Using queuing theory to describe steady-state runoff-runon phenomena and connectivity under spatially variable conditions

Authors

  • O. D. Jones,

    1. Department of Mathematics and Statistics, University of Melbourne, Parkville, Melbourne, Victoria, Australia
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  • G. J. Sheridan,

    Corresponding author
    1. Department of Forest and Ecosystem Science, University of Melbourne, Parkville, Melbourne, Victoria, Australia
    • Corresponding author: G. J. Sheridan, Department of Forest and Ecosystem Science, University of Melbourne, 221 Bouverie St., Parkville, Melbourne, Vic 3010, Australia. (sheridan@unimelb.edu.au)

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  • P. N. Lane

    1. Department of Forest and Ecosystem Science, University of Melbourne, Parkville, Melbourne, Victoria, Australia
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Abstract

[1] Runoff-runon occurs when spatially variable infiltration capacities result in runoff generated in one location potentially infiltrating downslope in an area with higher infiltration capacity. The runoff-runon process is invoked to explain field observations of runoff ratios that decline with plot length and steady-state infiltration rates that increase gradually with rainfall intensity. To illustrate the influence of spatial variability and runoff-runon on net infiltration and runoff generation, we use both (i) field rainfall simulation on soil with a high infiltration capacity and (ii) numerical rainfall-runoff-runon simulations over a spatially variable area. Numerical simulations have shown that the spatial variability of soil infiltration properties affects surface runoff generation; however, it has proven difficult to represent analytically the associated runoff-runon process. We argue that given some simplifying assumptions, the runoff-runon phenomenon can be represented by queuing theory, well known in the literature on stochastic processes. Using this approach we report simple analytic expressions derived from the queuing literature that quantify the total runoff under steady-state conditions from a spatially variable tilted 2-D plane, and the runoff produced by the area connected with the lower boundary of the plane under these conditions. These quantities are shown to be equal to the waiting time and the “sampled” busy period, respectively, of a first-in first-out queue with exponentially distributed arrivals and service times. The queuing theory model is shown to be consistent with field observations of runoff; however, the approach requires some simplifying assumptions and restrictions that may negate the benefits of the reported analytic solutions.

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.

2. Numerical Simulation of the Runoff-Runon Process

[8] Here we use a simple numerical simulation to illustrate how spatial variability and the runoff-runon process can result in the observed rainfall-runoff patterns described in section 1. Consider a tilted 1-D strip of land, divided into cells with spatially variable infiltration capacities. If the rainfall intensity over this strip exceeds the infiltration capacity in a given cell, then the excess will flow overland to the next cell downslope (here we neglect the effects of variable slope gradient and roughness and focus on the effects of variable infiltration capacity). Thus, in general, the water flowing into a cell is given by the sum of the rainfall onto that cell, and runon from the upslope cell. Any excess, after infiltration is taken into account, becomes runoff.

[9] Now consider a rectangular area split into a 50 × 50 cell grid, made up of 50 parallel strips (such as the one described earlier) each strip 50 cells in length, with randomly generated values for rainfall intensity and infiltration capacity assigned to each cell. In what follows we will use the term flow for the movement of a volume of water, in m3 h−1, and the terms rate, intensity, or flux for the movement of a volume of water per unit area, in mm h−1. For any given strip, label the cells 1, 2, … from the top and let Pk be the rainfall flow onto cell k and Ik be the infiltration flow for the kth cell (at capacity, in m3 h−1). Then along each strip (assuming parallel flow from top to bottom) we recursively calculate the runoff transfer flow Xk from the kth cell by

display math(1)

[10] Note that while it is natural to talk about rainfall and infiltration capacity in terms of rates or intensities, these runoff transfer equations are properly expressed in terms of flows. In the simulation, exponential distributions were used for the rainfall intensity and infiltration rate and were generated independently for each cell. The results are shown in Figure 1. Cells are shaded in proportion to the amount of runoff they produce: the darker the shading the greater runoff flow from the cell. The cells below the thick line are those “connected” to the lower boundary. A cell is “connected” to the lower boundary if there is an uninterrupted overland flow of water from it to the lower boundary. That is, on each strip the connected runoff is terminated above by a cell with zero runoff (shaded white in Figure 1).

Figure 1.

(a–f) Numerical simulations of runoff-runon on a 50 × 50 grid, each with mean infiltration capacity 1 mm h−1, and with the mean rainfall intensity taking values 0.25, 0.5, 0.75, 1, 1.25, and 1.5 mm h−1, respectively. For this simulation both rainfall intensity and infiltration capacity are exponentially distributed. Cells are shaded in proportion to the amount of runoff they produce. The cells below the thick line are those connected to the lower boundary by an uninterrupted overland-flow path.

[11] In Figures 1a–1f we illustrate the effect on runoff-runon as the rainfall intensity increases (from left to right, starting in the top row). It can be seen that the area connected with the downslope boundary grows (the area below the thick line). Note that considerable runoff connects with the lower boundary and is therefore delivered from the 50 × 50 grid, even when the mean rainfall intensity is less than the mean infiltration capacity (top row). Also note that when the rainfall intensity equals or exceeds the infiltration capacity (bottom row) there may still be a considerable portion of the grid that is not connected with the outlet.

[12] This simple numerical example illustrates why spatially averaged infiltration rates are observed to increase as an increasing function of rainfall intensity [e.g., Yu et al., 1998; Yu, 1999]. The shaded pixels in Figure 1 are producing runoff and therefore infiltrating at their maximum capacity. As the rainfall intensity increases, the shaded area increases as an increasing number of the pixels are infiltrating at maximum capacity. The net infiltration rate therefore increases with increasing rainfall intensity, until all the pixels are infiltrating at their maximum capacity.

[13] This numerical example also illustrates why infiltration rates are sometimes observed to increase with increasing plot length [Bryan and Poesen, 1989; Joel et al., 2002; Parsons et al., 2006; Gomi et al., 2008]. Only the shaded pixels below the thick line are delivering runoff across the lower boundary; runoff from the pixels above this line is re-infiltrated. In all of the examples shown in Figure 1, longer plots will on average contain more “unconnected” flow paths that re-infiltrate and do not reach the lower boundary, than will shorter plots, resulting in a higher net infiltration rate for longer plots, and a lower runoff ratio.

[14] The runoff flow and the connected area illustrated in Figure 1 were derived from numerical simulations based on equation (1) (i.e., the repeated random sampling of values for Pk and Ik from exponential distributions and the recursive use of equation (1) to calculate runoff). As to date, no analytical expression has been identified to represent this runoff-runon phenomenon under spatially variable conditions. The difficulty in representing the runon-runoff process analytically is that the output (i.e., net surface runoff) is a function of interactions between sequences of random variables (e.g., rainfall intensity and infiltration capacity), rather than scalar quantities, which limits the kind of mathematical approaches that can be used.

3. Field Observation of the Runoff-Runon Process

[15] It is argued that the runoff-runon phenomena described in the previous section and illustrated numerically in Figure 1 is a reasonable representation of the runoff processes that would occur in heterogeneous areas under steady-state rainfall. A more appropriate description of this process would require many replications of high-resolution measurements of the spatial variability of rainfall intensity and infiltration capacity, coupled with runoff measurements during rainfall. We do not undertake this robust evaluation in this paper, as it is not our objective to develop an empirical approach. Nevertheless, we propose a simple reality-check for the occurrence of the runoff-runon phenomena under natural conditions. We argue that this process should result in two observable patterns at the plot scale when mean rainfall intensity is lower than the mean infiltration capacity:

[16] 1. The plot infiltration rate should increase with the plot length, as longer plots will contain more “unconnected” flow paths that re-infiltrate and do not reach the lower boundary of the plot.

[17] 2. The plot infiltration rate should increase with rainfall intensity (up to a point), as the proportion of the plot where the point-scale infiltration capacity is exceeded will also increase.

[18] Plot rainfall simulation experiments were conducted, and the plot infiltration rate were measured as the difference between the rainfall rate and the runoff rate. Rainfall simulations were conducted in September 2007 in a burned mountainous forested area in SE Australia, described in Nyman et al. [2010]. The design of the rainfall simulator is described by Loch et al. [2001].

[19] In order to determine the relationship between plot length and infiltration rate, runoff was measured for three replicates of a 1.5 m wide and 2 m long plot subjected to a rainfall rate of 100 mm h−1 until a quasi steady state runoff rate was reached at approximately 30 min. The 2 m plot length was then reduced (from the lower boundary) in stages to 1.0, 0.5, and 0.25, and runoff measurements were repeated until quasi steady state was again reached for each of these plot lengths at the same rainfall rate. The relationship between rainfall intensity and infiltration rate was investigated using the above 0.25, 1.0, and 2.0 m plots subjected to an additional rainfall rate of 75 mm h−1 following the 100 mm h−1 rainfall event.

[20] The results of our experiments are shown in Figure 2. The data in Figure 2a show a positive relationship between the plot length and infiltration, with the infiltration rate approximately doubling from 20 to 40 mm h−1 as the plot length increases from 0.25 to 2.0 m. This result is consistent with point 1, abovementioned, where longer plots will contain more “unconnected” flow paths that re-infiltrate and do not reach the lower boundary, determining higher net infiltration rates.

Figure 2.

Observed relationships between mean plot-scale infiltration rate (mm h−1) under simulated rainfall and (a) plot length, and (b) rainfall intensity at several plot lengths (0.25, 1.0, and 2.0 m). Error bars shown are one standard error for three replicates.

[21] Figure 2b shows a positive relationship between the rainfall intensity and infiltration rate for each plot length utilized. This result is consistent with point 2 abovementioned, where point-scale infiltration capacity is exceeded in a larger number of points as rainfall intensity increases. Such results were also observed earlier at the plot scale [see Langhans et al., 2011]. These limited data suggest that the runoff-runon phenomenon illustrated in Figure 1 could be a reasonable representation of the real world.

4. Using Queuing Theory to Describe the Runoff-Runon Process

[22] First, we simplify our problem domain by considering a 1-D strip with width lx, divided into cells of length ly. This is equivalent to considering only a single strip within the 50 strips of cells illustrated in (Figure 1), and we invoke the same simplifying assumptions here. We number the cells inline image, starting at the top of the strip. Assume the strip is tilted and runoff flows into the next lower cell only. Let Xk be the flow of water from cell k to cell k + 1, in m3 h−1. Let pk be the rainfall rate and ik be the infiltration rate for cell k (both are fluxes, measured in mm h−1), assumed to be constant over time. If pk > ik, then runoff builds up down the length of the cell. Conversely, if pk < ik, then the runoff declines. Let the depth of water at the end of cell k be dk and let its speed be vk, then the volume of water leaving the block per unit time is inline image. If vk were constant, then we would have inline image. Let inline image be the flow of rain falling on block k, and let inline image be the maximum flow of water absorbed by block k (both in m3 h−1). As we saw in equation (1), inline image and for inline image.

[23] Importantly, equation (1) is also found in stochastic queuing theory, where it gives the waiting time in a single server first-in first-out (FIFO) queue. This queue is analogous to a single checkout at a shop, with one person serving, where customers are served in the order in which they arrive. Both the interarrival time between customers and the time it takes to serve a customer are random variables. If we let Pk be the (random) service time for customer k and let Ik be the (random) interarrival time between customers k and k + 1, then Xk is the (random) waiting time for customer k + 1, i.e., the time between arriving and service commencing. This waiting time is intuitively the sum of the waiting time of the kth arrival, Xk−1, the service time of the kth arrival Pk, minus the interarrival time Ik between Xk and Xk+1, as expressed in equation (1). Figure 3 provides an illustration of how the equation for the runoff rate from an area with spatially variable rainfall and infiltration (top schematic) can be represented by the equation for the waiting time in a single server queue (bottom schematic).

Figure 3.

An illustration of how the equation for the runoff flow from an area with spatially variable rainfall and infiltration (top schematic) can be represented by the equation for the waiting time in a single server queue (bottom schematic).

[24] The beauty of this equivalent interpretation of the runoff-runon equations is that it has already been extensively studied in the queuing literature, and solutions are available. However, this queuing literature has three major constraints when applied to runoff-runon process. First, it applies only to steady-state behavior, when it has been raining long enough for the infiltration rate to reach quasi-equilibrium. Second, we only have nice analytical results when the average rainfall is less that the average infiltration, and third, the theory only gives limiting/asymptotic results for runoff on a strip of infinite length.

[25] Let inline image, then when ρ < 1 we have results on the asymptotic distribution of Xk as k → ∞. We can interpret this asymptotic or limiting distribution as follows. Suppose that we had a strip of infinite length and we sampled the runoff X from one cell at “random”. The distribution of X is precisely the asymptotic distribution of the Xk.

[26] Most of the standard results on queuing theory can be found in Asmussen [2003], Prabhu [1998], or Kleinrock [1975a]. To obtain analytic results we need to make a number of assumptions about the random variables inline image and inline image. First, we suppose that they are time-invariant, i.e., pk and ik, and thus, Pk and Ik are independent of t. Next we suppose that they are mutually independent and identically distributed (i.i.d.) sequences. For the infiltration capacity this implies that there is no spatial correlation in the infiltration capacity at the scale of cells used in the model. Small-scale spatial correlation in the water infiltration capacity ik has been observed; thus, the validity of this assumption depends on the scales lx and ly: we need lx and ly large enough that the correlation between Ik and Ik+1 is negligible. Similarly, we are assuming that there is no spatial correlation in the rainfall intensity values at the scale of cells used in the model. As for the infiltration, small-scale spatial correlation in the rainfall rate pk has been observed, so to justify this assumption we again need lx and ly to be relatively large, compared to the correlation scale. Independence between Pk and Ik is perfectly reasonable; however, note that we are also assuming independence between Xk and Ik; that is, the infiltration flow ik is independent of surface water depth dk. This assumption is reasonable for the small range in runoff water depths dk at plot and hillslope scales.

5. The Mean Runoff Rate From a Runoff-Runon Process

[27] Let inline image and inline image (ρ is known as the “traffic intensity” in queuing theory). In what follows we write P and I for a generic Pk and Ik, respectively. The quantities below are all flows, measured in m3 h−1 and with mean and standard deviation proportional to lxly.

[28] If ρ < 1 then as n → ∞, Xn converges in distribution to some limit X. That is, the cumulative distribution function (cdf) Fn of Xn converges to the cdf F of some random variable X. We can write this as inline image. The form of the function F is not known in general (i.e., for P and I with any probability distribution). However, in the special case where both rainfall intensity and infiltration capacity are exponentially distributed, the form of F is well known. The runoff-runon process has been modeled by several authors assuming an exponential distribution for I [e.g., Hawkins and Cundy, 1987; Yu et al., 1997; Fentie et al., 2002; Kandel et al., 2005], although field observations commonly report a log normal distribution [Nielsen et al., 1973; Price, 1994; Loague and Kyriakidis, 1997]. We chose the exponential distribution here because it is right skewed, like the observations, yet analytically simple. Incoming rainfall at the spatial scale of interest here is presumably spatially uniform; however, vegetation spatially redistributes rainfall due to canopy interception and processes such as leaf drip and stemflow [Park and Cameron, 2008]. There are limited data quantifying the shape of the distribution of this redistributed rainfall under different vegetation structures, so here we assume an exponential distribution for reasons of analytic tractability. Analytic possibilities without these restrictive assumptions for P and I are explored in the discussion.

[29] Hereafter we therefore suppose inline image and inline image, so inline image and inline image. Then the cdf of the asymptotic runoff rate X is given by

display math(2)

[30] This distribution has the following interpretation: take a cell at random, then with probability ρ there is no runoff, but if there is runoff then the amount of runoff has an exponential distribution with mean inline image. This distribution is a surrogate for what one may observe at the lower end of a suitably long strip. The mean and the variance of X are easily calculated from F. We have

display math(3)

6. The Mean “Connected Length” of Runoff-Runon

[31] In addition to the runoff flow X, we are interested in the number of cells that contribute to this runoff, which we call the “connected length” (note this is not strictly a length, but a count of the number of contiguous cells contributing runoff). Let Mn be the number of cells that contribute to the runoff Xn from cell n, then

display math(4)

[32] Mn can also be interpreted in terms of a single server FIFO queue, with service times Pk and interarrival times Ik. Let Qk be the number of customers in the system just before the arrival of customer k + 1, then inline image. That is, the waiting time for customer k + 1 is zero if and only if there is no one in the system when he arrives. Thus, inline image, that is, Mn is the number of customers who arrived during the current busy period, observed just before the arrival of customer n + 1.

[33] The busy period of a queue is much less tractable than the waiting time, so to obtain results we need to make some relatively strong assumptions. Suppose again that inline image and ρ < 1. Let M be the limiting distribution of Mn then

display math(5)

[34] A proof is given in Appendix A.

7. From Strips to Planes

[35] The previous sections have described the asymptotic properties of a single strip. We consider now the aggregated asymptotic properties of a tilted plane, consisting of many parallel linear strips, as shown in Figure 1. A plane can be represented as a collection of adjacent strips placed perpendicular to a downslope boundary. The proposed plane model depends on the central limit theorem and assumes that adjacent strips are independent and that there are no lateral inflows or outflows from a strip.

[36] Let inline image be the runoff flow (m3 h−1) from the ith strip. By the central limit theorem, the sum inline image of m i.i.d. random variables inline image (of any distribution) with mean μ and finite variance σ2 is approximately normally distributed as m becomes large, with mean and variance 2. In our case we suppose that we have a tilted plane made up of m independent strips of width lx and length inline image. Let inline image be the width of the tilted plane. The total runoff Z (m3 h−1) at the lower boundary will be

display math(6)

where μX and inline image are the mean and variance of the runoff flow, given in equation (3). We stress that these approximations are valid as the number of cells per strip n → ∞ and the number of strips m → ∞. From equation (6) we have

display math(7)

[37] Here inline image and inline image have been scaled so that they do not depend on the cell size. We see that in this case the mean runoff Z from the plane is proportional to the width Lx of the hillslope, but the length Ly plays no part when n → ∞. The spatial variation is expressed by the factor ly in the mean, and the factor inline image in the standard deviation.

[38] We can think of the length ly as a system parameter that quantifies the spatial correlation scale of rainfall intensity and infiltration capacity in the direction of flow. It should be just large enough that the Pk and Ik appear to be uncorrelated.

[39] Following the approach described earlier for runoff from a plane, we can also calculate the connected area of a plane. That is, the contiguous runoff producing area connected to a lower boundary of a plane of width inline image. Let inline image be the number of cells contributing to the runoff from the ith strip. inline image will have mean and variance as given in equation (5). The total area contributing to runoff at the bottom of the plane will therefore be inline image (in m2). As before, assuming the strips do not interact, the central limit theorem gives us that for large m, Y will be approximately normally distributed. Let μM and inline image be the mean and variance of M (both dimensionless quantities), then

display math(8)

[40] As for the runoff, we have that the mean of Y is proportional to Lxly and the standard deviation is proportional to inline image.

[41] The capacity to represent the runoff flow and the connected area for a tilted plane only from the mean and variance of the runoff and connected area of the individual strips is an important result because the entire distribution of the runoff rate and connected area for a strip is not available in general.

8. Discussion

8.1. Representing Runoff-Runon Using Queuing Systems

[42] The aim of this paper was to evaluate if the runoff-runon process, which has in the past been represented and modeled numerically [e.g., Nahar et al., 2004; Morbidelli et al., 2006], could be usefully represented as a stochastic process. The results show that the process, under a set of limiting conditions, can be considered mathematically as a kind of queue. Both systems involve a recursive combination of two random sequences, and the stochastic equations capture the synergistic effects of these two sources of variability. Both systems are also stochastic processes that are bounded below at zero. In other words, the waiting time in the queue cannot be less than zero, just as the runoff rate cannot be less than zero. Queues are a stochastic process that has mathematical properties such as the mean waiting time and the mean length. These properties can be, and have been, under certain conditions, quantified analytically. By considering the runoff-runon process as a mathematical queue, we are able to analytically determine some useful properties of the runoff-runon process, such as the mean runoff flow and the mean connected length.

[43] There have been no previous attempts to represent the runoff-runon process as a queue; however, several authors have developed analytic expressions to represent the effect of the runoff-runon process on net runoff generation. In particular, Hawkins and Cundy [1987] were able to develop analytic expressions for the maximum and minimum bounds on net runoff for spatially variable conditions. This was achieved by recognizing that maximum net runoff is achieved when infiltration capacities are arranged from highest to lowest downslope, while the minimum net runoff rate occurs with the opposite arrangement (i.e., lowest to highest downslope). The queuing model introduced in this paper uses a different mathematical approach to Hawkins and Cundy [1987]; however, the objective is the same: to derive analytic expressions for the net runoff from a spatially variable area.

8.2. Model Simplifications and Assumptions

[44] The runoff-runon process described by equation (1) is a gross simplification of the actual runoff-runon process that occurs across natural soils and during natural rainfall conditions. These simplifications are imposed to enable the mathematics of queuing systems to be used to describe the runoff-runon process. However, these simplifications may detract from or negate the benefits of the simple closed-form analytic solutions described in this paper and are therefore examined in further detail here. Important simplifications include

[45] 1. The runoff-runon process is considered time-invariant, i.e., only steady-state conditions are considered.

[46] 2. The effect of surface conditions such as slope and surface roughness is neglected.

[47] 3. Only the case EP < EI is considered (mean rainfall intensity is less than mean infiltration capacity).

[48] 4. Slope length is considered infinite (there is no upper boundary condition).

[49] 5. Both P and I are assumed to be exponentially distributed.

[50] Considering only steady-state (time-invariant) conditions neglects the change in infiltration rate during the rainfall event. Steady-state infiltration rates are generally lower than initial infiltration rates into dry soil, and therefore, using steady-state values for I (rather than time varying rates) will result in an overestimate of the runoff rate. However, it should also be noted that using steady-state infiltration rates will underestimate the effects of spatial variability on runoff, which are greater for the smaller values of ρ likely at the beginning of infiltration into a dry soil.

[51] Considering only steady-state conditions also neglects the gradual increase in runoff that is observed during rainfall due to the time it takes for overland flow to travel over the length of the plot (commonly referred to as the time of concentration). Additional factors such as slope gradient and surface roughness will also affect the time of concentration. Neglecting these effects will also result in an overestimate of the runoff rate, especially when considering rainfall events with durations shorter than the time of concentration.

[52] Solutions to the analytic equations (equations (2)-(5)) are only available for the case where mean rainfall intensity is less than mean infiltration capacity (EP < EI), which is clearly restrictive from a practical perspective as one may expect that the largest runoff events are produced when the mean rainfall intensity is greater than the mean infiltration capacity. However, it is in the domain EP < EI where spatial variability exerts the greatest influence on runoff rates. As the mean rainfall intensity exceeds and increases above the mean infiltration capacity, the effect of spatial variability is diminished and the runoff rate approaches the difference between the mean rainfall intensity and the mean infiltration capacity.

[53] In addition, while it is reasonable to expect that most runoff is produced when EP > EI where the effects of spatial variability are diminished, it is interesting to note that inspection of runoff data collected from plots of different lengths (ranging from 0.5 to 60 m) under natural rainfall frequently shows a reduction in the runoff ratio with slope length, an effect consistent with the effects of spatial variability when EP < EI [e.g., Duley and Ackerman, 1934; Bren and Turner, 1979; Lal, 1983; Sharma, 1986; Lal, 1997; van De Giesen et al., 2000; Liu et al., 2000; Joel et al., 2002; Parsons et al., 2006; Gomi et al., 2008; Xu et al., 2009; Bagarello and Ferro, 2010; Moreno-de las Heras et al., 2010].

[54] These observations may be partly explained by the low mean intensity of most rainfall events. Dunkerley [2008] reviewed rainfall statistics from 26 studies with substantial records of natural rain revealing a mean rainfall rate of only 3.47 mm h−1 (standard deviation 2.38 mm h−1), which is considerably lower than the mean saturated hydraulic conductivities typically reported for a range of soil and land use types. For example, Elliot et al. [1989] used rainfall simulation on cropland soils at 32 locations across 20 states in the United States and in every case measured steady-state infiltration rates higher than 3.47 mm h−1. Sheridan et al. [2000] used rainfall simulation to measure steady-state infiltration rates for 32 different minesite soils and overburdens from Queensland, Australia, and report only two of these to have infiltration rates lower than 3.47 mm h−1, while rates for forested soils are typically reported to be in the range of tens to hundreds of millimeters per hour [e.g., Moore et al., 1986; Wilson and Luxmoore, 1988]. These data suggest that EP < EI for most rainfall events and equations (2)-(5) may assist in understanding the contribution of these rainfall events to total runoff.

[55] The analytic equations allow the connected length to grow indefinitely which is the equivalent of having an infinite slope length. This will result in an overestimate of runoff as rainfall intensity increases and the theoretical connected length of individual strips exceeds the finite distance from the lower boundary to the upper boundary. The magnitude of this possible overestimation will depend on the value of ρ and the length of ly relative to the slope length nly. The shaded connected cells in the numerical examples shown in Figure 1 illustrate the likely magnitude of this error for the case where the slope length is 50 times greater than the cell length. In this example, an overestimate of runoff will occur when a connected flow path exceeds 50 cells in length.

[56] Figures 1a–1d indicate that this error would be very small for ρ ≤ 1; for these cases <3 of the 50 flow lines are connected with the upper boundary. Therefore, given that we are only considering cases where EP < EI (simplification 3), any overestimation of runoff due to the lack of an upper boundary is likely to be insignificant, provided ly is small relative to the slope length nly. Alternatively, this error can be bounded above by noting that the when all cells are connected with the lower boundary, all cells are infiltrating at the maximum capacity, and the runoff rate from the grid therefore approaches mn(EP − EI). Setting this value as an upper bound for the runoff rate will limit the overestimation of runoff when ρ is large and/or ly is large relative to the slope length nly.

8.3. Model Application

[57] Textbook infiltration theory often involves the application of point-scale models such as the Green and Ampt [1911] model for the time evolution of infiltration using scalar values for infiltration parameters and ignoring spatial variability. Rainfall generated runoff can only occur when the rainfall intensity exceeds the infiltration capacity, so when using point-scale models (i.e., when P and I have a variance of zero) the runoff rate is either zero when ρ ≤ 1 or occurs at a rate proportional to the product of the area and the difference between the rainfall rate and the infiltration rate when ρ > 1;

display math(9)

[58] Figure 4 shows the difference in the relationship between ρ and the runoff rate given by equation (3) compared to equation (9). The dashed lines illustrate how with point-scale models (equation (9)) runoff production is initiated at ρ = 1 and increases linearly as ρ increases, at a rate that depends on the slope length.

Figure 4.

Runoff estimation compared using the spatially variable queuing system based model (solid line, equation (3)), and a point-scale model that assumes spatially homogeneous infiltration conditions for three slope lengths (dashed lines, equation (9)). Note that the plot of equation (3) is arbitrarily limited to ρ < 0.85 because for this equation the EX → ∞ as ρ → 1 (see text for discussion).

[59] In contrast, the solid line in Figure 4 illustrates how the queue model (equation (3)) predicts the production of runoff even when ρ < 1, and the gradual increase in runoff as PI. This pattern of runoff production is consistent with real-world observations such as those observed during rainfall simulation shown in Figure 2b, and with the field measurements in the studies listed earlier. These observed patterns are not well represented by point-scale models like equation (9) that predict a threshold shift from zero runoff at ρ = 1.

[60] Many authors have reported that runoff ratios under natural rainfall are observed to decrease with increasing slope length [e.g., Bryan and Poesen, 1989; Joel et al., 2002; Parsons et al., 2006; Gomi et al., 2008]. This phenomena can also be observed in the rainfall simulation data presented in Figure 2a, showing infiltration rates (in mm h−1) increasing with slope length. When conditions are in the domain ρ < 1, increasing the slope length will increase the plot area but according to equation (3) does not increase the runoff rate (because slope length is not a variable in equation (3)); hence, the estimated runoff ratio is reduced. A volumetric runoff equation that does not include slope length is counterintuitive; however, Ronan [1986] observed this effect when they measured natural runoff for 10 years from forested 20 m long runoff plots in southeast Australia. They concluded that the “volume harvested (from a plot) will depend on the length of the collection gutter, but not on the area above the gutter.” Bren and Turner [1979] utilized multilength natural rainfall-runoff plots in forests and reported similar conclusions.

[61] Slope length could be neglected in the estimation of runoff in both these studies because it is likely that ρ < 1 for most of the measured rainfall-runoff events due to the characteristically high infiltration capacities [Moore et al., 1986] and high spatial variability [Nyman et al., 2010] of these well-structured forest soils. However, for conditions where ρ ≥ 1, slope length is important as equation (5) shows that the theoretical mean connected length EM (in the absence of an upper boundary) approaches infinity, whereas in reality the slope length nly constrains the connected length and thus the total contributing area. For this reason, in Figure 4 we arbitrarily truncate the plot of equation (3) at ρ < 0.85 as the EX → ∞ as ρ → 1, which is not realistic for finite slopes. We therefore suggest two qualitative domains: when ρ < 1 spatial variability is important for the estimation of runoff, while slope length is less important, and ρ > 1 where spatial variability is less important and slope length increases in importance. We argue that the queue model (equation (3)) provides a better conceptual explanation for observed runoff generation patterns for ρ < 1, while equation (9) is more reasonable for ρ > 1. A single analytic mathematical expression that accounts for the role of spatial variability for ρ < 1, and the role of slope length when ρ > 1 is beyond the scope of this paper, and is the subject of ongoing research. Inspection of Figure 4 suggests that such an expression would transition from equation (3) on the left and asymptote toward one of the linear equations on the right, depending on the slope length.

[62] In deriving the probability distribution for runoff F(x) it was assumed that both rainfall intensity and soil infiltration capacity were exponentially distributed. The distribution of infiltration capacity is generally reported to be log normal, while the spatial distribution of rainfall intensity at the soil surface beneath vegetation is not well known. If inline image but P is allowed to have any distribution, then the famous Pollaczek-Khintchine formula gives the characteristic function of X, from which we can directly obtain the mean and variance of X [Kleinrock, 1975b, S2.3]:

display math(10)

[63] It is much more difficult to obtain exact analytic results (such as given in equations (3), (5), and (10)) when neither P nor I is exponentially distributed. However, in the absence of exact analytic results, some approximations can be found in the literature. For example, both Marchal [1976] and Whitt [1993] derive approximations for EX (the mean runoff rate) with no restrictions on the type of distribution for P or I. These approximations may enable future development of this modeling approach with less restrictive assumptions for the distribution of P and I.

[64] The simplifications and assumptions discussed earlier impose serious limitations on the general applicability of queuing models to the real-world problem of estimating runoff-runon from heterogeneous areas. Despite these obvious limitations, the recognition of queuing theory as a branch of mathematics with possible solutions to the mathematically complex runoff-runon problem may open up new avenues in the search for analytic closed-form solutions to the rainfall-runoff problem. Future research efforts should focus on addressing the limitations discussed earlier.

9. Conclusion

[65] The spatial variability of rainfall intensity and infiltration capacity affects surface runoff generation. This is because runoff generated in one location can potentially infiltrate downslope in a location with a higher infiltration capacity (the runoff-runon phenomena). In the past, the net effect of this process could only be explored using numerical simulations. In this paper we argue that the runoff-runon process observed under spatially variable conditions can be represented using queues from the stochastic processes literature. Using this new approach, we were able to derive analytic, closed-form expressions to describe steady-state runoff flows under spatially variable conditions (equation (7)). We were also able to quantify the runoff area connected to a downslope boundary by contagious runoff pathways (equation (8)). It is important to note however that queuing theory imposes mathematical constraints on our representation of the runoff-runon problem, which may detract from or negate the benefits of these simple closed-form analytic solutions shown earlier. The use of queues to help the description of the runoff behavior of heterogeneous areas is a topic of ongoing research.

Appendix:

[66] Here we derive equation (5) for the mean and variance of M. Let Q be the limiting distribution of the Qn. We have that inline image [Kleinrock, 1975a, equation 3.27] and inline image.

[67] Let B be the length of a busy period, and B* be the length of a randomly sampled busy period, where the probability of choosing a busy period is proportional to its length. Then inline image and we have [Kleinrock, 1975a, equation 5.157]

display math

[68] Given that M > 0, let B* be the length of the corresponding sampled busy period, then inline image. Thus,

display math

[69] We can use this to calculate the mean and variance of M. It is known [Kleinrock, 1975a, equation 5.156] that B has probability generating function

display math

[70] Taking derivatives at (1) we get after a little algebra,

display math

[71] Hence, we have for the expected value of M

display math

and for the variance

display math

Acknowledgments

[72] This project was funded by the Department of Sustainability and Environment, Victoria, and by Melbourne Water. Thanks to Andrew Western and Peter Hairsine for early discussions and inspiration on the spatial variability problem. Thanks to Christopher Sherwin for production of figures in this manuscript. Thanks also to the three anonymous reviewers whose suggestions significantly improved this manuscript.

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