be described adequately by a soil hydraulic model that is a closed-form equation with a certain number of param-Parametric pedotransfer functions (PTFs), which predict parameeters, e.g., Brooks and Corey or van Genuchten equation. ters of a model from basic soil properties are useful in deriving continuous functions of soil properties, such as water retention curves. The A parametric approach is usually preferred to singlecommon method for deriving parametric water retention PTFs in-point regression (predicting water retention at a specific volves estimating the parameters of a soil hydraulic model by fitting potential), as it yields a continuous function of the (h) the model to the data, and then forming empirical relationships berelationship. Water retention at any potential can be estitween basic soil properties and parameters. The latter step usually mated. Many soil-water transport models only require utilizes multiple linear regression or artificial neural networks. Neural the parameters of the soil hydraulic models as inputs, network analysis is a powerful tool and has been shown to perform thus the predicted parameters can be used directly to better than multiple linear regression. However neural-network PTFs run them. are usually trained with an objective function that fits the estimated The usual steps in deriving parametric PTFs are fitting parameters of a soil hydraulic model. We called this the neuro-p method. a soil hydraulic model to individual water-retention The estimated parameters may carry errors and since the aim is to be able to estimate water retention, it is sensible to train the network data, estimating the parameters of the model, and formto fit the measured water content. We propose a new objective func-ing empirical relationships between basic soil properties tion for neural network training, which predicts the parameters of and parameters. The latter step can be achieved by varithe soil hydraulic model and optimizes the PTF to match the measured ous mathematical methods, e.g., multiple linear regresand observed water content, we called this neuro-m method. This sion (Wö sten et al., 1995), or artificial neural networks method was used to predict the parameters of the van Genuchten (ANN) (Schaap et al., 1998). model. Using Australian soil hydraulic data as a training set, neuro-m The most widely used soil hydraulic model is the van predicted the water retention from bulk density and particle-size dis-Genuchten function (van Genuchten, 1980): tribution with a mean accuracy of 0.04 m 3 m Ϫ3. The relative improvement of neuro-m over neural networks that was optimized to fit the Abbreviations: AIC, Akaike's information criterion; ANN, artificial neural network; MD, mean deviation; MR, mean residuals; neuro-m, neural network PTF with objective function that matches the mea-B. Minasny, Australian Cotton Cooperative Research Centre, Departsured values; neuro-p, neural network PTF with objective function

Various calibration approaches have been proposed for determining profiles of apparent soil electrical conductivity (ECa) or soil electrical conductivity of a saturated soil paste extract (ECe) using an EM38 instrument. One of the most promising of these, the established‐coefficients approach, was selected for calibration of the EM38 to some irrigated cotton (Gossypium hirsutum L.)‐growing soil located in the Edgeroi district of the lower Namoi valley, northern NSW, Australia. However, the fitted salinity profiles were locally erratic, although global trends could be recognized. This method was compared with simple linear regression applied to the data from each depth increment separately. This was also unsatisfactory, and in fact, was not obviously different from the established‐coefficients method on the data. An alternative and statistically more rigorous approach was used in which a logistic curve is fitted to each calibration hole (the logistic profile model). This mixed random and fixed effects nonlinear model predicts ECe from measurements generated by an electromagnetic (EM) instrument (EM38) held in the horizontal mode of operation EM0,H The logistic profile model fitted the data well, with no obvious patterns in the residuals, and depended on far fewer parameters than the two alternative methods. In addition, unlike the established‐coefficients approach, the logistic model provided meaningful prediction errors.

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Some possible approaches to the aggregation and disaggregation of soil data and information are presented as an opener to the more detailed discussion. The concepts of hierarchy, grain, extent, scale and variability are discussed. Slight modifications to the Hoosbeek-Bryant scheme to deal with spatial and temporal scales and various types of quantitative models are suggested. Approaches to aggregation or upscaling are reviewed. The contributions of representative elementary volume (REV), variograms, fractal theory, multi-resolution analysis using wavelets, critical point phenomena, renormalisation groups and transfer functions are discussed followed by a brief presentation of some ecological approaches including extrapolation by lumping, extrapolation by increasing model extent and extrapolation by explicit integration. A clear distinction must be made between additive and nonadditive variables. The scaling of the former is much less problematic than the latter. Corroboration of any approach by testing against the aggregated values seems problematic. Methods of disaggregation or downscaling including transfer functions, mass-preserving or pycnophylactic methods are also discussed. In order to make quantitative advances, nested sampling or reanalysis of data in land information systems to obtain variance information over a complete range of scales is required. Finally an appeal is made for work to begin on a quantitative scale-explicit theory of soil variation.

Fractal models of soil structure can be used to predict the scaling properties of associated transport coefficients. For gas diffusion, the structure of the soil pore space is relevant, while the structure of the solid matrix is most implicated in heat conduction. In fractal soil structures, the magnitude of the relevant diffusivities can be written in the generic form , where D(r) is a length‐dependent diffusion coefficient, A is the normalization coefficient, r is the Pythagorean length, and ϕ is a structure‐dependent constant. The dependence of ϕ on structure has been described elsewhere; however, the influence of structure on the magnitude of A has not been previously elaborated. Here, we determine the functional dependence of A on the structural parameters of the soil. The heterogeneity and connectivity, as quantified by the mass fractal dimension (Dm) and spectral dimension (d), respectively, and porosity are estimated from sections of undisturbed soil cores. For these soil structures, we demonstrate that the magnitude of the thermal and gas diffusivities is more sensitive to the porosity than to the scale dependency inherent in fractal structures. A methodology is developed and applied to rank the predicted thermal and gas diffusivities for the soil structures studied.