Geoscience Reference
In-Depth Information
2Basics
The propagation of electromagnetic waves are described by Maxwell's equations
(Jackson
1998
). For a non-conducting, neutral medium like the troposphere these
equations are
∇
·
(ε
E
)
=
0
,
(1)
∇
·
B
=
0
,
(2)
=−
∂
B
∂
∇
×
E
t
,
(3)
=
με
∂
E
∂
∇
×
B
t
.
(4)
where
E
and
B
are the electric field and magnetic field vectors, respectively,
ε
the
μ
electric permittivity,
the magnetic permeability. Assuming that the spatial and
μ
ε
temporal variations in
are small, the equations can be combined into forming
a wave equation for the electric field.
and
2
E
∂
n
2
2
E
∂
=
με
∂
c
2
∂
2
E
∇
=
(5)
t
2
t
2
/
√
ε
0
μ
0
is the speed of light in vacuum and
n
is called the refractive
index. A similar expression for the magnetic field can also be derived.
It is clear from Eq. (
5
) that in order to describe the propagation of a radio wave we
need to know the refractive index
n
. In the neutral atmosphere of the Earth
n
is very
close to one, thus it is more convenient to use the so called refractivity instead. The
refractivity
N
(in “N-units”, mm/km, or ppm) is related to the refractive index by
=
where
c
1
10
6
N
=
(
n
−
1
)
·
.
(6)
In general the refractivity is a complex number. It can be divided into three parts
N
(ν)
−
iN
(ν).
N
=
N
0
+
(7)
In case the spatial and temporal variations of
N
are small, i.e. the variations over
one wavelength or one period are negligible, the effect on the propagation of electro-
magnetic waves caused by the real and the imaginary parts of the refractivity can be
considered separately. For the signals of space geodetic techniques traveling through
the atmosphere, this separation is a reasonable assumption since the wavelengths
are shorter than a few decimeters. The real part of the refractivity (
N
0
+
N
(ν)
)
causes refraction and propagation delay of signals traveling through the atmosphere.
It consists of a frequency-independent (non-dispersive) part
N
0
and a frequency-
dependent (dispersive) part
N
(ν)
.
