### Error due to Refraction and Curvature

Hi,

As we know, in Geodetic Surveying we have to consider the error due to the curvature of the earth and the refraction too. Reason is because a larger area is surveyed in Geodetic Surveying, more than about 256 km2. In such large areas, the error due to curvature of the earth has to be considered to calculate the linear distances and also in case of the angular measurements.

We have to consider the refraction error too, because here we are dealing with the large distances and in order to get the correct results we have to apply these correction.

• What is the Error due to Curvature?
To understand this simple concept of error due to curvature of the earth, first you have to understand the shape of the earth and the methods and instruments which we employ for calculating these distances.
When we do leveling with Theodolite or Autolevel, the line of sight first is set horizontal. Then we measure the vertical angle to the target and by applying some trigonometrical formulae we can calculate the vertical distance of the target from that horizontal line.

Error due to curvature comes into play, because in the cases of long distances, the horizontal line and level line do not coincide. Level line is a curved line, parallel to the level surface, but the horizontal lines goes straight.
This means that the vertical distance of that target from the level line is going to be larger than the distance which we calculate from the horizontal line. Please refer the figure given above, the amount of correction depends upon the magnitude of the horizontal distance between the target and the instrument station.

• What is Error due to Refraction?
Error to refraction can be understood easily once you understand the phenomenon which takes place when light passes from one density system to another density system. Refraction is nothing but the phenomenon by which when light travels from a denser media to the lighter media, it deflects away from the normal to the plane of the media.

Phenomenon occurs vice-versa when light travels from lighter media to denser media. This phenomenon has to be considered in the calculation of the distances in case of Geodetic Surveying. Suppose a man is taking the observation of the top of a hill from a point, which is at a considerable down far vertical distance from it to change the density of the air.

We know that density of air decreases with the height, this will effect your line of sight. The line of sight of a person who is taking observation to a point at a quite higher distance, will be a curved path, because light will continuously change its direction due to the continuous change in the density of the air.
Imporant question to ask is, how does it effect our observations? Well, the observed angle will be to high in case of taking the observation of an elevated object and the observed angle will be small in case of taking observations to an object in depressions.

• Correction for the Error due to Curvature and Refraction:
There are a numbers of textbooks, which explains the procedure to calculate the correction for the refraction and curvature.

Formulae:

Curvature correction, Cc = - 0.07849.D^2 meter
Refraction Correction, Cr = 0.01121.D^2 meter
Combined Correction C = Cc+Cr= -0.06728. D^2  meter;   here D is in kilometer