Infiltration models for hydrological applications are developed for rational forms of the soil hydraulic conductivity and moisture retention functions. The models are based on the traveling and kinematic wave solutions of the governing equation that proved to be highly accurate in specific flow conditions. The approximate solutions describe the one-dimensional infiltration into a semi-infinite medium for uniform initial conditions under either a time-dependent flux or a constant head boundary condition. They capture the profile development from early to late time accurately due to the proper derivation of the wetting front speed in terms of the time-varying surface boundary potential and flux rate. Algebraic expressions of theoretical and practical importance are derived in terms of well-defined soil water parameters and include the effect of the antecedent moisture content in a specific fashion. Practical results include simple expressions for the surface moisture variation for variable rainfall patterns, time-to-ponding expressions and infiltration equations that account explicitly for the initial condition, formulas for the depth of the wetting zone, and algebraic equations for parameter estimation of the soil hydraulic parameters by inverse analysis.
 Infiltration is important in a broad range of applications, including irrigation, aquifer recharge, hydrologic budget, and climate change studies. It is a key component in the hydrological cycle, and it has a significant impact in flood forecasting models and water budget calculation. The incorporation of physically based infiltration expressions leads to a better quantification of the infiltration component in hydrologic models and a more reliable prediction of the runoff hydrograph. This work is an attempt to make useful contributions in that direction.
 The infiltration process is represented by a highly nonlinear parabolic partial differential equation known as Richards' equation. A series of papers dealing with exact nonlinear solutions to the Richards' equation has been published over the past two decades [e.g., Sander et al., 1988; Broadbridge and White, 1988; Warrick et al., 1990; Kühnel et al., 1990; Warrick et al., 1991; Triadis and Broadbridge, 2010]. The solutions are derived for constant boundary conditions and they are relatively complex to use since they are in parametric form and implicit in depth. In the present study, approximate solutions are presented that are simpler in form and adapt easily to changing boundary conditions. They consist of the kinematic wave approximation where the second-order diffusive term is ignored and the traveling wave approximation where the local time derivative in a moving coordinate system is neglected. They are validated against a numerical solution and an exact solution derived for the case of infiltration under a variable flux. The approximate solutions turn out to be highly accurate for specific flow conditions.
 The objective of the present study is to develop practical and theoretically sound expressions for hydrological applications that preserve the inherent nonlinearity of the infiltration phenomena while being simple and flexible. For hydrologic applications, proper expressions for the variation of the surface moisture content, the time to ponding, the infiltration rate, and the runoff rate must be available. Water balance models must be able to predict the infiltration and the redistribution component with sufficient accuracy while being simple to use at coarse time scales. Hydrologic models employ empirical equations to quantify some of these variables and the primary limitation of these equations is that the parameters are not physically well-defined.
 In the present work, simple expressions of theoretical and practical importance in hydrology and irrigation are presented. The results are stated in terms of well-defined soil water parameters and include the effect of the antecedent moisture content in an explicit fashion. These include simple expressions for the moisture content variation at the land surface for variable rainfall rates, time-to-ponding expressions and infiltration equations that account explicitly for the initial condition, formulas for the depth of the wetting zone, and algebraic equations for parameter estimation of the soil hydraulic parameters by inverse analysis. The above expressions describe the main components of hydrologic models and can lead to better prediction than models that assume instantaneous redistribution and neglect the effect of antecedent moisture content or capillary impacts.
 The paper is organized as follows. Section 2 introduces the governing nonlinear partial differential equation of one-dimensional nonsteady infiltration in transformed dimensionless form. Section 3 extends the exact solution of infiltration and redistribution to the case of a piecewise constant flux rate at the soil surface with a piecewise uniform initial condition. Sections 4 and 5 present the traveling and kinematic wave solutions, respectively, which constitute the basis from which infiltration models of hydrological interest are developed. Section 6 evaluates the solutions under various conditions and illustrates the effect of the hydraulic conductivity parameter and the antecedent moisture condition. Section 7 introduces algebraic models for the variation of the surface water potential with rainfall intensity, the time to ponding for predicting the onset of runoff, and the subsequent infiltration rate after ponding. Section 8 presents algebraic equations for parameter estimation of the soil hydraulic parameters by inverse analysis and section 9 concludes the study.
2.1. Governing Equation
 The nonlinear Richards' equation describing water flow in the unsaturated zone is [Richards, 1931]
where ψ = the pressure head [L], = the volumetric moisture content [–], = the unsaturated hydraulic conductivity [L/T], s = source or sink term (e.g., water uptake) [1/T], z = the vertical distance (positive downward) [L], and t = the time [T]. The superscript * denotes that the variables are in dimensional form.
Here, θ is the normalized water content, is the saturated hydraulic conductivity, and is the residual moisture content whereby . The soil hydraulic diffusivity is an inverse quadratic function of the water content [Fujita, 1952]
The implied conductivity-pressure head relationship can be derived from the definition of the soil water diffusivity using the condition that at , where is the air-entry pressure head. Thus, one obtains an exponential dependence of the hydraulic conductivity on the soil water pressure head
The inverse length parameter is a measure of the relative importance of capillarity and gravity forces on water movement, and the parameter ke accounts for the pressure shift below which the hydraulic conductivity k decays exponentially as expressed by (5).
 The nonlinearity of the diffusivity and hydraulic conductivity is quantified by the shape factor ν. For , the D and k functions become linear and as v departs from zero, the nonlinearity increases. Figure 1 is a plot of the hydraulic conductivity function k for various values of the parameter v, and it also highlights the normalized moisture content θ values for selected values of the dimensionless pressure head for ke = 1. For large values of v, the shape parameter has a significant effect on the variation of k. The practical range of v is found to be restricted between 0.5 and 1. The value of v for a typical nonlinear soil is around , while for , the soil is highly nonlinear, and for , the soil is considered mildly nonlinear. Suitable values of v for a wide range of soils can be found in the work of White and Broadbridge  in the form of the parameter C where . The coefficient of the diffusivity (4) is also set to highlight the delta function character of D when v approaches 1 since the limiting values of both (2) and (4) yield a delta-function model for both and D similar to the Green-Ampt model. This choice of k and D enables us to study the effect of the nonlinearity on the moisture movement and can be considered as a fitting approach for extending the linear theory into the nonlinear one.
2.2. Dimensionless Transformed Form
 Using a particular case of the Kirchhoff transformation
The soil water relationships in terms of the transformed pressure head u become
Introducing the following dimensionless variables:
The governing equation becomes in dimensionless form
The above dimensionless system is in terms of four independent intrinsic soil parameters: the saturated hydraulic conductivity , the capillary length parameter , the shape parameter of the hydraulic conductivity function v, and the air-entry pressure head . The length ls and time ts scale used for the dimensionless variables are in terms of the common parameters and :
This dimensionless system permits easy comparison with the solution of other models, e.g., linear and Burgers, since they are based on the same scaling variables and are independent of the flow condition, whether it is flux-or head-controlled infiltration.
 An alternative scaling that is most commonly used is in terms of the soil sorptivity [Philip, 1957b] or the diffusivity possibly with a numerical factor. The macroscopic length scale ls and the associated time scale ts are then defined by
The dimensional sorptivity is a function of the flow condition and depends on the soil water content. It is defined in an integral form and is usually solved numerically. For the selected diffusivity function (4), it has an analytical form for the concentration boundary condition only (see equation (71)).
2.3. Initial and Boundary Conditions
 The initial condition consists of the specification of the initial moisture content in the whole flow domain at . In terms of the transformed potential u, the initial condition becomes . For a piecewise constant pressure profile over a defined depth Z0 with ua as the upper layer potential and ub as the background potential, the initial condition is given by
where U is the unit-step function defined as U(x) = 1 for x ≥ 0 and U(x) = 0 for x < 0. It should be noted that the initial moisture content is arbitrary and need not be equal to the residual moisture content that appears in the scaling factor (10).
 For a piecewise constant rainfall rate stepped over arbitrary time intervals, the applied flux can be expressed by
The boundary condition at the soil surface is then given by
where is the dimensionless flux distribution, positive for infiltration and negative for evaporation. The dimensionless cumulative flux is defined by
For a prescribed head at the soil surface, the boundary condition is where is the temporal variation of the pressure head at Z = 0. For shallow ponded conditions with negligible depth of ponding, ut = 1 (). For the far-field condition in semi-infinite domains, the dependent variable u can be normalized such that the boundary condition at infinity vanishes, i.e., by defining u′ = u − u∞ where . The flux is then given by (14) but with replaced by .
3. Exact Solution
 An exact solution of (9) is possible only for the case of constant flux boundary conditions. The nonlinear partial differential equation is reduced after three successive transformations [Kirchhoff, 1894; Storm, 1951; Hopf, 1950] to the linear diffusion equation. The particular case of infiltration and redistribution for a piecewise constant flux rate (13) at the soil surface with piecewise uniform initial condition (12) is presented below using the Green's function method [Greenberg, 1971]. The solution is an extension of previously published exact solutions for constant rate infiltration with zero initial condition [Sander et al., 1988] and for moisture redistribution with uniform [Kühnel et al., 1990] and nonuniform initial conditions [Sander et al., 1991]. It is also analogous to the solutions derived for an alternate form of k [Warrick et al., 1990; Warrick et al., 1991].
Hence the solution of (19) satisfies also the linear diffusion equation
The initial and boundary conditions are similarly transformed to give
where is the transformed rate. Using the method of Green's function, the solution of (20) is
where is the appropriate Green's function for this type of boundary condition (22):
and is the derivative of the Green's function at ,
3.2. Particular Solutions
3.2.1. Constant Rainfall
 For a constant rainfall and a piecewise uniform distribution of transformed pressure potential as defined by (12), the solution is obtained directly from (23) using (21) and (22):
where , , , and . The function is defined by
For a uniform initial condition , the bracket term cancels out. The solution in terms of u is given by (18) noting that the derivative of v is requisite, and the depth corresponding to the intermediate variable z is obtained from the integration of (16) using (18) and (22),
3.2.2. Variable Rainfall
 For a piecewise uniform distribution of transformed pressure potential as defined by (12) and a piecewise constant flux rate stepped over arbitrary time intervals as defined by (13), the solution in terms of u at time is given by (18) and (26). The solution for is derived similarly using the initial and boundary conditions evaluated from the solution of the previous time interval at . The relationship between v0 and v1 is obtained by equating the depth Z at time using equation (28) and the equivalent expression for . Hence
where . The initial and boundary conditions for the interval are
The potential for the interval is obtained from (23) using (30) and (31),
The integral at large values of ζs becomes computationally difficult to evaluate especially given that v0 in the integrand is not a constant or a simple function. It can be divided into two integrals at a sufficiently large value of z such that the asymptotic form of the integrand is used [cf. Warrick et al., 1991]. The transformed pressure is evaluated from the equivalent expression of (18) for , i.e., . The solution for the subsequent pulse in the interval can be obtained in a similar fashion and the result is expressed as a sum of two terms, one that is a function of the instantaneous rainfall rate and the other that accounts for the previous infiltration history.
4. Traveling Wave Approximation
 The traveling wave approximation assumes that the form of the solution can be expressed in terms of where is the propagation speed. This in effect reduces Richards' equation to an ordinary differential equation that can be integrated analytically for simple descriptors of k. The traveling wave solution was first obtained by Philip [1957a], who called it as the profile at infinity, and was further discussed by Parlange . Additional traveling wave solutions have appeared in the literature and they all assume that the traveling wave velocity is constant [e.g., Broadbridge and White, 1988; Witelski, 2005; Zlotnik et al., 2007]. In the present work, a time-varying traveling wave velocity is derived for either a specified potential or flux at the soil surface. One of the benefits of the solution is that it captures accurately the profile development from early to late time.
4.1. General Solution
 Using a transformation variable and , equation (9) becomes
Neglecting the local time derivative in the moving coordinate system as it becomes smaller than the remaining terms at sufficiently large times, one obtains an ordinary differential equation,
The solution of (34) gives the shape of the traveling wave profile. Setting , separating and integrating, one obtains
Using the far-field condition , one finds
A second integration of with respect to an arbitrary reference point yields
The constant of integration can be determined by matching the large time traveling solution with an early time solution at a reference point [Philip, 1957a]. Alternatively, one can evaluate the constant by equating the cumulative infiltration with the water storage [Broadbridge and White, 1988]. Selecting the top boundary as the reference point, ur = ut at and noting that , one obtains
Equation (39) yields the depth Z for a given degree of wetting θ using (38). The solution is not complete until the traveling wave velocity and the top boundary potential ut are defined.
4.2. Traveling Wave Velocity
 The traveling wave velocity is estimated by minimizing (34) over the soil domain,
Integrating once and applying the definition of the flux (14) at both boundaries, one gets
Equation (41) demonstrates that varies with time until the surface potential approaches asymptotically the value of the flux rate or vice versa. At large times, and the traveling wave speed becomes constant equal to , which is actually the value used by previous investigators [e.g., Broadbridge and White, 1988; Witelski, 2005; Zlotnik et al., 2007]. The dynamic wave speed does not result in a constant-in-shape linearly translating traveling wave profiles as obtained with and thereby offers an accurate simulation of the infiltration process from early time onwards.
4.3. Surface Flux
 A correct estimate of requires an accurate evaluation of ut and . The flux or surface potential ut can be estimated by equating the cumulative flux rates crossing the soil boundaries with the moisture content storage within the soil. The mass balance equation is expressed by
The left-hand side of (42) is the difference of the cumulative flux rates at the boundaries of the domain and the right-hand side is actually the integral of the moisture content θ over depth Z when expressed in terms of u. The integration over u requires a change of integration variable through , which is the inverse of p(35) with A and defined by (36) and (41), respectively. For a uniform initial condition ui = ub, equation (42) yields
where is the dimensionless cumulative infiltration (15). The flux rate at the surface can be obtained from (43) for uniform background conditions. Substituting (38) and (41) with into (43) and solving for , one finds
where , with , , and is the net cumulative infiltration. Replacing and integrating for with and , one gets a nonlinear algebraic equation in :
Solving for iteratively, one can then determine from (44). The infiltration law (44) is a form of a logistic function and equation (45) has the exact functional form of the three-parameter infiltration equation of Parlange et al.  that is discussed further below. Equation (45) bears also resemblance to the Green-Ampt equation when one expands the exponential term in a two-term Taylor series for small values of the net cumulative infiltration .
4.4. Surface Potential
 Setting the surface flux rate in (44) equal to the applied flux rate , and solving for the potential value at the soil surface, one finds
where and are the normalized surface potential and applied flux rate, respectively, and . For the asymptotic traveling wave velocity given by , equation (47) simplifies to
4.5. Nonuniform Initial Profile
 In principle, the above integration procedure can be carried out for any initial profile. However, the traveling wave approach is most suitable for uniform or piecewise uniform profiles since it assumes that the traveling wave profile is uniquely determined from the potential and flux values ahead and behind the wetting front (41). The accuracy of the traveling wave solution deteriorates when there is a significant transient dynamics due a highly nonuniform initial moisture distribution or redistribution. The TW approach may be applicable to layered potential profiles only if the evolution of the wetting front is broken up into as many parts as the number of layers.
 For a finite medium with an initial profile at equilibrium with the water table, the TW model can be applied in the wetting zone delineated by u(Zf) ≥ ui(Zf) after iterations on the depth of the wetting zone Zf, the background potential ub, and the resulting wave speed . The traveling wave solution is obtained in an iterative fashion because the wave speed is dependent on the nonuniform background potential ub and flux rate . Starting with the value of the initial potential at the land surface, one determines the traveling wave speed using (41) and the traveling wave profile using (39). The intersection of the calculated traveling wave profile and the steady initial profile gives an improved estimate of the background potential ub and, consequently, the wave speed . An updated profile is then calculated that provides a very good approximation of the potential profile only in the zone above the influence of the water table and until such times when the moisture movement is dominated equally by gravity and capillarity due to the near presence of the water table.
5. Kinematic Wave Approximation
 Kinematic wave models of infiltration and drainage have been applied to unsaturated flow [e.g., Sisson et al., 1980; Parlange, 1982; Smith, 1983; Charbeneau, 1984] and macropore flow [e.g., Germann and Beven, 1985]. The kinematic wave approach assumes that the flux q is only dependent on k. It considers only the influence of gravity and neglects the capillary pressure influence. Unlike the traveling wave approach, the diffusive term in the governing equation is herein neglected. It is therefore an appropriate approximation for gravity-dominated unsaturated flows. For constant flux boundary conditions, the kinematic wave solution is simple and reasonably accurate. For constant concentration condition ut = 1, the kinematic wave approximation yields the piston flow solution.
5.1. General Solution
 The kinematic wave approximation consists of neglecting the second-order diffusive term in (9) and replacing the nonlinear coefficient by a parameter c,
Equation (49) is a linear first-order partial differential equation that describes the propagation of u at a kinematic wave velocity c. Only the first-order term is shown since higher-order terms reduces the accuracy of the kinematic wave approximation significantly, especially at the toe of the wetting front or at any significant change in the boundary condition.
 The general solution of (49) for a given top boundary value and initial profile ui(Z) at is obtained through the Laplace transform method:
The first term on the right-hand side stands for the initial condition at time Ti, the second term stands for the time-varying surface potential distribution, the last term handles the plant uptake, and U is the unit-step function.
 The kinematic wave solution is a good approximation only if the kinematic wave celerity c is appropriately estimated and the potential at the top boundary is properly determined. In previous applications of the kinematic wave approximation [e.g., Smith and Hebbert, 1983], an instantaneous adjustment of the potential at the surface to the changing steady rate pulses is assumed to occur. In sections 5.2 and 5.3, the kinematic wave velocity and the time-varying soil surface potential are derived for the two flow conditions: infiltration and redistribution.
5.2. Kinematic Wave Celerity
 There are two different ways of estimating the kinematic wave celerity c depending on the flow conditions: infiltration or redistribution. In infiltration mode, a wetting front forms, the kinematic waves are self-sharpening, and moving at a constant velocity c, while in the drainage and redistribution case, the waves are self-spreading, and the tracking velocity of each potential value is different (c is not constant) being greater with increasing potential. In between the two phases, the soil water profile is therefore a combination of self-spreading profiles (drainage waves), constant profiles, and self-sharpening profiles (sharp wetting fronts).
 In the redistribution mode, the classical method of characteristics is used to rewrite equation (49) into two ordinary differential equations that describe the variation of u (as a function of ) along the characteristics defined by
The above solution is valid if the characteristics emanating from the boundary do not cross, which is the case in the redistribution mode.
 In infiltration mode, kinematic shock waves develop because faster waves (higher u) overcome slower waves. In the latter case, the rate of movement of the wetting front can be estimated from the integral representation of the governing equation [Charbeneau, 1984]. Alternatively, one can obtain the same result by minimizing the difference between the original and the approximate form of the governing equation
where ut and ub are the leading and background (initial) potential, respectively. Note that the kinematic wave speed (53) is similar to the asymptotic form of the traveling wave speed . For a constant flux boundary condition at the soil surface, the parameter c as given by (53) produces a good approximation when . For variable rainfall rates, c is computed assuming that the rainfall rate at any time consists of a single mean pulse equal to where is the cumulative rainfall until time . Another estimate of c that conserves mass can be obtained from mass balance considerations using (42). Both approaches offer a good approximation for variable flux infiltration with the implication that the whole front moves at a mean speed cm.
5.3. Surface Potential
 The kinematic wave solution (50) is made of two or more distinct parts, each part being valid in a particular zone of the solution domain. The first zone is under the influence of the initial condition only, and the second zone carries the effect of the upstream boundary condition. Therefore, the value of the potential at the soil surface has a direct influence on the accuracy of the kinematic wave approximation.
5.3.1. Infiltration Mode
 For a characteristic emanating from the surface boundary at time T0, the potential is constant along the characteristics line. Combining the boundary condition at the top (14) and the governing equation (49) with , one obtains
Solving and substituting the condition u(Ti) = ui0 with expressed by (13) and Ti < T1, one gets
Equation (55) requires a priori the value of the potential ui0 at the soil surface. The linear diffusive solution for a flux boundary condition [Basha, 1999] when evaluated at an early time value Ti = 0.05 gives a good estimate. The small-time approximation for T ≤ 0.1 of the linear diffusive solution can be expressed by
The value of the potential during the following pulse T1 ≤ ≤ T2 is obtained by successive integration of (54),
Equation (57) is a function of the surface potential at the start of the second pulse ui1 = uI(T1), which is evaluated from (55) with c = c0 and in (53). A slightly more accurate estimate of the surface potential during the following pulse T1 ≤ < T2 is obtained when ui1 is evaluated from (55) with c = c1 and noting that the surface potential is then discontinuous, i.e., uI(T1) ≠ uII(T1).
5.3.2. Redistribution Mode
 In redistribution, the surface potential is obtained similarly from (54) where the kinematic wave celerity is now given by (51),
Separating and integrating using u(Ti) = ui, one obtains
where and . The solution of (59) gives the surface potential u for any time during the drainage phase.
5.4. Kinematic Wave Model
Equation (50) is the kinematic wave model (KW) for rainfall infiltration, whereby the surface potential distribution is given by (55)–(57) or (59), and the initial profile is computed at time Ti from the linear diffusive solution [Basha, 1999, equation (31)] expressed in terms of with is a scaling factor given by (56) to correctly capture the initial diffusive effects. For variable rainfall, the KW solution is normally applied in a sequential procedure, tracking the boundary values and the initial profile separately until the next pulse application at which time the celerity value is updated and the profile is propagated accordingly. The celerity of the drainage wave being greater than the speed of the wetting front results in the merging of the drainage and wetting fronts and the subsequent reduction of the leading moisture content. The reduction is roughly computed based on water balance considerations within a time step. The tracking procedure of the leading and trailing waves along with the merging process of the draining and wetting fronts is rather cumbersome for a rainfall event consisting of more than two pulses.
 A simpler kinematic procedure for multipulse rainfall is to assume a single wave profile moving at an optimum celerity determined from water balance considerations. The single wave profile implicitly assumes that the various kinematic wave profiles have been combined as they would have done if capillary effects were not neglected. The largest error occurs at the time of a step change in flux at the soil surface. This approach is more useful and easier to apply than the sequential tracking of the kinematic wave solution for variable pulse applications.
6. Application and Results
6.1. Models Evaluation
 The accuracy of the approximate semi-infinite infiltration models is assessed through a comparison with the exact solution for the case of flux boundary condition and with the numerical solution with a very fine grid for the head boundary condition. The exact solution is evaluated from (18) using (26) for the first pulse and (32) for the subsequent pulses. The traveling wave solution is computed from (39) using (38), the traveling wave speed (41), and the surface flux rate (44) or potential value (47). The kinematic wave solution is given by (50) using the kinematic wave velocity (53), the potential value at the soil surface (e.g., (55), (57), or (59)), and the linear diffusive solution at Ti = 0.05 for the initial profile (56).
 The traveling wave (TW) and the kinematic wave (KW) models are expressed in terms of one or two dynamically varying parameters that are predetermined at a selected time value. The key parameter in the TW model is the traveling wave velocity , which is in terms of the set boundary values and the calculated ones. For a fixed potential boundary condition, the corresponding value of the flux is calculated from (44) as a function of time, while for a given flux rate at the soil surface, the corresponding potential value is estimated from (47) as a function of time. The background values of u and are set to be equal in a semi-infinite medium, i.e., .
 The KW model is a function of the kinematic wave speed and the potential at the soil surface. The kinematic wave speed is a function of the leading potential, which is set equal to the mean flux rate for flux-controlled infiltration and to the surface potential for head-controlled infiltration. For flux-controlled infiltration, the potential at the surface can be calculated as a function of time and the KW model provides a reasonable approximation of the infiltration front. For head-controlled infiltration, the corresponding flux for the fixed surface potential cannot be estimated as a function of time since the KW model predicts an instantaneous adjustment to the asymptotic value of . Hence, the KW model cannot capture the essential dynamics of the process for sudden ponding conditions.
6.2. Potential Profiles
Figure 2 compares the TW and KW models against the exact solution for the case of rainfall infiltration with variable rates as expressed by (13), whereby and at the time periods defined by T1 = 0.2 and T2 = 0.5. The initial condition is one of uniform potential equal to where , or an equivalent uniform moisture content from (7). The hydraulic conductivity parameter is that corresponds to a typical nonlinear soil. Figure 2 shows the transformed pressure profile at three different times with the first two marking the end of the first two rainfall pulses. Therefore, Figure 2a is the profile for a constant rainfall infiltration, Figure 2b shows the profile after an increase in the rainfall rate, while Figure 2c depicts the profile during moisture redistribution.
 At the earliest time point, the TW solution with an updated wave velocity (41) is a very good approximation while the one with the constant traveling velocity (TWc) is significantly in error near the land surface. The KW solution is a better approximation than TW near the land surface because the surface potential in the KW model (55) is accurately evaluated for T > 0.05. The TW solution does not deteriorate for small time and small flux as previously found by Broadbridge and White  because the propagation speed is herein varying with the potential and flux at the soil surface (41). For constant rainfall rates, the TW model and the exact solution are in close agreement at all times.
 At = 0.5 the TW solution reflects best the variation in the potential profile. The KW solution is an adequate approximation for varying rainfall rates if a proper value of the kinematic wave velocity and initial surface potential are used. The kinematic wave velocity is here determined from (53) with where is the mean rainfall pulse. The surface potential is obtained from (55) (rather than (57)), with an updating of and cn with each pulse while keeping ui0 and Ti = 0.05 fixed. At = 0.7 the error in the KW and TW predictions is larger because both models do not handle accurately the effect of a sudden drop in flux rates. For a sudden decrease in the flux rate, the TW approximation neglects the resulting transient change while the KW approximation neglects the ensuing diffusive gradient. However, the error in the various approximations decreases with time if the boundary conditions remain constant. For ≥ 1 the two models converge to the exact solution with KW being the farthest from the exact solution.
Figure 3 is a plot of the numerical and approximate solutions at for infiltration at a constant potential ut = 1 in a semi-infinite medium for selected values of v and uniform moisture profiles. The front is sharper for lower values of or higher values of v, and it propagates deeper for higher initial moisture content since the available pore space is smaller. The TW model is an excellent model as it provides a very good fit for all values of . The accuracy is due to the correct estimation of the flux rate at the soil surface (44) and thereby the traveling wave speed . The KW model predicts the profile at = 0.25 by tracking the linear diffusive profile at = 0.05 and the top boundary value for > 0.05, which is represented here by the curve with the constant value of u = 1. The resulting profile is similar in shape to a piston flow profile that is moving at a constant speed of .
7. Infiltration Models for Hydrological Applications
 Water balance models and watershed hydrological modeling require the accurate estimation of the moisture content variation at the soil surface, the time to ponding for runoff generation, and the infiltration rate in the preponding and postponding stages. The KW and TW models allow the derivation of simple formulae for each of these infiltration variables.
7.1. Surface Water Potential
 Soil moisture is a key variable in surface hydrology and land-atmosphere interactions because of its role in the water and energy balances at the land surface. The time-dependent equations below are all expressed in terms of the water potential u from which the surface moisture content can be obtained using (7). The expressions also provide the means to evaluate the soil hydraulic parameters from the measured soil surface pressure or moisture content.
 For a constant rate rainfall in a semi-infinite medium with a uniform initial moisture profile, the exact time dependence of the surface water potential is obtained from (18) and (26),
where , and are defined after (26). The limiting forms gives the linear solution for and for . For varying rainfall, the exact surface potential is evaluated from (26) and (32). The TW estimate of the time variation of the surface potential is obtained from (47), while equations (55)–(57) and (59) provide the KW estimate where the wave celerity cn is calculated from (53) with ut set equal to , the actual rainfall rate (as opposed to the mean rate used for potential profile computations). For the first pulse, the initial value of the surface potential is evaluated from the linear solution (56) at a very early time value Ti = 0.005.
Figure 4 shows the rise and fall of the surface water potential in a semi-infinite domain in response to the same varying rainfall rate as the one used for Figure 2. The surface potential ut increases from the given initial value to its asymptotic value, which is equal to the current rainfall pulse . Higher values of v lead to a sharper increase in ut and to a faster equilibrium time. Both the KW and TW models predict the initial rise of the potential accurately. For a sudden increase in the succeeding pulse at = 0.2, the KW predictions of the ensuing rise in u is very good. The TW model predicts a sudden rise in the potential at = 0.2 followed by a more gradual rise. The sudden increase is due to the abrupt increase in the value of as it is directly dependent on the applicable flux value at the surface. This is expected since the TW model neglects the unsteady effects of an abrupt increase or decrease in the surface boundary condition (). The limitation of the TW model is best seen for the sudden drop in rainfall rate at time = 0.5. On the other hand, the KW model includes the nonsteady term and provides a good approximation of the surface potential variation in the redistribution phase ( > 0.5). The simple exponential KW formula (55)–(57) is a very attractive one for surface potential calculation. It is only function of the kinematic wave speed c, the applicable rainfall rate , and the start-up potential ui0. It provides accurate results for the surface values ut or with minimal computation, although the drop in u in the drainage phase is given by a nonlinear algebraic equation (59).
7.2. Time to Ponding
 The time to ponding is the time beyond which surface runoff and erosion occurs. Its estimation is necessary for a proper design of irrigation systems and the prediction of the onset of flood. There are few expressions for time to ponding in the literature [e.g., Parlange and Smith, 1976; Broadbridge and White, 1987]. Most assume a time-averaged or steady rainfall rate and do not account for the antecedent moisture content directly.
 The ponding time Tp is obtained by substituting u = 1 and Z = 0 in the various models. The exact value for time to ponding for constant rainfall intensity is obtained by setting u = 1 in (60) and solving for T. The TW model offers a simpler time to ponding expression for any pattern and antecedent moisture content. Setting ut = 1 and in equation (47) and solving for , which is a surrogate of time , one gets
where is the value of the time-varying rainfall rate at ponding time Tp and . Equation (61) is based on the dynamic traveling wave speed . For the asymptotic constant speed , equation (48) gives
The KW time to ponding expression can also be derived from (55),
where . A similar expression can also be derived from the exact surface potential equation (60) for large values of and , i.e., for large v, large rainfall rates, or dry conditions, and is valid for ,
In the foregoing, it is implicitly assumed that ke = 1; a slight amendment to the algebra is required for ke ≠ 1.
However, the TW expression (61) is the most theoretically correct of all the above simple approximations (61)–(65). It is in terms of well-defined soil parameters rather than generic soil-dependent parameters. The TW expression is also applicable for any value of v > 1/3 rather than near the extreme value of . It is also explicitly dependent on the soil antecedent water content rather than implicitly or indirectly through either the sorptivity or a scaling factor. The dependence on the antecedent soil water condition ub is quadratic in (61) while it is linear in all the other simplified models (62)–(64). The approximation of (61) for large rainfall rates leads to a hyperbolic equation with the right-hand side equal to , which is similar to the Green-Ampt time to ponding expression. Equation (61) also allows the derivation of Tp for any time-dependent rainfall rate. In particular, a linearly or exponentially varying rainfall rate can yield a closed-form expression.
Figure 5 presents the time to ponding variation for constant and varying rainfall rates in a semi-infinite domain with a uniform initial moisture profile. As expected, a wetter initial profile leads to a shorter time of ponding with the curves shifting downward. A similar effect is also found for nonlinear soils with higher v values producing earlier time to ponding. The effect of v on Tp becomes significant for . For , the Tp curve (not shown) is slightly shifted upward from the curve.
 The exact solution allows the quantification of the effect of a varying rainfall rate and a nonuniform initial profile on the time to ponding. Figure 5 depicts the time to ponding for a two-pulse rainfall (curve marked with square symbols), whereby the first pulse is for ≤ 0.2 and the second pulse takes on the values shown on the horizontal axis, i.e., . For variable rainfall rates, the time to ponding for a large subsequent rainfall rate is near the pulse time switch value ( = 0.2). The variation in Tp is significant only for those flux rates () that produce an earlier time to ponding. A conservative estimate of the time to ponding for variable rainfall rates can then be obtained from the single pulse solution by substituting for ui the surface potential at the end of the previous pulse.
 Also shown in Figure 5 is the effect of nonuniform initial conditions on the time to ponding (curve marked with plus symbols), whereby the initial profile is made of two layers: a wetter topsoil with ua = exp(−1) over a depth of 0.5 and ub = exp(−2) elsewhere. One notices that at large rainfall rates the nonuniformity of the initial potential distribution has no effect on the time to ponding. The time-to-ponding curve merges with the one that corresponds to ui of the topsoil for . At smaller rainfall rates, the effect of a two-layered initial profile is apparent and, as the rainfall rate decreases from , the time to ponding varies from the curve corresponding to the top layer to the one equivalent for the bottom layer. The explanation is that redistribution of moisture in the wetter top layer occurs at a faster rate than ponding at the soil surface for .
 Both the KW and TW models are excellent approximations for gravity-dominated flows, i.e., for large v and small ub. The TW model yields accurate results for nonlinear soils () and it slightly overestimates Tp for decreasing v and increasing ub. The KW model works best for dry conditions , whereby the flow is gravity dominated assuming that the start-up value ui0 is found from the linear diffusive solution. The TW formula (61) offers a simple and accurate approximation to the time to ponding and performs better than the KW formula in wet uniform profiles since the KW model cannot capture the diffusive effects. The TW formula works well for varying rainfall rates in case of increasing wetness, but it is likely to deteriorate in accuracy if interim dryer periods exist since a decreasing rainfall rate may lead to hysteresis that the governing equation cannot handle in its present form.
7.3. The Three-Parameter Infiltration Equation
 The infiltration rate and the cumulative infiltration equation are of practical importance in quantifying the infiltration component in rainfall-runoff models. They are also of use in providing a theoretical framework for estimating the hydraulic parameters of the soil. Several simplified infiltration equations have been proposed in the literature and many of these represent inadequate physical detail. Some of these equations are now commonly applied in hydrologic modeling and have been included in popular watershed hydrology models. The TW model produces an equation for the infiltration rate (44) and an associated equation for the cumulative infiltration (45). The TW model is the only one of use because the KW model neglects the dominant effect of the capillary gradients. The infiltration equations are directly dependent on the antecedent moisture profile and are function of well-defined soil parameters that can be measured independently. They are obviously an advantage over empirical infiltration equations with parameters of ambiguous physical significance that are determined from curve fitting.
 The expression for has the exact functional form of the three-parameter infiltration equation of Parlange et al. . The latter equation is based on the assumption of a delta-function model for and D while equation (45) is based on well-behaved functions for k and D and includes the delta-function models as a special case (). Although both are based on the exponential dependence of k on pressure ψ, the starting point and the derivation of both expressions follow a different approach and some of the parameters mean different things. Parlange et al.  used a quasi-analytical method, whereby the shape of the moisture profile is assumed a priori and the conservation of mass with certain simplifying assumptions is then used to predict specific features of infiltration. The same result can also be obtained using a flux-concentration relationship [Smith et al., 2002], which is more invariant than the profile shape and is fairly insensitive to the precise form of D and k. The model of Parlange et al.  is a function of three parameters: the sorptivity, the conductivity at saturation, and a third parameter that can be related to k and seems to be not sensitive to soil structure, being the same value for clay and sand. It is an interpolation parameter whose limit corresponds to a different assumption on the behavior of k near saturation. The extreme values of this empirical parameter produce either the Green-Ampt or the Talsma and Parlange  infiltration models. The physical meaning of the third parameter was later considered as being slightly ambiguous [Haverkamp et al., 1990] and was removed at the expense of introducing another parameter that accounts for the pressure shift near zero. The effect of the initial condition is also taken care of indirectly through the sorptivity term, which is assumed to respond linearly to variations in the initial saturations.
 In the present traveling wave formulation, the derivation of (45) required no assumption on either the shape of the profile or on the flux-concentration relationship. The only assumption in the TW model is that the local time derivative of u in a moving coordinate system can be neglected (). Equation (45) is also function of four independent parameters with a well-defined meaning, , v, ks, and ke, that coalesce into a fewer number of parameters. The third ad hoc parameter of Parlange et al.  is herein replaced by the parameter , where is defined in terms of v (shape parameter of k), and ke (pressure shift of k), and accounts directly for the effect of the antecedent moisture condition. For a saturated soil surface , simplifies to .
 A very good approximation for all values of with ke = 1 is obtained by expanding (45) for large values of v and small values of ub,
 An explicit approximation of (45) can be derived for large values of in series of square root time T1/2,
The small-time approximation of the flux rate is
Equation (68) is an excellent approximation for ≤ 0.5 with an increase in accuracy as v decreases from 1 while equation (69) applies well for short to intermediate times. The two-term version of equation (69) shows resemblance to the Philip [1957b] equation and includes a constant that lies between 1/3 () to 2/3 () for ub = 0 and . The constant varies within a lower range for ke > 1 and within a higher range for nonzero initial conditions (ub > 0) noting that the linear model gives 1/2 for ub = 0 and ke = 1, while Burgers model gives .
 The large time approximation of is
where is given by the time explicit equation (68). Equation (70) is explicit in time and is accurate for all relevant values of . It is anchored at both ends by the small and large time estimates. Both the infiltration rate and the cumulative infiltration equations capture the expected short time and long time behavior, namely, q is asymptotic to ks at large times and the difference approaches a constant value A equal to as → ∞. The asymptotic value A is reached at ≥ 0.1 for and at lower values for higher v and wet initial profiles.
 The TW infiltration model (45) with predicts values for that are slightly larger than the correct ones. The discrepancy is primarily due to the assumption of a uniform initial profile that is not true by the time traveling waves develop. The error is negligible with flux boundary conditions because the mass balance equation (42) is correctly evaluated when the flux is imposed at the surface. A correction of the parameter a for head boundary conditions can be estimated by matching the capillary component of the infiltration rate with the exact nonlinear solution for constant potential absorption. The gravity-free solution can be adapted from the solution by Fujita  and can be used to derive the absorption flux rate and cumulative absorption
where is the solution of
The coefficient of in in (71) is by definition the dimensionless sorptivity So and the relationship to the dimensional sorptivity is . An approximate inversion of (72) is given by Broadbridge and White :
Equating the coefficient of −1/2 in (69) and (71), one gets
Figure 6a presents the infiltration rate for selected values of v and for uniform initial conditions. As expected, a wetter initial profile and higher v values lead to lower infiltration rates. The effect of v or on the flux rates can be easily inferred from (69). The rate of decrease is similar as all curves are parallel except for very wet initial conditions where diffusive effects dominate thereby producing a less rapid decline in the infiltration rate. Figure 6b presents the cumulative infiltration corresponding to the infiltration rates shown in Figure 6a. As expected, drier initial profiles or lower v values lead to higher infiltration volumes. The TW model performs very well for all relevant values of v and the time-explicit approximation of (70) is a very good one for all values of time. The rates of increase at large time are all equal to T while the intercept of each curve is given by A, i.e., the second term of (70). The intercept A is in terms of the soil parameters that might prove useful for parameter estimation.
7.4. Postponding Infiltration
 The above infiltration equations are all instantaneous ponding expressions. For hydrological applications, an equation that takes into account the infiltration volume before ponding occurs is more appropriate. The infiltration model in the postponding stage can be derived in a simple form from the TW solution since it can handle the changing boundary conditions at the soil surface. The postponding cumulative infiltration is expressed by
where , being the cumulative infiltration at the time of ponding Tp. The postponding infiltration rate is still given by (44). In the rainfall hiatus period, the drop in the surface potential can be computed according to the KW model (59) or the exact expression (60). The reinfiltration rate in the following rainfall event is then calculated as for the previous rainfall event with the sole modification of the background potential distribution. One can assume that the updated background potential ub is the value of the surface potential ut when rainfall begins again. This assumption would slightly overpredict the ponding time and the cumulative infiltration at ponding. However, the error is well under the accuracy expected in infiltration studies. This procedure is simpler than the proposed method by Smith et al. , whereby the potential or moisture profile is assumed to be invariant in shape (similarity profile) and the variation of the surface moisture is obtained by numerically solving a nonlinear differential equation.
 The common procedure for adapting the instantaneous expression to postponding stages is based on the time-compression approach (TCA) [e.g., Liu et al., 1998; Smith et al., 1993]. There are basically two major assumptions involved in its use. One is the assumption that at ponding the rainfall intensity is equal to the potential infiltration rate while, in principle, ponding occurs when the soil surface reaches the saturated moisture content regardless of the rainfall intensity. The second assumption is that the cumulative infiltration equation that was derived for ponding conditions can also apply even when there is an interval of nonponding by using an appropriate shift in the time scale, i.e., the time-to-ponding shift.
 The TCA assumption that ponding occurs when is justified in the traveling wave approximation since the relationship between the cumulative flux and the flux rate at the soil surface (44) gives exactly the same expression as the one for the time to ponding (61) when one substitutes for . Hence, there is a one-to-one relationship between the infiltration rate and the cumulative infiltration in both the preponding and postponding stages of infiltration though the value of a is slightly different in each stage. This result partly explains the success of the time compression approximation in modeling postponding infiltration with a preponding flux-concentration relationship. Based on the limitations of the traveling wave solution, this assumption is acceptable as long as the soil is nonlinear () and the water table is deep.
 The second assumption that a simple modification of the time variable in the instantaneous ponding expression is sufficient for postponding calculations is strictly correct for infiltration equations linear in time. One can show that an expansion of the logarithmic term in (45) would result in a linear equation in time with small remainder terms. The second assumption is therefore acceptable simply because the disregarded terms are negligible. However, the use of the TCA method is unwarranted if one can use the correct infiltration equation for the postponding stage (75) since both approaches have the same level of mathematical complexity and predict similar results.
7.5. Depth to Wetting Front
 The wetting front position is of interest when wetting is desired to a given depth. The common approximation of the wetting front advance is based on a piston-type water content profile and is derived from mass balance to be , which is also the same result as obtained from when is the asymptotic constant value. The error of this approximation is in excess of 20% for large v and small values of ub. An accurate and explicit equation for the depth of penetration Zf can be derived from equation (39) by substituting for the midrange value w = (wt + wb)/2,
where either or is set while the other is evaluated from (44) or (47), respectively. Equation (76) gives a very accurate estimate of Zf as shown in Figure 2 for flux-controlled infiltration () and in Figure 3 for head-controlled infiltration (ut = 1).
7.6. Gravity Time
 One characteristic time in infiltration is the gravity time, which is defined as the time at which the effect of gravity is expected to be as great as that of capillarity. The gravity time according to Philip  is where is the dimensional sorptivity. An alternate expression for Tg can be deduced from the flux definition that consists of two terms: the capillary component () and the gravity component (u). By definition, the time at which these two components are equal is the gravity time. Hence, for constant flux infiltration , the gravity time is when the surface potential is equal to . For constant head infiltration ut, it is when the surface flux is . The presence of ub in both formulations implies that the magnitude of both processes is measured relative to an existing background condition ub.
 Using the TW solution, one can find an estimate of Tg that satisfies these two conditions. For constant flux infiltration , setting ut = ug in equation (47) and solving for , one gets
For constant potential infiltration with ut = 1, the gravity time is obtained from equations (44) and (45) with ,
The above two conditions are defined at the surface Z = 0. At any depth Z, the travel time is to be added.
8. Parameter Estimation
 The derivation of algebraic relationships for estimating the hydraulic parameters of the soil is one of the essential goals of analytical investigations. Two complementary approaches are presented below: one for ponding conditions with a negligible ponding depth and the second for constant rainfall infiltration at subponding flux rates. The two approaches should give estimates that must not differ substantially if the assumed soil hydraulic properties apply.
8.1. Constant Potential Method
 Inverse analysis is best carried out by fitting the infiltration equation (45) to the measured Q data over the complete time range. Since the infiltration function is an integral property of the flow, this approach provides more reliable estimates as it matches the integral properties over a moisture content range. However, a quick estimate of few of the parameters can be obtained with the help of the small and large time approximations of the relevant results. At short times, a plot of the cumulative flux versus yields a slope equal to (see equation (68)), which is equal to the dimensionless sorptivity So (see equation (71) or (74)). At large times, the slope of versus is 1 and approaches a constant A equal to (see equation (70)).
 In dimensional terms, the short-time property yields the sorptivity
The sorptivity determines the amount of moisture absorbed in the absence of gravitational effects and is measured in a horizontal infiltration experiment under a constant supply potential. If an experimental measurement of the sorptivity is independently available, an estimate of α can then be obtained from the slope
The large time Q relationship becomes in dimensional terms
The slope is and the intercept is in terms of v and the antecedent moisture content through (46). Thus, they can be useful in obtaining an estimate of the parameters v and ko from infiltrometer data assuming that the saturated and residual water content are available based on the soil features.
 The wetting front depth gives another relationship between the soil water parameters and the cumulative infiltration for a uniform background moisture content value. Equation (76) relate the depth of penetration and the net cumulative infiltration through (44) with the soil parameters α, ko, and v for uniform background conditions ub.
8.2. Constant Flux Method
 One can also use a constant flux infiltration experiment with a rainfall simulator to measure some of the soil properties. The approach has been used to relate the wetting front penetration and the equilibrium surface moisture content with the soil water parameters [e.g., Clothier et al., 1981; White and Broadbridge, 1988].
Equation (76) provides the relationship at various time values between the depth of wetting , the applied flux rate , and the surface potential with the soil parameters α, ks, and ν. Here the variable is a surrogate of time and the surface moisture content at sufficiently large time can be taken as the equilibrium value (7),
Equation (82) gives a second relationship between the measured surface moisture content and the nonponding applied flux rate as a function of the hydraulic conductivity parameters. Although equation (82) matches the soil property at a single moisture content value, equation (76) matches the solution over a moisture content range.
 If the cumulative storage of moisture in a finite depth is available using time domain reflectometry [Parkin et al., 1992], the relationship (45) can then be used since the net cumulative infiltration is in fact equal to the dimensionless storage within a soil column. The time to ponding expression can also be used for parameter estimation if observed times to ponding are available.
 The present work has developed several results of theoretical and practical interest. Simplified solutions of Richards' equation were derived using the kinematic and traveling wave approximations. The solutions are applicable for any uniform initial condition rather than scaled with respect to a particular initial value. They have allowed further analytical exploration of the key infiltration processes, namely, the potential variation at the soil surface, the time to ponding, the infiltration rate, and the cumulative infiltration in the postponding period, in addition to the formulation of the depth of wetting zone for irrigation purposes, and algebraic equations for parameter estimation of the soil hydraulic parameters by inverse analysis.
 The main finding of this study is the correct derivation of the traveling wave solution that reflects accurately the features of the transient infiltration process from early time onwards rather than at large times only (profile at infinity). The correction is due to the proper formulation of the traveling wave speed that is time-dependent and varying with the potential or flux at the soil surface. Similarly, the kinematic wave solution has been formulated to account for the transient variation of the surface potential value in flux-controlled infiltration. One key result of the traveling wave solution is the three-parameter infiltration equation of sound physical basis. The derived infiltration equation is similar in form to the three-parameter infiltration equation of Parlange et al.  but is in terms of well-defined soil parameters that can be measured independently. Another significant result is the derivation of the time-to-ponding expressions from the various solutions, which are similar in form to the Parlange and Smith  equation, but include the shape of the hydraulic conductivity function and the effect of the antecedent moisture content in an explicit fashion. The shape parameter v encompasses soil properties ranging from linear (constant diffusivity) to Green-Ampt-like (delta function diffusivity). One advantage of the various expressions is that the required parameters may be estimated without much difficulty.
 The TW model preserves the inherent nonlinearity of the infiltration phenomenon while offering the flexibility and simplicity required in practical hydrologic applications. The accuracy of the TW solution increases with increasing nonlinearity of the soil and is appropriate for most practical applications with either flux-controlled or head-controlled boundary condition in the zone beyond the influence of the water table. It is most effective for the computation of the surface potential, the time-to-ponding, the infiltration rate, and the cumulative infiltration. The TW model provides simple expressions for the time to ponding for variable rainfall rates and allows the sequential simulation of the infiltration process for uniform initial conditions in both the atmosphere-controlled phase (flux infiltration) and the soil-controlled phase (head infiltration). The KW model is only applicable in gravity-dominated flows and is most effective in the calculation of the surface potential. It can model moisture profile propagation under flux-controlled infiltration for nonsteady rainfall rates but its main drawback is that it cannot capture the dynamic response when capillarity effects are significant.
 Although both TW and KW models assume homogeneous domains, they can still be useful for heterogeneous spatial domains since most approaches for handling soil heterogeneity assume parallel noninteracting homogeneous soil columns in which a simple point infiltration model is applied with selected stochastic parameters. The present infiltration and redistribution models can also be used in nonideal conditions as input to more complicated models. They portray well the essential dynamics of the flow process and they are therefore suitable as modules in complex watershed models.
 The suggestions of the editor, John Selker, associate editor, Jasper Vrugt, and the anonymous reviewers of WRR improved the earlier version of the manuscript.