THEODOLITE SURVEY

Objectives:

1. To make a traverse by measuring included angle (using theodolite) and sides by direct measurement (using tape).

2. To plot the plan of the given area by Tacheometric surveying.

Instruments required:

1. Transit Theodolite

2. Stadia rod

3. Ranging rods

4. Pegs and hammer

5. Tape

6. Prismatic compass

7. Plumb bob

Theory

Traversing

Traversing is that type of survey in which a number of connected lines form the framework and the directions and lengths of the survey lines are measured with the help of an angle or direction measuring instrument and a tape or chain respectively. When the lines form a circuit which ends at the starting point or point of the known position or coordinate, it is known as a closed traverse. If the circuit ends elsewhere, it is said to be an open traverse. The closed traverse is suitable for locating the boundaries of lakes, and for the survey of large areas. The open traverse is suitable for surveying a long narrow strip of land as required for a road or canal or the coast line.

Traversing by direct observation of angles

In this method, the angles between the lines are directly measured by a theodolite. The method is, therefore, most accurate in comparison to the chain traversing or compass traversing or plane table traversing methods. The magnetic bearing of any one line can also be measured if required and the magnetic bearing of other lines can be calculated. The angles measured at different stations may be included angles.

An included angle at a station is either of the two angles formed by the two survey lines meeting there. The method consists simply in measuring each angle directly from a backside in the preceding station. Both face observations must be taken and both the verniers should be read. Included angles can be measured clockwise or counter clockwise but it is better to measure all angles clockwise, since the graduations of the theodolite circle increase in this direction. The angle measured clockwise from the back station may be interior or exterior depending up on the direction of progress round the survey.

Plotting a traverse survey:

There are two principal methods of plotting a traverse survey:

1. Angle and distance method

In this method, distances between stations are laid off to scale and angles or bearings are plotted by protractor. This method is suitable for the small surveys, and is much inferior to the coordinate method in respect of accuracy of plotting. The ordinary protractor is seldom divided more finely than 10' or 15' which accords with the accuracy of compass traversing but not of theodolite traversing. A good form of protractor for plotting survey lines is the large circular cardboard type, 40 to 60 cm in diameter.

2. Co-ordinate system

In this method, survey stations are plotted by calculating their coordinates. This method is by far the most practical and accurate one for plotting traverses or any other extensive system of horizontal control. The biggest advantage in this method of plotting is that the closing error can be eliminating by balancing, prior to plotting.

Consecutive coordinate: Latitude and Departure

The latitude of a survey line may be defined as its coordinate length measured parallel to an assumed meridian direction i.e. true north or magnetic north or any other reference direction. The departure of a survey line may be defined as its coordinate length measured at right angles to the meridian direction. The latitude (L) of the line is positive when measured northward or upward and is termed as northing. The latitude is negative when measured southward or downward and is termed as southing. Similarly, the departure (D) of the line is positive when measured eastward or right and is termed as easting and is negative when measured westward or left and is termed as westing.

Fig. 1.1. Latitude and Departure

Thus, in fig. 1.1, the latitude and departure of the line AB of length l and reduced bearing θ are given by

L = + l cos θ

And D = + l sin θ

Independent coordinates:

The coordinates of traverse stations can be calculated with respect to a common origin. The total latitude and departure of any point with respect to a common origin are known as independent coordinates or total coordinates of the point. The two reference axes in this case may be chosen to pass through any of the traverse station bur generally a most westerly station is chosen for this purpose. The independent coordinates of any point may be obtained by adding algebraically the latitudes and departures of the lines between that point and the origin.

Closing error:

If a closed traverse is plotted according to the field measurements, the end point of the traverse will not coincide exactly with the starting point, owing to the errors in the field measurements of angles and distances. Such error is called closing error (Fig. 1.2). In a closed traverse, the algebraic sum of the latitudes (i.e. ∑ L) should be zero and the algebraic sum of the departures (i.e. ∑ D) should be zero. The error of closure for such traverse may be ascertained by finding ∑ L and ∑ D, both of these being the components of error e parallel and perpendicular to meridian.

Fif. 1.2. Closing error

In fig. 1.2,

Closing error e = AA' = (∑ L)2 + (∑ D)2

The direction of closing error is given by

tanδ = ∑D∑L

Adjustment of the angular error:

Before calculating latitudes and departures, the traverse angles should be adjusted to satisfy the geometrical conditions. In a closed traverse, the sum of interior angles should be equal to (2N − 4) × 90°. If the angles are measured with the same degree of precision, the error in the sum of angles may be distributed equally to each angle of the traverse. If the angular error is small, it may be arbitrarily distributed among two or three angles.

Balancing the traverse:

The term balancing is generally applied to the operation of applying corrections to latitudes and departures so that ∑ L =0 and ∑ D = 0. This applies only when the survey forms a closed polygon. Common methods of balancing the traverse are

1. Bowditch's method

The basis of this method is on the assumptions that the errors in linear measurements are proportional to l where l is the length of the line. The Bowditch's rule, also termed as the compass rule, is mostly used to balance a traverse where linear and angular measurements are of equal precision. The total error in latitude and in departure is distributed in proportion to the lengths of the sides.

The Bowditch's rule is:

Correction to latitude (or departure) of any side

= Total error in latitude (or departure) × Length of that sidePerimeter of traverse

i.e, CL = ∑L . l∑l

And CD = ∑D . l∑l

Where CL = correction to latitude of any side

CD = correction to departure of any side

∑L= total error in latitude

∑D= total error in departure

∑l = length of the perimeter

l = length of any side

2. Transit method

The transit rule may be employed where angular measurements are more precise than the linear measurements. According to this rule, the total error in latitudes and in departures is distributed in proportion to the latitudes and departures of the sides. It is claimed that the angles are less affected by corrections applied by transit method than by those by Bowditch's method.

The transit rule is:

Correction to latitude (or departure) of any side

= Total error in latitude (or departure) ×Latitude (or Departure) of that sideArithmatic sum of Latitudes (or Departures)

i.e., CL = ∑L . L∑|L|

and CD = ∑D . D∑|D|

Where L = latitude of any side

D = departure of any side

∑|L| = arithmetic sum of latitudes

∑|D| = arithmetic sum of departures

3. Graphical method

For rough survey, such as compass traverse, the Bowditch's rule may be applied graphically without doing theoretical calculations. Thus, according to the graphical method, it is necessary to calculate latitudes and departures etc. However, before plotting the traverse directly from the field notes, the angles or bearings may be adjusted to satisfy the geometrical conditions of the traverse.

Fig.1.3. Graphical adjustment of traverse

4. The axis method

This method is adopted when the angles are measured very accurately, the corrections being applied to lengths only. Thus, only directions of the line are changed and the general shape of the diagram is preserved.

Fig. 1.4. Axis method of balancing traverse

According to this method,

Correction to any length = that length × 12 length of closing errorLength of axis of adjustment

Tacheometric surveying:

Tacheometry (or Tachemetry or Telemetry) is a branch of angular surveying in which the horizontal and vertical distances of points are obtained by optical means as opposed to the ordinary slower process of measurements by tape or chain. The method is very rapid and convenient. Although the accuracy of Tacheometry in general compares unfavorably with that of chaining, it is best adopted in obstacles such as steep and broken ground, deep ravines, stretches of water or swamp and so on, which make chaining difficult or impossible. The accuracy attained is such that under favorable conditions the error will not exceed 1/1000, and if the purpose of a survey does not require greater accuracy, the method is unexcelled. The primary object of Tacheometry is the preparation of contoured maps or plans requiring both the horizontal as well as vertical control. Also, on surveys of higher accuracy, it provides a check on distances measured with the tape.

Instruments for tacheometry

An ordinary transit theodolite fitted with a stadia diaphragm is generally used for tacheometric survey. The stadia diaphragm essentially consists of one stadia hair above and the other an equal distance below the horizontal cross-hair, the stadia hairs being mounted in the same ring and in the same vertical plane as the vertical and horizontal cross-hairs.

Fig.1.5. Stadia diaphragm

Stadia method for tacheometric measurement

This is the most common method in tacheometry. In this method, observation is made with the help of a stadia diaphragm having stadia wires at fixed or constant distance apart. The readings on the staff corresponding to all the three wires are taken. The staffs intercepts, i.e. the difference of the readings corresponding to top and bottom stadia wires will therefore, depend on the distance of the staff from the instrument. When the staff intercepts is more than the length of the staff, only half intercept is read. For inclined sights, readings may be taken by keeping the staff either vertical or normal to the line of sight.

Principle of stadia method

The stadia method is based on the principle that the ratio of the perpendicular to the base is constant in similar isosceles triangles.

In Fig. 1.6, let two rays OA and OB be equally inclined to the central ray OC. Let A2B2, A1B1 and AB be the staff intercepts,

Then OC2A2B2 = OC1A1B1 = OCAB = constant = OC2 × AC

= 12 tan β2 = 12 cot β2

Fig. 1.6.

Fig. 1.7. Principle of stadia method

Let A, C and B = the points cut by the three line of sight corresponding to the three wires.

Also, a, c and b = top, axial and bottom hairs of the diaphragm.

ab = i = interval between the stadia hairs (stadia interval)

AB = s = staff intercept

d = distance of the vertical axis of the instrument from O

D = horizontal distance of the staff from the vertical axis of the instrument

M = center of the instrument corresponding to the vertical axis

Now, from figure,

FCAB = OFa'b' = fi

Or, FC = fi AB = fi s

Also, distance from the axis to the staff is given by

D = FC + ( f + d ) = fi s + ( f + d ) = k s + C

Above equation is known as the distance equation.

The constant k = fi is known as the multiplying constant or stadia interval factor and the constant C = ( f + d ) is known as the additive constant of the instrument.

Using a Plano-convex lens called anallatic or anallactic lens at the vertical axis (at point M), the values of k and C are made 100 and 0 for simplicity of calculation.

Distance and elevation formula

Fig. 1.8

We know, distance of the staff from the instrument is given as

D = k A'B' + C

= k s cos α + C

Then, the horizontal distance is given as

H = D cos α

H = k s cos2 α + C cos α

Also, elevation of the line of sight from the instrument is

V = k s cos α sin α + C sin α

V = k s sin 2α + C sin α

Thus, elevation of the staff station is given by

R.L. of staff station Y = R.L. of instrument station X + H.I. + V − h

Observations and Calculations:

Table 1.1. Linear Measurement of Survey Lines

S. N. Line Forward Measurement

(f) m Backward Measurement

(b) m Mean length

(x = f + b2 ) m

Discrepancy

(e = | f - b |) m Precision

( p = ex )

1 AB 40.594 40.614 40.604 0.020 12030

2 BC 25.896 25.902 25.899 0.006 14316

3 CD 41.364 41.358 41.361 0.006 16893

4 DE 28.732 28.734 28.733 0.002 114366

5 EA 42.486 42.492 42.489 0.006 17081

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