Determining Soil Hydraulic Properties from Tension Infiltrometer Measurements: Fuzzy Regression

numerical instability can be problems. Therefore, the nonlinear least-squares regression method introduced Tension infiltrometer measurements have been used to measure by Logsdon and Jaynes (1993) has been widely used, steady-state infiltration rates at applied tensions. The measurements owing to its simplicity. They estimated saturated hycan be used to determined soil hydraulic properties through linear or nonlinear regression. Such regression methods are often based on draulic conductivity (Kfs) and inverse macroscopic capila few imprecise field measurements, and the traditional regression lary length scale ( ) from steady-state infiltration rate analysis may not yield valid estimates and reliable predictions. The as a function of applied water tension through nonlinear objective of this study is to introduce fuzzy linear regression as an least-squares regression. Classical linear or nonlinear realternative to statistical regression analysis in determining hydraulic gression assumes that the measurement errors are norproperties from tension infiltrometer measurements. Using a tension mally distributed and independent of each other. Since infiltrometer, in situ steady-state infiltration rates [q∞(h )] were meaone needs a lot of samples to determine a probability sured at six different tensions (h ) between 3 and 22 cm of water on distribution, linear or nonlinear regression requires at silty loam and clay loam soils. Hydraulic properties (i.e., field saturated least 8 to 30 measurements or observations to obtain hydraulic conductivity, Kfs, and inverse macroscopic capillary length valid estimates of hydraulic parameters (i.e., Kfs and ) scale, G) and their confidence intervals were estimated following the fuzzy least-square linear regression procedure with the minimum (Bardossy et al., 1990). However, as commonly found fuzziness criterion and linear least-squares method (traditional statistiin most field experiments in hydrology, the number of cal method). Both calculation procedures yielded the same mean steady-state infiltration measurements for a site is limhydraulic properties. A comparison between fuzzy and statistical ln ited and varies from three to six due to the higher cost of q∞(h ) relationship indicated that the confidence bands resulting from longer field times. Moreover, infiltration measurements both procedures enveloped all the measurement points, but fuzzy under tension are subject to different kinds of uncertainregression estimates offered a tighter fit around the midpoint values ties. These arise from (a) measurement errors due to (least-square estimates). Fuzzy linear regression is more reliable and human and instrument imprecision, (b) an enormous may be used as a complement or an alternative to statistical linear spatial variability of soil hydraulic properties (Russo regression analysis for determining hydraulic properties from tension infiltrometer measurements. and Bouton, 1992), and (c) the presence of macropores that make the model invalid. Under these circumstances classical linear or nonlinear regression may not yield S hydraulic conductivity and inverse macroscopic valid estimates for parameters. In particular, a conficapillary length scale are important hydraulic propdence band estimated with a few data points is very erties for the prediction of water flow and solute transwide and may not provide information that is useful for port in soil. Tension infiltrometers have become valupredictive purposes. Fuzzy regression technique may be able instruments that offer a simple, fast, and convenient a helpful tool to overcome these problems. It has been means of determining these hydraulic properties in situ used in hydrology (Bardossy et al., 1990), paleoclimatic (Ankeny et al., 1991; Zhang, 1997), macroporosity (Wilresearch (Boreux et al., 1997), radial tree-growth modelson and Luxmoore, 1988; Bodhinayake et al., 2004a, ing (Boreux et al., 1998), solute transport (Uddameri, 2004b) but methodology varies. 2004), and prediction of the partition coefficient of perA number of calculation procedures exist for determinsistent organic pollutants (Uddameri and Kuchanur ing hydraulic properties from tension infiltrometer mea2004). Nevertheless, fuzzy linear regression has not been surements, such as the sorptivity method (Smettem and utilized to the best of our knowledge to estimate hydrauClothier, 1989), the two tension method (Ankeny et al., lic properties from tension infiltrometer measurements. 1991), the multi-tension/disc size method (Logsdon and Furthermore, there has been no analysis of the estimated Jaynes, 1993), the inverse procedure (Šimunek and van parameters’ uncertainty obtained from regression analGenuchten, 1996, 1997; Schwartz and Evett, 2003), and ysis and its effect on prediction. early time analysis (Zhang, 1997). Inverse procedures The objective of this study is to investigate whether allow estimation of parameters and their uncertainty in fuzzy linear regression (Tanaka et al., 1982) would result a statistical sense. However, inverse procedures require in smaller parameter uncertainty than classical regresmore measurements (usually one has to measure transion and to examine the effect of the uncertainty of the sient state infiltration rates) and non-uniqueness and estimated parameters on predicting effective porosity from the estimated hydraulic parameters. Dep. of Soil Science, University of Saskatchewan, Saskatoon, SK, Canada. Received 18 Jan. 2005. *Corresponding author (Bing.Si@ THEORY usask.ca). Linear regression method (Statistical Method) Published in Soil Sci. Soc. Am. J. 69:1922–1930 (2005). Soil Physics Gardner (1958) proposed the exponential dependence of doi:10.2136/sssaj2005.0022 hydraulic conductivity, K, on soil suction or tension, h: © Soil Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA K(h) Kfs exp( Gh) [1a] 1922 Published online October 27, 2005