In this paper, we examine the possibility of introducing a single stochastic scaling factor α, derived from macroscopic Miller similitude, to describe the spatial variability of soil hydraulic properties. Most of the information available allowed only a conventional statistical analysis of the scaling factors derived from different soil properties. The field studies of (Nielsen, Biggar, and Erh (1973)) and (Russo and Bresler (1981)) were suitable also for more detailed structural analyses. Results of these analyses suggested that the spatial structure of the α-set derived from the hydraulic conductivity function K(?) is different from that of the α-set derived from the water retentivity function h(?), reflecting the different spatial structures of the K(?) and the h(?) functions. Consequently, the statistical relationship between the uncorrelated residuals of the two α-sets was rather weak. For the Hamra field of (Russo and Bresler (1981)), the use of relative hydraulic properties to estimate the scaling factor sets considerably improved the correlation between the α-sets, which had essentially the same spatial structure but slightly different variances.

In this study, where the soil hydraulic properties are assumed to be described by the model of (Brooks and Corey (1964)), analytical expressions for the variances of the two different α-sets indicated that (1) both α-sets are dependent on the range of water saturation that is used to estimate them, (2) the correlation between the two sets will improve in media with a wide pore-size distribution, and (3) the two sets will be identical if and only if the relative hydraulic conductivity function K_r(h_r) is described by a deterministic function, K_r(h_r) = h_r^-2. This result suggested that, in general, a second scaling factor for K_r is required for media that are not characterized by this single deterministic relationship.

A more general K_r(h_r) relation, defined by K_r = h_r^-?;, was introduced using ?; as a second stochastic variable. In this representation, the α scaling factor for K_r is defined by K_r/K_r^* = α^?; instead of α² as in macroscopic Miller similitude. For the Hamra field, the resultant new α-set was identical to the α-set derived from the relative retentivity function. For the Panoche field, using the values of ?; to estimate the scaling factor from the relative hydraulic conductivity function considerably improved the correlation and the similarity between the two α-sets, but did not render them identical. The results of our analysis suggest that, for transient water flow, describing the spatial variability of K(?) and h(?) requires at least three stochastic variates: K_s, α, and ?;.