On variational downscaling, fusion, and assimilation of hydrometeorological states: A unified framework via regularization

Improved estimation of hydro-meteorological states from downsampled observations and background model forecasts in a noisy environment, has been a subject of growing research in the past decades. Here, we introduce a unified variational framework that ties together the problems of downscaling, data fusion and data assimilation as ill-posed inverse problems. This framework seeks solutions beyond the classic least squares estimation paradigms by imposing a proper regularization, expressed as a constraint consistent with the degree of smoothness and/or probabilistic structure of the underlying state. We review relevant smoothing norm regularization methods in derivative space and extend classic formulations of the aforementioned problems with particular emphasis on land surface hydro-meteorological applications. Our results demonstrate that proper regularization of downscaling, data fusion, and data assimilation problems can lead to more accurate and stable recovery of the underlying non-Gaussian state of interest with improved performance in capturing isolated and jump singularities. In particular, we show that Huber regularization in the derivative space offers advantages, compared to the classic solution and Tikhonov regularization, for spatial downscaling and fusion of non-Gaussian multi-sensor precipitation data. Furthermore, we explore the use of Huber regularization in a variational data assimilation experiment while the initial state of interest exhibits jump discontinuities and non-Gaussian probabilistic structure. To this end, we focus on the heat equation motivated by its fundamental application in the study of land surface heat and mass fluxes. D R A F T May 9, 2013, 6:31pm D R A F T EBTEHAJ & FOUFOULA-GEORGIOU: REGULARIZED DOWNSCALING, FUSION AND ASSIMILATION X 3