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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
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
 
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