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Consequently, by taking the Fourier transform of (
5.102
) and using the fact that
k
2
F
FT
{−
F
(
x
)
;
k
}=
(
k
),
(5.107)
we obtain the quantity
k
2
α/
2
Y
k
2
2
α/
2
Y
)
=
ξ(
FT
−
+
λ
(
x
)
;
=
+
λ
(
k
k
).
(5.108)
Therefore the Fourier transform of the solution of (
5.102
)is
ξ(
k
)
Y
(
k
)
=
)
α/
2
.
(5.109)
k
2
2
(
+
λ
Thesolutionto(
5.102
) can be written, using the inverse Fourier transform, as
ξ(
e
−
i
k
·
x
1
k
)
)
α/
2
d
2
k
Y
(
x
)
=
(5.110)
2
(
k
2
+
λ
2
(
2
π)
R
2
or explicitly taking the inverse Fourier transform in (
5.110
)[
12
] allows us to write the
spatial convolution integral
x
x
)
1
−
α/
2
λ
K
1
−
α/
2
(λ
−
x
)
d
2
x
,
Y
(
x
)
=
ξ(
(5.111)
2
α/
2
π(α/
2
)
1
−
α/
2
|
−
x
|
x
R
2
where
K
ν
(
z
)
is the modified Bessel function of the second kind in the Euclidean norm
x
2
|
|
=
y
2
x
+
,λ>
0 is a scale parameter controlling the spatial range of the kernel
and
2 is the parameter determining the level of smoothness of the statis-
tical fluctuations in the solution. Using the lowest-order approximation to the Bessel
function,
ν
=
1
−
α/
2
z
e
−
z
K
ν
(
z
)
∼
,
(5.112)
yields
e
−
λ
|
x
|
λ
(
1
−
α)/
2
2
(
1
+
α)/
2
√
π(α/
x
−
x
)
d
2
x
,
Y
(
x
)
=
x
|
(
3
−
α)/
2
ξ(
(5.113)
2
)
|
x
−
2
R
indicating an exponentially truncated inverse power-law potential that attenuates the
influence of remote random variations on a given location. It is clear that (
5.113
) can be
expressed in terms of a fractional integral.
The power spectrum for the spatial process (
5.113
) is given by the Fourier transform
of the autocorrelation function
d
2
xe
i
k
·
x
C
S
(
k
)
=
(
x
),
(5.114)
R
2
which, using the solution (
5.111
) and the statistics for the fluctuations (
5.104
), yields
1
1
S
(
k
)
=
k
2
2
α
.
(5.115)
2
(
2
π)
+
λ
Equation (
5.115
) is the power spectrum of a hyperbolic spatial response to the Gaussian
random field and it is evidently an inverse power law for
k
λ.
This is the kind of