Linear streamflow and subsurface runoff in arbitrary basins under Poisson point rainfall.

Abstract:

A novel stochastic model for the streamflow and subsurface runoff within a watershed is formulated and explicitly solved. The model is based on the linearized momentum/mass balance equations, and explicitly relates the transport of water between links of the river network and their surrounding hillslopes at the event time scale. The precipitation input, specified at the hillslope scale, is a steady marked Poisson point process with storm intensities of arbitrary distribution. A stochastic differential equation for the joint evolution of streamflow and runoff at every link of the river network is explicitly solved, and the associated invariant distribution characterized. The results explicitly show how the geometry of the river network, the storage dynamics of rivers and hillslopes, and the probabilistic properties of the rainfall field, conspire to shape the steady hydrological response of the watershed, along with its associated uncertainty. As an application, new formulas for the n-th moment of the streamflow are derived, as well as exact asymptotics of extreme discharge events. Consequently, the model offers insights about the long-term effects of a changing precipitation regime, over the streamflow distribution on an arbitrary watershed.