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and also the characteristic probability distribution of the precipitation intensity gradients
such as that displayed in Fig. 3 a.
It can be shown that the solution of ( 10 ) obtained via ' 2 -norm regularization (i.e.,
S ð x Þ¼ L k 2 ) is equivalent to the Bayesian maximum a posteriori (MAP) estimator where
the transformed variable Lx is well explained by a Gaussian distribution. On the other
hand, considering S ð x Þ¼ L k 1 , the ' 1 -norm regularized solution of ( 10 ), i.e., the solution
of Eq. ( 11 ), is the MAP estimator where Lx is well explained by the multivariate Laplace
distribution (the generalized Gaussian family with a = 1). In other words, the ' 1 -regu-
larization implicitly assumes that the probability of Lx goes as exp k L k 1
(Lewicki
and Sejnowski 2000 ; Ebtehaj and Foufoula-Georgiou 2013 ). We note that for the storm of
Fig. 2 , the estimated tail parameter a is 0.85 (see Fig. 3 ), which denotes that the pdf of Lx
goes as exp k L k a , where kk a ¼ R i ¼ 1 x j a . This value of a implies that the Laplace
distribution (a = 1) is only an approximation of the true distribution of the analyzed
precipitation (see Fig. 3 b for comparison), making thus the proposed ' 1 -norm regulariza-
tion solution only an approximate solution in a statistical sense. Finding a solution via
regularized inverse estimation that satisfies a prior probability for (Lx) with a \ 1 requires
solving a non-convex optimization, which may suffer from local minima and may be hard
to solve for large-scale problems. For this reason, we limit our discussion to the ' 1 -
regularization recognizing the slight sub-optimality of the solution for precipitation
applications but also its superiority relative to the Gaussian assumption about the rainfall
derivatives.
3 Working with an Unknown Downgrading Operator (H)
In the above formulation of the downscaling problem as an inverse problem, the down-
grading operator H is assumed to be linear and known a priori. A mathematically con-
venient form for the downgrading operator is to assume that it can be represented via a
linear convolution followed by downsampling. In other words, one may assume that the
low-resolution (LR) observation is obtained by applying an overlapping box (weighted)
averaging over the HR field and keeping one observation only, typically at the center, per
averaging box (downsampling). However, the downgrading operator is not generally
known in practice and its characterization might be sensor-dependent. Also often, this
operator is highly nonlinear (e.g., the relationship between the radiometer-observed
brightness temperature and the precipitation reflectivity observed by the radar) and its
linearization may introduce large estimation errors. This nonlinearity may also pose severe
challenges from the optimization point of view and may give rise to a hard non-convex
problem with many local minima (Bertsekas 1999 ).
To deal with the problem of an unknown downgrading operator, Ebtehaj et al. ( 2012 )
proposed a dictionary-learning-based methodology that allows to implicitly incorporate the
downgrading effect via statistical learning without the need to explicitly characterize the
downgrading operator. In this methodology, the downgrading operator is being learned via
a dictionary of coincidental HR and LR observations (e.g., in practice, TRMM-PR, and
ground-based NEXRAD or TMI and NEXRAD). The methodology is explained in detail
by Ebtehaj et al. ( 2012 ) and is only briefly summarized herein.
In simple terms, the idea is to reconstruct a HR counterpart of the LR rainfall field based
on learning from a representative data base of previously observed coincidental LR and HR
rainfall fields (e.g., TRMM-PR and NEXRAD observations). As is evident, due to different
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