Geoscience Reference
In-Depth Information
Combining (7.15), (7.16) and the Maxwell's equations (7.5) gives
1
ε
(
z
)
∂ε
(
z
)
∂z
∂Φ
(
r
)
∂z
2
Φ
(
r
)
+
k
0
(
z
)
ε
(
z
)
Φ
(
r
)=0
,
∇
−
(7.17)
2
Ψ
(
r
)+
k
0
(
z
)
ε
(
z
)
Ψ
(
r
)=0
.
∇
(7.18)
For a harmonics we get
1
ε
(
z
)
+
k
z
(
z
)
Φ
(
z
;
k
τ
)=0
,
∂
∂z
∂Φ
(
z
;
k
τ
)
∂z
ε
(
z
)
(7.19)
∂
2
Ψ
(
z
;
k
τ
)
∂z
2
+
k
z
(
z
)
Ψ
(
z
;
k
τ
)=0
,
(7.20)
where
k
z
=
k
0
ε
(
z
)
,
k
τ
=
k
x
+
k
y
.
The horizontal electric and magnetic components should be continuous
on the discontinuity
z
=
z
i
of dielectric permeability
ε
(
z
)
.
Hence, we find
boundary conditions for potentials
k
2
⊥
−
1
ε
(
z
)
∂Φ
∂z
{
Φ
}
z
=
z
i
=0
,
=0
,
(7.21)
z
=
z
i
∂Ψ
∂z
{
Ψ
}
z
=
z
i
=0
,
=0
,
(7.22)
z
=
z
i
where
{
Φ
}
z
=
z
i
=
Φ
(
z
i
+0)
−
Φ
(
z
i
−
0), etc.
Admittance Matrix
Electric and Magnetic Admittances
Let us define spectral admittances
Y
(
e,m
)
and impedances
Z
(
e,m
)
for the elec-
tric and magnetic modes as
b
(
e
y
(
z
;
k
τ
)
E
(
e
x
(
z
;
k
τ
)
b
(
e
x
(
z
;
k
τ
)
E
(
e
y
(
z
;
k
τ
)
1
Z
(
e
)
(
z
;
k
τ
)
=
Y
(
e
)
(
z
;
k
τ
)=
−
,
=
(7.23)
b
(
m
y
(
z
;
k
τ
)
E
(
m
)
b
(
m
x
(
z
;
k
τ
)
E
(
m
)
1
Z
(
m
)
(
z
;
k
τ
)
=
Y
(
m
)
(
z
;
k
τ
)=
−
=
.
(7.24)
(
z
;
k
τ
)
(
z
;
k
τ
)
x
y
They can be expressed in terms of potentials:
ik
0
ε
(
z
)
Φ
(
z
)
∂Φ
∂z
−
1
Y
(
e
)
(
z
)=
−
,
(7.25)
ik
0
Ψ
−
1
(
z
)
∂Ψ
1
Y
(
m
)
(
z
)=
−
∂z
.
(7.26)
h
give a definition of the spectral
impedance (or of the surface admittances ) on the ground surface
Relations (7.23) and (7.24) for
z
=
−
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