Surveying I.
Lecture 7.
Distance observations
Distance observations
Different ways to measure the distance
between points:
• tapes;
• optical methods, e.g. using the stadia lines in the
surveyor’s telescope;
• physical methods: electrooptical distance
measurements (EDM)
Measuring distances with tapes
How long is a 30-meter-long tape?
Error sources:
• graduation error (1m on the tape is not equal to the
standard 1 m);
• expansion caused by the pulling force;
• thermal expansion.
The effects must be taken into account to
measure the distances accurately!
The standardization of tapes
The tape should be compared to a standard length.
Therefore standard baselines have been created.
20 10
0
-10 -20
-20 -10
0
10 20
30 00
0 00
Tape
standard baseline
length = 29.998m
The standardization of tapes
Nominal length of the tape = 30.000m.
20 10
0
-10 -20
-20 -10
0
10 20
30 00
0 00
Tape
standard baseline
length = 29.998m
The position of the two ends of the tape should be
read on the scales, thus the true length of the tape
can be computed.
LL=+4.5 mm
LR=+3.2 mm
True length = LL+LR+Lbaseline=30,006 mm
The standardization of tapes
The standardization must be carried out using a
standard pulling force, and the temperature should
also be recorded to compute the effect of thermal
expansion later.
The true length is valid for the
standardization temperature only!
True length = LL+LR+Lbaseline=30,006 mm (T=18 °C)
The effect of pulling force and temperature
The effect of pulling force:
• when the same pulling force is used during the
measurements as during the standardization, then the
effect of this error source is eliminated.
The effect of temperature change:
• the temperature during the measurements must be
recorded;
• the thermal expansion of the tape can be computed
using the equation of thermal expansion of linear
objects:
 L    L  T  T0 
where  is the expansion coefficient of the material
of the tape (steel: =1.1x10-5 1/°C).
Physical Methods of Distance Observations
Indirect way of distance observation: the distance is
measured by the observation of other physical
variables.
Usually these methods are based on electromagnetic
signals:
• carrier signal – carries the measuring signal between
the points;
measuring signal – the carrier is modulated with this
signal, and this is used for the distance observations
Two parts of the electromagnetic spectrum are used:
• microwave (wavelengths are in the order of
centimeters)
• electrooptical (wavelengths are in the order of
micrometers – visible light or infrared rays
Electrooptical Distance Measurements (EDM)
Two different realization:
• time pulse measurements
• the carrier is modulated with a pulse (short
peak), and the travel time of the signal
between the station and the target is
measured.
• the speed of the carrier must be known
• phase angle measurements
• the phase angle must be measured at the
two endpoints;
• the distance is linked with the phase
difference in the signals;
• usually the phase difference of less than a full
cycle can be measured;
• the speed of the carrier must also be known;
The realization of EDM
Receiver
Transmitter
A transmitter is place on the station, and a receiver on
the target.
The carrier is modulated with the measurement
signal, and transmitted by the instrument to the
target.
The receiver receives the signal, demodulates it from
the carrier, and uses the measurement signal to carry
out the measurement.
The realization of EDM
The distance can be computed using the phase
difference or the travel time. However time
synchronization is necessary to be able to measure
these quantities.
We use a reflector at the target, and the receiver is
placed in the instrument together with the
transmitter:
Reflector
Transmitter
Receiver
Time pulse measurement
The instrument emits a pulse, which propagates to the
target, where the reflector reflects it back to the
instrument, where the receiver receives the signal.
The time-of-flight (t) is measured. Thus the covered
distance can be computed:
2 D  t  D 

t
2
Where u is the velocity of light in the atmosphere.
Phase angle measurement
Reflector
Transmitter
Receiver
The signal travels the length of 2D.
Let’s suppose that u and the frequency ( f ) of the
measurement signal are known.
In this case the signal travels during a full cycle:
 
u
f
Where  is the wavelength of the signal.
Phase angle measurement
d
Receiver
Transmitter
N full wa ves = N
2D
The observed phase angle can be used to compute
the residual distance (d):

2

d

Thus the distance:
2D  N  d

D  N

2

d
2
The equation of phase measurements
D  N


2
d
2

D  N


2
 
2 2
The effective wavelength of EDMs:
 eff 

2
The realization of phase measurements
D  N

2

 
2 2
Method of constant measurement frequency:
 is measured and the residual distance (d) is
computed.
Method of variable measurement frequency:
The measurement frequency is changed so, that 
equals zero. In this case the residual distance is zero
as well.
The determination of N
D  N

2

 
2 2
When the effective wavelength is shorter than the
observed distance, the observation is ambiguous (N
is not known), but accurate.
When the effective wavelength is larger than the
observed distance, the observation is unambiguous
(N=0), but inaccurate due to the long wavelength.
The determination of N
N2 full waves = N2
2D
Two different frequencies are used for the
measurements:
• 1 is larger than the range of the EDM
• 2 provides the higher accuracy
d2
Receiver
Transmitter
N1= 0
d1
The determination of N
N2 full waves = N2
d2
Receiver
Transmitter
N1= 0
d1
2D
The accuracy of phase angle measurement is usually
2-3/10000.
Let’s see the first observation (1eff=2000m):
N=0, accuracy is 0.6 m
d1 = 843.5 m
The determination of N
Let’s see the second observation (2eff= 10m):
N 2  0,
 d
N 2  int  1
 2
 eff
  2  2 . 1765  d 2 
 2
2
2

  84


eff
 3 , 464 m

D  N 2  2 eff  d 2  840 m  3 . 464 m  843 . 464 m
accuracy
 2  3 mm
Error sources of EDMs
Instrumental and reflector constants
The transmitter is not aligned with the standing axis
of the instrument, or the point of reflection is not
aligned with the vertical of the target.
Instrumental offsets are automatically included in
the results, but it is not the case with the prism
offset.
Determining the reflector constant (c):
Set out three collinear points (A,B and C), and
measure the distances between AC in one set, and
in two sets (AB and BC)!
The computation of the reflector constant
A
C
B
True distances: DAB, DAC, DBC
Observed distances: (D)AB, (D)AC, (D)BC
D AC   D  AC  c
D AB   D  AB  c
since :
D AC  D AB  D BC ,
 D  AC  c   D  AB  c   D  BC
c   D  AC   D  AB   D  BC 
 c,
D BC   D  BC  c
Frequency error
Frequency error: the oscillators, which create the
measurement signal are stable, but a drift can occur
over a longer operational time.
A change in the frequency causes an error in the
wavelength, thus in the observed distance as well.
D  N

2

 
2 2
Instruments must be calibrated frequently, corrections
can be applied to the observed distances (k).
k  1
f0  f
f0

D  k  D  raw
Atmospheric correction
The wavelength depends on the velocity of the signal in
the atmosphere and the frequency.
 
u
f
The velocity of the signal depends on the refractive
index of the atmosphere:
u 
c

Where
c is the velocity of light in vacuo,
 is refractive index.
The refractive index depends mostly on the air pressure
and the temperature.
Atmospheric correction
The atmospheric correction:
atm . corr . mm / km    1, 0  T0  T   0 , 4  p 0  p 
The multiplicator of atmospheric corr:
m  1  atm . corr .mm / km 10
6
 1  T 0  T   0 , 4  p 0  p 10
6
The corrected distance observation (reflector constant,
frequency error, atmospheric correction):
D  c  k  m   D raw
The reduction of distance observations
Definition: the distance between two points is
defined by the distance between the two points
projected to a reference surface (e.g. the Mean Sea
Level).
Thus the measured distance should be reduced to this
reference surface.
Note: Later we’ll see that the distances should also
be reduced to the applied projection plane.
Reduction to the local horizon
The distance observations are usually measured on a
slope. Therefore they should be converted to
horizontal distances.
Slope distances can be:
• straight slopes,
• sections with different slopes.
Straight slope
(e.g. EDM)
Sections with diff. slopes
(e.g. tape)
Reduction to the local horizon
Sections with constant slopes can be converted to
horizontal distances:
B
s
A

h
d
If  is known: d=s cos,
if h is known: d=s-h2/2s.
Reduction to the Mean Sea Level
d AB
A’
H
d ’AB
R
B’
MSL
dAB is the horizontal
distance at the mean
elevation ( H ) of the two
points
d AB
d AB
O

R
RH
,

d AB  d AB  d AB
H
R
Thank You for Your Attention!