I understand and I know that these next parts may be a little awkward or perhaps very theoretical for some of you; as I also know that the calculation methods and the apparatus used in these first parts are part of the recent past, but we are represented on a global scale and I do not want to stop covering all the markets. I mean, I present very theoretical calculations, just so that you understand where the results come from and use instruments such as the optical level and theodolite, knowing that these same calculations are performed automatically by the total stations and the software programs of data collected by GNSS solutions. Let us not forget that in many countries of the world, the old theodolite is still the king of measurements of levels and distances.
Altimetry or leveling is intended to determine the vertical or level difference between several points. The height difference between two points is the difference between these points. The determination of level differences between two points is possible with the following methods:
(a) geometric leveling;
(b) trigonometric leveling.
Influence of Earth's Curvature and Atmospheric Refraction
In Fig. 1, in order to determine the difference in level between points A and B, in B a vertical sight, and A a properly leveled instrument, giving horizontal plane AH, corresponding to the apparent level surface, which will intercept the crosshair in a point C, not B, since the arc AB can not be determined by surveying apparatus.
It is evident that the substitution of the true level by the apparent level causes an error in determining the height of a point on the ground, which is called an error due to the curvature of the earth. The mistake made, by admitting that points A and C are level (apparent level), is the error EC = BC, called error due to earth curvature. This error can be calculated as long as the extent of alignment AC =D, since the radius of the earth is known.
D2 = (OB + BC) 2 - OA2 or D2 = (R + EC) 2 - R2
D2 = EC (EC + 2.R) and EC = D2 / (EC + 2.R)
Since the error is a very small amount in relation to the Earth's radius, without committing a sensitive error, disregard EC in the denominator, and the formula for calculating the error due to the curvature of the earth is:
EC = D2 / 2.R
In the practice of altimetric operations, the error due to the curvature of the Earth, due to the effect of the atmospheric refraction on the visual radius. When a point is made from one point to another, the visual radius atmospheric layers of different densities are refracted, following a curved trajectory, situated on the vertical visual plane, whose concavity is directed on the surface of the ground.
Therefore, point C, when viewed from A is seen in C ', giving rise to the error of refraction: ER = CC'. The surface AC 'is said optical level surface. This error is dependent on the temperature and the hygrometric state of the air, besides other local circumstances. Under normal conditions, the refractive error equation is as follows:
ER = 0.1306.EC with 0.1306 representing the average daily refractive curvature radius.
The correction to be made in determining the height of point B, seen from A, will be:
C = EC - ER
C = D2 / 2R - 0,1306.D2 / 2R = D2 / 2.R.(1 - 0,1306)
C = 0,43.D2 / R ou C = 6,8.10-8.DH2 (m)
C = 0,068.DH2 (Km)
Knowing the value of R (approximately 6,370 Km), one can determine the error due to the curvature of the earth and the atmospheric refraction, for any example: for a 100 m view, the error will be equal to 0.0007 m; for a view of 120 m, the error shall be equal to 0.0010 m; already for a 1000 m view, the error will be equal to 0.068 m. Thus, for distances smaller than 120 m the error due to the curvature of the earth and the refraction can be neglected because it is less than millimeter.
On the next two #partes, I will show you how we can calculate a geometric leveling and a trigonometric leveling.
See also #PART 0