2-D Niblett-Bostick magnetotelluric inversion - MTNet

J. RODRÍGUEZ et al.
Geologica Acta, 8(1), 15-30 (2010)
DOI: 10.1344/105.000001513
2-D Niblett-Bostick
28
oriented in the i direction. On the other hand, the horizontal
magnetic field satisfies the following differential equation
(A.16 )
The solution of this equation for a homogeneous half-
space is simply
(A.17)
Therefore, the horizontal electric field is given by
(A.18)
Since E for TM mode is (Ex,0,Ez) and for a homogeneous
half-space Ez is zero, we only need g 11 to compute δEx. The
expression for g 11 is given in Lee and Morrison (1980).
(A.19)
Combining equations (A14), (A15), (A18), and (A19),
we find the Fréchet derivative for TM mode:
(A.20)
2 Integral of Fréchet derivatives for TE and TM
modes
Here we find the integral of the Fréchet derivatives for
a typical cell. For TE mode we integrate equation (A11) in
the region xi≤x’≤xi+1, zj≤z’≤zj+1. That is
(B.1)
The result is
(B.2)
For TM mode we integrate equation (A20) in the same
region, the result is
(B.3)
The integrals ITE and ITM as defined by Eqs. (B2) and (B3)
are complex numbers. However, the real part of each one
represents the corresponding integral of the Fréchet derivate
of usau. The horizontal axis is divided into n elements, from
x1 to xm, and the vertical distance is divided into m elements,
from z1 to xm, then Eqs. (B2) and (B3) can be used to find the
integral of the Fréchet derivative in the rectangular region
defined by [xi, xi+1], and [zi, zi+1]. To take into account the
whole half-space we have to consider (-∞, x1] and [xn, +∞)
in the horizontal coordinate and [zm, +∞) in the vertical
coordinate. The first case can be computed by noting
that the corresponding sine transform for the argument
x-b with b→±∞ is zero. The second case can be easily
computed, considering that the corresponding exponential
in equations (B2) or (B3) is null. In all cases, the actual
computations are performed using the digital filters for sine
and cosine transforms of Anderson (1975). Further details
are available in the thesis of Esparza (1991).
3 Series and parallel impedances and their derivatives
The series-parallel transformation
The horizontal components of the electromagnetic
field are linearly related through a second order impedance
tensor (e.g., Swift, 1967). That is,
(C.1)
where Zxx, Zxy, Zyx and Zyy are complex numbers. Ex, Zy are the
orthogonal components of the electric field and Hx, Hy are
the corresponding magnetic field components. Following
the work of Romo et al. (2005), we use an alternative
representation defined by the following transformation
(C.2)
where
(C.3)
(C.4)
(C.5)
(C.6)
If we consider a 2-D case in which the magnetotelluric
responses are solely characterized by ZTM = Zxy and ZTE =
4
u
+
λ
.
|
)
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)
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
x
δσ
= ∫ E (A.15)
.
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)
4
.
)
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)
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
.
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)
5
11
g
x
E
δ
11
g
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
(
)
'
(
|
'
|
0
2
11
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
x
g
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1, 1
' '
i i j j
x x x z z z
+ +
≤ ≤ ≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TE
e
e
i
u
u
i
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TM
e
e
i
u
u
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
TE
I
TM
I
|
| a
σ
1
x
m
x
1
z
m
z
[ ]
1
, +
i
i x
x
[ ]
1
, +
i
i z
z
]
,
( 1
x
−∞
)
,
[ +∞
n
x
)
,
[ +∞
m
z
,
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
y
x
yy
yx
xy
xx
y
x
H
H
Z
Z
Z
Z
E
E
(C.1)
xx
Z
xy
Z
yx
Z yy
Z
x
E
y
E
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
)
'
(
|
'
|
0
2
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1
'
j j
z z +
≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
e
e
i
u
u
i
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
e
e
i
u
u
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
,
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
y
x
yy
yx
xy
xx
y
x
H
H
Z
Z
Z
Z
E
E
(C.1)
11
g
x
E
δ
11
g
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
(
)
'
(
|
'
|
0
2
11
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
x
g
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1, 1
' '
i i j j
x x x z z z
+ +
≤ ≤ ≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TE
e
e
i
u
u
i
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TM
e
e
i
u
u
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
TE
I
TM
I
|
| a
σ
1
x
m
x
1
z
m
z
[ ]
1
, +
i
i x
x
[ ]
1
, +
i
i z
z
]
,
( 1
x
−∞
)
,
[ +∞
n
x
)
,
[ +∞
m
z
,
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
y
x
yy
yx
xy
xx
y
x
H
H
Z
Z
Z
Z
E
E
(C.1)
xx
Z
xy
Z
yx
Z yy
Z
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
(
)
'
(
|
'
|
0
2
11
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
x
g
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1, 1
' '
i j j
x x z z z
+ +
≤ ≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TE
e
e
i
u
u
i
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TM
e
e
i
u
u
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
|
]
1
+
i
]
1
+
i
]
, 1
x
)
∞
)
+∞
11
g
x
E
δ
11
g
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
(
)
'
(
|
'
|
0
2
11
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
x
g
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1, 1
' '
i i j j
x x x z z z
+ +
≤ ≤ ≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TE
e
e
i
u
u
i
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TM
e
e
i
u
u
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
TE
I
TM
I
|
| a
σ
1
x
m
x
1
z
m
z
[ ]
1
, +
i
i x
x
[ ]
1
, +
i
i z
z
]
,
( 1
x
−∞
)
,
[ +∞
n
x
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
)
'
(
|
'
|
0
2
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1
'
j j
z z +
≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
e
e
i
u
u
i
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
e
e
i
u
u
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
,
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
y
x
yy
yx
xy
xx
y
x
H
H
Z
Z
Z
Z
E
E
(C.1)
11
g
x
E
δ
11
g
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
(
)
'
(
|
'
|
0
2
11
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
x
g
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1, 1
' '
i i j j
x x x z z z
+ +
≤ ≤ ≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TE
e
e
i
u
u
i
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TM
e
e
i
u
u
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
TE
I
TM
I
|
| a
σ
1
x
m
x
1
z
m
z
[ ]
, x
x
5
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
;
,
(
)
'
(
|
'
|
0
2
11
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
x
g
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
G
z
i
u
TM
(A.20)
1, 1
' '
i i j j
x x x z z z
+ +
≤ ≤ ≤ ≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
)(
(
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TE
e
e
i
u
u
i
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
π
=
+
−
+
−
∞
+
∫
]
[
)
2
(
2 )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
TM
e
e
i
u
u
I
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
TE
I
TM
I
|
| a
σ
1
x
m
x
1
z
m
z
[ ]
1
, +
i
i x
x
[ ]
1
, +
i
i z
z
]
,
( 1
x
−∞
)
,
[ +∞
n
x
)
,
[ +∞
m
z
,
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
y
x
yy
yx
xy
xx
y
x
H
H
Z
Z
Z
Z
E
E
(C.1)
xx
Z
xy
Z
yx
Z yy
Z
x
E
y
E
x
H
y
H
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
TM xy
Z Z
=
TE yx
Z Z
=
S
Z
P
Z
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
a
ρ
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
S
a
ρ
),
(
1
TE
a
TM
a
S
a
ρ
ρ
ρ ∂
+
∂
=
∂
(C.10)
x
H
y
H
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
TM xy
Z Z
=
TE yx
Z Z
=
S
Z
P
Z
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
a
ρ
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
S
a
ρ
x
H
y
H
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
TM xy
Z Z
=
TE yx
Z Z
=
S
Z
P
Z
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
a
ρ
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
S
x
H
y
H
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
TM xy
Z Z
=
TE yx
Z Z
=
S
Z
P
Z
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
a
ρ
),
(
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
x
H
y
H
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
TM xy
Z Z
=
TE yx
Z Z
=
S
Z
P
Z
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
a
ρ
1
4
].
)
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)
'.
'
)
'
,
'
(
)
'
,
'
(
)
,
,
(
)
,
( 13
12
11
dz
dx
z
x
z
x
g
g
g
z
x
dE x
δσ
= ∫ E (A.15)
.
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)
)
, z
E
.
)]
'
(
cos[
]
[
2
1
)
'
,
'
)
'
(
|
'
|
0
2
λ
−
λ
+
πσδ
−
=
+
−
−
−
∞
∫
d
x
x
e
e
u
z
x
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
2
Re[
'
)
2
(
0
2
λ
−
λ
πδ
=
+
−
∞
∫
d
x
x
ue
z
i
u
(A.20)
1
' j
z +
≤
'.
'
)
'
,
'
,
(
2
1
1
dz
dx
z
x
x
G
I
TE
x
x
z
z
TE
i
i
j
j
δ
= ∫
∫
+
+
(B.1)
×
−
+
λ
+
λ
+
λ
+
λ +
−
+
−
∞
+
]
[
)
2
)(
(
)
2
( )
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
e
e
i
u
u
i
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
λ
+
−
+
−
∞
+
∫
]
[
)
2
(
)
2
(
)
2
(
0
1 j
j
z
i
u
z
i
u
e
e
i
u
u
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
3
.
)]
'
(
cos[
]
[
)
'
(
|
'
|
λ
−
λ
+
+
−
−
−
d
x
x
e
e
z
z
u
z
z
u
(A.19)
].
)]
'
(
cos[
'
)
2
λ
−
λ
+
d
x
x
z
i
(A.20)
'.
'
)
'
,
'
,
( dz
dx
z
x
x
G TE
(B.1)
×
−
+
−
+
− +
]
[
)
)
2
(
)
2
( 1 j
j
z
i
u
z
i
u
e
e
.
)]]
(
sin[
)]
(
[sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
−
+ +
]
)
2
(
)
2 1 j
j
z
i
u
z
i
e
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)
.
)]
'
(
cos[
]
)
'
(
|
'
|
λ
−
λ
+
+
−
−
d
x
x
e
z
z
u
z
z
(A.19)
].
)]
'
(
cos[
'
λ
−
λ d
x
x
z
(A.20)
'.
'
)
'
,
'
,
( dz
dx
z
x
x (B.1)
×
−
+
−
+
− +
]
)
2
(
)
2
( 1 j
j
z
i
u
z
i
u
e
.
)]]
(
sin[
)]
(
sin[ 1 λ
λ
λ d
x
x
x
x i
i
−
−
− + (B.2)
×
−
+
−
+
]
)
2
(
1 j
j
z
i
u
z
e
.
)]]
(
sin[
)]
(
[sin[ 1 λ
−
λ
−
−
λ + d
x
x
x
x i
i (B.3)