Closure to “Improved Decomposition Solution to Green and Ampt Equation” by Sergio E. Serrano

Michel presents an interesting comment to the writer’s contribution Serrano 2003a . His Eqs. 1 – 4 represent a novel approximate solution to the Green and Ampt equation. The problem I see is that his solution falls into the category of successive numerical approximation schemes of the implicit Green and Ampt model, of which there are many available in the literature and as such suffers from the typical limitations of numerical approximations, for example, results are obtained after a series of numerical approximations, solutions are not general in analytic form, or solutions are implicit. The result presented by Serrano 2003a constitutes an explicit general analytical solution with remarkable accuracy for the set of applications required by Eq. 11 by Serrano 2003a . Nonetheless, the method proposed in the discussion by Michel is worth pursuing an analysis on the accuracy, convergence, and numerical stability that may prove useful in hydrologic applications. The discussion by Barry et al. raises many of the same questions as in their former discussion Barry et al. 2003 to the writer’s paper on the Green and Ampt equation Serrano 2001a . This writer replied in detail to those questions Serrano 2003b and the writer will not repeat those comments here. In summary, the writer demonstrated conclusively that the result by Serrano 2001a and its corollary by Serrano 2003a constitute the first explicit solutions of the Green-Ampt infiltration equation; they are neither a difficult to calculate and mostly unavailable Lambert W series approximation, or empirical approximations to the Lambert W series approximation. It was demonstrated that the empirical expressions proposed by Barry et al. 2003 are incorrect as shown by Serrano 2003b, Figs. 1, 2, and 3 . The results by Serrano 2001a and Serrano 2003a were obtained using a decomposition series Adomian 1994 , which is a systematic procedure to obtain analytical solutions to nonlinear equations without linearization, perturbation, or discretization see Serrano 1997, 2001b . In this case, decomposition yields a remarkably accurate result when the convergence criterion is satisfied. However, there are several characteristics that are obvious to anyone familiar with asymptotic series. Eq. 6 by Serrano 2003a follows directly from Eq. 5 , after a logarithmic series pattern is identified. This produces a general series, Eqs. 6–8 by Serrano 2003a , that obviously contain more terms in partial closure than Eq. 5 . The essence of the argument presented in the discussion by Barry et al. is that the improved solution, Eq. 9 by Serrano 2003a is not better than the original decomposition solution Eq. 15 by Serrano 2001a . They present two tables of numerical comparisons between the above equations and another approximation, the Lambert W series approximation, Eq. 7 by Serrano 2001a . The discussers call the Lambert W series approximation “the exact GA formula” and define the “maximum relative error” as the difference between the various approximations. This analysis is inappropriate because it does not compare with respect to the exact Green and Amp solution. This is possibly the reason why the discussers’ Tables 1 and 2 completely contra-