Fig. 1 The cumulative distribution function (cdf) given by the integral of the beta distribution function P defined in Eq. [1]. When the cdf is fitted to data of fractional solute leaching (sorted in descending order) with fractional sampling area, the slope P(z) at any point on the curve (represented by the straight line) equals the ratio of the infiltration area (A_i) over the outflow area (A_o). Hence, P(z) quantifies the severity of preferential flow for one stream tube. The coordinates define the fractions of the sampling plane (z) and the soil surface (cumulative probability density) occupied by streamtubes with stronger preferential flow than that corresponding to P(z)