Geoscience Reference
In-Depth Information
the Helmholtz equation for an electro-magnetic wave propagating through a slowly
varying medium (Iizuka
2008
; Wheelon
2001
). Using the Eikonal equation, we can
determine the ray path and the optical path length. In the following sub-sections, we
present a ray-tracing system that can be applied for tropospheric modeling and dis-
cuss some basic elements in tropospheric ray-tracing calculations through Numerical
Weather Prediction Models.
4.1.1 Eikonal Equation and Ray Path
To derive the Eikonal equation we start from Maxwell's equations (Eqs.
1
-
4
). It is
convenient to use the H-field
H
instead of the magnetic field
B
, where
H
is defined by
B
=
μ
H
.
(67)
We can consider a general time-harmonic field
e
i
(
k
0
L
(
r
)
−
2
πν
t
)
,
E
(
r
,
t
)
=
e
(
r
)
(68)
e
i
(
k
0
L
(
r
)
−
2
πν
t
)
,
H
(
r
,
t
)
=
h
(
r
)
(69)
where
L
(
r
)
is the optical path, and
e
(
r
)
and
h
(
r
)
are (complex) vector functions. The
wave number for vacuum (
k
0
) is defined as
2
πν
c
2
π
λ
0
,
k
0
=
=
(70)
where
λ
0
is the vacuum wavelength. By inserting these equations into Maxwell's
equations (Eqs.
1
-
4
) and assume that there are no free charges and zero conductivity,
we get after some calculations (Born and Wolf
1999
)
1
ik
0
∇
×
∇
L
×
h
+
c
ε
e
=−
h
,
(71)
1
ik
0
∇
×
∇
L
×
e
−
c
μ
h
=−
e
,
(72)
e
e
,
1
ik
0
·
∇
ε
ε
e
·
∇
L
=−
+
∇
·
(73)
h
h
.
1
ik
0
·
∇
μ
μ
h
·∇
L
=−
+
∇
·
(74)
For small vacuum wavelength, and therefore large wave number for vacuum (
k
0
)
∇
L
×
h
+
c
ε
e
=
0
,
(75)
∇
L
×
e
−
c
μ
h
=
0
,
(76)
