Geoscience Reference
In-Depth Information
Differentiating
(15.71)
with respect to
iκ
yields
+∞
e
−
iκr
dr.
∂iκF
11
∂iκ
∂κF
11
∂κ
1
2
π
u
2
r
∂f
∂r
=
=
−
(15.72)
−∞
The inverse of
(15.72)
is
+∞
e
iκr
∂κF
11
(κ)
∂κ
−
u
2
r
∂f
∂r
=
dκ.
(15.73)
−∞
Thus, the physical-space expression
(15.65)
r
2
∂f
∂r
g(r)
=
f
+
transforms to
1
2
∂
F
11
=
F
11
−
∂κ
κF
11
.
(15.74)
We can rewrite this as
F
11
−
.
κ
∂F
11
∂κ
1
2
F
11
=
(15.75)
Since under isotropy the statistics are invariant to rotation and reflection of the
coordinate system, we also have
F
11
=
F
22
=
F
33
=
F
11
=
F
33
=
F
22
,
(15.76)
F
11
=
F
22
=
F
33
.
Finally, under isotropy we can relate these one-dimensional spectra to
E
.By
definition
∞
F
11
(κ
1
)
=
φ
11
dκ
2
dκ
3
,
(15.77)
−∞
and using the isotropic form
(15.55)
for
φ
ij
gives
∞
4
πκ
4
κ
2
+
κ
3
dκ
2
dκ
3
.
E(κ)
F
11
(κ
1
)
=
(15.78)
−∞
=
2
πκ dκ
and
κ
2
+
κ
3
κ
2
κ
1
,wehave
=
−
∞
2
κ
3
κ
2
κ
1
dκ.
E(κ)
F
11
(κ
1
)
=
−
(15.79)
κ
1