Geoscience Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
h
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Fig. 4 A piecewise linear function f(t) with a slope f
(1)
= 1/h at the non-horizontal part. As it is easily
shown (see text), for this function, the
'
1
(total variation)-norm
1
is constant and independent of
f
ð
1
Þ
2
2
¼
1
=
h goes to infinity as h goes to zero (i.e., for a very steep gradient). As a
result, the
'
2
-norm solutions do not allow steep gradients, while the
'
1
-norm does
f
ð
1
Þ
h while the
'
2
-norm
singularities in a quite different way than the
'
2
-norm of S(f). It is important to demonstrate
this point as it plays a key role in the proposed downscaling scheme.
Let us consider a piecewise linear function:
8
<
:
0
t\
2
0
;
ð
1
h
Þ
t
h
1
h
1
2
Þ
t
2
f
ð
t
Þ¼
;
ð
1
h
ð
1
þ
h
Þ
;
ð
6
Þ
2h
1
2
1
;
ð
1
þ
h
Þ
\t
1
as shown in Fig.
4
. It can be shown that the smoothing norms associated with the
'
1
and
'
2
-
norms of f
(1)
(t) satisfy:
1
¼
Z
1
0
dt
¼
Z
h
1
h
dt
¼
1
f
ð
1
Þ
f
ð
1
Þ
ð
t
Þ
ð
7
Þ
0
while
2
¼
Z
dt
¼
Z
h
0
1
h
2
dt
¼
1
1
2
2
f
ð
1
Þ
f
ð
1
Þ
ð
t
Þ
h
:
ð
8
Þ
1
is independent of the slope
of the middle part of f(t) while the smoothing
'
2
-norm is inversely proportional to h and, as
such, it severely penalizes steep gradients (when h is small). In other words, the
'
2
-norm of
f
(1)
will not allow any steep gradients and will produce a very smooth solution. Clearly, this
is not desirable in solving an inverse problem associated with the reconstruction of small-
scale details in precipitation fields, such as in the hurricane storm shown in Fig.
2
.
0
It is observed that the TV smoothing norm S
TV
ð
f
Þ¼
f
ð
1
Þ
2.2 Discrete Representation
Writing Eq. (
1
) in a discrete form, the problem of downscaling amounts to estimating a
high-resolution (HR) state, denoted in an m-element vector as x
2
m
, from its low-
R
n
, where m
n. It is assumed that this LR counterpart
resolution (LR) counterpart y
2
R
