Error due to curvature : In case of small sight distance error due to the curvature are negligible, but if the sight distances are large, the error should be estimated and accounted for, as discussed below. However, the error can be minimized through balancing of sight or reciprocal observation.
With reference to Figure 16.1, the horizontal line of sight through an instrument set at L is L' x h. The level line passing through L' is L' x l . The correct staff reading at X is x l. Thus, horizontal staff reading at station X, x h is associated with an error x h x l due to curvature of the earth.
In Figure 16.2, PH is a horizontal line tangent at P to the level line along the mean radius, Rm of the earth. At station L, LH is the amount of departure of the horizontal line from level line and thus the error due to curvature of the earth (ec). This can be calculated from the triangle OPH in which
OH2 = OP2 + PH2
Or, (Rm + ec )2 = Rm2 + PH2
(Neglecting ec in the denominator as it is very small in comparison to Rm ).
Assuming, mean radius of the earth as 6367 Km, and D is the distance in Km from the instrument position to the staff station, the error due to the curvature of the earth is
ec = 0.0785 D2
It is subtractive in nature as curvature of the earth always provides increase in staff reading.
Error due to refraction: It varies with temperature, terrain and other atmospheric conditions. It is usually considered to be one seventh times but in opposite nature to the error due to curvature. To minimize this error, reciprocal observation at the same instant of time is required to be adopted.
In actual field condition, the line of sight through a level is not straight but it bends downward due to the refraction of rays of light as it passes through the intervening medium. Thus, reduces the error due to curvature of the earth by approximately 14%. With reference to Figure 16.1, the actual line of sight of the instrument set at L is thus L' x a. The observed staff reading at station X is x a. Thus, the compensation due to refraction is thus x h x a which is error due to refraction (er ) through intervening atmosphere. In Figure 16.2, HA is the error due to refraction (er ).