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although not prescribed in our method, arises as an emergent property. The answer to this
question is not obvious. Our preliminary results (not reported herein) demonstrate that
statistical scaling indeed arises in both the
'
1
-norm variational downscaling (VarD) and the
SPaD schemes. However, the power law exponents (of the variance of the wavelet coef-
ficients as a function of scale) and the variance of the wavelet coefficients at the smallest
scale (similar to the analysis in Perica and Foufoula-Georgiou
1996
) seem to be lower than
those of the original fields. This might be due to the fact that, although our scheme is able
to accurately capture, much better than other statistical schemes, the magnitude of the
infrequent localized large gradients in precipitation fields, it might under-produce the
variability of the smaller gradients, reducing thus the overall variance. This is an issue that
is currently explored both from a theoretical perspective and via simulation, as in most
applications one is interested to preserve both the localized extremes but also the overall
variance of the smaller magnitude fluctuations.
The work presented herein falls within a larger research direction of using variational
regularization approaches or equivalently, Bayesian MAP estimators with heavy-tailed priors
in the derivative domain, for estimation problems in hydro-climatology, such as downscaling,
multi-sensor data fusion, retrieval, and data assimilation (see Ebtehaj and Foufoula-Georgiou
2013
). A relatively small number of abrupt gradients within the field of interest or heavy-
tailed PDFs in the derivative domain are associated with the notion of sparsity, that is, the fact
that, when the state is projected in a suitable basis, most of the projection coefficients are close
to zero and only a few coefficients carry most of the state energy. Estimation problems of
sparse states (posed in an inverse estimation setting or in a variational setting of minimizing a
functional) require the use of
'
1
-norm regularization, which results from imposing extra
constraints on the solution to enforce sparsity. Motivated by the need to preserve sharp
weather fronts in data assimilation of numerical weather prediction models, an
'
1
-norm
regularized variational data assimilation methodology was recently proposed by Freitag et al.
(
2012
) and demonstrated in a simple setting using the advection equation for the state evo-
lution dynamics. In Ebtehaj and Foufoula-Georgiou (
2013
), data assimilation in the presence
of extreme gradients in the state variable was further analyzed using as illustrative example
the advection-diffusion equation that forms the basis of many hydro-meteorological prob-
lems, such as those dealing with the estimation of surface heat fluxes based on the assimilation
of land surface temperature (e.g., see Bateni and Entekhabi
2012
). Application of these new
non-smooth variational methodologies in real data assimilation problems, and also in com-
bining data assimilation with downscaling of the state, is only in its infancy and is certain to
occupy the geophysical community in the years to come.
Acknowledgments This work has been mainly supported by a NASA-GPM award (NNX10AO12G), a
NASA Earth and Space Science Fellowship (NNX12AN45H), and a Doctoral Dissertation Fellowship of the
University of Minnesota to the second author. The insightful comments of one anonymous referee are also
gratefully acknowledged.
References
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