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UNIT-3

TRIGONOMETRIC LEVELLING

Process of determining difference in elevations of stations from observed vertical angles and known

distances, which are assumed to be either horizontal or geodetic lengths at mean sea level. The vertical angles may be measured by means of ·an accurate theodolite and the horizontal distances may

either be measured (in the case of plane surveying) or computed (in the case of geodetic

observations).

We shall discuss the trigonometrical levelling under two heads:

(I) Observations fur heights and distances.

(2) Geodetical observations. In the first case, the principles of plane surveying will be used. It is assumed that the distances

between the points observed are not large so that either the effect of curvature and refraction n1ay be

neglected or proper corrections may be applied linearly to the calculated difference in elevation. Under this bead fall the various methods of angular levelling for. determining the elevations of

particular points such as top of chimney, or church spire etc. points measured is geodetic and is large.

The ordinary principles of plane surveying are not applicable. Tbe corrections for curvature and

refraction are applied in angular measure directly to the observed angles. The geodetical observations of trigonometrical levelling have dealt in second volume.

Case-1:

Base of the object accessible-

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Case – 2-

BASE OF THE OBJECT INACCESSIBLE :

INSTRUMENT STATIONS IN THE SAME VERTICAL PLANE AS THE ELEVATED OBJECT

If the horizontal distance between the instrument and the object can be measured due to obstacles

etc., two instrument stations are used so that they are in the same vertical plane as the elevated

object.

Procedure

1. Set up the theodolite at P and level it accurately with respect to the altitude bubble.

2. Direct the telescope towards Q and bisect it accurately. Clamp both the plates. Read the vertical angle a,.

3. Transit the telescope so that the line of sight is reversed. Mark the second instrument station R

on the ground. Measure the distance RP accurately. Repeat steps (2) and (3) for both . face observations. The mean values should be adopted.

4. With the vertical vernier set to zero reading, and the altitude bubble in the centre of its run,

take the reading on the s1aff kept at the nearby B.M.

5. Shift the instrument 10 R and set up the theodolite there. Measure the vertical angle a, 10 Q with both face observations.

6. With the vertical vernier set 10 zero reading, and the altitude bubble in the centre of its run,

take the reading on the staff kept at the nearby B.M.

In order to calculate the R.L. of Q. we will consider three cases :

(a) when the instrument axes at A and B are at the same level:

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(b) when they are at different levels but the difference is small:

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(c) when they are at very different levels:

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Base of the object inaccessible: Instrument stations not in the same vertical plane as the elevated

object (DOUBLE PLANE METHOD):

Procedure:

1. Set the instrument at P & level accurately measure the angle of elevation to Q.

2. Sight the point R with reading on horizontal circle as zero & measure the angle RPQ.

3. Take the Back sight at the BM.

4. Shift the instrument to R & measure the angle.

Thus, AQQ' is a vertical plane. Similarly, BQQ" is a vertical plane, Q" being the vertical projection of

Q on a horizontal line through B. PRQ, is a horizontal plane, Q, being the vertical projection of Q, and

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R vertical projection of B on a horizontal plane passing through horizontal angles and vertical angles measured at A and B respectively.

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Tacheometric surveying:

The telescope used in stadia surveying are of three kinds:

1. The simple external-focusing telescope.

2. The external-focusing anallactic telescope (Porro's telescope).

3. The internal-focusing telescope.

The 'tacheometer' (as such) has the advantage over the first and the third type due to the fact that the additive constant of the instrument is zero. However, the internal focusing telescope is becoming

more popular, though it has a very small additive constant.

A tacheometer must essentially incorporate the following features: 1. The multiplying constant should have a nominal value of 100 and the error contained in this

value should not exceed I in 1000.

2. The axial horizontal line should be exactly midway between the other two lines. 3. The telescope should be truly anallactic.

4. The telescope should be powerful having a magnification of 20 to 30 diameters.

DIFFERENT SYSTEMS OF TACHEOMETRIC MEASUREMENT

1. The stadia system (a) Fixed Hair method or Stadia method

(b) Movable Hair method, or Subtense method.

2. The tangential system.

3. Measurements by means of special instruments.

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The principle common to all the systems is to calculate the horizontal distance between two points A and B and their difference in elevation.

(a) Fixed hair method: 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 staff intercept, 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 intercept 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. This is the most common method in tacheometry

and the name 'stadia method' generally bears reference to this method.

(b) Subtense method. This method is similar to the fixed hair method except that the stadia hair

so as to set them against the two targets on the staff kept at the point under observation. Thus,

in this case, the staff intercept, i.e., the distance between the two targets is kept fixed while

the stadia interval, i.e., the distance between the stadia hairs is variable. As in the case of fixed hair method, inclined sights may also be taken.

(c) The tangential method. In this method, the stadia hairs are not used, the readings being taken

against the horizontal cross-hair. To measure the staff intercept, two paintings of the

instruments are therefore, necessary. This necessitates measurement of vertical angles twice for one single observation.

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.

Let two rays OA and OB be equally inclined to the central ray OC. Let A2, B2, A1, B1, and AB be the

staff intercepts.

(OC2 / A2B2 ) = (OC1, / A1B1) = (OC / AB) = constant k = ½ * cot β / 2

This constant k entirely depends upon the magnitude of the angle β· If β is made equal to 34' 22".64, the constant k = ½ cot l7' 11".32 = 100. In this case, the distance between the staff and the point 0 will

be 100 times the staff intercept. In actual practice, observations may be made with either horizontal

line of sight or with inclined line of sight. In the latter case, the staff may be kept either vertically or

normal to the line of sight. We shall first derive the distance-elevation formulae for the horizontal sights.

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Let A, C & B = points cut by 3 lines of sight corresponding to 3 wires.

b, c & a = top, axial & bottom hairs of diaphragm,

ab = i = interval between stadia hairs,

AB = s = staff intercept.

f = focal length of the objective,

f1, = Horizontal distance of the staff from the optical centre of the objective.

f2. = Horizontal distance of the cross-wires from 0.

d = Distance of the vertical axis of the instrument fro111 0.

D = Horizontal distance of the staff from the vertical axis of the instrument.

M = Centre of the instrument, corresponding to the vertical axis.

Since the rays BOb and AOa pass through the optical centre, they are straight so that triangle AOB

and aOb are similar. Hence

(f1 / f2) = (s / i)

Again, since f1, and f2, are conjugate focal distances, we have from lens formula,

1 / f = 1 / f2 + 1 / f1

Multiplying throughout by ff1 , we get f1, = (f1 / f2) * f + f.

Substituting the values of (f1 / f2) = (s / i) in the above, we get f1 = (s / i)* f + f

The horizontal distance between the axis and the staff is

D=f1+d

D = f/i*s + (f+d) D = K*S + C

Constant k = f / i is known as the multiplying constant or stadia interval factor Constant (f+ d) = C is known as the additive constant.

K= 100, C = 0

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Alternative Method:

Equation can also be derived alternatively, with reference to Fig in which the rays Bb' and Aa' passing

through the exterior principal focus

F, become parallel to the optical axis. The rays Aa and Bb pass through 0 and remain undeviated.

Since the stadia interval ab is fixed in magnitude, the points a' and b' are fixed. Again, since F is also

fixed, being the exterior principal focus of the objective, the angle AFB is fixed in magnitude. From similar triangles AFB and a' Fb' we have

FC / AB = OF / a'b' = f / i or FC = f / i * AB = f / i s

Distance from the axis to the staff is given by D·= FC + (f+d) = f / i * s + (f+d) = ks + C

Elevation of staff station = Elevation of instrument axis - Central hair reading

Determination of constants k and C

1st Method: In this method, the additive constant C =if+ Q) is measured from the instrument while the multiplying constant k is computed from field observations:

1. Focus the instrument to a distant object and measure along the telescope the distance between

the objective and cross-hairs. 1 / f = 1/ f1 + 1/ f2

Since f, is very large in this case, f is approximately equal to f2, i.e., equal to the distance of the

diaphragm from the objective. 2. The distance d between the instrument axis and the objective is variable in the case of

external focusing telescope, being greater for short sights and smaller for long sights. It

should, therefore be measured for average sight. Thus, the additive constant (f + d) is known.

3. To calculate the multiplying constant k, measure a known distance D1 and take the intercept s1on the staff kept at that point, the line of sight being horizontal. Using equation

D1 = k.s1 + C or

k = D1 – C / s1 For average value, staff intercepts, s2, s3 etc., can be measured corresponding to distance D2, D3 etc.,

and mean value can be calculated.

Note. In the case of some external focusing instruments, the eye-piece-diaphragm unit moves during focusing. For such instruments d "is constant and does not vary while focusing.

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2nd Method: In this method, both the constants are determined by field observations.

1. Measure a line, about 200 m long, on fairly level ground and drive pegs at some interval, say

50 metres.

2. Keep the staff on the pegs and observe the corresponding staff intercepts with horizontal sight.

3. Knowing the values of D and s for different points, a number of simultaneous equations can

be formed by substituting the values of D and s in equation. The simultaneous solution of successive pairs of equations will give the value of k and C and the average of these can be

found.

DISTANCE AND ELEVATION FORMULAE FOR STAFF VERTICAL: INCLINED SIGHT.

Let P = Instrument station

Q =.Staff station M = Position of instruments axis

0 = Optical centre of the objective

A, C, B =Points corresponding to the readings of the three hairs s = AB = Staff intercept

i =Stadia interval

a = Inclination of the line of sight from the horizontal L = Length MC measured along the line of sight

D = MQ' =Horizontal distance between the instrument and the staff

V =Vertical intercept, at· Q, between the line of sight and the horizontal line.

h = Height of the instrument r = Central hair reading

β = Angle between the two extreme rays corresponding to stadia hairs.

Draw a line A'CB' normal to the line of sight OC. ∟AA'C = 90' + β / 2, being the exterior angle of the ΔCOA '.

Similarly, from ΔCOB', ∟OB'C

= ∟BB'C = 90'- β / 2. Since β / 2 is very small (its value being equal to 17' 11" for k = 100), ∟AA'C and ∟BB'C may be

approximately taken equal to 900.

∟AA'C = ∟BB'C = 900

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(a) Elevation of the staff station for angle of elevation:

Elev. Of staff station = Elevation of instrument station + h + V – r.

(b) Elevation of the staff station for angle of depression: Elev. Of staff station = Elevation of instrument station + h - V – r.

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DISTANCE AND ELEVATION FORMULAE FOR STAFF NORMAL.

Case (a) Line of sight at an angle of elevation θ

AB = s = staff intercept

CQ = r = axial hair reading

MC = L = Ks + C

D = MC1 + C1Q1 = L cos θ + r sin θ = (ks + C) cos θ + r sin θ V = L sin θ = (ks + C) sin θ

Elevation of Q = Elev. of P + h + V – r cos θ

Case (b) Line of sight at an angle of depression θ

MC = L = Ks + C

D = MC1 - C1Q1 = L cos θ - r sin θ = (ks + C) cos θ - r sin θ V = L sin θ = (ks + C) sin θ

Elevation of Q = Elev. of P + h - V – r cos θ

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ANALLACTIC LENS

In the distance formula D = ks + C, s is proportional to (D - C) which is the distance between the staff

and the exterior principal focus of the objective.

Vertex of the measuring triangle fall at the exterior principal focus of the objective and not at the

vertical axis of the instrument.

Anallactic lens is provided in external focusing telescope and not in internal focusing telescope since

the latter is virtually anallactic due to very small additive constant.

Theory of Anallactic lens

D = ks = 100 s

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Anallactic Telescope: inclined sight

MC = L = k . A1B1 = ks cos θ

D = L cos θ = ks cos2 θ

V = L sin θ = ks cos θ sin θ = ks/2 sin 2 θ

R.L. of Q = R.L. of P + h + V - r

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The Subtense Method

PRINCIPLE OF SUBTENSE (OR MOVABLE HAIR) METHOD:

VERTICAL BASE OBSERVATIONS

In the stadia principle, the tacheometric angle is always a constant for a given telescope. The staff

intercept, which forms the base of stadia measurement, varies with the distance of the staff from the instrument. The principle of subtense method is just the reverse of it. In subtense measurement, the

base may be kept either horizontal or vertical. If the base is vertical, the method is known as vertical

base subtense method and the angle at F can be measured with the help of special diaphragm. If the

base is horizontal, the method is known as ‘horizontal base subtense method’ and the angle at F can be measured with the horizontal circle of the theodolite by the method of repetition.

For the staff at point P, the rays Aa' and Bb', passing through the exterior focus F of the objective, become parallel to the principal axis after refraction. The points a and b correspond to the positions

of stadia wires for this observation so that the lines of sight intersect the targets at A and B. Similarly,

the dashed lines show the corresponding optical diagram for another staff position at Q, the staff intercept being the same.

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Let AB = s = Staff intercept = distance between the targets ab = i = Stadia interval measured at the diaphragm

F = Exterior principal focus of the objective

M = Centre of the instrument

a',b' = Points on the objective corresponding to A and B.

From Δs a'b'F and FAB

FC / s = FO / a'b' = f / i or FC = f / I * s

D = FC+ MF= f / I * s + (f +d)

Thus the expression for the subtense measurement is the same as for the stadia method. The only

difference is that in this expression, s is fixed quantity while i is variable.

Due to this reason, the multiplying factor f / i varies with the staff position and is no longer

constant. The stadia interval i is measured with the help of micrometre screws having a pitch p.

Let m be the total number of the revolutions of the micrometre screw for the staff intercept s.

Then, i = mp. Substituting the value of i, we get

D = f / mp * s + (f+d) or D=Ks / m + C

where K = f / p = constant for an instrument and C = additive constant. If, however, e is the index error, expression reduces to

D = Ks / m – e + C

Expression for inclined sight:

D = K.s / m – e cos2 θ + C cos θ

V = K.s / m – e . Sin 2θ / 2 + C sin θ = D tan θ

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Horizontal Base Subtense measurement

AB = Horizontal base of length s

O = point of instrument

From Δ AOC,

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Accuracy depends on size of the angle. O is measured with the theodolite. Subtense bar is used to

provide base AB.

Merits and Demerits of Movable Hair Method.

Obsolete due to lack due to lack in the speed in the field.

Computations are not quicker.

Long sights can be taken with greater accuracy than in stadia method,

The term 'subtense method' is now more or less exclusively applied to horizontal base

subtense method.

THE SUBTENSE BAR

Suntense bar is used for measurements of comparatively short· lines in a traverse as subtense

base.

The length of the base is generally 2 metres (6 ft) or. 3 metres (10 ft).

The distance between the two targets is exactly equal to the length of the base.

Length of base may not vary due to temperature and other variations attached to invar rod.

The invar rod is supported at a number of points in a duralumin tube provided with a spirit

level. ·

The bar is centrally supported on a levelling head for accurate centring and levelling.

A clamp and slow motion screw is also provided to rotate the bar about its vertical axis.

Small telescope is provided at the centre of the bar to align it perpendicular to the line OC joining the theodolite station and the centre of the bar.

In order to make the equation 22.17 valid, the longitudinal axis of the subtense bar must be

perpendicular to the line OC.

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The difference in elevation between theodolite station 0 and subtense bar station C does not

affect the magnitude of the angle AOB, since the angle AOB is always measured on the horizontal circle of the theodolite.

Effect of Angular Error on horizontal Distance

For a given length (s) of the subtense base, D is inversely proportional to the angle β, hence the

negative error in the measurement of the angle will produce a positive error in the distance D and vice

versa. Let the angular error be δβ (negative), and the resulting linear error be δD

(positive).

S = Dβ = (D + δD) (β – δB)

D + δD / D = β / (β – δβ) or (D + δD) – D / D = β – (β – δβ) / β - δβ

δD / D = δB / β – δβ

δD = Dδβ / (β – δβ)

If δβ is positive, resulting error δD (negative) is given by

δD = Dδβ / (β + δβ)

If δβ is very small to β,

δD = Dδβ / β

Example 22.14: The stadia intercept read by means of a fixed hair instrumental on a vertically held

staff is 1. 05 metres, the angle of elevation being 50 361. The instrument constants are 100 and 0.3.

What would be the total number of rums registered on a movable hair instrument at the same station

for a 1.75 metres intercept on a staff held on the same point, the vertical angle in this case being 50 241 and the constants 1000 and 0.5 ?

Solution. (a) Observations by means of fixed hair instrument:

D = ks cos2 θ + c cos θ = 100 X 1.05 cos2 5° 36' + 0.3 cos 5' 36'

= 104.29 m.

(b) Observations by means of movable hair instrument:

D= K / n * s cos2 θ + C cos θ

104.29 = 1000 / n * 1.75 cos2 5° 24' + 0.5 cos 5' 24'

1734.5 / n = 103.8 or n = 1734.5 / 103.8 = 16.71

Example 22.16: The horizontal angle subtended at a theodolite by a subtense bar with vanes 3 m

apart is 12 '33". Calculate the horizontal distance between the instrument and the bar. Also find (a) the error of horizontal distance if the bar was 30 from being normal to the line joining the instrument and

bar stations; (b) the error of the horizontal distance if there is an error of 111 in the measurement of the

horizontal angle at the instrument station.

Solution. β = 12' 33" = 753"

D = 206265 / β * s = 206265 / 753 X 3 = 821.77 m. (a) The above distance was calculated on the assumption that the bar was normal to the line joining

the instrument and bar station. If, however, the bar is not normal, the correct horizontal distance is

given by

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D' = D cos β = 821.77 cos 30 = 820.64 m

Error= D' - D = 821.77 - 820.64 = 1.13 m

Ratio of error = e / D1 = 1.13 / 820.64 = 1 in 726

(b) If there is an error of 1" in the measurement of the angle at the instrument.

δD = Dδβ / β = 821.77 / 753 X 1 = 1.09 m.