Saturday, March 31, 2018

Trigonometric Leveling - Theory and Formulas for Calculations.


Trigonometric Leveling is the branch of Surveying in which we find out the vertical distance between two points with the help of some measurements of the vertical angles and the known distances. The known distances are either assumed to be horizontal or the geodetic lengths at the mean sea level(MSL). The distances are measured directly(as in the plane surveying) or they are computed as in the geodetic surveying.

The trigonometric Leveling can be done in two ways:

  1. Observations taken for the height and distances
  2. Geodetic Observations.
In the first, we can measure the horizontal distance between the given points if it is accessible.

We take the observation of the vertical angles and then compute the distances using them. If the distances are large enough then we have to provide the correction for the curvature and refraction and that we provide to the linearly to the distances that we have computed. 

In the second, i.e geodetic observations, the distances between the two points are geodetic distances and the principles of the plane surveying are not applicable here. The corrections for the curvature and refraction are applied directly to the angles directly.

Now we will discuss the various cases to find out the difference in elevation between the two Points.

(1) The two points are at known distance, i.e. The base of the object is accessible.

When the two points are at a known horizontal distance then we can find out the distance between them by taking the vertical angle observations.

If the vertical angle of elevation from the point to be observed to the instrument axis is known we can calculate the vertical distance using trigonometry.

Horizontal distance*tangent(vertical angle) = Vertical difference between the two.

If the points are at small distance apart then there is no need to apply the correction for the curvature and refraction else you can apply the correction as given below:
C= 0.06728D*D

Where D is the horizontal distance between the given two points in Kilometers.
but the Correction is in meters (m).

(2) The base of the object is not accessible :

(a) When the instrument is shifted to the nearby place and the observations are taken from the same level of the line of sight: 

In such case we have to take the two angular observations of the vertical angles. The instrument is shifted to a nearby place of known distance, and then with the known distance between these two and the angular observations from these two stations, we can find the vertical difference in distance between the line of sight of the instrument and the top point of the object.

(b) When the line of sights of the two instrument setting is different :

Here again there are two cases:

  1.  When the line of sights are at a small vertical distance which can be measured through the vertical staff readings. 
  2. When the difference is larger than the staff height.

(i) In first case, It is advised to apply the formula for the difference in the height of the top of the object from these two lines of sights. The difference in lines of sights is same as the staff readings difference, when the staff is kept at a little distance from these two points. So we can get the solution for the vertical distance easily.

(ii) In the second case, there is a need to put a vane staff at the first instrument station and the angle of elevation is measured from the second point of observation. This gives us the difference in the line of the sights between the two points of instrument station. Then again we do the same.

(c) When the instrument station and the top of object are not in same vertical plane:

In this case there is a need to measure at-least two horizontal angles of the horizontal triangle formed by the two instrument stations and the base of the object.

Again we will take the vertical angular observations from the two instrument stations also and then we can apply the sine rule to solve the horizontal distances of the triangle. With the help of these angles and the distances we can get the vertical distance between any two point(Instrument station and the top of object).




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