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Soil Science Society of America Journal 63:1934-1941 (1999)
© 1999 Soil Science Society of America

DIVISION S-8-NUTRIENT MANAGEMENT & SOIL & PLANT ANALYSIS

Equifinality and the Problem of Robust Calibration in Nitrogen Budget Simulations

Karsten Schulza, Keith Bevena and Bernd Huweb

a Dep. of Environmental Sciences, Lancaster Univ., Lancaster LA1 4YQ, UK
b Dep. of Soil Physics, Univ. of Bayreuth, D-95440 Bayreuth, Germany

k.schulz{at}lancaster.ac.uk


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Methods
 Results
 Discussion
 REFERENCES
 
Recent trends in predicting N dynamics of agricultural catchments have led to the development of increasingly complex simulation models. However, the parameter calibration of these models is often limited by the availability and temporal resolution of appropriate measurement data. This study addressed the problem of evaluating the predictive uncertainty of a simple N budget simulation model when applied across a winter period by using the Bayesian Generalized Likelihood Uncertainty Estimation (GLUE) methodology. Within a Monte Carlo simulation analysis, it was shown that parameter equifinality was obtained across wide areas of the model parameter space. Equifinality is used here in the sense that many different parameter combinations will produce similar good simulation results with respect to available calibration data. The GLUE methodology assigned a likelihood weight to each acceptable simulation from where uncertainty bounds for the predicted amounts of mineral N in a soil profile and for N drainage fluxes were calculated. It was demonstrated that the equifinality of different parameter sets results in large uncertainty in the predictions. This study also suggested that introducing more complexity into the process description is very unlikely to allow this uncertainty to be constrained.

Abbreviations: CDE, convective dispersion equation • GLUE, generalized likelihood uncertainty estimation • Nmin, mineral N


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Methods
 Results
 Discussion
 REFERENCES
 
THE INCREASING USE OF N FERTILIZER within the last few decades has led to a dramatic increase in the NO3 concentrations of many drinking-water wells in Germany. These levels often greatly exceed the European Community threshold value of 11.3 mg NO3–N L-1. Attempts have been made to reduce agricultural N inputs through governmental regulation and benefit payments to farmers. To support these N management strategies and to estimate potential N leaching risks, recent investigations have focused on the ability of N budget simulation models to predict N dynamics in the unsaturated soil zone. There are many N simulation models available; for an overview, see Engel et al. (1993). The models include simple balance calculations (e.g., Bach, 1987), box or lumped models with simplified transport mechanisms (e.g., Jones and Kiniry, 1986; Huwe, 1992; Ling and El-Kadi, 1998), and complex, physically based deterministic simulations. The latter describe water, heat, and solute transport processes by partial differential equations and explicitly formulate the complex N interactions of the soil–plant system (Wagenet and Hutson, 1987; Kersebaum, 1989; Vereecken et al., 1990; Huwe and Totsche, 1995).

Application of these models is often complicated by limited available information on parameter values and appropriate initial and boundary conditions. Typically, only external data from nearby weather stations, rough estimates of soil hydraulic and (micro-)biological properties, and observations of soil water contents and N concentrations for a few points in time exist. Even when more accurate and laborious measurements are available (e.g., for unsaturated hydraulic conductivity or N mineralization kinetics), they are generally only representative of the processes at the scale of a core, whereas effective parameter values for the patch or field scale are needed for the practical application of the model (e.g., Beven, 1996). Such effective values can normally only be derived indirectly from calibration against other measurements (usually time series of averaged water contents and mineral N [Nmin] concentrations), which themselves may be subject to error. When parameter values of a N simulation model are back-calculated from such measurements, the derived values and model predictions should be expected to be subject to large uncertainty. It has recently been shown elsewhere in the field of hydrology (Freer et al., 1996; Franks et al., 1997, 1998; Zak et al., 1997; Aronica et al., 1998; Lamb et al., 1998; Romanowicz and Beven, 1998) that, even for simplified model structures, there is a great possibility of overparameterization and consequent equifinality of parameter sets. Equifinality is used here in the sense that an equally good description of, for example, the observed Nmin dynamics may be achieved by a great number of models or parameter sets, all of which may be physically reasonable. There may be no clear single optimum parameter set (or model structure) and, consequently, it may be difficult to achieve a robust calibration, in the sense of being able to clearly identify a set of parameters that will produce accurate simulations across a range of different boundary conditions.

The objective of this study was to demonstrate the problem of equifinality in the calibration of a simple N budget simulation model which allowed for the calculation of Nmin contents in different depths and the prediction of N fluxes to the groundwater across a winter period. Also, a novel methodology to investigate parameter sensitivity as well as to estimate the associated predictive uncertainty of Nmin data and N fluxes is introduced.


   Methods
TOP
 ABSTRACT
 INTRODUCTION
 Methods
Results
 Discussion
 REFERENCES
 
GLUE Methodology
Traditional statistical parameter estimation procedures are based on the assumption that there is a correct model of the process of interest and that parameter estimation is concerned with maximizing the likelihood of a suitable function of the noise arising from errors in the observed data. These methods imply that as more data are made available, the likelihood function (surface) becomes increasingly peaked around the optimum parameter set, resulting in a more precise parameter determination.

However, this type of approach may not be generally appropriate when applied to environmental systems, where it is generally not possible to assume that any of the available models is correct nor that adding more data will necessarily refine the parameter estimation. Empirical studies of parameter response surfaces for a variety of models and likelihood measures have demonstrated that the assumption of a well-defined optimum rarely holds and often a wide range of parameter sets reproduce the available data in an acceptable way (Duan et al., 1992; Beven, 1993).

These studies led to the development of the GLUE method (Beven and Binley, 1992) as a way of dealing with multiple acceptable parameter sets (equifinality) within a Bayesian Monte Carlo framework. This method is used for the calibration and uncertainty estimation of models based on generalized likelihood measures and is an extension of the generalized sensitivity analysis of Spear and Hornberger (1980). In the application of GLUE a large number of model runs are made, each parameterized with random sets of parameter values chosen from uniform distributions across the range of each parameter. In the GLUE procedure, any prior knowledge about the expected joint distributions of parameter values can be reflected in a prior likelihood measure associated with the parameter set (rather than in the density of sampling). The acceptability of each run is assessed by some chosen likelihood measure (see below), calculated from a comparison of observed data and simulated responses. Model runs that only achieve a likelihood below a certain threshold may then be rejected as "nonbehavioral", that is, given zero likelihood and thereby removed from any further analysis. The likelihood measures of the retained runs are then rescaled so that their cumulative total is 1.0. The predicted output from the retained behavioral runs is weighted by the associated likelihood measure for that parameter set. At each time step they are ranked to form the cumulative distribution of the output variable from which chosen quantiles can be selected to represent the model uncertainty.

It should be noted that the GLUE method contains a number of subjective elements: in the choice of prior parameter ranges, in the choice of likelihood measure employed, and in the choice of threshold of acceptability. However, it does force those choices to be made explicitly. The Monte Carlo sampling requires a large number of computer runs, particularly for models with a large number of parameters, but the method is well suited to implementation on parallel computer systems that greatly facilitate Monte Carlo methods. The GLUE methodology does have the important advantages of being conceptually very easy to understand and easy to implement. It is readily extended to the use of multiple and fuzzy likelihood measures (e.g., Franks et al., 1998).

Model Description
In this analysis a simple mixing cell approach is applied to approximate the convective dispersion equation (CDE) to describe solute transport in the unsaturated zone and to simulate the N dynamics across a winter period. In this model, a one-dimensional unsaturated soil profile of the length D is divided into M cells of equal length. Every cell is characterized by an effective water storage capacity V*. The basic principle used to describe solute transport is the assumption that the soil water solution in a (soil) cell is completely mixed at every time. By further considering an infiltration rate that is constant with soil depth and across the whole simulation period, the temporal change of the N content in every cell i is described by:

(1)
where Ci is the amount of Nmin in cell i (kg N ha-1), t is the time (d), R is the infiltration rate and is the effective water storage capacity of the cells (mm), and V is the effective water storage capacity for the whole soil profile (mm). After integrating Eq. [1] and using the initial condition , the amount of Nmin in a single cell at any time is obtained by:

(2)

When solving Eq. [1] simultaneously for all cells i = 1, ..., M, by subsequently using Eq. [2], the amount of Nmin in every cell i is given by:

with

(3)
where Cj,0 is the initial amount of Nmin in cell j (kg N ha-1) and Ratm is the average atmospheric Nmin deposition (kg N ha-1 d-1). Thus, C0 represents the amount of Nmin per unit area for a rainfall of height V*.

If we further assume a spatial and temporal constant N mineralization rate Rmin in the upper p cells of the soil profile (corresponding with a mineralization depth ), Eq. [1] becomes:

(4)

where {varsigma} = Rmin/p is a constant N mineralization rate per cell (kg N ha-1 d-1) and Rmin is the N mineralization (kg N ha-1 d-1) in the mineralization zone of depth Dmin (mm). The solution of this system of equations is given by:

(5)




Besides simplicity and a minimal parameter requirement, the advantage of Eq. [5] in describing Nmin transport compared with numerical solutions is the much lower computational demand. Compared with analytical solutions of the CDE, it is much more flexible with regard to depth-dependent initial conditions. By analyzing the CDE in a discretized form (Huwe, 1992), a formal relationship between the length of the cells and the dispersivity ({lambda}) in the CDE can be obtained . Successful applications of this simple approach to simulate N dynamics across a winter period are described by Huwe (1992).

Clearly, the accuracy of the predictions of Eq. [5] will be constrained by the simplifying assumptions that are underlying it (such as the constant infiltration and mineralization rate for the simulation period and a constant water storage capacity with depth). However, it will be shown below that this simple model structure is able to reproduce available calibration data and the solutions rely on the correct specification of the effective values of model parameters.

Simulations
Meteorological and Nitrogen Data
Data sets from the Weiherbach catchment near Karlsruhe, Germany were used in this study (Plate, 1992). For the simulation period of 15 Oct. 1991 to 15 Mar. 1992, rainfall and latent heat fluxes were obtained from a nearby meteorological station. Daily values of net infiltration (the differences between rainfall and actual evapotranspiration) are shown in Fig. 1 ; the mean net infiltration rate across the simulation period (R, Eq. [1]) was 1.43 mm d-1.



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  		Fig. 1 Daily net infiltration R (differences between daily rainfall and evapotranspiration) during the simulation period

 
The soil type under investigation is classified as a Haprendoll (Soil Survey Staff, 1998) having a loamy soil texture. Summer barley (Hordeum vulgare L.) was grown in the summer 1991 and was harvested {approx}2 mo before the start of the simulation period. Single soil samples (100 cm3) in the 0- to 0.3-, 0.3- to 0.6-, and 0.6- to 0.9-m depths were taken at 10 locations (within an area of 20 m2) and then mixed to analyze soluble Nmin contents for each depth. Nitrate was analyzed by using the salicylate method (Höll, 1970). Ammonium was determined as proposed by Alef (1991). More details on the sampling and measurement procedure are given by Rohde (1992). Soil sampling and analysis of soluble Nmin were repeated 10 times within the simulation period so that a calibration data set as illustrated in Fig. 2 was obtained.



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  		Fig. 2 Measured mineral N contents (Nmin) in the 0- to 0.3-, 0.3- to 0.6-, and 0.6- to 0.9-m depths

 
Atmospheric dry and wet N depositions in the Weiherbach catchment were determined by means of bulk samples and analyzing for NO3–N and NH4–N (Beudert, 1997). The totals for 1991 and 1992 are given as 13.9 and 15.1 kg N ha-1 yr-1, respectively, but no data on a finer temporal resolution were available.

Model Parameter Ranges
In this study, by using Eq. [5] to simulate the N dynamics of a soil profile of 0.9-m length, the N content with depth as well as related N drainage fluxes are determined by the initial N content (derived from measured Nmin values in the three depths), the average net infiltration rate and five further model parameters. These are the number of cells used to discretize the soil profile (M), the effective water storage capacity of the soil profile (V), the soil depth to where N mineralization takes place (Dmin), the rate of atmospheric N deposition (Ratm), and the rate of N mineralization (Rmin). Although requiring fewer parameters than most of the contemporary N budget models mentioned above, we will demonstrate that this model structure still has enough flexibility to describe observed data.

To represent the uncertainty of the a priori parameter estimates and measurements, a range of physically, (micro-)biologically, or chemically realistic values based on either experience or observed variability and uncertainty in measurements was assigned to each of these parameters. Parameters and their ranges are summarized in Table 1 . Note that due to the presence of a reliable meteorological weather station, the average seasonal value of net soil water infiltration is fixed to the value derived from measurements as precipitation less actual evapotranspiration (Fig. 1). However, under normal conditions, especially when latent heat fluxes are estimated with simple prediction schemes, this parameter will also be subject to great uncertainties.


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  		Table 1 Parameters and ranges of the applied N budget model

 
Monte Carlo Sampling Strategy
A number of 50000 random sets of parameter values were generated from uniform distributions across the specified ranges of the five model parameters and three initial conditions (Table 1). Simulations were then run for each of these sets. While this seems to be a great number of runs, this still does not represent a dense probing of an eight-dimensional parameter space, and there is still a chance that more suitable parameter sets will be found by using more parameter realizations. The number of simulations required was determined from the sensitivity of the cumulative distributions of the predicted outputs (Nmin contents, N fluxes) to additional runs. The choice of uniform distributions was selected due to the lack of any specific prior knowledge. We also did not account for any possible covariation between the parameter values in these prior choices as there was no adequate justification for it. Some information on the covariation among parameters in fitting the available observations will appear in the posterior distribution of likelihood measures after conditioning the parameter sets within the GLUE procedure. If there is justification that certain parameter combinations are unrealistic, they may be rejected a priori.

Specification of Likelihood Weights
The likelihood measure employed in this study was a coefficient of determination equivalent to the widely used efficiency measure (Nash and Sutcliffe, 1970) defined by the proportion of the variance of the observations explained by the model:

(6)
where L(Y|{Theta}i) is the likelihood of simulating the data set Y given the parameter set {Theta}i, {sigma}2i is the variance of the residuals between measured and simulated variables for parameter set {Theta}i, given the set of observations Y, and {sigma}2obs is the variance of the observations. Here, a likelihood value was first calculated separately for all three soil depths and then averaged by equally weighting each depth.

When the likelihood weights of the retained simulation runs were calculated, they were then rescaled so that the sum of their totals was 1.0. They may be combined with some prior likelihoods associated with each parameter set using Bayes equation in the form

(7)
where L0 is a prior likelihood measure for the parameter set , L is the likelihood measure calculated with respect to observed data, L is the posterior likelihood of parameter set i conditioned on the simulation of Y, and C is a scaling constant. In the uniform prior case as used in this study, L0 is constant and Eq. [7] yields the same posterior likelihood as Eq. [6].

To determine the uncertainty bounds at each time of interest, the predicted output from the retained runs were likelihood weighted and ranked to form a cumulative distribution of the output variable from which chosen quantiles can be selected to represent the model uncertainty. Throughout this study the 5, 50, and 95% quantiles were used to characterize the determined distributions of the outputs.


   Results
TOP
 ABSTRACT
 INTRODUCTION
 Methods
 Results
Discussion
 REFERENCES
 
Equifinality and Sensitivity of Parameters
Scatter plots for the likelihood measure given in Eq. [6] are shown in Fig. 3 for five of the model parameters (see Table 1). These plots represent the projection of the multidimensional parameter response surface, as sampled by the Monte Carlo simulations, onto single parameter axes. The maximum values of model efficiency (Eq. [6]) obtained were of the order of 0.85. For the parameters Dmin, M, and Ratm, very good and very poor simulations were available throughout the whole or at least throughout wide parts of the chosen parameters ranges. The structure of the scatter plots for these three parameters seen in Fig. 3 suggested that the parameter response surface is very complex, with many smaller peaks, so that no single optimum parameter set could be determined. This emphasized the need to treat parameters as sets of values. Some of the parameters might show significant interactions, and one of the advantages of the GLUE methodology is that by treating the parameters as a set, the effect of such interactions will be reflected implicitly in the value of the likelihood measure associated with each set.



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  		Fig. 3 Scatter plots showing the range of model efficiency L (Eq. [6]) produced across the range of the five model parameters: V, effective water storage capacity of the soil profile (mm); Dmin, soil depth to where N mineralization takes place (mm); M, number of cells; Ratm, average atmospheric N deposition (kg N ha-1 d-1); and Rmin, average N mineralization rate (kg N ha-1 d-1)

 
The scatter plots for the parameter Rmin, the average N mineralization rate, and V, the effective water storage capacity of the soil profile, indicated that within the chosen model structure the range of possible values could be at least constrained by the available calibration data. By only considering good (behavioral) simulations with likelihood measures >0.5, and rejecting all others, the parameter values for Rmin could be limited to the range of 0.2 to 0.95 kg N ha-1 d-1. For V at least the lower bound could be constrained to parameter values higher than 150 mm, reflecting to some extent the characteristics of the loamy soil texture of the simulated field. But again, it has to be remembered that the selection of such a rejection threshold value is a subjective choice by the user. A lower rejection criteria would allow more model parameterization to be classified as behavioral. The consequences of this subjective choice for the predictions will be analyzed below.

The observed changes of the likelihood value within specific regions of the parameter range of individual parameters already revealed some evidence about their sensitivity. More information was obtained by an extension of the Generalized Sensitivity Analysis of Spear and Hornberger (1980). Here, the 50000 realizations were ranked according to their likelihood measure and then 10 performance classes were created by dividing the behavioral simulations (a rejection criteria <0.5 was chosen) into 10 groups with equidistant ranges of likelihood values. The cumulative distributions of five of the 10 performance classes for the five model parameters are given in Fig. 4 . Straight-line distributions suggested limited sensitivity of these parameters within the model, as can be seen for the parameters Ratm and M. The distributions for the parameters V and Rmin clearly differed from uniformity and also changed with changing performance class. Both distributions were narrower for better performance classes, and thus demonstrated a high parameter sensitivity with regard to model performance. Additionally, some sensitivity was obtained and displayed for the parameter Dmin, the maximum soil depth to where mineralization takes place. However, Fig. 4 demonstrates that for all five parameters, even for the best performance class, the parameter distributions were spread across wide ranges of the values considered.



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  		Fig. 4 Plots of sensitivity of performance class for five different model parameters. Class 1 is the lowest performance class, Class 10 is the highest (for clarity, only five of the ten classes are plotted). Parameters are: V, effective water storage capacity of the soil profile (mm); Dmin, soil depth to where N mineralization takes place (mm); M, number of cells; Ratm, average atmospheric N deposition (kg N ha-1 d-1); and Rmin, average N mineralization rate (kg N ha-1 d-1)

 
The conclusion of "nonsensitivity" for single model parameters strictly holds only when regarding the parameters separately. Values of these parameters may still be significant in the context of a particular parameter set, and multidimensional plotting techniques might reveal some more insight into parameter interaction. However, two and three dimensional scatter plots of the likelihood measure (not shown here) did not indicate any further structure within the parameter response surface.

Uncertainty Bounds of Mineral Nitrogen and Nitrogen Flux Predictions
Following the GLUE procedure as described above, uncertainty estimates were calculated for the Nmin values in the 0- to 0.3-, 0.3- to 0.6-, and 0.6- to 0.9-m depths (the observed variables) and the cumulative N recharge rates in the 0.9-m depth (a nonobserved variable). Figure 5 shows the resulting prediction quantiles (5, 50, and 95%) for the Nmin values in the three depths, when nonbehavioral model parameterizations were characterized by using a rejection criteria for the likelihood measure <0.5. Both, the means (50% quantile) as well as the uncertainty bounds (5 and 95% quantiles) for all depths, demonstrated that in general the model structure, even with its very simple assumptions, was able to reproduce measured Nmin values and that the derived uncertainty bounds encompassed almost all of the observed data. However, the calculated quantiles indicated that for all depths large uncertainties were associated with the predicted Nmin values. The uncertainty bounds for the end of the simulation period ranged from {approx}30 kg N ha-1 for the 0- to 0.3- and 0.6- to 0.9-m depths to {approx}40 kg N ha-1 for the 0.3- to 0.6-m depth and were of the same order as the observed values. Quantitatively the same results were achieved for the corresponding N fluxes at this site, with wide ranges of predicted cumulative N discharge flux (Fig. 6) . Predicted quantiles also showed that with increasing time the form of the cumulative N flux distribution became increasingly skewed and were far from being normal.



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  		Fig. 5 Uncertainty bounds for the predicted mineral N (Nmin) values in the 0–0.3, 0.3–0.6, and 0.6–0.9 m depths, using a rejection criteria for nonbehavioral simulations <0.5

 


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  		Fig. 6 Uncertainty bounds (5 and 95% quantiles) for the predicted cumulative mineral N (Nmin)-fluxes in the 0.9-m depth, using a rejection criteria for nonbehavioral simulations <0.5

 
The influence of the chosen rejection criteria is demonstrated in Fig. 7 . By using lower threshold values, more parameter sets are retained as behavioral and result in slightly wider uncertainty bounds for the predicted Nmin values in the 0.3- to 0.6-m depth. The effect of two other transformations of the likelihood values of accepted parameter sets is illustrated in Fig. 8 , to give an idea of the sensitivity of the results to the choice of likelihood measure used. While the first likelihood measure (L) was that used before (given by Eq. [6] with a rejection criteria for nonbehavioral simulations <0.5), the second measure used the same criteria but rescaled the likelihood values to 0 to 1 before weighting the outputs. Thus, parameter sets got less weight with decreasing likelihood value (compared with L). Dependent on the parts of the parameter ranges being more affected by lower likelihood values and on the way these ranges influence the outputs, different consequences for the uncertainty bounds may be achieved. For the Nmin values in the 0.3- to 0.6-m depth the rescaling resulted in a shift and a slight spread of the calculated uncertainty bounds. The third likelihood measure used L, but raised to the power of ten (L10). This had the effect of greatly increasing the relative likelihood of the best simulations, with the result that the uncertainty bounds are narrowed (but still contained all the observed data points).



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  		Fig. 7 Comparison of uncertainty bounds (5 and 95% quantiles) for the predicted mineral N (Nmin) values in the 0.3- to 0.6-m depth, using three different rejection criterias for nonbehavioral simulations

 


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  		Fig. 8 Uncertainty bounds (5 and 95% quantiles) for the predicted mineral N (Nmin) values in the 0.3- to 0.6-m depth using the likelihood measure as described in Eq. [6] (L) with a rejection criteria for nonbehavioral simulations <0.5, the same criteria, but rescaling the likelihood values to 0 to 1 after acceptance, and the same likelihood measure but taken to the power of 10 (L10)

 
In general, the predicted uncertainty bounds in this study indicated that an appropriate estimation of the spring Nmin value and corresponding N fluxes was almost impossible. This was the case, although spatially sampled Nmin data for calibration were available on a high temporal resolution and rainfall, evapotranspiration rates, and atmospheric N inputs were relatively well known compared with available data at other field sites.


   Discussion
TOP
 ABSTRACT
 INTRODUCTION
 Methods
 Results
 Discussion
REFERENCES
 
In this study a very simplified N budget model has been used to simulate both observed data of Nmin as well as N fluxes of an agricultural field during a winter period. We demonstrated that even with the simplifying assumptions of the layered box model with constant average infiltration and N mineralization rates, observed Nmin data could be well described and model efficiencies (proportion of explained variance) >0.85 were obtained for the available calibration data. However, the demonstrated equifinality of different parameter sets within a Monte Carlo analysis indicated that the model structure was still overparameterized in a system identification sense, and therefore a robust calibration was unlikely to be achieved. By weighting these behavioral parameter sets according to their associated likelihood, significant uncertainties were estimated for the predicted Nmin values and cumulative N discharge rates. These uncertainties should be taken into account in a risk-based decision-making process when assessing agricultural regulations to preserve groundwater quality.

The results also suggest that the problem of equifinality and induced uncertainties of predicted variables may increase when more complexity is incorporated into the modeling scheme. The addition of processes is often done with the intention to achieve better results or to have a more realistic description of the processes controlling N dynamics. Possible improvements which already have been implemented in many of the recently available N budget simulation models include the explicit use of time series of rainfall, temperature, and other climatic data to describe water and heat fluxes; temperature and water content dependent formulations of mineralization and other N transformation processes; consideration of multiple organic N pools with different kinetic characteristics; and differentiation of N species (NH4, NO3, N2O). The problem, of course, is to estimate appropriate parameter values for the additional process descriptions, especially when applied at specific sites and at much larger scales than the core scale of microbiological investigations. The model will then require effective parameter values at that scale, which can only be obtained by calibrating against any relevant available data. We expect that the calibration of a more complex model than used here would result in even more parameter sets that would reproduce the (limited) observational data well.

However, this expectation has to be proved explicitly. As model prediction errors result not only from improper estimation of (effective) parameter values, but also from an incorrect model structure, it might be possible that a structural enhancement could so improve accuracy as to overcome the additional uncertainty introduced by additional parameters. Therefore, the addition of complexity into a model structure (to describe for instance more dynamic time series of Nmin concentration due to microbiological activities or fertilization in summer periods or to use additional time series of the water contents for calibration) should be evaluated by using a before-and-after analysis within the type of uncertainty analysis presented here.

The GLUE methodology is also able to consider multiple model structures. The only requirement is that the likelihood measures used to express relative performance should be consistent. In that case, the problem of model choice can be posed within a framework of model rejection so that where multiple model structures and parameter sets are found to be behavioral, it may be possible to seek additional observations or measures of performance that would allow at least some of those models to be rejected. An additional problem that has not been fully addressed yet by the scientific community is the uncertainty of the Nmin calibration data themselves. Large spatial variability of Nmin values within short distances (Schmidhalter et al., 1992) will lead to the problem that even more parameter sets within a GLUE type analysis must be considered suitable when compared with these uncertain calibration data.


   ACKNOWLEDGMENTS
 
Research on which this paper is based has been supported by the German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF), project no. 02WA93810 and the German Research Foundation (DFG), grant no. Schu 1271/1-1.

Received for publication September 10, 1998.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Methods
 Results
 Discussion
 REFERENCES
 





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