Surveying I. (BSc)
Lecture 5.
Trigonometric heighting.Distance measurements, corrections and
reductions
Trigonometric levelling
AAABBB
AAABBB
ztzt
zdzdm
coscos
cotcot
Advantage: • the instrument height is not necessary;• non intervisible points can be measured, too.
Trigonometric heighting
Advantages compared to optical levelling:
• A large elevation difference can be measured over short distances;
• The elevation difference of distant points can be measured (mountain peaks);
• The elevation of inaccessible points can be measured (towers, chimneys, etc.)
Disadvantages compared to optical levelling:
• The accuracy of the measured elevation difference is usually lower.
• The distance between the points must be known (or measured) in order to compute the elevation difference
The determination of the heights of buildings
The horizontal distance is observable, therefore:
AAP zdm cot
AAPO zdlm cot
Determination of the height of buildings
Using the sine-theorem:
sinsin
180sinsinad
adAP
AP
sinsin
180sinsinad
adBP
BP
Determination of the height of buildings
Using the observations in pont B:
BBPBO
B zdlm cot
2
BA mmm
Trigonometric heightingThe effect of Earth’s curvature
The effect of Earth’s curvature:
Rd
Rd
dd ABABABABsz 222
tan2
Trigonometric heightingThe effect of refraction
Let’s introduce the refractive coefficient:
13,0R
k
Thus m can be computed:
rABAB zdd
zdm cot2
cot2
where:
Rd
kd
r 22
22
Trigonometric heightingThe combined effect of curvature and refraction
Note that the effects have opposite signs!
Trigonometric heightingThe combined effect of curvature and refraction
Rd
kzdm AB 2cot
2
Rd
sz 2
2
=r
Rd
kzdhm AB 21cot
2
l
The elevation difference between A and B (the combined effect of curvature and refraction is taken into consideration):
The fundamental equation of trigonometric heighting
The combined effect reaches the level of 1 cm in the distance of d 0,4 km = 400 m.
Determination of distancesDistance: is the length of the shortest path between the points
projected to the reference level
Determination of distancesDistance: is the length of the shortest path between the points
projected to the reference level
The distance at the reference level
can not be observed, therefore the
slope distance must be measured in
any of the following ways:
• It can be the shortest distance
between the points (t)
Determination of distancesDistance: is the length of the shortest path between the points
projected to the reference level
• The distance measured along the intersection of the vertical plane fitted to A and B, and the surface of the topography.
The distance at the reference level
can not be observed, therefore the
slope distance must be measured in
any of the following ways:
• It can be the shortest distance
between the points (t)
Reduction of slope distance to the horizontal planeThe slope distance is measured along the terrain
Suppose that the angle (i) between the li distance and the horizon is known, thus
iviiv ,, where:
i
iiv
m
2
2
,
iiiv cos, or:
i
im2
and:
ivvt ,l
Reduction of slope distance to the horizontal planeThe slope distance is measured between the points directly.
ztt fv sin
where:
When the elevation difference is known:
vfv tt
fv t
m
2
2
Determination of distance on the reference surface
Reduction of horizontal distance to the reference level (MSL)
RH
t
t
HRH
HRHHR
t
t
HRR
t
t
v
g
v
g
v
g
1
1
vvvvg tR
Httt
Thus the distance on the reference surface:
The reduction is:
vv tR
H
Distances can be measured directly, when a tool with a given length is compared with the distance (tape, rod, etc.)
Distances can be measured indirectly, when geometrical of physical quantities are measured, which are the function of the distance (optical or electronical methods).
Determination of distances
Standardization of the tapeHow long is a tape in reality?
The length of the tape depends on
• the tension of the tape, therefore tapes must be pulled with the standard force of 100 N during the observation and the standardization;
• the temperature of the tape, therefore the temperature must be measured during observations (tm) and during the standardization (tk), too.
, , .
Standardization of the tape
rl ddd The difference between the true length (l) and the baseline length (a) from a single observation:
The difference of the true length and baseline length from N number of repeated observations:
ndi
The real length of the tape
a
Corrections of the length observations
Standardization correction (takes into account the difference between the nominal and the true length):
k
where l is the standardized length and (l) is the nominal length
Temperature correction (takes into consideration the thermal expansion of the tape):
kmt tt C/101,1 5 (steel)
Thus the corrected length:
tk
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