Downscaling Satellite Precipitation with Emphasis on Extremes: A Variational ‘1-Norm Regularization in the Derivative Domain

The increasing availability of precipitation observations from space, e.g., from the Tropical Rainfall Measuring Mission (TRMM) and the forthcoming Global Precipitation Measuring (GPM) Mission, has fueled renewed interest in developing frameworks for downscaling and multi-sensor data fusion that can handle large data sets in computationally efficient ways while optimally reproducing desired properties of the underlying rainfall fields. Of special interest is the reproduction of extreme precipitation intensities and gradients, as these are directly relevant to hazard prediction. In this paper, we present a new formalism for downscaling satellite precipitation observations, which explicitly allows for the preservation of some key geometrical and statistical properties of spatial precipitation. These include sharp intensity gradients (due to high-intensity regions embedded within lower-intensity areas), coherent spatial structures (due to regions of slowly varying rainfall), and thicker-than-Gaussian tails of precipitation gradients and intensities. Specifically, we pose the downscaling problem as a discrete inverse problem and solve it via a regularized variational approach (variational downscaling) where the regularization term is selected to impose the desired smoothness in the solution while allowing for some steep gradients (called ‘1-norm or total variation regularization). We demonstrate the duality between this geometrically inspired solution and its Bayesian statistical interpretation, which is E. Foufoula-Georgiou (&) Saint Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] A. M. Ebtehaj Saint Anthony Falls Laboratory, Department of Civil Engineering, School of Mathematics, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] S. Q. Zhang A. Y. Hou NASA Goddard Space Flight Center, Greenbelt, MA, USA e-mail: [email protected] A. Y. Hou e-mail: [email protected] 123 Surv Geophys (2014) 35:765–783 DOI 10.1007/s10712-013-9264-9