The volume of catchment discharge that reaches a stream via the overland flow path is critical for water quality prediction, because it is via this pathway that most particulate pollutants are generated and transported to the stream channel, via surface erosion processes. When it rains, spatial variation in the soil infiltration rate leads to the formation and reabsorption of rivulets on the surface, and local topography determines the coalescence of rivulets.

We consider the question of how coalescence facilitates overland flow in two ways. Firstly we take a highly abstracted version of the problem, in which the drainage pattern is represented by a Galton-Watson tree. We show that as the rate of rainfall increases there is a distinct phase-change: when there is no stream coalescence the critical point occurs when the rainfall rate equals the infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the infiltration rate, and increasing the amount of coalescence increases the total expected runoff.

Secondly we fit a model for coalescing runoff to field data collected over a period of months from a burnt hillslope in the Victorian alps in SE Australia. The hillslope is discretised using a hexagonal lattice, and local topography is modelled by randomly perturbing the drainage pattern (which is derived from satellite elevation data). More perturbation corresponds to a rougher surface and more coalescence. The many latent variables mean the model has an intractable likelihood, however it is easily simulated, allowing us to fit the model using Approximate Bayesian Computation (ABC).

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