Geoscience Reference
In-Depth Information
Appendix A
Let us expand fluctuating variables in the Fourier series in spatial harmonics
with wavenumbers
q
:
⎛
⎞
⎛
⎞
δσ
ik
∂
(
δσ
ik
)
/∂x
i
∂
2
δϕ/∂x
i
∂x
k
∂
(
δσ
ik
)
/∂x
i
∂δϕ/∂x
k
δσ
ik
(
q
)
iq
i
δσ
ik
(
q
)
−
⎝
⎠
⎝
⎠
=
q
q
i
q
k
δϕ
(
q
)
iq
i
δσ
ik
(
q
)
iq
k
δϕ
(
q
)
exp
i
(
qr
)
.
(10. A.1)
Substituting these relations into (10.22), we get
q
σ
ik
q
q
i
q
k
δϕ
(
q
)exp
i
(
qr
)
iq
i
δσ
ik
(
q
)exp
i
(
qr
)
e
k
+
+
q
p
q
i
p
k
δσ
ik
(
q
)
δϕ
(
p
)exp
i
(
q
+
p
)
r
+
δσ
ik
q
q
i
q
k
δϕ
(
q
)exp
i
(
qr
)=0
.
It can be rewritten as
+
p
(
q
i
−
iq
i
δσ
ik
e
k
+
q
i
q
k
δϕ
(
q
)
σ
ik
p
i
)
p
k
δσ
ik
(
q
−
p
)
δϕ
(
p
)
+
q
i
q
k
δσ
ik
(
q
−
p
)
δϕ
(
q
)=0
.
From this we deduce
B
k
(
q
)
δϕ
(
q
)=
i
q
i
q
m
e
k
,
(10. A.2)
σ
im
where
B
k
(
q
) satisfies the equation
B
k
q
q
)
q
l
q
i
δσ
il
(
q
−
B
k
(
q
)=
−
δσ
ik
(
q
)
q
i
−
(10. A.3)
σ
mn
q
m
q
n
q
=0
If we make substitution for
δσ
ik
and
∂δϕ/∂δx
k
from (10. A.1), then the rela-
tionship (10.20) between the total current and perturbations of conductivity
and potential becomes
ip
k
δϕ
(
p
)exp
i
pr
δσ
ik
(
q
)exp
i
(
qr
)
p
j
i
=
σ
ik
e
k
−
q
i
q
=
σ
ik
e
k
−
δσ
ik
(
−
q
)
δϕ
(
q
)
q
k
Because, by definition,
f
(
r
)
d
r
1
V
f
(
r
)
=
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