Stationary properties of Runoff under stochastic rainfall: lessons from a linear model.

Abstract:

We explore a linear system of ordinary differential equations to study the hydrological response of a watershed under a stochastic rainfall regime at the hourly-to-daily time scales. The equations are linked according to the geometry of the river network which is modeled as an arbitrary binary tree. Spatial heterogeneity on the parameters is allowed, and semi-linear effects of scale can be included. With this conceptual model, we show that if the net rainfall is a stationary Poisson process, the runoff processes throughout the basin will achieve a stationary probability distribution for which we have a complete mathematical characterization. The invariant distribution is emerging property of the watershed-climate system at seasonal time scales. We identify key non-dimensional quantities that provide the link between the stochastic properties of rainfall, the geophysical properties of the watershed, and the uncertainty associated with discharge at the outlet. We illustrate this by writing the coefficient of variation of discharge in terms of that of rainfall, and non-dimensional quantities. We furthermore give necessary conditions for various type of statistical scaling of the distribution of discharge with basin area and Horton order on self-similar networks. Finally, we establish results characterizing the tails of the distribution of discharge with those of the rainfall intensity.