Validity of First-Order Approximations to Describe Parameter Uncertainty in Soil Hydrologic Models

system under study. Confidence intervals on the calibrated model parameters can be used to express the Model nonlinearity and parameter interdependence violate the use degree of uncertainty in these quantities, but calculation of a first-order approximation to obtain exact confidence intervals of of confidence intervals is far from straightforward beparameters in soil hydrologic models. In this study, the posterior distribution of parameters in soil water retention and hydraulic concause the solution of Richards’ equation is generally ductivity functions is examined using observed water retention data a highly nonlinear function of the model parameters and a laboratory transient multistep outflow experiment. Parameter (Christensen and Cooley, 1999). Without a realistic asuncertainties obtained with traditional first-order approximations and sessment of parameter uncertainty, it is not wise to ununiform grid sampling strategies were compared with those obtained dertake with any confidence tasks such as evaluating using the Metropolis algorithm, a Markov Chain Monte Carlo (MCMC) confidence limits on future hydrological responses or sampler. A diagnostic measure, based on multiple sequences generassessing the relationships between model parameters ated in parallel, was used to test whether convergence of the Metropoand easily measurable basic soil properties. lis sampler to the posterior distribution had been achieved. Most signifiA frequently employed approach, which is particucantly, as the Metropolis algorithm can cope with rough response surfaces larly popular in the area of vadose zone hydrology, is generated by the objective function used, it not only successfully infers the multivariate posterior probability distribution of the model paramto obtain confidence intervals of parameters by utilizing eters, but also provides valuable insights in parameter interdepena first-order approximation to the model function near dence in the full parameter space. its minimum (see Carrera and Neuman [1986] in groundwater hydrology; Kool and Parker [1988] in unsaturated soil water flow; Kuczera and Parent [1988] in rainfallN simulation models of water flow and sorunoff modeling). As this classical first-order approxilute transport in the vadose zone are important mation does not account for correlations between the tools in environmental research. The accuracy of predicparameter estimates, computed standard errors can aptions with these models heavily relies on accurate estipear too favorable (Hollenbeck and Jensen, 1998; Chrismates of the unsaturated soil hydraulic parameters. Betensen and Cooley, 1999). Indeed, case studies have cause direct assessment of the hydraulic parameters is shown that these linearly calculated confidence intervals generally impossible, estimation is usually based on fitare generally smaller than their corresponding nonlinear ting a numerical solution of the Richard’s equation to counterparts (Cooley, 1993; Christensen and Cooley, observations collected during an experiment. In recent 1999). In general, the first-order approximation yields years, much effort has been directed to finding the most reasonable results provided that the estimated paramelikely parameter set from experimental laboratory exter uncertainty does not extend beyond the range for periments, such as Multi-step outflow (Eching and Hopwhich a first-order approximation of the model equation mans, 1993; Van Dam et al., 1994), evaporation experiapplies. In soil hydrologic models, however, parameter ments (Šimunek et al., 1998a; Romano and Santini, 1999) interdependence and model nonlinearity violate the use and from field data (Abbaspour et al., 1999; Musters and of this first-order approximation to obtain exact confiBouten, 2000; Vrugt et al., 2001b,c; among many others). dence intervals of the parameter. Surprisingly, relatively little attention has been given to Alternatively, a robust but computationally intensive a realistic assessment of parameter uncertainty under method for the calculation of confidence intervals, conthe different types of experimental conditions. We share trasting the classical first-order approach, is the generathe recent opinion by Durner et al. (1997) that improved tion of contour plots (Toorman et al., 1992; Gribb, 1996; interpretation of parameter uncertainty can yield valuRomano and Santini, 1999; Vrugt et al., 2001a). This kind able information to enable a better judgment of the of exhaustive uniform grid sampling requires discretizing limits of our theoretical understanding of unsaturated the parameter space and computing the objective funcwater flow in soils. tion for each grid point, which is a rather primitive Parameter estimates obtained from calibrated soil hymethod. Recently, Abbaspour et al. (1997, 1999) used a drologic models are generally error-prone, because the similar kind of sampling strategy, entitled the Sequentual data used for calibration contain measurement errors Uncertainty Fitting algorithm (SUFI) for estimating suband because the model never perfectly represents the surface flow and transport parameters. However, as simple this approach might be, it requires massive computing resources for highly dimensioned parameter spaces. Jasper A. Vrugt and Willem Bouten, Institute for Biodiversity and For example, if we wish to sample, on average, a paramEcosystem Dynamics, Section Physical Geography, Univ. of Amsterdam, The Netherlands, Nieuwe Achtergracht 166, Amsterdam, 1018 WV. Received 19 Nov. 2001. *Corresponding author (j.vrugt@ Abbreviations: CV, coefficient of variation; MCMC, Markov Chain science.uva.nl). Monte Carlo; RMSE, root mean squared error; SCE, shuffled complex evolution; VG, van Genuchten. Published in Soil Sci. Soc. Am. J. 66:1740–1751 (2002).