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The Effect of the Spatial Sensitivity of TDR on Inferring Soil Hydraulic Properties from Water Content Measurements Made during the Advance of a Wetting Front

Ty P. A. Ferré^*,a, Henrik H. Nissen^b and Jirka imnek^c

^a Department of Hydrology and Water Resources, University of Arizona, AZ
^b Department of Environmental Engineering, Institute of Life Sciences, Aalborg University, Denmark
^c George E. Brown, Jr., Salinity Laboratory, USDA-ARS, Riverside, CA
* Corresponding author (ty{at}hwr.arizona.edu)

Received 15 May 2002.

	ABSTRACT

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Many measurement methods are considered to return point values of volumetric water content, water pressure, or other hydrologically relevant properties. However, many methods have spatially distributed measurement sensitivities, averaging a property of interest over some sample volume. In this investigation, we study the effects of the spatially distributed measurement sensitivity of time domain reflectometry (TDR) on the inversion of hydraulic properties from water content measurements. Specifically, a numerical analysis of the water breakthrough curves that would be measured by TDR probes of varying designs during the advance of a wetting front is presented. Numerical inversion of these water breakthrough curves is performed to estimate the soil hydraulic parameters. Time domain reflectometry probes with larger rod separations show less impact on the flow of water at the wetting front. However, these probes have more widely distributed spatial sensitivities, leading to more smoothing of the observed water breakthrough curve. The TDR-measured wetting front shape is more distorted for vertically emplaced probes than for horizontal probes. The optimal TDR probe configuration for inversion of hydraulic parameters from measurements recorded during the advance of a vertical wetting front has three closely spaced rods that lie in a common horizontal plane. The inversion results using this design show close agreement with known values and very small 95% confidence intervals of the inverted properties. This specific recommendation cannot be adopted generally for all TDR monitoring applications. Rather, we recommend that a similar analysis be performed for each specific monitoring application. While the results presented are specific to TDR responses, the same consideration should be given to all instruments with spatially distributed sensitivities.

Abbreviations: TDR, time domain reflectometry

	INTRODUCTION

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TIME DOMAIN REFLECTOMETRY has become a standard instrument for water content measurement since its introduction by Topp et al. (1980). Time domain reflectometry has been shown to give accurate water content measurements in a wide range of porous materials with little need for soil-specific calibration. In addition, the continual advance of TDR instrumentation and associated software has led to the ready availability of off-the-shelf TDR monitoring systems. Like other indirect water content measurement methods (e.g., neutron probes and capacitance probes), TDR measures some average water content in a volume of medium surrounding the probes. One unique aspect of TDR is the ease with which a user can alter the sample volume and spatial sensitivity of TDR probes. For the TDR method, this spatial sensitivity has been examined both analytically (Knight, 1992) and numerically (Knight et al., 1997). Further examination has shown the dependence of the TDR spatial sensitivity on the size and orientation of TDR probes (Ferré et al., 1998; Nissen et al., unpublished data). On the basis of these analyses, users can design probes for specific monitoring objectives (e.g., Nissen et al., 2002). For example, the number and relative position of the rods comprising the probe can be altered to achieve high-resolution measurements within small sample volumes adjacent to the rods or to return water content measurements that are representative of larger sample volumes farther from the rods. Despite these advances, there has been little consideration of the effects of this spatial sensitivity on the utility of TDR for monitoring hydrologic processes. Furthermore, the presence of the impermeable rods will have some effect on water flow. Therefore, an analysis of the mutual effects of the spatial distribution of dielectric permittivity, the spatial sensitivity of TDR probes, and the effects of TDR rods on water flow is necessary to choose the optimal probe design for any specific measurement application.

In this investigation, we study the optimal probe design to infer soil hydraulic properties from measurements made during the advance of a wetting front into initially dry sand. These conditions lead to a strong dielectric permittivity contrast across a sharp wetting front. Given that the spatial sensitivity of TDR probes depends upon the spatial distribution of dielectric permittivity, these are considered to be the most challenging conditions for representative water content measurement with TDR. We designed a numerical study to evaluate the effects of TDR probes and TDR spatial sensitivity on the numerical inversion of hydraulic properties. Initially, we used a numerical forward model of unsaturated water flow to simulate the advance of a wetting front past TDR rods of varying geometries. We then used a numerical analysis of the spatial sensitivity of TDR probes to predict the water content that would be measured as a function of time for different probe designs. Finally, we used numerical inversion of the TDR responses to estimate soil hydraulic properties, thereby identifying potential errors in hydraulic property inversion arising from TDR-measured water contents.

The objectives of this investigation are (i) to determine whether TDR probes cause significant disruption to flow during the advance of a wetting front, (ii) to determine whether numerical inversion of TDR-measured water contents leads to accurate estimates of soil hydraulic properties, and (iii) to determine whether any inversion errors that may result can be minimized by the correct choice of TDR probe design and orientation.

	METHODS

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HYDRUS-2D (imnek et al., 1999) was used to simulate the vertical advance of a wetting front into a two-dimensional cross-section by numerically solving the Richards' equation. The soil hydraulic properties (Table 1) were chosen to represent those of sand based on the ROSETTA (Schaap et al., 2001) database. The simulation domain was 0.05 m wide and 0.05 m deep. Initially, the pressure head was a constant -0.5 m throughout the domain. (Note that all pressure heads are expressed as meters of water.) The lateral boundaries were no flux. The upper boundary was held to a constant pressure head of 0 m, representing ponded infiltration, and the lower boundary was held to a constant -0.5 m. The simulation time was 60 s. TDR probes were placed within the transport domain as impermeable objects and were represented as circular no flow boundaries. "Horizontal" probes were placed such that all of the horizontal rods were located in a common horizontal plane, while "vertical" probes had horizontal rods located in a common vertical plane (Fig. 1). All of the rods were placed such that the center of the TDR probe was located at the center of the domain. Probes were chosen to represent very small probes like those that may be used in column experiments. As such, the probes were comprised of 0.0015-m-diameter rods. For comparison of probe geometries, "Two-rod" probes were represented as two parallel rods with the centers of the rods separated by 0.01 m. "Three-rod" probes had three equally spaced rods, with the outermost two rod centers separated by 0.01 m. To examine the effects of rod separation, further simulations were conducted for two-rod horizontal and vertical probes with separations of 0.005 and 0.02 m. The probes are described based on the number of rods, the orientation, and the separation of the outermost rods. For example, the two-rod horizontal probe with the rods separated by 0.01 m is referred to as the 2H(0.01) probe.

View this table: [in this window] [in a new window]   Table 1. The values of , n, and Ks used in the forward model (Actual), those determined through inversion of the water content at the center of the domain with no rods present as a function of time (No rods), those determined from the water content at the mid depth of the domain with 2H, 2V, and 3H TDR probes present (designated as "rods"), and those determined from inversion of the water contents measured with 2H, 2V, and 3H TDR probes.

View larger version (76K): [in this window] [in a new window]   Fig. 1. Finite element domain used to simulate the advance of a wetting front into a domain with an initial uniform pressure head of -0.5 m. The circular holes in the domain are no-flow boundaries to represent the obstruction to flow caused by a 2H(0.01) TDR probe. Schematic vertical cross-sections of 2V and 3H probes are shown as well. The rods lie in the horizontal plane for all of the probe designs.

An example of the finite element domain used to simulate water flow around a 2H(0.01) probe is shown on Fig. 1. An effort was made to make the finite element grid geometries used in all of the simulations as similar as possible. To achieve this, circles outlining all of the rods were included in all of the simulation domains. In total, 13 circles were used. Those circles that were not used to represent rods in any given simulation were filled with elements. Circles representing rods were impermeable holes in the grid. An observation point was placed at the midpoint of each probe, located at the center of the domain. Given that an observation point cannot be placed in the center of the domain for the three-rod probes, which have a rod centered at the domain center, the observation point for these probes was placed at a point equidistant from an outer and the inner rod.

The water content, (m³ m^-3), distribution within the domain was determined every 2.4 s for 1 min of simulation time, for a total of 26 water content distributions. The relative dielectric permittivity, K, distribution was determined from the water content distribution based on the linear form of the Topp et al. (1980) relationship presented by Ferré et al. (1996):

[1]

For each simulated time, the numerical analysis of Knight et al. (1997) was then used to determine the dielectric permittivity that would be measured for the given TDR rod distribution on the basis of the dielectric permittivity distribution at that time. This analysis is based on the determination of the spatial distribution of the energy of the electromagnetic field in the plane transverse to the direction of electromagnetic plane wave propagation. Knight (1992) showed that this distribution could be used to define the spatial sensitivity of a TDR probe in the transverse plane. We use the numerical analysis to determine the weighted average dielectric permittivity that would be measured by a TDR probe of a given configuration for a given dielectric permittivity distribution in the vertical plane. Equation [1] can be rearranged to define the "measured" water content based on this dielectric permittivity. The inversion capabilities of HYDRUS-1D (imnek et al., 1998) were subsequently used to determine the hydraulic parameters that best fit the 26 probe-measured water contents as a function of time. HYDRUS-1D uses the Levenberg–Marquardt optimization technique to minimize the sum of squared deviations of measured and computed variables. The soil hydraulic properties were defined using the van Genuchten (1980) formulations. The Brooks and Corey (1964) formulations, or a step function, could have been used to examine a very sharp wetting front, thereby increasing the expected errors. However, we chose to use the van Genuchten (1980) form to determine whether smoothing at a wetting front would greatly reduce errors due to TDR spatial sensitivity. For the inversion, the saturated water content, _s, the residual water content, _r, and the l parameter (Mualem, 1976) were held constant with values of 0.43, 0.045, and 0.5, respectively. These values are equal to those used for the forward flow simulations with HYDRUS-2D. The inversion optimized the values of the van Genuchten parameters (m^-1) and n, and the saturated hydraulic conductivity, K (m s^-1) for the relationships:

[2]

[3]

[4]

where h is the pressure head (m) and h_s is the air-entry value. The value of each of the fitted parameters that was used in the forward flow model was used as an initial guess for the inversion.

	RESULTS

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Limitations of One-Dimensional Inversion of Wetting Front Data
One objective of this investigation was to determine the impacts of the distributed spatial sensitivity of TDR on the monitoring of wetting fronts. Specifically, we investigated the inversion of hydraulic parameters from water content measurements collected during the advance of a wetting front. However, it should be recognized that, in general, inverse modeling does not necessarily guarantee unique values of inverted properties. Furthermore, the forward flow model used in these analyses (HYDRUS-2D) was not identical to the inverse model used (HYDRUS-1D). For example, HYDRUS-2D uses the finite element method for a spatial discretization of the Richards equation, while HYDRUS-1D uses finite differences. Therefore, we conducted an initial numerical experiment to identify errors and uncertainties associated with the inverse modeling of a wetting front given exact knowledge of the water content at a single observation point. An equally weighted 26-point objective function was formed from the water content time series at the observation node with no rods present (all circles filled with elements). The correct soil hydraulic properties were used as initial estimates in the inverse model. Thus, the objective of this study was not to determine the uniqueness of optimized parameters, but, rather, to determine how much information is lost when TDR-measured water contents are used to form the objective function. Finally, similar spatial discretization was used in both the HYDRUS-1D and HYDRUS-2D models, with the nodal spacing of 0.05 cm for HYDRUS-1D and an average nodal spacing of 0.08 cm for HYDRUS-2D with higher resolution near the rods. The time discretization, initial conditions, and boundary conditions were identical in both models.

The observed and fitted water content time series (Fig. 2) showed excellent agreement (R² = 0.99998), suggesting that there is very little error introduced by the combined use of HYDRUS-2D for forward modeling and HYDRUS-1D for inversion. (Note that the observations are based on the actual water content at the midpoint of the domain, not on the TDR-measured water content time series.) The fitted values of , n, and K_s are shown in Table 1 and are referred to as the "No rods" scenario. The inverted values were used to construct characteristic curves (Fig. 3). The actual soil characteristic curves (thick lines) and the no rod inversion (thin lines) are virtually identical. Furthermore, the 95% confidence intervals for the inverted parameters are very narrowly defined (Table 1). Confidence intervals of optimized parameters are calculated in HYDRUS-1D from knowledge of the shape of the objective function at its minimum, the number of observations, and the number of unknown fitted parameters. It is desirable that the estimated mean value be located in a narrow interval. Large confidence limits indicate that the results are not very sensitive to the value of a particular parameter. Given that the confidence intervals will be a function of the exact choice of parameters to fit, the reported values should not be taken to describe the absolute accuracy of the inversion of hydraulic parameters from TDR measurements. Rather, the relative values of the confidence intervals should be used to compare the suitability of several probe designs. The region between the dashed lines on Fig. 3 shows the full range of values describing the characteristic curves that lie between curves calculated with the 95% confidence intervals of the inverted van Genuchten (1980) parameters. The uppermost dashed line was formed using the highest values of , n, and K_s within the 95% confidence interval, and the lowermost line used the lowest values.

View larger version (12K): [in this window] [in a new window]   Fig. 2. The water content breakthrough curve at the midpoint of the domain with no rods present determined through forward modeling with HYDRUS-2D (Actual) and the water content breakthrough curve predicted (Fitted) based on optimized values for the saturated hydraulic conductivity and van Genuchten's (1980) and n parameters as determined by numerical inversion of the observed water contents with HYDRUS-1D.

View larger version (24K): [in this window] [in a new window]   Fig. 3. The inverted soil hydraulic functions based on optimization of the actual water content at the center of the two-dimensional domain with no rods present lie within a very narrow range (region between the dashed lines) that includes the correct functions (thick lines). The fitted functions (thin lines) produce an excellent estimate of the correct functions.

View larger version (14K): [in this window] [in a new window]   Fig. 4. Water content breakthrough curves at the mid-depth of the simulation domain with no TDR rods present and with 2H or 2V probes of varying rod separations.

View larger version (24K): [in this window] [in a new window]   Fig. 5. The inverted soil hydraulic functions based on optimization of the actual water content at the center of the two-dimensional domain with a 2H(0.005) probe present lie within a narrow range (region between the dashed lines) that includes the correct functions (thick lines). The fitted functions (thin lines) produce an excellent estimate of the correct functions.

View larger version (17K): [in this window] [in a new window]   Fig. 6. Simulated TDR probe responses compared with the actual water content breakthrough curve at the center of the domain during the advance of a wetting front without rods present. The time of arrival of the wetting front is identified as the time required to reach one-half of the total water content change, which is shown as the intersection of the horizontal dashed line with each breakthrough curve.

Hydraulic Properties Determined from TDR-Measured Water Contents
The spatially distributed sensitivity of TDR probes has a clear effect on the shape of water content breakthrough curves. Therefore, researchers must be cautious when making detailed interpretations of water breakthrough curves based on TDR measurements. This caution also applies to measurements made with other instruments that have spatially distributed sensitivities. However, it is unclear from direct observation of the water content breakthrough curves how the distortion of the wetting front shape will impact the interpretation of basic hydraulic properties derived assuming point water content measurements. To examine this, HYDRUS-1D was used to infer the values of , n, and K_s from the simulated probe responses for the 2V(0.01), 2H(0.01), 3H(0.01), and 2H(0.005) probes. The fitted values and the 95% confidence interval of the parameter values are shown in Table 1. The best-fit inverse models show good agreement with the simulated TDR-measured water breakthrough curves, giving R² values in excess of 0.96 for all probes. The 3H probe shows the smallest range between the upper and lower 95% confidence intervals for all of the estimated values. Even comparing the 2H(0.005) and 3H(0.01) probes shows that although these probes had nearly identical actual water content breakthrough curves at the probe midpoints, the larger sampling area of the 2H probe led to slightly greater smoothing of the water content breakthrough curve. This leads to less accurate hydraulic property inversions.

The interpreted water retention and hydraulic conductivity functions are shown in Fig. 7 and 8 for the modeled TDR probes. As in Fig. 3, the region between the dashed lines shows the range of hydraulic functions that lie within the 95% confidence interval of the inversion. The accuracy of the fit is much better and the width of the 95% confidence interval is much narrower for the inversion based on the water content at the midpoint of a domain including a 2H(0.005) probe (Fig. 3) than for the inversion based on the TDR-measured water contents for this case (Fig. 7 and 8). Similar comparisons can be made for the 2H(0.01), 2V(0.01), and 3H(0.01) probes based on the results reported in Table 1. These results demonstrate the direct impact of the spatial sensitivity of TDR on hydraulic property inversions. That is, for the relatively thin rods examined, we can conclude that the effects of flow disturbance by the probes is insignificant compared with the effects of the spatially distributed sensitivity of TDR. The uncertainty introduced into the inversion of hydraulic parameters because of the spatial averaging of TDR probes is minimized by the use of 3H probes. Furthermore, the predicted hydraulic properties based on measurements made with 3H probes agree most closely with the correct values. Compared with the 3H responses, equally sized 2H and 2V probes lead to unacceptably uncertain parameter estimates. For the 2V probe, the correct hydraulic conductivity function lies outside of the 95% confidence interval at low water pressures. As a result, this configuration is not recommended for monitoring the advance of a wetting front.

View larger version (29K): [in this window] [in a new window]   Fig. 7. Fitted log hydraulic conductivity based on water content observations made with 2H(0.01), 3H(0.01), 2V(0.01), and 2H(0.005) probes. The thick lines show the soil parameters used in the forward model. The region between the dashed lines on each panel shows the 95% confidence interval. The thin lines show the best-fit parameters.

View larger version (27K): [in this window] [in a new window]   Fig. 8. Fitted hydraulic functions based on water content observations made with 2H(0.01), 3H(0.01), 2V(0.01), and 2H(0.005) probes. The thick lines show the soil parameters used in the forward model. The region between the dashed lines on each panel shows the 95% confidence interval. The thin lines show the best-fit parameters.

	CONCLUSIONS

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	REFERENCES

TOP ABSTRACT INTRODUCTION METHODS RESULTS CONCLUSIONS REFERENCES

 

• Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrology Papers. Colorado State University, Fort Collins, CO.
• Ferré, P.A., J.H. Knight, D.L. Rudolph, and R.G. Kachanoski. 1998. The sample area of conventional and alternative time domain reflectometry probes. Water Resour. Res. 34:2971–2979.
• Ferré, P.A., D.L. Rudolph, and R.G. Kachanoski. 1996. Spatial averaging of water content by time domain reflectometry: Implications for twin rod probes with and without dielectric coatings. Water Resour. Res. 32:271–279.
• Knight, J.H. 1992. Sensitivity of time domain reflectometry measurements to lateral variations in soil water content. Water Resour. Res. 28:2345–2352.
• Knight, J.H., P.A. Ferré, D.L. Rudolph, and R.G. Kachanoski. 1997. The response of a time domain reflectometry probe with fluid-filled gaps around the rods. Water Resour. Res. 33:1455–1460.
• Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513–522.
• Nissen, H.H., P.A. Ferré, and P. Moldrup. 2002. Metal-coated printed circuit board time domain reflectometry probes for measuring water and solute transport in soil. Water Resour. Res. In press.
• Schaap, M.G., F.J. Leij, and M.Th. van Genuchten. 2001. Rosetta: A computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions. J. Hydrol. 251:163–176.
• imnek, J., M. ejna, and M.Th. van Genuchten. 1998. The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media. Version 2.0, IGWMC- TPS- 70. International Ground Water Modeling Center, Colorado School of Mines, Golden, CO.
• imnek, J., M. ejna, and M.Th. van Genuchten. 1999. The HYDRUS-2D software package for simulating two-dimensional movement of water, heat, and multiple solutes in variably saturated media. Version 2.0, IGWMC- TPS- 53. International Ground Water Modeling Center, Colorado School of Mines, Golden, CO.
• Topp, G.C., J.L. Davis, and A.P. Annan. 1980. Electromagnetic determination of soil water content: measurement in coaxial transmission lines. Water Resour. Res. 16:574–582.
• van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898.[Abstract/Free Full Text]

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