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Department of Soil Sciences, SLU, Box 7014, 750 07, Uppsala, Sweden
* Corresponding author (stephanie.roulier{at}mv.slu.se).
Received 14 February 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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To date, inverse methods have been widely applied in soil physics to derive soil hydraulic properties and in large-scale distributed hydrological models (Hopmans and Simunek, 1999; Madsen, 2003), but very little is currently known about the possibilities and potential problems of inverse modeling techniques applied to macropore flow problems. Durner et al. (1999) showed that the parameters of bimodal water retention and hydraulic conductivity functions assumed in some dual-permeability models could be determined by inverse modeling against measured water outflows from multistep outflow experiments. They claimed that the procedure was robust, leading to unique solutions with limited data (water outflows only), irrespective of the number of parameters included, providing the underlying model accurately represented the true soil hydraulic properties. Schwartz et al. (2000) attempted to estimate the parameters of a dual-permeability model by inverse modeling on steady-state bromide breakthrough experiments on a variably charged tropical soil, where the Br- ion could be considered as a weakly sorbed reactive solute. They encountered great difficulties in obtaining physically realistic estimates of two critical parameters, namely the dispersion coefficient in the micropores and the fraction of sorption sites in the macropores. They concluded that inverse procedures are problematic even for the simple case of steady water flow with four unknown parameters to estimate and were also pessimistic about the potential to estimate macropore flow parameters under transient conditions in the field. These findings highlight the need to investigate the feasibility of inverse procedures before applying them to actual data, to avoid the risk of identifying physically inappropriate parameter values. In particular, it remains to be seen whether robust estimation of model parameters regulating macropore flow is possible for transient leaching experiments with reactive solutes.
This study focused on the development and testing of an inverse procedure to derive soil hydraulic properties and adsorption and transformation parameters in simulation models which account for rapid nonequilibrium flow in soil macropores. The dual-permeability model, MACRO (Jarvis, 1994) was linked to the inverse modeling package SUFI (Abbaspour et al., 1997). A theoretical study was then performed using the combined modeling tool SUFI/MACRO to assess the data requirements for robust parameter estimation in macropore flow models. Generated "dummy data", representing transient state column leaching experiments for a tracer and a reactive solute and for two different degrees of macropore flow in the soil system, were used for this purpose. The consequences of data availability were investigated by excluding some "measured" output variables. Moreover, the effectiveness of simultaneous and stepwise procedures (Armstrong et al., 1996) and the impact of experimental and model errors were also evaluated.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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The simulation model MACRO was linked to the inverse modeling program SUFI. SUFI (Abbaspour et al., 1997, 1999) is a forward, sequential, and iterative parameter estimation procedure. The method starts with user-defined prior uncertainty domains, that is, a range of possible values, for the parameters to be fitted. Each uncertainty domain is divided into equidistant strata, and parameter values are defined by the first moment of each stratum. The MACRO model is run for every combination of parameter values, and the results of the simulations are compared with observed variables. The deviation between an observed variable and the corresponding simulated values is quantified by a user-defined objective or goal function. A critical value of the goal function, or tolerance, is then defined. Any parameter combination which gives values of the objective function above the tolerance is eliminated. This results in reduced uncertainty domains for each parameter. The next iteration consists of repeating the above steps with the reduced uncertainty domains. The procedure stops either when the goal function cannot be minimized anymore, or when it is not possible to reduce the uncertainty domains for the next iteration.
Data Generation
For testing and development purposes, numerically generated data sets are preferred to measured data sets because the true values of the parameters are known beforehand. Moreover, the measurement errors existing in any experimental data set can introduce bias into the analysis.
Lysimeter experiments are commonly used to infer water flow and/or solute transport and transformation parameters by inverse modeling (Schoen et al., 1999; Kätterer et al., 2001), especially undisturbed soil columns that allow field-like conditions. Data for an anionic tracer and a reactive solute were generated using MACRO. The lysimeter was 21 cm high. The dummy data were the result of a 1-mo hypothetical experiment where the solutes were applied at the surface of the lysimeter (the concentrations in the soil water in the upper 3 cm of the column were 5000 g m-3 for both solutes, and the sorbed concentration for the reactive solute was calculated internally in the model assuming equilibrium sorption), which was then subjected to natural rainfall conditions for 25 d (the total amount of rainfall at the end of the experiment was 108 mm, and the accumulated potential evaporation was 75 mm). The available data set consisted of (i) the accumulated water percolation, W (L); (ii) the leaching rate for both solutes, SLR (M L-2 T-1); and (iii) the resident concentration in each of seven 3-cm-thick layers at the end of the experiment, Cr (M L-2), for both solutes. The sampling for the water percolation and the solute leaching rate occurred on a daily basis. The solute leaching rate was chosen instead of the solute flux concentration because it gives more weight in the parameter estimation procedure to solute losses occurring during major flow events.
Two data sets were generated. The first was the result of a leaching experiment in a soil strongly affected by macropore flow. In the second, the solute transport through the soil was predominantly convective–dispersive. This was achieved in MACRO by modifying the effective diffusion pathlength d (mm) regulating mass exchange between the two pore domains. A large value of d implies a slow exchange of solute between the macroporosity and the soil matrix, resulting in a nonequilibrium flow, whereas a small value of d results in a fast mass exchange producing convective–dispersive transport. Figure 1 shows the generated dummy data for both cases with and without macropore flow, based on the parameter values shown in Tables 1 and 2.
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b; kinematic exponent, n*), (ii) solute dispersion and the exchange of solute between the micro- and macroporosity (dispersivity, Dv; mixing depth, zd; effective diffusion pathlength, d), and (iii) adsorption and transformation parameters (degradation rate coefficient, µref; fraction of sorption sites in the macropores, f; Freundlich exponent, n; sorption distribution coefficient, zf). These parameters were selected on the basis of fulfilling one or more of the following criteria: (i) difficulty and/or impossibility of direct measurement, (ii) large uncertainty in deriving parameter values from highly variable measured data and/or the uncertainty involved in extrapolating laboratory derived values to the field (e.g., degradation and sorption parameters) (Boesten, 2000), and (iii) large inherent model sensitivity to the parameter (Dubus and Brown, 2002).
The first iteration in SUFI uses a prior estimation of the parameter uncertainty domains (Table 2). Each domain was divided into four strata. For a given iteration MACRO was run with all possible combinations of the four values of each parameter. The resulting simulations were then compared with the dummy measured data, through a goal function expressed as
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The succeeding iterations in the inverse procedure start with a reduced uncertainty domain for each parameter, which was selected based on a critical tolerance Tcrit, defined as Tcrit = 2gmin, where gmin is the minimum value of the goal function for the current iteration. The selected value for the critical tolerance should strike a balance between being too "greedy", so that the reduction in the uncertainty domains from one iteration to the next is too large (Abbaspour et al., 2001), and a conservative approach where the number of iterations becomes impracticably large.
Both stepwise and simultaneous calibration procedures were investigated for the two cases studied (with and without macropore flow). The stepwise procedure (Armstrong et al., 1996) consisted of calibrating first the water flow parameters on the accumulated percolation, and then the solute transport parameters on the tracer concentrations. Finally the parameters controlling adsorption and degradation were estimated from the concentrations of the reactive solute. In the simultaneous procedure, the water flow and solute transport parameters were estimated simultaneously, from both the accumulated percolation and the tracer concentrations, and then the adsorption and transformation parameters were calibrated against the concentrations of the reactive solute. To evaluate the importance of data availability, both the stepwise and the simultaneous procedures were applied first to solute leaching rates only (assuming that resident concentrations were not measured). Subsequently, both leaching rates and resident concentrations for both solutes were considered as measured variables.
For each case, the calibrated values and the corresponding final uncertainty domains were compared to the true values of the parameters. For the case with macropore flow, the best procedure was then tested against a data set including simulated errors. Independent normally distributed random errors were added to each generated data point. Based on the general level of uncertainty expected, the errors were chosen to be 5% for daily water outflow, 10% for the tracer leaching rate and resident concentration, and 30% for the leaching rate and resident concentration of the reactive solute.
Response Surface Analysis
Constructing response surfaces is a useful way to investigate parameter sensitivity and the uniqueness of the solution in inverse modeling problems. One potential issue in the inverse modeling procedure used for MACRO in this study is the correlation between the diffusion pathlength d and the dispersivity Dv on the one hand and the kinematic exponent n* on the other hand. Theoretically, d and Dv might show positive correlation, since both parameters will tend to increase dispersion. Conversely, d and n* might show negative correlation, since they have opposing effects on macropore flow and transport. Thus, keeping all other parameters fixed at their true values, the response surfaces for the goal function in the (Dv, d) and (n*, d) planes were built. The prior uncertainty ranges for d, Dv, and n* (Table 1) were divided into 100, 35, 40, discrete intervals, respectively, and the model was run for each combination of parameters. The goal function was then calculated according to Eq. [1], on the basis of the accumulated percolation and the tracer resident concentration and leaching rate.
The behavior of the sorption coefficient zf, the degradation rate coefficient µref, the fraction of sorption sites f, and the Freundlich exponent n were studied in more detail, and the response surfaces of the goal function in the (µref, zf), (f, zf), and (n, zf) planes were built. The prior uncertainty domains of zf, µref, f, and n (Table 2) were divided into 40, 100, 90, and 40 discrete intervals, respectively. The model was then calibrated by comparing the simulated results with the resident concentration and leaching rate of the reactive solute for the case with macropore flow.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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b, and n*) lay outside the corresponding reduced uncertainty domain (Table 3). The estimated value of the kinetic exponent n* was clearly wrong, since this parameter has little effect when macropores are not functioning. On the other hand, the relative differences between the real and estimated values of Kb and b were small (1 and 14%, respectively), so the incorrect reduced uncertainty domain was the consequence of a too "greedy" strategy: if the critical tolerance is too small, fewer iterations need to be run, but the solution might not be optimal (Abbaspour et al., 2001). Although not shown here, the soil water contents and water discharge were accurately simulated, despite the errors in Kb, b, and n*. The next step in the stepwise procedure was the calibration of the solute transport parameters (Dv, zd, and d) against the tracer leaching rate and resident concentration, using the values of the water flow parameters previously calibrated. This second step gave a correct estimation of the dispersivity Dv and the diffusion pathlength d (Table 3). Thus, SUFI successfully identified the true value of d (1 mm), which lay at the lower limit of the prior uncertainty domain, reflecting equilibrium flow and transport. If the prior uncertainty domain had extended to much smaller values of d, then this parameter would have demonstrated a complete lack of sensitivity for values <1 mm. On the other hand, the mixing depth zd was strongly overestimated, with the true value lying outside the reduced uncertainty domain for this parameter. In the last step of the procedure, MACRO was calibrated against the reactive solute leaching rate and resident concentration, but none of the reduced uncertainty domains for the adsorption and transformation parameters included the corresponding true values (Table 3). Nevertheless, the estimated values for the degradation rate coefficient µref and the Freundlich exponent n were close to their true values (the relative difference between the true value and the estimated value was 0.5% for µref and 6% for n). The error in the reduced uncertainty domain might be explained by a too greedy strategy. On the other hand, the fraction of sorption sites f and the sorption coefficient zf were seriously in error. This might be the consequence of internal correlation between zf, n, and f, since each of these parameters affects the solute retardation due to adsorption.
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The results were not better when the water flow parameters (Kb, b, and n*) and the solute transport parameters (Dv, zd, and d) were calibrated simultaneously against the accumulated percolation and the tracer leaching rate and resident concentrations (Table 3). Only the saturated matrix volumetric water content b was estimated properly. The dispersivity Dv was underestimated, and the prior uncertainty domain for the mixing depth zd and the diffusion pathlength d were not reduced at all. The failure of the MACRO/SUFI tool here is the result of an ill-posed inverse problem, where the overparameterization of the model in the absence of macropore flow leads to problems with lack of sensitivity.
The simultaneous procedure did not differ from the stepwise procedure for the calibration of the reactive solute parameters. The degradation rate coefficient µref was properly estimated, and the estimated value of the Freundlich exponent n was similar to its true value, even though the true value lay outside the reduced uncertainty domain. On the other hand, the fraction of sorption sites f and the sorption coefficient zf were strongly overestimated for the reasons discussed above.
Case with Macropore Flow
The accumulated water percolation alone was not enough to estimate properly the parameters related to macropore flow. Indeed, the true values of the saturated matrix hydraulic conductivity and volumetric water content Kb and b, respectively, lay outside the reduced uncertainty domain estimated with the stepwise procedure (Table 4). However, the estimates were similar to the true values (relative differences of 9.8 and 0.9% for Kb and b, respectively). This indicates that the optimization strategy was too greedy; that is, the critical tolerance was too small. The two subsequent steps of the stepwise procedure were the calibration of solute transport parameters against the tracer flux and resident concentration, and then the calibration of the adsorption and transformation parameters against the flux and resident concentrations of the reactive solute. A correct estimation of all these parameters was obtained, except for the sorption coefficient zf (Table 4). However, the coefficient of uncertainty for the diffusion pathlength d, calculated as the reduced uncertainty domain divided by the true value of the parameter, remained rather high (83%). On the other hand, the small degree of uncertainty found for another critical parameter, the degradation rate coefficient µref, is very encouraging (Table 4). The true value of zf lay outside the reduced uncertainty domain, but the relative difference between the estimate and the true value was small (11%).
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The simultaneous procedure, where the goal function was calculated using the accumulated water percolation, and the solute leaching rate and the resident concentration for both solutes gave the best results (Table 4). All of the posterior uncertainty domains included the true values of the parameters, except for the sorption coefficient zf, which was not estimated properly. Moreover, the goal function was sensitive enough to the parameters to allow accurate estimation; the uncertainty in the estimated parameters varied between 0.43 and 70%, with seven parameters out of nine having a coefficient of uncertainty <10%. The largest uncertainty domains were obtained for the fraction of sorption sites f and the diffusion pathlength d (70 and 22%, respectively). Compared with the stepwise procedure, where the reduced uncertainty domains for Kb and b were not estimated properly, this result implies that not only data concerning water flow are necessary to calibrate the parameters regulating macropore flow, but information on solute transport is also needed. This means that in the case of macropore flow, transport information indirectly contains information on water flow that cannot be obtained from water flow data alone. It also confirms the need for an accurate estimate of the water flow and solute transport parameters for a proper estimation of adsorption and degradation parameters (Armstrong et al., 1996).
The simultaneous procedure without resident concentrations as measured data performed less well (Table 4). In contrast to the previous case, the true values of the saturated matrix water content b, the dispersivity Dv, and the fraction of sorption sites f lay outside the corresponding reduced uncertainty domains. However, the estimated values were similar to the true values. In this case, the reduced uncertainty domains were not estimated properly because the critical tolerance became too small, as the goal function based only on water flow and solute leaching rate was less sensitive to these parameters. The variation of the sorption coefficient zf did not affect the goal function, since its prior uncertainty domain was not reduced at all. Another consequence of excluding the resident concentration in the goal function was an increase in the uncertainty in the parameters that were correctly estimated (Table 4).
Impact of Experimental Errors
As the simultaneous procedure with both the tracer and reactive solute leaching rates and resident concentrations included in the calculation of the goal function performed the best, it was applied to data corrupted by errors. As for the case without errors, all the parameters except the sorption coefficient zf were estimated properly, and their true values were within the corresponding reduced uncertainty domains (Table 5). The main difference compared with the case without errors was a clear loss of accuracy. The coefficient of uncertainty increased considerably for all the parameters and was especially high for the degradation rate coefficient (294%). However, the estimated values were similar to the true values (the maximum relative difference being 25% for the kinematic exponent n*), which gave a good prediction of the dummy measured data (Fig. 2 and 3) Surprisingly, the underestimation of the sorption coefficient by more than 50% did not seem to affect the profile of resident concentration for the reactive solute (Fig. 3b). This might indicate a compensation of errors due to internal correlation between zf and the other parameters related to adsorption, especially the Freundlich exponent.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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The main weak point of the procedure was the inability of the tool to estimate the sorption parameters accurately, due to a combination of internal correlation and sensitivity issues. The procedure could be improved by increasing the amount of measured data, and especially by considering more than one solute resident concentration profile in the data used for calibration. In addition, the effective diffusion pathlength was found to be relatively insensitive with the experimental design investigated in this study. The reason for this is not clear, but design improvements, such as higher time resolution sampling and/or longer columns, may improve the identifiability of this important parameter.
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ACKNOWLEDGMENTS |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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