An Analytical and Numerical Study of Optimal Channel Networks

We analyze the Optimal Channel Network model for river networks using both analytical and numerical approaches. This is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations. The fractal character of river networks is reflected in the power law behaviour of various quantities characterising the morphology of the basin. In the context of a finite size scaling Ansatz, the exponents describing the power law behaviour are calculated exactly and show mean field behaviour, except for two limiting values of a parameter characterizing the dissipated energy, for which the system belongs to different universality classes. Two modified versions of the model, incorporating quenched disorder are considered: the first simulates heterogeneities in the local properties of the soil, the second considers the effects of a non-uniform rainfall. In the region of mean field behaviour, the model is shown to be robust to both kinds of perturbations. In the two limiting cases the random rainfall is still irrelevant, whereas the heterogeneity in the soil properties leads to new universality classes. Results of a numerical analysis of the model are reported that confirm and complement the theoretical analysis of the global minimum. The statistics of the local minima are found to more strongly resemble observational data on real rivers.Comment: 27 pages, ps-file, 11 Postscript figure

Variational Downscaling, Fusion and Assimilation of Hydrometeorological States via Regularized Estimation

Improved estimation of hydrometeorological states from down-sampled 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 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 proper regularization, which are constraints consistent with the degree of smoothness and probabilistic structure of the underlying state. We review relevant regularization methods in derivative space and extend classic formulations of the aforementioned problems with particular emphasis on hydrologic and atmospheric applications. Informed by the statistical characteristics of the state variable of interest, the central results of the paper suggest that proper regularization can lead to a more accurate and stable recovery of the true state and hence more skillful forecasts. In particular, using the Tikhonov and Huber regularization in the derivative space, the promise of the proposed framework is demonstrated in static downscaling and fusion of synthetic multi-sensor precipitation data, while a data assimilation numerical experiment is presented using the heat equation in a variational setting