Geoscience Reference
In-Depth Information
equivalent to assuming a Laplace prior distribution for the precipitation intensities in the
derivative (wavelet) space. When the observation operator is not known, we discuss the
effect of its misspecification and explore a previously proposed dictionary-based sparse
inverse downscaling methodology to indirectly learn the observation operator from a data
base of coincidental high- and low-resolution observations. The proposed method and ideas
are illustrated in case studies featuring the downscaling of a hurricane precipitation field.
Keywords Sparsity
Inverse problems
'
1
-norm regularization
Non-smooth
convex optimization
Generalized Gaussian density
Extremes
Hurricanes
1 Introduction
Precipitation is one of the key components of the water cycle and, as such, it has been the
subject of intense research in the atmospheric and hydrologic sciences over the past
decades. While it still remains the most difficult variable to accurately predict in numerical
weather and climate models, its statistical space-time structure at multiple scales has been
extensively studied using several approaches (e.g., Lovejoy and Mandelbrot
1985
; Lovejoy
and Schertzer
1990
; Kumar and Foufoula-Georgiou
1993a
,
b
; Deidda
2000
; Harris et al.
2001
; Venugopal et al.
2006a
,
b
; Badas et al.
2006
). These studies have documented a
considerable variability spread over a large range of space and timescales and an orga-
nization that manifests itself in power law spectra and more complex self-similar structures
expressed via nonlinear scaling of higher-order statistical moments (e.g., Lovejoy and
Schertzer
1990
; Venugopal et al.
2006a
). Stochastic models of multi-scale rainfall vari-
ability have been proposed based on inverse wavelet transforms (Perica and Foufoula-
Georgiou
1996
),
multiplicative
cascades
(Deidda
2000
),
exponential
Langevin-type
models (Sapozhnikov and Foufoula-Georgiou
2007
), among others.
The small-scale variability of precipitation (of the order of a few kms in space and a few
minutes in time) is known to have important implications for accurate prediction of
hydrologic extremes especially over small basins (e.g., Rebora et al.
2006a
,
b
) and for the
prediction of the evolving larger-scale spatial organization of land-atmosphere fluxes in
coupled models (Nykanen et al.
2001
). This small-scale precipitation variability, however,
is not typically available in many regions of the world where coverage with high-resolution
ground radars is absent or in mountainous regions where spatial gaps are present due to
radar blockage. It is also missing from climate model predictions that are typically run at
low resolution over larger areas of the world. As a result, methods for downscaling pre-
cipitation to enhance the resolution of incomplete or low-resolution observations from
space or numerical weather/climate model outputs continue to present a challenge of both
theoretical and practical interests.
To date, multiple passive and active ground-based (i.e., gauges and radars) and
spaceborne sensors (i.e., geostationary, polar and quasi-equatorial orbiting satellites) exist
that overlappingly measure precipitation with different space-time resolutions and accu-
racies. Sparsely populated networks of rain gauges provide relatively accurate point
measurements of precipitation continuously over time, while ground-based radars detect
precipitation in fine enough spatiotemporal scales (e.g., *6 min at 1 9 1 km) but over
limited areal extents. The ground-based radar data are among the most accurate and high-
resolution estimates of spatial rainfall. However, this source of information is subject to
