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parameter
h
. The peak of the spectrum is determined to be the fractal dimension, as
it should be. Here again we have an indication that the interstride-interval time series
describes a multifractal process. But we stress that we are only using the qualitative
properties of the spectrum for
q
<
0, due to the sensitivity of the numerical method to
weak singularities.
5.4
Spatial heterogeneity
5.4.1
Random fields
Random fields play a central role in the modeling of complex geophysical webs because
of the natural heterogeneity of oil deposits, hydrology, earthquakes, meteorology, agri-
culture and even climate change. In a geophysical context the smoothness of the random
field can be characterized in terms of the fractal dimension obtained from the scaling
behavior of the spatial autocorrelation. Lim and Teo [
12
] have considered the case in
which the spatial correlation strength has a hyperbolic form for large time lags. We
discuss their work and the complex phenomena it was designed to describe in this
section.
Let us consider a two-dimensional plane
x
=
(
x
,
y
)
on which a random function
Y
(
x
)
is defined as the solution to the fractional stochastic differential equation
2
α/
2
−
+
λ
Y
(
x
)
=
ξ(
x
),
(5.102)
where
α>
0
,
the Laplace operator in two dimensions is
2
2
≡
∂
x
2
+
∂
(5.103)
∂
∂
y
2
and the random field is zero-centered, delta-correlated and of unit strength with two-
dimensional Gaussian statistics
ξ(
x
)
=
x
).
ξ(
x
)
=
0
,
x
)ξ(
2
δ(
x
−
(5.104)
It probably bears stressing that what is discussed with regard to the solution to (
5.102
)
is readily generalized to dimensions of arbitrary size, but the notation is somewhat more
compact in two dimensions.
The solutions to the equations of motion in time were found using Laplace trans-
forms. The equations in space are typically dealt with by implementing Fourier
transforms, so we introduce the notation
F
e
i
k
·
x
F
d
2
x
(
k
)
=
FT
{
F
(
x
)
;
k
}≡
(
x
)
,
(5.105)
2
R
with the inverse Fourier transform denoted by
)
=
FT
−
1
F
x
≡
1
e
−
i
k
·
x
F
d
2
k
F
(
x
(
k
)
;
(
k
)
.
(5.106)
2
(
2
π)
2
R
