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measured. This travel time is then converted to a distance measurement by multi-
plying with the speed of light in vacuum. The atmosphere will introduce an error
in this distance since it will affect the propagation path of the signal and since the
propagation speed of the signal in the atmosphere is lower than the speed of light in
vacuum.
If the variations in the refractivity over the distance of one wavelength is negligible
we can use the geometric optics approximation. This means that the propagation of
an electromagnetic wave can be described as a ray. When calculating the propagation
time of the electromagnetic wave we thus only have to consider the refractivity along
the ray path. For the propagation of the signals used in space geodesy the wavelengths
are a few decimeters at most, thus in the Earth's atmosphere this approximation will
normally be valid. The electric path length L (propagation time divided by the speed
of light in vacuum) of a ray propagating along the path S through the atmosphere
will be
=
(
)
.
L
n
s
d s
(26)
S
The electric path will be longer than the geometric length G of a straight line
between the endpoints of the path for two reasons (see Fig. 4 ). Firstly, the propagation
velocity is lower in the atmosphere than in vacuum. Secondly, the path S taken by the
ray is, according to Fermat's principle, the path which minimizes L . The atmospheric
delay,
Δ
L , is defined as the excess electric path length caused by the atmosphere
ds G = 10 6
Δ L = L G =
N ( s ) d s + S G ,
(27)
where S is the geometric length of the actual propagation path of the ray. By dividing
the refractivity into hydrostatic and wet parts using Eq. ( 16 ) we get
n ( s ) d s G =
[ n ( s ) 1] d s +
S
S
S
S
10 6
10 6
Δ
L
=
N h (
s
)
d s
+
N w (
s
)
d s
+
S
G
= Δ
L h + Δ
L w +
S
G
,
(28)
S
S
S
G
e
e 0
A
e
Earth
Fig. 4 Path taken by a signal through the atmosphere. The signal will take the path with the
shortest propagation time ( S ). Since the signal propagates slower in the atmosphere than in vacuum,
the geometrical length of S will be larger than the straight path G
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