Geoscience Reference
In-Depth Information
f(κ
n
)
1
x
+
/
2
e
iκ
n
x
dx
N
N
sin
(κ
n
/
2
)
(κ
n
/
2
)
f(κ
n
)e
iκ
n
x
.
(6.3)
=
=
x
−
/
2
n
=−
N
n
=−
N
If we write this as
N
N
f
r
(x)
f
r
(κ
n
)e
iκ
n
x
f(κ
n
)T (κ
n
)e
iκ
n
x
,
=
=
(6.4)
n
=−
N
n
=−
N
we see that the
amplitude transfer function T(κ
n
)
for the running-mean operator is
f
r
(κ
n
)
sin
(κ
n
/
2
)
(κ
n
/
2
)
T(κ
n
)
=
f(κ
n
)
=
,
(6.5)
which is plotted in
Figure 6.1
.
Since sin
x/x
0, the averaging min-
imally affects the Fourier coefficients of wavelength large compared to
(those
with
κ
n
→
1as
x
→
1). The attenuation begins at
κ
n
∼
1 and is severe for
κ
n
1.
6.2.2 The generalization to spatial filtering
We introduced in
Chapter 3
a more general representation of space averaging (or
spatial
filtering
, as it is often called in the LES literature). In one dimension this
generalizes the local average of
Eq. (6.1)
to
∞
f
r
(x)
x
)f (x
)dx
,
=
G(x
−
(6.6)
−∞
with
G
called the
filter function
. This integral
is called the
convolution
of
f
and
G
.
Figure 6.1 The filter function,
Eq. (6.9)
, and transfer function,
Eq. (6.5)
,ofa
one-dimensional running-mean filter.
