Geoscience Reference
In-Depth Information
underlying physics, the shape and patterns of rainfall intensities, viewed in a storm-scale
field of view, might be drastically dissimilar. However, the small-scale patterns of rainfall
when viewed over smaller windows might be repetitive and ''similar'' within different
regions of the same storm or within different storms. Therefore, the central idea is to
(a) collect a representative set of coincidentally observed LR and HR rainfall fields, with
some similarities in their underlying physics; (b) zoom down into small-scale patterns
(patches) of the given LR rainfall field; (c) for each patch, find few but very similar LR
patches in the collected data base; (d) for those similar LR patches, obtain the corre-
sponding HR patches in the data base and then reconstruct the HR counterparts of the LR
patch of interest based on an optimality criterion; and (e) repeat this procedure for all
possible patches and obtain a HR estimate for the observed LR rainfall field.
To be more specific, let us consider that the representative training set of N coincidental
pairs of LR and HR rainfall fields are denoted by
Z
l
N
and
Z
i
h
N
i
¼
1
, respectively. As
previously explained, for each patch y
l
of the given LR rainfall field, we need to find a few
very similar patches in
Z
l
N
i
¼
1
i
¼
1
, where similarity is defined in terms of localized rainfall
fluctuations and not in the mean values of the rainfall patches. To this end, all of the LR
fields are projected (i.e.,
Z
l
!Z
i
h
0
) onto a redundant orthogonal basis (called feature
space) to capture the rainfall local fluctuations including horizontal and vertical edges (i.e.,
zonal and meridional) and curvatures. This was performed by Ebtehaj et al. (
2012
) via an
undecimated orthogonal Haar wavelet, which basically performs a high-pass filtering in
each direction using first- and second-order differencing. Then, all of the constituent
patches of the transformed LR fields in the data base were extracted, vectorized in a fixed
order, and then stored as columns of a matrix W, the so-called empirical LR-dictionary.
Clearly, for each coincidental pair
Z
l
; Z
i
h
, a set of ''residual fields'' can be formed by
subtracting the LR fields from their HR counterparts via
R
i
h
¼Z
i
h
Q
Z
l
, where Q is a
readily available interpolation operator (e.g., a nearest-neighbor or bilinear, bicubic
interpolator). Notice that, these residual fields contain the rainfall variability and high-
frequency (fine spatial-scale) components that are not captured by the LR sensor and need
to be recovered. Therefore, all of the constituent patches r
h
of the residual fields can also be
collected, vectorized in a fixed order, and then stored in the columns of a matrix U, the so-
called HR-dictionary. Note that, by the explained construction, the empirical LR and HR
dictionaries share the same number of columns while there is a one-to-one correspondence
between them. In other words, while the columns of the W contain LR rainfall features, the
columns of the U contain the corresponding HR residuals, needed for the reconstruction of
the HR field.
The premise is that the local variability of any LR patch y
l
, denoted by y
l
, in any storm
can be well approximated by a linear combination of the elements of the LR dictionary as
follows:
y
l
¼
Wc
þ
v
;
ð
12
Þ
where c is the vector of representation coefficients in the LR dictionary and v
N
0
;
R
ð
Þ
denotes the estimation error that can be well explained by a Gaussian density.
By analyzing a sample of 100 storms over Texas, it was documented by Ebtehaj et al.
(
2012
) that the vector of representation coefficients c in the LR dictionary is very sparse. In
other words, any desired local rainfall variability in the given LR field can be approximated
by a linear combination of only a few columns of the LR empirical dictionary (of the order
