolites, micrometers are provided for reading horizontal and vertical graduated circles.

Technical terms

1. Vertical axis: The axis about which the theodolite may be rotated in a horizontal plane is called vertical axis. Both upper and lower plates may be rotated about the vertical axis.
2. Horizontal axis: The axis about which the telescope along with the vertical circle of a theodolite may be rotated in a vertical plane is called horizontal axis. It is also called as transverse axis.
3. Line of collimation: The line that passes through the intersection of the cross hairs of the eyepiece and optical centre of the objective and its continuation is called line of collimation. The angle between the line of collimation and the line perpendicular to the horizontal axis is called error of collimation. The line passing through the eyepiece and any point on the objective is called line of sight.
4. Axis of telescope: The axis about which the telescope may be rotated is called axis of telescope.
5. Axis of the level tube: The straight line that is tangential to the longitudinal curve of the level tube at its centre is called axis of the level tube. When the bubble of the level tube is central, the axis of the level tube becomes horizontal.
6. Centreing: The process of setting up a theodolite exactly over the ground station mark is known as centreing. It is achieved when the vertical axis of the theodolite is made to pass through the ground station mark.
7. Transiting: The process of turning the telescope in a vertical plane through 180° about its horizontal axis is known as transiting. The process is also sometimes known as reversing or plunging.
8. Swing: A continuous motion of the telescope about the vertical axis in the horizontal plane is called swing. The swing may be in either direction, i.e., left or right. When the telescope is rotated in clockwise (right) direction, it is known as right swing. If it is rotated in the anticlockwise (left) direction, it is known as left swing.
9. Face left observations: When the vertical circle is on the left of the telescope at the time of observations, the observations of the angles are known as face left observations.
10. Face right observations: When the vertical circle is on the right of the telescope at the time of observations, the observations of the angles are known as face right observations.
11. Changing face: It is the operation of changing the face of the telescope from the right to left and vice versa.
12. A measure: It is the determination of the number of degrees, minutes and seconds or grades contained in an angle.
13. A set: A set of horizontal observation of any angle consists of two horizontal measures, one on the face left and the other on the face right.
14. Telescope normal: A telescope is said to be normal when its vertical circle is to its left and the bubble of the telescope is up.
15. Telescope inverted: A telescope is said to be inverted or reversed when its vertical circle is to its right and the bubble of the telescope is down.

Fundamental lines of a transit

The fundamental lines of a transit are as follows:

1. The vertical axis

 Figure 27.17The axis of plate bubble 

1.  The fundamental lines of a transit
2. The line of collimation which is also sometimes called the line of sight
3. The horizontal axis
4. The bubble line of telescope bubble or altitude bubble (Figure 27.17).

Adjustments of a theodolite

The adjustments of a theodolite are of two types

1. Temporary adjustments
2. Permanent adjustments

Temporary adjustments

The adjustments which are required to be made at every instrument station before making observations are known as temporary adjustments. The temporary adjustments include mainly the following:

1. Setting up the theodolite over the station
2. Levelling of the theodolite
3. Elimination of parallax

Setting up

The operation of setting up of a theodolite includes the centreing of the theodolite over the ground mark and also the approximate levelling with the help of tripod legs.

Centreing The operation by which the vertical axis of the theodolite, represented by a plumb line, is made to pass through the ground station mark is called centreing.

The operation of centreing is carried out in the following steps:

• Suspend the plumb bob with a string attached to the hook fitted to the bottom of the instrument to define the vertical axis.
• Place the theodolite over the station mark by spreading the legs well apart so that the telescope is at a convenient height.
• The centreing may be done by moving the legs radially and circumferentially till the plumb bob hangs within 1 cm of the station mark.

It is necessary to ensure that the tripod is approximately levelled before centreing is done. The approximate levelling may be done by eye judgement.

Levelling of the theodolite

The process of making the vertical axis of the theodolite truly vertical is known as levelling. After having levelled approximately and centred accurately, accurate levelling is done with the help of plate levels (Figure 27.18).

The following steps are involved in levelling with a three-screw head:

• Turn the horizontal plate until the longitudinal axis of the plate level is approximately parallel to a line joining any two levelling screws.
• Bring the bubble to the centre of its run by turning both foot screws simultaneously in opposite directions.
• Turn the instrument through 180° in azimuth.
• Note the position of the bubble. If it occupies a different position, move it by means of the same foot screws to the approximate mean of the two positions.
• Turn the theodolite through 90° in azimuth so that the plate level becomes perpendicular to the previous position.
• With the help of the third foot screw, move the bubble to the approximate mean position already indicated.
• Repeat the process until the bubble retains the same position for every setting of the instrument in azimuth.

Elimination of parallax

An apparent change in the position of the object caused by the change in position of the observer’s eye is known as parallax. In a telescope, parallax is caused when the image formed by the objective is not situated in the plane of the cross hairs. Unless the parallax is removed, accurate bisection and sighting of objects become difficult.

 

 

Figure 27.18 Levelling of a theodolite with three-screw head

 

Elimination of parallax may be done in two ways as discussed below:

• Focusing the eyepiece: To focus the eyepiece for distinct vision of cross hairs, either hold a white paper in front of the objective or sight the telescope towards the sky. Move the eyepiece in or out until the cross hairs are seen sharp and distinct.
• Focusing the objective: After cross hairs have been properly focused, direct the telescope towards a well-defined distant object and intersect it with a vertical wire. Focus the objective till a sharp image is seen. Removal of parallax may be checked by moving the eye slowly to one side. If the object still appears intersected there is no parallax.

If on moving the eye laterally, the image of the object appears to move in the same direction as the eye and the observer’s eye and the image of the object are on the opposite sides of the vertical wire, then the image of the object and the eye are brought nearer to eliminate the parallax. This parallax is called far-parallax.

If, on the other hand, the image appears to move in a reverse direction to the movement of the eye and the observer’s eye and the image of the object are on the same side of the vertical wire, this parallax is known as near-parallax. It is removed by increasing the distance between the image and the eye.

Permanent adjustments

The permanent adjustments include:

1. Adjustment of the horizontal plate level
2. Adjustment of the horizontal axis
3. Adjustment of the telescope
4. Adjustment of the telescope level

Adjustment of the horizontal plate level

With this adjustment, the axis of the plate levels is made perpendicular to the vertical axis of the theodolite. This is necessary since the vertical axis should remain truly vertical for accurate measurement of vertical and horizontal angles. To test this, the theodolite must be set on a firm ground. Clamp the lower plate and turn the upper plate until the plate level becomes parallel to any pair of foot screws. Bring the bubble to the centre of its run by turning the foot screws. Now rotate the instrument about the vertical axis through 180°. If the bubble remains central, the vertical axis of the theodolite is perpendicular to the axis of the plate level.

Adjustment of the horizontal axis

With this, the horizontal axis is made perpendicular to the vertical axis. The object of this adjustment is to ensure that the line of sight revolves in a vertical plane perpendicular to the horizontal axis. This adjustment is very necessary for prolonging straight lines, by making observations on one face only.

Adjustment of the telescope

This adjustment includes adjustment of the horizontal hair and adjustment of the vertical hair. The object of adjustment of the horizontal hair is to bring the horizontal hair of the eyepiece into the horizontal plane through the optical axis. If the horizontal hair does not lie in the horizontal plane through the optical axis, vertical angles will have error. It is particularly required when the instrument is used in levelling operations. It has no effect in the measurement of horizontal angles.

The object of adjustment of the vertical hair is to make the line of collimation perpendicular to the horizontal axis. This adjustment is necessary for measuring horizontal angles between points at different elevations and also for prolonging the lines by making observations on one face only.

Adjustment of the telescope level

The object of this adjustment is that the line of collimation should remain horizontal when the bubble of the level tube, fitted on the telescope, is brought at the centre of its run. This adjustment is essential when the theodolite is used as a level and also when vertical angles are observed.

27.10.5 Measurement of horizontal angles

27.10.5.1 Direct method of measuring the angle

To measure the horizontal angle between BA and BC the following procedure is adopted (Figure 27.19):

1. Set up, centre and level the theodolite over the ground point B.
2. Loosen the upper plate, set the vernier to read zero and clamp the upper plate.
3. Loosen the lower plate and swing the telescope until the left point A is sighted. Tighten the lower clamp. Accurate bisection of the arrow held on station A is done by using the lower tangent screw. Read both the verniers and take the mean of the readings.
4. Unclamp the upper plate and swing the telescope in clockwise direction until the point C is brought in the field of view. Tighten the upper clamp and bisect the arrow on station C accurately using the upper tangent screw.
5. Read both the verniers and take the mean of the readings. The difference of the means of the readings to stations C and A is the required angle ABC.
6. Change the face of the instrument and repeat the whole procedure. The measure of the angle is again obtained by taking the difference of the means of the readings to C and A on face right.
7. The means of the two measures of the angle ABC on the two faces is the required angle ABC.

Measurement of angle by method of repetition

Let ABC be the required angle between sides BA and BC to be measured. For accurate and precise work, the method of repetition is generally used. In this method, the value of the angle is added several times mechanically and the accurate value of the angular measure is obtained by dividing the accumulated reading by the number of repetitions (Figure 27.20).

 

 

Figure 27.19 Measurement of horizontal angles

 

 

Figure 27.20 Repetition method

 

To measure a small horizontal angle ABC, the following procedure is adopted:

1. Keeping the face of the instrument left, centre and level it accurately over the ground point B.
2. Set the vernier to read zero. Loosen the lower plate and swing the telescope in azimuth to sight the left-hand point A. Using the lower tangent screw, bisect the point A accurately.
3. Read both the verniers and take the mean of the two readings.
4. Loosen the upper plate and swing the telescope in clockwise direction until point C is brought in the field of view. Using the upper tangent screw bisect the point C accurately.
5. Read both the verniers and take the mean of the readings. The difference of the mean readings for points C and A gives the approximate value of the angle.
6. Unclamp the lower plate and turn the telescope in clockwise direction until point A is again sighted. Clamp it and bisect it accurately with the lower tangent screw.
7. Loosen the upper plate and swing the telescope in clockwise direction and again bisect point C accurately using the upper tangent screw. The verniers will now read double the value of the angle ABC.
8. Repeat the process until the angle ABC is repeated the required number of times, say 5 times.
9. Read both the verniers. The accumulated reading is obtained by taking the difference of the two mean readings to stations C and A.
10. Divide the accumulated angle by the number of repetitions to get the correct value of the angle ABC on face right.
11. The mean of the two values on the angle obtained on face left and face right gives the required value of the angle ABC.

Measurement of the angle by reiteration method

This method is generally adopted when several angles having a common vertex are to be measured. In this method, angles are measured successively, starting from a reference station and closing on the same station. Making observations on the starting station twice provides a check on the sum of all angles around a station. The sum should invariably be equal to 360°. This method is sometimes known as direction method of observation of the horizontal angles.

Let the instrument station be O whereas A, B, C, D and E are the stations sighted for measuring angles AOB, BOC, COD, DOE and EOA. To measure the angles by reiteration the following steps are involved (Figure 27.21):

1. Centre the theodolite accurately over the ground station mark and level it.
2. Bisect a well-defined distant station A using the lower clamp and make the vernier to read zero degrees.
3. Unclamp the upper plate, swing the theodolite clockwise and bisect B accurately using the upper tangent screw.
4. Read both the verniers and take the mean of the readings.
5. Similarly, bisect the stations C, D, E, etc. successively and finally the starting station A. In each case, read both the verniers and take the mean of the readings.
6. Calculate the included angles by taking the differences between two consecutive readings.

  

1. Figure 27.21 Reiteration method
2. Transit the telescope, swing the instrument in an anticlockwise direction and make observations on the face right to get the measure of each angle.
3. The mean of the two measures of each angle is the correct value of the angle.

Measurement of vertical angles

A vertical angle is defined as the angle subtended by the inclined line of sight and the horizontal line of sight in the vertical plane. If the point sighted is above the horizontal axis of the theodolite, the vertical angle is known as an angle of elevation, and if it is below it is known as an angle of depression (Figure 27.22).

To measure a vertical angle subtended by the station B at the instrument station A, the following steps are involved:

1. Set up the theodolite over the ground station mark A and level it.
2. Set the zero of the vertical vernier exactly in coincidence with the zero of the vertical scale using the vertical clamp and vertical tangent screw. Check whether the bubble of the altitude level is in the centre of its run. If not, bring it to the centre by means of the clip screw. In this position the line of collimation of the telescope is horizontal and the verniers read zero.  
3. Figure 27.22 Measurement of vertical angles
4. Loosen the vertical circle clamp and move the telescope in the vertical plane until the station B is brought in the field of view. Use the vertical circle tangent screw for accurate bisection.
5. Read both the verniers of the vertical circle. The mean of the two vernier readings gives the value of the vertical angle.
6. Change the face of the instrument and make observations in a similar way.
7. The average of the two values is the required value of the vertical angle.

Measurement of vertical angle between two stations at different elevations

The measurement of the vertical angle between two stations at different elevations may be made as follows:

1. Measure the vertical angle of the higher station as explained earlier. Let it be α.
2. Measure the vertical angle of the lower station. Let it be β.
3. The required vertical angle between the stations may be calculated by finding the algebraic difference between the two readings, assuming the angles of elevation as positive and the angles of depression as negative.

Sources of error in theodolite work

The sources of error in the theodolite work may be broadly divided into three categories:

1. Instrumental errors
2. Personal errors
3. Natural errors

Instrumental errors

The theodolites are very delicate and sophisticated surveying instruments.

In spite of the best efforts during manufacturing, perfect adjustment of the fundamental axes of the theodolite may not be possible. Instrumental errors may be further subdivided as discussed below:

Error due to imperfect adjustment of the plate level

If the plate bubbles are not adjusted properly, the vertical axis of the instrument does not remain vertical even if the plate bubbles are at the centre of their run. Non-verticality of the vertical axis introduces errors in the measurements of both the horizontal and vertical angles. This error can be eliminated only by levelling the instrument carefully, with the help of the altitude or telescope bubble before starting the observations.

Error due to line of collimation not being perpendicular to the trunnion axis

If the line of collimation of the telescope is not truly perpendicular to the trunnion axis, it generates a cone when it is rotated about the horizontal axis. This introduces errors in horizontal angles measured between stations at different elevations. This error may be eliminated from the measured angle by taking the average of the two values of the horizontal angles measured on both the faces.

Error due to horizontal axis not being perpendicular to the vertical axis

If the horizontal axis is not perpendicular to the vertical axis, the line of collimation does not revolve in the vertical plane, when the telescope is raised or lowered. Due to this imperfect adjustment, errors are introduced in both the horizontal and vertical angles. The magnitude of the error depends on:

• The angle between the horizontal axis and the vertical axis.
• The vertical angle of the station sighted.
• Elevations of the stations sighted. It is considerable if the stations sighted are at different elevations.

For elimination of the error, observations must be made on both the faces. This is because the average of the two values of the horizontal angle observed on both the faces is equal to the correct value of the angle.

Error due to non-parallelism of the axis of the telescope level and line of collimation

If the axis of the telescope level is not parallel to the line of collimation, an error is introduced in the vertical angle, because the zero line of the vertical verniers does not represent the true line of reference. The error can be eliminated by taking the mean of the two observed values of the angle, one with the telescope normal and the other with the telescope inverted.

Error due to eccentricity of inner and outer vertical axes

If the centre of the graduated circle plate does not coincide with the centre of the vernier plate, the angle recorded by either of the verniers is incorrect. To eliminate the error due to this source, observe both the verniers and take the mean value.

Error due to eccentricity of verniers

If the line joining the zeros of the horizontal plate verniers does not pass through the centre of the vernier plate, an error in the measured horizontal angles is introduced. The error may be eliminated by taking the mean of the two values by reading both the verniers.

Personal errors

This includes the following two categories of errors:

1. Errors of manipulation
2. Errors of sighting and reading

Errors of manipulation

This includes errors as explained below:

• Inaccurate centreing: If the centre of the theodolite does not coincide with the ground station mark, the horizontal angles measured will be in error, known as centreing error. The magnitude of the error depends upon the distance between the theodolite centre and the ground station mark, the direction and distance of the station sighted, etc.It may be noted that the error due to centreing cannot be eliminated unless accurate centreing is done. Also, the error due to defective centreing varies inversely as the length of sights.
• Error due to inadequate levelling: Inaccurate levelling introduces a serious error in the horizontal angles when the stations sighted are at considerable height differences. This error is similar to the error due to non-adjustment of the plate levels. If the stations sighted are at the same level, the error is small. For elimination of the error, accurate levelling should be done with the help of altitude bubble or telescope bubble which is generally more sensitive.
• Error due to manipulation of the wrong tangent screw: An inexperienced surveyor generally commits mistakes of using wrong tangent screws. It must be noted that manipulation of the upper tangent screw changes the graduated circle reading whereas manipulation of the lower tangent screw swings the theodolite without changing the readings.

Errors due to sighting and reading

These errors may arise due to the following reasons:

• Inaccurate bisection of signals: If the signal erected at the station sighted is not clearly visible, due to vegetative cover or intervening ground, the observer may bisect the signal wrongly. This introduces an error whose magnitude varies inversely with the length of sights. It may be eliminated by sighting the signal clearly and always at its lowest portion.
• Non-verticality of signals: If the signal is not truly vertical, an error is introduced. This error is inversely proportional to the length of sight. This error may be eliminated by erecting the signal truly vertical and also bisecting its lowest portion.
• Error due to parallax: If the objective and eyepiece are not properly focused before bisecting the station mark, this error is introduced. The error may be eliminated by properly focusing the eyepiece and objective before bisecting the station mark.

Natural errors

The errors included in this category are the errors occurring due to higher temperature, strong wind, blazing hot sun and unequal settlement of the tripod.