Geoscience Reference
In-Depth Information
Differentiating (15.71) with respect to 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 1 )
=
φ 11 2 3 ,
(15.77)
−∞
and using the isotropic form (15.55) for φ ij gives
4 πκ 4 κ 2 +
κ 3 2 3 .
E(κ)
F 11 1 )
=
(15.78)
−∞
Figure 15.2 shows the geometry of this integral. Since 2 3
=
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 1 )
=
(15.79)
κ 1
 
Search WWH ::




Custom Search