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,* Henrik H. Nissen, and Jirka Šimůnek
ABSTRACTamination 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sensitivity on the size and orientation of TDR probes(Ferré et al., 1998; Nissen et al., unpublished data).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 therods 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 thisspatial sensitivity on the utility of TDR for monitoringhydrologic processes. Furthermore, the presence of theimpermeable 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 onwater 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 thenused a numerical analysis of the spatial sensitivity of
TDR probes to predict the water content that would bemeasured 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 thecorrect choice of TDR probe design and orientation.

Many measurement methods are considered to return point valuesof volumetric water content, water pressure, or other hydrologically
relevant properties. However, many methods have spatially distributed measurement sensitivities, averaging a property of interest oversome sample volume. In this investigation, we study the effects of thespatially distributed measurement sensitivity of time domain reflectometry (TDR) on the inversion of hydraulic properties from watercontent 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 reflectometryprobes with larger rod separations show less impact on the flow ofwater at the wetting front. However, these probes have more widelydistributed 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 inversionof hydraulic parameters from measurements recorded during the advance of a vertical wetting front has three closely spaced rods thatlie 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.

T

ime 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 offthe-shelf TDR monitoring systems. Like other indirect
water content measurement methods (e.g., neutron probes
and capacitance probes), TDR measures some averagewater content in a volume of medium surrounding theprobes. One unique aspect of TDR is the ease with which
a user can alter the sample volume and spatial sensitivityof TDR probes. For the TDR method, this spatial sensitivity has been examined both analytically (Knight,
1992) and numerically (Knight et al., 1997). Further ex-

Ty P.A. Ferré, Department of Hydrology and Water Resources, University of Arizona, AZ; Henrik H. Nissen, Department of Environmental Engineering, Institute of Life Sciences, Aalborg University,
Denmark; and Jirka Šimůnek, George E. Brown, Jr., Salinity Laboratory, USDA-ARS, Riverside, CA. Received 15 May 2002. *Corresponding author (ty@hwr.arizona.edu).
Published in Vadose Zone Journal 1:281–288 (2002).

Abbreviations: TDR, time domain reflectometry.

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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 thewater contents measured with 2H, 2V, and 3H TDR probes.†
Scenario
ActualNo rods2H(0.01) rods
2V(0.01) rods
3H(0.01) rods
2H(0.005) rods
2H(0.01)
2V(0.01)3H(0.01)2H(0.005)

␣

n

Ks

R2

m⫺1
14.5
14.4 (13.9–4.8)
14.5 (14.0–15.0)
15.1 (14.4–15.7)
14.0 (13.0–15.0)
13.9 (12.9–15.0)
12.1 (9.08–15.0)
16.0 (12.2–19.8)
14.5 (13.0–16.1)
12.1 (10.2–13.9)

2.68
2.70 (2.66–2.74)
2.69 (2.62–2.77)
2.67 (2.61–2.72)
2.74 (2.60–2.87)
2.75 (2.62–2.88)
2.44 (2.15–2.73)
2.06 (1.83–2.29)
2.54 (2.38–2.69)
2.59 (2.38–2.80)

m s⫺1 ⫻ 10⫺5
8.25
8.13 (7.88–8.38)
8.23 (7.87–8.58)
8.46 (8.10–8.82)
7.98 (7.36–8.60)
7.92 (7.28–8.57)
7.32 (5.18–9.46)
8.95 (6.85–11.1)
8.57 (7.53–9.60)
7.16 (5.81–8.51)⬎0.999⬎0.999⬎0.999⬎0.999⬎0.999
0.992
0.968
0.998
0.997

† The values shown in parentheses are the 95% confidence intervals of the parameter values as reported by HYDRUS-1D.

METHODSHYDRUS-2D (Šimůnek et al., 1999) was used to simulate
the vertical advance of a wetting front into a two-dimensionalcross-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 and0.05 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 wereno 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 timewas 60 s. TDR probes were placed within the transport domain
as impermeable objects and were represented as circular noflow 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 placedsuch that the center of the TDR probe was located at thecenter of the domain. Probes were chosen to represent verysmall 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 centersof the rods separated by 0.01 m. “Three-rod” probes had
three equally spaced rods, with the outermost two rod centersseparated 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, theorientation, 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.
Nissen et al. (unpublished data) found that 3V probes cangive rise to the simultaneous transmission of multiple waves
traveling at different velocities if the dielectric permittivity
varies between two adjacent rods. This precludes the application of travel time measurement to determine the volumetricwater content. As a result, this probe design is not recommended for monitoring the advance of a wetting front. Therefore, we considered only 2H, 2V, and 3H probes.
An example of the finite element domain used to simulate
water flow around a 2H(0.01) probe is shown on Fig. 1. Aneffort was made to make the finite element grid geometries
used in all of the simulations as similar as possible. To achievethis, 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 wereimpermeable holes in the grid. An observation point was placedat the midpoint of each probe, located at the center of thedomain. Given that an observation point cannot be placed in
the center of the domain for the three-rod probes, which havea rod centered at the domain center, the observation point
for these probes was placed at a point equidistant from anouter 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, fora total of 26 water content distributions. The relative dielectric
permittivity, K, distribution was determined from the watercontent distribution based on the linear form of the Topp etal. (1980) relationship presented by Ferré et al. (1996):

K 0.5 ⫽ 8.47␪ ⫹ 1.56

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 areshown as well. The rods lie in the horizontal plane for all of theprobe designs.

[1]

For each simulated time, the numerical analysis of Knight etal. (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 thattime. This analysis is based on the determination of the spatial
distribution of the energy of the electromagnetic field in theplane 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 thetransverse 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

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dielectric permittivity distribution in the vertical plane. Equation [1] can be rearranged to define the “measured” watercontent based on this dielectric permittivity. The inversion
capabilities of HYDRUS-1D (Šimůnek et al., 1998) were subsequently used to determine the hydraulic parameters thatbest fit the 26 probe-measured water contents as a function oftime. HYDRUS-1D uses the Levenberg–Marquardt optimization technique to minimize the sum of squared deviations ofmeasured 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 tothose 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:␪ ⫽ ␪r ⫹␪ S ⫺ ␪rfor h ⬍ hs, and(1 ⫹ |␣h|n)m␪ ⫽ ␪S for h ⱖ hs
Se ⫽

[2]
␪ ⫺ ␪r␪S ⫺ ␪ r

K(h) ⫽ KSSe0.5[1 ⫺ (1 ⫺ Se1/m)m]2

[3]
[4]

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

RESULTS
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, weinvestigated the inversion of hydraulic parameters fromwater 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 timeseries at the observation node with no rods present (allcircles filled with elements). The correct soil hydraulic283

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 wasused 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 wereidentical in both models.
The observed and fitted water content time series(Fig. 2) showed excellent agreement (R 2 ⫽ 0.99998), suggesting that there is very little error introduced by the
combined use of HYDRUS-2D for forward modelingand 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 watercontent time series.) The fitted values of ␣, n, and Ks
are shown in Table 1 and are referred to as the “No rods”
scenario. The inverted values were used to constructcharacteristic curves (Fig. 3). The actual soil characteristic curves (thick lines) and the no rod inversion (thinlines) 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 thenumber of unknown fitted parameters. It is desirable
that the estimated mean value be located in a narrowinterval. 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 functionof the exact choice of parameters to fit, the reported
values should not be taken to describe the absoluteaccuracy of the inversion of hydraulic parameters from
TDR measurements. Rather, the relative values of theconfidence intervals should be used to compare the suitability of several probe designs. The region between thedashed lines on Fig. 3 shows the full range of values

Fig. 2. The water content breakthrough curve at the midpoint of thedomain with no rods present determined through forward modelingwith HYDRUS-2D (Actual) and the water content breakthroughcurve 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 watercontents with HYDRUS-1D.

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Fig. 3. The inverted soil hydraulic functions based on optimization of the actual water content at the center of the two-dimensional domain withno 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.

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 Ks within the 95% confidence interval,
and the lowermost line used the lowest values.

Effects of TDR Rods on Wetting Front Advance
The water content time series at the center of thedomain with no rods present is shown as a thick line in
Fig. 4. The water content time series at the center of thedomain with different 2H and 2V probes are shown todemonstrate the influence of the TDR rods on water
movement at the center of the domain. Most of theprobes have little observable effect on the water content
breakthrough curve. A narrowly spaced 2H probe causesslightly faster breakthrough because of acceleration of
the water between the rods. As would be expected fromtheir similar geometries, a 3H probe (not shown) has
an identical response as a 2H probe with one-half theoutermost rod spacing; for example, a 3H(0.01) probe
has the same effect as a 2H(0.005) probe. A narrowly
spaced 2V probe causes a slight delay in wetting front
arrival due to “shadowing” of the probe midpoint by
the upper rod. Probes with larger rod separations show
a much smaller effect on water movement at the probe
midpoint.

The hydraulic function determined from the 2H(0.005)
water content breakthrough curve is shown in Fig. 5,
and the inversion results are included in Table 1. The
impacts of the rods on the wetting front have little effecton the best-fit interpreted hydraulic functions, whichstill show excellent agreement with the actual values.
The only observable effect is an increase in the widthof the 95% confidence envelope, with the results of the2H probe slightly better than those of the 2V(0.005)
probe (results not shown).

The TDR-Measured Wetting Front Advance
The TDR-measured water content at any given time
will be a function of the specific water content distribution around the rods at that time. Nissen et al. (unpublished data) found that the numerical approach of
Knight et al. (1997) could be used to describe the spatial
sensitivity of TDR probes in the presence of sharp dielectric boundaries. To better test the applicability of
the numerical model of TDR sensitivity, they examined
the response of probes near fluid–fluid interfaces. In
this investigation, we applied this numerical analysis toinclude the effects of TDR rods on the distribution ofwater in a soil, as discussed above. The simulated responses of four TDR probe configurations are compared with the actual water content at the center of thedomain with no rods present in Fig. 6. Three of thewww.vadosezonejournal.org

285

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 ofvarying rod separations.

Fig. 5. The inverted soil hydraulic functions based on optimization of the actual water content at the center of the two-dimensional domain witha 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.

probes have identical rod diameters and outer rod separations. The fourth probe, 2H(0.005), is included for
direct comparison with the 3H(0.01) probe. The 2V
probe shows a markedly different response than the 2H

or 3H probes. This probe shows an early first arrival ofwater and the most dramatic delay in the arrival of thewetting front (defined here as the time to reach one-halfof the final water content change, shown as a horizontal

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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 shownas the intersection of the horizontal dashed line with each breakthrough curve.

dashed line). All of the 2H and 3H probe responses
show the same time of arrival of the wetting front. A
3H probe with the same outer rod separation shows lesseffect of spatially distributed sensitivity than a 2H probewith the same separation. However, the response of a3H probe and a 2H probe with the same distance between adjacent rods is very similar [i.e., 3H(0.01) vs.
2H(0.005) probes].
The TDR responses can be explained based on anunderstanding of the sample areas of TDR probes (Ferré
et al., 1998; Nissen et al., unpublished data). Specifically,
3H probes have very restricted sample areas, with theprobe sensitivity concentrated in the region immediately
adjacent to the central rod. In contrast, a 2H probe withthe same outer rod separation has a sample area thatis more evenly distributed in an approximately ellipticalregion surrounding the probe rods. In a homogeneous
medium, a 2V probe has a sample area that is identicalto that of a 2H probe of the same dimensions, rotatedby 90⬚. However, Nissen et al. (unpublished data) foundthat if the dielectric permittivity varies between the rods,
the sample area of a 2V probe is strongly biased toward
the region of lower dielectric permittivity. This bias
arises because all of the energy must flow through boththe high and low dielectric permittivity regions, but thepotential gradient is much higher through the low dielectric permittivity region than through the high dielectric
permittivity region. Given that the energy can be defined based on the square of the gradient of the potential, the energy, and therefore the probe sensitivity, isconcentrated within the low dielectric permittivity region. One result of this is that while horizontal probes
will measure the arithmetic average of the dielectric
permittivities ahead of and behind a sharp wetting front
that intersects the probe midpoint, a vertical probe willmeasure the harmonic mean dielectric permittivity (Ferré
et al., 1996). As a result, 2V probes will overweight thedry region between the wetting front and the lower rod,
giving a lower water content when the wetting front islocated between the rods. On the basis of their sample
areas, it is expected that the 3H probe will preserve
more of the sharpness of the wetting front in the watercontent breakthrough curve than will a 2H with the sameouter rod separation. Furthermore, horizontal probes
are expected to show the correct time of arrival of a
step function wetting front. The modeling results demonstrate that this result extends to a more dispersed wetting front as well. Meanwhile, the 2V probe will sense
the arrival of the wetting front before the wetting front
reaches the probe midpoint because the upper rod islocated above the probe midpoint. Then, the changing
spatial sensitivity of the 2V probe with further advanceof the wetting front gives rise to a complex water content breakthrough curve. Nissen et al. (unpublished data)
found that a 2V probe does not sense a sharp dielectric
permittivity boundary until both rods are in contact
with the higher dielectric permittivity region. This may
suggest that the inversion of the 2V results could be improved by considering the measurement point to belocated at the center of the lowest rod. In fact, the shapeof the wetting front is so badly distorted by this probe
configuration that this approach gives only marginallybetter inversion results (not shown).

Hydraulic Properties Determinedfrom 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 thewater 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, andKs 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 parametervalues are shown in Table 1. The best-fit inverse models
show good agreement with the simulated TDR-measured water breakthrough curves, giving R 2 values inexcess of 0.96 for all probes. The 3H probe shows thesmallest 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 showsthat although these probes had nearly identical actual
water content breakthrough curves at the probe midpoints, the larger sampling area of the 2H probe led toslightly greater smoothing of the water content breakthrough curve. This leads to less accurate hydraulicproperty inversions.
The interpreted water retention and hydraulic conductivity functions are shown in Fig. 7 and 8 for themodeled 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 thewidth of the 95% confidence interval is much narrower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.

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 thicklines 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.

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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), and3H(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 ofthe 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, thepredicted hydraulic properties based on measurements
made with 3H probes agree most closely with the correct
values. Compared with the 3H responses, equally sized2H 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, thisconfiguration is not recommended for monitoring theadvance of a wetting front.

CONCLUSIONS
Numerical analysis of the response of TDR probes tothe advance of a wetting front shows that the distributed
spatial sensitivity of TDR probes leads to smoothing of
the observed wetting front. The observed wetting front
shape is more distorted for vertically emplaced probes
than for horizontal probes. The distortion caused byhorizontal probes decreases with a decrease in the rod
separation. However, with decreases in the rod separation, the impact of the rods on the flow of water increases. In practice, this effect can be reduced by the
choice of relatively small TDR rods. The optimal TDR
probe configuration for inversion of hydraulic parameters from wetting front monitoring was a narrowly spaced
3H design. Despite the effects of this probe on water
flow, the highly concentrated spatial sensitivity of thisprobe leads to minimal smoothing of the water content
breakthrough curve. This result suggests that 3H probeswith a very small rod separation will capture the watercontent breakthrough curve most accurately, leading to
optimal hydraulic parameter inversion. However, thisconclusion must be balanced by the tendency of probeswith very small rod spacings to be highly sensitive tothe effects of disturbance during insertion of rods into
the medium or during packing of the medium aroundthe rods. Given this practical concern, we recommendthat a larger separation, on the order of five times therod diameter, be used with 3H probes to monitor theadvance of a wetting front. This specific recommendation for the application of TDR monitoring to wetting
fronts cannot be adopted generally for all monitoring
applications. Rather, we recommend that a similar analysis be performed for each specific monitoring application. Direct examination is especially warranted forprobes with components that would be expected to produce more disturbance to flow (e.g., rods mounted on
substrates) or rods with shapes that may lead to nonuniform concentration of probe sensitivity (e.g., flat plateswith sharp corners). In theory, the method of analysis
presented could also be used to examine the responseof very thin rods, with diameters approaching the grain
size of the surrounding material. However, this will require a more accurate representation of the distributionsof grains, water movement, and variations in the dielectric permittivity of pore water at such small scales. Finally, while the results shown here are specific to TDR
responses, the same consideration should be given to
all instruments with spatially distributed sensitivities.
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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.
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