## 1. Introduction

### 1.1. The Runoff-Runon Process

[2] Estimating infiltration-excess overland flow is important because this runoff process is linked to hillslope erosion processes that are in turn linked to land degradation and water quality issues. Predicting surface runoff often involves assuming a homogeneous set of soil hydrologic properties for the area of interest and applying a rainfall-runoff model. In many cases this homogeneous assumption works well; however, sometimes patterns of observed surface runoff are poorly explained using this approach.

[3] For example, the spatially averaged infiltration rate is often observed to increase as an increasing function of rainfall intensity [e.g., *Yu et al*., 1998; *Yu*, 1999]. This is inconsistent with a homogeneous assumption, which should result in a threshold shift from zero runoff, to the uniform generation of runoff across the homogeneous area. It is assumed that this observation results from spatial variability in soil infiltration properties, such that an increasing proportion of the soil area is infiltrating at its maximum capacity as rainfall intensity increases, asymptoting to a maximum value.

[4] Many authors have also reported that runoff ratios are observed to decrease with increasing slope length [*Gomi et al*., 2008; *Bryan and Poesen*, 1989; *Joel et al*., 2002; *Parsons et al*., 2006]. This is also inconsistent with a homogeneous assumption, which should not result in scale-dependent runoff, and it is assumed that spatial variability in soil infiltration properties contributes to this observation. As a result, during rainfall not all locations will necessarily be generating runoff, and runoff generated in one location may infiltrate downslope in an area with higher infiltration capacity. This is known as the runoff-runon process.

[5] Several authors have explored the importance of the runoff-runon process [*Nahar et al*., 2004; *Morbidelli et al*., 2006]. *Yu* [1999] reported that a rainfall-runoff model accounting for runoff-runon on natural hillslope plots performed better than a model assuming homogeneous conditions. *Nahar et al*. [2004] used numerical modeling to show that for soils with moderate to high mean saturated hydraulic conductivity relative to rainfall, the runoff-runon process plays an important part in determining the total surface runoff from a spatially variable area. These conditions are typical in temperate forests, where saturated conductivity values are often high [e.g., *Nyman et al*., 2010], and are common in many other landscapes for the majority of rainfall events [*Dunkerley*, 2008]. When the rainfall intensity is very high relative to the infiltration capacity of the soil, spatial variability is likely to be unimportant as most locations will be infiltrating at, or close to, their capacity. This paper focuses on the former case (i.e., high mean infiltration capacity relative to the mean rainfall intensity in a spatially variable area).

### 1.2. Stochastic Processes and Queuing Theory

[6] A stochastic process is a sequence of random variables, usually considered in time. Systems that are characterized by “arrivals” and “departures,” such as the evolution of queues, are often well described using a branch of stochastic processes known as queuing theory. Arrivals do not have to be customers, they can be anything; for example, these kinds of models have been used to characterize the water storage in a dam [*Prabhu*, 1998] or in the soil profile [*Milly*, 1993], given random inputs and outputs of water. Furthermore, arrivals do not have to occur in time, they can occur in space; in fact, any arbitrary index is possible.

[7] In the long run, stochastic processes such as queues can be characterized by a set of asymptotic properties. An asymptotic property can be thought of as what you might see on average if you were to “drop in” on the process at random. Typical asymptotic properties of interest are the length of the queue and the waiting time in the queue. In the dam example above, one asymptotic property is the probability distribution of the water level over long time periods. Asymptotic properties are always probabilistic (e.g., distributions, means, variances, and other moments). Queues are parameterized by the interarrival times between “customers” and service times of “customers”. In this paper the mathematics of queuing systems is used derive analytical equations to describe the surface runoff characteristics of spatially variable areas where runoff-runon processes are occurring.