Geoscience Reference
In-Depth Information
relates to the high-resolution (HR) state via a linear downgrading (e.g., a linear blurring
and/or downsampling
1
) operator H
2
m
n
as follows:
y
¼
Hx
þ
v
;
R
ð
9
Þ
where v
N
0
; ð Þ
is a zero-mean Gaussian error with covariance R. Due to the fact that
the dimension of y is less than that of x, the operator H is a rectangular matrix with more
columns than rows and thus solving problem (
9
) for x is an ill-posed inverse problem (an
under-determined system of equations with many solutions). As discussed above, we seek
to impose a proper regularization to make the inverse problem well posed.
Following the developments presented above in a continuous setting and replacing f
(1)
with a discrete approximation derivative operator L, the choice of the smoothing
'
2
-norm
regularization for S(x) becomes
L
k
2
while for the
'
1
-norm becomes
L
k
1
, where in
discrete space
kk
2
¼
R
i
¼
1
x
j
2
and
kk
1
¼
R
i
¼
1
x
j
.
Thus, the solution (HR state x) can be obtained by solving the following regularized
weighted least squares minimization problem:
;
1
2
k
R
1
þ
kS
ð
x
Þ
x
¼
argmin
x
y
Hx
ð
10
Þ
k
It is clear that the smaller the value of k, the more weight is given to fitting the
observations (often resulting in data over-fitting), while a large value of k puts more weight
into preserving the underlying properties of the state of interest x, such as large gradients.
The goal is to find a good balance between the two terms. Currently, no closed form
method exists for the selection of this regularization parameter and the balance has to be
obtained via a problem-specific statistical cross validation (e.g., Hansen
2010
). Note that
the problem in (
10
) with S
ð
x
Þ¼
L
k
1
is:
;
1
2
k
R
1
þ
k L
k
1
x
¼
argmin
x
k
y
Hx
ð
11
Þ
that is, a non-smooth convex optimization problem as the regularization term is non-
differentiable at the origin. As a result, the conventional iterative gradient methods do not
work and one has to use greedy methods (Mallat and Zhang
1993
) or apply the recently
developed non-smooth optimization algorithms such as the iterative shrinkage thresholding
method (Tibshirani
1996
), the basis pursuit method (Chen et al.
1998
,
2001
), the con-
strained quadratic programming (Figueiredo et al.
2007
), the proximal gradient-based
methods (Beck and Teboulle
2009
), or the interior point methods (Kim et al.
2007
). In this
work, we have adopted the method suggested by Figueiredo et al. (
2007
).
2.3 Geometrical Versus Statistical Interpretation of the
'
1
-Norm Regularized
Downscaling
As was discussed in the introduction, the motivation for introducing a new downscaling
framework lies in the desire to reproduce some geometrical but also some statistical
features of precipitation fields. Specifically, the question was posed as to how a down-
scaling scheme could be constructed that can reproduce both the abrupt localized gradients
1
Here, by downsampling, we mean to reduce the sampling rate of the rainfall observations by a factor
greater than one.
