Geoscience Reference
In-Depth Information
Table 7.1.
Characteristic wave and medium parameters
X ≈ Y
k
0
km
−
1
√
ε
m
5
×
10
1
(night)
τ
(s)
h
(km)
c
A
(km/s)
2
×
10
−
5
≈
10
2
≈
10
3
3
×
10
2
5
×
10
3
(day)
1
σ
g
s
−
1
√
ε
a
√
ε
g
H
(km)
Σ
g
(km/s)
10
5
−
10
8
3
×
10
2
5
×
10
2
≈
10
8
<
5
Assume that a layer with a constant conductivity
σ
g
lies on a perfect con-
ductor at a depth
H
. The thin conductive ionosphere is located, as before,
at the height
h
. The magnetosphere is modeled by a uniform half-space with
a constant Alfven velocity
. T
he model contains several characteristic spatial
scales:
h
,
H
,
d
g
,
l
m
,(
k
0
√
ε
a
)
−
1
and
k
−
1
. The sense of the first two parame-
ters is evident.
d
g
=(Re
k
0
√
ε
g
)
−
1
is
the
ground skin depth. Th
e w
avelength
in the magnetosphere
l
m
=2
π
(
k
0
√
ε
m
)
−
1
=2
πc
A
/ω
.2
π
(
k
0
√
ε
a
)
−
1
is the
wavelength in the atmosphere and
k
−
1
is the horizontal wave-scale.
Table 7.1 shows the characteristic scales of the wave and the medium. One
can see that
k
0
√
ε
g
k
0
√
ε
a
|
k
0
√
ε
m
H
−
1
<h
−
1
.
|
and
The skin depth and thickness of the underlying half-space is compared on
periods
T
=10
−
9
H
2
σ
g
,
where
H
is in km and
σ
g
in s
−
1
.
Analytical Properties of R and T Matrices
The behavior of the ground fields and signals reflected into space is deter-
mined by the analytical properties of
R
(
k
)and
T
(
k
) in the complex plane
of wavenumbers
k
=
k
x
.
The first thing to pay attention to is the presence of branch points (see
Fig. 7.3):
•
In a model '
ope
n' at
H
=
∞
(se
e F
ig. 7.2 ) there are four branch points
k
0
√
ε
m
and
k
3
,
4
=
k
0
√
ε
g
;
k
1
,
2
=
±
±
•
In the two-layer model with a layer of the finite thickness underla
in
by a
perfect conductor (Fig. 7.2), only two points remain:
k
1
,
2
=
k
0
√
ε
m
.
In order to isolate univalent branches of the functions under investigation
their behavior will be considered on a many-sheeted Riemann surface with
the number of sheets determined by the number of branch points.
Construct a Riemannian surface for the 1-st model of the Earth as a half-
space with finite conductivity
σ
g
. Draw four cuts on plane
k
from the branch
points to infinity. Im
κ
s
=Im
κ
g
= 0 on cuts (see Fig. 7.3). Numeration the
sheets is shown in Table 7.2.
±
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