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methods, while it can be shown that the solution of problem (11) is zero for all
k
L
T
H
T
R
1
y
1
.
As discussed before, VarD assumes the downgrading operator H to be known. In our
case, we used as H the ''true'' operator, i.e., the same operator we used to coarse grain the
HR (1 9 1 km) reflectivity field to the LR (8 9 8 km) one. It is observed that the VarD
downscaled field has a smoother appearance than the original field (it does not have the
1 9 1 km pixelized appearance of the original HR field), which is not unexpected given
that the
'
1
-regularization promotes smoothness in the solution while allowing for some
steep gradients as demonstrated in the illustrative example of Fig.
4
. A one-dimensional
cross section shown in Fig.
5
e confirms this observation and shows that the downscaled
field is much closer to the true field compared to the LR field.
Suppose now that the true filter H is not known and only the LR field is given without
guidance as to what ''filtering'' the sensor did to the HR field to return the LR observations.
As discussed in the previous section, and in Ebtehaj et al. (
2012
), we demonstrated that this
filter can be ''learned'' implicitly and locally using coincidental high- and low-resolution
images available for a number of similar storms. In that study, a sample of 100 HR summer
storms over Texas was used to construct a set of coincidental LR storms (using again a
simple box averaging and a downsampling operator). This hundred storm sample was then
used to compute the LR and HR dictionaries, which formed the basis of the SPaD method
as explained in the previous section. This same dictionary was used herein to recover the
1 9 1 km HR rainfall field of the Claudette storm from 8 9 8 km observations. The
results are shown in Fig.
5
d.
In general, it is expected that the SPaD method will outperform the VarD method when
the operator H is not known at all or is locally varying, due, for example, to instrument
range effects or cloud interference or different performance of an instrument in low- versus
high-resolution rainfall intensities. However, it is noted that, since in our data base the LR
and HR fields relate to each other with a simple box averaging operator H (by construc-
tion), we expect that the dictionary-learning SPaD downscaling will perform comparably
to the VarD method. Extra information in SPaD will be gained by the localized nature of
the estimation methodology, which might reproduce extra high-frequency (small-scale)
features, obtained from the available dictionaries that may not be recovered in the VarD
approach.
To more quantitatively compare the two downscaled fields to the true underlying HR
field and to each other, we compare in Fig.
6
the PDF of the derivatives in the horizontal
direction in terms of their q-q plot (quantiles of the variable of interest vs. standard normal
quantiles). We observe that both methods are able to reproduce the heavy tails of the PDF
of the precipitation gradients, which are much thicker than those of the Gaussian PDF, and
thus, both methods are able to reproduce high gradients in the HR recovered field. VarD is
seen to slightly outperform SPaD in reproducing high positive gradients, not surprisingly
since, in VarD, the H operator was customized to this specific storm, while, in SPaD,
information from a suite of other storms was also used.
Turning our attention to the preservation of the statistics of the precipitation field itself,
we show in Fig.
7
the comparison of the PDFs of the LR rainfall field with that of the true
HR field and the downscaled fields. We recall that although the preservation of the thicker-
than-Gaussian (Laplace) tails in the PDF of precipitation intensity gradients is explicitly
incorporated in the
'
1
-norm VarD downscaling methodology, no explicit preservation of
the extreme rainfall intensities themselves is accounted for. However, it is clear from
Fig.
7
that VarD performs satisfactorily in reproducing extreme rainfall intensities in the
