Numerical Evalua on of Uncertainty in Water Reten on Parameters and E ff ect on Predic ve Uncertainty

where Se is the eff ective saturation, h is the pressure head, θ is volumetric water content, θs and θr are saturated and residual volumetric water contents, respectively, and α and m (n = 1 − 1/m) are water retention parameters related to the water entry pressure and soil pore size distribution, respectively. Th e water retention parameters are usually estimated from water retention data obtained from core samples, and accurately estimating these parameter values has been an active research fi eld for many years (Yates et al., 1992). Due to their spatial variability, the water retention parameters are treated as random variables in stochastic subsurface hydrology. Probability density functions of the parameters are required for evaluating their uncertainty and its propagation through unsaturated fl ow and solute transport models (Christiaens and Feyen, 2001; Avanidou and Paleologos, 2002; Zhou et al., 2003; Lu and Zhang, 2004; Chen et al., 2005; Boateng, 2007). Th e parameter estimates and the PDFs can be obtained in two ways: direct methods of fi tting the water retention data (e.g., Meyer et al., 1997; Schaap and Leij, 1998; Hollenbeck and Jensen, 1998; Christiaens and Feyen, 2000, 2001; Vrugt and Bouten, 2002; Børgesen and Schaap, 2005; Ye et al., 2007a; Chirico et al., 2007) and indirect methods of calibrating the Richards equation (Yeh and Zhang, 1996; Hughson and Yeh, 2000; Wang et al., 2003; Abbaspour et al., 2004; Minasny and Field, 2005). We developed a direct method of estimating the PDFs for measuring the uncertainty of the water retention parameters and for evaluating the eff ect of the uncertain parameters on the predictive uncertainty of unsaturated fl ow and contaminant transport. Many methods have been developed for estimating the water retention parameters and their associated estimation uncertainty. Among them, the least square (LS) method is the most widely used due to its simplicity and fl exibility. Th e LS method has been implemented in the RETC (retention curve) software (van Genuchten et al., 1991; Yates et al., 1992), and the accuracy of the LS estimates is measured by a covariance matrix. Th e ML method incorporates measurement errors in a rigorous manner and can evaluate the adequacy of model fi t (Hollenbeck and Numerical Evalua on of Uncertainty in Water Reten on Parameters and Eff ect on Predic ve Uncertainty