Geoscience Reference
In-Depth Information
of 3-5 elements). To impose this sparsity (called ''group sparsity'') in solving (
12
) for c,
the solution needs to be constrained via an
'
1
-norm regularization as follows:
:
1
2
2
R
1
þ
k
kk
1
y
l
Wc
c
¼
argmin
c
ð
13
Þ
Using the representation coefficients obtained from (
13
), one can recover the corre-
sponding residual fields (the details missed by the LR sensor) as follows:
r
¼
Uc
: ð
14
Þ
Having the estimated residual fields, the HR patch can be obtained as x
¼
Qy
l
þ
r.
Applying the same estimation methodology for all of the patches of the given LR rainfall
field, we can recover the entire HR rainfall field (see Ebtehaj et al.
2012
). The most
important implication of the above framework is that we characterized the pair of
ð
W
;
U
Þ
empirically without explicit access to the structure of the downgrading operator H, which
is the main advantage of this dictionary-based rainfall downscaling method versus the
previously explained approach. Since advantage was taken of the rainfall group sparsity
(and also implicitly of the sparsity of the precipitation fields themselves), the dictionary-
based downscaling methodology was termed SPaD.
4 Results from a Case Study
To demonstrate the proposed downscaling methodology, we have chosen a specific tropical
storm, hurricane Claudette, which occurred in July 2003. Claudette began as a tropical
wave in the eastern Caribbean on July 8, 2003 and moved quickly westward to the Gulf of
Mexico. It remained a tropical storm until just before making landfall in Port O'Conner,
Texas, when it quickly strengthened to a category 1 hurricane. Although Claudette pro-
duced moderate rainfall across southern Texas, peaking at approximately 6.5 inches
(165 mm), it maintained a tropical storm intensity for over 24 h after landfall with winds
gusting to 83 mph (134 km/h) at Victoria Regional Airport, Texas. The storm caused
excessive beach erosion and damages estimated at 180 million dollars. For this storm, we
have available data from a NEXRAD station in Houston, Texas, for which a snapshot at
11:51:00 (UTC) on July 15, 2003 is shown in Fig.
2
.
The issues we want to examine here are the following: (1) the ability of the proposed
variational downscaling (VarD) scheme to reproduce the steep gradients in precipitation
intensities as evidenced by reproducing the tails of the PDF of intensity gradients; (2) the
effect of an unknown kernel (smoothing and downsampling operation imposed on the true
HR field by a sensor) on the downscaling scheme performance using the proposed
methodology; (3) a comparison of the VarD method with a local dictionary-based meth-
odology based on sparse representation (SPaD) as discussed in the previous section; and
(4) insights into the ability of the proposed VarD methodology and SPaD to reproduce not
only the extreme gradients but also the extreme rainfall intensities, i.e., the tails of the
rainfall intensity probability distribution functions (PDFs).
The original HR data at 1 9 1 km (Fig.
5
a) were downgraded to 8 9 8km LR
observations via a coarse-graining filter consisting of a simple box averaging of size 8 9 8
followed by downsampling with a factor of 8 (i.e., keeping one observation per box of
8 9 8 km). The resulting LR field is shown in Fig.
5
b and is considered to be the field that
would be available to us from a satellite sensor. Figure
5
c, d shows the results of
