The Neuro-m Method for Fitting Neural Network Parametric Pedotransfer Functions Budiman Minasny* and Alex. B. McBratney ABSTRACT 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 parame- eters, e.g., Brooks and Corey or van Genuchten equation. ters of a model from basic soil properties are useful in deriving continu- ous functions of soil properties, such as water retention curves. The A parametric approach is usually preferred to single- common 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 be- relationship. Water retention at any potential can be esti- tween 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 form- to 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 vari- the soil hydraulic model and optimizes the PTF to match the measured ous mathematical methods, e.g., multiple linear regres- and observed water content, we called this neuro-m method. This sion (Wo ¨ 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 -3 . The relative improve- ment of neuro-m over neural networks that was optimized to fit the (h) = r +  s - r (1 + | h| n ) m [1] parameters (neuro-p ) is 13%. Compared with a published neural net- work PTF, the new method is 30% more accurate and less biased. where  r and  s are the residual and saturated water content,  is the scaling parameter, n is the curve shape P edotransfer functions (Bouma, 1989), predictive factor and m is an empirical constant, which can be functions of certain soil properties from other easily, related to n by m = 1 - 1/n. routinely, or cheaply measured properties, have recently Attempts have been made to correlate the parameters become a popular topic in soil science research. Differ- of the van Genuchten equation (or other soil hydraulic ent types of function have been developed to predict models) to basic soil properties (Vereckeen et al., 1989), either physical or chemical properties of the soil. Re- however many researchers have encountered difficulties search areas include formulating a better physical model (Tietje and Tapkenhinrichs, 1993; Scheinost et al., 1997). (Arya et al., 1999), finding the most influential soil prop- Van den Berg et al. (1997) suggested that this could be erties as predictive variables (Timlin et al., 1999), group- caused by interdependency amongst the parameters. ing soil into classes to minimize the variance of prediction The problems are also likely because the model does (Pachepsky and Rawls, 1999), and developing alterna- not always fit the data, and overparametization of the tive methods to derive or fit the PTFs (Scheinost et al., model (too many parameters to fit over a limited water- 1997). Most pedotransfer functions have been devel- retention data). Hence the fitted parameters may carry oped to predict soil hydraulic properties, especially wa- errors and bear no significant physical meaning. To ter retention curves. This is mainly as the response to overcome this problem, Van den Berg et al. (1997) sug- the urgent need for soil hydraulic properties as inputs gested the following approach: fit the model to observed to soil-water models. data, apply multiple regression analysis to one of the In probably the first research of its kind, Bloemen parameters, fit the model again by fixing the parameter (1977, 1980) derived the relationships between parame- calculated from the regression, continue to fit a regres- ters of the Brooks-Corey model and particle-size distri- sion to another parameter and repeat the process until bution. This approach is now called parametric PTFs. all the parameters are fitted. Alternatively, Scheinost For the water-retention curve, we can assume that the water content, , and the potential, h, relationship can 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, Depart- sured values; neuro-p, neural network PTF with objective function ment of Agricultural Chemistry and Soil Science, Ross St. Building that matches the parameters; P 2 , mass particles 2 mm; P 2–20 , mass A03, The University of Sydney, NSW 2006, Australia. Received 6 particles 2 to 20 mm; P 20–2000 , mass of particles 20 to 2000 mm; PTF, Mar. 2001. *Corresponding author (budiman@acss.usyd.edu.au). pedotransfer function; RMSD, root mean squared deviation; RMSR, root mean squared of residuals; SSR, sum of square residuals. Published in Soil Sci. Soc. Am. J. 66:352–361 (2002). 352