2-D Niblett-Bostick magnetotelluric inversion - MTNet

Geologica Acta, 8(1), 15-30 (2010)
DOI: 10.1344/105.000001513
J. RODRÍGUEZ et al. 2-D Niblett-Bostick
27
1 Fréchet Derivatives
In this appendix we develop formulae for the 2-D Fréchet
derivatives of apparent conductivity for a homogeneous
half-space. We assume e ivt for time dependence in the
fields. The definition of apparent conductivity in terms of
the fields for the TE mode is:
(A.1)
perturbing the conductivity distribution s(x,z) produces
perturbations on both sides of this equation. We have
(A.2)
We will need Ey(x,z) and Hx(x,z). To find Ey(x,z) we
have to solve the following differential equation:
(A.3)
for the case of a homogeneous half-space. The solution is
simply
(A.4)
where δ is the skin depth, given by
(A.5)
In what follows, all the distances are normalized by the
skin depth. Hx(x,z) can be found using Maxwell’s equation
,3E = -iωm0H, the result is
(A.6)
Using equation (A4), the horizontal magnetic field can
be written as
(A.7)
To find δEy(x,0) we use the scattering relation
(Hohmann, 1971; Weidelt, 1975) which, for TE mode,
reduces to
(A.8)
In this equation, G is the y component of the electric
field at (x,0) produced by a line source of alternating
current located at (x’,z’) and parallel to the y axis. The
expression for G can be found in Wait (1962) and is
(A.9)
with u 2 = l 2 + 2i. To find δHy(x,0) we take perturbations
on both sides of equation (A6), the result is
(A.10)
Finally, using equations (A2), (A8) and (A10), we find
the Fréchet derivative in terms of the normalized distances.
For TE mode:
(A.11)
where
(A.12)
For TM mode apparent conductivity is given by
(A.13)
Taking perturbations on both sides of this equation, we find
(A.14)
In the above equation, since δHy is zero, we have to
find δEx only. For this we use the scattering relations given
by Lee and Morrison (1985):
(A.15)
gij is the j-component of the electric field in (x,z) produced
by a 2-D oscillating electric dipole located in (x’, z’) and
appeNDiX
3
.
]
[
sgn
2
1
2
1 )
(
)
(
1
)
1
(
⎭
⎬
⎫
⎩
⎨
⎧
+
+
= +
−
=
=
≠
+
∑
∑ i
t
l
j
l
n
n
l
ij
N
i
j
t
i
I
w
T σ
σ (16)
n
n
k ,...,
, −
=
}.
]
)
1
[(
sgn{
2
1
2
1 )
(
)
(
)
(
1
)
i
t
l
j
l
ij
n
n
l
t
j
ij
N
i
j
I
w
T
T
b +
+
−
+
= +
−
=
=
≠
∑
∑
σ
β
σ (17)
,
)
(
)
(
)
(
)
(
)
(
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,
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1
2
1
1
2
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T
z
z
a
a
a
a
a
a
δ
δ
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2
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1
),
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i
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a
,
|
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0
,
(
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0
,
(
|
|
|
2
0
x
E
x
H
y
x
a
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σ = (A.1)
].
)
0
,
(
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0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
x
x
y
y
a
a
δ
δ
σ
σ
δ −
−
= (A.2)
,
0
)
,
(
0
2
=
−
∇ y
y
E
z
x
i
E σ
ωµ (A.3)
,
)
0
(
)
,
(
2 δ
z
i
y
e
E
z
x
E
−
= (A.4)
.
2
0
2
σ
ωµ
δ = (A.5)
H
0
µ
3
.
)
(
)
(
)
( t
l
j
l
n
l
t
j w +
−
=
∑
= σ
σ (15)
.
]
[
sgn
2
1
2
1 )
(
)
(
1
)
1
(
⎭
⎬
⎫
⎩
⎨
⎧
+
+
= +
−
=
=
≠
+
∑
∑ i
t
l
j
l
n
n
l
ij
N
i
j
t
i
I
w
T σ
σ (16)
)
1
( +
t
)
(t
j
n
n
k
n
,...,
,
1
2
1
−
=
+
=
}.
]
)
1
[(
sgn{
2
1
2
1 )
(
)
(
)
(
1
)
1
(
i
t
l
j
l
ij
n
n
l
t
j
ij
N
i
j
t
i
I
w
T
T
b +
+
−
+
= +
−
=
=
≠
+
∑
∑
σ
β
σ
σ (17)
,
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
1
2
1
1
2
2
2
1
T
T
T
T
T
T
z
z
a
a
a
a
a
a
δ
δ
δ
σ
δ
σ
σ
−
−
>=
< (18)
>
)
,
( 2
1 z
z
2
,
1
),
(
707
.
0 =
= i
T i
a
δ
2
1 z
z
NEXES
,
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
y
x
a
ωµ
σ = (A.1)
)
, z
x
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
x
x
y
y
a
a
δ
δ
σ
σ
δ −
−
= (A.2)
)
,
( z
x
y
)
,
( z
x
x
)
,
( z
x
y
,
0
)
,
(
0
2
=
−
∇ y
y
E
z
x
i
E σ
ωµ (A.3)
,
)
0
(
)
,
(
2 δ
z
i
y
e
E
z
x
E
−
= (A.4)
.
2
0
2
σ
ωµ
δ = (A.5)
)
,
( z
x
x
H
i
E 0
ωµ
−
=
×
3
)
( l
j
l
n
l
j +
−
=
∑
.
]
[
sgn
2
1
2
1 )
(
)
(
1
)
1
(
⎭
⎬
⎫
⎩
⎨
⎧
+
+
= +
−
=
=
≠
+
∑
∑ i
t
l
j
l
n
n
l
ij
N
i
j
t
i
I
w
T σ
σ (16)
n
n
k ,...,
,
1
−
=
}.
]
)
1
[(
sgn{
2
1
2
1 )
(
)
(
)
(
1
)
1
i
t
l
j
l
ij
n
n
l
t
j
ij
N
i
j
t
I
w
T
T
b +
+
−
+
= +
−
=
=
≠
+
∑
∑
σ
β
σ (17)
,
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
1
2
1
1
2
2
2
1
T
T
T
T
T
T
z
z
a
a
a
a
a
a
δ
δ
δ
σ
δ
σ
σ
−
−
>=
< (18)
>
)
2
,
1
),
( =
i
T i
a
δ
S
,
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
y
x
a
ωµ
σ = (A.1)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
x
x
y
y
a
a
δ
δ
σ
σ
δ −
−
= (A.2)
,
0
)
,
(
0
2
=
−
∇ y
y
E
z
x
i
E σ
ωµ (A.3)
,
)
0
(
)
,
(
2 δ
z
i
y
e
E
z
x
E
−
= (A.4)
.
2
0
2
σ
ωµ
δ = (A.5)
H
0
ωµ
3
.
)
(
)
(
)
( t
l
j
l
n
n
l
t
j w +
−
=
∑
= σ
σ (15)
.
]
[
sgn
2
1
2
1 )
(
)
(
1
)
1
(
⎭
⎬
⎫
⎩
⎨
⎧
+
+
= +
−
=
=
≠
+
∑
∑ i
t
l
j
l
n
n
l
ij
N
i
j
t
i
I
w
T σ
σ (16)
n
n
k ,...,
−
=
}.
]
)
1
[(
sgn{
2
1
2
1 )
(
)
(
)
(
1
i
t
l
j
l
ij
n
n
l
t
j
ij
N
i
j
I
w
T
T
b +
+
−
+
= +
−
=
=
≠
∑
∑
σ
β
σ (17)
,
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
1
2
1
1
2
2
2
1
T
T
T
T
T
T
z
z
a
a
a
a
a
a
δ
δ
δ
σ
δ
σ
σ
−
−
>=
< (18)
2
,
1
),
( =
i
T i
,
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
y
x
a
ωµ
σ = (A.1)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
x
x
y
y
a
a
δ
δ
σ
σ
δ −
−
= (A.2)
,
0
)
,
(
0
2
=
−
∇ y
y
E
z
x
i
E σ
ωµ (A.3)
,
)
0
(
)
,
(
2 δ
z
i
y
e
E
z
x
E
−
= (A.4)
.
2
0
2
σ
ωµ
δ = (A.5)
H
0
3
.
)
(
)
(
)
( t
l
j
l
n
n
l
t
j w +
−
=
∑
= σ
σ (15)
.
]
[
sgn
2
1
2
1 )
(
)
(
1
)
1
⎭
⎬
⎫
⎩
⎨
⎧
+
+
= +
−
=
=
≠
+
∑
∑ i
t
l
j
l
n
n
l
ij
N
i
j
I
w
T σ (16)
n
n,...,
}.
]
)
1
[(
sgn{
2
1 )
(
)
(
)
(
1
i
t
l
j
l
ij
n
n
l
t
j
ij
N
i
j
I
w
T
T
b +
+
−
+ +
−
=
=
≠
∑
∑
σ
β
σ (17)
,
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
1
2
1
1
2
2
2
1
T
T
T
T
T
T
z
z
a
a
a
a
a
a
δ
δ
δ
σ
δ
σ
−
−
>= (18)
2
,
1
=
i
,
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
y
x
a
ωµ
σ = (A.1)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
x
x
y
y
a
a
δ
δ
σ
σ −
−
= (A.2)
,
0
)
,
(
0
2
=
−
∇ y
y
E
z
x
i
E σ
ωµ (A.3)
,
)
0
(
)
,
(
2 δ
z
i
y
e
E
z
x
E
−
= (A.4)
.
2
0
2
σ
ωµ
δ = (A.5)
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
z
z
u
z
z
u
(A.9)
i
2
)
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
z
z
u
z
z
u
(A.9)
i
2
1 ∂
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
(A.9)
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
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,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
(A.9)
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
(A.9)
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
(A.9)
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
(A.9)
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
(A.9)
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
4
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
(A.9
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
).
,
(
1
)
,
(
0
z
x
E
z
i
z
x
H y
x
∂
∂
=
δ
ωµ
(A.6)
.
)
0
(
2
)
,
(
2
0
z
i
x
e
E
i
i
z
x
H
−
−
=
δ
ωµ
(A.7)
)
0
,
(x
E y
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
'
,
'
;
0
,
(
)
0
,
( dz
dx
z
x
z
x
E
z
x
x
G
x
E y
y
δσ
=
δ ∫ (A.8)
)
0
,
(x
( ', ')
x z
,
)]
'
(
cos[
]
[
2
)
'
,
'
;
,
(
|
'
|
)
'
(
0
0 λ
−
λ
+
λ
+
λ
−
π
ωµ
−
=
−
−
+
−
∞
∫
d
u
x
x
e
e
u
u
i
z
x
z
x
G
z
z
u
z
z
u
i
u 2
2
2
+
= λ
)
0
,
(x
H x
δ
.
|
)
,
(
1
)
0
,
( 0
0
=
∂
∂
= z
y
x
z
x
E
z
i
x
H δ
δ
ωµ
δ (A.10)
],
)]
'
(
cos[
2
Re[
0
2
λ
−
λ
πδ
= λ
∞
∫
d
x
x
G
G
TE
TE
(A.11)
.
)
2
( '
)
2
( z
i
u
TE
e
u
i
G
+
−
λ
+
λ
λ
+
+
λ
= (A.12)
.
|
)
0
,
(
)
0
,
(
|
|
|
2
0
x
E
x
H
x
y
a
ωµ
σ = (A.13)
].
)
0
,
(
)
0
,
(
)
0
,
(
)
0
,
(
Re[
|
|
2
|
|
x
H
x
H
x
E
x
E
y
y
x
x
a
a
δ
δ
σ
σ
δ −
−
= (A.14)
y
H
δ
x
E
δ
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
ij
g
( ', ')
x z
.
1
0
2
ρ
ρ
σ
ωµ ∇
×
∇
−
=
−
∇ y
y
y
H
H
i
H (A.16)
.
)
0
(
)
,
(
2 z
i
y
e
H
z
x
H
−
= (A.17)
.
2
)
0
(
)
,
(
2 z
i
x
e
i
H
z
x
E
−
=
σδ
(A.18)
( )
,0,
x z
E E
z
E
3