PRINCIPLE OF EQUALISING BACKSIGHT AND FORESIGHT DISTANCES

PRINCIPLE OF EQUALISING BACKSIGHT AND FORESIGHT DISTANCES

   In levelling, the line of collimation should be horizontal when the staff readings are taken. Again the fundamental relation is that the line of collimation should be exactly parallel to the axis of the bubble. So, when the bubble is at the centre of its run, the line of collimation is just horizontal. But sometimes the permanent adjustment of level may be disturbed and the line of collimation may not be parallel to the axis of the bubble. In such case, due to the inclination of the line of collimation, cross in levelling are lickly to occur. But it is found that if the backsight and foresight distances are kept equal, then the error due to the inclination of the collimation line is automatically eliminated.

Case I – When the line of collimation inclined upwards Let A and B be two points whose true difference of level is required. The level is set up at O, exactly midway between A and B (Fig. – L.16).

Let θ = angle of inclination of collimation line.

   Aa = true reading

  Aa₁ = observed staff reading on A

Error = Aa₁ - Aa = aa₁ = D tanθ

So true reading Aa = Aa₁ - aa₁ = Aa₁ - D tanθ     ………..(1)

Similarly, Bb = true reading

Bb₁ = observed staff reading on B

Error = Bb₁ - Bb = bb₁ = D tanθ

So true reading Bb = Bb₁ - bb₁ = Bb₁ - D tanθ    ………….(2)

From (1) and (2),

True difference of level between A and B = Aa – Bb

                                               = Aa₁ - D tanθ - Bb₁ + D tanθ

                                               = Aa₁ - Bb₁

Thus, this is seen that the error due to inclination of the collimation line is completely eliminated and the apparent difference is equal to the true difference.

Case II – When the line of collimation inclined downwards The staff readings on A and B are taken after setting the level at O. Suppose the readings are a₂ and b₂ (Fig. – L.17).

Here,   Aa = true staff reading

  Aa₂ = observed staff reading on A

Error = Aa - Aa₂ = aa₂ = D tanθ

So true reading Aa = Aa₂ + aa₂ = Aa₂ + D tanθ     ………..(1)

Similarly, Bb = true reading

Bb₂ = observed staff reading on B

Error = Bb - Bb₂ = bb₂ = D tanθ

So true reading Bb = Bb₂ + bb₂ = Bb₂ + D tanθ    ………….(2)

From (1) and (2),

True difference of level between A and B = Aa – Bb

                                                     = Aa₂ + D tanθ - Bb₂ - D tanθ

                                                     = Aa₂ - Bb₂

Thus, this is seen that the error due to inclination of the collimation line is completely eliminated and the apparent difference is equal to the true difference.

So, always remember that the level should be placed exactly midway between backsight and foresight in order to eliminate any error.

CORRECTION TO BE APPLIED

(1). Curvature correction: For long sights, the curvature of the earth effect staff readings. The line of sight is horizontal, but the level line is curved and parallel to the mean spheroidal surface of the earth (Fig. – L.18).   

   The vertical distance between the line of sight and level line at a particular place is called the curvature correction. Due to curvature, objects appear lower than they really are.

Curvature correction always subtractive (i.e. negative) 

Let   AB = D = horizontal distance in kilometers.

       BD = Cc = curvature correction

       DC = AC = R = radius of earth

       DD’ = diameter, considered as 12742 km

From right angle triangle ABC (Fig. – L.19).

  

     BC² = AC² + AB²

     (R + Cc) ² = R² + D²

or  R² + 2RCc + Cc² = R² + D²

or  2R X Cc = D²     (Cc² is neglected as it is very small in comparison to the

                                 diameter of the earth)

Curvature correction   Cc = D²/2R

Cc = (D² x 1000)/12742 = 0.0785D² m   (negative)

Hence,   True staff reading = observed staff reading – curvature correction.

(2). Refraction correction: Rays of light are refracted when they pass through layers of air of varying density. So, when long sights are taken, the line of sight refracted towards the surface of the earth in a curved path. The radius of this curve seven times that of the earth under normal atmospheric conditions. Due to the effect of refraction, objects appear higher than they really are. But the effect of curvature varies with the atmospheric conditions.

However, on an average, the refraction correction is taken one-seventh of the curvature correction.

               Cr = 1/7 x D²/2R

Refraction correction, Cr = 1/7 x 0.0785D² = 0.0112D² m (positive)

Refraction correction is always additive (i.e. positive)

True reading = observed staff reading + refraction correction

(3). Combined correction: The combined effect of curvature and refraction is as follows:

Combined correction = curvature correction + refraction correction

                               = -0.0785D² + 0.0112D²

                               = -0.0673D² m

So, combined correction always subtractive (i.e. negative)

True reading = observed staff reading - combined correction

Combined correction may also be expressed as

                          D²/2R – 1/7 x D²/2R = 6/7 x D²/2R   (negative)

(4). Visible horizon distance

Let    AB = D = visible horizon distance in kilometers (Fig. – L.20).

Considering curvature and refraction corrections,

                       h = 0.0673D²

                       D = √(h/0.0673)

(5). Dip of horizon

AB = D = tangent of earth at A

BD = horizontal line perpendicular to OB

θ = dip of horizon

The angle between the horizontal line and the tangent line is known as the dip of the horizon. It is equal to the subtended by the arc AC at the centre of the earth (Fig. – L.21).

Dip θ = arc AC/radius of earth     in radians

θ = D/R, in radians  (Taking AC approx, equal to AB)

 

Hear, D and R must be expressed in the same unit.

(6). Sensitiveness of the bubble: The term sensitiveness in the context of a bubble means the effect caused by the deviation of the bubble per division of the graduation of the bubble tube.

   Sensitiveness is expressed in term of the radius of the curvature of the upper surface of the tube or by an angle through which the axis is tilted for the deflection of one division of the graduation.

Determining sensitiveness: consider Fig. L.22. Suppose the level was set up at O at a distance D from the staff at P. The staff reading is taken with the bubble at extreme right end (i.e. at E). Say it is PA. Another staff reading taken with the bubble at the extreme left end (i.e. at E₁). Let it is be PB  

Let,   D = distance between level and staff

        S = intercept between the upper and lower sights,

        n = number of divisions through the bubble is reflected,

        R = radius of curvature of the tube,

        θ = angle subtended by arc EE₁, and

        d = length of the division of the graduation, expressed in the same unit

              as D and S.

Movement of centre of bubble = EE₁ = nd

Triangle OEE₁ and ACB are similar.

Here,           Rθ = arc EE₁

or                θ = EE₁/R = nd/R   …….(1)   (as arc EE₁ = chord EE₁)

Again           EE₁/R = S/D   (height of triangle OEE₁ may be considered as R)

or                 nd/R = S/D      ……………. (2)

From (1) and (2),    

                   θ = nd/R = S/D   …………(3)

Therefore,      R = (nd x D)/S

Let                θ’ = angular value for one division in radians

                     θ’ = θ/n = S/D x 1/n radians

                    

                     θ’ = S/(D x n) x 206265 secs. (1 radian = 206265 secs.)

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