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DIVISION S-1—SOIL PHYSICS

Robust Estimation of the Generalized Solute Transfer Function Parameters

Dep. of Environmental Sciences and Land Use Planning, Université catholique de Louvain (UCL), Croix du Sud, 2 Bte 2, B-1348 Louvain-la-Neuve, Belgium

* Corresponding author (javaux{at}geru.ucl.ac.be)

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
A method is presented to mathematically characterize a wide range of solute transport processes in soils from time-domain-reflectometry (TDR)-based breakthrough curve (BTC) measurements. To do this, we combined the flexible generalized transfer function model (GTF) with the definition of the time normalized resident concentrations C^rt*. The GTF is a four-parameter flexible transfer function able to describe both the convection-dispersive (CD) and the stochastic-convective (SC) process of dispersion in a soil. In addition, it allows other dispersion processes typical for heterogeneous soils to be modeled. To obtain robust estimations of the GTF model parameters, closed-form expressions of the GTF time normalized resident concentrations C^rt* and temporal moments of C^rt* were defined. These expressions allow the transport parameters to be estimated from TDR-based BTCs without relying on problematic probe calibrations and mass recovery assumptions. Since GTF is a four-parameter model, the robustness of the parameterization procedure was tested by fitting the C^rt* of the GTF model to numerically generated, error-contaminated BTCs. Using the least-square optimization technique, robust estimates of the GTF parameters could be obtained from TDR observations at two different soil depths. Application of the proposed method was also tested for a real undisturbed sandy subsoil and compared with the convective lognormal transfer (CLT) and CD models. For this case, it was shown that the GTF model improved greatly the goodness of prediction of the BTCs. The proposed method offers a new powerful tool for analyzing transport processes of nonreactive solutes in soil.

Abbreviations: BTC, breakthrough curve • CD, convective dispersive • CLT, convective lognormal transfer • CV, coefficient of variation • ETFM, extended transfer function model • GTF, generalized transfer function • LSO, least-squares optimization • MA, moments analysis • pdf, probability density function • RMSE, root mean squared error • SC, stochastic convective • SD, standard deviation • TDR, time domain reflectometry

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
SOLUTE TRANSPORT has often been modeled using linear transfer function theory (Jury, 1982). Using this theory, the solute breakthrough at any depth and time is given by the convolution of the top boundary input function, describing the solute application, and the solute impulse response function. This latter function encodes the governing transport process, which is specific for a given soil. The commonly used impulse response functions in solute transport studies are based on the CD and the CLT. By means of two soil specific solute transport parameters, they allow the prediction of solute dispersion in space and time for a range of relatively simple boundary conditions.

However, solute transport in a natural soil cannot always be conceptualized as either being a CD or a SC process. Structural heterogeneity may generate heterogeneous flow modifying the governing flow concept when solute migrates through the soil (Nissen et al., 2000; Seuntjens et al., 1999; Snow et al., 1994; White et al., 1986).

To improve the flexibility of the two classical transfer functions, Liu and Dane (1996) proposed a simple extended transfer function model (ETFM) in which the travel time variance increases proportionally with the depth to the power 2a, where a is an adjustable constant representing the degree of horizontal solute mixing. Recently, Zhang (2000) presented a flexible GTF in which relationships between travel time moments are:

[1]

with ₁ and ₂ (similar to the ETFM a) are parameters of the solute travel time moments, z, the depth, and l, the reference depth. In addition, the GTF model can also be used to characterize solute transport processes in heterogeneous soils such as those in which the mean travel time increases non linearly with depth and those in which the dispersivity is a scale dependent function of the travel distance with any power value. The GTF is thereby a comprehensive transfer function model allowing the solute transport process in heterogeneous soils to be formalized in a synoptic way.

Yet, a major inconvenience of the GTF model formalism is the addition of supplementary parameters to describe the transport process, which adds an additional burden on the parameter identification process and supplementary uncertainty in the parameter estimates (Vanderborght et al., 1997). Therefore, it is of paramount importance to develop robust parameter estimates if the GTF model is to be used in practical applications.

Identification of solute transport parameters is often done by the analysis of BTCs. Recently, great progress has been made in identifying transport processes by applying TDR technology to measure solute concentration (see, e.g., Vanclooster et al., 1995). Major advantages of using TDR technology for characterizing solute transport are related to the possibility of easily applying it to undisturbed soil material and of measuring transport with a high spatiotemporal resolution. This latter property is a prerequisite for validating transport concepts for undisturbed soil. A major problem in the earlier studies of solute transport with TDR was the need for a soil and layer specific calibration equation, relating signal attenuation to the resident solute concentration of an ionic tracer (Mallants et al., 1996; Vanclooster et al., 1994; Vogeler et al., 1997). However, to avoid this latter calibration problem, Vanderborght et al. (1996) introduced the time-normalized resident concentration in terms of the two-parameter CLT solute transport model, which allows the CLT parameters to be directly identified from TDR output readings. Jacques et al. (1998) extended the approach for the two-parameter CD model, but a general approach for the GTF model is presently lacking.

The objective of this paper is to present robust parameter identification procedures for estimating the GTF parameters from TDR-based breakthrough experiments. To do this, we first developed closed-form expressions of the travel depth probability density function (pdf) for the GTF model, which were then used to obtain closed-form expressions of the time normalized resident concentrations and their temporal moments. These expressions can further be used to obtain estimates of the GTF parameters either by least squares optimization or by moment analysis. Apparent velocities and apparent dispersivities can then be expressed in terms of the GTF parameters to physically describe the dispersion process. The well-posed parameter identification problem and the generality of the GTF modeling approach is analyzed in the last part of the paper. It is shown (i) that the GTF C^rt* expressions are appropriate generalizations of the earlier published CD and CLT expressions; (ii) that the GTF parameter estimation approach from moment and least-squares analysis is well posed, and (iii) that the method can be easily applied for modeling solute transport in a real, nondisturbed sandy soil.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
Travel Time and Depth Probability Density Function of the Generalized Transfer Function Model
Transfer function theory is now widely used to describe soil solute transport phenomena (Jury and Roth, 1990). This theory is based on the linearity of the solute transport process, and encodes soil specific solute transport within a specific solute transfer function. Solute fluxes leaving the soil profile are then obtained by convoluting the solute input function with the transfer function. Zhang (2000) proposed a generalized solute transfer function in terms of the solute travel time pdf. The integral of this pdf measures the probability that a solute particle added at time t₀ will reach a given depth z in a time smaller than or equal to t. The equation shows the same form as the well-known CLT function:

[2]

where f^f (t,z) is the flux type travel time pdf, and µ_z and _z depth-dependent model parameters that scale with depth as follows:

[3a]

[3b]

where µ_l and _l are the values of the parameters and µ at the reference depth l. The variables _µ and are auxiliary variables dependent on µ_l, _l, ₁, and ₂, the four solute transport parameters of the GTF model. Equations linking those parameters and variables are given by Eq. [23] and [28] in Zhang (2000) and reported hereafter in Table 1, third column.

[4]

Combining Eq. [2] with the parameter definitions [3a] and [3b] in Eq. [4] yields, after mathematical simplification, the travel depth pdf of the GTF model:

[5]

with factor a, a constant, defined as:

[6]

Through this equation, the definition of _z and µ_z (Eq. [3]) and of _µ and (see Table 1), Eq. [5] depends on four soil-characteristic parameters for solute transport in soil: ₁, ₂, µ_l and _l.

Time Normalized Resident Concentrations for the GTF Model
Solute transport parameters can be obtained from steady-state BTC experiments in initially solute free soil, subjected to a Dirac pulse solute application at the soil surface. When ionic solute tracers are used, TDR can be used to monitor the time course of the resident concentration at a given depth during the BTC experiment. Considering horizontally installed TDR probes, the resident concentration in the TDR sampling volume during a steady-state BTC experiment is equal to

[7]

where Z*(z,t) is the difference between the actual and initial impedance load of the TDR trace at infinity, and k*, a calibration constant, dependent on the probe configuration and the soil status. Uncertainty in the value of the probe constant may have considerable impact on the identification of the solute transport process (Mallants et al., 1996). To avoid the problematic calibration of the TDR-probe constant during solute transport experiments, we use the time-normalized resident concentrations resulting from a Dirac solute flux input C^rt*, as suggested by Vanderborght et al. (1996):

[8]

The advantage of using such a definition becomes apparent if we insert the definition of the residential concentration measured by TDR probes (Eq. [7]) in Eq. [8]: the time-normalized resident concentration becomes directly related with the output of TDR signal without needing any calibration:

[9]

In fact, C^rt*(z,t) is the instantaneous percentage of total solute mass which will pass through the sampling window of the probes.

Now, let's divide the numerator and denominator of Eq. [8] by .M₀ with , a fraction of the total mass M₀ added at the top of the soil, which may be 1 depending on the area of the horizontal surface covered by the sampling window and the mass recovery in each sampling region. By simplifying the factor and since f^r(z,t) = C^r(z,t)/M₀, we obtain the definition:

[10]

Since the percentage of the total mass disappears, Eq. [10] is really independent from this initial mass M₀. For the GTF, the denominator of Eq. [10] can be found by integrating Eq. [5] by variable transformation (Appendix 1). It yields:

[11]

Substitution of Eq. [5] and [11] in Eq. [10] yields, after some simplifications,

[12]

with µ_z and _z defined by the Eq. [3a] and [3b] and a by the Eq. [6]. The variable, ₁ is a characteristic soil parameter. It should be emphasized that the time-normalized resident concentrations are completely independent from the total mass of solute crossing the point z, that is, no assumptions have been made on the mass recovery, or on the representativity of the sampling window in front of the horizontal plane crossing the soil column.

Moments of the Generalized Transfer Function
Moments can be calculated from the C^r, C^f, and C^rt* based BTCs (Jacques et al., 1998). Given the uncertainties on C^r because of the calibration of the TDR probes and the bad spatial resolution of TDR data, travel-depth moments are not very useful for obtaining adequate solute transport parameters. For the C^f based solute travel-time, Zhang (2000) gave closed-form expressions for the two first moments of the GTF model. A general expression can also be found, involving all the GTF travel time moments, by using the general expression for travel time moments for the CLT formulation (Appendix 2):

[13]

Obviously, if ₁ and ₂ equal 1, we obtain the moments of the CLT general travel-time moments (Eq. [2.77] in Jury and Roth, 1990). If the C^rt* pdf is used, one can define the moments of the time-normalized depth resident concentration as Jacques et al. (1998):

[14]

Inserting Eq. [12] in Eq. [14] and using the same variable transformation as used in Appendix 1 (Eq. [A3]), we obtain:

[15]

This expression can also be written as a function of _l and µ_l, using the definitions of _z and µ_z (Eq. [3a] and [3b]). By definition, for the CLT process ₁ = 1 and ₂ = 1, which results in:

[16]

This equation is consistent with the definitions of the two first temporal moments of C^rt* given by Jacques et al. (1998).

Parameter Identification of the Generalized Transfer Function
A first method to obtain the solute transport parameters is by moments analysis (MA). With this method calculated time moments of the experimental C^rt* curves are inserted in the analytical definitions of Eq. [14] through [16]. As the GTF model has four arguments, a system of at least four independent equations is required to identify the unknown parameters. However, given the nonlinearity of the system of equations, closed-form expressions for the model parameters cannot be derived from the moment expressions, and inverse modeling procedures have to be used. As suggested by Jury and Sposito (1985), Eq. [29] to [32], we used least-squares minimization to solve the nonlinear system of equations. A second method is by least-squares optimization (LSO), fitting the solution of Eq. [12] directly to the experimentally determined C^rt* BTC. However, fitting a nonlinear four-parameter function to the single monomodal C^rt* curve may be ill posed. In addition, as already shown by Jury and Roth (1990) for identifying governing transport mechanisms, observations are needed at least at two different soil depths. We therefore propose to apply LSO simultaneously for, at least, two C^rt* curves measured at two different depths. Actually, this requirement is no problem given that most of the TDR-based BTC experiments already use more than two curves. Nevertheless, since the model is able to describe a wide range of dispersion processes evolving with depth, fitting the model to all the depths together will result in only one parameter set which should have a better predictive power.

Advantages and drawbacks of MA versus LSO have already been discussed, by Jury and Sposito (1985) among others. They analyzed the effect of the shape of the observed BTC on parameter uncertainties. They suggested that, if a few observations alter significantly the shape of the BTC, the LSO procedure, by averaging the error, would not conserve sufficiently the shape of the BTC. In this case, MA could appear more stable. Yet, bias is also introduced in MA, since each concentration point C^rt* is implicitly weighted by t^N (Eq. [14]), which implies that the MA gives more weight to the asymptotic long-time concentrations. Ellsworth et al. (1996) concluded that LSO would be more appropriate if predictions of concentrations at multiple space-time intervals have to be made, whereas MA would be more appropriate if asymptotic behavior is of concern. The latter is typically the case when soil based BTC are used to predict larger scale transport.

Physical Significance of the Generalized Transfer Function Parameters
Despite the fact that the GTF is a transfer equation, one may define expressions for the apparent velocity and dispersivity in terms of the GTF parameters µ_l, _l, ₁, and ₂. Such definitions allow the transport processes to be characterized in terms of soil physical properties and experimental boundary conditions.

Apparent velocity. By definition, the apparent solute transport velocity is:

[17]

Inserting the definition of the first temporal moment (Eq. [13] with N = 1) in Eq. [17] yields the apparent velocity in terms of the model parameters. Apparent velocity definitions for the GTF, CD, and CLT models are given in Table 1.

A particular feature of the GTF model formalism, as compared with the CD and CLT model, is that the apparent velocity may evolve monotonically with depth, depending on the value of ₁. That means that the ₁ parameter is therefore not a soil-dependent parameter but rather depends on the water-flow boundary conditions such as the top flux rate or the bottom pressure head. If ₁ > 1, v_a decreases with depth; in case ₁ < 1, v_a increases. No change in the velocity with depth is assumed when ₁ is equal to 1 as for the CLT or the CD model. Local depthwise changes of velocities occur for instance when the water content is not uniform with depth (Bejat et al., 2000), or when analyzing transport in a layered soil (Snow et al., 1994).

Under some assumptions, ₁ can be linked with the water-content profile. If the actual apparent velocity profile v_a(z) is known or derived from water-content profile in the absence of preferential flow, ₁ can be fixed before other parameter optimization from BTCs. This is done by fitting the definition of v_a for the GTF (Table 1, third column) to actual v_a(z) and letting ₁ and the exponential term as two fitting parameters.

Apparent dispersivity _a(z). By definition, _a is equal to (Simmons, 1982):

[18]

which for the GTF model yields (Appendix 3):

[19]

A comparison of the GTF model formulations with the CD and CLT formulations can be performed on the basis of the apparent velocity and dispersivity. Closed-form expressions for the CV, the apparent dispersivity and velocity for the three models are given in Table 1.

Transport regimes in terms of GTF model parameters. Zhang (2000) classified the transport processes in function of the difference ₁ - ₂: (i) if ₁ - ₂ = 0, that is, var[ln(t)] is constant with the travel distance, the process is SC; (ii) if the difference is negative, var[ln(t)] increases with travel-distance, the process is scale-dependent; (iii) if this difference is positive, var[ln(t)] decreases with travel-distance, as in the CD model (₁ - ₂ = 0.5).

In the same way, transport processes can be classified in terms of the relationship between dispersivity and depth (Eq. [19]). In Fig. 1 , we plotted the evolution of the apparent dispersivity in function of the depth, defined by the Eq. [19], for different values of ₁ - ₂. The reference depth l was taken equal to 100, and _l and µ_l were respectively, equal to 0.2 and 4. If ₁ - ₂ = 0.5, the apparent dispersivity does not change with distance like in the CD model. Below this value, the dispersivity grows with travel distance (as observed, e.g., by Nissen et al., 2000). Zhang (2000) showed that, for ₁ - ₂ = 0.25, dispersivity obeyed the Neuman universal scaling relationship in a saturated porous media. If the difference equals 0, the process is SC and the dispersivity increases linearly with the depth. Finally, if the difference is larger than 0.5, the apparent dispersivity decreases with depth. Even though this latter case is not explainable by classical theories, a local decrease in apparent dispersivity with depth because of local heterogeneity has sometimes been observed (see, e.g., Khan and Jury, 1990).

View larger version (33K): [in this window] [in a new window]   Fig. 1. Influence of 1 - 2 on the apparent dispersivity in function of the depth. Values in the graph represent the a calculated from Eq. [19] with reference depth l = 1 m, l = 0.2, and µl = 10.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

By means of Eq. [12], we generated C^rt* BTC at four different depths numbered from 1 to 4 (0.1-, 0.4-, 0.7-, and 0.85-m depth) for a virtual soil having the following GTF parameters: ₁ = 0.7, ₂ = 0.7, µ_l = 3.6, and _l = 0.32 with l, the reference depth, being equal to 1 m. On those time-series, we added a realistic noise. As no calibration is performed on the reflection coefficient time series, the only source of random error comes from the direct measurements of the impedance. Heimovaara et al. (1995) reported an average standard deviation (SD) for the reflection coefficient of 4.4 x 10^-3 [-]. As we divide the output signal of the TDR by an integral usually >1 (Eq. [9]), SD of the C^rt* curve should be lower. We created a random, normally distributed noise, centered on 0 and with a SD of 8 x 10^-4, that we added to our GTF-simulated BTCs. The effect of the noise on the generated data at the four depths is shown in Fig. 2 .

View larger version (34K): [in this window] [in a new window]   Fig. 2. Error contaminated data simulated with Eq. [11] at four different depths (0.1, 0.4, 0.7, and 0.85 m, l = 1 m) and a realistic noise of standard deviation of 8 x 10-4 [-].

[20]

with N defined above and i being between 1 and 4. Minimization of Eq. [20] was done with a simple local Gauss-Newton optimization algorithm programmed in a Matlab environment.

Least-Squares Optimization
To check the sensitivity of the LSO to the TDR sampling depth, we analyzed results for five possible sets of two BTCs selected at the four different depths. In a final scenario, we identified parameters considering all BTCs simultaneously. A local search Levenberg-Marquardt algorithm, programmed in a Matlab environment, was used to carry out the LSO. Initial guesses for all the parameters were set equal to one. The response surface of the objective function in the parameter space was analyzed to identify possible parameter correlation. The considered objective function was set equal to:

[21]

with b the vector of GTF parameters, C^rt* (z_i, t_i) the real, and C^rt* (z_j, t_i, b) the simulated time-normalized resident concentration at time t_i and depth z_i, and with parameters given by b. Linear confidence intervals were estimated using the procedures presented by Donaldson and Schnabel (1986).

To compare optimization results, the root mean square error (RMSE) was calculated as follows:

[22]

Laboratory Experiment
The GTF model identification procedure was further tested and compared with other models on a real dataset of BTCs collected on a sandy, heterogeneous, undisturbed subsoil. A 1-m high, 0.8-m i.d. monolith was sampled from a quarry at Mont-Saint-Guibert, Belgium, at a depth of 10 m in the unsaturated zone following the method described by Vanclooster et al. (1995). Three TDR and temperature probes were inserted at 0.15-, 0.3-, and 0.45-m depths (Levels 1, 2, and 3). After establishing a steady-state unsaturated water flow (flux = 18.1 cm d^-1), a 1-h pulse of 15.95 g L^-1 CaCl₂–loaded water was added at the top of the sand column. Subsequently, tracer-free water was applied to leach the saline tracer. The TDR impedance load was monitored by the three TDR probes at regular times and transformed to C^rt* by the definition (Eq. [9]).

The CLT, CD, and GTF C^rt* equations (Table 2, second column) were first calibrated at Levels 1 and 2 to derive the parameters. Using the calibrated parameters, predictions were made for the Level 3 data and compared with real data at this depth. The goodness of the calibration and prediction was qualified by means of the lack-of-fit mean square (Whitmore, 1991):

[23]

with b, the optimal parameter set, K, the number of parameters, N, the number of observation levels, and M_i, the number of observations at the i level. ^rt* (z_j, t_i, b) is the simulated time-resident concentration and C^rt* (z_j, t_i, b) is actual measurement.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

View larger version (31K): [in this window] [in a new window]   Fig. 3. Numerical simulation of a CDE and a CLT process at the 0.3-, 0.6-, and 1-m depth fitted by the GTF solution (Eq. [11]). The CDE parameters: D = 5 cm h-2 and v = 1 cm h-1, fitted parameters are 1 = 1, 2 = 0.49, µl = 3.81, and l = 0.435. The CLT parameters: µl = 3.73, l = 0.480, fitted GTF parameters are exactly the same with 1 and 2 equal to 1.

View this table: [in this window] [in a new window]   Table 3. Optimized GTF parameters and root mean square error obtained by moments analysis (MA) applied on different datasets of generated breakthrough curve (BTC) contaminated with errors. First column gives the numbers of the two BTCs used for the MA. Breakthrough curve 1 is at 0.1-m depth, BTC 2 at 0.4-m, BTC 3 at 0.7-m, and BTC 4 at 1-m depth. Initial guesses: µl = 1, l = 1, 1 = 1, 2 = 1.

Parameters resulting from the two-depth MA have a very low RMSE but a relatively bad parameter determination, which confirms our previous remark. Adding a third depth does not improve the identification for the 1/2/3 and 1/2/4 scenarios. However, when time series at all the depths are taken simultaneously into account, parameter estimates are acceptable.

It appears that the observation depth seems to be more important than the number of time series. Scenarios including time-series observed at the first depth generally have a poor-parameter estimation, which is because of the important relative error on the variance at shallow depths.

In conclusion, it appears that MA suffers from stability problems when error contaminated data are used. Moments analysis should therefore only be considered when observations at many depths are available or as initial guesses for a further LSO.

Least-Squares Analysis
Table 4 summarizes the results of the least-squares analysis of the numerically generated BTC experiment. First of all, we observe that the four parameters are well estimated in all cases and that the confidence intervals are quite small. This is a very promising feature, especially given the important noise that was considered on the measurement signal. The scenario that considers all BTC simultaneously results in the smallest confidence intervals.

View this table: [in this window] [in a new window]   Table 4. Optimized generalized transfer function (GTF) parameters with their confidence intervals, root mean square error (RMSE), and determination coefficients (r2) obtained by LSO applied on different generated error-contaminated breakthrough curve (BTC). First column gives the position of the time series used for the LSO. Breakthrough curve 1 is obtained at the 10-cm depth, BTC 2 at the 40-cm depth, BTC 3 at the 70-cm, and Probe 4 at the 100-cm depth. Initial guesses: µl = 1, l = 1, 1 = 1, 2 = 1.

View larger version (69K): [in this window] [in a new window]   Fig. 4. Surface plots of the objective function related to the error-contaminated Crt* at 0.1- and 0.4-m depth. The cross indicates the actual parameters (1 = 0.7, 2 = 0.7, µl = 3.6, and l = 0.32).

View larger version (75K): [in this window] [in a new window]   Fig. 5. Objective function of error-contaminated Crt* from the six BTCs couples with actual parameters: 1 = 0.7, 2 = 0.7, µl = 3.6, and l = 0.32. Comparison of response surfaces of parameters 1 and 2 in function of the dataset used. The two figures labeling each subplot give the position of the two BTC used. Figure 1, 2, 3, and 4 correspond respectively to 0.1-, 0.4-, 0.7-, and 0.85-m depths.

View larger version (30K): [in this window] [in a new window]   Fig. 6. Comparison of experimental BTCs at the 0.15- and 0.3-m depth in an undisturbed sandy subsoil and fitted results of the CD, CLT, and GTF models.

View larger version (27K): [in this window] [in a new window]   Fig. 7. Comparison of experimental BTC at the 0.45-m depth in an undisturbed sandy subsoil and predictions of the CD, CLT and GTF models.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

The C^rt* pdf was fitted to the theoretical solutions for the CD and CLT transport model and gave excellent results. This means that with real data, the transport process can directly be identified by observing the ₁ and ₂ fitted parameters.

Furthermore, relationships between GTF parameters and the apparent dispersivity and velocity were defined. The difference between ₁ and ₂ characterizes the evolution of the apparent dispersivity with depth while ₁ provides information on the variation of solute velocity with depth.

Given the increased number of parameters in the GTF model, the stability and robustness of this model have been tested. Using error contaminated numerical experiments, we applied MA and least-squares analysis to obtain the four GTF parameters. Moments Analysis was shown to be strongly influenced by the random error and gave relatively poor results when limited observation depths were considered. Unique parameters could only be obtained when moments of order -2 to 2 at the four observation depths were considered. However, a more powerful optimization algorithm and addition of constraints on the ₁ parameter could be used to improve the results.

Using the LSO technique, robust estimates of the GTF parameters were obtained from TDR observations at two different soil depths. Objective function analysis emphasized the effect of the observation depth on parameters determination. It also allowed the correlation structure between ₁ and µ_l, and between ₂ and _l to be characterized.

Comparison between CLT, CD, and GTF models was also performed on a real undisturbed sandy subsoil. For this case, it was shown that the GTF model predicted better the BTC.

To sum up, it was shown that the increase in the number of parameters in the GTF does not constitute a major drawback for parameter estimation from TDR-based BTCs, and the larger number of parameters is obviously an advantage to better characterize more complex solute transport behavior. So, we recommend using the GTF time normalized resident concentrations formulation to analyze, by LSO technique, the BTCs obtained from TDR experiments to reduce the uncertainty because TDR calibration problems and characterize transport processes in soils in a generic way.

	APPENDIX

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
Appendix 1: Time Integration of the Travel-Depth Probability Density Function
To obtain the integration of the travel depth pdf, we take Eq. [5] multiply and divide by t:

[A1]

Introducing the variable

[A2]

and defining:

[A3]

Inserting the substitutions Eq. [A3] in [A1], we obtain:

[A4]

Taking the integral and posing w = y - _z, we have:

[A5]

Considering that the integrand of the first term of this equation is 0 (integral of an odd function) and the integrand of the second part is 1, one obtains Eq. [11].

Appendix 2: General Expression of the Generalized Transfer Function Travel Time Moments
By definition, the GTF and CLT formulations are the same (Eq. [2]). The definition of the general travel-time moments for a CLT is (Eq. [2.77] p. 44 in Jury and Roth, 1990):

[A6]

Inserting the definitions of µ_z and _z (Eq. [3]), it yields:

[A7]

By definition, _µ and are related to ₁ and ₂ by (Eq.[23] and [28] in Zhang, 2000):

[A9]

Putting Eq. [3], [A8], and [A9] in Eq. [A7] yields Eq. [13].

Appendix 3: Coefficient of Variation and Apparent Dispersivity
The definition of the coefficient of variation of the travel time is (Butters and Jury, 1989):

[A10]

Taking the definition of variance and mean (Eq. [11] in Zhang, 2000) and introducing Eq. [3] in [A10], it yields:

[A11]

Plugging Eq. [A9] and [A11] into the definition of the apparent dispersivity (Eq. [18]) yields:

[A12]

which is the definition of the apparent dispersivity at depth z after a time t in function of the GTF parameters.

	ACKNOWLEDGMENTS

 
This research was partly funded by the Belgian Fond National de la Recherche Scientifique. The authors are grateful to Guido Rentmeesters for expert technical assistance.

Received for publication January 22, 2002.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 

• Bejat, L., E. Perfect, V.L. Quisenberry, M.S. Coyne, and G.R. Haszler. 2000. Solute transport as related to soil structure in unsaturated intact soil blocks. Soil Sci. Soc. Am. J. 64:818–826.[Abstract/Free Full Text]
• Butters, G.L., and W.A. Jury. 1989. Field scale transport of bromide in unsaturated soil. 2. Dispersion modeling. Water Resour. Res. 25:1583–1589.
• Donaldson, J.R., and R.S. Schnabel. 1986. Computational experience with confidence regions and confidence intervals for nonlinear least squares. In D.M. Allen (ed.) Computer Science and Statistics. Elsevier Science, New York.
• Ellsworth, T.R., P.J. Shouse, T.H. Skaggs, J.A. Jobes, and J. Fargerlund. 1996. Solute transport in unsaturated soil: Experimental design, parameter estimation, and model discrimination. Soil Sci. Soc. Am. J. 60:397–407.[Abstract/Free Full Text]
• Heimovaara, T.J., A.G. Focke, W. Bouten, and J.M. Verstraten. 1995. Assessing temporal variations in soil water composition with time domain reflectometry. Soil Sci. Soc. Am. J. 59:689–698.[Abstract/Free Full Text]
• Jacques, D., D.J. Kim, J. Diels, J. Vanderborght, H. Vereecken, and J. Feyen. 1998. Analysis of steady state chloride transport through two heterogeneous field soils. Water Resour. Res. 34:2539–2550.
• Jury, W.A. 1982. Simulation of solute transport using a transfer function model. Water Resour. Res. 18:363–368.
• Jury, W.A., and K. Roth. 1990. Transfer functions and solute movement through soil: Theory and applications. Birkhäuser, Basel, Switzerland.
• Jury, W.A., and G. Sposito. 1985. Field calibration and validation of solute transport models for the unsaturated zone. Soil Sci. Soc. Am. J. 49:1331–1341.[Abstract/Free Full Text]
• Khan, A.U.H., and W.A. Jury. 1990. A laboratory study of the dispersion scale effect in column outflow experiments. J. Contam. Hydrol. 5:119–131.
• Leij, F.J., and J.H. Dane. 1991. Solute transport in a two-layer medium investigated with time moments. Soil Sci. Soc. Am. J. 55:1529–1535.[Abstract/Free Full Text]
• Liu, H.H., and J.H. Dane. 1996. An extended transfer function model of field-scale solute transport: Model development. Soil Sci. Soc. Am. J. 60:986–991.[Abstract/Free Full Text]
• Mallants, D., M. Vanclooster, N. Toride, J. Vanderborght, M.T. Vangenuchten, and J. Feyen. 1996. Comparison of three methods to calibrate TDR for monitoring solute movement in undisturbed soil. Soil Sci. Soc. Am. J. 60:747–754.[Abstract/Free Full Text]
• Nissen, H.H., P. Moldrup, and R.G. Kachanoski. 2000. Time domain reflectometry measurements of solute transport across a soil layer boundary. Soil Sci. Soc. Am. J. 64:62–74.[Abstract/Free Full Text]
• Seuntjens, P., D. Mallants, Ch. Cornelis, and P. Geuzens. 1999. Grid lysimeter study of non-reactive solute transport through distinct multi-layered Spodosols using TDR and wick samplers. p. 284–294. In J. Feyen and K. Wiyo (ed.) Proc. Symp. Modelling of transport processes in soils. Leuven. 24–26 November 1999. Wageningen.
• Simmons, C.S. 1982. A stochastic-convective transport representative of dispersion in one-dimensionnal porous media systems. Water Resour. Res. 18:1193–1214.
• Snow, V.O., B.E. Clothier, D.R. Scotter, and R.E. White. 1994. Solute transport in a layered field soil-experiments and modeling using the convection-dispersion approach. J. Contam. Hydrol. 16:339–358.
• Toride, N., F.J. Leij, and M.Th. van Genuchten. 1995. The CXTFIT code for estimating transport parameters from laboratory or field tracer experiments. U.S. Salinity Lab., Riverside, CA.
• Vanclooster, M., C. Gonzalez, J. Vanderborght, D. Mallants, and J. Diels. 1994. An indirect calibration procedure for using TDR in solute transport studies. p. 215–226. In Proceedings of the International Symposium on Time Domain Reflectometry in Environmental Infrastructure, and Mining Applications. Evanston, IL. 7–9 Sept. 1994. Spec. Publ. SP19-94, U.S. Department of Interior, Bureau of Mines, Washington, DC.
• Vanclooster, M., D. Mallants, J. Vanderborght, J. Diels, J. Vanorshoven, and J. Feyen. 1995. Monitoring solute transport in a multi-layered sandy lysimeter using time domain reflectometry. Soil Sci. Soc. Am. J. 59:337–344.[Abstract/Free Full Text]
• Vanderborght, J., D. Mallants, M. Vanclooster, and J. Feyen. 1997. Parameter uncertainty in the mobile-immobile solute transport model. J. Hydrol. (Amsterdam) 190:75–101.
• Vanderborght, J., M. Vanclooster, D. Mallants, J. Diels, and J. Feyen. 1996. Determining convective lognormal solute transport parameters from resident concentration data. Soil Sci. Soc. Am. J. 60:1306–1317.[Abstract/Free Full Text]
• Vogeler, I., B.E. Clothier, and S.R. Green. 1997. TDR estimation of the resident concentration of electrolyte in the soil solution. Aust. J. Soil Res. 515–526.
• White, R.E., J.S. Dyson, Z. Gerstl, and B. Yaron. 1986. Leaching of herbicides through undisturbed cores of a structured clay soil. Soil Sci. Soc. Am. J. 50:277–283.[Abstract/Free Full Text]
• Whitmore, A.P. 1991. A method for assessing the goodness of computer simulation of soil processes. J. Soil Sci. 42:289–299.
• Zhang, R.D. 2000. Generalized transfer function model for solute transport in heterogeneous soils. Soil Sci. Soc. Am. J. 64:1595–1602.[Abstract/Free Full Text]

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