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(a)
(b)
0.5
0
Obs.
Fitted ( =0.85)
Gaussian
Laplace (
α
-2
0.4
α
=1)
-4
0.3
-6
0.2
-8
0.1
-10
0
-10
-5
0
5
10
-10
-5
0
5
10
c
h
c
h
Fig. 3 a Histogram of the derivatives in the horizontal direction of the hurricane snapshot shown in Fig.
2
.
The derivative coefficients are obtained by the Sobel operator that produces a second-order discrete
approximation of the field derivative. b Same histogram plotted on a log-probability scale showing the
empirical PDF (circles), the fitted generalized Gaussian PDF with parameter a = 0.85, the Gaussian PDF
(a = 2.0), and the Laplace density (a = 1.0) for comparison. Note that the assumption of a Laplace density
for the rainfall derivatives is theoretically consistent with the proposed
'
1
-norm variational downscaling
framework
regularization approach, which imposes constraints on the specific degree of smoothness
(regularity) of the precipitation fields. The proposed regularization is selected to allow the
preservation of large gradients while at the same time impose the desired smoothness on
the solution. The paper is structured as follows. In Sect.
2
, the need for regularization is
explained with special emphasis on a total variation regularization scheme (
'
1
-norm in the
derivative space) in order to reproduce steep gradients and to preserve the heavy-tailed
structure of rainfall. In this Section, the statistical interpretation of the variational
'
1
-norm
regularization is also explained. In particular, it is elucidated that the downscaled rainfall
fields obtained via
'
1
-norm regularization in the derivative domain is equivalent to the
Bayesian maximum a posteriori (MAP) estimate with a Laplace prior distribution in the
precipitation derivatives, a special case of the generalized Gaussian distribution p
ð
x
Þ/
exp
ð
k
j
x
j
a
Þ
with a = 1 (Ebtehaj and Foufoula-Georgiou
2011
). Section
3
presents insights
into the problem of an unknown downgrading observation operator or kernel that ''con-
verts'' the high-resolution rainfall to the lower-resolution observations and discusses an
alternative methodology, dictionary-based sparse precipitation downscaling (SPaD),
developed in (Ebtehaj et al.
2012
). In Sect.
4
, we present a detailed implementation of our
variational downscaling (VarD) methodology in a tropical (hurricane) storm and compare
the results of VarD with those of the SPaD method. Finally, concluding remarks and
directions for future research are presented in Sect.
5
.
2 Precipitation Downscaling as a Regularized Inverse Problem
2.1 Basic Concepts in the Continuous Space
Consider the true state (or signal) f(t) that is not known but is observed indirectly via a
measuring device, which imposes a smoothing on the original state and returns the
observation g(s). Let f(t) and g(s) relate via the following linear transformation:
