## Indirect Estimation of Water Retention and Hydraulic Conductivity Functions

Much work is being directed toward the evaluation of the soil water-retention and hydraulic-conductivity functions from related soil physical properties. The increasingly complex computer models employed in environmental studies require a large amount of input data, especially those characterizing the soil from the hydraulic viewpoint, which in turn are notoriously difficult to determine. Therefore, when simulating hydrological processes in large areas, the possibility of deriving hydraulic parameters from soil data (such as bulk density, organic-matter content, and percentage of sand, silt, and clay), which are relatively simple to obtain or already available, is highly attractive. This task is carried out by using the pedotransfer functions (PTFs), which transfer basic soil physical properties and characteristics into fixed points of the water-retention function or into values of the parameters describing an analytical 9 (h) relationship [27]. Unsaturated hydraulic conductivity characteristics then usually are evaluated by PTF predictions of hydraulic conductivity at saturation [28]. PTFs appear to provide a promising technique to predict soil hydraulic properties, and they are highly effective for deriving soil water-retention characteristics; however, there is still some debate in the literature on the accuracy and reliability of unsaturated hydraulic conductivity evaluated by this predictive method [29]. To date, most of the research relating to PTFs usually have been directed toward comparisons between measured and estimated hydraulic properties for different types of soils [30], but a few studies have investigated the effects of PTF predictions on some practical applications [31].

More recently, many authors have come to be interested in the feasibility of simultaneously estimating the water-retention and hydraulic-conductivity functions from transient flow experiments by employing the inverse-problem methodology in the form of the parameter optimization technique. By using this approach, only a few selected variables need to be measured during a relatively simple transient flow event obtained for prescribed but arbitrary initial and boundary conditions. Data processing assumes that the soil hydraulic properties 9 (h) and K(9) are described by analytical relationships with a small number of unknown parameters, which are estimated by an optimization method minimizing deviations between the real system response measured during the experiment and the numerical solution of the governing flow equation for a given parameter vector. Assuming homoskedasticity and lack of correlation among measurement errors, the optimization problem reduces to a problem of nonlinear ordinary least squares. For soil hydrology applications, however, the observations usually consist of quantities (e.g., water contents, pressure potentials, water fluxes) that show differences in measurement units and accuracy. Thus, a less restrictive reasonable hypothesis accounting for this situation is that one assumes uncorrelated errors but unequal error variances among the different variables. The method becomes a weighted least-squares minimization problem. Another advantage of the parameter estimation methods is that they also can provide information on parameter uncertainty.

Such a methodology is suitable for characterizing hydraulically a soil either in the laboratory (employing, for example, multistep outflow experiments [32] or evaporation experiments [33]) or in the field (generally by inversion of data from transient drainage experiments [34, 35]). In many cases, the inverse-problem methodology based on the parameter estimation approach allows experiments to be improved in a way that makes test procedures easier and faster. The variables to be measured, the locations of the sensors, the times at which to take measurements, as well as the number of observations used as input data for the inverse problem, may exert a remarkable influence on reliability of parameter estimates. Therefore, the experiment should be designed to ensure that the relevant inverse problem allows solution without significantly compromising accuracy in parameter estimates. In fact, parameter estimation techniques are inherently ill-posed problems. Ill-posedness of the inverse problem is associated mainly with the existence of a solution that can be unstable or nonunique, as well as to the fact that model parameters can be unidentifiable. Even if problems relating to ill- posedness of the inverse solution can arise, parameter optimization techniques are undoubtedly highly attractive for determining flow and transport parameters of soil and have proved to be effective methods, especially when a large amount of data needs to be analyzed.

## Growing Soilless

This is an easy-to-follow, step-by-step guide to growing organic, healthy vegetable, herbs and house plants without soil. Clearly illustrated with black and white line drawings, the book covers every aspect of home hydroponic gardening.

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