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DIVISION S-1-SOIL PHYSICS

Estimating Soil Hydraulic Properties During Constant Flux Infiltration

Inverse Procedures

Soil Science Dep., Univ. of Saskatchewan, Saskatoon, SK, Canada S7N 5A8

kachanoski{at}admin.usask.ca

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Analysis Results and discussion Summary REFERENCES

 
There is a need for accurate and cost-effective methods to estimate the hydraulic properties of soils. Past work indicated measurements of a single hydraulic response will not necessarily result in unique and stable estimates of hydraulic parameters when the number of unknowns is more than two. Prior information regarding the parameters or additional measurements are needed for the estimation problem to be well posed. However, accurate prior information is seldom available due to variations of the hydraulic properties in space and time. This paper presents a method for estimating hydraulic properties from simultaneous measurements of soil water storage to a fixed depth as a function of time during constant flux infiltration, and steady-state pressure head readings using vertically installed multi-purpose time domain reflectometry probes (MTDR). Multi-purpose TDR probes have a porous steel cup at their ends allowing soil water storage and to be simultaneously measured at the same location. Our parameter estimation is formulated by an inverse procedure which combines a weighted nonlinear least square method with analytical solutions for soil water content and pressure head as functions of depth and time during one dimensional infiltration. We analyze the possibility of using water storage data combined with the initial and steady-state pressure head readings for the purpose of estimating soil hydraulic properties. The uniqueness problem was analyzed by studying the behavior of response surfaces. The combination of water storage measurements during constant flux infiltration with an initial and a steady-state pressure head reading yielded unique and stable solutions of the inverse problem. The utility of the parameter estimation procedure is demonstrated using experimental and theoretical data.

Abbreviations: BW, Broadbridge and White (1988) • MTDR, multi-purpose time domain reflectometry probes

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Analysis Results and discussion Summary REFERENCES

 
IN EFFORTS TO BETTER MONITOR AND MANAGE the migration of chemicals in the vadose zone, scientists and engineers over the past several decades have developed analytical and numerical models describing the movement of water and chemicals into and through the unsaturated zone. These models have become indispensable tools for quantifying and integrating transport processes in the unsaturated soil zone. The application of these models to field-scale flow and transport problems relies heavily on the quality of the model parameters, especially the unsaturated hydraulic properties.

For accurately describing soil unsaturated hydraulic conductivity, K(), and soil pressure head, (), relationships over a wide range of soil water contents, , the equations of Brooks and Corey (Brooks and Corey, 1966), van Genuchten and Mualem (van Genuchten, 1980) and Broadbridge and White (Broadbridge and White, 1988) are good choices (van Genuchten and Nielsen, 1985; White and Broadbridge, 1988). These equations have four to five independent parameters, which vary considerably among soils.

Inverse procedures can be used to estimate the parameters of hydraulic functions (Dane and Hruska, 1983; Zachman et al., 1981; Kool et al., 1987, 1988; Russo et al., 1991; Simunek and van Genuchten, 1996). For given initial and boundary conditions, Richards' equation can be solved with appropriate analytical or numerical methods. Particular constitutive functions for the hydraulic properties are assumed. The parameters in these hydraulic functions are estimated by minimizing the difference between the predicted and observed hydraulic responses, such as pressure head, , water content , flow rate, or other flow attributes. The approach is attractive because few restrictions are imposed upon the experimental conditions, allowing relatively simple experimental designs.

A serious problem encountered in the estimation of hydraulic functions stems from their overparameterization, or inclusion of unidentifiable parameters. For example, Parker et al. (1985) indicated that the inverse problem for a one-step outflow experiment is non-unique when the number of unknown parameters is three and only outflow volume is measured. Russo et al. (1991) concluded the same for ponded infiltration when only infiltration rate is recorded. Measurement of cumulative infiltration rate from a tension infiltrometer is also not enough information to obtain unique estimation of saturated hydraulic conductivity K_s, and two other shape parameters for van Genuchten equations (Simunek and van Genuchten, 1996). To solve the identification problem, additional information regarding the parameters must be provided. There are usually two types of information: (i) additional measurements of one or more response variables, such as measurements as in the case of Parker et al. (1985), or another set of experiments (van Dam et al., 1992; Eching and Hopmans, 1993; Parkin et al., 1995b); (ii) prior information about the hydraulic parameters, such as the saturated hydraulic conductivity K_s and one of the shape parameters (Russo et al., 1991).

Prior information about the parameters is a very effective method to remove nonuniqueness. Prior information usually takes the form of an initial guess of the parameter values and range based on easily measured soil properties. The actual value of the parameter identified in the inverse problem can be different from this initial guess. However, this pre-specified and approximate value constrains the parameter within the range. An inverse problem is always unique if accurate prior information about all the parameters is available and an appropriate estimation procedure is adopted (Abaspour et al., 1997). Prior information about saturated water content _s can be obtained through bulk density, and soil particle-size distribution. Accurate prior information about the shape parameters and K_s is seldom available, because the parameters are highly variable in field conditions. It is usually easier to take additional measurements of a response variable such as than to have accurate prior information about these hydraulic parameters.

Not all additional response measurements are useful for improving well-posedness of an inverse problem. The usefulness of additional measurements depends on its sensitivity to the hydraulic parameters, independence of the existing measurements, and measurement error. For example, water content, pressure head can be added to measurement of transient cumulative infiltration rate from the disk tension infiltrometer for inverse parameter identification (Simunek and van Genuchten, 1997). The effort needed to take the measurements is also an important factor. Pressure head is a good choice for additional measurements if only water flux or water content have been measured (Kool and Parker, 1988; Parker et al., 1985). The standard method to measure is to install a tensiometer at a spatial location different from the soil water measurements. This may introduce significant error in spatially varying soils (Tseng and Jury, 1993).

Recently, vertically installed multi-purpose TDR probes developed by Baumgartner et al. (1994) were used to measure simultaneously soil water storage, W, and at the same spatial location during constant flux infiltration (Si et al., 1999). This effectively alleviates the error from measurements taken at different locations. The approach of Parkin et al. (1995b) was used to measure directly the field average K(). The approach used the spatial variability of vertical soil water flux under constant water application at different rates. Multipurpose TDR probes were used to measure field average () after steady-state had been reached. Despite the simplicity, these methods have the following disadvantages (Parkin et al., 1995b; Si et al., 1999): (i) multiple water application rates are required. (ii) only field average K() and () are obtained. (iii) The method does not make use of transient information on the depth integrated shape of the wetting front collected from water storage measurements at each TDR probe.

The objectives of this paper are to extend the method of Si et al. (1999) and estimate the hydraulic parameters at a single location (TDR probe) from multipurpose TDR measurements of water storage W and pressure head during a single constant flux infiltration experiment. The method was formulated as an inverse problem and the full set of transient water storage measurements are used in the estimation process. The need for measurements, to guarantee unique parameter estimates are examined from the response surfaces for different combinations of parameter spaces. Analytical solutions for transient soil water storage, W(t), and under constant flux infiltration (Parkin et al., 1992) are used with the realistic hydraulic function of Broadbridge and White (1988). The utility of the parameter estimation procedure is demonstrated using experimental and theoretical data.

	Theory

TOP ABSTRACT INTRODUCTION Theory Analysis Results and discussion Summary REFERENCES

 
Statement of the Estimation Problem
A typical curve of soil water storage as a function of time, W(t), to a fixed depth as measured by TDR during constant flux infiltration is shown in Fig. 1 . This curve can be partitioned into three distinct pieces of information. The first piece is the water storage during the period that the wetting front remains within the length of the TDR probe. In this period, assuming approximate one dimensional flow, the water storage increases linearly with time. The rate of change of W(t) during this time is equal to the local infiltration rate at this location. The second piece of information is the shape (curvature) of the soil water storage curve from the time when the wetting front just reaches the end of TDR rod to the time when the wetting front completely passes the TDR rod. This section contains the depth integrated transient shape of the wetting front and represents the nonlinearity of water flow in soil during constant flux infiltration. The third piece of information is the water storage value at steady state.

View larger version (15K): [in this window] [in a new window]   Fig. 1 Delineation of soil water storage curve into different regions (phases): Phase 1 (water storage increases linearly with time); Phase 2 (water storage breakthrough phase); and Phase 3 (steady-state flow phase)

In this paper, the saturated hydraulic conductivity (K_s), inverse capillary length scale (), a shape parameter (C) introduced by Broadbridge and White (1988), and the saturated water content (_s) are treated as unknowns. The purpose is to estimate K_s, , C, and _s given the measurements of the full W(t) curve, a priori information about _s and the easily measured at the initial condition and the final steady-state condition. Here, prior information about _s is used, because the prior _s is easy to obtain. Our intention is not to analyze the influence of prior information on the parameter estimation, but rather to use as much information as possible in the estimation problem based on the Bayesian philosophy (Box and Tiao, 1973). The influence of additional measurements of steady-state is examined.

Parameterization of Soil Hydraulic Properties
Broadbridge and White (BW) (1988) and Sander et al. (1988) independently developed an analytical solution for constant flux infiltration. The BW solution is based on the following parameterization of hydraulic conductivity, K() (cm h^-1) and diffusivity function, D() (cm² h^-1)

(1)

(2)

where and . _s and _r are the saturated and residual water content (m³ m^-3), respectively. K_s (m s^-1) and (m^-1) are the saturated hydraulic conductivity and inverse capillary length scale (Philip, 1985), respectively. The parameter C is a shape constant introduced by BW. By definition,

(3)

we have,

(4)

By assuming ₀ as zero as did Broadbridge and White (1988), substitution of Eq. [1] and Eq. [2] into Eq. [4] and integration yields

(5)

Forward Problem
Richards' equation used to describe one-dimensional nonhysteretic flow in homogeneous soils is given by

(6)

where (z,t) is the volumetric water content, z (m) is the depth being positive downward and t (s) is the time. The initial and boundary conditions considered here are

(7)

(8)

where R is the application rate (m s^-1) at the soil surface and _i is the initial soil water content (m³ m^-3). Utilizing Eq. [1] and Eq. [2], through a series of transforms (i.e., Kirchhoff, Hopf and Cole, and Storm transforms), BW derived an analytical solution as

(9)

and

(10)

where is a parameter connecting Eq. [9] and Eq. [10], u(,t) is given by Eq. [43] of BW, and

(11)

By change of variable of integration, Parkin et al. (1992, 1995a) derived analytical solutions for water storage for infiltration and drainage, respectively. Using the unified solution of Warrick et al. (1990) for the soil water content profile, a unified water storage solution for both infiltration and drainage can be obtained as following,

(12)

Equation [12] provides the basic formula for estimation of soil hydraulic parameters.

Formulation of the Inverse Problem
The inverse problem is to obtain parameter vector by best fitting Eq. [12] to measured water storage W(L,t)*. This is repeated under the constraint of prior information about _s and additional measurements of steady-state (Eq. [5]). To obtain a good estimate of a particular hydraulic parameter, we need to define an estimator. A good estimator minimizes the discrepancy between the measurement and predicted response, while best reflecting the hydraulic properties of the medium. A general, intuitively appealing and theoretically sound estimator is the maximum a posteriori estimator, which incorporates additional measurements and prior information into the estimator (Bard, 1974). In this way, the obtained parameters are guaranteed physically meaningful and may convert a degenerate equation into a nondegenerate case (Bard, 1974). Assuming the measurement errors asymptotically follow multi-variate normal distributions, the likelihood function, L(ß|y*), can be formulated as

(13)

where n is the number of observations, det[ ] indicates determinant,

and ; the values of transient water storage W, and pressure head are those predicted by Eq. [12], and Eq. [5] at times associated with the measurements; G_w, GC, and G_p are the covariance matrices of W, , and prior information, respectively. The independence between W, , and prior information are assumed, as indicated by zero nondiagonal terms in the covariance matrix. This is a reasonable assumption, because: (i) the measurements of water storage and hydraulic head are taken by different instrumentation and measured at different times; (ii) the prior information is usually a good guess from other sources of information or an approximation according to characteristics of the solution. Therefore, the measurement error associated with the hydraulic head or the water storage are not the dominant factor controlling the error of prior information. Since the logarithm is a monotonic increasing function of its argument, the value of ß that maximizes L(ß) also maximizes log (L(ß)). Since log L is frequently a simpler function than L, the maximum likelihood estimator is obtained by minimizing the negative log of the a posterior likelihood function (Bard, 1974),

(14)

Assuming uniform measurement error (constant variance) and prior information available only for _s, minimizing Eq. [14] is equivalent to minimizing the following weighted nonlinear least squares estimator (Carrera and Neuman, 1986)

(15)

where ²_W, ², and ² are the variances of measurement error in W, of measurement error in , and of estimation error in _s, respectively. The inverse problem is to minimize S with respect to ß given W*, * and prior information about _s, ^*_s. Usually, _w, , and are unknown, and could be treated as unknown parameters in minimizing Eq. [15]. However, this introduces more parameters and uncertainty in the inverse problem, which is not recommended. Therefore, _w, , and are selected empirically from other sources of information or experience (Bayesian philosophy), which is subjective. A smaller means a heavier weight on the measurement and the resulting parameters fit the measurement better at the cost of fitting W or prior information worse. Too much weight on one kind of measurement is a waste of information of the other kinds. Therefore, improper selection of weights will lead to an ill-posed inverse problem. An inverse problem is ill posed if (i) the solution does not exist, (ii) is not unique, or (iii) not stable, which means small change in the response can cause large change in the parameters. For this problem, the possibility of ill-posedness is reduced because:

1. As shown by Broadbridge and White (1988), the BW form is fairly realistic and capable of incorporating soil properties ranging from those of the weakly nonlinear Burgers' equation to those of a highly nonlinear Green-Ampt-like model. Thus, the parameterization is simple enough to yield a well-posed problem but complex enough to capture the salient features of the change of pressure head and hydraulic conductivity or diffusivity with water content.
   
2. Three (K_s, , _s) of four parameters in the BW model have clear physical meaning, thus more information about the parameters can be projected into the model according to the properties of the soil, such as organic mater, bulk density, etc.
   
3. Adopting an advanced technology such as the multipurpose TDR probe allowing repeated measurements, should reduce measurement error of the response.
   
4. An exact analytical solution is used and the model error or the numerical error due to discretization in space and time should be minimized.
   
5. Flux boundary conditions are easy to control (Si et al., 1999).
   
6. Local application rate, R*, can be calculated from the solution characteristics: The rate of increase in water storage at initial stage equals the local water flux. In this way, the local infiltration rate can be obtained, and one unknown is reduced from the inverse problem.
   

	Analysis

TOP ABSTRACT INTRODUCTION Theory Analysis Results and discussion Summary REFERENCES

 
Sensitivity Coefficients
To find the approximate range of applied infiltration rates over which the parameters have the maximum sensitivity to the water storage W(L,t). The sum of the sensitivity coefficients over time at specific infiltration rate R was calculated using the finite difference method as follows (Yeh, 1986).

(16)

Where e_j is the jth unit vector and m is the number of measurements. In this paper, only the sensitivity coefficients of as a function of R were calculated and the numerical calculations were carried out using the software package Mathcad (version 6, Mathsoft, Inc.).

Uniqueness and Stability Analysis
To investigate the question of nonuniqueness, the global properties of the prediction error were examined. If the prediction error surface has a single minimum point, the solution is unique. If it has more than one minimum point, the solution is nonunique, and additional information must be added to resolve the indeterminacy. When the number of unknown parameters is two, it may be possible to investigate the shape of the surface by graphical techniques (Menke, 1989). Thus, the uniqueness of the inverse problem was evaluated from the two-dimensional response surfaces of the objective function as a function of pairs of soil hydraulic parameters. This contour analysis applies only to the case where there are two unknown parameters. Because of our human limitation of understanding in higher dimensions, the uniqueness analysis for problems with three unknowns or more is usually carried out in two dimensions, by fixing one or more of the other parameters. However, the information provided by the contours are nonconclusive regarding the well-posedness, but conclusive regarding the ill-posedness of an inverse problem. To confirm the above results obtained by analyzing response surfaces, we use the Statistical Analysis System (SAS/Stat User's Guide, vol. 2, 1994) procedure NLIN to numerically find the global minimum of the objective functions for different scenarios. To analyze the cases where there are more than two unknowns, the above graphical analysis is not conclusive and the problem has to be solved several times with five randomly selected initial parameter guesses, to have confidence. A program named FIT_BW.sas written in SAS language was used to carry out the numerical inversion. The program makes use of a modified Gauss-Newton nonlinear optimization methods. Because of the complicated nature of the derivatives of water storage with respective to the parameters K_s, , and C, a numerical derivative was used.

Simulated Soil Water Storage for a Sandy Soil
A simulated case for a sandy soil with a uniform initial condition of and a flux boundary condition of is used as an example. Parameter values for C, K_s, , and _s, and _r for a typical fine sandy soil are assumed to be 1.24, 1.38 x 10^-5 m s^-1, 8.0 m^-1, 0.41 m³ m^-3, and 0.06 m³ m^-3, respectively. Water storage data was generated at a time interval of 0.15 h from 0 to 7.5 h using Eq. [12] for a TDR probe of length . The generated data is error-free, in the sense that it is identical to the predicted data. The generated water storage vs. time is shown in Fig. 2 for error-free water storage measurement.

View larger version (16K): [in this window] [in a new window]   Fig. 2 Calculated soil water storage vs time for error-free measurements (solid curve) and measurements with error (standard deviation, = 0.0026 m) (dashed line)

For the sandy soil example, the error response surface as a function of parameter pairs was calculated according to Eq. [14], while keeping other parameters at the true value. The calculations were carried out with and without error in the water storage measurements. Response surface contour lines were drawn by Surfer (Golden software, 1995) using the inverse distance interpolation algorithm. Response surface were also calculated with prior average _s estimated at 0.41 m³ m^-3 with standard deviation = 0.09 m³ m^-3 (Carsel and Parish, 1988). Finally, response surfaces were calculated assuming either a final steady-state measurement was available: , or both a final and an initial are available. We assumed that the ² is three times bigger than ²_W. Therefore, all the unknowns in Eq. [15] are known and the response surface for different scenarios can be constructed.

Field Experiment
The experiment was conducted at Borden, Ontario, Canada. A detailed description can be found in Si et al. (1999). Briefly, the experimental surface area was covered with a greenhouse to prevent effects of wind, precipitation, and evaporation. Within the sampling area, 50 multi-purpose (length = 0.2 m) TDR probes for a given depth were installed every 0.15 m in a 7.5-m-long transect. A hanging track system with spray nozzles was used to provide uniform application of water along the transect. The nozzles were installed in a linear array at 0.1-m intervals with their spray axis aligned perpendicular to the transect. This produced a uniform narrow wetted area of approximately 0.2 by 2 m². The nozzle array was attached 0.5 m above the soil surface to a commercially available, programmable water application system designed for greenhouses (model DCA, Monorail Boom Spray System, Waterford, ON, Canada). The system has a single pressure regulator, electric solenoid valve, and microprocessor unit attached to a hanging track and conveyor belt. The hanging track is mounted to the top of the greenhouse and allows the nozzle array to go smoothly back and forth along a straight line (9-m long) with programmable delay time at either end of the line. The water can be turned on or off, at either end of the line. The system produces a uniform wetted area 2-m wide by 9-m long centered over the TDR instrumented transects.

Soil water content was measured using the TDR method of Topp et al. (1980). The readings were taken manually from the display screen of two pre-calibrated Tektronix (1502 C) metallic cable tester by four operators. The readings were taken just prior to the start of water application and every 5 to 30 min depending on infiltration rate and rate of change of . After the wetting front was beyond the 80-cm depth and all measurements indicated little or no change with time, the pressure head measurements were taken using two tensimeters (Soil Measurement System, Tucson, AZ). The measurements were taken for application rates = 5.5 x 10^-7, 2.5 x 10^-6 and 9.1 x 10^-6 m s^-1. Infiltration experiments were conducted for six water application rates. After each application, the profile was allowed to drain. We also took measurements at the initial condition before we started an infiltration experiment for application rate = 1.7 x 10^-5 m s^-1.

Estimated K() and () functions using the inverse procedure and average water storage measurements W* for the single application rate = 7.2 x 10^-6 m s^-1 were compared with those estimated from steady-state measurements from all six application rates.

	Results and discussion

TOP ABSTRACT INTRODUCTION Theory Analysis Results and discussion Summary REFERENCES

 
Sensitivity Analysis
The sensitivity of the objective function to the hydraulic parameter, K_s, C, and is shown in Fig. 3 . The maximum sensitivity for K_s occurs when the water application rate . The sensitivity to K_s increases linearly for R 0.25 K_s and then starts to plateau with almost maximum sensitivity for R 0.5 K_s and then decreases again for R > 0.8 K_s. Sensitivity to C and is maximal at approximately , and decreases with higher or lower application rates. At approximately , there is very good sensitivity to all three parameters. Thus for sandy soils, a constant application rate equal to approximately 0.4 K_s is likely optimal for hydraulic parameter estimation.

View larger version (20K): [in this window] [in a new window]   Fig. 3 Sensitivity of water storage W with respect to (a) Ks, (b) C, and (c) to the change of relative application rate R/Ks

Uniqueness and Stability Analysis
Figure 4 shows the response surfaces of the objective function S(W) for the three different parameter planes, K_s–C, K_s–, and C– assuming only error-free water storage data are available. The K_s– response surface shows a single well defined minimum, at the true parameter values (Fig. 4a). Since the objective function is the negative logarithm of the joint probability density of K_s and at a given value of C the contour line also reflected the reciprocal of the probability of the combination of the two parameters. The width of the contour reflects the spread of the probability distribution about the mean of the parameter. Since the contour is elliptical, the uncertainty associated with is bigger than that of the K_s. Thus, W is more sensitive to K_s than to . In addition, the contour ellipses are tilted. Thus, large values of K_s are more probable if is large. This suggests and K_s are positively correlated. Contour of the response curve in the C– plane are also tilted ellipses, suggesting a large will also likely have a corresponding large C (Fig. 4c). The response surface in the K_s–C plane has a well defined valley which starts at high K_s and low C values and extended linearly through nearly the entire parameter space (Fig. 4b). Thus, the parameters C and K_s cannot be estimated simultaneously from the water storage measurements only.

View larger version (86K): [in this window] [in a new window]   Fig. 4 Contours of S for error-free water storage data as a function of (a) Ks and , (b) Ks and C, and (c) and C

View larger version (79K): [in this window] [in a new window]   Fig. 5 Contours of S for water storage data contaminated by an error of = 0.0026 m as a function of (a) Ks and , (b) Ks and C, and (c) and C

View larger version (70K): [in this window] [in a new window]   Fig. 6 Contours of S with an additional measurement for error-free water storage data as a function of (a) Ks and , (b) Ks and C, and (c) and C

View larger version (79K): [in this window] [in a new window]   Fig. 7 Contours of S with an additional measurement for water storage data with an error of = 0.0026 m as a function of (a) Ks and , (b) Ks and C, and (c) and C

When an additional measurement of initial is added, the parameters converge to the true values with relative error <2% even for the four parameter case (Table 1). The simultaneously measured pressure head and water content data at two states uniquely defines the general shape of the water retention curve, and the saturated hydraulic conductivity becomes the main unknown parameter in the inverse problem. The correlations between K_s and , K_s and C were also reduced. However, the correlation between C and is still high, suggesting more information such as prior information on K_s would produce a better estimate.

Inversion of In Situ Data
We conclude by illustrating the performance of our proposed methodology to the field experiment. Average water storage data from 50 MTDR probes is shown in Fig. 8, Fig. 9 . With only water storage data, the inverse solution converged to acceptable results, but the estimation errors are very large (Table 2) and cannot be calculated for .

View larger version (16K): [in this window] [in a new window]   Fig. 8 Measured and predicted water storage vs. time (L = 0.2 m; R = 9.1 x 10-6 m s-1)

View larger version (8K): [in this window] [in a new window]   Fig. 9

With water storage and two measurements, the estimated parameters were again very close to the steady-state values measured by Si et al. (1999) and have small asymptotic standard errors (Table 2). The correlation matrix of estimated parameters is better than the single case.

The estimated parameters in Table 2 were used as input to Eq. [12] to predict water storage as a function of time for application rate (Fig. 8). The coefficient of determination for predicted vs. measured water storage was and the standard deviation of prediction error for the average water content over the probe length is 0.0036 m³ m^-3. The prediction error is within the measurement error suggested for TDR (Topp et al., 1980).

The results for all measurement scenarios can be further improved with prior information about K_s, or additional measurements using a different water application rate. In this paper, only the hydraulic parameters of the BW form were estimated by taking advantage of the analytical solution of constant flux infiltration which assumes the BW form. Parameters for other forms such as van Genuchten and Brooks and Corey can be used with numerical or quasi-analytical solutions. The procedure proposed in this paper can also apply to the drainage process, since the primary equation (Eq. [12]) holds for both infiltration and drainage.

The analysis was based on the assumption that the measurement error is random. The three-term equation of Topp et al. (1980) may systematically underestimate or overestimate the soil water content from TDR measured dielectric constant. The effect of this systematic bias on the estimation of hydraulic parameters from measured soil water storage should also be examined.

The variance of measurement error or prior information has significant influence on the well-posedness of an inverse problem. If the variance of prior information is much bigger than the measurement error of soil water storage, the inclusion of prior information would be of little use for improving the well-posedness of an inverse problem. In addition, if the measurement error is much bigger, the inverse (conditioning) method would not be able to improve the precision of the parameters based on prior information. In this paper it was assumed that prior information on the approximate field average _s can be obtained from bulk density, particle-size distribution, or at the very least a hand texture determination. However, the results in this paper are very conservative. The variance of measurement error used for soil water storage and the variance of prior information about _s were estimated for a wide range of soils. A good TDR calibration equation and prior _s is likely much more accurate than we have assumed. Therefore, better uniqueness and stability of the inverse problem and more accurate parameter estimates can be expected in most field application. The correlation matrix in Table 2 will hold only approximately in the vicinity of the true parameter values where the linear approximation applies (Bard, 1974).

The proposed method assumes that flow is one-dimensional locally and the flow domain is semi-infinite, which implies that R is constant locally. This assumption is supported by previous work (Parkin et al., 1995; Si et al., 1999) and an excellent linear relationship between water storage and early time (R² > 0.96). Thus, the uncertainty associated with R is assumed to be very small.

Maximum sensitivity for simultaneous estimation of all parameters occurred for a water application rate R 0.5 K_s. However, this is only a guide for experimental development and not a requirement. Some information about the value of K_s for a site is usually known a priori, or the very least can be estimated from a hand texture determination on-site. Thus, it should be possible to choose an appropriate value of R. In addition, more than one rate can be applied. Si et al. (1999) used multiple rates and took measurements during only Phases I and III (Fig. 1) to determine the field average hydraulic parameters. The use of one additional water application rate (analysis not given) and similar measurements results in unique estimates of all parameters. In addition, as the amount of information increases, the requirement of prior information about _r, _s, and K_s is reduced. An analysis of combinations of several application rates and measurement types is beyond the scope of this paper. Finally, after steady-state has been reached it is possible to stop the water application (i.e., setting R = 0), and use the same analytical solution (Eq. [12]) to predict subsequent drainage. This is equivalent to the classic instantaneous profile method for hydraulic parameter estimation. Si and Kachanoski (2000) have examined measured vs. predicted water storage during drainage, using hydraulic parameters estimated from only infiltration measurements. They modified the analytical solution (Eq. [12]) to include a discrete change in (i.e., a Haines jump given by ) to account for hysteresis. Thus, it is also possible to combine the method presented in this paper with drainage measurements and obtain estimates of hydraulic parameters optimized for both infiltration and drainage, including the effect of hysteresis.

	Summary

TOP ABSTRACT INTRODUCTION Theory Analysis Results and discussion Summary REFERENCES

 
Measurements from MTDR probes combined with inverse procedures provides an accurate, fast, and nondestructive way to estimate hydraulic properties of the soil during constant flux infiltration. Water storage and measurement are made in the same volume, consequently, reducing the error associated with the spatial variability in the horizontal direction. Water storage measurements only and a priori _s during one-rate constant infiltration resulted in unique estimates of only two hydraulic parameters (K_s, ) in the BW form, because of the interactive nature between K_s and C in the solution. If C can be estimated, then reasonable estimator of K_s and are possible. Water storage combined with a steady-state and a priori _s gave unique estimates of all hydraulic parameters, but the inverse solution may not be stable with large measurement errors. Water storage and a priori _s combined with an initial and steady-state pressure head measurement resulted in unique and stable estimates of all hydraulic parameters.McLaughlin Townley 1996; SAS Institute 1994; Tikhonov Arsenin 1977

Received for publication May 5, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION Theory Analysis Results and discussion Summary REFERENCES

 

• Abaspour K.C., van Genuchten M.Th., Schulin R., Schläppi E. A sequential uncertainty domain inverse procedure for estimating subsurface flow and transport parameters. Water Resour. Res. 1997;33:1879-1892.
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• Box G.E.P., Tiao G. Bayesian inference in the statistical analysis. Reading, MA: Addison-Wesley, 1973.
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