The prediction of the time to ponding following commencement of rainfall is fundamental to hydrologic prediction of flood, erosion, and infiltration. Most of the studies to date have focused on prediction of ponding resulting from simple rainfall patterns. This approach was suitable to rainfall reported as average values over intervals of up to a day but does not take advantage of knowledge of the complex patterns of actual rainfall now commonly recorded electronically. A straightforward approach to include the instantaneous rainfall record in the prediction of ponding time and excess rainfall using only the infiltration capacity curve is presented. This method is tested against a numerical solution of the Richards equation on the basis of an actual rainfall record. The predicted time to ponding showed mean error ≤7% for a broad range of soils, with and without surface sealing. In contrast, the standard predictions had average errors of 87%, and worst-case errors exceeding a factor of 10. In addition to errors intrinsic in the modeling framework itself, errors that arise from averaging actual rainfall records over reporting intervals were evaluated. Averaging actual rainfall records observed in Israel over periods of as little as 5 min significantly reduced predicted runoff (75% for the sealed sandy loam and 46% for the silty clay loam), while hourly averaging gave complete lack of prediction of ponding in some of the cases.
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 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.
 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  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  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 . The main result is that accounting for soil surface sealing has a greater effect on infiltration than accounting for soil spatial variability.
 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  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  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.  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  for the unsealed (mulched) soil surface case.
 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.
 During infiltration under shallow ponding (i.e., where infiltration is not strongly affected by the depth of ponding) the infiltration capacity rate, fcap, decreases due to the decrease of the hydraulic head gradient resulting from the advancement of the wetting front. The infiltration capacity curve, fcap(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 fcap(t), is a function of the pattern of water application.
 Once tp 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 < tp, F(t) is equal to the cumulative rainfall, R(t) = r(t)dt. One may define the cumulative infiltration capacity, Fcap (t) = fcap (t)dt, and a compression reference time, tcr, which is the time required to produce the same cumulative infiltration volume under shallow ponding conditions from t = 0. Thus F(tp) = R(tp) = Fcap(tcr). Once tp and tcr are known, f(t) for continued ponding can be evaluated as fcap(t − t0), with t0 = (tp − tcr). It is evident that the TCA requires an accurate estimate of tp and tcr. As was true for tp, the available expressions for estimation of tcr assume either constant or time-averaged wetting rate [Brutsaert, 2005] which limits the practical utility of this approach.
where r(t) is the observed rainfall rate which is required to be at most slowly varying close to tp; rp is the rainfall rate at tp; A is a linear function of the initial water content assumed constant with depth in the soil profile; Ks is the saturated hydraulic conductivity; and β is a parameter found to be close to 2. Parlange and Smith  proposed an alternative expression which requires one less parameter that can be applied for any rainfall pattern for which rp < Ks:
 These expressions are implicit functions where the unknown variable is tp, 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).
 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 tp 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 tp estimates.
 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 tp 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 tp 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  and Govindaraju et al. , the method can be directly applied to space-averaged infiltration when spatial variability in soil and rainfall are accounted for.
2. Direct Method for Estimating tp and Infiltration After Ponding
 We seek an explicit method to compute the time to ponding using arbitrarily time varying rainfall rate. Until ponding
 Thereafter, in accordance with the framework of TCA/IDA [Smith et al., 2002; Brutsaert, 2005], it is assumed that for the period of ponding, the actual infiltration rate is a one-to-one function of cumulative infiltration. On the basis of numerical simulations, this assumption has been shown to be valid in the cases of homogeneous and layered soil profiles [Smith, 1990], as well as for sealed soil profiles [Mualem and Assouline, 1996; Assouline and Mualem, 2001]. Adopting this assumption we may now define the postponding infiltration rate as
 Therefore, at the moment ponding occurs,
 The time to ponding, tp, is thus the time when the condition r(R) = fcap(Fcap) is fulfilled:
 Since the function Fcap is obtained directly from the known fcap and R is measured, equations (5) and (6) are in essence equivalent and allow direct calculation of tp.
 This calculation is valid for any rainfall pattern. Many mathematical expressions are available for fcap(t) [e.g., Green and Ampt, 1911; Kostiakov, 1932; Horton, 1940; Philip, 1957a; Smith and Parlange, 1978; Parlange et al., 1999], most of which can be readily adapted to represent fcap(Fcap). Alternatively, simple expressions can be fitted to measured fcap data. For the cases where the rainfall patterns are simple and r(R) and t(R) can be described mathematically, closed form expressions of tp can be developed based on equation (6). For example, fcap(Fcap) can be described by means of
where β is a soil-dependent parameter. Using equation (6) for a constant rainfall intensity case leads to
 Similarly, but now describing fcap(Fcap) by means of
where γ is a soil-dependent parameter, leads to
 For convenience time and rainfall rate can be rescaled following Brutsaert :
 The infiltration rate after ponding is given in equation (4). The difference between the actual rainfall rate and the infiltration rate is referred to as the rainfall excess representing the potential runoff rate, q:
 The only assumption made here is that after ponding the infiltration rate is a unique function of cumulative infiltration. It does not require, for example, any of the assumptions made to describe the fcap(t) relationships that led to the previous expressions of tp [Smith et al., 2002; Brutsaert, 2005]. This method does not require specific computation of the soil properties Ks and S, though these might be fit to the fcap(t) function, or conversely fcap(t) could be computed if the soil properties are known. It is further noteworthy that this method is valid for any time-varying rainfall pattern. The infiltration after ponding predicted by equation (4) does not require the evaluation of the compression reference time, tcr, avoiding additional sources of errors. Of interest in this study is the feature that the method allows for direct computation of the possible impact of the time aggregation of rainfall data on infiltration, ponding time, and runoff estimates.
 Three soil types with widely differing hydraulic properties were selected: Sharon sandy loam (SL); Ruhama loam (L); and Atwood silty clay loam (SCL). These soils were chosen because they are well characterized for infiltration, including having hydraulic properties of the respective seal layers that develop during exposure to rainfall [Assouline and Mualem, 1997; Assouline, 2004].
 The data of the rainfall event chosen to represent the temporal variation of the rainfall intensity is presented in Figure 1. This event occurred on 18 December 2003 at Ramat Hacovesh, central Israel, and represent typical events during the rainy season in this region of semiarid climate. It was measured using a tipping bucket, for which each tip was calibrated to correspond to 0.1 mm of rainfall. We employ the first 60 min of the storm during which rainfall intensity varied from 2.0 to 120.0 mm/h. For the analysis of temporal averaging these data were then aggregated into 5 min, 15 min and 60 min averages.
 Infiltration into undisturbed soil profiles, representing the case where the soil surface is protected from the raindrop impacts by mulch, and into soil profiles over which a seal layer had formed were simulated using HYDRUS-1D [Simunek et al., 2005]. The van Genuchten  expression for the water retention curve, and Mualem's  model for the hydraulic conductivity function were used. In the case of the sealed soil profiles, the surface hydraulic properties used were those determined by Assouline and Mualem  which employ the Brooks and Corey  expression for the retention curve. The parameters for the van Genuchten  model were determined by fitting retention data over the 0 to −200 cm capillary head range (Table 1). The seal layer thickness was taken to be 4.0 cm, and the depth of the simulation domain was 100 cm, which was sufficient to assure that the wetting processes did not come in contact with the lower boundary. The upper boundary condition switched from a Neuman condition to a Dirichlet condition after ponding. The lower boundary was assumed to be a free drainage boundary. Initial capillary head of −100 cm was applied to the whole profile. Time discretizations were as follows: initial time step, 0.5 min; minimum time step, 0.014 min; and maximum time step, 1.0 min. HYDRUS-1D was also used to generate the infiltration capacity curves, fcap(t), for the different soils and surface conditions. In that case the upper boundary condition was a Dirichlet condition for the whole simulation. The resulting fcap(t) curves for the three soils and the two soil surface conditions are depicted in Figure 2, and present a wide range of soil infiltrability. The model of Philip [1957b] was fitted to fcap (t) to estimate the sorptivity, S.
Table 1. Parameters of the Hydraulic Properties of the Mulched (Undisturbed) Soils and of Their Respective Seal Layersa
S2, cm2 min−1
Ks, cm min−1
θs, m3 m−3
θr, m3 m−3
S is the soil sorptivity; Ks, the saturated hydraulic conductivity; θs and θr, the saturated and residual water content; and α and n, the parameters in the van Genuchten  expression for the water retention curve.
Atwood silty clay loam (SCL)
Ruhama loam (L)
Sharon sandy loam (SL)
4. Results and Discussion
4.1. Direct Method for Estimating tp
 The performance of the TCA method is illustrated in Figure 3 for the three soils and the case of the sealed profiles as they all produce runoff for the rainfall event. The ponding time, tpcal and the compression reference time, tcrcal are estimated according to the expressions suggested by Brutsaert :
where S is the sorptivity; Ks is the saturated hydraulic conductivity of the seal; p is the mean rainfall intensity prior to ponding; and a and b, the fitting parameters of Philip [1957b] equation to fcap (t) (with a = S/2). Note that equation (13) follows from equation (2) if rp is replaced by p. Values of variables obtained using the simulation model will be identified by subscript “sim” while those calculated by either traditional or the proposed method will be identified by subscript “cal”. In Figure 3 the cumulative measured rainfall depth, R(t), and the simulated cumulative infiltration, Fsim(t) curves are becoming distinct at the simulated ponding time tpsim. Also depicted in Figure 3 are the cumulative Fcap(t) and the cumulative Fcap(t − t0cal) curves, with t0cal = (tpcal − tcrcal). According to the TCA method, (1) the estimated ponding time, tpcal, is at the intersection between the R(t) and Fcap(t − t0cal) curves, and (2) the cumulative infiltration after ponding should be represented by Fcap(t − t0cal). It can be seen that the TCA predictions conclusively failed for the SL and the L soil cases. In the first one, it overestimated tpsim with an error of 64%, and in the second, it both underestimated tpsim (error of −291%) and yet overestimated the cumulative infiltration after ponding. For the SCL soil case, characterized by the lowest hydraulic conductivity and infiltrability, the TCA method also underestimated tpsim (error of −277%) and overestimated the cumulative infiltration after ponding, as found for the L soil, but the differences are smaller.
 The proposed direct method to estimate the ponding time (equation (6)) is illustrated in Figure 4. Three curves are shown in Figure 4 (top): the rainfall intensity versus the cumulative rainfall, r(R); the simulated infiltration capacity curve versus the cumulative infiltration capacity, fcap(Fcap); and the actual simulated infiltration curve versus the cumulative infiltration, fsim(Fsim) corresponding to the rainfall event. In Figure 4 (bottom), the inverse function of cumulative rainfall versus time, t(R), is plotted, with the abscissa on the same scales between Figures 4 (top) and 4 (bottom). The cumulative depth at which fcap(Fcap) and r(R) (the two dashed curves) intersect (Figure 4, top) is translated into the estimate of the ponding time, tpest, through the t(R) curve (Figure 4, bottom). The verification of this estimate is carried out using the simulated fsim(Fsim) curve (solid line). The simulated ponding time tpsim, representing the “exact” solution, can be determined from the intersection between fsim(Fsim) and r(R) (Figure 4, top) and the t(R) curve (Figure 4, bottom). It can be seen that the accuracy of the suggested direct method is good, even when ponding occurred during the steep increase of rainfall intensity.
 Graphical presentation of the procedure makes the implementation, which would typically be carried out numerically, easily understood (Figure 4). The graphical construction is as follows: (1) plot, against the same cumulative depth axes, fcap(Fcap) and r(R); (2) plot the t(R) relationship using the same x axis as in the previous plot and align the origins; (3) use the first plot to determine the value of R at which r = fcap; (4) scanning down to the second plot, determine the time corresponding to the previously determined R value, which is tp.
 Following this methodology, the occurrence and timing of ponding is presented for all of the cases considered in this study (Figure 5). It is immediately apparent that ponding will occur for the mulched SCL and for all the sealed surface cases. Estimates of the expected amount of runoff are also readily obtained (equation (12)).
4.2. Effect of Time Interval for Rainfall Intensity Aggregation on Infiltration and Runoff
 The suggested approach also offers the possibility to estimate the effect of temporal variability of rainfall on ponding and runoff. This is demonstrated by considering the effect of representing the rainfall event discussed above using temporally averaging over 5, 15, and 60 minute intervals. In Figure 6, the fcap(Fcap) for the mulched and the sealed SCL soil cases (those that produced runoff using the continuous time record) are plotted along with the r(R) curves. When the soil is mulched, no runoff is predicted using the 60-min and the 15-min time intervals, while some (but still far less than with the complete data set) is predicted based on the 5-min time averaged data. When the soil surface is sealed, runoff is produced for all the rainfall data sets; however the time averaging interval affects the estimated ponding time and the expected amount of runoff produced.
 The effect of the averaging interval is also apparent in the time evolution of the capillary head at the soil surface during rainfall (Figure 7). A monotonic power-like increase of head is seen in the case where the rainfall intensity is constant. When the temporal variability of rainfall intensity is accounted for, the evolving head is no longer either smooth or monotonic, with significant dependence on the time averaging applied, with most dramatic discrepancies during the brief high-intensity periods of the storm. As the time averaging interval decreases, a higher peak value of head is simulated. This can be of importance when infiltration during subsequent rainfall events has to be considered since accounting for temporal variability of rainfall can affect the initial conditions in the soil profile at the consecutive rainfall event. For the sealed condition (Figure 7, bottom) the constant rainfall intensity leads to head at the surface ≤ −20 cm and no runoff, of the more resolved rainfall intensities produce ponding (head=0) and hence runoff, though the time to ponding is different and related to the averaging time interval. The differences at low rainfall intensities are larger for the sealed than for the mulched soil. The impact on subsequent rainfall events may also be expected to be greater in sealed soils.
 The simulated cumulative infiltration curves corresponding to the case depicted in Figure 7 (bottom) are shown in Figure 8 along with the cumulative rainfall for the different averaging time intervals. For the constant rainfall intensity, the two lines are linear and coincide exactly, as expected since no runoff was simulated. As the averaging time interval decreases, the departure of the cumulative infiltration from the cumulative rainfall increases and more runoff is “produced” for the same rainfall event.
 To summarize, the effects of the time averaging interval on the estimated ponding time and total runoff (Figure 9) may be presented relative to the values obtained using the numerical simulation. Employing an averaging time of 60 min always underestimated ponding time, with an error of around 50% for the loam soil case (Figure 9, top). The averaging time interval of 5-min allows a relatively accurate estimate of tp, while the effect of the intermediate time averaging interval of 15-min varies with the soil type. For the sealed SL and SCL soils, it causes to a strong overestimation of tp, the error in the case of the SCL soil being of 100%. For the sealed loam soil, it underestimates it, and the error is around 25%.
 The effect on the relative total runoff (Figure 9, bottom) is consistent in greater underestimation with increasing averaging time. The trend, however, is soil type and soil surface dependent. The impact of the time averaging interval is negligible for the sealed Atwood soil but huge for the sealed sandy loam or the mulched Atwood soil. It appears to be convex for the soils with the higher conductivity (infiltrability), and concave for the lower-conductivity soils. In both cases, the errors can be substantial: for the mulched SCL soil, only 46% of the simulated runoff are produced by the 5-min averaged rainfall data, and 75%, for the sealed SL soil, while for the L soil, 78% of the simulated total runoff is produced by the 15-min averaged data, and 87%, when the 5-min averaged data are used.
 The scaled tp+(rp+) values, using equation (11), that were estimated using the simulated infiltration curves were compared to those resulting from the new method (equation (6)) and to those obtained from the traditional approach (equation (13)) (Figure 10, top). The calculated tp+(rp+) curve (equation (13)), which assumes a constant rainfall intensity prior to ponding, underestimates ponding time for all but one case, typically with error on the order of a factor of 10. The mean error was computed according to:
where k is the number of points in the sample. The mean error was 87% for the calculated tp+ using equation (13) and dropped to 7% for the suggested method (equation (6)). The relationship given by equation (13) is best up to rp+ = 20.0 where errors are less frequently less than a factor of 10, but above this value the predictions are essentially without predictive significance.
 The scaled t0+ values estimated by the direct method (equations (6) and (14)) and calculated using equations (13) and (14) are plotted versus the scaled corresponding simulated values (Figure 10, bottom). The comparison between the calculated and simulated t0+ values makes clear the dramatically improved performance of the direct method presented here to estimate tp while accounting for temporal variability of rainfall intensity, in which ME (equation (15)) for the new method is 10%, while the traditional method yields ME of 56% for the cases considered here.
5. Summary and Conclusions
 We have proposed an accurate and practical method of using a comparison of the integrated rainfall versus the integrated infiltration capacity to predict the time to ponding and the subsequent excess rainfall. The method builds directly on a wealth of previous related work, but with several significant improvements. The calculations themselves are greatly simplified, since the method is explicit, therefore not requiring iterative estimation of the time to ponding. The approach also allows direct calculation of excess rainfall. Finally, the method requires no particular pattern to rainfall events, while most previous analytical methods were quite restrictive in this aspect. The comparison of the predictions of this simple approach to precise numerical simulations provides striking support for this approach over traditional TCA with respect to the accuracy of the predictions obtained. While the time to ponding using the suggested method had a mean error of 7%, the traditional methods had errors that include complete lack of prediction of ponding, and in many cases, predictions of the time from start of rainfall to ponding that were in error by a factor of 10. It is clear there are conditions in which the two approaches would be indistinguishable in performance (e.g., constant rainfall rate). Though our testing of the comparative performance of the new method to the standard TCA was by no means comprehensive, considering only a very small set of rainfall patterns, it appears that the precision afforded justifies serious reconsideration of use of the techniques that have heretofore been standard. Further exploration of possible errors for specific rainfall-soil conditions is needed to gain reliable assessment of the accuracy of the proposed model, with particular emphasis on the wide range of rainfall patterns observed in the diverse climatic systems where such an approach might be applied.
 An area of further research need that is particularly evident is that of the affect of antecedent soil moisture on time to ponding and infiltration. Clearly rainfall often occurs in sequences of closely spaced events, thus resolution of this issue is critical to the advancement of these concepts to many rainfall records. Can one simply integrate the water content over the characteristic capillary length scale of the soil and employ this as effective cumulative infiltration with the method we have proposed? The success of this approach thus far suggests that the issue of antecedent water content may well be amenable to an analytical approach, in which case quite accurate prediction of runoff to rainfall might be tractable as well.
 This study was performed during a joint visit of the authors to the EPFL, Lausanne. The authors thank the School of Architecture, Civil and Environmental Engineering (ENAC) and the directors of the sponsoring laboratories (M. Parlange, D. Or, and A. Barry) for their support and outstanding hospitality.