Geoscience Reference
In-Depth Information
Z
1
K
ð
s
;
t
Þ
f
ð
t
Þ
dt
¼
g
ð
s
Þ
0
;
t
1
;
ð
1
Þ
0
where K(s, t) is a known kernel, which downgrades the true state by damping its high-
resolution components and making it smoother. The problem of recovering f(t) knowing
the observation g(s) and the kernel K(s, t) is a well-studied inverse problem, known as the
Fredholm integral equation of the first kind. Inverse problems are by their nature ill-posed,
in the sense that they do not satisfy at least one of the following three conditions: (1)
existence of a solution, (2) uniqueness of the solution, and (3) stability in the solution, i.e.,
robustness to perturbations in the observation. It can be shown that the above inverse
problem is very sensitive to the observation noise, since high frequencies are amplified in
the inversion process (so-called inverse noise) and they can easily spoil and blow up the
solution (see Hansen
2010
). In this sense, even a small but high-frequency random per-
turbation in g(s) can lead to a very large perturbation in the estimate of f(t). This is relevant
to the problem of reconstructing small-scale features in precipitation fields (downscaling)
from low-resolution noisy data, when the noise can be of low magnitude but high fre-
quency, e.g., discontinuities in overlapping regions of different sensors or instrument noise.
Therefore, naturally, if we define the distance between the observations and the true
state by the following residual Euclidean norm:
R
ð
f
Þ¼
Z
1
0
2
K
ð
s
;
t
Þ
f
ð
t
Þ
dt
g
ð
s
Þ
;
ð
2
Þ
then minimizing R(f) alone does not guarantee a unique and stable solution of the inverse
problem. Rather, additional constraints have to be imposed to enforce some regularity (or
smoothness) of the solution and suppress some of the unwanted inverse noise components
leading to a unique and more stable solution. Let us denote by S(f) a smoothing norm,
which measures the desired regularity of f(t). Then, obtaining a unique and stable solution
to the inverse problem amounts to solving a variational minimization problem of the form
n
o
;
R
ð
f
Þ
2
þ
k
2
S
ð
f
Þ
f
ð
t
Þ¼
argmin
f
ð
3
Þ
The value of k (called the regularization parameter) is chosen as to provide a balance
between the weight given to fitting the observations, as measured by the magnitude of the
residual term R(f), and the degree of regularity of the solution measured by the smoothing
norm S(f). Common choices for S(f) are
'
2
-norms of the function f(t) or its derivatives, i.e.,
2
¼
Z
1
2
2
S
ð
f
Þ¼
f
ð
d
Þ
f
ð
d
Þ
ð
t
Þ
dt
;
d
¼
0
;
1
;
...
ð
4
Þ
0
where f
(d)
denotes the dth order derivative of f. Another smoothing norm of specific interest
in the present study is the
'
1
-norm of the gradient of f, that is,
S
TV
ð
f
Þ¼ kk
1
¼
Z
1
dt
;
f
ð
1
Þ
ð
t
Þ
ð
5
Þ
0
known as the Total Variation (TV) of the function f(t). Both the S(f) and S
TV
(f) norms yield
robust solutions with desired regularities but the S
TV
(f) penalizes local jumps and isolated
