The stochastic runoff-runon process: Extending its analysis to a finite hillslope O.D. Jones a,⇑ , P.N.J. Lane b , G.J. Sheridan b a School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia b School of Ecosystem and Forest Sciences, University of Melbourne, VIC 3010, Australia. article info Article history: Received 31 July 2014 Received in revised form 20 June 2016 Accepted 25 June 2016 Available online 14 July 2016 This manuscript was handled by K. Georgakakos, Editor-in-Chief, with the assistance of Venkat Lakshmi, Associate Editor Keywords: Infiltration excess runoff Overland flow Stochastic runoff-runon process Single server queue Finite hillslope abstract The stochastic runoff-runon process models the volume of infiltration excess runoff from a hillslope via the overland flow path. Spatial variability is represented in the model by the spatial distribution of rain- fall and infiltration, and their ‘‘correlation scale”, that is, the scale at which the spatial correlation of rain- fall and infiltration become negligible. Notably, the process can produce runoff even when the mean rainfall rate is less than the mean infiltration rate, and it displays a gradual increase in net runoff as the rainfall rate increases. In this paper we present a number of contributions to the analysis of the stochastic runoff-runon pro- cess. Firstly we illustrate the suitability of the process by fitting it to experimental data. Next we extend previous asymptotic analyses to include the cases where the mean rainfall rate equals or exceeds the mean infiltration rate, and then use Monte Carlo simulation to explore the range of parameters for which the asymptotic limit gives a good approximation on finite hillslopes. Finally we use this to obtain an equation for the mean net runoff, consistent with our asymptotic results but providing an excellent approximation for finite hillslopes. Our function uses a single parameter to capture spatial variability, and varying this parameter gives us a family of curves which interpolate between known upper and lower bounds for the mean net runoff. Ó 2016 Elsevier B.V. All rights reserved. 1. Introduction 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. Two of the key properties determining this vol- ume are the rainfall rate and the infiltration rate. In natural sys- tems both these rates are variable in both space and in time. Suppose that our hillslope is divided into cells. If the rainfall rate exceeds the infiltration rate in a given cell, then the excess will flow overland to the next cell downhill. Thus the water flowing into a cell is given by the sum of the rainfall and runon from the cell above. Any excess, after infiltration is taken into account, becomes runoff. The resulting system is highly non-linear, because runoff is truncated below at zero. Nahar (2003) showed that for soils with moderate to high mean saturated conductivity relative to rainfall rate, the runoff-runon process plays an important part in determining the total overland discharge for a hillslope. These conditions are typical in temperate forests, where saturated con- ductivity values are usually high, and are common in many other landscapes for the majority of rainfall events (Dunkerley, 2008). Because of the complexity of the problem, models that incorpo- rate both spatial and temporal variability have, to date, been anal- ysed using numerical simulation methods. Our interest is in analytic solutions. The most common simplification made in this context is to neglect spatial variability and model rainfall and infil- tration as a function of time only. This can be attributed to the early development of analytical expressions for the temporal change in infiltration rate at a point (Green and Ampt, 1911). For catchment scale predictions these point-scale results have gener- ally been scaled up by optimizing the infiltration parameters using catchment or hillslope runoff time-series data. As a result of this scaling process, the parameters lose their physical meaning (e.g. see discussion by Grayson et al., 1992). A recent alternative is the stochastic runoff-runon process introduced by Jones et al. (2009) and developed in Jones et al. (2013) and Harel and Mouche (2013, 2014). The stochastic runoff-runon process allows for spa- tial variability but assumes temporal stationarity. It does however admit analytic asymptotic solutions, with parameters that retain their physical meaning. In this paper we pay particular attention http://dx.doi.org/10.1016/j.jhydrol.2016.06.056 0022-1694/Ó 2016 Elsevier B.V. All rights reserved. ⇑ Corresponding author. E-mail address: odjones@unimelb.edu.au (O.D. Jones). Journal of Hydrology 541 (2016) 677–688 Contents lists available at ScienceDirect Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol AUTHORS COPY