Surveying I. (BSc) Lecture 5. Trigonometric heighting. Distance measurements, corrections and reductions.

Surveying I. (BSc)

Lecture 5.

Trigonometric heighting.Distance measurements, corrections and

reductions

How could the height of skyscrapers be measured?

? ?

The principle of trigonometric heighting





zdhmhm cot

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

The distance is not observable.







Using the sine-theorem:

sinsin

180sinsinad

adAP

AP

sinsin

180sinsinad

adBP

BP



AAPOA zdlm cot


Using the observations in pont B:

BBPBO

B zdlm cot

2

BA mmm

Trigonometric heightingThe effect of Earth’s curvature



RdAB

The central angle:


The tangent-chord angle is equal to /2.


The effect of Earth’s curvature:

Rd

Rd

dd ABABABABsz 222

tan2

Trigonometric HeightingThe effect of refraction

Trigonometric HeightingThe effect of refraction

2ABd

ABzdm cot

dzdm ABcot

2cot

2dzdm AB

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



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)



• 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

Surveying I. (BSc) Lecture 5. Trigonometric heighting. Distance measurements, corrections and reductions.

Documents

effect of refraction

trigonometric levelling

elevation difference

height of buildings

reference level slide

heights of buildings

combined effect of curvature

shortest distance