b) 3. Trigonometric Leveling
Trigonometric leveling is based on the resolution of a right triangle. In this leveling, the level difference is determined in an indirect way, by means of triangles situated in vertical planes passing through points whose level difference is computed. The accuracy is lower when compared to geometric leveling, of the order of some decimeters, on the other hand, has a higher yield, that is, a rapid advance. The slope angles of the terrain are measured using theodolite. Trigonometric leveling is employed when determining the difference level between two accessible points, separated by great distance, or when one has an accessible point and the other one is inaccessible. In these cases, the intersection process is conjugated with trigonometric resolutions. In this case, to measure the vertical distances, it is counted with the aid of the telescopic road.
Determination of the level difference
Trigonometric leveling is based on the tangent value of the slope angle of the terrain, since the value of this trigonometric function always represents the level difference meter of horizontal distance measured in the ground between the points considered.
Thus, determining the horizontal distance (DH) between the points under study and the angle (α), the level difference (DN) is calculated by applying the following formula:
DN = DH.tg α, deduced from figure 1
tg α = BB '/ AB' ∴ tg α = DN / DH ∴ DN = DH.tg α
Desiring to determine the difference in level between the topographic points A and B of the terrain profile shown in figure 2, proceed as follow:
Z is the zenith angle;
i is the vertical angle;
hi is the measure of the geometric center of the telescope to the topographic point;
FM is the reading in the sights;
DN is the level difference between points A and B; and
DH is the horizontal distance between points A and B.
With the theodolite (or with a total station) parked in A, the aim is placed vertically in B, measured the height where the horizontal reticule of the telescope intercepts the crosshair and the vertical angle of the concerned.
The vertical angle can be from the zenith to the line of sight, when the theodolite has its vertical limb zeroed in zenith; or from the horizon to the line of sight, when the theodolite has its vertical limb zeroed in the horizon.
The most modern theodolites, for the most part, are zeroed in the zenith, and the vertical angles is called the "zenith angle".
From Fig. 2 one can deduce:
DNAB + FM = hi + DH.tg i
DNAB = DH.tg i + hi - FM
in the case of theodolite measure zenith angles:
DNAB = DH.cotg Z + hi - FM, where i = 90º - Z.
Due to the replacement of the true level by the apparent level, when leveling, as already seen, an error occurs due to the Earth's curvature and refraction atmospheric pressure. The correction to be made in the measures carried out, as already shown, is:
C = 0.068.DH2 (Km)
Thus, the formula for calculating the level difference between two points in the trigonometric level becomes the following:
DNAB = DH.cotg Z + hi - FM + C
However, in short visas, up to 250 meters, we can have curvature and refraction.
Trigonometric leveling of polygonal and other applications
The spreadsheet below contains the field observations of a trigonometric leveling the ABCD open traverse. The dimensions of A and D are known, being 150 m and 135.28 m, respectively. The goal is to calculate the compensated dimensions of vertices B and C.
Calculation of the correction due to the error of curvature and refraction:
C (m) = 0.068.DH2 (Km)
C = 0.068 (0.52435) 2 = 0.01869 ≅ 0.02
C = 0.068. (0.73246) 2 = 0.036 ≅ 0.04
C = 0.068. (0.63124) 2 = 0.027 ≅ 0.03
Calculation of the level difference:
DNAB = DHAB.cotg Z + hi - FM + C
Determining the dimension of an inaccessible point
Considering Figure 3, let P be the point whose quota we want to determine, with the base AB. With the theodolite we measure the horizontal angles a and b and the angles zenitals Z1, Z2 and Z3. The lengths D1 and D2 are obtained from the relations:
The level differences will be obtained by the following formulas:
The calculation of the coordinates of points B and P, as a function of the quotient of A, which is known, is done in the following way:
CB = CA + DNAB
CP = CA + DNAP
See also #PART 3