Geoscience Reference
In-Depth Information
Reflection R and Transformation T Matrices
Let us introduce the admittance matrix
Y
Y
a
=
Y
11
,
Y
12
Y
21
Y
22
which relates horizontal components of the electric
E
and magnetic
b
fields:
b
1
b
2
z
=
−
0
=
Y
a
E
1
E
2
z
=
−
0
.
(7.11)
The total field above the ionosphere is represented as
b
(
r
)=
b
(
i
)
(
r
)+
b
(
r
)
(
r
)
,
where
b
(
i
)
(
r
) is the magnetic component of the incident wave;
b
(
r
)
(
r
)isthe
reflected field.
In order to find the relations between
b
(
i
)
(
r
)and
b
(
r
)
(
r
) as well as between
b
(
i
)
(
r
) and the field
b
(
g
)
(
r
) on the ground surface, we shall represent the fields
being expressed by their Fourier transforms. Then the relation between the
corresponding Fourier harmonics of the magnetic field on the ground and
above the ionosphere in the general case may be written
b
(
g
)
(
k
)=
T
(
k
)
b
(
i
)
τ
b
(
r
τ
(+0
,
k
)=
R
(
k
)
b
(
i
)
(+0
,
k
)
,
(+0
,
k
)
.
τ
Here
R
(
k
)and
T
(
k
) are reflection and transformation matrices for harmon-
ics with horizontal wavenumber
k
. Vectors
b
(
i
τ
(+0
,k
)and
b
(
r
τ
(+0
,k
)arethe
amplitudes of the horizontal magnetic field in the incident and the reflected
wave above the ionosphere. The transformation matrix
T
Σ
defined as a ratio
of the horizontal ground magnetic field to the total horizontal wave magnetic
component is
b
(
g
)
=
T
Σ
b
τ
.
One can find the matrix
T
by relating the amplitude of the horizontal mag-
netic fields on the ground to the amplitude of the initial wave
b
(
i
τ
(+0) by
b
(
g
)
=
T
Σ
1
+
R
b
(
i
τ
.
T
=
T
Σ
1
+
R
.
7.3 Atmospheric and Ground Fields
Here we shall obtain and analyze expressions for coecients
R
(
k
)and
T
(
k
).
We begin with the investigation of fields in the atmosphere and on the ground,
which enables us to connect the electric and magnetic fields observed on the
ground surface and to obtain specific expressions for admittances included in
boundary condition (7.11).
Search WWH ::
Custom Search