Downscaling satellite precipitation with emphasis on extremes : A 1 Variational ` 1 - norm regularization in the derivative domain

The increasing availability of precipitation observations from space, e.g., from the Tropical Rain7 fall Measuring Mission (TRMM) and the forthcoming Global Precipitation Measuring (GPM) mission, has 8 fueled renewed interest in developing frameworks for downscaling and multi-sensor data fusion that can 9 handle large data sets in computationally efficient ways while optimally reproducing desired properties of 10 the underlying rainfall fields. Of special interest is the reproduction of extreme precipitation intensities and 11 gradients, as these are directly relevant to hazard prediction. In this paper, we present a new formalism 12 for downscaling satellite precipitation observations which explicitly allows for preservation of some key ge13 ometrical and statistical properties of spatial precipitation. These include sharp intensity gradients (due to 14 high intensity regions embedded within lower intensity areas), coherent spatial structures (due to regions of 15 slowly varying rainfall), and thicker than Gaussian tails of precipitation gradients and intensities. Specifi16 cally, we pose the downscaling problem as a discrete inverse problem and solve it via a regularized variational 17 approach (VarD) where the regularization term is selected to impose the desired smoothness in the solution 18 while allowing for some steep gradients (called `1-norm or total variation regularization). We demonstrate 19 the duality between this geometrically-inspired solution and its Bayesian statistical interpretation which is 20 equivalent to assuming a Laplace prior for the precipitation intensities in the derivative (wavelet) space. 21 When the observation operator is not known, we discuss the effect of its misspecification and explore a pre22 viously proposed dictionary-based sparse inverse downscaling (SPaD) methodology to indirectly learn the 23 observation operator from a data base of coincidental high and low-resolution observations. The proposed 24 method and ideas are illustrated in case studies featuring the downscaling of a hurricane precipitation field. 25