## 1. Introduction

[2] When water is applied to the soil surface, it infiltrates until the application rate exceeds the soil-limited infiltration rate, when ponding occurs at the soil surface, and runoff and erosion can be initiated. The ability to estimate accurately when initial ponding occurs and how much runoff is produced is important in civil and agricultural engineering, and is essential for the proper design of irrigation systems, rain harvesting reservoirs, and hydraulic structures at the level of the watershed.

[3] Infiltration is a complex phenomenon controlled by a series of factors. In principle, local infiltration is ruled by the actual hydraulic properties of the soil profile, the rainfall intensity, and the water content distribution with depth. These basic factors hold when one extends the analysis to infiltration in locally nonuniform soil profiles or spatially varying systems at the scale of the field or the watershed. A large body of research has shown that spatial variability of soil properties affect infiltration at such scale [*Russo and Bresler*, 1982; *Sivapalan and Wood*, 1986; *Saghafian et al.*, 1995]. The effect of local heterogeneity within the soil profile on infiltration was also demonstrated for layered or nonuniform soils, mainly relying on the *Green and Ampt* [1911] approach [*Childs and Bybordi*, 1969; *Beven*, 1984; *Selker et al.*, 1999; *Chu and Marino*, 2005]. A special case of soil nonuniformity is when a seal layer develops at the soil surface due to the raindrop impacts [*Assouline*, 2004]. Infiltration through such nonuniform soil profiles was also modeled [*Hillel and Gardner*, 1970; *Ahuja*, 1983; *Parlange et al.*, 1984; *Baumhardt et al.*, 1990; *Assouline and Mualem*, 1997]. Recently, *Chu and Marino* [2005] have presented a modified Green and Ampt model that deals with infiltration into layered soils under unsteady rainfall. In their model the time to ponding can be identified only if all the infiltration process is solved step by step and the cumulative infiltration computed according to the rainfall time discretization. The combined effect of soil spatial variability and profile heterogeneity on infiltration was studied by *Assouline and Mualem* [2002]. The main result is that accounting for soil surface sealing has a greater effect on infiltration than accounting for soil spatial variability.

[4] Spatial and temporal variability in rainfall or water application rates also affect infiltration. A constant rate supply of water may well represent sprinkler irrigation, however temporal variability is ubiquitous in rainfall with clear influence on runoff and erosion estimates [e.g., *Agnese and Bagarello*, 1997; *Wainwright and Parsons*, 2002; *Frauenfeld and Truman*, 2004; *Strickland et al.*, 2005; *Govindaraju et al.*, 2006]. *Agnese and Bagarello* [1997] found that the temporal resolution required for the accurate prediction of infiltration was strongly dependent on the soil type, and that its effect was practically negligible for soils with either high or low permeability. *Wainwright and Parsons* [2002] concluded that overland flow models that account for run-on infiltration underpredict runoff when the mean rainfall intensity is used instead of time-varying rainfall intensity. Efforts are now invested in modeling infiltration under variable rainfall intensity. *Govindaraju et al.* [2006] suggested a semianalytical model to compute the space-averaged infiltration at hillslope scale when spatial variability in both soil property and rainfall intensity are accounted for. The soil spatial heterogeneity is characterized by a lognormal distribution of the saturated hydraulic conductivity, while the rainfall spatial heterogeneity is simulated by a uniform distribution between two extreme rainfall intensities. At each location, the soil saturated hydraulic conductivity and the rainfall intensity was assumed to remain constant during the rainfall event. The results of this model are in agreement with those of *Assouline and Mualem* [2002] for the unsealed (mulched) soil surface case.

[5] On the basis of this literature review, we focus, in this paper, on the processes of local infiltration and ponding occurrence for variable water application rates at the surface of both a homogeneous soil and a heterogeneous one represented by a sealed profile.

[6] During infiltration under shallow ponding (i.e., where infiltration is not strongly affected by the depth of ponding) the infiltration capacity rate, *f*_{cap}, decreases due to the decrease of the hydraulic head gradient resulting from the advancement of the wetting front. The infiltration capacity curve, *f*_{cap}(*t*), can be thus considered a soil characteristic with dependence on the initial soil water content profile, which can be relatively easily characterized under laboratory or field conditions. When water is applied at a prescribed rate, for example under low rainfall intensity or sprinkler or drip irrigation, all of the supplied water infiltrates into the soil until ponding occurs, whence the actual rate of infiltration, *f*, is controlled by the soil infiltration capacity until the application rate falls below it. The temporal history of the actual infiltration *f*(*t*), unlike *f*_{cap}(*t*), is a function of the pattern of water application.

[7] The importance of the infiltration process in soil, hydrology, and environmental sciences had led to considerable literature dealing with experimental observations, and theoretical, analytical, numerical and empirical modeling of infiltration [e.g., *Clothier*, 2001; *Warrick*, 2002; *Smith et al.*, 2002; *Hillel*, 2004; *Brutsaert*, 2005; *Hopmans et al.*, 2006]. Analytical and empirical mathematical expressions have been proposed to provide a quantitative description of *f*_{cap}(*t*) and *f*(*t*) [e.g., *Green and Ampt*, 1911; *Kostiakov*, 1932; *Horton*, 1940; *Philip*, 1957a; *Smith and Parlange*, 1978; *Parlange et al.*, 1999]. From these results expressions have been derived to estimate the time when ponding occurs, *t*_{p} [e.g., *Chow et al.*, 1988; *Kutilek and Nielsen*, 1994; *Parlange et al.*, 1999; *Smith et al.*, 2002; *Brutsaert*, 2005]. Although being theoretically valid for unsteady rainfall, most of its practical applications (1) have assumed that water is supplied at a constant rate or consider the time-averaged rate of supply until ponding, (2) have neglected the effect of raindrop impact on the soil surface when a bare soil is exposed to high-energy rainfall, and (3) do not account for the antecedent water distribution. These restrictions do not allow accurate representation for many situations.

[8] Once *t*_{p} is evaluated, the second important need is prediction of *f*(*t*) after ponding, essential for prediction of processes governed by runoff (e.g., floods and erosion). Methods widely used are the time compression approximation (TCA) [*Brutsaert*, 2005], or the infiltrability-depth approximation (IDA) [*Smith et al.*, 2002]. The TCA was introduced in the 1940s [*Sherman*, 1943; *Holtan*, 1945] and has been applied widely [e.g., *Reeves and Miller*, 1975; *Sivapalan and Milly*, 1989; *Kim et al.*, 1996]. It relies on the assumption that infiltration rate after ponding is a unique function of the cumulative infiltration volume, *F*. For t < *t*_{p}*, F*(*t*) is equal to the cumulative rainfall, *R*(*t*) = *r*(*t*)*dt*. One may define the cumulative infiltration capacity, *F*_{cap} (*t*) = *f*_{cap} (*t*)*dt*, and a compression reference time, *t*_{cr}, which is the time required to produce the same cumulative infiltration volume under shallow ponding conditions from *t* = 0. Thus *F*(*t*_{p}) = *R*(*t*_{p}) = *F*_{cap}(*t*_{cr}). Once *t*_{p} and *t*_{cr} are known, *f*(*t*) for continued ponding can be evaluated as *f*_{cap}(*t* − *t*_{0}), with *t*_{0} = (*t*_{p} − *t*_{cr}). It is evident that the TCA requires an accurate estimate of *t*_{p} and *t*_{cr}. As was true for *t*_{p,} the available expressions for estimation of *t*_{cr} assume either constant or time-averaged wetting rate [*Brutsaert*, 2005] which limits the practical utility of this approach.

[9] On the basis of numerical calculations, *Smith* [1972] and *Smith and Chery* [1973] suggested an implicit computation of *t*_{p}:

where *r*(*t*) is the observed rainfall rate which is required to be at most slowly varying close to *t*_{p}; *r*_{p} is the rainfall rate at *t*_{p}; *A* is a linear function of the initial water content assumed constant with depth in the soil profile; *K*_{s} is the saturated hydraulic conductivity; and *β* is a parameter found to be close to 2. *Parlange and Smith* [1976] proposed an alternative expression which requires one less parameter that can be applied for any rainfall pattern for which *r*_{p} < *K*_{s}:

where *S* is the soil sorptivity. *Broadbridge and White* [1987] developed an expression similar to equation (2) for *t*_{p} for the case of rainfall events characterized by a linear increase in *r* with *t*. Insight on the physics leading to these expressions can be found in the literature on infiltration [e.g., *Clothier*, 2001; *Warrick*, 2002; *Smith et al.*, 2002; *Hillel*, 2004; *Brutsaert*, 2005; *Hopmans et al.*, 2006].

[10] These expressions are implicit functions where the unknown variable is *t*_{p}, and require that soil sorptivity and hydraulic conductivity be known. This is a significant constraint for field conditions where heterogeneity, anisotropy, and/or preferential flow make these parameters difficult to obtain. One related point is the effect of surface condition on soil properties and consequently, on infiltration and runoff. When a bare soil surface is exposed to rainfall, the energy of the raindrop impacts lead to soil surface sealing. This process can significantly reduce the infiltration rate, and consequently the time to ponding [*Assouline*, 2004]. Here the influence of surface sealing is evaluated through comparison with an unsealed soil surface (denoted herein as “mulched,” since this would typically occur only if the surface was mechanically protected from raindrop impact).

[11] The above mentioned expressions were developed in a context where rainfall data were available mainly on a daily basis, for which taking the rainfall intensity to be constant was reasonable. In the past decade the use of electronically recording tipping bucket, radar rainfall estimates, and desdrometers has made high temporal resolution rainfall data widely available. It is therefore timely to have a simple method for estimating *t*_{p} that can readily be applied to complex rainfall patterns. It is further of considerable interest to study the effect of the time-averaging interval on ponding and runoff estimates to understand how the rainfall reporting interval affects *t*_{p} estimates.

[12] Infiltration, time to ponding, and runoff generation are considered as they manifest in three soils simulated to have been exposed to a natural rainfall event with highly variable intensity. The specific focus here is put on estimation of *t*_{p} considering (1) the effect of the time interval for averaging rainfall intensity data and (2) the effect of the soil surface sealing. A simple, direct method for estimating *t*_{p} for any pattern of temporal variation in rainfall intensity is presented in comparison to direct numerical simulations. The study does not intend to be comprehensive, with the important considerations of initial water content distribution, hysteresis in water retention, hydrophobicity and soil swelling being not included. However, relying on the results of *Assouline and Mualem* [2002] and *Govindaraju et al.* [2006], the method can be directly applied to space-averaged infiltration when spatial variability in soil and rainfall are accounted for.